Research on Undergraduate Mathematics Education Bibliographic Database SIGMAA on RUME 21 March 2001 1. {ach:howcu} P. S. III Ache. How curriculum choice affects student understanding and confidence: A comparison of business calculus reform models. PhD thesis, Texas A&M University, 1996. This study was designed to explore the relationship between student understanding of calculus and confidence in that understanding in relation to different calculus reform models. Students in two sections of business calculus courses at a state university in Texas were asked to rate how confident they were in their responses on three different exams during the semester of instruction. In addition to the confidence ratings, accuracy scores were assigned to each response. This set of data was used to answer two broad research questions. The first question was: How does the type of curriculum experienced affect student understanding (both conceptually and procedurally) of differentiation and integration and confidence in that understanding? The second question examined how this relation might change throughout the course of instruction. It was found that accuracy, and hence to some degree understanding, was necessary in order for students to be confident in their work on each exam. It was also found, however, that showing signs of understanding did not necessarily imply confidence in that understanding. Students exposed to a more radical reform model were not only more confident in their understanding of differentiation, but were also more accurate on conceptual questions. This was not the case on integration items. Students exposed to the reform model were not as confident, nor as accurate on either procedural or conceptual items on the exams. However, the time devoted to integration during the course of the semester was very limited, and thus, inferences about student understanding of integration were influenced by this lack of time. Lastly, it was determined that confidence in ability on mathematical tasks for college students changed very little over the course of the semester. For the most part, students that began the course confident, or not confident, in their abilities remained so throughout the semester. Students whose confidence did change usually felt less confident in their understanding at the end of the semester than at the beginning. 2. {akk-kle-wuu:alice} C. Akkoc, J. Klemmack, and J. Wuu. Alice in solution land: A student project to demonstrate the effect of singularities on the solution space of a first-order. Mathematics and computer education, page 270, 1998. A project on ordinary differential equations (ODE) assigned to a class leads students to self-discovery of interesting behavior in the trajectories of a representative ODE. The geometry of the solution space of a linear first-order ODE is examined using mostly computer graphics and some analysis. The motivation is the curiosity of a child discovering things for the first time, as Alice would probably do if she found herself in the solution space of a differential equation. 3. {ale-dea:group} D. Alexander and L. DeAlba. Groups for proofs: collaborative learning in a mathematics reasoning course. Primus, 7(3):193-207, September 1997. Abstract (quoted from paper): `In the Mathematics Reasoning course at Drake University students learn how to construct a proof. To this end, they study logic, set theory, functions, and relations. This article describes the course and the authors' efforts to introduce collaborative, small-group mixed-ability activities into the classroom. We offer our observations on the ease, difficulties, rewards, and demands involved in implementing a collaborative learning environment. We conclude by summarizing student and instructor attitudes towards these collaborative activities.' 4. {ale:aninv} E. H. Alexander. An investigation of the results of a change in calculus instruction at the University of Arizona. PhD thesis, The University of Arizona, 1997. The results of the change in Calculus instruction at the University of Arizona in 1991, 1992, and 1993 were examined using three complementary methods. A survey of students (45) who took calculus during this period was administered, and analyzed for attitudinal differences between those who took traditional and those who took reform calculus. There were no statistically significant differences in reported attitude. Volunteers (14) were solicited from those who had been freshmen during the change to participate in interviews. These interviews included students taught by each method, and were analyzed by using concept maps to determine if there is a difference in retained knowledge. Although consortium (reform) students showed slightly improved retention, the differences were not statistically significant. University computerized grade records were used to determine if there was a difference between students who took consortium calculus and those who took the traditional course. Both retention and grades in subsequent calculus-dependent mathematics, science, and engineering courses were examined. A pattern of comparisons emerged which showed that consortium students somewhat outperformed traditional students. The patterns were indicative of better teaching and cannot be directly attributed to the materials. There is good evidence that the consortium students were not at a disadvantage in subsequent course work. This research should be of interest to teachers of calculus, and those involved in calculus reform. The techniques and computer programs for analysis of large data sets for performance differences in subsequent (dependent) course work can be useful for comparing different instructors, procedures, or materials in large institutions. 5. {ali:towar} D. Alibert. Towards new customs in the classroom. For the Learning of Mathematics, 8(2):31-43, June 1988. The author reports on an experimental teaching method, set in a constructivist framework, for first-year university mathematics. The method is characterized in terms of new `customs' for the classroom: (1) accepting uncertainty in the learning process; (2) addressing proofs to other students rather than to the teacher; (3) introducing new mathematical tools only when they appear necessary for the solution of some problem; (4) calling students' attention to the fact that they construct their own knowledge by a process of reflection. Dialog in the classroom follows a `scientific debate' model in which students, prompted by a well-crafted question from the teacher, propose conjectures and either validate or disprove them. The methodology involves careful planning of class sessions in which questions are chosen for their perceived ability to elicit the responses from the students that the researchers want to encourage. The classes are recorded and analyzed with regard to the number of students who participate and how much of the content of the session has been enunciated by the students rather than by the teacher alone. Bases on the analysis, appropriate changes in the teacher's behavior are made as the course progresses. 6. {ali-tho:resea} D. Alibert and M. Thomas. Research on mathematical proof, volume 11 of Mathematics Education Library, pages 215-230. Kluwer Academic Publishers, Boston, 1991. The authors examine existing research into mathematical proof and its presentation. Specifically, they include research by Fischbein, Movshovitz-Hadar, Tall, Steiner, and Dreyfus & Eisenberg on the types of proofs preferred by students, research by Vinner and Fischbein on cognitive obstacles to understanding proofs, a method of proof exposition formalized by Leron called structural proof, and a cooperative learning classroom technique used by the Grenoble school called scientific debate. 7. {alk:theef} H.M. Alkhateeb. The effect of using graphics calculators on students' attitudes towards mathematics and students' achievement in introductory calculus. PhD thesis, Ohio University, 1995. The objective of this study was to investigate the effect of using graphics calculators on students' attitudes towards mathematics and students' achievement in introductory calculus. Subjects involved in this study were college students in two introductory calculus classes offered at Ohio University in the Fall, 1994. One class was given access to graphics calculators and was encouraged to use them in solving problems throughout study. The other class was not given access to graphics calculators and was not encouraged to use them. To measure a change in attitude, an instrument entitled, 'Attitude Toward Mathematics, Scale Form B' as presented by Suydam in measuring attitude toward mathematics was used at the beginning and at the end of the study. An analysis of the dependent t-test on the attitude scores indicated no significant change in group means in the attitude of students in either of the two groups. Also, analysis of the attitude differences between the groups, using an analysis of covariance, showed no significant differences between the two groups on the attitude scores. Students' achievement in calculus was measured by performance on a instructor-made calculus achievement test, given at the end of the study. A preliminary evaluation test (a pretest) in college algebra was administered at the beginning of the course and used as a covariate measure to adjust for the initial differences in the analysis of covariance used to examine the difference in achievement between the group using graphics calculators and the group not using graphics calculators. There was no significant difference between the groups in calculus achievement. 8. {all:acomp} G. H. Allen. A comparison of the effectiveness of the Harvard calculus series with the traditionally taught calculus series. PhD thesis, San Jose State University, 1995. This study compared six Harvard Calculus students with six students from traditionally taught calculus, all of whom had completed a year of calculus during the 1993-1994 school year. The students were interviewed and tested on their feelings about the course and text as well as their understanding of key calculus topics. Analysis of the interviews and tests reveals that the students from the Harvard Group have a better attitude about mathematics as well as a better understanding of calculus. The Harvard Group consistently performed better than the traditional group on questions regarding continuity, differentiation, and integration. Limits was the only area in which the traditional group performed better, although neither group did very well on this topic. The results from this study clearly indicate that Harvard Calculus is a worthy alternative to the traditionally taught calculus. It is recommended that other Mathematics Departments offer a reform-style calculus. 9. {all:astud} N. Allen. A Study of Metacognitive Skill as Influenced by Expressive Writing in College Introductory Algebra Classes (Writing to Learn). PhD thesis, The Louisiana State University and Agricultural and Mechanical College, 1991. (This annotation is quoted from the dissertation abstrace.) `Writing in the mathematics classroom has previously received anecdotal support for its benefits to the learner and to the instructor, and limited quantitative benefits in problem-solving ability toward mathematics. This study examined the effect of expressive writing on self-awareness and would suggest quantitative support that writing is beneficial in promoting student ability to assess the correctness of work. If metacognitive skills are a necessary condition for successful mathematics performance, the use of writing may provide the process for attaining these essential skills.' 10. {alo:coll} R. A. Alo. A collaborative learning approach for undergraduate numerical mathematics. Paper presented at the Conference on Computers in the Undergraduate Curricula, Claremont, California, June 18-20; (ERIC Document Reproduction Service No. ED084157). Described is an undergraduate numerical analysis course organized around projects and tasks assigned to student teams. Most teams had five students within which the student with the most computer programming experience assumed the leadership role. The leaders' responsibility extended to distribution of work assignments and coordination of group interaction. Intragroup cooperation, leadership or lack of leadership and assignment of individual final grades are among the topics and problems discussed. 11. {alv-93:calc} L. Alvarez and Others. Calculus instruction at new mexico state university through weekly themes and cooperative learning. Primus, 3(1):83-98, 1993. Working in groups on weekly themes, students discover calculus concepts through assigned readings and written reports. Provides an outline of the course content for two courses and how the courses are organized. An appendix contains the complete text of two themes on curve sketching and applications of the definite integral. 12. {amit-vin-pme-90} M. Amit and S. Vinner. Some misconceptions in calculus: Anecdotes or the tip of an iceberg? In G. Booker, P. Cobb, and T. N. de Mendicuti, editors, Proceedings of the 14th International Conference of the International Group for the Psychology of Mathematics Education, volume 1, pages 3-10, CINVESTAV, Mexico, 1990. 13. {ami-vin:somem} M. Amit and S. Vinner. Some misconceptions in calculus: Anecdotes or the tip of an iceberg? In Proceedings of the Annual Conference of the International Group for the Psychology of Mathematics Education with the North American Chapter 12th PME-NA, volume 1, pages ???-???, Mexico, 1990. International Group for the Psychology of Mathematics Education. To be added later. 14. {ant:anint} T.L. Anthony. An introduction to linear algebra: A curricular unit for pre-calculus students. PhD thesis, Rice University, 1995. Matrices are important mathematical tools that facilitate the process of organizing and manipulating data. In this work, the matrix operations of addition, subtraction, scalar multiplication, and matrix multiplication are built logically from the intuition of the students and their knowledge of real numbers. From this knowledge, the concepts of inverses, determinants, and consistency and inconsistency of linear systems of equations are formed. Interesting applications of matrices in the areas of Markov chains, curve fitting, and eigenpairs are included and are not beyond the comprehension of pre-calculus students when they are presented carefully. Pre-calculus students can also appreciate many of the numerical challenges that can be encountered when real-world problems are solved; therefore, we include a discussion of some of these topics. 15. {app:writi} A. Applebee. Writing and reasoning. Review of Educational Research, 54(4):577-596, 1984. The article contains a review of the literature on writing and reasoning. It includes reports on several empirical studies. 16. {arcavi-rcme-98} A. Arcavi, C. Kessel, L. Meira, and J. P. III Smith. Teaching mathematical problem solving: An analysis of an emergent classroom community. CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education III, 7:1-70, 1998. 17. {arm:amult} S. M. Armstrong. A multivariate analysis of the dynamics of factors of social context, curriculum, and classroom process to achievement in calculus at the community college. PhD thesis, The University of Rochester, 1997. Since 1987 the National Science Foundation has awarded multi-million and multi-year grants for the renewal of the undergraduate calculus curriculum. A primary focus of these initiatives is to effect a change in the high failure rate in calculus. Despite these curriculum reform initiatives the failure remains high, especially among students in community colleges. The current research investigated the relationship between the factors of social context, classroom process, and curriculum to the achievement in calculus at the community college level. The primary interest was to determine the characteristics of the students who are successful in calculus. A secondary interest was to evaluate the effect of the curriculum of the Calculus Consortium based at Harvard University (the most used of the calculus reform initiatives) on the achievement of students in community colleges. The research utilizes survey and outcome data from calculus students drawn from a stratified sample of community colleges in New York and New Jersey. Two-thirds of the institutions used the Harvard (CCH) material, while the other institutions used a traditional curriculum. The survey data was supplemented with in-class participant observation by the investigator who participated in two of the calculus courses (one in each curriculum). The quantitative data was analyzed using parametric and non-parametric procedures. The variables were evaluated for overall effect and for effect by curriculum. The findings from this study, in a path analysis design, indicate that students' algebraic ability was the strongest predictor of success in the outcome variable (final course grade). The results also indicate that students who are successful are more likely to have: a strong algebra background, a positive attitude towards mathematics, taken their pre calculus courses in high school, and had positive engagement in the calculus course. The data did not support the calculus reform assumptions that students with low pre-calculus backgrounds can succeed in calculus with the aid of a graphing calculator. The findings also suggest that non-Asian minority students were more likely to be hindered than helped by enrollment in a reform curriculum. 18. {art:cogni} M. Artigue. Cognitive difficulties and teaching practices. In G. Harel and E. Dubinsky, editors, The concept of Function: Aspects of Epistemology and Pedagogy. Mathematical Association of America, Washington, DC, 1992. To be added later. 19. {art-men-vie:somea} M. Artigue, J. Menigaux, and L. Viennot. Some aspects of students' conceptions and difficulties about differentials. European journal of physics, page 262, 1998. To be added later 20. {art-new-90:impl} A. F. Artzt and C. M. Newman. Implementing the standards. cooperative learning. Mathematics Teacher, 83(6):448-452, 1990. Reviewed are the basic principles of cooperative learning including a rationale for its use and the formation of cooperative learning groups in the classroom. Examples of the application of this teaching method to mathematics teaching are discussed. 21. {asi-bro-dev-dub-mat-tho-96:fram} M. Asiala, A. Brown, D. DeVries, E. Dubinsky, D. Mathews, and K. Thomas. A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6:1-32, 1996. The authors detail a research framework with three components and give examples of its application. The framework utilizes qualitative methods for research and is based on a very specific theoretical perspective that was developed through attempts to understand the ideas of Piaget concerning reflective abstraction and reconstruct them in the context of college level mathematics. For the first component, the theoretical analysis, the authors present the APOS theory. For the second component, the authors describe specific instructional treatments, including the ACE teaching cycle (activities, class discussion, and exercises), cooperative learning, and the use of the programming language ISETL. The final component consists of data collection and analysis. 22. {asi-bro-kle-mat-98:dev} M. Asiala, A. Brown, J. Kleiman, and D. Mathews. The development of students' understanding of permutations and symmetries. International Journal of Computers for Mathematical Learning, 3:13-43, 1998. The authors examine how abstract algebra students might come to understand permutations of a finite set and symmetries of a regular polygon. They give initial theoretical analyses of what it could mean to understand permutations and symmetries, expressed in terms of APOS. They describe an instructional approach designed to help foster the formation of mental constructions postulated by the theoretical analysis, and discuss the results of interviews and performance on examinations. These results suggest that the pedagogical approach was reasonably effective in helping students develop strong conceptions of permutations and symmetries. Based on the data collected as part of this study, the authors propose revised epistemological analyses of permutations and symmetries and give pedagogical suggestions. 23. {asi-cot-dub-sch:thede} M. Asiala, J. Cottrill, E. Dubinsky, and K. Schwingendorf. The development of student's graphical understanding of the derivative. Journal of Mathematical Behavior, 4:399-431, 1997. In this study the authors explore calculus students' graphical understanding of a function and its derivative. An initial theoretical analysis of the cognitive constructions that might be necessary for this understanding is given in terms of APOS. An instructional treatment designed to help foster the formation of these mental constructions is described, and results of interviews, conducted after the implementation of the instructional treatment, are discussed. Based on the data collected as part of this study, a revised epistemological analysis for the graphical understanding of the derivative is proposed. Comparative data also suggest that students who had the instructional treatment based on the theoretical analysis may have more success in developing a graphical understanding of a function and its derivative than students from traditional courses. 24. {asi-dub-mat-mor-oct-97:stu} M. Asiala, E. Dubinsky, D. Mathews, S. Morics, and A. Oktac. Student understanding of cosets, normality and quotient groups. Journal of Mathematical Behavior, 16(3):241-309, 1997. Using an initial epistemological analysis from Dubinsky, Dautermann, Leron and Zazkis (1994), the authors determine the extent to which the APOS perspective explains students' mental constructions of the concepts of cosets, normality and quotient groups, evaluate the effectiveness of instructional treatments developed to foster students' mental constructions, and compare the performance of students receiving this instructional treatment with those completing a traditional course. 25. {asp:thero} L. N. Aspinwall. The role of graphic representation and students' images in understanding the derivative in calculus: Critical case studies. PhD thesis, The Florida State University, 1994. Calls for reform in the way that calculus is taught stress the importance of instruction focused on graphic as well as analytic representations of functions and derivatives. The value of calculus lies in its potential to reduce complex problems to simple rules and procedures. However, students taught only rules and procedures often emerge from calculus classrooms without the ability to analyze graphs and lack an understanding of the conceptual foundations of the slope of a tangent line. Study based solely on analytic representations of functions and their derivatives often produces only procedural understanding. In this study, two undergraduate calculus students were confronted with graphic representations for functions and their derivatives and asked to produce graphs that represented their images - their unique internal representations. Their attempts to provide external representations of their images provided the data for the study. The purposes of the study are two-fold: (1) to contrast the different mathematical understandings of these two students that have been revealed as a result of analyses of their graphic constructs for the derivative function and (2) to present the consequences of an instructional strategy based on graphic representation for functions and derivatives. The study demonstrates that graphic instructional representations for functions and their derivatives, and students' concomitant images, have the potential for producing a richer understanding than that achieved by analytic study alone. Stimulated by graphic instructional representations, students form and can utilize mental images to construct understanding of the calculus derivative and to demonstrate their unique internal mathematical representations. 26. {aye-dav-dub-lew-88:com} T. Ayers, G. Davis, E. Dubinsky, and P. Lewin. Computer experiences in the teaching of composition of functions. Journal for Research in Mathematics Education, 19(3):246-259, 1988. Students from two sections of a college mathematics lab (n=13) who were given computer experiences to help induce reflective abstraction scored higher on a test of their understanding of functions and compositions than students from another section (n=17) who were taught according to traditional methods. The comparison was based on questions intended to indicate whether reflective abstraction had taken place. 27. {azc:insta} C. Azcarate. Instantaneous speed: Concept images at college students' level and its evolution in a learning experience. In Proceedings of the 15 th Conference of the International Group for the Psychology of Mathematics Education (PME), volume 1, pages ???-???, Assisi, Italy, 1991. International Group for the Psychology of Mathematics Education. To be added later. 28. {azz:writi} A. Azzolino. Writing as a tool for teaching mathematics: the silent revolution, volume 53 of NCTM Yearbooks, chapter 11, pages 92-100. National Council of Teachers of Mathematics, 1990. The author presents numerous strategies for using writing assignments in the mathematics classroom. 29. {badd-cog-95} A. Baddeley. Working memory. In M. S. Gazzaniga, editor, The Cognitive Neurosciences, pages 755-764. MIT Press, Cambridge, MA, 1995. 30. {bak-coo-tri:devel} Baker, Cooley, and Trigueros. Development of students' calculus graphing schema. In ??, pages ???-??? The Association of Research in Undergraduate Mathematics Education, 1998. To be added later. 31. {bak-coo-tri:thesc} B. Baker, L. Cooley, and M. Trigueros. The schema triad - a calculus example. Submitted for publication. In this paper the authors report on the ways students try to solve a non-routine mathematical problem which involves graphing a function given certain properties such as particular limits associated with the function, continuity, and first and second derivative information. In order to analyze the interview data the authors extend the theory of schema development within the APOS theory. They found that students had to deal with two different schemas and that, for a particular student, those schemas could be at different stages of development. The first schema relates to the real line and the handling of information given in overlapping intervals; the second is the schema for the various properties of the function. Building on work of Piaget and Garcia as well as Clark, Cordero, et. al. (1997), the authors develop a double triad theoretical framework in terms of the interaction between these two schemas and classify the students in these two triad levels. They find student responses at each of the nine possible levels except one: the trans property - intra interval level. In addition to the theoretical development, the authors pinpoint specific difficulties students had with this problem: understanding the graphical meaning of the second derivative, imagining a continuous curve that has a cusp, drawing points of inflection, and relating continuity to differentiability. 32. {ban:compu} B. W. Banks. Computer corner. a rich differential equation for computer demonstrations. College Mathematics Journal, 21:45-50, 1990. Presents an example using a computer to illustrate concepts graphically in an introductory course on differential equations. Discusses the algorithms of the computer program displaying the solutions to an equation and the inclination field of the equation. (YP) 33. {bar-nar:techn} F. Barber and J. Narayan. Technology, cooperative learning, and assessment in the teaching of ordinary differential equations. PRIMUS, 4:337-346, 1994. Reports on the use of technology to enhance the teaching of ordinary differential equations, gives examples of laboratory activities using cooperative learning, and discusses assessment of student learning. MacMath, TI-81 graphing calculators, and Maple were used in this course. 34. {bar-win:harve} J. Barna and J. Winstead. Harvey mudd college: Technology integration offers unique opportunities for undergraduates. T.H.E. Journal, 2:105-108, 1993. Describes undergraduate projects at Harvey Mudd College (California) that use advanced laboratory equipment and procedures normally reserved for graduate students. Examples are given in experimental biology (e.g., digital imaging and DNA analysis), in physics (e.g., using satellites to study earthquake faults), and in mathematics (e.g., teaching differential equations). 35. {bar-slo-tah-mac-sea-pop-rei-haw-hew-bal-and:teach} T. Barnard, A. Slomson, D. Tahta, J. Mackernan, J. Searl, S. Pope, D. Reid, A. Haworth, D. Hewitt, B. Ball, and J. Anderson. Teaching proof. Mathematics Teaching, 155:6-39, June 1996. The report Tackling the Mathematics Problem, published by the London Mathematical Society, the Institute of Mathematics and its Applications, and the Royal Statistical Society referred to a `changed perception of what mathematics is--in particular of the essential place within it of precision and proof.' The editors of Mathematics Teaching invited each of the authors to comment on the role of precision and proof in mathematics and in mathematics education. Thus the article takes the form of a series of essays, one by each of the authors. 36. {bar-tal:cogni} T. Barnard and D. Tall. Cognitive units, connections, and mathematical proof. In Proceedings of the 21st International Conference for the Psychology of Mathematics Education, volume 2, pages 41-48, Lahti, Finland, 1997. International Group for the Psychology of Mathematics Education (PME). ED416083. The authors describe a study in which they used clinical interviews with eighteen students from three different stages in the mathematics curriculum representing high school and university mathematics. The interviews focused on the proofs of the irrationality of root 2 and root 3. Quoting from the paper abstract: `In this paper a theory is suggested involving cognitive units which can be the conscious focus of attention at a given time and connections in the individual's cognitive structure that allow deductive proof to be formulated. Whilst elementary mathematics often involves sequential algorithms where each step cues the next, proof also requires a selection and synthesis of alternative paths to make deductions.' 37. {bar:asses} J. Barnett. Assessing student understanding through writing. Primus, 6(1):77-86, March 1996. (This annotation is quoted from the paper abstract.) `Writing assignments which allow lower division students to express their understanding of mathematics in their own terms provide a more informative view of student difficulties than those which require communication in formal mathematical terms. The former can reveal student insights, as well as student difficulties. A specific type of writing assignment designed to encourage the students' logical analysis skills is described. Some instructional consequences and practical concerns which arose from the use of writing as an assessment tool in the author's classes are considered.' 38. {bar:graph} S. D. Barton. Graphing calculators in college calculus: An examination of teachers' conceptions and instructional practice. PhD thesis, Oregon State University, 1995. The study examined classroom instructional practices and teacher's professed conceptions about teaching and learning college calculus in relationship to the implementation of scientific-programmable-graphics (SPG) calculators. The study occurred at a university not affiliated with any reform project. The participants were not the catalysts seeking to implement calculus reform, but expressed a willingness to teach the first quarter calculus course with the SPG calculator. The research design was based on qualitative methods using comparative case studies of five teachers. Primary data were collected through pre-school interviews and weekly classroom observations with subsequent interviews. Teachers' profiles were established describing general conceptions of teaching calculus, instructional practices, congruence between conceptions and practice, conceptions about teaching using SPG calculators, instructional practice with SPG calculators, and the relationship of conceptions and practice with SPG calculators. Initially, all the teachers without prior experience using SPG calculators indicated concern and skepticism about the usefulness of the technology in teaching calculus and were uncertain how to utilize the calculator in teaching the calculus concepts. During the study the teachers became less skeptical about the calculator's usefulness and found it effective for illustrating graphs. Some of the teachers' exams included more conceptual and graphically-oriented questions, but were not significantly different from traditional exams. Findings indicated the college teachers' conceptions of teaching calculus were generally consistent with their instructional practice when not constrained by time. The teachers did not perceive a dramatic change in their instructional practices. Rather, the new graphing approach curriculum and technology were assimilated into the teachers' normal teaching practices. No major shifts in the role of the teachers were detected. Two teachers demonstrated slight differences in their roles when the SPG calculators were used in class. One was a consultant to the students as they used the SPG calculators; the other became a fellow learner as the students presented different features on the calculator. Use of the calculator was influenced by several factors: inexperience with the calculator, time constraints, setting up the classroom display calculator, preferred teaching styles and emphasis, and a willingness to risk experimenting with established teaching practices and habits. 39. {bec:effec} C. E. Beckman. Effect of computer graphics use on student understanding of calculus concepts. PhD thesis, Western Michigan University, 1988. Student understanding of selected calculus concepts as developed through use of a Cartesian coordinate graphical representation system were investigated. Subjects (N = 163) enrolled in first-semester calculus sections at Western Michigan University participated in one of four treatment conditions: Graphics (G), exposure to a computer-graphically-developed conceptual course; Graphics Plus (G+), exposure to the same course as G subjects plus provision of computer graphics software and related supplemental assignments; Standard 1 (S1), exposure to a graphically-developed, conceptual course; and Standard 2 (S2), exposure to a traditional skill-oriented course. Two investigations were undertaken. In Investigation 1, comparisons were made between G and G+ sections on student: (a) understanding of Cartesian graphs, including the ability to use graphs in understanding calculus concepts, and (b) attitudes toward the use of graphs. In Investigation 2, comparisons were made between G, G+, S1, and S2 sections on student: (a) performance on routine applied, routine symbolic, and nonroutine symbolic questions; (b) performance on the departmental final exam and its subscales: (c) changes in attitudes toward mathematics; and (d) attitudes toward the course. Prior calculus experience was used as a blocking variable for cognitive measures on two levels, prior and no prior experience. Multivariate analysis with covariates, precalculus competency and attitudes toward mathematics, were performed for cognitive variables. [$\chi\sp2$] tests were conducted for affective variables. For Investigation 1, no significant differences (p [$<$].05) were detected between the G and G+ sections. G+ subjects' scores on cognitive variables were slightly higher than those of G subjects, suggesting that further study is warranted. Attitudes pertaining to the use of graphs were overwhelmingly positive. For Investigation 2, significant differences favored G subjects over S2 subjects on nonroutine symbolic questions. Questionable significant differences were detected for routine questions. Attitudes toward the course and mathematics were generally positive. Retention rates were much higher for the conceptually-developed sections than for the technique-oriented section. Results suggest that developing calculus concepts through the use of a graphic representation system, especially as presented through computer graphics, can positively affect student understanding and interest without necessarily, negatively influencing skill acquisition. 40. {bei-jon-wel:incre} J. Beidleman, D. Jones, and P. Wells. Increasing students' conceptual understanding of first semester calculus through writing. Primus, 5(4):297-316, December 1995. (This annotation is quoted from the paper abstract.) `Since calculus is highly symbolic in nature, students often try to get through calculus by manipulating the symbols without understanding the meaning of such symbols (i.e. having a procedural but not a conceptual understanding of the topics in calculus). A variety of writing assignments were implemented in one section of a traditional first semester calculus course in an attempt to increase the students' understanding of the ideas, procedures, and concepts of calculus. This paper outlines and describes the philosophy, structure, and outcomes of the course and discusses finding and implications. An appendix of sample writing assignments follows.' 41. {bei:writi} B. C. Beins. Writing assignments in statistics classes encourage students to learn interpretation. Journal of Educational and Behavioral Statistics, 20(3):161-164, 1993. The author studied three levels of writing emphasis on four sections of introductory statistics for psychology majors. The first section was considered traditional-emphasis, the second and fourth sections were considered moderate-emphasis, and the third section was considered high-emphasis. He compared the four groups on three segments of the final assessment: computational, conceptual, and interpretative. The high-emphasis class scored significantly better than one of the moderate-emphasis classes on the computational segment. There was no significant difference among the three groups on the conceptual segment. He found that greater emphasis on writing during class resulted in higher scores on the interpretive portion of the assessment. That is, on interpretive items, the high-emphasis students scored better than the moderate-emphasis students who in turn scored better than the traditional-emphasis students. 42. {ber:use} A. D. Jr. Berard. The use of small axiom systems to teach reasoning to first-year students. Primus, 2(3):265-277, 1992. Uses small axiom systems to teach logic and reasoning to first year mathematics students. Incorporates a collaborative technique to train students to write formal arguments in a non intimidating atmosphere by applying logic informally to small axiomatic systems. Provides examples of student-designed systems. 43. {ber:str} K. F. Berg. Structured cooperative learning and achievement in a high school mathematics class. Paper presented at the Annual Meeting of the American Educational Research Association, Atlanta, GA. (ERIC Document Reproduction Service No. ED364408). This study of college-bound 11th graders assessed the feasibility and effectiveness of instruction that used a structured cooperative learning technique. The students worked in dyads with scripts that contained two learning situations with two roles: (1) explainer and checker; and (2) solver and checker. Both students then worked on summary questions and homework. Verbal interaction influenced learning and appeared to be a mediator of the effects of student characteristics on achievement. Specifically, the study focused on two questions: (1) Can an effective program using dyadic studying techniques be designed for a high school course in higher mathematics; and (2) When high school students are trained to use a dyadic studying strategy for learning from their text, what is the nature of their verbal interaction and does this interaction change over time? Two groups were compared using the same texts, tests, and teacher. Both questions were answered affirmatively and supported statistically. The study concluded that: (1) students can be expected to respond positively to the experience and to work cooperatively and productively together; and (2) 94% of the time students had on-task interaction. Numerous tables contain specific statistical information. Contains 47 references. 44. {blu-kir:prefo} W. Blum and A. Kirsch. Preformal proving: examples and reflections. Educational Studies in Mathematics, 22:183-203, 1991. Inspired by Branford's and Wittmann's three levels of proving (experimental, intuitional or preformal, and formal) the authors argue that, while experimental proofs are not really proofs at all, it is possible for preformal proofs to be valid. In fact, they argue that more mathematics in the classroom should be done on a preformal level. The authors define preformal proof as `a chain of correct, but not formally represented conclusions which refer to valid, non-formal premises.' They emphasize that rigorous and formal are not synonymous, and that correct preformal proofs are valid, rigorous proofs. They elaborate on this definition and illustrate it with a preformal proof that a non-trivial solution to the differential equation [$f^{\prime} = f$] has no zeros. Other examples are also given and analyzed. 45. {boal-esm-99} J. Boaler. Participation, knowledge and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40:259-281, 1999. 46. {bon:symbi} M. V. Bonsangue. Symbiotic effects of collaboration, verbalization, and cognition in elementary statistics. In J. J. Kaput and E. Dubinsky, editors, Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results. Mathematical Association of America, 1994. The author studied the effect of collaborative learning on community college students' achievement in introductory statistics. He compared two sections of the course, one taught using cooperative learning techniques for small-group problem solving and one taught using a traditional lecture format. Student achievement was measured on four common course examinations. The researcher found that the collaborative learning class did significantly better than the traditional class on the second, third, and fourth exams. There was no significant difference between the two groups on the first exam. 47. {bor-ros:journ} R. Borasi and B. Rose. Journal writing and mathematics instruction. Educational Studies in Mathematics, 20:347-365, 1989. The authors conducted a teaching experiment in which students in an algebra course kept journals. An analysis of the journals and the students' evaluations of the journal activity was undertaken, revealing benefits to students and teachers of journal writing. The paper emphasizes the value of the student-teacher dialog created by the journal activity. Numerous examples of student journal entries illustrate the authors' conclusions. 48. {bran-cmesg-88} L. Brandau. The power of mathematical autobiography. In L. Pereira-Mendoza, editor, Proceedings of the 1988 Annual Meeting of the Canadian Mathematics Education Study Group, pages 143-159, Winnipeg, Manitoba, Canada, 1988. 49. {bre-hir-77:effe} S. M. C. Brechting and C. R. Hirsch. The effects of small group-discovery learning on student achievement and attitudes in calculus. MATYC Journal, 11(2):77-82, 1977. Compared were the effects of two modes of instruction in the calculus: small group discovery and traditional lecture-discussion. The discovery mode was more effective in producing successful achievement in areas of manipulative skills, there were no differences in achievement as measured by a concepts test. 50. {dubi-esm-92} D. Breidenbach, E. Dubinsky, J. Hawks, and D. Nichols. Development of the process conception of function. Educational Studies in Mathematics, pages 247-285, 1992. 51. {bre-dub-haw-nic-92:dev} D. Breidenbach, E. Dubinsky, J. Hawks, and D. Nichols. Development of the process conception of function. Educational Studies in Mathematics, 23:247-285, 1992. The authors argue that APOS theory, and how it applies to the concept of function, points to an instructional treatment, using computers, that results in substantial improvements in the understanding of function for many students. The students appear to develop a process conception of function and are able to use it to perform certain mathematical tasks. 52. {bre:howsh} D. M. Bressoud. How should we introduce integration? College Mathematics Journal, 23(4):296-98, 1992. Teaching the concept of integration differs depending on which of four perspectives is used to introduce the topic. Presents a method based on the historical development of the use of integration that introduces integral as antiderivative. Discusses examples of differential equations used in the development and ways to connect this to the other perspectives. 53. {bre:whydo} D. M. Bressoud. Why do we teach calculus? American Mathematical Monthly, 99(7):615-617, 1992. Discusses two answers to the question of why we teach calculus in the college mathematics curriculum: (1) calculus is used in real mathematical applications across a variety of disciplines; and (2) the historical development of calculus exposes students to the foundation of the scientific world view. 54. {bri-bur-mar-mcl-ros:thede} J. Britton, T. Burgess, N. Martin, A. McLeod, and H. Rosen. The Development of Writing Abilities. Macmillan, London, 1975. This book is often quoted in the literature on writing in mathematics education. The authors classify writing as to function. Their categories include expressive writing, which records feelings and thoughts, and transactional writing, which is used to inform, persuade or instruct. 55. {bro-ral:impac} P. A. Brosnan and T. G. Ralley. Impact of calculus reform in a liberal arts calculus course. In Proceedings of 17 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Columbus, OH, 1995. This report describes the changes in a freshman-level calculus course that occurred as a consequence of adopting the Harvard Consortium Calculus text. The perspective is that of the lecturer. The course is intended as an introduction to calculus for liberal arts students, that is, students who will not be expected to use calculus as a mathematical tool in their area of major study. The perceptions of the instructor about global changes that occurred in the course include that learning was different, the material was appropriate to students' needs and level of sophistication, students came away with different attitudes about mathematics, the use of the graphing calculator opened up new ways of understanding and representing mathematics, the use of cooperative groups is an important technique in promoting student involvement in learning, and the increased use of writing was critical to students' learning conceptually rather than mechanically. 56. {brou-trans-97} G. Brousseau. Theory of Didactical Situations in Mathematics. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. Translated into English and editted by N. Balacheff, M. Cooper, R. Sutherland, and V. Warfield. An editted and annotated collection of translations of Guy Brousseau's seminal papers on didactique. The introduction gives the details of a classroom lesson to illustrate the facets of the didactical contract idea; the first chapter is a more theoretical and deeper discussion of aspects of classroom interaction. Subsequent chapters address adidactical situations, epistemological obstacles, problems, and didactical engineering; problems with teaching decimal numbers; the didactical contract: the teacher, student and milieu; objects, usefulness, and difficulties of didactique for the teacher. 57. {bro-dev-dub-tho-97:learn} A. Brown, D. DeVries, E. Dubinsky, and K. Thomas. Learning binary operations, groups, and subgroups. Journal of Mathematical Behavior, 16(3):187-239, 1997. The authors examine how abstract algebra students might come to understand binary operations, groups, and subgroups. Using APOS theory, they give preliminary analyses of what it could mean to understand these topics. They describe an instructional treatment designed to help foster the formation of mental constructions postulated by the theoretical analysis, and discuss the results of interviews and exam performance which suggest that the instruction was successful. Based on the data collected, they propose revised epistemological analyses of these topics, and give some further pedagogical suggestions. 58. {bro-tho-tol-99:nature} A. Brown, K. Thomas, and G. Tolias. The nature and development of preservice elementary teachers' understanding of divisibility. The manuscript is in preparation. The authors report on an examination of prospective elementary teachers' understanding of the concept of multiples, with a particular focus on the least common multiple. Students' understanding is examined using the APOS theory. 59. {bro:thebe} M. L. Brown. The behavioral and attitudinal differences between math students in a contextualized classroom learning environment and math students in a traditional classroom learning environment at a mid-size, rural community college. PhD thesis, Miami University, 1997. This dissertation explored the use of a contextualized teaching method compared to a traditional method of instruction in four pre-calculus courses at an Upstate New York community college in order to investigate the relative effectiveness of each style. The researcher administered pre- and post-tests to ascertain students' beliefs and perceptions regarding math and conducted classroom observations, interviews with both the students and three professors and an analysis of the textbooks used. Two of the classes were taught using a traditional approach and two used a contextualized approach. Three of the four professors involved agreed to participate in interviews and observations with the researcher. The fourth professor allowed his students to participate in the study; however, he declined to participate further in the role as instructor. Results were analyzed by calculating the change in percentage points in the students' pre- and post-tests regarding their perceptions about math. Additionally, the student interviews, instructor interviews, classroom observations, and comparisons of the two textbooks used in teaching the course were analyzed. It was the researcher's assumption at the onset of this study that a contextualized method of instruction could be a pivotal approach to changing the community college's pedagogy as the college prepares its diverse student population for the new workforce paradigm. However, the findings of this study did not suggest that the contextualized pedagogy is as strong as the researcher originally thought it would be. The results of this study indicate the limitations of this investigation and possible directions for future research. Recommendations based on this study include the following: the need for a larger group in future studies; the need for studying a variety of settings of community colleges to obtain a more ethnically and geographically diverse population of students; the need to study the role of the instructors in overall student outcomes; and the need to take into account the learning environments and experiences of the students prior to the community college setting. All played an important role in the outcomes of this study. It will also be important to establish control groups in some future studies to more clearly ascertain which specific pedagogical components impact the students' attitudes and behaviors as related to the workplace. 60. {bru:acomp} M.R. Brunett. A comparison of problem-solving abilities between reform calculus students and traditional calculus students. PhD thesis, The American University, 1995. The purpose of this study was to evaluate the effects of CCH calculus instruction on problem solving achievement with the students enrolled in CCH sections as the experimental group and students enrolled in traditional curriculum sections as the control group. The major objective of the study was to determine whether the CCH or the traditional calculus students demonstrate greater problem solving achievement. Related objectives were to determine the specific strategies being used by the students who demonstrate better problem solving abilities, and what components of the students' problem solving procedures are consistent with demonstrated strengths or weaknesses in problem solving. The study was conducted with thirty-one students in the control group, receiving traditional instruction, and forty-one students in the experimental group, receiving CCH instruction. Each group worked the same five posttest problems as part of their final exam. The groups' performance on these problems was compared using a linear regression model, at the [$\alpha$] =.05 significance level, with the postscore as the dependent variable, the prescore as the predictor variable and the instructional curriculum as the independent variable. It was shown that calculus students who received traditional instruction performed significantly better on a problem solving posttest than calculus students who received CCH instruction. Using the Mann-Whitney Rank Test, it was shown that calculus students who received traditional instruction used significantly more problem solving techniques on an applied maximum/minimum problem than calculus students who received CCH instruction. From the results of the Fisher Exact Test for 2 x 2 Tables, it was shown that the course completion rate, with a letter grade of 'C' or better, was not significantly different for the two groups. Recommendations for further research were outlined. One recommendation was the inclusion of an interview component so that the researcher can ask subjects to provide a discussion of calculus concepts and their solutions to problems. A second recommendation was to track the success of students from the two instructional groups in subsequent mathematics and mathematics related courses. 61. {bun:astud} S. Bunch. A study of the development of written discourse competency within a graduate community. PhD thesis, Peabody College for Teachers of Vanderbilt University, 1994. (This annotation is quoted from the dissertation abstract.) `This naturalistic study explored the written discourse development of four doctoral students in a mathematics education program at a research university. The purpose of the study was to understand what the students had to learn in order to become competent users of the discourse, how they learned these things, and the role the local community of professors and graduate students played in the students' initiation into the larger discourse community of mathematics education researchers. Data were collected through participant observation, informal interviews, and document collection over a 5-month period. Data analyses were conducted using the constant-comparative method. Analyses revealed 12 competencies these students needed to develop in order to become proficient writers of their professional discourse.' 62. {bur:prior} D. T. Burkam. Prior calculus knowledge and self-selected tracking in college calculus. PhD thesis, The University of Michigan, 1994. Students in a traditional beginning college calculus course who have never studied calculus are often at a disadvantage: Their course performance is typically lower than their peers who studied calculus previously. In light of this previous exposure, or 'calculus,' gap-the difference in course performance in beginning college calculus between those students with and without prior calculus knowledge-special sections of beginning calculus are available at a large mid-western university to students who enter college without any previous exposure to calculus. The aim of offering such 'self-selected tracking' is to provide a homogeneous educational atmosphere wherein students need not be concerned that they are in any way disadvantaged because they are approaching an entirely new mathematical subject. How effective are these special sections (termed 'novice' sections) in accommodating the diversity of students' mathematical preparation? Do they offer viable, alternative peer environments for certain students? Or does the resulting academic grouping merely isolate the less-prepared students from their more-prepared peers, resulting in high-track/low-track classrooms with the frequently seen harmful effects on performance and motivation for students in the low-track setting? This dissertation employs a multi-level statistical methodology-Hierarchical Linear Modelling-which simultaneously distinguishes between and estimates the individual- and classroom-level effects. In addition, the method evaluates whether or not individual-level relationships (e.g., relationships linking gender and prior calculus background with course performance) are constant across classrooms. The results suggest that such a curricular option does not lead to improved performance. Even after controlling for incoming mathematics skill level and prior calculus experience, average student performance is lower in the novice sections. In addition, the calculus-gap widens in the novice sections. Consequently, the novice environment is neither an effective nor an equitable option for students in beginning college calculus. 63. {atk:astud} Jr. C. D. Atkins. A study to produce guidelines for evaluating calculus reform projects. PhD thesis, North Carolina State University, 1994. The purpose of the study was to produce guidelines for evaluating small-scale calculus reform projects. The study developed from the perceived need for a source of reference for prospective evaluators of calculus reform projects who are relatively inexperienced in program evaluation. The study accessed three primary sources of information, (a) literature on the calculus reform movement, (b) literature on program evaluation, and (c) field experiences with three undergraduate calculus classes at North Carolina State University. The literature provided insights into the nature of the calculus reform movement and how an evaluation of a calculus reform project can be planned. Although there is much diversity among calculus reform projects, they often address similar goals. Evaluation should be comprehensive, addressing evaluation questions that pertain to both the implementation and effects of a project. The evaluation should focus on the aspects of a project that relate to the goals of calculus reform that are incorporated into design. The classes involved in the field study were sections of the university Analytic Geometry and Calculus II course. Two of the sections were calculus reform projects. Evaluation questions were established for the calculus reform projects. This was followed by an investigation into approaches to addressing the evaluation questions with accessible sources of information. Evaluative information was collected from a variety of sources which included interviews, observations, course materials, records, questionnaires, surveys, and samples of students' work. Recommendations were proposed for using these sources of information in an evaluation of a calculus reform project. The study concludes with a chapter that presents guidelines and strategies for performing evaluations of calculus reform projects. The guidelines and strategies are given for each phase of an evaluation. Descriptions are given of how they might be applied to the evaluation of an exemplary class. The exemplary class is idealized, founded on the experiences with the calculus reform classes at the university. 64. {car:becom} L. Carlin. Becoming average: Factors influencing persistence of high-achieving college students in science and engineering programs. PhD thesis, University of Washington, 1997. Although a number of studies of achievement motivation have investigated factors related to persistence and academic achievement, it is not clear what role these factors play in long-term persistence, such as completing a college degree. The goal of this longitudinal study was to determine the influence of academic self-efficacy, academic self-concept, and reasons for pursuing a goal on long-term persistence. One-hundred-thirty first-year university students planning to major in math- or science-related degrees responded to a series of questionnaires relating to self-efficacy judgments in their first-quarter calculus, chemistry, and English or social science courses. They made normative self-concept judgments in math, science, English, and general academic ability at the beginning and end of their first year. These responses, in addition to reasons given for choosing their major, were compared by sex, and with first- and second-year persistence. Only science self-concept emerged as a consistent measure of persistence for men and women. Women's persistence was also positively related to advice (from high school teachers or parents) and negatively related to job-related concerns. There was no difference in performance (GPA) or persistence rates between men and women. These unexpected findings are discussed in relation to long-term academic persistence and math- and science-related career decisions for men and women. 65. {car-98:cross} M. Carlson. A cross-sectional investigation of the development of the function concept. Research in Collegiate Mathematics Education III, CBMS Issues in Mathematics Education, 7:114-162, 1998. In this study the author investigates students' development of the function concept as they progress through undergraduate mathematics. An exam measuring students' understandings of major aspects of the function concept was developed and administered to students who had just received A's in college algebra, second-semester honors calculus, or first-year graduate mathematics courses. Follow-up interviews were conducted with five students from each of these groups. The data analysis procedure takes into account APOS theory as well as other frameworks researchers have used to classify students' conceptual views of function. The author reaches a number of conclusions, including agreement with Breidenbach, et al. (1992) that students' understanding of functions was improved as a result of engaging in construction activities. 66. {carl-rcme-98} M. P. Carlson. A cross-sectional investigation of the development of the function concept. CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education III, 7:114-162, 1998. 67. {car:theco} G. Carmona. The concept of tangent and its relationship with the concept of derivative. PhD thesis, Instituto Tecnologico Autonomo de Mexico, 1996. Selected students who had finished two calculus courses, or who were at the end of their studies were given a questionnaire and interviewed. APOS decompositions of the concepts of tangent and derivative based on the results of Tostado (1995) were used. It was found that some of the students' conceptions are very resistant to change even when they are taught using computers or other non-traditional methods. Moreover, even if the students do change, after the calculus courses their original conceptions tend to reappear when they loose contact with formal mathematics. 68. {car:dataa} V. M. Carson. Data analysis to explore the sampling distribution of the sample mean. PhD thesis, Georgia State University, 1995. The author researched college students' understanding of the sampling distribution of the sample mean. Students used graphing calculators to simulate sampling from various non-normal distributions. The study focused on their construction, interpretation, and understanding of histograms resulting from the sampling activities. Carson's results were mixed; some students developed appropriate conceptions while other did not. She suggests increasing students' experiences with data analysis. 69. {cas:visua} T. F. Castillo. Visualization, attitude, and performance in multivariable calculus: Relationship between use and nonuse of graphing calculator. PhD thesis, The University of Texas at Austin, 1997. Graphing calculator experience has become more and more secondary knowledge for students completing a high school education. In that spirit, visualization, attitude, and overall performance effects that could be related to the graphing calculator were observed in a traditional lecture/discussion multivariable calculus course. The investigation was conducted the fall semester of 1996 as a quasi-experiment consisting of four sections of participants: two sections from San Antonio College, one section from Palo Alto College, and one section from The University of Texas at San Antonio. Students in the treatment group were given ten supplemental exercises to perform with the TI-85 graphing calculator to enhance the visualization of three dimensional points and surfaces. The remaining two sections (nontreatment group) followed the same curriculum guide without the use of the graphing calculator. The dependent variables were visualization, attitude, and performance. Assessment of these variables was through changes in pretest surveys, posttest surveys, and student scores on homework and/or quizzes, three in-class examinations, and a comprehensive final. Pretest and posttest surveys, respectively, were in terms of a Mathematical Processing Instrument, the Fennema-Sherman Mathematics Attitudes Scales, and a 'general knowledge of Calculus II' examination. The final comprehensive examination, an exit examination for most mathematics and science students, was edited for commonality by the researcher and the four participating instructors. Results indicated that the treatment and nontreatment groups had no statistically significant difference [$(/alpha = .05)$] in their mathematical processing preference. The four sub-scales used from the Fennema-Sherman Mathematics Attitudes Scales also did not register a statistically significant difference between groups. Assigned grade for overall performance between the treatment and nontreatment groups was statistically significant at [$\alpha = .05$] in favor of the treatment group. Recommendations for future research include: implementation of a TI-92 graphing calculator in calculus with open or closed laboratory; measurement of the variation in completion and success ratio of students in multivariable calculus with or without technology; development of a more sensitive set of attitude questions for a multivariable calculus class; and measurement of attitudinal behavior in younger generation instructors implementing graphing calculators versus younger generation instructors using computer-algebra-systems. 70. {cha:exper} B. L. Chance. Experiences with authentic assessment techniques in an undergraduate introductory statistics course. In American Statistical Association Proceedings of the Section on Statistical Education, pages 36-44, 1996. The author evaluated the use of student journals in introductory statistics. Students in one section were required to keep journals while students in the other section were not. The journals provided students with an opportunity to reflect on class activities, write chapter summaries, ask questions, and make connections to topics outside the course. Overall student achievement and course satisfaction were the same for both groups of students. However, there was more variability among the journal-writing group, suggesting that better students developed deeper understanding whereas weaker students became overwhelmed by the requirement and gave up on the course. 71. {cha:highs} D. Chazan. High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24:359-387, 1993. (This annotation is quoted from the paper abstract.) `Concerns about the use of computer-aided empirical verification in geometry classes lead to an investigation of students' understandings of the similarities and differences between the measurement of examples and deductive proof. The study reports in-depth interviews with seventeen high school students from geometry classes which employed empirical evidence. The analysis focuses on students' reasons for viewing empirical evidence as proof and mathematical proof simply as evidence.' 72. {chi-cogsci-81} M. T. H. Chi, P. Feltovich, and R. Glaser. Categorization and representation of physics problems by experts and novices. Cognitive Science, 5:121-152, 1981. 73. {cif:repre} V. Cifarelli. Representation processes in mathematical problem solving. In Proceedings of Annual Meeting of the American Educational Research Association, volume ?, pages ???-??, Atlanta, GA, 1993. American Educational Research Association. This study examines the construct of problem representation and the processes used by learners to construct or modify problem representations in problem-solving situations. Students (n=14) from freshman calculus courses at the University of California at San Diego participated in videotaped interviews in which they were asked to think aloud as they encountered dilemmas in solving certain tasks. The videotape protocols for each participant were analyzed and results were reported in the form of case studies. The case studies were considered as a group for the purpose of generalizing results. Three sections summarize the findings of the study. First, the levels of solution activity that the students were inferred to achieve during the interview are summarized. Second, the results of a pair of case studies are presented to illustrate individual differences in solvers' ability to construct representations. Third, the results are discussed in more general terms, including a comparison to other research findings about representation. Results indicate that: (1) traditional views of representation need to be reconsidered, (2) the process of representation appears more dynamic than previously thought, and (3) the solvers' use of increasingly abstract levels of solution activity suggests the need to address qualitative aspects of mathematical performance. Contains 23 references. 74. {cla-cor-cot-cza-dev-joh-tol-vid:const} J. Clark, F. Cordero, J. Cottrill, D. J. DeVries B. Czarnocha, D. St. John, G. Tolias, and D. Vidakovic. Constructing a schema: The case of the chain rule. Journal of Mathematical Behavior, (4):345-364, 1997. Based on an initial description (genetic decomposition) of how the chain rule concept may be learned, an attempt to interpret student interview data using APOS was made. The insufficiency of this alone led to an extension of the APOS theory to include a theory of schema development based on ideas of Piaget and Garcia. The Piagetian triad is suggested as a mechanism for describing schema development in general, and the chain rule is used as an example. The triad of the intra-, inter- and trans- levels of schema development provides the structure for interpreting the students' understanding of the chain rule and classifying their responses to interview questions about the chain rule. The results of this data analysis allowed for a revised epistemological analysis of the chain rule. 75. {cla-dev-lit-mel-mor-sch-vid-99:story} J. Clark, D. DeVries, G. Litman, M. Meletiou, S. Morics, K. Schwingendorf, and D. Vidakovic. A story of progress: Research in undergraduate mathematics education. Manuscript submitted for publication,1999. To make the point that when a group of researchers adheres to a particular framework and theoretical perspective over a period of time, considerable progress can be made both in understanding how students learn mathematics and in building pedagogy on that understanding, the authors report on the collective works of a group of researchers who have adopted APOS as the basis for research studies of college students' cognitive development and understandings of mathematical concepts. This paper provides a brief overview of APOS theory followed by a sample discrete mathematics lesson on learning (single-level) quantification. This lesson is offered as an example of how research rooted in the APOS perspective has led to pedagogical strategies. The paper then summarizes some of the research, the instructional materials, and the methods that have been developed over the past decade using this framework and theoretical perspective. The research described is in the areas of student understandings of concepts in pre-calculus, calculus, abstract algebra, statistics, and discrete mathematics. 76. {cla-hem-stj-tol-vak:stud} J. Clark, C. Hemenway, D. St. John, G. Tolias, and R. Vakil. Student attitudes toward abstract algebra. Primus, to appear. The authors report on one study of a research and curriculum development program in abstract algebra. The instructional treatment is based on APOS theory and places special emphasis on computer programming activities and cooperative learning. Students from both this and more traditional courses were interviewed about their impressions of the course and abstract algebra in general. Their responses favored the computer/cooperative learning approach in many ways, even though the content of this course was at least as rigorous and demanding for them as that of the more traditional courses. 77. {cla-mat:succ} J. Clark and D. Mathews. Successful students' conceptions of mean, standard deviation and the central limit theorem. Manuscript in preparation, 1999. The authors present analyses based on APOS of audio-taped clinical interviews with college freshmen immediately after they completed an elementary statistics course and obtained a grade of ``A''. The authors find that APOS is a useful way of describing students' understanding of mean, standard deviation, and the Central Limit Theorem. In addition, they conclude that traditional instruction in statistics does not help students make the appropriate mental constructions. In particular, traditional instruction seems to inhibit students from moving from a process to an object conception of standard deviation, and that it is very difficult for students to move beyond a strong process image of standard deviation. 78. {cla-way-ste:probi} D. Clarke, A. Waywood, and M. Stephens. Probing the structure of mathematical writing. Educational Studies in Mathematics, 25:235-250, 1993. (This annotation is quoted from the article abstract.) `This paper examines one mode of mathematical communication: that of student journal writing in mathematics. The focus of the discussion is a study of four years' use of journal writing in mathematics involving approximately 500 students in Grades 7 through 11 in a particular Victorian secondary school. The evaluation of the experimental use in one school of journal writing in mathematics provides a powerful demonstration of the link between language and mathematics and suggests a relationship between students' mathematical writings and their perceptions of mathematics and mathematical activity.' 79. {coa:first} J. E. Coakley. First-year experiences in mathematics: A follow-up study of able high school students. PhD thesis, Harvard University, 1990. This thesis is a follow-up survey of the experiences of 47 graduates from one high school during their first semester in a variety of colleges. The study focuses primarily on their attitudes and perceptions in mathematics. The research literature has documented that psycho-social variables such as perceived usefulness and confidence dimensions in mathematics influence student enrollment in mathematics. Developmental psychologists have hypothesized that females may learn math differently from males. The study has three major goals: to examine levels of achievement and attitudes among a sample of 12th grade calculus students; to explore the experience of these same students in their first semester mathematics courses through a follow-up survey; and to examine in detail the experiences in the perceptions and enrollment patterns of a subgroup of young women as they report on their attitudes about studying mathematics, first in high school and later in the first year of college. The method for this study is a descriptive analysis of data from the high school transcript including SAT-M scores, final grades, and exams on the achievement of these calculus students during their senior year in high school. Additional data came from two questionnaires designed to assess their high school and college mathematics attitudes. To supplement this data, individual in-depth interviews were conducted with ten female students in high school about their learning experiences in mathematics. Data analysis consisted of focusing initially on comparing the high school questionnaires with the college survey and then exploring gender differences in the high school and college data, including the interviews. The major findings from this study showed that these high-achieving students who demonstrated many similarities in attitudes and achievement in high school differed in their perceptions and pursuit of mathematics as undergraduates. Eighty-five percent of the males compared to 51preference for courses that were more useful and relevant to their lives. Such information may be useful in designing mathematics curricula for students, both female and male, who are disenfranchised in mathematics. 80. {con-90:fro} J. Conciatore. From flunking to mastering calculus: Treisman's retention model proves to be ``too good'' on some campuses. Black Issues in Higher Education, 6:5-6, 1990. This paper describes the development of a model for improving the calculus achievement of minority group college students currently used by 25 institutions of higher learning. Utilizes group study and an ``honors class,'' rather than a remedial approach. 81. {con-vil:writi} P. Connolly and T. Vilardi, editors. Writing to Learn Mathematics and Science. Teachers College Press, New York, NY, 1989. Here is the table of contents: Foreward: The Ordinary Experience of Writing, Leon Botstein Writing and the Ecology of Learning, Paul Connolly Writing and Mathematics: Theory and Practice, Barbara Rose Using Writing to Assist Learning in College Mathematics Classes, Marcia Birken Writing to Learn Science and Mathematics, Shelia Tobias Reflections on the Uses of Informal Writing, Alan Marwine Writing Is Problem Solving, Russel W. Kenyon Locally Original Mathematics through Writing, William P. Berlinghoff Writing and the Teacher of Mathematics, David L. White and Katie Dunn Writing `Microthemes' to Learn Human Biology, Kathryn H. Martin The Synergy between Writing and Mathematics, David Layzer Exploring Mathematics in Writing, Sandra Keith Writing to Learn: An Experiment in Remedial Algebra, Richard J. Lesnak Writing as a Vehicle to Learn Mathematics: A Case Study, Arthur B. Powell and José A. López Writing in Science Education Classes for Elementary School Teachers, Mary Bahns The Advanced Writing Requirement at Saint Mary's College, Joanne Erdman Snow Qualitative Thinking and Writing in the Hard Sciences, Willaim J. Mullin What's an Assignment Like You Doing in a Course Like This? Writing to Learn Mathematics, George D. Gopen and David A. Smith On Preserving the Union of Numbers and Words: The Story of an Experiment, Erika Duncan They Think, Therefore We Are, Anneli Lax Writing and Reading for Growth in Mathematical Reasoning, Hassler Whitney The Dignity Quotient, Dale Worsley Is Mathematics a Language?, Vera John-Steiner A Mathematician's Perspective, Reuben Hersh 82. {coo:evalu} L.A. Cooley. Evaluating the effects on conceptual understanding and achievement of enhancing an introductory calculus course with a computer algebra system. PhD thesis, New York University, 1995. This study examined the effects on achievement and conceptual understanding of integrating a computer algebra system, CAS, into an introductory calculus course. The researcher sought to determine whether students in a CAS enhanced calculus course developed a higher level of conceptual understanding of key concepts (limit, derivative, instantaneous rate of change, integral, maximum and minimum, and curve sketching) than students in the traditional calculus course. The researcher also sought to determine whether students in the CAS enhanced calculus course had higher achievement than students in the traditional class on standard calculus problems. Two calculus classes were studied at New York University, a large, urban, private university. One class was enhanced with a computer component which included laboratories written for Mathematica, a computer algebra system, on MacIntosh computers. The other class was taught in the traditional manner, without technology. Background data were collected from both classes at the beginning of the semester. Both classes completed a conceptual exam at the end of the semester to measure conceptual understanding of the six calculus concepts; limit, derivative, instantaneous rate of change, integral, maximum and minimum, and curve sketching. Five students from each class were interviewed at the end of the semester and discussed various calculus questions. The two groups of students were very similar in their background characteristics. The only significant difference was that a larger percentage of students in the technology group had previously completed a high school calculus course. Therefore, previous completion of a calculus course was used as a covariate to compensate for this difference. Students registered for the sections through normal registration procedures. Hence, these two groups are a good representation of students studying calculus at a large, urban, private university. The students in the technology group scored significantly higher in three of the six conceptual areas: limit, derivative, and curve sketching. The non-technology group did not score higher in any of the conceptual areas. The overall, total conceptual scores were also significantly higher for the technology group. The technology group also scored significantly higher on the traditional calculus questions. This shows that these students did not suffer any loss of computational skills, which is a fear often voiced by educators who are against the use of technology in the mathematics classroom. Since the technology group scored significantly higher on both the conceptual examination and the traditional calculus questions, it appears that having the conceptual basis may aid students in the algorithmic processes. 83. {cor-sol:actos} F. Cordero and M. Sol?s. Actos visuales y anal?ticos en el entendimiento de la ecuaciones diferenciales lineales. In R. FarfÀn, editor, Actas de la Und?cima Reuni?n Latinoamericana de MatemÀtica Educativa. Grupo Editorial Iberoam?rica, primera edici?n, 1997. The authors report on a research project about understanding linear differential equations using analytic and visual acts based on mental constructions as described in APOS theory. 84. {cor:limit} B. Cornu. Limits. In D. Tall, editor, Advanced Mathematical Thinking. Kluwer, Boston, 1991. To be added later. 85. {cot:acano} C. P. Cotten. A canonical correlation analysis of the cognitive variables of mathematics achievement and critical thinking and the affective variables of attitude toward mathematics, confidence in learning mathematics, and effectance motivation in learning mathematics. PhD thesis, The University of Southern Mississippi, 1992. The purpose of this research was to provide information on the effectiveness of utilizing certain learner characteristics as predictors of critical thinking skills and mathematics achievement in college students. Subjects in the study were four-year university students enrolled in College Algebra, Calculus II, and Calculus for Business Majors. Critical thinking skills were measured by the Watson-Glaser Critical Thinking Appraisal and the level of mathematics achievement was determined by the college mathematics grade-point average. Attitude Toward Mathematics, Confidence in Learning Mathematics, and Effectance Motivation in Learning Mathematics were assessed using the Fennema-Sherman Mathematics Attitude subscale scores. By utilizing canonical correlation analysis, a collection of demographic data and cognitive and affective variables were organized into sets of multiple criterion and multiple predictor variables. The analysis allowed the investigation of the possible predictive quality of certain learner behaviors and the interrelationships among these behaviors. The results of the study indicated that from 16sets can be explained. Several educational considerations emerged from the analysis: (1) the gender and age of a college student are not significant predictors of mathematics achievement and critical thinking skills, (2) Confidence in Learning Mathematics is a significant predictor of Effectance Motivation in Mathematics, (3) college mathematics grade-point average is a significant predictor of Confidence in Learning Mathematics and Effectance Motivation in Mathematics, (4) critical thinking skills in college students are not significant predictors of college mathematics achievement, and (5) the Watson-Glaser total score and the subtests of Interpretation and Deduction are significant predictors of Confidence in Learning Mathematics, Attitude Toward Success in Mathematics, and Effectance Motivation in Mathematics. 86. {cot-99:stud} J. Cottrill. Students' understanding of the concept of chain rule in first year calculus and the relation to their understanding of composition of functions. PhD thesis, Purdue University, 1999. This is a follow-up study to Clark, Cordero, et. al (1997). The author finds that the triad mechanism describes the observations of student behaviors and can be used to develop instruction to help students make certain mental constructions. It presents more detailed descriptions of the intra-, inter-, and trans- levels of the development of the chain rule schema than were given in Clark, Cordero, et. al (1997). 87. {cot-dub-nic-sch-tho-vid:under} J. Cottrill, E. Dubinsky, D. Nichols, K. Schwingendorf, K. Thomas, and D. Vidakovic. Understanding the limit concept: Beginning with a coordinated process schema. Journal of Mathematical Behavior, 15(2):167-192, 1996. The authors suggest a new variation of the dichotomy between dynamic or process conceptions of limit and static or formal conceptions. They also propose explanations of why these conceptions are so difficult for students to construct. 88. {cou:writi} J. Countryman. Writing to Learn Mathematics. Heinemann, Portsmouth, NH, 1992. 89. {cra:foste} P. Crawford. Fostering reflective thinking in first-semester calculus students. PhD thesis, Western Michigan University, 1998. This study focuses on the fostering of reflective thinking in students in a reform calculus course through completion of homework assignments incorporating reflective tasks, and the effect of these assignments on student understandings of calculus and conceptions of mathematics. The study, conducted in Fall 1997, involved two sections of first-semester calculus at a large midwestern university and used quantitative (n = 25, 18) and qualitative (n = 7) techniques. Homework assignments incorporating reflective tasks included asking students to compare and contrast textbook ideas; to write about how obstacles were overcome as they attempted exercises; to develop concept maps organizing and relating course material; and to explain, in writing, strategies regarding specified tasks. Analysis of covariance with pretest achievement scores as covariate was used to analyze student performance on four examinations by section. Student responses at the beginning and end of the semester to an inventory of mathematical conceptions were analyzed by section using a two-sample t-test. Audiotaped 'think aloud' problem sessions were conducted with selected treatment section students and analyzed by category of thought using time-line graphs, which provided detail on thinking used during problem solving unavailable from in-class examinations. No significant differences in adjusted means were determined on the four examinations. Inspection of regression lines of examination scores and intersection points revealed an interaction between treatment and precalculus achievement. Students scoring at the 12th percentile had better achievement on Exam 1 than control students. Students had better achievement than the control students at the 28th percentile for Exam 2, at the 32nd percentile for Exam 3, and at the 44th percentile for the Final examination. As the semester progressed, an increasing number of students appeared to benefit from the treatment. The 'think aloud' problem sessions supported this benefit of treatment. By the end of the semester, students exhibited categories of reflective thinking, such as Direction of Thinking, which were virtually absent at the beginning of the semester and exhibited more repetition and variety in their categories of thought. 90. {cro:dista} D. J. Crowe. Distance education utilizing two-way interactive television and special features. A 4th year evaluation study: Perceptions by students and parents. PhD thesis, Wayne State University, 1990. Rural school districts have been searching for alternatives which would permit greater parity with larger urban and suburban districts to provide advanced high school courses to their students. Telecommunications technologies such as two-way interactive television have opened up opportunities for rural school districts to coordinate schedules and to share resources, thereby providing an expansion of curricular offerings and educational opportunities for students. Measuring perceptions and opinions of the various groups piloting involvement with this medium can help in the future development of teacher and student training models. Technology aside, effective training and use of two-way interactive television is key to a successful program. If educators concerned with small rural schools are interested in realizing this vision, and there is ample evidence that they are, then substantial changes must occur in how teachers and students in small rural schools are trained. This study analyzed the perceptions of parents and students who have participated in Providing Academics Cost Effectively (PACE) project's two-way interactive instructional program centered in Cheboygan, Otsego and Presque Isle Intermediate School District. It investigated the responses of each group of participants and observers in relation to the present project which is in its fourth year of operation. The population studied consisted of all students currently enrolled in the PACE two-way interactive television courses at the various sites. PACE courses being offered to students via two-way interactive television included advance courses in Calculus/Statistics, French, German, and a general course, Media/Electronics. Parents or guardians of the students enrolled were also surveyed. Results of data analysis disclosed that parents and students did not differ in their opinions toward the use of two-way interactive television in distance education at the high school level. While both groups were positive toward the use of two-way interactive television, parent opinions appear more favorable than those of students who indicated higher levels of reservation with this system. Parents were not in the classroom and did not experience first hand the technical problems that seemed to be the focus of complaints by students in their subject comments located in the end of the survey. 91. {cul:theca} E. Culotta. The calculus of education reform. Science, 255(5048):1060-1062, 1992. Discusses, analyzes, and provides anecdotes about the calculus reform movement, in general, and the experimental, undergraduate calculus classes at Duke University, in particular. 92. {cun-smi:amath} R. S. Cunningham and D. A. Smith. A mathematics software database update. College Mathematics Journal, 18(3):242-247, 1987. Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. 93. {cur:under} C. R. Curjel. Understanding vector fields. American Mathematical Monthly, 97(6):524-527, 1990. Presented are activities that help students understand the idea of a vector field. Included are definitions, flow lines, tangential and normal components along curves, flux and work, field conservation, and differential equations. 94. {dahl-hous-esm-97} R. P. Dahlberg and D. L. Housman. Facilitating learning events through example generation. Educational Studies in Mathematics, 33:283-299, 1997. 95. {dav-89:coop} N. Davidson. Cooperative learning and mathematics achievement: A research review. Cooperative Learning, 10(2):15-16, 1989. The current status of research and development on cooperative learning in mathematics is discussed in an interview with Neil Davidson. The focus is on significant achievement differences favoring small group procedures. Key issues discussed here are cooperative learning and problem-solving skills, and students' need for some kind of reward. 96. {dav-89:coo} N. Davidson. Cooperative learning in mathematics. Cooperative Learning, 10(2):2-3, 1989. Small group cooperative learning can be applied with all age levels of mathematics students. A community of learners actively working together enhances each person's mathematical knowledge, proficiency, and enjoyment. Classroom procedures for cooperative mathematics lessons are outlined. 97. {dav-90:coope} N. Davidson, editor. Cooperative Learning in Mathematics: A Handbook for Teachers. Addison-Wesley Publishing Company, Inc., 2725 Sand Hill Rd., Menlo Park, CA 94025, 1990. Small group cooperative learning provides an alternative to both traditional whole-class expository instruction and individual instruction systems. The procedures described in this volume are realistic, practical strategies for using small groups in mathematics teaching and learning with methods applicable to all age levels, curriculum levels, and mathematical topic areas. Included are: (1) an introduction and overview to orient the reader; (2) problem solving and exploration with manipulative materials in groups of four; (3) a bilingual integrated mathematics/science program addressing classroom status; (4) team learning approaches based upon individual accountability and team recognition; (5) a general conceptual model of cooperative learning with a detailed discussion of its basic components; (6) three computer-based cooperative learning strategies for classroom use; (7) various learning activities for the initiation of a cooperative classroom setting; (8) procedures for group problem solving and inquiry in algebra, geometry, and trigonometry; (9) a narrative on cooperation in heterogeneous group mathematics in the Netherlands; (10) group interactions in algebra and calculus using computers for problem solving; (11) free exploration and guided discovery in cooperative groups with examples from calculus; (13) issues affecting the use of cooperative learning in mathematics with emphasis on teachers' decision making and factors affecting implementation; and (14) appendices that include information about the sponsoring organizations, annotated survey responses from classroom teachers, and annotated listing of 38 resource materials with contact person. 98. {dav-kro:over} N. Davidson and D. L. Kroll. An overview of research on cooperative learning related to mathematics. Journal for Research in Mathematics Education, 22:362-365, 1991. Increased use of cooperative learning methods is a visible change mathematics education in the last decade. Research questions on cooperative learning concerning different models employed, their effectiveness compared to traditional methods of instruction, their effects on student achievement, and cognitive and affective benefits gained during student learning are reviewed. 99. {dav-ole:how} N. Davidson and P. W. O'Leary. How cooperative learning can enhance mastery teaching. Educational Leadership, 47(5):30-33, 1990. Transforms the debate over cooperative learning and Hunter's mastery teaching model by illustrating how both approaches reinforce each other. Mastery teaching synthesizes the most rewarding aspects of traditional expository instruction, while cooperative learning breathes life into that teaching by inviting both students and teachers to become idea ``coproducers.'' Includes 17 references. 100. {dav:visua} P. Davis. Visual theorems. Educational Studies in Mathematics, 24:333-344, 1993. The author argues that our view of mathematics should include visual theorems which he defines in such a way as to include: (1) all the results of elementary plane and solid geometry that appear to be intuitively obvious; (2) all the theorems of calculus (or of the higher mathematical disciplines) that have an intuitively geometric or visual basis; (3) all graphical displays (hand-drawn or otherwise) from which certain pure or applied mathematical conclusions can be derived almost by inspection; (4) graphical results of computer programs which the brain organizes coherently in a certain way. 101. {dav:theth} R. Davis. The theoretical foundations of writing in mathematics classes. Journal of Mathematical Behavior, 12:295-300, 1993. The author responds to the article ``Writing for Conceptual Development in Mathematics,'' by Richard G. Shepard in Journal of Mathematical Behavior, 12, 287-293 (1993). 102. {dea:teach} E. Dean. Teaching the proof process: a model for discovery learning. College Teaching, 44:52-55, Spring 1996. The article describes a model for teaching proof called the `Super Student' model used by the author in number theory, abstract algebra, geometry and real analysis classes. The six phases of the model are open, brainstorm, instantiate, convince, reflect, and extend. The model is somewhat similar to the problem solving plans of Polya and Schoenfeld. Classroom strategies involving the model are discussed. 103. {debell-pme-98} V. A. DeBellis. Mathematical intimacy: Local affect in powerful problem solvers. In S. Berenson et al., editor, Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume 2, Raleigh, North Carolina, 1998. 104. {debell-gold-pme-99} V. A. DeBellis and G. A. Goldin. Aspects of affect: Mathematical intimacy, mathematical integrity. In O. Zaslovsky, editor, Proceedings of the 23rd International Conference of the International Group for the Psychology of Mathematics Education, volume 2, pages 249-256, Haifa, Israel, 1999. 105. {dee-89:coo} R. Dees. Cooperative mathematics lesson plans. Cooperative Learning, 10:32-40, 1989. The article presents several classroom-tested model cooperative mathematics activities, ranging from simple to highly structured, for all grade levels. Some of the activities are creating and solving problems from the newspaper; playing team-building games; exploring with fractions; and working on grouping and informal computation using a jar full of candy kisses. 106. {dee-91:role} R. Dees. The role of cooperative learning in increasing problem-solving ability in a college remedial course. Journal of Research in Mathematics Education, 22(5):409-421, 1991. Students (n=105) enrollee in a college remedial mathematics course participated in a one-semester experiment to determine whether cooperative learning would help students increase their problem-solving skills in mathematics. Results indicated significant differences in favor of students using cooperative learning in solving word problems in algebra and geometry. 107. {defr-rcme-96} T. C. DeFranco. A perspective on mathematical problem-solving expertise based on the performance of male Ph.D. mathematicians. CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education II, 6:195-213, 1996. 108. {den:histo} D. Dennis. Historical perspectives for the reform of mathematics curriculum: Geometric curve drawing devices and their role in the transition to an algebraic description of functions. PhD thesis, Cornell University, 1995. This thesis establishes the following three claims: (1) Experience with mechanical curve drawing devices played a role of fundamental historic and conceptual importance of in the development of analytic geometry, algebraic symbolism, calculus, and the notion of functions. (2) Two secondary students of mathematics benefited from their experiences with physical curve drawing devices, and that both the geometric and algebraic analysis of such devices raised, for them, crucial epistemic issues, the consideration of which led them to engage in a more balanced dialogue between the physical world and symbolic mathematical language. (3) A mathematical discussion of the tangents, areas, and arclengths associated with many curves need not be deferred until calculus, and that, quite the contrary, an understanding of the semiotic importance of calculus comes from seeing it as a language constructed from physical experiences. Such an understanding depends upon being able to correlate symbolic manipulations with independently verifiable geometric and physical experience. Such experience can be readily gained from the use of mechanical curve drawing devices, and from simulations of such devices using available dynamic geometry computer applications. The first and third claims are established through a detailed analysis of the genetic epistemology of analytic geometry and calculus. This analysis proceeds from original historical documents mostly from seventeenth century European sources. Based upon historical curve drawing techniques, both physical replicas of devices and computer simulations are used to explore the properties of curves. The second claim is established through a series of clinical interviews with two high school students. The students worked with replicas of three different physical curve drawing devices. Their conceptions, inventions, beliefs, and modes of expression are analyzed from videotapes made during these interviews. 109. {MR97m:01024} D. Dennis and J. Confrey. The creation of continuous exponents: a study of the methods and epistemology of John Wallis. In Research in collegiate mathematics education, II, pages 33-60. Amer. Math. Soc., Providence, RI, 1996. The subject of the article is connected to issues in mathematics education that are stimulated in part by Imre Lakatos' philosophical writings. Lakatos wanted to look beyond the logical formalist presentation of mathematics and to uncover the underlying methodology whereby new mathematics is actually produced. His concept of rational reconstruction, of idealized historical case studies, was employed as a critical tool to accomplish this goal. Although Dennis and Confrey are sympathetic to Lakatos' position, they favour a more fundamental place for history in understanding mathematics, as a forum for what they term "multiple forms of representation". They contrast the logical formalist approach of modern mathematics with what they characterize as the more practical and "empirical" nature of mathematics in the seventeenth and eighteenth centuries. The core of the paper is an account of Wallis' Arithmetica infinitorum (1655), including a very nice exposition of his original derivation of the famous Wallis formula for [$\pi/2$]. In the conclusion, the authors observe that "The methods of investigation outlined in this paper have been largely purged from our mathematics curriculum. These mathematical results are now presented to students in a formal logical setting that came about in the nineteenth and twentieth centuries." (The modern derivation of Wallis' formula, not given in the paper, involves recursive expressions for the integral of [$\sin\sp{2n}\theta$] on the interval from 0 to [$\pi/2$].) They argue persuasively that a student of mathematics can learn much from a study of the older methods and results. Historians may take exception to the characterization of mathematics in the period 1600-1800 as empirical/practical and mathematics after 1800 as logical/formalist. There were definite rational principles and points of view in the older period, and there are also pragmatic elements in modern mathematics. On the other hand, the idea of multiple forms of representation does seem a very useful concept, and points to the existence of a deep underlying epistemological relativism in the historical development of mathematics. 110. {dori-icmi-98} J. L. Dorier, J. Pian, A. Robert, and M. Rogalski. A qualitative study of the mathematical knowledge of French prospective maths teachers: Three levels of practice. In Pre-Proceedings of the ICMI Study Conference on the Teaching and Learning of Mathematics at University Level, pages 118-122, National Institute of Education, Singapore, 1998. 111. {dou:today} R. G. Douglas. Today's calculus courses are too watered down and outdated to capture the interest of students. Chronicle of Higher Education, 34(19):B1, 3, 1988. Calculus provides the language for expressing the differential equations that govern change and also the methods for solving them. In order to insure that more Americans qualify for science-related careers, the way calculus is taught must change. 112. {dre-eis:conce} T. Dreyfus and T. Eisenberg. Conceptual calculus: Fact or fiction? Teaching Mathematics and Its Applications, 9(2):63-67, 1990. Discusses a concept-oriented calculus course as an alternative to a traditional skill-oriented calculus course. Two main ideas of conceptual calculus are visual conceptualization and receptive argumentation. Includes typical examination questions. 113. {dub:teacI} E. Dubinsky. Teaching mathematical induction i. Journal of Mathematical Behavior, 5:305-317, 1986. (This annotation is quoted from the paper abstract.) `A prototype version of a novel approach to teaching mathematical induction was used in a small class of eight college students. The instructional treatment is based on a Piagetian theory of learning abstract mathematical concepts in which the learner uses reflective abstraction to construct new schemas out of old ones in a hierarchy that ultimately reaches the desired concept. The treatment uses certain computer experiences in an attempt to induce the student to make the appropriate reflective abstractions. The method is seen to be reasonably effective and several areas of possible improvement are indicated.' 114. {dub-87:teach} E. Dubinsky. Teaching mathematical induction i. Journal of Mathematical Behavior, 6(1):305-317, 1987. A prototype version of a novel approach to teaching mathematical induction was used in a small class. The instructional treatment was based on an early version of the APOS theory of learning abstract mathematical concepts in which the learner uses reflective abstraction to construct new schemas out of old ones in a hierarchy that ultimately reaches the desired concept. The treatment uses certain computer experiences in an attempt to induce the student to make the appropriate reflective abstraction. The method is seen to be reasonably effective and several areas of possible improvement are indicated. 115. {dub-89:teach} E. Dubinsky. On teaching mathematical induction ii. Journal of Mathematical Behavior, 8:285-304, 1989. This paper is a continuation of Dubinsky and Lewin (1986) and Dubinsky (1987). Here the author details two classroom experiments in which a theoretically-based instructional approach using computer experiences with SETL and ISETL was implemented. Students seem to develop a more positive attitude toward making induction proofs. They are totally successful in solving straight-forward problems. When presented with more difficult, unfamiliar problems, they tend to set up most problems correctly and it is usually clear that the students know how to use induction and intend to do so, although some difficulties with specific proofs persist. 116. {dub:teaII} E. Dubinsky. Teaching mathematical induction ii. Journal of Mathematical Behavior, 8:285-304, 1989. The author proposes a genetic decomposition of induction, outlines an instructional approach involving SETL and ISETL, and describes the results when the approach is tried with two discrete mathematics classes. 117. {dub-91:constr} E. Dubinsky. The constructive aspects of reflective abstraction in advanced mathematics. In L. P.Steffe, editor, Epistemological Foundations of Mathematical Experiences. Springer-Verlag, New York, 1991. The author presents a brief discussion of a developing theory of mathematical knowledge and its acquisition. He also describes specific methods of construction that he has observed in students. He presents an analysis of induction, quantification, and function that have been studied using this point of view. 118. {dub-1991:refl} E. Dubinsky. Reflective abstraction in advanced mathematical thinking. In D. Tall, editor, Advanced Mathematical Thinking. Kluwer, Dordrecht, 1991. The author makes the case that the concept of reflective abstraction can be a powerful tool in the study of advanced mathematical thinking, that it can provide a theoretical basis that supports and contributes to our understanding of what this thinking is, and suggests how we can help students develop the ability to engage in it. 119. {dub:alear} E. Dubinsky. A learning theory approach to calculus. In Z. Karian, editor, Symbolic computation in undergraduate mathematics education, MAA Notes, volume 24, pages 48-55. Mathematical Association of America, 1992. The author outlines the APOS theory of how people can learn mathematical concepts. He then discusses some of the choices about teaching that seem to follow from the beliefs about learning to which this theory has led him. In particular, he discusses how computers can be used in teaching and learning. 120. {dub-94:theo} E. Dubinsky. A theory and practice of learning college mathematics. In A. Schoenfeld, editor, Mathematical Thinking and Problem Solving. Erlbaum, Hillsdale, 1994. The author examines two dichotomies and looks at ways to build syntheses between two apparently disparate notions. The two dichotomies examined are that of research and development, and beliefs and choices. As part of the examination, the APOS theoretical perspective involving the mental construction of processes and objects is presented. 121. {dub-95:pro} E. Dubinsky. A programming language for learning mathematics. Communications on Pure and Applied Mathematics, 48:1-25, 1995. The author gives a brief history of the development of a pedagogical strategy for helping students learn mathematical concepts at the post-secondary level. The method uses ISETL to implement instruction designed on the basis of APOS theory. ISETL is described in some detail and examples are given of the use of this pedagogy in abstract algebra, calculus, and mathematical induction. 122. {dub:techn} E. Dubinsky. Technology tips: Is calculus obsolete? Mathematics Teacher, 88(2):146-148, 1995. Gives an example of one danger of using graphing software without an accompanying mathematical analysis by illustrating what might happen if a student were capable of dealing with the drawing of the complete graph of a function only by using a technological tool. 123. {dub:onlea} E. Dubinsky. On learning quantification. Journal of Computers in Mathematics and Science Teaching, 16(2/3):335-362, 1997. In this study the author examines students' learning of universal and existential quantification. The instruction in the course was based on the theoretical analysis of quantification found in Dubinsky, Elterman and Gong (1988) and was designed to assist students to make effective mathematical constructions in their minds by making these constructions on a computer using a programming language. Results from written questions suggest that when the pedagogical approach described is used, students can develop some understanding of quantification and the ability to work with it, even when the particular problems they are given are difficult. 124. {dub-97:learning} E. Dubinsky. On learning quantification. Journal of Computers in Mathematics and Science Teaching, 16(2/3):335-362, 1997. In this study the author examines students' learning of universal and existential quantification. The instruction in the course was based on the theoretical analysis of quantification found in Dubinsky, Elterman and Gong (1988) and was designed to assist students to make effective mathematical constructions in their mind by making these constructions on a computer using ISETL. Results from written questions suggest that when the pedagogical approach described is used, students can develop some understanding of quantification and the ability to work with it, even when the particular problems they are given are difficult. 125. {dub-deu-ler-zaz-94:learn} E. Dubinsky, J. Dautermann, U. Leron, and R. Zazkis. On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3):267-305, 1994. The authors present one of the first systematic investigations of students' construction of the concepts of group, subgroup, coset, normality and quotient group. Using APOS theory, the authors make general observations about learning these specific topics, the complex nature of ``understanding'', and the role of errors and misconceptions. 126. {dub-elt-gon-88:stud} E. Dubinsky, F. Elterman, and C. Gong. The student's construction of quantification. For the Learning of Mathematics, 8(2):44-51, 1988. In this paper, the authors detail a proposed genetic decomposition for the concept of quantification. The observations are taken from an informal study of a Discrete Mathematics class where quantification was a major topic and the instructional treatment used computer experiences with SETL, the programming language on which ISETL is based. 127. {dub-elt-cat:thest} E. Dubinsky, F. Elterman, and C. Gong. The student's construction of quantification. For the Learning of Mathematics, 8(2):44-51, June 1988. After an introductory discussion of APOS theory, the authors discuss the objects and processes which might make up a learner's quantification schema and propose a genetic decomposition of the concept of quantification. 128. {dub-har-92:nat} E. Dubinsky and G. Harel. The nature of the process conception of function. In G. Harel and E. Dubinsky, editors, The concept of function: Aspects of epistemology and pedagogy. Mathematical Association of America, 1992. The authors examine interviews with 13 students who have gone through an instructional treatment based on APOS theory and which involved ISETL programming activities to see how far beyond an action conception and how much into a process conception each student was at the end of the instruction. The authors find that the process conception of function is very complex and examine the data through a number of facets: restrictions students possess about what a function is, the severity of the restrictions, students' ability to construct a process when none is explicit in the situation, and their confusion with one-to-one. 129. {dub-lew-86:refl} E. Dubinsky and P. Lewin. Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness. Journal of Mathematical Behavior, 5:55-92, 1986. The authors formulate the beginning of a theory of learning abstract mathematical concepts at the post-secondary level (a precursor to APOS theory) by interpreting Piaget's epistemology, focusing on the model of equilibration and the concept of reflective abstraction. They then use the concepts of mathematical induction and compactness to elucidate the theoretical ideas and show that it is possible to arrive at coherent genetic decompositions of fairly sophisticated concepts. 130. {dub-yip:predi} E. Dubinsky and O. Yiparaki. Predicate calculus and the mathematical thinking of students. Paper presented at the DIMACS Symposium on Teaching Logic and Reasoning, Rutgers University, July 25-26, 1996. Available at http://www.cs.cornell.edu/Info/People/gries/symposium/symp.htm., 1996. This paper reports on ongoing research on quantification which is being conducted following the framework developed in Asiala, et. al. published in Research in Collegiate Mathematics Education, II, 1996. The authors present a preliminary report of a study in which 42 students from two universities filled out a questionnaire probing their understanding of quantification. The questionnaire consisted of 11 statements. Here is an example: `Someone is kind and considerate to everyone. True or False? Why?' A preliminary analysis of the questionnaire data is presented. Using APSU theory, the authors describe a genetic decomposition of quantification. They describe an instructional treatment modeled on the ACE cycle for teaching quantification. The instructional treatment makes use of the computer program ISETL. Subsequent papers will report on related interview data and the effectiveness of the instructional treatment. 131. {MAA44} Ed Dubinsky, David Mathews, and Barbara E. Reynolds, editors. Readings in Cooperative Learning for Undergraduate Mathematics. MAA Notes 44. Mathematical Association of America, Washington, DC, 1997. Seventeen papers that relate to the use of cooperative learning activities in undergraduate mathematics courses are contained within this collection. Papers were selected by the staff of Project CLUME (Cooperative Learning in Undergraduate Mathematics Education) and organized into categories pertaining to constructivism and the teachers' role, research and effectiveness, and implementation issues. Each paper is preceded by comments prepared by respected professionals in the field of undergraduate mathematics, and each poses some questions that may be helpful in initiating discussion of the article and its implications for practice. Papers include: ``Teachers and Learning Groups: Dissolution of the Atlas Complex (Finkel and Monk); ``Reducing Student Costs and Enhancing Student Learning (Part II), Restructuring the Role of Faculty" (Guskin); ``Collaborative Learning: Shared Inquiry as a Process of Reform" (MacGregor); ``A Framework for Research and Curriculum Development in Undergraduate Mathematics Education" (Asiala, Brown, DeVries, Dubinsky, Mathews, and Thomas); ``Small-Group Learning and Teaching in Mathematics-A Selective Review of the Research" (Davidson); ``When Does Cooperative Learning Increase Student Achievement?" (Slavin); ``The Controversy Over Group Rewards in Cooperative Classrooms" (Graves); ``Do Students Learn More in Heterogeneous or Homogeneous Groups?" (Good and Marshall); ``Within-Class Grouping: A Meta-Analysis" (Lou, Abrami, Spence, Poulsen, Chambers, d'Apollonia); ``Restructuring the Classroom: Conditions for Productive Small Groups" (Cohen); ``Teaching Problem Solving through Cooperative Grouping (Part I): Group Versus Individual Problem Solving" (Heller, Keith, Anderson); ``Learning the Concept of Inverse Functions in a Group Versus Individual Environment" (Vidakovic); ``Social Skills for Successful Group Work" (Johnson and Johnson); ``Teaching Problem Solving through Cooperative Grouping (Part 2): Designing Problems and Structure Groups" (Heller and Hollabaugh); ``Treating Status Problems in the Cooperative Classroom" (Cohen, Lotan, Catanzarite); ``The Small-Group Discovery Method in Secondary- and College-Level Mathematics" (Davidson); ``Constructing Calculus Concepts: Cooperation in a Computer Laboratory" (Dubinsky and Schwingendorf); and ``Annotated Bibliography of Science, Mathematics, Engineering and Technology (SMET) Resources in Higher Education" (Cooper and Robinson). 132. {duv:stude} S. Duvall. Students' understanding of parameters in a reform differential equations course (a pilot study). In ??, pages ???-??? The Association of Research in Undergraduate Mathematics Education, 1998. To be added later. 133. {eas:abrid} K. Easley. A bridge to calculus: A textbook for the study of functions and graphs. PhD thesis, Texas Woman's University, 1997. A course covering functions and graphs was instituted in the summer of 1995 by the mathematics department of Texas Woman's University. The purpose of the course was to prepare students for the fall semester integrated calculus-physics course. The university was participating in a National Science Foundation Grant which had objectives of integrating calculus and physics curricula, using technology in the classroom, and developing students' teaming skills. A textbook which incorporated these elements could not be obtained. Therefore, the faculty combined pieces of several sources to provide material for the summer 'Bridge to Calculus' course. This paper is the result of an attempt to create a text that not only covered the pre-calculus subjects of functions and graphs, but also instructed students on the use of the TI-82 graphing calculator in exploring functions and graphs so they might become proficient in its use. The text incorporated teaming elements into its method of presentation, and supplied a number of team projects in order to provide students with an opportunity to develop teaming skills. 134. {dos:confr} J. A. Dossey Ed. ? In Confronting the Core Curriculum: Considering Change in the Undergraduate Mathematics Major. Proceedings of the West Point Core Curriculum Conference in Mathematics, volume ?, pages ???-???, New York, USA, 1998. Mathematical Association of America. To be added later. 135. {edw:stude} H. Edwards. Student surveys and calculus reform. Primus, 3(2):133-140, 1993. Reports results of a survey of demographics and work and study habits of college freshmen mathematics students. No statistically significant correlation was found between work hours and grade outcome. 136. {pad:calcu} III E.E. Padgett. Calculus I with a laboratory component. PhD thesis, Baylor University, 1994. The purpose of this study was to compare the calculus achievement of a group of students which used a computer laboratory to the achievement of a group which used the traditional lecture method. Two separate sections of first-semester calculus were evaluated. The students in the section which used the laboratory participated in five laboratory events in place of five traditional classroom lecture periods. The computer laboratory used the Maple mathematics software program. A common final examination, a 26-question multiple choice test, was used to measure student achievement in calculus. An attitude survey was also administered at the beginning and the end of the course. The result of the final examination was that the students in the traditional section performed significantly better. A detailed question-by-question analysis offers insight into the similarities and differences between the sections. In fact, in the topics presented directly in the laboratory, the students in the section which did use the computer performed slightly better. Also, the students who used the computer laboratory scored slightly higher on questions which were studied in the laboratory than on the questions which were covered only during lecture sessions. The literature lists many calculus reform projects. In general, the computer projects which are implemented as supplemental instruction produce student benefit, while those which seek to replace the traditional calculus direct instruction show little or no effect. The Mathematical Association of America(MAA) has produced a great deal of literature concerning the teaching of calculus. One particular MAA concern is the reduction of the number of topics in the curriculum. Other literature analyzes laboratories, curriculum development, and the instructor's role in calculus. Use of the computer offers a unique opportunity for greater understanding of calculus concepts, but it is unlikely that the first-semester calculus course can successfully add a computer element without the loss of student performance in another topic. If the computer is used in calculus instruction, it should be used in a supplementary role. 137. {ellis-00} D. B. Ellis. Becoming a master student. Houghton Mifflin, New York, NY, 2000. 138. {ell:theef} M. J. Ellison. The effect of computer and calculator graphics on students' ability to mentally construct calculus concepts. PhD thesis, University of Minnesota, 1993. This study investigated the evolving cognitive development of students' concept images of the derivative in an instructional environment enhanced by TI-81 graphing calculators and the computer software A Graphic Approach to the Calculus (Tall, 1991). Technology was integrated into two Calculus and Analytic Geometry classes, facilitating a focus on multiple representations of calculus concepts and the connections between them. Qualitative, case study, methodology was used. Three one-hour task-based interviews were conducted with each of ten subjects selected from the two classes. Selection criteria used in choosing the subjects included gender, spatial ability, prior calculus experience, and precalculus background. Interview data were supplemented by pretests, posttests, unit exams, homework assignments, lab reports and exit surveys. Pretests, posttests, and exit surveys from the general class populations were examined to confirm or qualify findings from the ten case studies. Five characteristics of a mature concept image of the derivative were defined and used as a framework to describe and analyze each subject's evolving concept image. They also provided structure for a cross case analysis and a comparison of the case studies to the general class populations. Separate case reports for each of the ten subjects are presented along with a multicase analysis. The data indicated that the technologically enhanced instructional environment did positively affect students' ability to mentally construct an appropriate concept image of the derivative. The majority of the subjects constructed concept images that included most of the key components included in the derivative concept image definition. There were, however, notable exceptions. While all ten subjects could distinguish between graphs of functions and their derivatives by the end of the semester, only six could adequately sketch the graph of a fairly complicated derivative from a graph of its parent function. When presented with the graph of a derivative function, only four could adequately draw conclusions about characteristics of the parent function. Although eight subjects retained formal definition of the derivative, only five could link the formal definition of the derivative with a visual image of the limiting slope of secant lines. All ten subjects gained proficiency with symbolic differentiation. 139. {eme:theef} G.L. Emese. The effects of guided discovery style teaching and graphing calculator use in differential calculus. PhD thesis, The Ohio State University, 1993. The two major objectives of the study were to verify that students can discover a significant portion of differential calculus and to investigate the effects of the use/non-use of graphing calculators and the instructional technique (lecture/discussion or guided discovery style teaching). The development of interactive graphing technology resulted in a renewed interest in discovery learning since it is a very effective tool for student experimentation and discovery. The research design was a three group experimental study in a university freshmen differential calculus course with the following group specifications. Group 1: Use of graphing calculators and discovery approach, Group 2: Use of graphing calculators without discovery, Group 3: Traditional instruction. In the discovery section, part of the new material was covered using worksheets, where a sequence of questions/problems led to the new concept, relationship or technique. Students worked in groups, pairs or individually. They could get help from the hint-sheet, solution-sheets, their classmates and/or from the instructor. According to a questionnaire Group 1 students completed after the final exam, students found the answer on their own to 47questions on the worksheets where the answer was not previously known to them. They found the answer to an additional 22from the hint-sheets, from classmates or the instructor. Over 75students found the answer to the majority of such questions with or without hint. Most of the students (8830the discovery style teaching is a viable alternative to traditional teaching for at least part of the new material. No statistically significant differences were found on the computational, conceptual, and transfer skills parts of the pretest, nor on the following background variables: placement level, the year in which students took the placement test, their precalculus grade and the year in which they took precalculus. Analyses of covariance were used for student achievement comparisons. The scores on the corresponding subtest of the pretest served as covariates. Students' time spent on the course and the extent to which students worked with their classmates outside of class were compared. Statistically significant differences were not found between the groups on any of these variables. No instructional method proved superior to the others on comparison. 140. {enr:visua} G. Enrique. Visualization in the calculus class: relationship between cognitive style, gender, and use of technology. PhD thesis, The Ohio State University, 1994. The present study investigated the relationship between college students' preferred mode of processing mathematical information-visual or nonvisual-and their performance in calculus classes with and without technology. Students elected one of three different versions of an introductory differential calculus course: (a) using graphing calculators, (b) using the computer algebra system Mathematica [$/sp/circler,$] or (c) using no technology. A total of 139 students participated in the research volunteering from eight sections using graphing calculators, two sections using Mathematica[$/sp/circler,$] and eight sections using no technology. Presmeg's Mathematical Processing Instrument (MPI) was used to determine students' visual processing preference. The research questions were: (1) What is the relationship between preferred mode of processing mathematical information and performance in each of three different approaches to the calculus course? (2) Does gender interact with visual processing preference in relation to students' performance in three different approaches to the calculus course? (3) Does the calculus approach taken change students' preferred mode of processing mathematical information by the end of the course? (4) How do students of different visual processing preferences interact with technology? It was found that the MPI scores were normally distributed. Visualizers, nonvisualizers, and students of the harmonic type were found in every calculus approach. The following conclusions, which apply to the participants of this study, may be made on the basis of the findings: (a) students who are nonvisualizers obtain significantly better scores than visualizers in both the calculus sections using no technology and the sections using Mathematica, (b) there are no significant differences in the calculus scores obtained by visualizers and nonvisualizers in the sections using graphing calculators, (c) there are no significant sex-related differences in the degree of visualization or in the calculus performance of the students, (d) there are no significant interactions between sex, visualization preference, and calculus performance, (e) there is no significant change in the students preferred mode of processing mathematical information by the end of the course, (f) the students' visual orientation observed during task-based interviews and the software tools they use correspond to the degree of visuality indicated by the MPI. 141. {enr-pme:visua} G. Enrique. Visualization and students' performance in technology-based calculus. In Proceedings of 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH, 1995. The relationship between college students' preferred mode of processing mathematical information-visual or nonvisual-and their performance in calculus classes with and without technology was investigated. Students elected one of three different versions of an introductory differential calculus course - using graphing calculators, using the computer algebra system Mathematica, or using no technology. A total of 139 students participated in the research. Presmeg's Mathematical Processing Instrument (MPI) was used to determine students' visual processing preference. The interactions of students of different visual processing preferences with the software Mathematica were also investigated using task-based interviews. Results from the sections using graphing calculators suggest that appropriate uses of technology may equally benefit students of different cognitive styles. 142. {epp:thero} S. Epp. The role of proof in problem solving, pages 257-269. Lawrence Erlbaum Associates, Hillsdale, NJ, 1994. The author begins by examining in detail the thought processes that an undergraduate might go through in answering three elementary questions in abstract mathematics. In each example, the role of deductive reasoning in the problem solving process is evident. She then engages in a more general discussion of problem solving in the context of determining the truth value of a statement of the form: `For all (elements in some domain), if (hypothesis) then (conclusion).' Again, the role of deductive reasoning in the discovery process is evident. The next section of the paper is devoted to a discussion of obstacles to students' understanding of predicate and propositional logic. These obstacles include the fact that terms like `if' are often used in everyday language in ways that are different from the mathematically precise language of theorems. Sometimes (in both everyday language and mathematical prose) biconditionals are implied rather than explicitly stated. Often in mathematical prose, quantification is implied rather than explicit. The final section of the paper contains references to research in mathematics education and cognition which has bearing on students' ability to use deductive reasoning. She concludes with implications for classroom teaching. 143. {ern:mathe} P. Ernest. Mathematical induction: a pedagogical discussion. Educational Studies in Mathematics, 15:173-189, 1984. (This annotation is quoted from the paper abstract.) `It is observed that many students have difficulty in producing correct proofs by the method of mathematical induction. The notion of a correct proof by this method is analyzed mathematically. Subsequently, the topic is analyzed into behavioral skills and subjected to a conceptual analysis. Common misconceptions of induction are considered, with recommendations for their remediation. Finally, criteria for the analysis and evaluation of textbook treatments of induction are evolved and applied to a selection of texts.' 144. {est:graph} K. A. Estes. Graphics technologies as instructional tools in applied calculus: Impact on instructor, students, and conceptual and procedural achievement. PhD thesis, University of South Florida, 1990. The purpose of this research was to examine the effect of implementing graphics calculator and computer technologies as instructional tools in Applied Calculus. The research had three major components: the impact of the technology on the instructor, on the student, and on conceptual and procedural achievement gains. Action research methods were used to collect data about the impact of the technology on the instructor. The investigator kept detailed anecdotal notes and analyzed them for type and frequency of occurrence. Survey methods were used to collect data about the impact of the technology on the student. Five surveys, which recorded students' responses to the technologies used in class, were analysed as to type and frequency of response. Four categories, instructional design problems, syllabus-schedule outcomes, computer-peripheral difficulties, and environmental difficulties, identified the impact on the instructor. Most of the problems in the last three categories had to do with initiating technology into the classroom, and were resolved. Problems with instructional design and syllabus-schedule outcomes were explored, but not completely resolved by this research project. Determining effective uses of technology as an instructional tool requires further action research. The student survey data indicated that for the most part students believed that the calculator and computer technologies were helpful in their learning, but only if the student understood how to use the technology. Students indicated a preference for the hands-on calculator technology over the computer demonstrations. Most students indicated that they liked the interaction among algebraic, graphic, and tabular viewpoints and would have liked to have had the opportunity to learn College Algebra in this manner also. The third component employed a pretest, treatment, posttest design. Group equivalence was measured by the CLEP Test of College Algebra. For the experimental group, calculator and computer technology played a major role in class lectures and assignments. The control group lectures were traditional lectures. Throughout the semester, students were given conceptual questions on each unit test, and at the end of the semester, students were given an exit exam including both conceptual and procedural questions. When compared with the control group, taught by traditional methods, the experimental group scored significantly higher on conceptual achievement while there was no significant difference in procedural achievement. The calculator and computer technologies appeared to positively impact conceptual achievement. 145. {lau:stude} A. D. Lauten et al. Student understanding of basic calculus concepts: Interaction with the graphics calculator. Journal of Mathematical Behavior, 13(2):225-237, 1994. Describes five college and two high school students' understandings of function and limit in a graphics calculator-based environment and identifies instances where students' understanding seems to have been influenced by the availability of a graphing calculator. 146. {joh-84:circ} D. W. Johnson et. al, editor. Circles of Learning, Cooperation in the Classroom. Association for Supervision and Curriculum Development, Alexandria, VA, 1984. Cooperative learning processes have been rediscovered and are being used throughout the country on every level. The basic elements of cooperative goal structure are positive interdependence, individual accountability, face-to-face interaction, and cooperative skills. The teacher's role in structuring cooperative learning situations involves clearly specifying lesson objectives, placing students in productive learning groups and providing appropriate materials, clearly explaining the cooperative goal structure, monitoring students, and evaluating performance. For cooperative learning groups to be productive, students must be able to engage in the needed collaborative skills. Cooperative skills and academic skills can be taught simultaneously. The implementation of collaborative professional support groups among educators. Both the success of implementation efforts and the quality of life within most schools depend on teachers and other staff members cooperating with each other. Support for the program takes as careful structuring and monitoring as does cooperative learning. 147. {whe:radic} G. H. Wheatley et al. Radical constructivism as a basis for mathematics reform. In Proceedings of 17 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Columbus, OH, 1995. This paper describes the use of radical constructivism as a basis for curriculum reform in university mathematics courses and reports on research conducted on two of the courses developed, one in geometry and one in problem solving. The theoretical underpinnings of the project are described along with the implications for course design and instruction. Finally, results from qualitative research conducted on the courses are presented. The courses were found to foster intellectual autonomy, challenge students to rethink mathematics from a conceptual rather than procedural perspective, promote confidence in their mathematics knowledge, become more positive mathematics learners, and make connections among algebra, geometry, and calculus concepts. 148. {ste:calcu} H. Stephen et al. Calculus: An active approach with projects. Primus, 3(1):71-82, 1993. Discusses a pedagogical approach to calculus based on the question: What kinds of problems should students be able to solve? Includes a discussion of types of problems and curriculum threads for such a course. Describes a projects-based calculus with examples of projects and classroom activities. 149. {eyl:rlmoo} J. W. Eyles. R. L. Moore's Calculus course. PhD thesis, The University of Texas at Austin, 1998. R. L. Moore (1882-1974) distinguished himself not only as a research mathematician, but also as a professor of mathematics at the University of Texas from 1920 until he retired at age 86 in 1969. In 1964, the Mathematical Association of America produced the film 'Challenge in the Classroom: The Methods of R. L. Moore,' which credits Moore as one of the most prolific producers of research mathematicians in history. In 1973, the University of Texas named its new Physics-Mathematics-Astronomy building Robert Lee Moore Hall in honor of, according to Moore's student R. L. Wilder, 'not just Moore's eminence as a research mathematician but his achievements as a great teacher.' The 'Moore Method' is described by noted mathematician and author Paul Halmos as the 'right way to teach anything and everything.' The past decade can be characterized as a time of initiation of broad reform in mathematics education from primary grades through high school up to the college level through what has come to be known as The Calculus Reform Movement. Proponents of mathematics education reform emphasize the change in how mathematics is taught is at least as important as the change in which mathematics is taught. Some of the proposed teaching practices resemble, if not model, the teaching practices used by R. L. Moore a half century ago. While the 'Moore Method' has been well publicized, his pedagogical techniques are described in the context of graduate level and upper division mathematics courses. Although the fact that Dr. Moore taught calculus on a regular basis is often mentioned in the literature, it is not clear how he employed his techniques in this typically non-theorem-proving course. The purpose of this research was to describe Dr. Moore's calculus course and to investigate how well it met objectives of the current reform movement. Two descriptions of Moore's calculus course are presented, one edited from notes he made, but never published, and a second drafted from interviews with former students. A comparison of final grades in a physics course indicates that Moore's approach to calculus instruction was not detrimental to later study in applied areas. 150. {eylo-lynn-rer-88} B. S. Eylon and M. C. Lynn. Learning and instruction: An examination of four research perspectives in science education. Review of Educational Research, 58:251-301, 1988. 151. {fau:anexp} V. G. Faurot. An exploration into the effects of mathematical knowledge, beliefs, and emotions on task performance by university calculus students. PhD thesis, University of Oregon, 1993. The purpose of this study is to explore how calculus students' mathematical knowledge, beliefs about the symbolic nature of mathematics, and emotions affect their performance on mathematical tasks involving calculus. Eight undergraduate students from the University of Oregon participated in this study. Seven subjects were enrolled in Calculus for Business and Social Science and one subject was enrolled in Introduction to Methods of Probability and Statistics. All subjects were trained in methods of think-aloud and worked on two tasks using these methods. The two tasks required understanding of the derivative. One task, the traditional task, required facility with algebraic reasoning and the other task, the nontraditional task, required facility with graphical reasoning. All subjects were interviewed. In the traditional task clear patterns arose from the data: (a) The presence and absence of mathematical knowledge affected the degree of successful completion of the task, (b) Less proficient calculus students believed that decimal results indicated calculation errors, (c) More proficient calculus students did not believe that mathematical symbols represented procedures, (d) Calculus students, who perceived that they should be able to complete the task successfully, experienced frustration when they could not, and (e) Less proficient calculus students who believed themselves successful experienced initial anxiety followed by relief upon completing the task. In the nontraditional task, the initial interpretation of the task had the greatest effect on performance. Mathematical knowledge had some effect on task performance but disassociating knowledge from performance proved difficult in some cases. No patterns emerged from beliefs about the symbolic nature of mathematics. One gender difference emerged: Women reported feeling anxious about the nontraditional task whereas men did not. The performance groupings on the two tasks were not identical. Possible explanations include (a) symbols replace thinking, (b) different types of symbols are used in different ways, and (c) symbols interfere in thinking. The data also indicated that inconsistencies exist between the subject's perception of success and the reality of their success. This study has implications for the teaching and learning of calculus. 152. {fau-kno:effec} L. V. Fausett and C. Knoll. Effective use of teaching assistants in first year calculus. Primus, 1(4):407-414, 1991. Compares the effectiveness of four different calculus instructional formats based on quantitative measures of student performance, adjusted for differences in student preparation for calculus. Student performance is shown to improve when instruction is presented in a format with faculty involvement. 153. {fen:diagn} F. Fennell. Diagnostic teaching, writing and mathematics. Focus on Learning Problems in Mathematics, 13(3):39-50, Summer 1991. The article describes a `pen pal' project in which pre-service teachers were paired with elementary school students. 154. {fer-geu:anove} J. Ferrini-Mundy and K. Geuther. An overview of the calculus curriculum reform effort: Issues for learning, teaching, and curriculum development. American Mathematical Monthly, (7):627-635, 1991. Following a short history of calculus reform efforts, this article discusses curriculum development and research on student learning with respect to the concepts of function, limits and continuity, the derivative, and the integral; the availability and influence of new technologies on curriculum development; the consideration of the role of the teacher; and research directions. 155. {fer-gra:resea} J. Ferrini-Mundy and K. Graham. Research in calculus learning: Understanding of limits, derivatives, and integrals. In Proceedings of Special Session on Research in Undergraduate Mathematics Education at the annual meeting of the American Mathematical Society and the Mathematical Association of America, volume ?, pages ???-???, San Francisco, California, 1991. Mathematical Association of America and AMS. To be added later. 156. {fer-lau:conne} J. Ferrini-Mundy and D. Lauten. Connecting research to teaching: Learning about calculus learning. Mathematics Teacher, (2):115-121, 1994. Discusses research findings related to students' ability to make connections between analytical (symbolic) and graphical representations of functions in calculus. Describes graphing tasks and typical student interpretations. Implications for teaching are suggested. (Contains 17 references.) 157. {fin:first} K. Finlow-Bates. First year mathematics students' notions of the role of informal proof and examples. In Proceedings of the 18th International Conference for the Psychology of Mathematics Education, volume ?, pages 344-351, Lisbon, 1994. International Group for the Psychology of Mathematics Education (PME). ED383537. Abstract (quoted from paper): `This paper presents the results of a series of video-taped interviews held to investigate first year mathematics students' notions of informal proof. In the context of their usual classroom experience of working in small discussion groups, students were asked to rank four solutions to a familiar problem on closure of a set under addition. The solution consisting of an informal proof followed by some examples was ranked higher than any of the other solutions by all the interviewees. Although failing to refer to the informal proof as a `proof', or comment on its role in convincing the reader of the truth of the initial conjecture, students consistently used relevant words such as `explanation' and `clarification' to describe some of its functions.' 158. {fin-ler-mor:asurv} K. Finlow-Bates, S. Lerman, and C. Morgan. A survey of current concepts of proof held by first year mathematics students. In Proceedings of the 17th International Conference for the Psychology of Mathematics Education, volume 1, pages 252-259, Tsukuba, Japan, 1993. International Group for the Psychology of Mathematics Education (PME). ED383536. Abstract (quoted from paper): `This paper presents results of a questionnaire concerning concepts of proof held by first year undergraduates. Students were asked to comment on arguments which were presented to them as `proofs.' Their reasons for judging the validity of arguments, rather then being based on logic or rigor, mirrored Hanna's (1989) description of the practicing mathematicians' criteria for accepting theorems. Three modes of judgement have been identified: empirical, logical, and aesthetic. Problem areas for students included following the chain of reasoning, using mathematical concepts and knowledge, and making sense of mathematical language.' 159. {fis-ked:proof} E. Fischbein and I. Kedem. Proof and certitude in the development of mathematical thinking. In Proceedings of the Sixth International Conference for the Psychology of Mathematics Education, volume 6, pages 128-131. Psychology of Mathematical Education (PME), 1982. The authors seek to answer the question, `Does the high-school student clearly understand that a formal proof of a mathematical statement confers on it the attribute of a priori, universal validity? Or will he assume that more checks are always desirable in order to evaluate the validity of the theorem?' To address this question the authors conducted a study involving 397 students enrolled in three high schools. They found that most of the subjects felt the need for supplementary checks of an already proven statement, even if they have stated that they agree with the proof of the statement. 160. {fis:compr} M A. Fisher. Comprehension of graphically presented data. PhD thesis, The University of Oklahoma, 1991. This study examines the nature of the process of comprehending quantitative data displayed graphically. The Pinker-Kosslyn model of graph comprehension shows that both perceptual and categorization processes are important to graph comprehension. These processes are examined in terms of mathematical experience and precollege mathematics achievement, measured by Math ACT. Three different levels of college mathematics students were examined (Intermediate Algebra, Calculus II and an upper division mathematical Applied Statistics course). The research questions were (1) whether there were differences between levels of mathematical experience in graph reading skills, (2) whether there were differences in views of graph typicality for different levels of mathematical experience, and (3) whether different graph types were read differently by subjects of different levels of mathematical experience. The first result showed that there are differences between groups in pattern description skills while there were no differences in value reading skills. It also showed that there was no significant correlation between Math ACT and either graph reading skill. The second result, analyzed the typicality data using multidimensional scaling and cluster analysis showed that there were differences in views of typicality for the groups of mathematical experience examined, with the group with the most experience having the most complex configurations and clusters and the group with the least experience having much more simple configurations and clusters. These clusters showed an increasing attention to orientation of the graph as a factor in typicality rating with increasing mathematical experience. The third result showed that there were differences in graph reading ability for different types of graphs. In fact, the value reading skill results show that the graphs that were associated together in the clusters from the typicality data are read equally well. However, these graphs were not grouped in terms of perceptual complexity. Thus categorization or schema selection is seen to play an important role in graph comprehension. However, perceptual complexity is shown not to have as important a role in graph comprehension as the preliminary study suggested (McKnight and Fisher, 1991a, 1991b). 161. {fis:acomp} M. B. Fiske. A comparison of the effects on student learning of two strategies for teaching the concept of derivative. PhD thesis, The Ohio State University, 1994. Two secondary school calculus classes were taught the concepts of differentiability and the derivative using two distinct organizing principles. One class (N = 27), the local linearity group, received instruction based on the idea of local straightness, a differentiable function under uniform magnification is locally straight. The second class (N = 28), the secant-tangent group, received instruction based on the limiting process of a secant approaching a tangent. Both classes were taught by the same teacher, explored the same examples and nonexamples of differentiable functions, and used computer software extensively. It was hypothesized that the local linearity students would achieve a different and more complete understanding of the concepts of differentiability and the derivative than the secant-tangent students. Instruments consisted of a pretest and posttest designed to measure student understanding of differentiability and the derivative, student interviews, and a taxonomy to observe classroom roles and learning activities. An analysis of covariance model used treatment, either local linearity or secant-tangent approach, as the independent variable, pretest score as the covariate, and posttest score as the dependent variable. The model demonstrated significant interaction between the pretest and treatment. The research hypothesis was not confirmed. On the basis of the pretest, the secant-tangent group began instruction with less understanding of concepts underlying differentiability and the derivative than the local linearity group. The two groups achieved similar overall results on the posttest. The local linearity group demonstrated stronger understanding of the concepts of slope, rate of change, derivative and differentiability, and symbolic differentiation than the secant-tangent group. The secant-tangent group was better than the local linearity group at sketching the graphs of derivatives of functions. Student interviews confirmed the quantitative results that students in both groups achieved a strong visual representation of the derivative as a function. Analysis of transcripts obtained using the taxonomy of classroom roles and learning activities confirmed equality of instruction between the two groups. The classroom was teacher-centered with the teacher functioning as an explainer and task setter, students synthesizing procedures and concepts, and students actively engaged in investigative activities. 162. {foe:calcu} Foehl. Calculus, core curricula, and critical thinking. Primus, 3(2):141-150, 1993. Proposes a calculus curriculum combining formative knowledge, mathematical foundations, and instrumental knowledge in mathematics. Discusses each of these components, the organization of a core calculus course, and the use of problem solving in calculus instruction. 163. {fol:asses} M. E. F. Foley. Assessment of higher-order thinking in mathematics: The definite Integral. PhD thesis, Texas A&M University, 1992. This study was designed to examine open-ended assessment items which target conceptual, procedural, or strategic knowledge of the definite integral. Two research questions were considered. The first question was: To what extent does student performance on each item reflect the targeted knowledge type? The second question targeted student knowledge: What do students know about the definite integral? Students from both honors and engineering calculus classes were given open-ended assessment items, followed by interviews. It was found that at one point in students' growth in procedural knowledge over the semester, performance on procedural questions was masked by strategic concerns. By the end of the course, however, procedural knowledge was again displayed in response to procedural questions. Student responses to conceptual questions revealed not only the conceptual links which the students had formed but also the links which were missing. Because responses to open-ended questions are written in natural language form, care must be taken not to read more into and answer than is there. The strongest conceptual links were between integral and area, and between integral and antiderivative. The weakest conceptual link was between integral and limit of Riemann sum. Strategic items did yield a path to an answer, although students generally displayed only one path, and not all possible strategies were found. 164. {for:analy} C. A. Forbes. Analyzing the growth of the critical thinking skills of college calculus students. PhD thesis, Iowa State University, 1997. This research study analyzed the growth of the cognitive critical thinking skills of college calculus students from six sections of Calculus 165 at Iowa State University. For one semester subjects from three experimental sections learned calculus in a consistently facilitated active learning environment. Subjects from three comparison sections learned the same material in a traditional passive learning environment. Fidelity of treatment-i.e. adherence to an active learning format, was verified by direct observations. The main difference between any active and passive learning environment is the amount of time spent on instructor activities (passive learning) versus the amount of time spent on student activities (active learning). Results from two separate observation sessions for sections from both the treatment and the comparison group, clearly showed a dramatic difference in the type of learning climate that existed. At the beginning of the semester the students were informed of the opportunity to be a part of the study. During the second week of the semester subjects from both the experimental and the comparison group took Form A of The California Critical Thinking Skills Test: College Level (CCTST) as a pretest. During the last week in the semester subjects from both groups took Form B of the CCTST as a post test. Analysis of paired statistics for the post test means indicated that the growth in the subjects' critical thinking (CT) skills did not occur in the predicted direction. Possible explanations could be the small size of the study (in a statistical sense), and the short duration (one semester) of treatment. The findings do not detract from the ecological validity of the study. The study presents a strong consensus definition of CT, it shows that treatment fidelity involving an active learning component is possible, and the study points to research questions for further study. The limitations of the study can be easily rectified in future studies, and the extensive evidence in the literature suggests that these studies will yield significant results. Hopefully these results will provide the impetus for positive educational change at the micro-level in settings where educators and learners interact. 165. {tall-mis-96} Robin Foster and David Tall. Can all children climb the same curriculum ladder? Mathematics in School, 25(3):8-12, 1996. ERIC Acc. No. EJ530010. Concludes that mathematics uses symbols both as processes and concepts. The mathematically oriented student develops flexible ways of using them, but less successful students cling to the security of known procedures to get answers that are less suitable for thinking than flexible symbols which can also be considered as mathematical objects to be compared, related, and operated upon. 166. {fox:teach} T. Fox. Teacher change in a reform calculus curriculum: Conceptual development of the integral. In ?, volume ?, pages ???-??? The Association of Research in Undergraduate Mathematics Education, 1998. To be added later. 167. {fra:theco} E. J. Francis. The concept of limit in college calculus: Assessing student understanding and teacher beliefs. PhD thesis, University of Maryland College Park, 1992. The calls for reform in college calculus have urged increasing conceptual understanding of the topics within calculus. Yet current calculus instruction typically emphasizes procedural skills and leads to less than desirable levels of success. The purpose of this study was to better understand the procedural versus conceptual controversy surrounding the limit concept in college calculus by investigating the following questions: (1) How well do students understand the concept of limit in college calculus? That is, what procedural limit skills can be demonstrated and what level of conceptual knowledge is evident? What are the common errors made by students? (2) What are teachers' pedagogical beliefs regarding the nature of learning calculus and the role that limits play within calculus? (3) How does teachers' pedagogical content knowledge concerning limits compare with student performance on limit questions? A test of procedural and conceptual limit understanding was developed and admininstered to 199 students enrolled in eleven different second or third semester calculus courses in seven different four-year colleges and community colleges. Students scored significantly better on procedural questions than on those dealing with conceptual concerns. The format of the question, whether symbolic, tabular, graphic, or abstract had differing effects on problem difficulty. A survey of teacher beliefs regarding limits was developed and fifty-seven instructors from nine different two-year and four-year colleges provided data. The teachers sampled were evenly divided on whether procedural or conceptual knowledge ought to be the primary focus of calculus instruction. Teachers were able to accurately predict which of two given problems would be more difficult for students in just over half of the examples. The researcher concluded that the area of testing holds much potential as an untapped avenue for increasing conceptual understanding in calculus. 168. {fre-mur:histo} M. Freeman and M. Murphy. History, mathematics, writing: an experience in which the whole is greater than the sum of its parts. Mathematics and Computer Education, 26(1):15-20, Winter 1992. The article describes the use of writing assignments in the `History of Applied Mathematics' course at the University of Houston-Downtown. 169. {fri:under} S.D. Frid. Undergraduate calculus students' language use and sources of conviction. PhD thesis, University of Alberta (Canada), 1992. The purpose of this study was to investigate student learning in introductory, undergraduate calculus from a constructivist perspective. Students' language use and sources of conviction were the focus of analysis. Sources of conviction refer to how one determines mathematical truth and validity. Three undergraduate calculus classes were involved, with students taught by one of three instructional approaches: technique-oriented, concepts-first and infinitesimal instruction. Interviews with 17 students provided data on students' language use and sources of conviction. Instructor interviews, classroom observations and textbook analyses provided description of each instructional approach. Student interview data revealed the existence of three groups of students who differed in their sources of conviction. These groups were named Collectors, Technicians, and Connectors. Collectors exhibited the highest degree of external sources of conviction, using teacher or textbook presentations as means by which to determine truth or validity. Their calculus conceptualizations were constructed as a collection of isolated, relatively unconnected statements, rules and procedures. Technicians based truth and validity upon their knowledge of the logical, organized structure of calculus, and constructed their conceptualizations as a logical organization of statements, rules and procedures. Connectors exhibited the highest degree of internal sources of conviction, displaying a sense of personal understanding of calculus. Their conceptualizations were constructed as a network of connections between various aspects of calculus and between calculus and themselves. Analysis of students' language use indicated they used pre-calculus and everyday language, visually and physically oriented language, and procedural language knowledge in their calculus responses. Except for Connectors, students did not make extensive use of symbols to interpret or explain calculus ideas. Students who received infinitesimal instruction used infinitesimal language related to notions of infinitesimal closeness and infinite magnification of a curve. The study revealed the existence and characteristics of three types of calculus learners, and there was some relationship between these groups and competency in calculus. The study also revealed that students used language as a source of conviction. Finally, students' perceptions of their own learning were revealed as an unexpected but important element in the nature of their sources of conviction and construction of calculus conceptualizations. 170. {fri:resea} M. L. Friedman. Research: The sparse component of calculus reform: Part 2: An investigation of calculus instruction. Primus, 3(2):113-123, 1993. Describes the behavior of five calculus professors and their students based on observations of calculus classes. Results showed that the dominant mode of instruction was lecture and students were usually not actively engaged in doing mathematics. 171. {edw:curre} T. g. Edwards. Current reform efforts in mathematics education. In ERIC/CSMEE Digest. ERIC Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, Ohio, 1994. The current reform effort in mathematics education has its roots in the decade of the 1980's and the national reports that focused attention on an impending crisis in education, particularly in mathematics and science. Within this context, dozens of individual reform efforts have been initiated in recent years. Many have focused on the development of new curricula, others on teacher enhancement, some on both. Still others have taken the use of technology in mathematics instruction as their central theme. The projects listed in this digest are but a small sample of current efforts, but they serve to illustrate the diversity of programs nationwide. Programs discussed in this digest include : Connected Mathematics Project, Adventures of Jasper Woodbury, Maneuvers with Mathematics, Mathematics in Context, Quantitative Reasoning Project, University of Chicago School Mathematics Project, Atlanta Math Project, Teaching to the Big Ideas, Integrating Science and Mathematics Teaching for Middle School Underrepresented Students, Math Matters, New York City Mathematics Project, Project IMPACT, Quantitative Understanding-Amplifying Student Achievement and Reasoning; Cognitively Guided Instruction, Delaware Teacher Enhancement Project, Math Learning Center, Calculator and Computer Precalculus Project, Computer Intensive Algebra, Empowering Teachers in Computer-Intensive Environments, Geo-Logo, Graphing Calculator-Enhanced Algebra Project, and Calculus Curriculum Project. 172. {gan:integ} A. Ganguli. Integrating writing in developmental mathematics. College Teaching, 37(4):140-142, Fall 1989. The author describes a study involving two sections of a developmental algebra class. No writing activities were incorporated into one section which was designated the control. In the experimental section brief in-class writing assignments were made. The final exam performance of the experimental group was better, on average, than that of the control group. 173. {gan:writi} A. Ganguli. Writing to learn mathematics: enhancement of mathematical understanding. The AMATYC Review, 16(1):45-51, Fall 1994. (This annotation is quoted from the paper abstract.) `Twenty-five college students enrolled in an intermediate algebra class were asked to complete five-to-seven-minute in-class writing assignments about mathematical concepts, procedures for solving problems, etc., along with solving algebraic problems in a ten-week lecture class. Analysis of the written responses at the beginning and at the end suggests that writing assignments helped remedial students to think mathematically.' 174. {gar:reten} B.E. Garner. Retention of concepts and skills in traditional and reformed Applied Calculus. PhD thesis, University of Maryland College Park, 1998. A fundamental question is currently being asked throughout the collegiate mathematics education community: 'How can we help students understand and remember calculus better?' There seems to be general dissatisfaction with the knowledge and abilities of students who have completed calculus courses. Reformers in the calculus arena are striving to change instruction to help students understand better and remember longer what they have learned. The Calculus Consortium based at Harvard (CCH) recently published new textbooks for applied calculus which embody a major switch in the philosophy of calculus teaching. The CCH texts, in which applications are the central motivation and not coincidental afterthoughts, emphasize concepts more than symbol manipulation and encourage student-driven discovery of fundamental ideas. Is this reformed way of teaching applied calculus more effective than the traditional method? Which method leads to better long-term understanding and ability? The purpose of this study was to shed light on these questions by characterizing and comparing the skills and conceptual understandings of students of traditional and reformed methods several months after they completed their applied calculus course. A sample of 108 students of applied calculus (57 reformed, 51 traditional) who completed their course in April of 1997 were given a written test in November of 1997 to assess their conceptual understandings and computational skills. Sixteen of these students (8 traditional, 8 reformed) were interviewed to ascertain more about their conceptual understandings as well as their motivation, commitment and attitudes with respect to their applied calculus courses. Test results indicate that although there was no significant difference in overall performance between the two groups, students of the reformed method performed better on conceptual problems, while students of the traditional method performed better on computational problems. Interview results indicate that of the two groups, reformed course students were more confident in their ability to explain derivatives. Reformed course students mentioned graphs and applications more, and they also were more inclined to use estimation techniques than traditional course students. The traditional course students had a clearer idea of the connection between the derivative and the integral. 175. {gas:stude} J. L. Gaston. Student reluctance/difficulty with calculator use in community college mathematics courses. PhD thesis, Columbia University Teachers College, 1990. Certain students in remedial and non-remedial Pre-calculus classes exhibited a reluctance to use calculators and/or difficulty with calculator use in classes where such was permitted. The purpose of this study was (1) to identify, via the analysis of responses given by a population of urban community college Pre-calculus students to a researcher-designed questionnaire, those students who were reluctant to use calculators and/or had difficulty with calculator use in mathematics courses where such use was permitted; (2) to identify, via the aforementioned questionnaire responses and interviews of the students identified in (1), the reasons why these students were reluctant to use calculators and/or had difficulty with calculator use and (3) to identify, based upon the analyses in (2), pedagogical strategies which might be used to alleviate any difficulty with calculator use by this student population. The analyses of questionnaires and followup interviews of students who had exhibited a reluctance to use calculators and/or had difficulty with calculator use indicated that most of these students had poor attitudes toward the use of calculators in the mathematics classroom; had little experience with calculator use; had little perception of the usefulness of calculators; and/or were unable to achieve the successful integration of appropriate levels of mathematical competency and calculator competency. The overall results of this research investigation were discussed with other mathematics educators in order to propose pedagogical strategies which might alleviate the problems common to students who feel reluctant to use calculators and/or have difficulty with calculator use. These recommendations included the use of both calculator use of both calculator and non-calculator assessment instruments; the use of pedagogical strategies which stress not only the understanding of mathematical concepts but the appropriate use of calculators to efficiently perform related problem-solving tasks; and for fair time restraints on tests which permit calculator use. 176. {gav:agend} M. K. Gavin. A gender study of students with high mathematics ability: Personological, educational, and parental variables related to the intent to pursue quantitative fields of study. PhD thesis, The University of Connecticut, 1997. It is well documented that more males than females enter and pursue mathematically related career fields. Research has generally examined gender issues concerning mathematics majors and related career goals as an integral part of majors and careers in the sciences. However, an examination of the distribution of women in these fields presents a picture of uneven advancement. Women are clustered in the life sciences with far fewer in physical sciences, mathematics, engineering, and computer science. Using data from the National Education Longitudinal Study of 1988 (NELS:88), this study examined personological and educational characteristics of females and males identified as having high ability in mathematics. These data consist of a sample of 24,599 students from 1,052 schools throughout the nation who completed surveys in eighth, tenth, and twelfth grades. Gender similarities and differences were explored using descriptive and inferential statistics. Findings from this study revealed no gender differences with respect to performance or participation in mathematics courses. Males scored significantly higher on the verbal section of the SAT test, while no gender differences were found on the mathematics section. Also, males rated usefulness of mathematics significantly higher than females. In addition, significant differences were found between parental levels of education and expectation. The more educated the parent, the greater the expectations were for the child's educational goals. Logistic regression analyses were performed to predict the gender of students who intend to pursue a quantitative field. The odds ratios indicated that SAT verbal scores and teacher emphasis on further study in mathematics were significant predictors for males, while credits in calculus and SAT mathematics scores were significant predictors for females. Analyses also revealed that high mathematics ability females who intend to pursue a quantitative field were more likely to consider mathematics as useful to their future and had more credits in calculus than high mathematics ability females who do not intend to pursue a quantitative field. 177. {geo:impac} J. H. Georgakakos. Impact of differential college environments on the science reasoning ability of community college students: A matriculation study. PhD thesis, University of California, Riverside, 1995. The ability of college environmental factors to influence science reasoning (SR) ability of full- and parttime community college students in 55 classes across the curriculum was examined. Subjects (Ss) were pre- and posttested for SR ability at the beginning and end of fall semester 1991 using the Collegiate Assessment of Academic Proficiency (CAAP) SR Test; they were also surveyed for demographic information. Additional input data were obtained from the ACT CAAP Test Folder and the college computer. Three-tiered probing of the curriculum employed five samples: college-wide (N = 843), science-wide (N = 494), and department level-biology (N = 271), chemistry (N = 190), and physics (N = 66), with data for each analyzed using Input-Environment-Outcome multiple regression methodology (see Astin, 1991); in addition, analysis of covariance was performed on the chemistry and physics sample data. With pertinent student inputs and initial SR ability controlled, broad-level (significant at each tier), semi-broad (at two tiers), and unique (one level only) environmental effects were revealed. Positive broad-level impacts were found for science courses, college courses in general, and semester GPA; lower SR was predicted for Ss who tended to rate themselves highly on improved intellectual and improved social self-confidence (so-called, H-H Ss). Semi-broad findings showed, among science units taken, physics units were most conducive to SR growth. Greater SR gains were observed, also, as Ss took more total units, although a negative impact was found for units of history. Unique positive effects were observed for psychology units (college-wide); taking calculus or taking a course in art, music, or theater (biology sample); and in general, within each department, for taking more-advanced sciences-especially courses with a laboratory component. Results were discussed in terms of Astin's theory of student involvement as well as cognitive and other factors related to SR. It is concluded students may realize greater SR gains if they become more actively engaged in their academic programs, particularly as these programs become more deeply enriched with science-related subject matter. However, before adequate improvement in SR performance can be realized, certain prerequisite academic experiences and skills need to be established. 178. {geo:reaso} E. A. George. Reasoning with visual representations: students' use of diagrams, figures, and graphs in solving problems on the advanced placement calculus examination. PhD thesis, University of Pittsburgh, 1997. Visual representations play an important role in mathematical problem solving. The purpose of this dissertation study was to examine when and how mathematically capable high school students used visual representations in solving problems. Specifically, students' written solutions to selected problems presented on the 1996 BC level Advanced Placement Calculus Examination were analyzed with respect to visual representation use and problem-solving success. Visual representation use was defined as the modification of given diagrams or the construction of new diagrams. Through both quantitative and qualitative analyses, similarities and differences in the frequency of visual representation use and in the descriptions of the diagrams created were identified for subgroups of students based on gender, overall performance level on the examination, and degree of success in solving specific problems. Analyses of 600 students' written solutions revealed significant gender and performance level differences in the frequency of visual representation use. Overall, females used diagrams more often than did males, and more high scorers than low scorers drew diagrams in their solutions. Further analyses of the diagrams created by 180 students identified greater diversity in the diagrams students modified than in the diagrams they constructed. Gender differences were more pronounced than performance level differences. Females were more likely to create complex visual representations by modifying given diagrams, and males tended to construct relatively simple new diagrams. Students who used visual representations and were successful in problem solving highlighted essential features and demonstrated flexibility in modifying and constructing diagrams. The frequency with which students drew diagrams and the relationship of diagram use and problem-solving success point to the important role that reasoning with visual representations plays in solving calculus problems. The diversity of diagrams students created and the gender differences identified in visual representation use and problem-solving success present challenges and opportunities for textbook authors and teachers in fostering effective visual representation use in mathematical problem solving. 179. {gib:advan} D. Gibson. Advanced calculus students' use of visual representations in the creation of mathematical proofs. PhD thesis, University of Kentucky, 1996. The author investigated the ways in which twelve advanced calculus students used visual representations in the process of constructing proofs. The data was obtained from three task-based interviews in which students worked proofs and a final interview in which students described their proving behavior. The ways in which students used visual representations were classified, and the reasons that the visual representations assisted the students with the problem-solving process were analyzed. The author suggests that instructors in advanced mathematics courses might help their students by including instruction in the use of visual representations and in translating from visual representations to verbal and/or symbolic ones. 180. {gib:stude} D. Gibson. Students' Use of Diagrams to Develop Proofs in an Introductory Analysis Course, volume 7 of Issues in Mathematics Education, pages 284-307. American Mathematical Society, 1998. Abstract (quoted from the paper) `This study investigated the strategy of drawing diagrams to develop proofs. Task-based interviews with students in an introductory analysis (advanced calculus) course revealed that these students used diagrams to perform the following subtasks of proof development: (a) understand information, (b) judge the truthfulness of statements, (c) discover ideas, and (d) write out ideas. Typically, using diagrams helped students complete subtasks that they were not able to complete while working with verbal/symbolic representations alone. Diagrams aided students' thinking by corresponding more closely to the part of their understanding with which they were operating at the time and by reducing the burden that proving placed on their thinking.' 181. {vonglas-cog-89} E. Von Glasersfeld. Cognition, construction of knowledge, and teaching. Synthese, 80:121-140, 1989. 182. {goe:theco} M. Goetting. The College Student's Understanding of Mathematical Proof. PhD thesis, University of Maryland College Park, 1995. (This annotation is quoted from the dissertation abstract.) `This study investigated students' understanding of proof--the arguments they find convincing, those they accept as valid proofs, and the relationships they maintain between these two ideas. Forty university volunteers from three populations were interviewed; elementary education students, secondary mathematics education students, and students taking an upper-level mathematics course. In the interviews, students were presented with mathematical propositions and arguments. They were asked to rate both the degree to which they felt the proposition was true and the degree to which they felt the supplied argument constituted a valid mathematical proof. ...The results from the interviews provide evidence for three different understandings of proof. The first group (primarily elementary education students) understood proof as a supporting argument that is not necessarily conclusive. The second group understood proof is a tool for verifying conjectures conclusively. The third group (primarily secondary education students) understood proof as an explanation or classroom exercise for which conclusive verification is necessary but sometimes not sufficient. This group was characterized by student who rejected as valid proofs examples of existence and/or counterexamples because the arguments were not generalized, did not investigate the extent to which the statement was true, or lacked cosmetic features they associated with proofs. ...' 183. {goe-kah:surpr} A. Goetz and J. Kahan. Surprising results using calculators for derivatives. Mathematics-Teacher, (1):30-33, 1995. Attempts to answer and generalize the question: When is the numerical derivative obtained on the graphing calculator greater than the actual derivative, and when is it smaller? Discusses symmetric difference. 184. {gol:integ} D. Goldberg. Integrating writing into the mathematics curriculum. Two Year College Mathematics Journal, 14(5):421-424, November 1983. The author argues that writing exercises should be part of mathematics courses and provides some suggestions for getting started and giving feedback. 185. {gop-smi:whats} G. Gopen and D. Smith. What's an assignment like you doing in a course like this?: writing to learn mathematics. The College Mathematics Journal, 21(1):2-19, January 1990. This article describes an experimental calculus course at Duke University in which writing was a main component. The article deals with misconceptions which students have about the nature of writing and of mathematics -- misconceptions which become obstacles which get in the way of students writing to learn. A number of common problems with student writing are treated in detail, with examples. Reader expectation theory is discussed. Examples of student writing are included along with instructor comments based on the reader expectation theory model. Peer review and double submission are also discussed as techniques to help students improve their writing. 186. {gor:calcu} S. P. Gordon. Calculus must evolve. Primus, 3(1):11-18, 1993. Calculus must evolve or face the prospect of becoming irrelevant. The minimum level of classroom technology now available requires us to rethink the content of our calculus courses. Proposes using graphing calculators and computer algebra systems to include the following topics: local linearity, optimization problems, families of curves, and approximation techniques to evaluate definite integrals. 187. {gra-90:coo} T. Graves. Cooperative learning and academic achievement: A tribute to david and roger johnson, robert slavin, and shlomo sharan. Cooperative Learning, (10):13-16, 1990. An overview is presented of the work and theories of three teams of researchers who have focused on cooperative learning. Cooperative learning has become the outstanding example of an educational innovation in which practice is informed by research; the collective leadership of these teams has advanced its use. 188. {gra-91:cont} T. Graves. The controversy over group rewards in cooperative classrooms. Educational Leadership, (48):77-79, 1991. Suggests ways to minimize the negative effects of extrinsic group rewards in cooperative classrooms, explains how to use intrinsic rewards, and outlines conditions calling for extrinsic rewards. The ``social rewards'' of working cooperatively probably enhance intrinsic motivation and are among the great advantages of employing cooperative learning strategies 189. {tall-esm-99} Eddie Gray, Marcia Pinto, Demetra Pitta, and David Tall. Knowledge construction and diverging thinking in elementary and advanced mathematics. Educational Studies in Mathematics, 38(1-3):111-133, 1999. Note - Special Issue - Forms of Mathematical Knowledge: Learning and Teaching with Understanding; ERIC Acc. No. EJ597945. Considers the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum, such as children's arithmetic shows divergence in performance. Explains how students cope with the transition to advanced mathematical thinking in different ways, leading once more to a diverging spectrum of success. 190. {tall-jrme-94} Eddie M. Gray and David Tall. Duality, ambiguity, and flexibility: A ``proceptual" view of simple arithmetic. Journal for Research in Mathematics Education, 25(2):116-140, 1994. ERIC Acc. No. EJ491793. Discusses duality between process and concept in mathematics, ambiguity of symbolic notation, and flexibility in thought processes of successful students. Content analysis and empirical evidence suggest a qualitatively different kind of mathematical thought in more able compared to less able elementary students in arithmetic. 191. {gri-sch:teach} D. Gries and F. Schneider. Teaching math more effectively, through calculational proofs. The American Mathematical Monthly, 102(8):691-697, October 1995. The authors advocate the teaching of proof by means of a formal approach using an equational logic. 192. {guc:thero} A. Guckin. The Role of Mathematics Informal Focussed Writing in College Mathematics Instruction. PhD thesis, University of Minnesota, 1992. (This annotation is quoted from the dissertation abstract.) `The purpose of this study is to investigate the use of mathematics informal focussed writing (MIFW) within college mathematics instruction. ...Four subinvestigations of the main study investigate: (a) the effect of writing treatment with the MIFW upon mathematics test and assignment scores; (b) MIFW scores and their relation to mathematics and NMW (nonmathematical writing) scores; (c) the comparison of the MIFW scores of students in college elementary algebra, college intermediate algebra, trigonometry, and calculus classes; and (d) the mathematical content and errors within the MIFW. ...' 193. {gur-92:lib} K. Gura. Liberal arts mathematics: Probability and calculus. Primus, 2(2):155-164, 1992. Presents one model for a liberal arts mathematics course that combines probability and calculus. Describes activities utilized in the course to heighten students' interest and encourage student involvement. Activities include use of visualization, take-home tests, group problem solving, research papers, and computer usage with DERIVE computer software. 194. {gut:mathe} J. Gutbezahl. Mathematics performance and under performance: Effects of gender and confidence. PhD thesis, University of Massachusetts, 1996. Boys and men tend to do better in math class and to have higher math confidence than their female classmates. It has been hypothesized that low confidence is a precursor to poor performance. Because of this, a great deal of effort has been expended on raising our students' confidence. As a result of this, students in the United States are more confident of their math ability than students anywhere else in the world. Despite this, math performance remains low, for both male and female students. American students seem to interpret this low performance as an indication of limited ability. To change this interpretation, I told some students in introductory calculus classes that their prior math failures were due to low effort and that increased effort should lead to success in college. To make this information more believable, it was embedded in a personality profile that had been generated specifically for the student, and given a scientific sounding name: Talent/Motive Disjunction (TMD). Students who were told that they had TMD did significantly better in calculus than students who were not told they had TMD. This increase in performance was not the result of increased confidence. Students in the TMD condition were no more confident at the end of the semester than students in the No TMD condition. This suggests that changing attributions about success and failure may be effective in improving our students' performance even if confidence raising is not. 195. {hac:theef} L.D. Hackett. The effects of writing in an applied calculus course: An analysis of performance and errors. PhD thesis, The American University, 1998. This research study investigated the performance on a departmental final of students who wrote about their errors and misconceptions in complete sentences using correct mathematical terminology compared to students who did not write in complete sentences about their errors and misconceptions. The subjects in this study were primarily freshmen in six fall semester sections of Applied Calculus at a major university. The researcher taught the experimental and first control sections, while the four other sections were taught by four full-time faculty members. The statistical data was gathered from the departmental common final which had been given to all sections simultaneously. The experimental group was required to correct problems by identifying the errors made, describing the appropriate procedure to solve the problem using correct mathematical terminology, vocabulary, and complete sentences, and by providing a complete corrected solution with verification. The first control group's only requirement was to provide a complete corrected solution with verification. The remaining four sections of Applied Calculus, the second control group, had no requirements for correcting mistakes on their examinations. Both quantitative and qualitative methodologies were employed to analyze and describe the achievements of the students. For the quantitative analysis, the performance of the students was evaluated using the Wilcoxon rank sum test on the final scores from the departmental common final. From the statistical analyses, the experimental group performed on average significantly better than the second control group on the departmental common final. The experimental group also performed on average significantly better than the first control group on the departmental common final. There is statistical evidence the students in the first control group performed on average better than the second control on the departmental final. The qualitative portion of the study indicated the students in the experimental group who did through and complete error analyses did not repeat their conceptual errors on the final. The experimental and first control groups' test correction analyses indicated those who had conceptual errors on the final had a pattern of either incomplete or incorrect error analyses. 196. {ham:theas} D. M. Hamm. The association between computer-oriented and noncomputer- oriented mathematics instruction, student achievement, and attitude towards mathematics in Introductory Calculus. PhD thesis, University of North Texas, 1989. The purposes of this study were (a) to develop, implement, and evaluate a computer-oriented instructional program for introductory calculus students, and (b) to explore the association between a computer-oriented calculus instructional program, student achievement on three selected calculus topics, and student attitude toward mathematics. An experimental study was conducted with two groups of introductory calculus students during the Spring Semester, 1989. The computer-oriented group consisted of 32 students who were taught using microcomputer calculus software for in-class presentations and homework assignments. The noncomputer-oriented group consisted of 40 students who were taught in a traditional setting with no microcomputer intervention. Each of three experimenter-developed achievement examinations was administered in a pretest/posttest format with the pretest scores being used both as covariate and in determining the two levels of student prior knowledge of the topic. For attitude toward mathematics, the Aiken-Dreger Revised Math Attitude Scale was administered in a pretest/posttest format with the pretest scores being used as a covariate. Students were also administered the MAA Calculus Readiness Test to determine two levels of calculus prerequisite skill mastery. An ANCOVA for achievement and attitude toward mathematics was performed by treatment, level, and interaction of treatment and level. Using a.05 level of significance, there was no significant difference in treatments, levels of prior knowledge of topic, nor interaction when achievement was measured by each of the three achievement examination posttests. Furthermore, there was no significant difference between treatments, levels of student prerequisite skill mastery, and interaction when attitude toward mathematics was measured, at the.05 level of significance. It was concluded that the use of the microcomputer in introductory calculus instruction does not significantly effect either student achievement in calculus or student attitude toward mathematics. 197. {han:chall} G. Hanna. Challenges to the importance of proof. For the Learning of Mathematics, 15(3):42-49, November 1995. (This annotation is quoted from the author's introduction to the paper.) `An informed view of the role of proof in mathematics leads one to the conclusion that proof should be part of any mathematics curriculum that purports to reflect mathematics itself, and furthermore that the main function of proof in the classroom reflects one of its key functions in mathematics itself: The promotion of understanding. Yet developments in both mathematics and mathematics education have now caused the very place of proof in the teaching of mathematics to be called into question. Examining these developments, this paper concludes that proof is alive and well in mathematical practice, and that it continues to deserve a prominent place in the mathematics curriculum. This paper also argues that the most important challenge to mathematics educators in the context of proof is to enhance its role in the classroom by finding more effective ways of using it as a vehicle to promote mathematical understanding.' 198. {han-jah:proof} G. Hanna and H. Jahnke. Proof and application. Educational Studies in Mathematics, 24:421-438, 1993. (This annotation is quoted from the paper abstract.) `This paper outlines an epistemological conception which attempts to relate the formal aspects of mathematical proof to its pragmatic dimensions. In addition to the key concept of application, the paper makes use of several concepts from the domain of analytical philosophy to present a view of proof that might best be categorized as a dialectical one. A number of implications fro teaching are discussed.' 199. {han:mathe} Gila Hanna. Mathematical proof, volume 11 of Mathematics Education Library, pages 54-61. Kluwer Academic Publishers, 1991. The author discusses the forms which proofs take in everyday mathematical practice and the social context in which they are evaluated referring to ideas advanced by Lakatos, Kitcher, Tymoczko, and Davis. Implications for pedagogy are suggested. 200. {har-rhe:journ} J. Harchelroad and D. Rheinheimer. Journal writing: an analysis of its effectiveness in a college-level developmental mathematics class. Research and Teaching in Developmental Education (RDTE), 9(2):55-63, Spring 1993. The authors report on a study involving students in a developmental algebra class. At the end of each class period a control group of students did practice exercises while a treatment group (from the same class) wrote journal entries. On a posttest, the control group did better than the journal writers; however, the posttest was highly computational in nature, with few word problems, and no opportunities for written responses to questions. 201. {har:compa} W. J. Hardin. Comparison of four instructional approaches and mathematics background on students' conception of limits. PhD thesis, Syracuse University, 1997. Limits are central to calculus. They are what separate analysis from algebra. Unfortunately, many students find limits to be a difficult and confusing topic. In this dissertation, I attempt to see how different ways of teaching the concept of limits affect students' conceptions of limits as well as their computational proficiency. To accomplish my goals, I taught four different recitation sections, each with a different instructional approach. The treatments were conducted for a total of three weeks. The instructional approaches were: paper-and-pencil computational, graphics calculator-based, paper-and-pencil conceptual, and computer-based. All the groups except the paper-and-pencil computational group emphasized conceptual knowledge first. I found that emphasizing conceptual knowledge (except for the graphics calculator group) resulted in students that were stronger in conceptual knowledge, while emphasizing computation yielded students stronger in computation. A notable exception was the paper-and-pencil conceptual group, that was at the top of the groups on scores, on both conceptual and procedural problems. In the instructional activities, the students in the paper-and-pencil conceptual group were asked to compute parameters like slopes of tangents themselves. This may give some indication that forcing the students to process the information may cause the connections to become more explicit. During the investigation, I developed software to help students visualize the concept of limits. This software let students dynamically control important parameters. In this way, the software acted like a virtual manipulative. I also enhanced a classification model for students' conceptions that was developed by Williams (1991). Additionally, I attempted to classify errors based on a classification model by Movshovitz-Hadar, Zaslavsky and Inbar without too much success. Finally, I developed a set of activities for each of the instructional approaches. The activities were similar between the pencil-and-paper conceptual and the computer-based sections but different for the calculator and the paper-and-pencil computational groups. These activities rely on the development of exposing, discrepant and resolving events to bring the notion of limits to light. The activities were designed so that the computational group did not have discussions, instead they practiced computational problems. However, in the activities, the conceptual groups were asked to make conjectures and were designed as the foundation of class discussions. 202. {har:facto} G. Harel. Factors in learning linear algebra. In Proceedings of the 16 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Baton Rouge, Louisiana, 1994. International Group for the Psychology of Mathematics Education. North American Chapter. To be added later. 203. {har-sow:stude} G. Harel and L. Sowder. Students' proof schemes: results from exploratory studies, volume 7 of Issues in Mathematics Education, pages 234- 283. American Mathematical Society, 1998. The authors report on research which is part of the PUPA (`Proof Understanding, Production, and Appreciation') project. The focus of the paper is on mapping students' schemes of mathematical proof. Their methodology consisted of a sequence of teaching experiments in number theory, college geometry, and linear algebra, supplemented by a case study of a junior-high school student studying geometry and calculus. Defining a person's proof scheme as `what constitutes ascertaining and persuading for that person' the paper suggests three major categories of proof schemes: external conviction proof schemes, empirical proof schemes, and analytical proof schemes. The authors subdivide each of these schemes into several subcategories. These subcategories are not proof techniques (induction, direct proof, indirect proof, etc.), but rather what the authors believe are non-mutually-exclusive cognitive stages in students' mathematical development. Episodes from the teaching experiments are presented to illustrate each subcategory. 204. {har-tal:thege} G. Harel and D. Tall. The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1):38-42, February 1991. The authors define formal abstraction as the abstraction of specific properties of one or more mathematical objects to form the basis of the definition of a new abstract mathematical object. They note that the process of formal abstraction is a powerful tool for the mathematician, but it is very difficult for the learner to accomplish. To help students overcome this difficulty they propose a teaching method in which students are given prototypical examples of the new abstract concept. It is hoped that the student will come to see these prototypes as typical of a wider range of examples making up an abstract concept. A student who achieves this way of thinking about the prototypes is said to have engaged in generic abstraction. During a three-year teaching experiment in linear algebra the authors developed three principles to aid students in the process of abstraction: (1) the entification principle (for a student to abstract a mathematical structure from a model, the student must see the elements of that model as objects); (2) the necessity principle (each idea must be presented in such a way that the students see it as necessary); and (3) the parallel principle (the instructional activities within a concrete model should parallel the processes which will apply within the abstract structure). The authors note that after generic abstraction has occurred, formal abstraction must still take place for full understanding. However, they argue that if generic abstraction has occurred first, the process of formal abstraction will involve less cognitive strain than would otherwise be necessary. 205. {harel-tall-91} Guershon Harel and David Tall. The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics - An International Journal of Mathematics Education, 11(1):38-42, 1991. ERIC Acc. No. EJ433624. The terms generalization and abstraction are used with various shades of meaning by mathematicians and mathematics educators. Introduced is the idea of ``generic abstraction" that gives the student an operative sense of a mathematical concept and provides a passage point in the process toward formal abstraction. 206. {har:build} D. K. Hart. Building concept images: Supercalculators and students' use of multiple representations in calculus. PhD thesis, Oregon State University, 1991. This study investigated how the use of supercalculators (HP 28S and HP 48SX) affects students' conceptual understanding of differential and integral calculus. Students (n = 324) from 12 institutions throughout the United States studied an experimental curriculum emphasizing multiple representations (symbolic, numerical, graphical). (The curriculum was developed through the Oregon State University Calculus Project supported by the National Science Foundation.) In particular, these students were taught graphical and numerical methods for analyzing and solving problems with the aid of supercalculators. The specific research questions concerned: student's representational knowledge and concept image (representational facility, connection among representations, management of representations) and student's calculator usage and interpretation of calculator results (management of the calculator, conflict resolution and confidence in the calculator). The theoretical framework for the study is a concept image theory put forward by Tall and Vinner (1981). Paper-and-pencil tasks were administered to the experimental students as well as 30 students from traditional calculus classes (18 from Oregon State University and 12 from a parallel class at a project site). Audio-taped task-based interviews were conducted with 33 experimental students and 31 traditional students. Results indicated that: (1) Experimental students showed greater facility with graphical and numerical representations and exhibited better ties among the three representations than traditional students. (2) Individual students do show definite preferences for certain representations but different factors influence their choices. (3) More evidence of compartmentalization was observed among the traditional students than among the experimental students. (4) Grades do not appear to be a good predictor of the quality of the connections among the representations. (5) Students use of the calculator is closely tied to their management of representations. (6) Students who lack confidence in their symbol manipulation skills appear to use the calculator more readily than those who are confident in their symbol manipulation skills. (7) When a device (machine or a formula) is used to perform a computation in a routine fashion, those are results students look at least critically. (8) Students' confidence in graphical information appearing on the screen is tied to having a priori information. In addition, the role of the instructor appears to be particularly important in terms of management of representations and of the calculator. 207. {har:aconc} E. Hart. A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory, volume 33 of MAA Notes, pages 49-62. The Mathematical Association of America, 1994. The author begins with a comprehensive review (from 1967 to the time of publication of the paper) of the literature concerning proof at the undergraduate level. He concludes from the literature review that there is no consistent body of research pointing to effective methods for teaching proof and that there is a need for more cognitive-based research founded upon a coherent theoretical context. The balance of the paper is a description of a study, conducted by the author, of the proof-writing performance of twenty-nine college mathematics majors. The students came from three abstract algebra classes at three different levels: beginning undergraduate, advanced undergraduate, and graduate. The students were given six proofs to complete. Based on their responses, the students were classified into four operationally-defined levels of understanding. The proof processes used by the students and errors committed by the students were analyzed in relation to these levels of understanding. The analysis emphasizes issues of representation, general versus domain-specific strategies, and metacognition. 208. {har:conce} B.J. Hartter. Concept image and concept definition for the topic of the derivative. PhD thesis, Illinois State University, 1995. The study investigated the extent to which a student's concept image of the derivative matches the concept definition. Since the formal definition of the derivative in most calculus textbooks relies on both the concept of function and the concept of limit, the study also investigated the relationship between a student's prior understanding of functions and limits and the degree of match of his/her concept image of the derivative and the concept definition of the derivative. Additionally, the research attempted to determine the relationship between using multiple representations and the student's concept image of the derivative. The study focused on the understanding of the derivative developed by eight students, enrolled in an introductory, undergraduate calculus course. This study was an observational, qualitative study that was informed by examination and interview data. An initial survey and interview provided information used to categorize each student's prior understanding of the concepts of function and limit. After instruction, the student's conceptual and procedural understanding of the concept of the derivative were both analyzed. The data show that few students developed a conceptual understanding of the derivative which included the notion that the behavior of a derivative is that of a function representing a rate of change. Generally, however, those students with a more robust understanding of functions exhibited a stronger match between concept image and concept definition. The findings also suggest that perhaps a strong graphical understanding of the concept of function is the most critical component of this development. Additionally, the presentations through multiple representations appear to enable the students to develop a more complete concept image of the derivative, as do specific exercises involving proportionality, graphic interpretation, and numerical spreadsheets. 209. {has:calcu} S. Hassani. Calculus students' knowledge of the composition of functions and the chain rule. PhD thesis, 1998. Calculus plays a central role in the mathematical education of three-quarters of a million students annually. However, recent research into the nature of students' understanding of the concepts underlying the calculus has shown significant gaps between their conceptual understanding of the major ideas of calculus and their ability to perform procedures based on these ideas. The present study is a combination of qualitative and quantitative investigations of first-year undergraduate students' understanding of function composition and its role in applying the chain rule. Students' understandings are examined from graphical, numerical (tabular), and algebraic/symbolic points of view. Since the concept of composition of functions and the chain rule meet in the classical application area known as related rates, information was collected on student performance in this area as well. In order to better understand the knowledge and abilities underlying students' understanding and performance in these areas, data was also collected about the students' algebraic manipulation skills, their knowledge of functional relationships, and their knowledge of the Pythagorean theorem and related geometric measurements. Following an analysis of both quiz and test results and a careful analysis of interviews conducted with students, several important findings emerged. The study revealed that first-year undergraduate calculus students have a very meager understanding of the concept of composition of functions. Furthermore, the results showed that first-year undergraduate calculus students' ability to explain or apply the chain rule is significantly related to their algebraic manipulative skills and their general knowledge of function concepts and function composition. 210. {hau:growt} G. S. Hauger. Growth of knowledge of rate in four precalculus students. In Proceedings of Annual Meeting of the American Educational Research Association, volume ?, pages ???-???, Chicago, IL, 1997. American Educational Research Association. Several studies have shown the difficulties students encounter in making sense of situations involving rate of change. This study concerns how students discover errors and refine their knowledge when working with rate of change. The part of the study reported here concerns the responses of four precalculus students to a task which asked them to sketch a distance-time graph showing slowing down then speeding up. These four students drew the same incorrect graph. This report is about how they discovered and corrected the error. Two general conclusions from this study are that students use a variety of resources to address rate of change, and that slope and changes over intervals are both powerful ways for precalculus students to think about rate of change. An instructional implication of this study is that calculus and precalculus teachers should provide opportunities for students to use their knowledge of slope and changes over intervals to construct knowledge of rate of change. Teachers should notice the knowledge students use to make sense of situations and help students use that knowledge to construct new mathematical knowledge. Contains 41 references. Appendices contain statements of tasks and graphs. 211. {hau:highs} G.S. Hauger. High school and college students' knowledge of rate of change. PhD thesis, Michigan State University, 1998. Rate of change is important from historical, academic, and everyday points of view. But many students struggle with learning rate of change and especially the derivative in calculus. One way to think about rate of change considers three perspectives and three types of reasoning. This dissertation attempted to discover what knowledge of rate of change precalculus, calculus, and postcalculus students have, how this knowledge differs among these three groups, and if the three perspectives and types of reasoning provide an interpretive frame for understanding students' knowledge of rate of change. In this study knowledge was taken as strategies and learning as strategy construction. Hence I was interested in what strategies precalculus, calculus, and postcalculus students used to solve tasks involving rate of change, how strategy use differed among these three groups of students, and if these strategies seemed consistent with the three perspectives and types of reasoning. Twelve precalculus, fifteen calculus, and ten postcalculus students were given the same set of tasks in individual audio and video taped interviews. Students' written work and verbal explanations were used to identify strategies they used to solve the problems, classify strategies in terms of the three perspectives and types of reasoning, and compare and contrast strategies used by the three groups. Nineteen different strategies were used. Many strategies, like 'delta d' and 'visual slope', were used by each of the three groups and most students used more than one strategy on each task. Precalculus students' use of many strategies showed they have knowledge that can be used to solve tasks involving rate of change and that may support their learning of the derivative in calculus. Calculus and postcalculus students' use of these same strategies suggests they are based on knowledge that is shared by all three groups of students and may therefore serve an important role in coming to know rate of change. Some strategies were used primarily by some groups but not others. Calculus and postcalculus students used slope of tangent line but no precalculus student did. Although all three groups used average rate of change to address rate of change at point, precalculus students used long and medium sized intervals before the point, calculus students used short intervals before the point, and postcalculus students used intervals straddling the point. Over the course of the tasks, precalculus students came to use shorter intervals before the point, calculus students used the shortest available intervals before the point and some straddle intervals, and postcalculus students relied more and more on straddle intervals. Interval based strategies were dominant in addressing rate of change at a point, usually via average rate of change. Some numerical and algebraic strategies were helpful in exploring differences between additive and multipicative comparison of changes over intervals. 212. {hazz-zazk-pme-97} O. Hazzan and R. Zazkis. Constructing knowledge by constructing examples for mathematical concepts. In E. Pehkonen, editor, Proceedings of the 21st International Conference of the International Group for the Psychology of Mathematics Education, volume 4, pages 299-306, Gummerus, Finland, 1997. 213. {heg:astud} S. J. Hegedus. A study of the metacognitive behaviour of mathematics undergraduates in solving problems in the integral calculus. PhD thesis, University of Southampton (United Kingdom), 1998. This thesis analyses the metacognitive behaviour of mathematics undergraduates in order to understand more about their behaviour in solving single and multi-variable integration. The project offers a framework for other researchers to use in analysing metacognitive behaviour for the purposes of understanding more about students' thought processes in solving single and multi variable integration, as well as in other domains of mathematics. This is for the purpose of aiding educational practitioners rather than providing any pedagogical strategy, per se. Situated in the field of Advanced Mathematical Thinking (see Tall, 1991 for summary), the study was designed using Schoenfeld's (1985a) method of think-aloud verbal transcripts and protocol analysis, to investigate the mathematical thinking of 1 [$\sp[\rm st]$] year undergraduates. Three groups of students were studied over a period of six months. The model was adapted through a 3-stage empirical process by allowing interventions by the researcher and analysing their impact on the students' thought processes. The study concentrated on self-regulatory metacognitive behaviour including Reflection, Organisation, Monitoring and Extraction, which developed a ROME model of analysis. The results of the study offered more mathematical interpretations of the students' self-regulatory behaviour solving single and multi variable integral problems. Through this mathematical efficacy in Calculus problem solving has been discussed in terms of ROME. The development of the ROME model of analysis has the potential to be used to analyse metacognitive behaviour of problem-solvers in other fields of mathematics. 214. {hel:integ} M. Helfgott. Integrated calculus. PhD thesis, Montana State University, 1997. This research addressed the question of whether there were differences in achievement between students that followed an integrated approach to calculus that integrated mathematics and physics, compared to students that followed a non-integrated approach. The subjects in the study were the 151 students that completed Calculus II (second semester calculus intended mainly for engineering students) at Montana State University-Bozeman during the fall of 1996. There were a total of five sections with five different instructors. All the sections used the Harvard Calculus book and took common exams. Three sections were assigned to the experimental group, which followed the integrated method, while two sections acted as the control group. Both groups covered the main topics of chapters 6-10 of the Harvard Calculus book. The instructors in the experimental group stressed problems about applications to physics, as well as the conceptual and computational aspects of calculus. In addition, students in this group received enrichment notes that supplemented the textbook. The instructors in the control group also stressed the conceptual and computational aspects of calculus as well as applications to physics. However, the control group did not delve as deeply into these applications and did not have the support of the enrichment notes. Analysis of Covariance (ANCOVA), with Calculus I scores and SAT-math scores acting as covariates, was the technique of choice to compare methods with regard to Calculus II and Physics I scores. Physics I is the first semester calculus-based physics course. ANCOVAS were also used with gender as a factor, and when students take Physics I as a factor (not yet, concurrently with Calculus II, or before Calculus II). For interaction analyses, two-way analyses of variance were employed once students were categorized into three groups according to their scores in Calculus I, SAT-math, and Calculus II. Students in the integrated group did significantly better in Calculus II. Interaction was found when Physics I scores were analyzed, with method and SAT-math groups as factors. Students with high mathematical aptitude in the integrated group scored significantly better than students with high mathematical aptitude in the non-integrated group, when Physics I scores were analyzed. No other interactions were detected. Furthermore, there were no differences in Calculus II achievement according to when students took Physics I. No differences in achievement according to gender were found either. On the basis of the findings of this study, an integrated approach to the teaching of second semester calculus is recommended. 215. {her-hit:artic} A. Hernandez and F. Hitt. Articulations between the settings, numeric, algebraic and graphic related to the differential equations. In Proceedings of the 16 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Baton Rouge, Louisiana, 1994. International Group for the Psychology of Mathematics Education. North American Chapter. To be added later. 216. {her:provi} R. Hersh. Proving is convincing and explaining. Educational Studies in Mathematics, 24:389-399, 1993. (This annotation is quoted from the paper abstract.) `In mathematical research, the purpose of proof is to convince. The test of whether something is a proof is whether it convinces qualified judges. In the classroom, on the other hand, the purpose of proof is to explain. Enlightened use of proofs in the mathematics classroom aims to stimulate the students' understanding, not to meet abstract standards of `rigor' or `honesty.' 217. {her-laz-90:int} R. Hertz-Lazarowitz. An integrative model of the classroom: The enhancement of cooperation in learning. In ??, volume ??, page ??, Boston, MA, 1990. American Educational Research Association. Cooperative learning aims to enhance students' on task interactive behaviors in the classroom. Observation in Israeli elementary schools has indicated that interactive behavior of students in their learning sequence holds potential for quality cooperation and help among children, but that teachers lack the skills to structure learning tasks that will enhance a high level of cooperation. Based on prior research an integrative model of the classroom was developed, and detailed developmental stage taxonomy was suggested to describe, explain, guide, and predict students' cooperative interaction in all types of classroom structures. The model is based on six related dimensions: (1) the physical organization of the classroom; (2) the nature of the learning task; (3) the instructional mode of the teacher; (4) the communication pattern of the teacher; (5) student social-learning behaviors; and (6) students' academic-cognitive behavior. This model and the research that followed indicate that variation in learning task structure and teacher behavior are the main factors in shaping students' behaviors. The inclusion of cooperative learning as part of the daily experience in the classroom is proposed. 218. {hod-mor:explo} T. Hodgson and P. Morandi. Exploration, explanation, formalization: a three-step approach to proof. Primus, 6(1):49-57, March 1996. The paper outlines a three-step approach to proof suggested by Mason, Burton, and Stacey, and discusses the authors' implementation of the approach in a transition course. The three steps are (1) convince yourself (exploration), (2) convince a friend (explanation), and (3) convince an enemy. The last step involves a formal, rigorous proof. The authors note that the first two steps are often omitted from instruction on proof, and they suggest that these steps are important in fostering an students' understanding of the role of proof in mathematics and in helping students develop the ability to write formal proofs. 219. {hon-tho:using} Y. Y. Hong and M. Thomas. Using the computer to improve conceptual thinking in integration. In Proceedings of the Conference of the International Group for the Psychology of Mathematics Education, volume 3, pages ???-???, Lahti, Finland, 1997. International Group for the Psychology of Mathematics Education. To Be added later. 220. {hoo:theef} D. E. Hooley. The effects of supplemental computer-assisted linear algebra instruction that provides geometric representations and additional applications. PhD thesis, The University of Iowa, 1988. The purpose of this study was to determine the effects on student achievement of supplemental computer-assisted linear algebra instruction that provides geometric representations and additional applications and to describe its operation and impact. Data were collected from the records of 444 students (31 who had participated in the supplemental instruction and 413 who had not), in order to analyze the effects of the supplemental instruction on course grade linear algebra. Stepwise regression procedures were used to form a linear model which predicted linear algebra grade from college grade point average, calculus grades, and ACT mathematics scores. This model accounted for 38the variance in linear algebra grade. Participation in the supplemental instruction did not add significantly to the ability to predict linear algebra grade after the other variables entered the model. A researcher-developed test of geometric representations of elementary linear algebra concepts was administered to students in the supplemental laboratory in spring and fall semesters of 1987 and to an equal number of regular linear algebra students, 36 subjects in all. There were significant initial group differences on background variables so an analysis of covariance was carried out using college grade point average as covariate. Significant differences favored the group which participated in the supplemental laboratory on additional understanding of geometric representations of concepts and the use of these geometric concepts in solving problems. Instruction in the supplemental laboratory was observed. The presentation of additional geometric representations of elementary linear algebra concepts took approximately 21the teaching of linear algebra rated all of the geometric representations presented as moderately to highly useful for understanding mathematical concepts in further study. The presentation of additional applications of linear algebra, consisting of 2- and 3-dimensional graphics, took less than 11mathematical concepts necessary for a complete 3-dimensional graphic program. Suggestions for improvements in the laboratory were gathered from participants. Recommendations for the laboratory and suggestions for future research are discussed. 221. {hoy:thecu} C. Hoyles. The curricular shaping of students' approaches to proof. For the Learning of Mathematics, 17(1):7-16, February 1997. The author discusses proof in the U.K. National Curriculum and describes a national study undertaken to survey 15-year-old students' concepts of proof. Results of the study are analyzed and the paper contains numerous examples of student responses to problems and questions on the survey. Particular attention is given to the role of the National Curriculum in shaping students' concepts. 222. {hug-sin-mcc-wai-gar-jag:mathe} S. Huggett, M. Singer, J. McConnell, I. Wain, T. Gardiner, and J. Jagger. Mathematics and proof. Mathematics Teaching, 158:20-27, March 1997. This article is a collection of essays written in response to the essay `What's the point of proof?' by Jonathan MacKernan included as part of the article `Teaching Proof' which appeared in the June 1996 issue of Mathematics Teaching. 223. {james} W. James. The principles of psychology. Holt, New York, NY, 1910. 224. {joh-joh-90:soc} D. W. Johnson and R. T. Johnson. Social skills for successful group work. Educational Leadership, (47):29-33, 1990. People do not know instinctively how to interact effectively with others. For cooperation to succeed, students must get to know and trust one another, communicate accurately and unambiguously, accept and support one another, and resolve conflicts constructively. A seven-step recommended procedure is outlined. 225. {sjj-coop-98} D. W. Johnson, R. T. Johnson, and K. A. Smith. Cooperative learning returns to college: What evidence is there that it works? Change, 30:26-35, 1998. There is a rich theoretical base for cooperative learning. Three interrelated types have been developed (formal, informal, cooperative base groups) that provide a framework for effective college teaching. However, too much emphasis is placed on developing the skills of individuals and too little on creating learning communities within which achievement of all students is enhanced. 226. {joh:gradu} E. L. Johnson. Graduate teaching assistants' beliefs about teaching mathematics. PhD thesis, The University of Nebraska - Lincoln, 1998. In many schools graduate teaching assistants represent undergraduates' primary mathematics instructional contact. Nevertheless, GTAs often have varying knowledge of and interest in mathematics education. The study explored the relationship between GTAs' beliefs in their potential impact on student achievement, their self-concept, and their attitudes toward certain targeted pedagogical issues. Additionally, this study investigated the relationship between these GTAs' beliefs and student achievement. The sample for this study consisted of 71 mathematics GTAs conducting a calculus recitation or teaching an introductory mathematics course at the University of Nebraska-Lincoln, Purdue University, and the University of Kansas during the 1997-1998 academic year. Descriptive analysis, correlations, analysis of variance, multivariate analysis of variance, and multiple regression analysis were conducted to answer the research questions. Results indicated that student achievement was inversely related to UNL GTAs' external efficacy dimension. GTAs' sense of current educational status was related to student achievement. Belief variables internal, task frequency, confidence, reform, and status of current education as measured by this study were related. Reported task frequency was the significant predictor of GTAs' confidence, internal efficacy dimension, and sense of instructional efficacy for all GTAs. GTAs' confidence significantly predicted their sense of the status of current education, while GTAs' confidence and their sense of the internal efficacy dimension significantly predicted their reported task frequency. Several demographic variables related to graduate teaching assistants' beliefs. Multivariate analyses results indicated a difference in all GTAs' belief variables according to school, but follow-up univariate analyses were unable to pinpoint these differences. Multivariate analyses indicated a difference in all GTAs' belief variables according to career goals, but univariate follow-up analyses did not yield any significant differences. Multivariate analyses revealed differences among UNL GTAs' belief variables according to the demographic variable gender. Follow-up univariate analyses revealed female GTAs scored higher than male GTAs on both the internal efficacy dimension and reported task frequency. Recommendations of this study included development of a GTA professional development training addressing the relationship between instructors' beliefs with classroom behavior, students' beliefs, and student achievement, grounded in self-analysis of efficacy beliefs. 227. {sha:thein} III J.R. Shackleford. The influence of teaching strategies on college students' acquisition of calculus content. PhD thesis, Vanderbilt University, 1992. The purpose of this research was to determine correlations between teaching practices exhibited by teachers of differential calculus and student learning demonstrated by the students in the classes of the subject teachers. Ten teaching practices were selected based on earlier research performed in college and junior high mathematics classes. Portable computers were taken into the classrooms of 21 volunteer teachers. Data concerning teacher activity were captured at sixty second sampling intervals. Counts of teacher questions asked, correct student answers to teacher questions, and student questions asked during the 60 second interval were entered at the end of each interval. Students were administered a calculus pretest at the beginning of the semester, and a parallel form calculus posttest at the end of the differential calculus portion of the semester. Gain scores (posttest score minus pretest score) were determined for the students in the classes. The gain scores (corrected for entry level) were averaged for the students in each teacher's class to determine a gain score for each teacher in the sample. Nineteen of the observed teachers turned out enough students on the posttest to be included in the sample. A correlation analysis was conducted to correlate the teaching practice variables with the gain scores for each teacher. Additionally, the four most effective teachers (in terms of corrected gain scores) were compared to the four least effective teachers on each of the ten teaching practice variables. Finally, a regression analysis was performed to determine the ideal percentage of correct student answers to teacher questions. Time spent in interactive problem solving and the number of teacher questions asked were both significantly correlated with student learning as measured by gain scores. Additionally, the regression analysis showed that the ideal percentage of correct student answers to teacher questions was 83in line with previous mathematics (non calculus) research. This suggests that there is some generalizability across course boundaries and a positive effect from involving students actively in processing important concepts. The implications of this study for future research entail the use of the identified significant teaching practices in experimental research involving comparisons of the student achievement gains for students taught by (experimental group) teachers trained in the use of these practices and students taught by (control group) teachers who do not employ them. 228. {jur-zei:theef} M. Jurdak and R. Zein. The effect of journal writing on achievement in and attitudes toward mathematics. School Science and Mathematics, 98(8):412-419, December 1998. (This annotation is quoted from the paper abstract.) `A teaching experiment was conducted to investigate the effect of journal writing on achievement in and attitudes toward mathematics. ...Subjects were selected from first intermediate students (11-13 years) attending the International College, Beirut, Lebanon ...The journal writing group (JW) received the same mathematics instructions as the no-journal-writing (NJW) group, except that the JW group engaged in prompted journal writing for 7 to 10 minutes at the end of each class period, three times a week, for 12 weeks. The NJW group engaged in exercises during the same period. The results of ANCOVA suggest that journal writing has a positive impact on conceptual understanding, procedural knowledge, and mathematical communication but not on problem solving, school mathematics achievement, and attitudes toward mathematics.' 229. {kai:demon} M. Kaiser. Demonstrating proof by contrapositive and contradiction through physical analogs. School Science and Mathematics, 93(7):369-372, November 1993. The author gives several examples from physical science illustrating the use of proof by contradiction. 230. {MR97d:00013} J. Kaput, A. H. Schoenfeld, and E. Dubinsky, editors. Research in collegiate mathematics education. II. American Mathematical Society, Providence, RI, 1996. The ten papers in this collection include the following, which are being reviewed individually: David Dennis and Jere Confrey, The creation of continuous exponents: a study of the methods and epistemology of John Wallis (33-60); Rina Zazkis and Ed Dubinsky, Dihedral groups: a tale of two interpretations (61-82). 231. {MAA33} James J. Kaput and Ed Dubinsky, editors. Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results, MAA Notes Number 33. Mathematical Association of America, Washington, DC, 1994. Contains nine research papers presented at the Joint Special Session on Research in Undergraduate Mathematics Education at the annual meeting of the American Mathematical Society and the Mathematical Association of America (San Francisco, California, January 1991). Paper titles are: ``The Teaching and Learning of College Mathematics: Current Status and Future Directions" (Joanne Rossi Becker, Barbara J. Pence); ``Even Good Calculus Students Can't Solve Nonroutine Problems" (John Selden, Annie Selden, Alice Mason); ``Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals" (Joan Ferrini-Mundy, Karen Graham); ``A Conceptual Analysis of the Proof-Writing Performance of Expert and Novice Students in Elementary Group Theory" (Eric W. Hart); ``Cognitive Obstacles to the Learning of Calculus: A Kruketskiian Perspective" (F. Alexander Norman, Mary Kim Prichard); ``The Role of Emotion: Expert and Novice Mathematical Problem-Solving" (Frances A. Rosamond); ``Understanding How Students Acquire Concepts Underlying Sets" (Nancy Hood Baxter); ``Symbiotic Effects of Collaboration, Verbalization, and Cognition in Elementary Statistics" (Martin V. Bonsangue); ``Multiple Representations for Functions" (Albert A. Cuoco). 232. {kat:using} V. Katz. Using the history of calculus to teach calculus. Science and Education, (3):243-249, 1993. A historical approach to the teaching of calculus provides the students with a better understanding of the material than the standard approach and helps as well to introduce them to the relationship between mathematics and other aspects of our culture. Describes in some detail a course in calculus which is based on such an approach. 233. {kee-ste:coope} C. M. Keeler and R. K. Steinhorst. Cooperative learning in statistics. Teaching Statistics, 16(3):81-84, 1994. The authors studied university students' success with cooperative learning in elementary statistics. The study involved one section each semester for three semesters. The first semester students were taught using a traditional lecture format. The following two semesters, cooperative learning techniques were incorporated into the course. Pairs of students worked together for three to five minutes to answer questions posed during a break in the lecture. These questioning breaks occurred every ten to fifteen minutes throughout the class. The researchers found that the students in the cooperative classes had better course averages than students in the previous lecture course. They also found that fewer students withdrew from the cooperative sections. 234. {ken:effec} P.A. Kenney. Effects of supplemental instruction (SI) on student performance in a college-level mathematics course. PhD thesis, The University of Texas at Austin, 1988. This experimental study examined the effects of participation in a Supplemental Instruction (SI) program on student performance in a first-semester calculus course for business and economics majors. SI is an academic support program which incorporates study techniques into the framework of an academic course. According to Blanc, DeBuhr, and Martin (1983), students who experienced Supplemental Instruction had higher course grades, semester grade point averages, and rates of reenrollment than did nonparticipants. The experimental portion of the present study was conducted within the discussion sections of two large lecture classes of business calculus. This researcher served as a teaching assistant to students (n = 51) in two discussion sections; she adopted the role of Supplemental Instruction (SI) leader to students (n = 50) in another two discussion sections. The experimental design included controls for instructor effects and experimenter bias. Data from the experimental situation were analyzed using multiple linear regression and chi-square techniques. Results showed that the SI students received higher course grades in first-semester business calculus and earned higher semester grade point averages than their non-SI counterparts. However, there was no evidence that participation in the SI program affected the frequency of D and F grades or course withdrawals. The experiment revealed that the SI program seemed to operate in conjunction with other factors, including mathematics aptitude, prior academic achievement, and discussion section attendance. To investigate any residual effects from the SI program, the 101 students were tracked for an additional semester. Results from the follow-up study showed that students who had experienced SI had a pattern of fewer F grades in and withdrawals from the second-semester business calculus course; however there was no significant difference between the mean final course grades. Results from the experimental situation and the follow-up study had implications for future Supplemental Instruction research and implementation. Additional controlled experimental studies on the effects of SI should be conducted along with research on the role and behavior the SI leader. Suggestions for program implementation included using the discussion section as a location for SI as well as incorporating SI techniques as part of the training program for university teaching assistants. 235. {ken:anexa} P.D. Kent. An examination of variables that may influence achievement in high school calculus students. PhD thesis, The Ohio State University, 1990. during the 1981-82 school year, U.S. high school calculus students performed at the median level attained by students in the other countries participating in the SIMS. Performance gains since the First International Mathematics Study given in 1964 were slight. Thus, U.S. high school calculus students, a more select group than from the other countries, were scoring lower than students in seven other countries. In order to seek ways to improve the achievement level of high school calculus students, nine factors were examined that may influence achievement. These factors were chosen based upon other research in mathematics education. No research could be found that examined only high school calculus students and achievement. One high school in each of five central Ohio public school districts agreed to be part of the study, involving seven teachers and 204 students. Each student was given one test form of the achievement test developed for SIMS, an attitude survey from SIMS, and some biographical questions. The data were examined in terms of: teachers of higher-achieving classes versus teachers of lower-achieving classes, higher-achieving classes versus lower-achieving classes, higher-achieving students versus lower-achieving students and gender. The analysis was by frequency distributions, t-tests, and ANOVAs. Of the nine factors, three were not statistically different between groups: teacher enthusiasm, hours spent per week on all homework, and plan to take the AP-Test. Two factors were statistically different for all groups: attitude towards mathematics and plans to take more years of mathematics. The other four factors: semester grade, previous grade, use of calculator in the classroom, and hours spent on mathematics homework per week, were not statistically different for all groups. The results of this study were: (1) more experienced teachers of calculus had higher-achieving classes than less-experienced calculus teachers; (2) a negative relationship between positive attitude towards mathematics and score on the achievement test existed; and (3) higher-achieving students planned to take more years of mathematics than lower-achieving students. 236. {kier-aera-95} T. Kieren, L. G. Calvert, D. A. Reid, and E. Simmit. Coemergence: Four enactive portraits of mathematical activity. Paper presented at AERA, ERIC #ED390706, 1995. Team research is important in studying cognition as enactive. This paper contains four different pieces of research directed toward the evidences and artifacts of two students in Canada engaging in a sustained mathematical activity. These four portraits of mathematical cognition in action consider the conversation in which the activity occurs; the structures manifested in the beliefs of these students about mathematics; the patterns of reasoning in action; and the dynamical growth or changes in the mathematical understanding of this pair of students. These approaches and pieces of research can be observed as coemergent. An introductory paper, "Enactivism and Education, Especially Mathematics Education" is followed by four research papers: "A Portrait of Mathematical Conversation" (Lynn Gordon Calvert); "A Portrait of the Coemergence of Reasoning" (David A. Reid); "A Portrait of Beliefs in Action" (Elaine Simmt); and "A Pathway Portrait of Mathematical Understanding in Inter-Action" (Thomas E. Kieren). Appendixes contain verbal snapshots and writing samples of the two students. 237. {kin:theef} D. P. Kinney. The effect of graphing calculator use and the Lesh Translation Model on student understanding of the graphical relationship between function and derivative in a nonrigorous calculus course. PhD thesis, University of Minnesota, 1997. The purpose of this study was to examine student understanding of the graphical relationship between function and derivative in a nonrigorous calculus course. With the increasingly widespread use of graphing calculators, which can readily generate the graphs of functions and derivatives, graphical representations are being more widely used in calculus. The Lesh Translation Model suggests that student understanding of function and derivative can be described by students' ability to make translations between and within modes of representation such as algebraic equations, graphs, and words. Students in the investigator's experimental group received explicit instruction in the use of the TI-85 graphing calculator along with performing translations involving function and derivative. The Lesh Translation Model provided the theoretical framework for the construction of these translations. Nimerovsky's work with 'resemblance' and 'variational' approaches was incorporated into the instruction for the experimental group. The investigator's control group received explicit instruction in the use of the TI-85 graphing calculator, but no instruction related to translations. Quantitative results showed that students who received instruction in performing translations among various representations significantly outperformed those who did not receive instruction when asked to perform translations among functions and derivatives, despite both groups using graphing calculators. There was no measurable difference in students' ability to perform routine symbolic manipulations. The experimental group tended to use more sophisticated strategies when performing translations than the control group. Students in the control group generally had difficulty interpreting the properties of graphs, especially the graphs of derivatives. Students who use calculators still must learn to correctly interpret the graphs of functions and derivatives, and when both are graphed simultaneously, develop strategies for understanding the relationship between the graph of a function and its derivative. Graphing calculator use when combined with instruction performing translation can lead to significantly increased ability to perform translations and a richer, more appropriate concept image of function and derivative than graphing calculator use without instruction. 238. {kle:acomp} T. J. Klein. A comparative study on the effectiveness of differential equations instruction with and without a computer algebra system. PhD thesis, Peabody College for Teachers of Vanderbilt University, 1993. This study examines the use of the computer algebra system (CAS) Mathematica in traditional differential equations classes. One objective of the study was to determine if the use of Mathematica as a classroom demonstration tool affects achievement in solving differential equations. A second objective was to determine if the use of Mathematica in this way affects student attitudes toward computers, Mathematica, and differential equations. The last objective was to determine if students would use Mathematica for differential equations if student use were optional. The study was conducted in four sections of an elementary differential equations course, two using a CAS and two using traditional instruction. Effectiveness of instruction was determined by measuring achievement levels on examinations of relevant material. Student use of Mathematica and attitudes toward computers, Mathematica, and differential equations were measured with student questionnaires and interviews. Student use of Mathematica was not required during the treatment period. It was found that Mathematica as a demonstration tool did not improve achievement in solving differential equations. Both experimental classes' computer attitudes changed significantly, but the classes did not attribute these changes to Mathematica. Students attitudes concerning Mathematica were mixed, and few students used Mathematica outside of class when it was available to them. 239. {kle:rigor} I. Kleiner. Rigor and proof in mathematics: a historical perspective. Mathematics Magazine, 64(5):291-314, December 1991. In tracing the history of mathematical proof, the author highlights three themes: (1) the perceived validity of a proof is a reflection of the overall mathematical climate at any given time; (2) the causes of transition from less rigor to more rigor (or vice versa) were, in general, not aesthetic or epistemological; there were good mathematical reasons for such changes; (3) every tightening (or relaxation) of the standard of rigor created new problems having to do with rigor. 240. {kno:anana} C. A. Knoll. An analysis of the effectiveness of a remedial precalculus course on the calculus achievement of science and engineering majors at a private technological university. PhD thesis, Florida Institute of Technology, 1990. This study examines the effectiveness of a formal remedial precalculus course when applied to science and engineering students prior to entry into the first calculus course. All entering freshmen (without transfer credit) in science and engineering disciplines were tested with the Calculus Readiness Test published by the Mathematical Association of America. A treatment group of 142 students requiring remediation took a one quarter precalculus course. After the precalculus course, these students took a parallel version of the Calculus Readiness Test. A quasi-experimental design was employed to determine whether the precalculus course improved student achievement in the first calculus course. Results were evaluated using both final exam scores and course grades. The control consisted of 37 students who should have taken the precalculus course but did not. Additionally, the possibility of an interaction effect on calculus grades between taking the precalculus course and Scholastic Achievement Test (SAT) verbal scores was examined. Three t-tests were employed to determine: (1) if the remedial precalculus course improved the algebra and trigonometry skills; (2) whether remedial precalculus enhanced student performance on the calculus final exam (30 pairs of students, seven in the control group did not take the final) and (3) if remedial precalculus improved the students' calculus course grades (37 pairs of students). Additionally, analysis of covariance (with all students in the treatment and control groups) was employed to determine whether remedial precalculus improved student performance on the calculus final exam and course grade and to determine whether or not an interaction effect on calculus finals and grades exists between taking the precalculus course and SAT verbal scores. The results of this study indicate that those students who took the precalculus course scored significantly higher on the second Calculus Readiness test and had higher calculus final exam scores and grades than those students who should have taken the precalculus course but did not (p [$<$].05). Although the treatment effect on Calculus I grades was only 7the treatment group averaged nearly one standard deviation above the control group on the Calculus I final exam. However, no interaction effect on calculus final exam scores or grades between taking the precalculus course and SAT verbal scores was detected. Additionally, a calculus grade prediction model was constructed using multiple regression. High school GPA, SAT math scores and the Calculus Readiness Test scores contributed significantly (p [$<$].05) for a total of 15% variance. 241. {kra:class} D. P. Kraines and et al. Classroom computer capsule. College Mathematics Journal, (2):160-162, 1991. This article describes a calculus lesson that illustrates the nature of cycles in simple systems of nonlinear differential equations through the use of the Lotka-Volterra predator-prey model as incorporated in the computer software package, Phaser (version 1.0). 242. {kra:promo} P.A. Kraus. Promoting active learning in lecture-based courses: Demonstrations, tutorials, and interactive tutorial lectures. PhD thesis, University of Washington, 1997. The research described in this dissertation focuses on improving student learning in introductory physics courses in which most of the instruction is conducted in a lecture setting. We began this investigation by examining the effectiveness of a very common medium for engaging students intellectually-the lecture demonstration. Results obtained early in the study suggested that many lecture demonstrations, as they are typically shown, do not assist students in the development of a functional understanding of the concepts that the demonstrations are intended to elucidate. Therefore, as the research progressed, we shifted the emphasis away from demonstrations toward a broader examination of student learning in a lecture-based course. We found that the guided inquiry approach that had proved effective in the small-group tutorials developed by the Physics Education Group could be successfully adapted for use in large lecture courses. Research on student understanding in the calculus-based physics course was used to guide the design of several interactive tutorial lectures. The development process is described in the context of several examples that include the following topics: two-dimensional motion, forces and Newton's laws, and magnets and charge. 243. {kru:image} L. Krussel. Image structures and reification in advanced mathematical thinking: The concept of basis. In Proceedings of the 16 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Baton Rouge, Louisiana, 1994. International Group for the Psychology of Mathematics Education. North American Chapter. To be added later. 244. {kuh-mcc-sch:along} J. Kuhn, G. McCabe, and K. Schwingendorf. A longitudinal study of the c4l reform program: Comparisons of c4l and traditional students. Submitted for publication. The authors present results of a statistical comparison between 205 students who took the course Calculus, Concepts, Computers and Cooperative Learning (a reform course designed using APOS theory) and 4431 students who took a traditional calculus course at Purdue University. The data consists of grades and the numbers of calculus and non-calculus mathematics courses taken by each student. The reform course students earned higher grades in calculus courses, were as adequately prepared for math courses beyond calculus as well as all other academic courses, took more calculus courses, and took about the same number of non-calculus mathematics courses as the traditionally taught students. 245. {lak:proof} I. Lakatos. Proofs and Refutations. Cambridge University Press, 1976. This classic work in the philosophy of mathematics was first published in 1963 as a series of articles in the British Journal for the Philosophy of Science. Its format is a hypothetical dialog among a teacher and his students regarding the Euler-Descartes formula: V-E+F=2. The teacher presents a `proof,' the students find counterexamples to statements in the `proof,' the teacher produces a revised `proof' (or a revised statement of the conjecture), and the cycle begins again. In this way Lakatos applies his epistemological analysis to informal mathematics, mathematics as it is practiced daily by mathematicians and students. The dialog mirrors the actual history of the Euler-Descartes conjecture which is communicated to the reader via footnotes to the text. According to Hanna (Educational Studies in Mathematics, 24: 421-438, 1993), Proofs and Refutations exerted a strong influence on mathematics education, as it helped to swing the pendulum of the philosophy of mathematics away from formalism. 246. {lav:anass} B. Laverne. An assessment of knowledge for solving proportion-related problems. PhD thesis, The Catholic University of America, 1994. Theories of knowledge (e.g., Anderson, 1983) hypothesize that human memory consists of networks of related pieces of information (i.e., declarative and procedural knowledge). In problem solving, these knowledge types are embedded in a third knowledge cluster (i.e., schema knowledge). Schema knowledge consists of a collection of well-connected facts, features, algorithms, skills, and/or strategies. Marshall (1991) depicts problem solving knowledge with a knowledge graph including individual pieces of information (i.e., nodes) and their connections (arcs). This study (1) used these theories of memory, to develop such a graph for the knowledge required for solving proportion-related problems and then (2) assessed the degree to which learners possessed a well-connected body of domain knowledge for solving these problems. Specifically, in a three phase assessment procedure, using college students from three successive course levels this research: (1) identified underlying knowledge for solving proportion-related problems; (2) assessed the knowledge nodes possessed by learners; (3) assessed the connectivity between these nodes; (4) created inferred student knowledge graphs; (5) made predictions about performance; and (6) verified predictions by collecting and analyzing subjects' talk-aloud protocols. As hypothesized, the results of the first assessment revealed a significant difference in the number of knowledge nodes possessed by the algebra and precalculus groups. Additionally, fewer subjects than expected in the algebra group possessed the equivalent-ratio knowledge node. The reverse occurred for the calculus group. For the algebra group, fewer subjects than expected possessed the proportion knowledge node. For the precalculus and calculus groups, the reverse occurred. Based on their scores on the first assessment, subjects were ranked and placed in three groups (i.e., low, mid, high). Supporting the hypothesis, the findings in the second assessment revealed a significant difference between the low and high groups in the degree of connectivity of knowledge. However, contrary to the prediction, knowledge graphs for the high group deviated significantly in number of nodes and arcs when compared to the posited knowledge graph. Lastly, predictions about performance were more accurate for subjects with well connected knowledge graphs than those with fragmented ones. 247. {law:anana} G. S. Lawrence. An analysis of the algebraic competencies and other characteristics which affect success in developmental mathematics courses on the college level. PhD thesis, North Carolina State University, 1988. One purpose of this investigation was to identify those entrance level algebraic competencies needed to successfully complete freshman developmental mathematics courses at North Carolina State University. Secondly, there was an attempt made to determine if a relationship exists between success in Basic Algebra (MA 115) and success in the subsequent mathematics course, Algebra and Trigonometry (MA 111). A residual function of the study was to examine the relationship between the algebra and trigonometry course and other math and science courses such as Calculus I and Chemistry I. Success in MA 115 was defined as a final grade of A or B, while success in MA 111 was defined as a final grade of A, B, or C. The other variables are: High school grade point average (HSGPA), Algebra Placement Test score (ALG), Scholastic Aptitude Test-Mathematics score (SAT-M), Scholastic Aptitude Test-Verbal score (SAT-V), Trigonometry Placement Test score (TRIG), RACE and SEX. An investigation using factor analysis and canonical discriminant function analysis revealed several of the algebraic competencies on the Mathematics Placement Test as discriminating variables for success in MA 115 and MA 111. They are: algebraic expressions (simple and rational), exponents and radicals, solution of linear equations, solution of linear inequalities, solution of quadratic equations, and solution of systems of equations. A significant relationship between success in MA 115 and the student's overall high school grade point average was found to exist. However, there is no significant relationship between sex, race, or the interaction of the two and success in MA 115. These findings were the same with regard to MA 111. Success in MA 115 does affect success in MA 111. When two more variables, success in MA 115 and high school math total grade point average, were added, the findings indicated that these last two variables are highly significant in determining success in MA 111. A significant relationship was found between the grade received in MA 111 and the grade received in Chemistry 101 at North Carolina State University. However, students who initially placed in Calculus I (MA 102) performed better in Chemistry 101 than those who were required to take MA 111 prior to Calculus. (Abstract shortened with permission of author.) 248. {leg-91:coll} A. LeGere. Collaboration and writing in the mathematics classroom. Mathematics Teacher, (84(3)):166-171, 1991. Described are classroom strategies chosen to elicit greater involvement by students in the learning process and to furnish opportunities for practice in critical thinking. The advantages and disadvantages of this teaching approach are discussed. 249. {leg:linke} J. M. Leggett. Linked case studies of the dissemination of the emerging scholars programs in three community colleges. PhD thesis, The University of Texas at Austin, 1997. This study examined the dissemination attempts of three community colleges in adapting an Emerging Scholars Program (ESP) on their campuses. Using linked case studies, the study focused on the description of the role of faculty and administrators in implementing the Emerging Scholars Program. Of key significance was the perceived need for the ESP, the problem to be addressed, the combined effort used within the institution to adapt the program, how dissemination occurred, and the results obtained. The Emerging Scholars Program evolved from the work of Dr. Philip Uri Treisman, a University of California at Berkeley mathematician, who was intrigued by the success of Asian American students and the lack of success of African American students in freshman calculus. Treisman's study led to the development of a set of strategies and a framework for addressing the persistent under-performance of African American, Hispanic, and Native American students in introductory collegiate mathematics courses. The ESP is an academic excellence program with six characteristic elements. The Emerging Scholars Program focuses on students' strengths rather than their weaknesses. The existence of several well-established Emerging Scholars Programs at four-year institutions has permitted numerous studies on the program. In community colleges, however, the Emerging Scholars Program is still in its infancy, and additional research is needed. The dissemination of the ESP has evolved from a grass roots effort conducted by the Charles A. Dana Center at the University of California at Berkeley to its current broad dissemination through building connections among mathematicians to encourage underrepresented minority students to seek careers in mathematics. A major finding of this study of the dissemination of the ESP in three community colleges is that underrepresented minority and other students who participated in ESP workshops successfully completed courses in mathematics and the sciences consistently at a success rate up to a grade higher than non-ESP students. Other findings are: (1) the ESP model cannot be translated in its entirety across all sectors of higher education; (2) real creativity is needed to establish an ESP at a community college; (3) stable funding is key in establishing and institutionalizing an ESP; (4) institutional planning, evaluating, and customizing are required in establishing an Emerging Scholars Program; and (5) only a small number of students will be served. 250. {leo:proof} B. Leonard. Proof: the power of persuasion. Mathematics Teacher, 90:202-205, March 1997. The author uses three specific examples of lessons from his classroom to argue that: (1) `If we incorporate the notion of convincing to a greater extent in our proofs, we may well affect the impact of our presentation.' (2) `Sometimes an extreme example or counterexample will do more to convince a student of a certain conclusion than a formal recitation of symbolic language.' (3) `Although we were taught that `using a particular number is not a proof ...,' I suggest that the rigor is not always dependent on the symbols but rather on the generality of the idea conveyed.' (4) `...we might frequently plan to create a `thirst' for the result in the mind of the student before offering a proof of same.' (5) `Learning to give back formal symbols from memory is perhaps not as effective as developing an ability to recognize a logical flow of ideas or the absence thereof.' 251. {ler:struc} U. Leron. Structuring mathematical proofs. The American Mathematical Monthly, 90:174-185, March 1983. Applying the idea of `structured programming' from computer science to the art of presenting proofs to students, the author advocates organizing the proof presentation into levels of generality, using a `top-down' approach. This method of communicating proofs is illustrated with four detailed examples from number theory, calculus, high-school geometry, and and linear algebra. The author concludes with some suggested learning activities. 252. {ler:heuri} U. Leron. Heuristic presentations: the role of structuring. For the Learning of Mathematics, 5(3):7-13, November 1985. Applying the idea of `structured programming' from computer science to the art of presenting proofs to students, the author suggests two heuristics for communicating proofs. These heuristics, to be incorporated into the presentation of the formal proof, are (1) prefacing a formal proof with a brief, intuitive overview and (2) (for a proof requiring a construction) using the constraints of the problem to search for the form of the object to be constructed and then using that form to define the object (rather than just producing the object out of the blue and arguing that it works). The author illustrates these heuristics with a proof of the Cantor-Bernstein Theorem of set theory. The ideas of `structured programming' come into play as the proof presentation is organized into levels of generality, using a `top-down' approach in which the heuristics are applied at each level. 253. {lev-90:mat} B. N. Levy. A mathcad exploration: Hunting for hidden roots. Mathematics Teacher, (83(9)):704-708, 1990. Discussed is the use of a program called MathCAD which allows students to solve higher order polynomial equations and geometry problems. The use of cooperative solving is emphasized. Included are graphs and part of a printout generated while solving problems with MathCAD. 254. {lid:calcu} D. C. Lidstone. Calculus students' understanding of dynamic situations. PhD thesis, Simon Fraser University (Canada), 1992. The questions I wished to investigate were: (1) What do calculus students understand about the concepts of change, and rate of change? (2) How does this contribute to their mastery of formal course material? I used a variety of non-standard tasks to promote unpractised responses from the subjects. The tasks addressed their understanding of dynamic situations in descriptive, graphical, numerical, and algebraic settings. These settings were used to describe situations involving the relative motion of two vehicles, the height of water flowing into an assortment of containers, the populations of two microbe cultures, the weights of two groups of children, and various situations represented only by mathematical functions. The results of the study indicated that all the subjects had a good intuitive sense of dynamic situations but that connections between this intuitive sense and the formal constructs to study such situations were, generally, quite varied. Each of the three portraits showed that a different aspect of the notion of rate played a dominant role in the students' understanding of this concept. All subjects showed some difficulty distinguishing between the behaviour of a function and the values of the function. This seemed to be intimately connected to developing facility with the notion of negative rate. The subjects' conceptual command seemed to be quite independent of their computational command. Many of the activities in the interviews seemed to promote conceptual command, and the connections between subjects' intuition and formal mathematical notions inherent in studying dynamic situations. (Abstract shortened by UMI.) 255. {low:thede} R. J. Lowinger. The development and validation of measures of students' abilities to learn procedural and conceptual mathematical material for the prediction of performance in college calculus. PhD thesis, State University of New York at Albany, 1996. The purpose of the study was to: (1) provide measures of students' abilities to learn procedural and conceptual mathematical material; (2) determine the extent to which students' abilities to learn procedural and conceptual mathematical material are related with each other, and to other measures which may be relevant to performance in learning complex problem solving in college mathematics; and (3) determine the extent to which students' abilities to learn procedural and conceptual mathematical material are related to their ability to master standard mathematical exercises in first-year college calculus. A test instrument was developed consisting of four subscales to measure abilities for procedural acquisition (P1), procedural application (P2), understanding mathematical symbols and operations (C1), and understanding conceptual rationales for mathematical procedures (C2). A sample of thirty-three first-year college calculus students was used in the test development process; an additional sample of forty-four students drawn from four intact first-year college calculus classes provided the sample for examining the inter-relationships of the measures and their correlations with students' performance on routine exercises on their college calculus final examinations. The overall test reliability was alpha =.84; the subscale reliabilities were: P1, alpha =.48; P2, alpha =.65; C1, alpha =.65; C2, alpha =.51. Interrater reliability was.94. The correlation between P1 and P2 was relatively high (r =.57); the correlation between C1 and C2 was.48. C1 was highly correlated with both P1 (r =.73) and P2 (r =.74). Correlations of C2 with P1 and P2 were considerably lower (r =.40 and r =.39, respectively). Correlations of procedural and conceptual abilities with performance on calculus exercises in order of magnitude were C1 (r =.42), P2 (r =.40), P1 (r =.34), and C2 (r =.10). Other variables that correlated significantly with calculus performance were prior high school mathematics grades as well as mathematics and verbal SAT scores. Students' strategies studying for their final examination did not correlate positively with calculus performance. The results of multiple regression analyses showed that a combination of the combined procedural subtests and grades in third-year high school mathematics accounted for 36C1 and grades in third-year high school mathematics also accounted for 36the variance in calculus performance. 256. {tur:conce} P. Turegano M. Concepts pertaining to measurement and the learning of infinitesimal calculus. PhD thesis, Universidad Del Pais Vasco/Euskal Herriko Unibertsitatea (Spain), 1994. In principle, the purpose of our research has been to find a model inside the mathematical context (a definition of integral alternative to that of Riemann) so that we can use it in the elaboration of a didactic proposal which allows us to introduce Secondary Education students into the definite integral on a conceptual level. This proposal, tested on students who have not been initiated in the study of infinitesimal calculus, would allow us, on the one hand, to analyse the difficulties which they find with our proposal, and, on the other hand, to determine if the learning difficulties, which have already been stated in different studies, can be solved. It would allow us as well to identify the images of the concept of definite integral formed in the minds of students. Therefore, in our research there are two phases which are clearly differentiated: (1) the first phase occurs in the field of mathematical ideas, and (2) the second phase takes place in the field of educational research. Naturally, each phase has different objectives and different ways of achieving them. We think that the originality of this research lies, in the first place, in the fact that we have not found any previous studies whose aim was the learning of the concept of integration as a first introduction to infinitesimal calculus, taking the calculus of plane areas as a starting point; in the second plane, in the use of a statistical method (factor analysis of correspondences) for classifying students who are to be interviewed according to three subjects which, in our opinion, have a direct influence on the concept of definite integral, and which allow us to analyse the learning difficulties and the students' development in these subjects; finally, in that the analysis of the answers has been made using the data obtained both in written tests and interviews. We therefore have not used any previous method of classifying the students. 257. {ma-jrme-99} X. Ma. A meta-analysis of the relationship between anxiety toward mathematics and achievement in mathematics. Journal for Research in Mathematics Education, 30:520-540, 1999. 258. {ma-kish-jrme-97} X. Ma and N. Kishor. Assessing the relationship between attitude toward mathematics and achievement in mathematics: A meta-analysis. Journal for Research in Mathematics Education, 28:26-47, 1997. Describes a study designed to assess the magnitude of the relationship between attitude toward mathematics and achievement in mathematics. The study employs a meta-analysis to integrate and summarize the findings from primary studies. 259. {mac:inves} J. E. Mackin. Investigation of selected outcomes of the dynamic physics learning environment: Understanding of mechanics concepts and achievement by male and female students. PhD thesis, The Pennsylvania State University, 1998. The study investigated the Dynamic Physics learning environment to determine its effectiveness in promoting understanding of mechanics concepts and achievement as well as the differences between male and female students in understanding of physics concepts and progress in the course. The Dynamic Physics learning environment was chosen for research since it offered an opportunity to study an introductory calculus based physics that incorporated recommendations, goals, and strategies for effective learning outlined in recent reports on physics education. Data were collected using pretests and posttests, course assessments, surveys, students' evaluations, and midterm feedback interviews. The data from the pretest, posttest, and course assessments were analyzed using a difference of means t test ([$p<0.05$]). Data from evaluations and interviews were analyzed to identify central themes in students' response. Students made above-average normalized gains in conceptual understanding from pretest to posttest when compared with a range of normalized gains from physics education studies. Analysis of interviews and evaluations of the course indicated students perceived they were understanding physics concepts in this learning environment and realized this understanding was important background for future courses. Students suggested that connections to real world applications, the format of the course, and collaborative learning were factors that helped them to understand physics concepts by making them more meaningful. Students' responses suggested that teaching strategies used in the course not only helped them to develop understanding, but confidence in their ability to learn physic. The sample of female students had significantly lower average scores in the pretest and posttest compared to male students; however, the sample of female students made the same or higher average gains in conceptual understanding when compared to the male students. Additional findings included that groups with one or more female member earned higher averages in group assessments than all male groups. Female students' perceptions of physics changed during the course; female students indicated an increase in relating personal experiences and real world experiences to the study of physics. The study indicated the Dynamic Physics learning environment offered the opportunity to learn for both male and female students and promoted understanding of physics concepts. 260. {mac-92:idea} S. MacLeod. Ideas in practice: Writing the book on fractions. Journal of Developmental Education, (16(2)):26-28, 1992. Describes a project in which students in a community college basic mathematics class worked collaboratively to write their own book on fractions. Students reinforced their math and cooperative skills, gained confidence, organized and revised their ideas, and created their own examples. Relates teacher and student responses to the project. 261. {mad:writi} C. Madigan. Writing across the curriculum resources in science and mathematics. Journal of College Science Teaching (JCST), 16:250-253, February 1987. This article consists of an annotated bibliography of resources for writing across the curriculum. 262. {mal:astru} J. R. Malpass. A structural model of self-efficacy, goal orientation, worry, self-regulated learning, and high-stakes mathematics achievement. PhD thesis, University of Southern California, 1994. The study investigatd self-regulated learning, self-efficacy, goal orientation, and worry on a sample of high school students in an Advanced Placement Program in mathematics. The study's objectives were to (1) determine if goal orientation and self-efficacy (motivation) are integral parts of self-regulated learning, and (2) document the relationships between self-regulated learning, motivation, worry, and math achievement. Subjects consisted of 144 students in grades 10-12. The measurement instrument was a modified version of O'Neil, Sugrue, Abedi, Baker, and Golan's (1992) self-regulation questionnaire; with added scales for self-efficacy, and learning and performance goal orientation, developed for this study. Students indicated how they thought or felt during the 'high-stakes' calculus exam. All scales had reliabilities above.65. An exploratory factor analysis, using principal components analysis and varimax rotation, extracted three components: self-regulated learning; self-efficacy, learning goal orientation and worry; and performance goal orientation. The initial confirmatory factor analysis was performed on the hypothesized model with less than satisfactory results. The objective that hypothesized goal orientation and self-efficacy were integral to self-regulated learning was not validated. However, a post hoc confirmatory factor analysis was successful: [$\chi$]2(22, N = 144) = 32.12, p [$<$].08; likelihood ratio = 1.46; all fit indices were greater than.90. Modifications included eliminating the nonsignificant performance goal orientation path coefficient to self-regulated learning and the insertion of path coefficients from self-efficacy to both worry and math achievement. The study concluded that (1) self-efficacy and math achievement are moderately and positively correlated, and both are moderately and negatively correlated with worry; (2) neither self-efficacy nor performance goal orientation significantly affected self-regulated learning, and neither of the two goal orientations affected math achievement; (3) worry had a significant negative affect on both self-regulated learning and math achievement; and (4) self-regulated learning significantly and positively affects math achievement. Implications are: (1) a reliable statistical basis for a structural model of self-regulated learning, and (2) a sound statistical verification of the components of self-regulated learning advocated by various researchers. 263. {mam:pupil} J. Mamona-Down. Pupils' interpretations of the limit concept: A comparison study between greeks and english. In Proceedings of the Annual Conference of the International Group for the Psychology of Mathematics Education with the North American Chapter 12th PME-NA, volume 1, pages ???-???, Mexico, 1990. International Group for the Psychology of Mathematics Education. To be added later. 264. {mang-cog-93} B. Mangan. Taking phenomenology seriously: The ``fringe'' and its implications for cognitive research. Consciousness and Cognition, 2:89-108, 1993. 265. {fipse-93} Dora Marcus and et al., editors. Lessons Learned from FIPSE Projects II. Fund for the Improvement of Postsecondary Education, Washington, DC, 1993. Describes 30 college and university programs funded by the Fund for the Improvement of Postsecondary Education from 1989 to 1991. Each description includes information on program purpose, project activities, major insights and lessons, project continuation, and available information. The first group of 10 are programs focused on assessment and include an assessment resource center, area concentration achievement testing with curricular evaluation, computers and college writing, assessment seminars, New Pathway Curriculum impact evaluation, liberal education model assessment, college-wide measures toward general education goals, comprehensive assessment in academic disciplines, and a regional assessment network. Another group of four programs address college teaching: professional development, medical scholars, and database and online service orientation. Nine projects address curriculum and teaching in the disciplines including laboratory education, undergraduate mathematics, economic curricula, scientific thinking, French language and culture, case study physics, music theory, biology instruction, and freshman chemistry. Two programs address general education. Three projects involve teacher education and two programs address ethics instruction. The following institutions are included: University of Tennessee; Austin Peay State University (Tennessee); City University of New York; Harvard University (Massachusetts); Miami University (Florida); State University of New York; Winthrop College (South Carolina); University of California; Ohio State University; Salem State College (Massachusetts); Clemson University (South Carolina); Denison University (Ohio); Dickinson College (Pennsylvania); Tufts University (Massachusetts); University of Maryland; New Mexico State University; Northwestern University (Illinois); University of Oregon; University of Rhode Island; University of North Texas; Indiana University of Pennsylvania; Northern Virginia Community College; Union College (New York); University of Connecticut; and Saint Cloud State University (Minnesota) 266. {mar-90:cha} G. Marshall. A changing world requires changes in math instruction. Executive Educator, (12):23-24, 1990. In response to our technological society, the National Council of Teachers of Mathematics recommends decreasing drill and reliance on multiple-choice tests and increasing problem solving and real-world applications. Other recommendations include simulation software to provide mathematical challenges, collaborative learning, and alternative assessment methods. 267. {marshall-95} S. P. Marshall. Schemas in Problem Solving. Cambridge University Press, New York, NY, 1995. 268. {mar:calcu} T. S. Martin. Calculus students' abilities to solve geometric related rate problems and their understanding of related geometric growth factors. PhD thesis, Boston University, 1997. The purpose of the study was twofold: to assess subjects' abilities to compute related rates of change for continuously changing geometric figures, and to characterize robust and weak solvers' understanding of related discrete changes in geometric figures. A written test was used to determine how first-year calculus students perform on geometric related rate problems. A second written test was used to determine which of the following steps were most related to successful solution of geometric related rate problems: sketch the situation and label the variables, summarize the problem statement, recall relevant geometric formula, implicitly differentiate the geometric formula, substitute specific values of the variables into the related rate equation and solve for the desired rate, interpret and report results, or solve an auxiliary problem. A clinical interview was used to gather evidence of subjects' understanding of the relationships between discrete multiplicative changes in geometric figures. This evidence, in conjunction with data from the first two instruments, was used to develop profiles of six subjects identified as robust or weak solvers of geometric related rate problems. Overall scores on geometric related rate problems were low (sample mean-44subscore means (17sketch and label, and summarize the problem statement, respectively, whereas, the highest subscore mean (78step, recall a relevant geometric formula. The steps identified as most related to successful performance on geometric related rate problems were: implicitly differentiate, substitute and solve, and solve an auxiliary problem. Robust solvers had more advanced understanding of geometric growth factors than did weak solvers on a developmental scale of understanding, suggesting that understanding of related discrete changes may be connected to the ability to solve problems relating continuous changes. The stages identified in the developmental scale overlapped and extended the stages of understanding of related geometric growth factors proposed by Piaget. Diagrams developed for the present study were powerful tools for representing subjects' stages of understanding as well as the amount and type of support received during the interviews. 269. {mar-har:proof} W. Martin and G. Harel. Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1):41-51, 1989. (This annotation is quoted from the paper abstract.) `This study asked 101 preservice elementary teachers enrolled in a sophomore-level mathematics course to judge the mathematical correctness of inductive and deductive verifications of either a familiar or an unfamiliar statement. For each statement, more than half the students accepted an inductive argument as a valid mathematical proof. More than 60% accepted a correct deductive argument as a valid mathematical proof; 38% and 52% accepted an incorrect deductive argument as being mathematically correct for the familiar and unfamiliar statements, respectively. Over a third of the students simultaneously accepted an inductive and a correct deductive argument as begin mathematically valid.' 270. {mason-spen-esm-99} J. Mason and M. Spence. Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 28:135-161, 1999. 271. {mat-mcd-str:under} D. Mathews, M. A. McDonald, and K. Strobel. Understanding sequences: A tale of two objects. In preparation. APOS was used to examine students' cognitive construction of the concept of sequence. The authors show that students tend to construct two distinct cognitive objects and refer to both as a sequence. One construction, which the authors call SEQLIST, is what one might understand as a listing representation of a sequence. The other, which they call SEQFUNC, is what one might interpret as a functional representation of a sequence. As the connections between these two entities becomes stronger, and the students reflect on these connections, they begin to understand sequence as a single cognitive entity and SEQLIST and SEQFUNC as mathematical representations of this entity. In this paper the authors detail the construction of SEQLIST and SEQFUNC by the students, and characterize the connections between them through the triad model of schema development introduced by Clark, Cordero, et. al. (1997). 272. {mat:using} J. H. Mathews. Using computer symbolic algebra to solve differential equations. Mathematics and Computer Education, (3):174-182, 1989. This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. 273. {mat-91:group} S. M. Mathews. Group problem solving in the college mathematics classroom. Primus, 1(4):430-442, 1991. Describes the mechanics of group work in the college mathematics classroom specifically group formation, preliminary class work, class and group discourse, individual and group assignments, and impact on test taking. Includes examples from a first-semester calculus course. 274. {mat:acomp} N.L. Ott. Matthews. A comparison of traditional and reform styles in teaching Applied Calculus. PhD thesis, The University of Oklahoma, 1998. This study compared the use of reform methods in the teaching of applied calculus at a large comprehensive public university during the Spring 1997 semester. Fifty-nine students, mostly freshmen and sophomores enrolled in two sections of the second semester of a two-semester sequence of applied calculus. The experimental section used a reform textbook, graphing calculators, and small group activities. The control section used a traditional textbook, scientific calculators, and lectures. Common examination questions were used to compare the two groups. Students in the experimental section scored significantly better on conceptual questions, and showed no significant difference on computational question. Students in the experimental section had better affective responses to questions about the usefulness of mathematics and their ability to solve mathematical problems, especially non-routine problems. 275. {rot:facto} A. M. Roth Mcduffie. Factors that influence college mathematics professors in the process of implementing reform-based instruction. PhD thesis, University of Maryland College Park, 1998. Fundamental changes in teaching and learning have been proposed for mathematics education in the United States. Achieving the goals of reform at the college level would not only enhance the learning of college students in general, but pre-service teachers would experience the type of teaching and learning environments that we hope they will create in their own classrooms. This dissertation synthesizes the current knowledge base about changing instructional practice by presenting a framework of the factors that influence teachers as they attempt to change instructional practice. The potentially influential factors include: instructional materials; in-service programs; summer workshops; interaction with and reaction from peers, administrators, students, and/or educational researchers; and institutional policies. The dissertation reports an investigation of the practices of two university mathematics professors in the process of changing teaching mathematics at a pre-calculus level. The purpose of the research was to gain understanding about: (1) how the two mathematics professors enact the goals for reform; and (2) what factors influence the mathematics professors as they attempt to implement changes in their instructional practice. The two professors were studied through a qualitative case study methodology. Data sources of the semester-long study included interviews, classroom observations, and a student questionnaire. Using software for qualitative data analysis, the data were analyzed through methods of constant comparison and analytic induction to construct patterns of similarities and differences in the participants' perceptions. The primary factors that influenced the professors in this study included: students; key colleagues interested in change; broader collegial networks; time; and institutional policies. Of these factors, the students seemed to be the most powerful influence on the professors' in implementing reform-based strategies. This in-depth investigation expands on the understanding developed in the framework mentioned above, provides an opportunity for educators to reflect on their experiences, and informs future research and practice as we continue to advance our efforts in reform. 276. {mcg:explo} C. K. Mcgraw. Exploring the mathematical paths students follow in high school And college. PhD thesis, Syracuse University, 1996. A logistic regression model was developed to examine the probability of college students' participation in calculus. The goal was to determine which, if any, of several variables from each student's high school experience played a role in the student's participation in a calculus sequence in college. That is, the goal was to answer the following question-What is the relationship between past participation and performance in high school mathematics and foreign language courses and future participation in mathematics courses; in particular, which factors contribute to students' participation in a calculus or noncalculus sequence in college? The subjects were undergraduates from Syracuse University (103 men and 110 women), who attended New York State Regents curricula high schools. These students typically represent 30undergraduate population at the university. Data were collected from both the high school and college transcripts, anonymously. Variables included SAT Verbal and Quantitative Scores, overall grade point-average (GPA), total years of high school mathematics studied, total years of Regents mathematics studied, grade in math-nine, grade in math-ten, grade in math-eleven, grade in pre-calculus, grade in calculus, grade in general math twelve, sum of all mathematics grades, participation in pre-calculus or calculus in high school, total number of different foreign languages studied, total years of foreign language study, total years in most deeply studied foreign language, mean foreign language grade, and sum of all foreign language grades. Univariate analysis of the high school variables indicated gender differences for students in the sample and, as a result, separate logistic regression models were developed for each gender. For both sexes, the final models include participation in either pre-calculus or calculus courses in high school as significant predictors of taking a calculus sequence in college. The women's model included overall grade-point average in high school, while the men's model included overall performance in math courses in high school. 277. {mck:theef} C. McKnight. The Effects of Journal Writing on the Beliefs, Attitudes, and Achievement of Elementary Education Students in a Content Course in Mathematics. PhD thesis, The University of Oklahoma, 1991. (This annotation is quoted from the dissertation abstract.) `This study examines the possible effects that journal writing may have on the beliefs, attitudes, and achievement of elementary education students in a content course in mathematics. The method utilized in this study was to administer journal writing in two sections of the content course Mathematics for Elementary Education Students. In both sections, cognitive journal entries were stressed, while only one of the sections also emphasized affective entries. Due to spontaneous affective crossovers in the cognitive only group, the design was ineffective as a control, making the interpretation of the results difficult. ... Significant gains were seen in both groups' abilities to give better explanations when describing the borrowing process, as well as in the use of correct borrowing methods. ...The student questionnaires provided evidence of attitude and belief changes which can be linked to the journaling experience.' 278. {mee:diffi} D. Meel. Difficulties with related rates problems:examining the obstacles. In ?, volume ?, pages ???-??? The Association of Research in Undergraduate Mathematics Education, 1998. To be added later. 279. {mee:acomp} D.E. Meel. A comparative study of honor students' understandings of central calculus concepts as a result of completing a calculus and mathematica or a traditional calculus curriculum. PhD thesis, University of Pittsburgh, 1995. This study compared the understandings of two groups of third semester honors calculus students, one of which completed the Calculus & Mathematica (C&M) curriculum [$ (n=16)$] and the other a traditional calculus (TRAD) curriculum [$ (n=10)$]. Three instruments were utilized to examine students' understandings of limit, differentiation, and integration. The first instrument, administered to the entire sample, required students to give written responses to ten items assessing understanding in the three topic areas. In addition, a problem-solving interview and an understanding interview were conducted with volunteers from each sample to examine in more detail students' problem-solving approaches and to gain further insight into student understandings of limit, differentiation and integration. Calculators and computers were available for use during the interviews but not during the written test. Analysis of students' responses to the written test revealed significant between-group differences, favoring the TRAD students, on students' understanding of the limit, on conceptually-oriented items, and on items presented without figures. A qualitative analysis of students' responses revealed few significant differences but many interesting similarities, including a dynamic view of the limit, some difficulties with the concept of differentiation, and an understanding of properties of of integration. The problem-solving interviews revealed that the C&M students were more successful and tended to be more flexible in solving problems than were their TRAD counterparts. In the understanding interviews, students of both curriculums conceived the concepts of limit and differentiation in ways that did not capture their entire essence. However, with respect to integration, the students of both curriculums appeared to display understandings compatible with understanding the formal definition of integration. In general, results suggest that neither curriculum does a strong job of developing formalized understandings of the important underlying concepts of calculus in these honors students. In fact, neither curriculum appears to provide the correct mixture of technology infusion, student independence, authoritative guidance, computational skill, and structure necessary to prepare students to either succeed in upper division mathematics courses or to take their position in society. 280. {mei-ris:writi} J. Meier and T. Rishel. Writing in the Teaching and Learning of Mathematics, volume 48 of MAA Notes. The Mathematical Association of America, 1998. In this very practical book, the authors discuss how to make writing an effective instructional strategy and why writing experiences enhance learning. The chapter titles follow: 1. Getting Started 2. Using Small Assignments 3. How to Salvage a Bad Assignment 4. Integrating Assignments into the Curriculum 5. Grading Essays 6. Learning from the Writing Faculy 7. Knowing the Audience 8. Prewriting and Writing 9. Learning from the Cognitive Faculty 10. Talking Through the Problem - Conferences 11. What `Guzinta' a Good Major Project 12. Presenting Major Projects 13. Learning of Limits: Limits of Learning: A Case Study on the Impulse Toward Narrative in Mathematics 14. A Rationale for Writing 281. {met:writi} C. Mett. Writing in mathematics -- evidence of learning through writing. The Clearing House, 62:293-296, March 1989. The author describes the writing assignments which she uses in her classes. She includes examples of student writing. 282. {mic:under} E. Michener. Understanding understanding mathematics. Cognitive Science, 2:361-383, 1978. The author begins by proposing an epistemology of mathematical knowledge in which the items in a content domain are classified into the categories results, examples, and concepts. For each of these sets a relation is described which organizes it; for example, the organizing relation for results is logical support. For each item in a set, there are dual items -- items from the other two sets which are involved in stating, discussing, motivating, illustrating, generalizing, constructing, extending, and/or proving it. Those types of examples, results, and concepts which play a significant role in the understanding of a domain are classified: start-up examples, reference examples, model examples, counterexamples, mega-principles, counter-principles, basic results, key results, culminating results, transitional and technical results. The point is made that like problem-solving, understanding is an active process, and the author gives a list of questions (in the spirit of Polya's `How to Solve It' questions) which can be used to probe and prompt understanding. Finally, the author reports on a freshman seminar on the theory of eigenvalues in which these ideas were used as an organizational model. 283. {mil:teach} D. Miller. Teacher benefits from using impromptu writing prompts in algebra classes. Journal for Research in Mathematics Education, 23(4):329-340, 1992. (This annotation is quoted from the paper abstract.) `The purpose of this study was to examine the benefits to teachers who used impromptu writing prompts in first- and second-year algebra classes. The study attempted to answer two questions: (a) What can teachers learn about their students' understanding of school mathematics from reading their responses to in-class, impromptu writing prompts; and (b) Are instructional practices influenced as a result of reading students' responses to in-class, impromptu writing prompts? An interpretive research methodology was used to collect and analyze data. The researchers concluded that the teachers' assessment of students' understanding of school mathematics was enhanced by reading their students' responses to impromptu writing prompts, and that the teachers in the study perceived that their instructional practices were influenced as a result of reading their students' responses to impromptu writing prompts.' 284. {mon:probl} J. Monaghan. Problems with the language of limits. For the Learning of Mathematics, (3):20-24, 1991. Presents the portion of a larger study of A-level British students understandings of calculus that deals with ambiguities inherent in the phrases ``tends to,'' ``approaches,'' ``converges,'' and ``limit.'' Responses to two questionnaires indicate that the four phrases generate everyday connotations that are at odds with the mathematical meanings. 285. {moo:makin} R. Moore. Making the transition to formal proof. Educational Studies in Mathematics, 27:249-266, 1994. (This annotation is quoted from the paper abstract.) `This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs. Two preliminary studies and the main study were conducted in undergraduate mathematics courses at the University of Georgia in 1989. The students in these courses were majoring in mathematics or mathematics education. The data were collected primarily through daily nonparticipant observation of class, tutorial sessions with the students, and interviews with the professor and the students. An inductive analysis of the data revealed three major sources of the students' difficulties: (a) concept understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing. Their difficulties with concept understanding are discussed in terms of a concept-understanding scheme involving concept definitions, concept images, and concept usage. The other major sources of difficulty are discussed in relation to this scheme.' 286. {mov:stimu} N. Movshovitz-Hadar. Stimulating presentation of theorems followed by responsive proofs. For the Learning of Mathematics, 8(2):12-30, June 1988. The author analyzes the structured proof and guided discovery methods of presentation of theorems and proofs. In contrast, an alternative method, the stimulating responsive method is defined and discussed. This method involves provoking an element of surprise on the part of the learner by the way in which the theorem is presented. This surprise is the result of a perceived gap between what is known or expected and what is stated. The role of proof, then, is to bridge this gap. The author advocates two styles of proof presentation, gap-bridging and generic example assisted for helping students to bridge the gap. Arguing that the guided discovery method is teacher-motivated, the author asserts that the stimulating responsive method is student- motivated, that it avoids some of the pedagogical risks associated with the guided discovery method, and that it fosters better attitudes toward mathematics. 287. {mow:fatme} P. Mower. Fat men in pink leotards or students writing to learn algebra. Primus, 6(4):308-323, December 1996. (This annotation is quoted from the paper abstract.) `This paper presents the finding from a study of the learning experiences of a group of undergraduates enrolled in a writing intensive college algebra course. The curriculum of the course included the usual algebra content; however, the daily teaching and learning practices in this course involved extensive writing to learn activities and journal writing. Many of the writing activities were successful in terms of facilitating student comprehension of mathematical content and could easily be incorporated into any math course. Overall the study revealed that the use of words allowed the student writers to gain ownership of the algebraic content.' 288. {mye:anexp} K. N. Myers. An exploratory study of the effectiveness of computer graphics and simulations in a computer student interactive environment in illustrating random sampling and the central limit theorem. PhD thesis, The Florida State University, 1990. The author studied the impact of computers on community college students' understanding of two statistics concepts, random sampling and the central limit theorem. Students in one class used a computer to investigate these concepts while the other class studied the same content through a traditional lecture. She found that the computer users scored significantly higher on a test of concepts, but noted no significant difference between the two groups on a test of applications. A retention test given to both groups three weeks later showed no significant difference between the two classes. 289. {dav-90:sta} et. al N. Davidson. Staff development for cooperative learning in mathematics. Journal of Staff Development, (11):12-17, 1990. The article discusses four models of staff development for cooperative learning in mathematics: graduate seminar with weekly meetings; workshop enhanced by support systems and follow-up; school-based staff development with monthly meetings; and projects involving collaboration between university and school district personnel. The article lists 10 important conclusions about staff development. 290. {mtnov98} National Council of Teachers of Mathematics. Focus Issue on the Concept of Proof, volume 91, November 1998. Here is a list of articles dealing with proof in this issue: Proof by Contradiction and the Electoral College, Charles Redmond, Michael P. Federici, and Donald M. Platte Can Computers Be Used to Teach Proofs?, Judith A. Silver Types of Students' Justifications, Larry Sowder and Guershon Harel Ideas for Developing Students' Reasoning: A Hungarian Perspective, Anita Szombathelyi and Tibor Szarvas Sharing Ideas about Teaching Proving, David A. Reid A Unified Framework for Proof and Disproof, Susanna S. Epp Characterizing Students' Understandings of Mathematical Proof, Eric J. Knuth and Rebekah L. Elliott Proof in Modern Geometry, Stanley P. Izen On Proofs and Their Performance as Works of Art, Greisy Winicki-Landman Prove It!, Amy A. Prince 291. {nay:unite} S. Nayer. United States and Russian calculus achievement examinations: A comparison of student performance. PhD thesis, Columbia University Teachers College, 1994. The purpose of the study was to compare the student performance of the Advanced Placement Calculus course and the Soviet secondary school course in calculus and elementary functions by: (1) Comparative analysis of the respective syllabi to determine correlation and treatment of concepts and methods. (2) Comparison of scores earned by American students on the Soviet entrance examination and by Soviet students on the Advanced Placement Calculus Examination. In order to permit unbiased comparisons, Soviet and American students each sat for both examinations-the Advanced Placement Calculus Examination and the Soviet entrance examination. The study was able to answer such questions as: (1) To what extent are the topics offered in Soviet or American courses the same or different? (2) How do the times spent on topics differ in Soviet and American schools? (3) How do the Advanced Placement Calculus Examination in the USA and the College Entrance Examination in Russia differ? (4) On what topics do Soviet students seem stronger than American students and vice versa? (5) Are differences in examination results explainable in terms of curricular emphasis? (6) What gender interactions are present in the Soviet and American data? Based on the findings of this study, some recommendations for mathematics teaching in the United States and Russia were made. 292. {nctm-2000} Principles and Standards for School Mathematics, Electronic Final Version. National Council of Teachers of Mathematics, [$<$]http://standards.nctm.org/document/index.htm[$>$], 2000. 293. {nem-rub:stude} R. Nemirovsky and A. Rubin. Students' tendency to assume resemblances between a function and its derivative. In Proceedings of Annual Meeting of the, volume ?, pages ???-???, Chicago, IL, 1991. American Educational Research Association. This study was designed to determine students' abilities and difficulties in articulating the relationship between function and derivative. High-school students were presented 15 problems during two 75-minute interviews in which they were asked to construct functions experimentally in three different contexts: motion, fluids, and number-change. In each of the environments the students utilized tools, ranging from computer software to manipulative materials, that enabled them to generate functions. Analysis of the interviews indicated that students had the tendency to assume resemblances between the behavior or appearance of the function and its derivative in all three contexts. Three types of cues that activated resemblances in the predicted functions were identified: syntactic, semantic, and linguistic. Approaches that students used to move from the derivative to the function and vice-versa are described. A case study of one 17-minute episode of a student working with an airflow device describes the evolution of one such approach. The report concluded that the construction of such an approach was complex and that tools to explore the mathematical ideas helped frame the interviewer/student discourse through which the approach was developed. 294. {new:stude} J.C. Newell. Student experiences in a first semester university calculus course: A study using ethnographic methods. PhD thesis, Simon Fraser University (Canada), 1994. The purpose of this study was to gain and accurately depict an understanding of students' experiences in a first year calculus course. Ethnographic methods were used in order to produce a naturalistic description of the student experiences. The fieldwork was conducted in a first semester calculus class at Simon Fraser University during the Fall Semester, 1991. Data acquisition consisted primarily of fieldwork and interviews with the instructor and six of the students. Several conclusions arise from this study. The students perceive this course as an obstacle they must overcome before being permitted to take the courses they want. The course is not well coordinated with the high school curriculum; it assumes skills, knowledge and mathematical sophistication that are seldom taught in the secondary schools. The course tries to accomplish too many things; mathematical rigour, computational techniques, fundamental concepts, problem solving, and the use of scientific calculators are blended together into a bewildering melange. There is too much material for the time allotted; many topics are dealt with superficially. The pace of the course allows no latitude for student illness, fatigue or personal upsets. In summary, this study raises issues not yet discussed in the calculus reform debate and puts a personal face on many issues already presented. (Abstract shortened by UMI.) 295. {nig:thero} P. Nigam. The role of visualization in teaching undergraduate mathematics: A multi case study of teachers' perceptions and practices. PhD thesis, Syracuse University, 1998. It has been documented that students learn mathematics in different ways and through different learning styles. In particular, it has been found that some students prefer to understand and learn mathematics visually. Other researchers have determined that visual methods may be used by problem solvers in mathematics even in solutions that are deemed non-visual. Thus, the visual context appears to play a significant role in learning mathematics. But are instructors of mathematics aware of and responsive towards these needs? It is this question that I attempted to address in this study. More precisely, in this study, I examined the use of visual strategies in the teaching of mathematics. My intention was to study the use of such strategies from the instructors' perspective. Thus, I selected five instructors in the mathematics department of a university who were teaching different courses in calculus. I observed several classes that they taught during one semester of the course and conducted interviews with them in order to understand their perceptions of teaching and mathematics. We also discussed several other issues related to teaching such as assessment and the role of visualization in teaching and learning mathematics. These observations and interviews delineated for me the teaching approaches of these instructors, and more specifically, how they used visual strategies in order to solve problems of instruction. An examination of the assessment tools used by the instructors with regard to the visual strategies used therein completed my examination of the instructors' individual case studies. I identified four primary visual teaching strategies and attempted to categorize the use of each of them for each instructor. My findings from this study indicated that while instructors used a wide range of visual instructional strategies, frequently blending several of them in one teaching episode, they often downplayed the role of such strategies in their classrooms. Based on the study and the research in other similar areas, I made several suggestions for teaching and research in mathematics education. 296. {nor-pri:cogni} F. A. Norman and M. K. Prichard. Cognitive obstacles to the learning of calculus: A kruketskiian perspective. In Proceedings of Special Session on Research in Undergraduate Mathematics Education at the annual meeting of the American Mathematical Society and the Mathematical Association of America, volume ?, pages ???-???, San Francisco, California, 1991. Mathematical Association of America and AMS. To be added later. 297. {web-cul-83:group} N.Webb and L.Cullian. Group interaction and achievement in small groups: Stability over time. American Educational Research Journal, (20):411-423, 1983. The relationships among (a) student and group characteristics, group interaction, and achievement in small groups in junior high school mathematics classrooms and (b) the stability of these relationships over time were investigated. Interaction in the group was a potent predictor of achievement. 298. {pme-98-vol2} Alwyn Olivier and Karen Newstead, editors. Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (22nd, Stellenbosch, South Africa, July 12-17, 1998). Volume 2. International Group for the Psychology of Mathematics Education, 1998. Papers in Volume 2: 1. ``Learning Algebraic Strategies Using a Computerized Balance Model" (J. Aczel); 2. ``Children's Perception of Multiplicative Structure in Diagrams" (B. Alseth); 3. ``A Discussion of Different Approaches to Arithmetic Teaching" (J. Anghileri); 4. ``A Model for Analyzing the Transition to Formal Proofs in Geometry" (F. Arzarello, C. Micheletti, F. Olivero, O. Robutti, D. Paola); 5. ``Dragging in Cabri and Modalities of Transition from Conjectures to Proofs in Geometry" (F. Arzarello, C. Micheletti, F. Olivero, O. Robutti, D. Paola, G. Gallino); 6. ``The Co-Construction of Mathematical Knowledge: The Effect of an Intervention Program on Primary Pupils' Attainment" (M. Askew, T. Bibby, M. Brown); 7. ``Didactical Proof: Should We Teach It to Physics Students?" (R. R. Baldino); 8. ``Lacan and the School's Credit System" (R. R. Baldino and T. C. B. Cabral); 9. ``Which Is the Shape of an Ellipse? A Cognitive Analysis of an Historical Debate" (M. G. Bartolini Bussi and M. A. Mariotti); 10. ``Children's Understanding of the Decimal Numbers through the Use of the Ruler" (M. Basso, C. Bonotto, P. Sozio); 11. ``Construction of Multiplicative Abstract Schema for Decimal-Number Numeration" (A. R. Baturo and T. J. Cooper); 12. ``Classroom-Based Research to Evaluate a Model Staff Development Project in Mathematics" (J. Rossi Becker and B. J. Pence); 13. ``Some Misconceptions Underlying First-Year Students' Understanding of `Average Rate' and of `Average Value'" (J. Bezuidenhout, P. Human, A. Olivier); 14. ``Operable Definitions in Advanced Mathematics: The Case of the Least Upper Bound" (L. Bills and D. Tall); 15. ``Beyond `Street' Mathematics: The Challenge of Situated Cognition" (J. Boaler); 16. ``The `Voices and Echoes Game' and the Interiorization of Crucial Aspects of Theoretical Knowledge in a Vygotskian Perspective: Ongoing Research" (P. Boero, B. Pedemonte, E. Robotti, G. Chiappini); 17. ``Children's Construction of Initial Fraction Concepts" (G. Booker); 18. ``Graphing Calculators and Reorganization of Thinking: The Transition from Functions to Derivative" (M. C. Borba and M. E. Villarreal); 19. ``Pre-Algebra: A Cognitive Perspective" (G.M. Boulton-Lewis, T. Cooper, B. Atweh, H. Pillay, L. Wilss); 20. ``The Right Baggage?" (M. Briggs); 21. ``Learner-Centered Teaching and Possibilities for Learning in South African Mathematics Classrooms" (K. Brodie); 22. ``Researching Transition in Mathematical Learning" (T. Brown, F. Eade, D. Wilson); 23. ``Metaphor as Tool in Facilitating Preservice Teacher Development in Mathematical Problem Solving" (O. Chapman); 24. ``Restructuring Conceptual and Procedural Knowledge for Problem Representation" (M. Chinnappan); 25. ``The Structure of Students' Beliefs towards the Teaching of Mathematics: Proposing and Testing a Structural Model" (C. Christou and G. N. Philippou); . and 1. ``Abstract Schema versus Computational Proficiency in Percent Problem Solving" (T. J. Cooper, A. R. Baturo, S. Dole); 2. ``Implicit Cognitive Work in Putting Word Problems into Equation Form" (A. Cortes); 3. ``Three Sides Equal Means It Is Not Isosceles" (P. Currie and J. Pegg); 4. ``Making Sense of Sine and Cosine Functions through Alternative Approaches: Computer and `Experimental World' Contexts" (N. Lobo Da Costa and S. Magina); 5. ``Teacher and Students' Flexible Thinking in Mathematics: Some Relations" (M. M. M. S. David and M. da Penha Lopes); 6. ``The Influence of Metacognitive and Visual Scaffolds on the Predominance of the Linear Model" (D. De Bock, L. Verschaffel, D. Janssens); 7. ``To Teach Definitions in Geometry or Teach To Define?" (M. De Villiers); 8. ``Student thinking about Models of Growth and Decay" (H. M. Doerr); 9. ``Analysis of a Long Term Construction of the Angle Concept in the Field of Experience of Sunshadows" (N. Douek); 10. ``On Verbal Addition and Subtraction in Mozambican Bantu Languages" (J. Draisma); 11. ``Teachers' Beliefs and the 'Problem' of the Social" (P. Ensor); 12. ``From Number Patterns to Algebra: A Cognitive Reflection on a Cape Flats Experience" (C. B. A. Felix); 13. ``Affective Dimensions and Tertiary Mathematics Students" (H. J. Forgasz, G. C. Leder); 14. ``Social Class Inequalities in Mathematics Achievement: A Multilevel Analysis of TlMSS South Africa Data" (G. Frempong); 15. ``Context Influence on Mathematical Reasoning" (F. Furinghetti, D. Paola); 16. ``What Students Think about the Median and Bisector of an Angle in the Triangle, What They Say and What Their Teachers Know about It" (H. Gal); 17. ``Levels of Generalization in Linear Patterns" (J. A. Garcia-Cruz, A. Martinon); 18. ``The Evolution of Pupils' Ideas of Construction and Proof Using Hand-Held Dynamic Geometry Technology" (J. Gardiner, B. Hudson); 19. ``Cognitive Unity of Theorems and Difficulty of Proof" (R. Garuti, P. Boero, E. Lemut). ERIC Acc. No. ED427970 299. {ols:first} D. Olson. First-year students love calculus proofs. Primus, 7(2):123-128, June 1997. Abstract (quoted from paper): `The author presents a method to help students learn the architecture of proofs and suggests a writing assignment for assessment. The method includes a short presentation of the proof followed by reconstruction of the proof by student groups.' 300. {oma-sca-90:comp} C. O'Malley and E. Scanlon. Computer supported collaborative learning: Problem solving and distance education. Computer and Education, (15):127-136, 1990. Three studies of cooperative problem solving among university students examined how to design effective computer-based support for collaborative learning in distance education. The first was a questionnaire study of 150 Open University students who had participated in three different courses in physics and mathematics. Its aim was to determine students' participation in and preferences for cooperative work in their courses. The main finding was that although students express a preference for working alone, many participated in collaborative work in their courses and regarded such activities as helpful. More detailed investigation was needed to determine the nature of collaborative activities in which these students engaged and the ways in which they found them helpful. An observational study of group activities at summer school and a comparison of the use of two different kinds of interface for supporting synchronous cooperative problem solving were then conducted. The characteristics of successful divisions of labor observed in noncomputer-based activities suggested that one issue related to designing computer support for such problem solving is how to design appropriate tools and representations for joint activity. 301. {one:aneva} E. N. O'Neal. An evaluation of an interdisciplinary science course designed to help at-risk students. PhD thesis, University of Pennsylvania, 1995. This report describes an evaluation of the new interdisciplinary science course that is part of the Prefreshman Program at the University of Pennsylvania. The objective of this course is to help the most at-risk entering freshmen who desire to major in science or engineering to be more academically prepared for the rigors of college course work. The course was offered for the first time in August, 1993 to 42 students. These students met one of the following criteria: (1) regularly admitted matriculants with a predictive index (PI) less than or equal to 2.4, indicating they are at high academic risk; (2) Philadelphia residents with a PI less than or equal to 2.7; and (3) special admission students who are at risk and who may be admitted on account of special interests including socioeconomic reasons, being faculty/staff children or first-generation college attendees, diversity reasons, or other academic concerns. Mathematics, chemistry, and physics were integrated to show the interrelatedness of the disciplines while at the same time teaching fundamental problem-solving skills necessary for all mathematics and science courses at the University. After observing the course daily during August 1993, to understand the content, methodologies, and interactions of the faculty and students, two comparison groups were created for the study. The first comparison group, identified as group 2, consisted of 296 students who had declined the invitation to attend the program, but were selected with the same criteria as the attendees, i.e. they had the lowest predictive indices upon matriculating at the University. To control for the self-selection process to attend or not to attend, a second comparison group (group 3) of 975 students was created. These students were not invited to attend the Prefreshman program because they barely missed the cutoff point for selection. Their predictive indices were among the lowest of the entering freshman ('C' range), although not as low as those of the invitees. This study compared enrollment rates in calculus for both semesters of the freshman year, group mean GPAs in mathematics each semester, and cumulative GPAs at the end of the year. In addition, interviews were conducted with four students from each group throughout the year. The interviews solicited information on study habits, specific comments on their mathematics and science courses, and general comments about the University of Pennsylvania. Enrollment rates in entry-level mathematics courses were higher for program attendees that for the comparison groups, but their successful course completion rates were lower. Both group 1, the program invitees, and comparison group 2 lagged slightly behind group 3 in academic performance the entire year. It was discovered through the interview data that program attendees used academic support services more often than students in the comparison groups and were much more inclined to attend their professors' and TAs' office hours for help. They preferred collaborative learning as opposed to individualistic approaches to their math and science work. These same students also preferred studying in the shared workspaces compared to students in the comparison groups who preferred studying in their dorms. (Abstract shortened by UMI.) 302. {orz:anact} M. Orzech. An activity for teaching about proof and about the role of proof in mathematics. Primus, 6(2):125-139, June 1996. A dialog from The Mathematical Experience by Davis and Hersh forms the cornerstone of a lab in a linear algebra in which students reflect on what constitutes a mathematical proof. 303. {oso:about} F.C. Osorio. About the heritage in calculus textbooks: A definition of integral or the fundamental theorem of calculus. In Proceedings of the 11 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume 1, pages ???-???, New Brunswick, New Jersey, 1989. International Group for the Psychology of Mathematics Education. North American Chapter. To be added later. 304. {ott:mathe} M. Otte. Mathematical knowledge and the problem of proof. Educational Studies in Mathematics, 26:299-321, 1994. (This annotation is quoted from the paper abstract.) `The paper presents some examples and reflections intending to hint at the role of formal thought in the process of knowledge growth. It argues that there is no division of labor according to which certain modes of human cognition are associated with certain tasks and certain cognitive roles exclusively. In this connection, the paper claims that he subject matter of mathematical activity is represented within the system of activity by many different means. Mathematics differs in fact from logic in as much as a principle of heterogeneity or of flexible `means-objects-relationships' is valid. Formalization in contrast brings forward a principle of homogeneity -- that like follows like. Every subject matter requires principles homogeneous with itself. The paper tries to draw some conclusions from this difference with respect to the role of formalization within human cognitive development.' 305. {pal-joh-89:jig} J. Palmer and J. T. Johnson. Jigsaw in a college classroom: Effect on student achievement and impact on student evaluations of teacher performance. Journal of Social Studies Research, (13):34-37, 1989. Examines a cooperative learning technique called Jigsaw that was used to determine whether college-level students taught by this method scored higher on a posttest than students who were not. Results showed no significant difference between those taught by the Jigsaw technique and those who were not. 306. {pap:anexa} A. J. Papakonstantinou. An examination of high school students' understanding of the concept of function. PhD thesis, University of Houston, 1992. The mathematical concept of function causes learning difficulties for many students. The study's purpose was to assess high school students' knowledge of the definition of function and to relate this knowledge to students' ability to give examples and nonexamples of functions. The results were related to variables of gender, ethnicity, grade earned in the most recently completed mathematics course, curricular level, and number of mathematics courses completed beyond first-year Algebra. The sample consisted of 552 students from Geometry, second-year Algebra, Pre-Calculus and Calculus classes from two urban schools. Students completed a questionnaire of six open-ended questions. Students' responses to 'What is a function?' were categorized according to a hierarchical classification system. ANOVA results revealed significant differences (p [$<$] 0.05) in student definitions of function between curricular levels, ethnic groups, grades earned in the most recently completed mathematics course, and number of mathematics courses completed beyond first-year Algebra. Students reported prototypical examples. Chi-square results indicated significant differences (p [$<$] 0.05) between curricular levels, grades earned in the most recently completed mathematics course, and number of mathematics courses completed beyond first-year Algebra for students' choices of representation to express nonexamples and non-mathematical examples of function. Ethnicity was significant in students' choice of representation for nonexamples. Students' examples and nonexamples of functions and their justifications were analyzed for correctness and classified to report if students' justifications were consistent with their reported definition. Results indicated that less than 20justifications were consistent with students' definitions. ANOVA results indicated significant differences (p [$<$] 0.05) between curricular levels, ethnic groups, grades earned in the most recently completed mathematics course, and number of courses completed beyond first-year Algebra for students' correct justifications of their examples and nonexamples. There was a moderate, positive, statistically significant correlation between variables reporting points earned for examples and nonexamples and highest definition category reported. High school students' understanding of the function concept, although incomplete, increases with curricular level, years of study, and grades earned in mathematics courses. These findings are consistent with NAEP and point to the need for examination of how the function concept is introduced to students. 307. {par:impac} V. W. Parks. Impact of a laboratory approach supported by 'Mathematica' on the conceptualization of limit in a first calculus course (Computer Clgebra System). PhD thesis, Georgia State University, 1995. Purpose. The focus of this study was an instructional environment that promoted a laboratory approach to learning calculus supported by the computer algebra system Mathematica. Students were in a computer laboratory that fostered active learning and collaboration-contributing factors to constructing mathematical concepts. The study examined student achievement as the concept of limit developed from an intuitive level to a formal definition and to making connections between the definition of limit and the graph of a function. Also investigated were changes in students' dispositions towards mathematics and technology. Methods. Two groups of students participated in the study. One group attended class in a computer laboratory where Mathematica was accessible during class for use by the instructor and students and was available outside class. The other group of students met in a traditional classroom. The syllabus was the same in both courses. Instruments for evaluating students' understanding of the concept of limit were the same in the two classes. The investigator assessed changes in students' view of mathematics and technology. Statistical tests and qualitative methods were used to analyze the data. An investigator journal and student interviews provided the basis for the qualitative analysis. Results. Both groups performed well on the instruments used to assess knowledge of the limit concept. Findings of the study revealed a significant difference in the groups in one aspect related to understanding a formal definition of limit. The responses to the instruments did not reveal changes in the groups' view of mathematics and technology. Conclusions. This study supports use of a laboratory approach and technology to experiment and investigate concepts in calculus. In the experimental section, the availability of Mathematica promoted use of a variety of problem-solving strategies. Decomposing the limit concept, using a constructivist instructional approach, and encouraging active participation from students through experimentation and investigation prepared the students to move on in the calculus sequence with a foundation for further exploration and inquiry. 308. {par:acomp} B.D. Parnell. A comparison of advanced placement calculus students' and other students' success in the first two semesters of college calculus. PhD thesis, Auburn University, 1993. The purpose of this study was to compare the achievement of students in the first and second semesters of college calculus who took (1) Advanced Placement Calculus in high school, (2) Non-AP Calculus in high school, (3) Precalculus as the last mathematics course in high school, and (4) Precalculus in college with no calculus background. The departmental examination score and the final grade in the first semester of college calculus and the final grade in the second semester of college calculus were used as achievement measures. The sample for the study of the first semester of college calculus consisted of 348 students; 214 of the students in the first semester sample also completed the second semester of college calculus, and, thus, were included in the second semester study. The data were analyzed with one-way analysis of variance and with analysis of covariance in order to detect differences in the mean scores of the four groups. The American College Testing (ACT) mathematics score was used as the covariant for the first semester study, and the first semester departmental examination scores and final grades were used as covariants in the second semester study. The results of the statistical analyses for the first semester study indicated that Advanced Placement Calculus students achieved significantly higher scores than the other three groups. The Non-AP Calculus students achieved significantly higher scores in first semester college calculus than the students who took Precalculus as the last course in high school. There were no significant differences in the remaining comparisons of group achievement. For the study of the second semester of college calculus, the data indicated that there were no significant differences in achievement in the four groups. It appeared that the effect of taking calculus in high school did not last through the end of second semester calculus. 309. {pen:relat} B. Pence. Relationships between understandings of operations and success in beginning calculus. In Proceedings of 17 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Columbus, OH. In an effort to examine the impact of the changes being made at San Jose State University (California) in the calculus curriculum, multiple measures were collected and analyzed. This study focuses on the relationship between performance on a pretest and the class grade. Through written responses on the pretest, a belief and knowledge profile for each student was constructed. Students were grouped according to their answers on an item which asked them to graph 2, x, x squared, and 2 to the x power. Profiles of student perceptions and knowledge were consistent within groups and varied across groups. Results showed that the concept of multiplication was not well understood and was closely related to success in first semester calculus. Multiplication was itself still a process, and in some cases, this process produced multiple concept images within cognitive neighborhoods. 310. {phi-cre:devel} E. Phillips and S. Crespo. Developing written communication in mathematics through math penpal letters. For the Learning of Mathematics, 16(1):15-22, February 1996. The authors describe a teaching experiment in which the first author's fourth grade class communicated with the second author's class of pre-service teachers via a series of penpal letters. The article includes numerous examples of the penpal dialog between the participants. The authors noted improvements in the quality of writing (both of the fourth-graders and their penpals) during the course of the study. The letters also served as a vehicle for problem solving and mathematical reasoning. Study of the letters provided the authors with otherwise unavailable insights into the attitudes of their students. 311. {pho-91:four} C. Phoenix. A four-strategy approach used to teach remedial mathematics in a freshman year program. Community-Review, (7):45-52, 1991. Describes the use of student verbalization and immediate feedback, cooperative learning, a concept/discovery-based approach, and creative classroom activities in a remedial mathematics class for first-year students at Medgar Evers College. Compares student achievement with that in other sections of the same course. Includes sample problems. 312. {pir-sch:mathe} S. Pirie and R. Schwarzenberger. Mathematical discussion and mathematical understanding. Educational Studies in Mathematics, 19:459-470, 1988. 313. {polya-57} G. Pólya. How to Solve It. Princeton University Press, Princeton, NJ, 2nd edition, 1957. 314. {por:commu} J. H. Da Costa Portela. Communicating mathematics through the internet: A qualitative case study. PhD thesis, Texas A&M University, 1997. The purpose evolved naturally to fill a void and sought to answer questions left unanswered: to describe the instructor's rationale for using the Internet as a vehicle to deliver and learn mathematics, and to describe how mathematics graduate students reacted to the instruction delivered through the Internet. Stated perceptions of mathematics graduate students concerning the potential of the Internet as a vehicle to deliver instruction were also analyzed. The context for this study was a college classroom. The primary emphasis of the course was the application of computers to facilitate the communication of mathematics. The students and instructor were physically located in the same mathematics computer laboratory and the course content, the syllabus, the class activities, and the homework were delivered to students through the Internet. The following topics were included in the syllabus: UNIX operating system, electronic mail, MapleV, LaTeX, strategies to search information on the Internet, and creation of a personal home-page. The stated role of the instructor was to facilitate the understanding of new concepts and to help students with topics and problems they encountered, such as calculus problems or computer-related problems. Data were collected from observations in the mathematics computer laboratory, from semistructured interviews using electronic-mail, and from semistructured open-ended interviews. These data sources were coded and sorted with a hypertext application to do a content analysis, which permitted searching for categories reconstructed using relationships, interpretations, and inferences. The interpretation of interviews, observations, and documents revealed that face-to-face class meetings in a close physical proximity were considered essential for success by the students. Students concluded that major advantages of being connected to the Internet included the ability to: (a) e-mail the instructor at any time; (b) concentrate on the subject matter instead of taking notes; (c) look back at the work from previous classes and access it at any time from any place; (d) work at one's own pace; (e) participate in classroom interactions; (f) learn by doing; (g) receive individual help from the instructor without holding the whole class back; and (h) immediate access of sites which were related to the assignments. 315. {por-mas:theef} M. Porter and J. Masingila. The effects of writing to learn mathematics on the types of errors students make in a college calculus class. In Proceedings of the 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th PME-NA), volume 17, pages 3-8, Columbus, OH, 1995. North American Chapter of the International Group for the Psychology of Mathematics Education. (This annotation is quoted from the paper abstract.) `This study examined how engaging calculus students in Writing to Learn Mathematics affected the types of conceptual and procedural errors that the students made on their examinations. Students in two sections of an introductory college calculus course in Fall 1994 were the respondents in this study. We used Hiebert and Lefevre's (1986) characterization of conceptual knowledge as a framework to guide our examination of students' conceptual knowledge. To analyze the errors the students made, we developed a classification system and used some of the ideas and methods of Movshovitz-Hadar, Zaslavsky and Inbar (1987).' 316. {por:theef} D. T. Porzio. The effects of differing technological approaches to calculus on students' use and understanding of multiple representations when solving problems (problem solving). PhD thesis, The Ohio State University, 1994. This research examined how students from (a) Calculus & Mathematica, (b) a traditional calculus course, and (c) a calculus course where graphics calculators were used extensively to emphasize graphical representations differ in their abilities to use and understand connections between multiple representations when solving calculus problems. Two classes from each course participated in the research. Thirty-six students from the classes, twelve per course, participated in individual interviews. Quantitative and qualitative data were collected to make comparisons among students on their use of graphical, numerical, and symbolic representations. Research questions for this study investigated the relationship between the instructional approach students' experienced and (a) changes in their initial preferences for different representations, (b) their abilities to use different representations when solving calculus problems, and (c) their abilities to make connections between representations in the context of problem situations. The theoretical framework developed for this research incorporated theories put forward by Hiebert and Carpenter (1992) and Dubinsky (1991) concerning internal networks of represented knowledge and concerning reflective abstraction, respectively. Results indicated Calculus & Mathematica students were better able to use different forms of representations, particularly ones involving combinations of different representations, and were better able to make connections between representations than graphics calculator or traditional students, and that there was little difference between the latter two groups of students in their abilities to use and make connections between representations. Findings provide evidence that students are better able to use, and make connections between representations when the instructional approach they experience emphasizes different representations and has students solve problems specifically designed to explore, establish, or reinforce connections between representations. This study suggests that the solving and interpreting problems specifically designed to help students examine connections between different representations of the same concept facilitates the development of these connections and that students' understanding of calculus is not necessarily improved by simply adding a technological component to the existing curriculum. Overall, the findings provide encouraging evidence in support of calculus instruction, like that in Calculus & Mathematica, that emphasizes appropriate use of technology, multiple representations of concepts, and interpreting and solving problems. 317. {por:effec} D. T. Porzio. Effects of differing technological approaches on students' use of numerical, graphical and symbolic representations and their understanding of calculus. In Proceedings of Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th), volume ?, pages ???-???, Columbus, OH, 1995. North American Chapter of the International Group for the Psychology of Mathematics Education. This study sought to gather empirical evidence of the effectiveness of calculus instruction like that used in the Calculus and Mathematica project by examining and comparing the effects of three different instructional approaches to calculus on students' (n=100) abilities to use and understand connections between numerical, graphical, and symbolic representations when solving calculus problems. Data were collected using classroom observations, pre- and posttest instruments, and 36 student interviews. Analysis of the data indicated that: (1) Calculus and Mathematics students were better able to use and to recognize and make connections between different representations than the other students; (2) graphics calculator students were proficient at using graphical representations but had some trouble using symbolic representations and recognizing and making connections between graphical and symbolic representations, even though the use of these representations was stressed during their course; and (3) traditional students were the least proficient at using graphical representations and had the most difficulty recognizing and making connections between different representations. An appendix contains the posttest instrument. 318. {pow-pie-ram:resea} A. Powell, E. Pierre, and C. Ramos. Researching, reading, and writing about writing to learn mathematics: pedagogy and product. Research and Teaching in Developmental Education, 10(1):95-109, Fall 1993. (This annotation is quoted from the paper abstract.) `The collaborative work of a mathematics professor and two undergraduate students to produce an annotated bibliography about writing to learn mathematics offers a new research paradigm. The authors discuss this paradigm, procedural and affective aspects of compiling and annotating the bibliography, and criteria for selecting bibliographic entries. At the end of the article, the authors present their annotated bibliography.' 319. {pra:anexp} M. Pratt-cotter. An exploratory investigation on the impact of reform Precalculus on the concept of function. PhD thesis, Georgia State University, 1998. Statement of the Problem. Changes in calculus have generated a need for changes in courses typically deemed as preparation for it, such as precalculus. There exists a need to investigate whether innovations being implemented in a reform precalculus class are achieving the desired results. Development of the function concept is essential to success in calculus; therefore, this topic will be the focus of the investigation. Changes with regard to students' views toward mathematics will also be examined. Method. The methodology employed in this study used both qualitative and quantitative methods to investigate differences resulting from two approaches to precalculus. Two sections of precalculus were examined; one was reform precalculus and the other was traditional precalculus. The reform precalculus section had 22 students, whereas the traditional had 30 students. Instruments used to collect data include the View of Mathematics Inventory (VMI), student information sheets (INFO), student journals (JOURNALS), multiple choice math test (MPRE and MPOST), the final exam (FINAL), and a function focused item (FUNCTIONS). Other data was gathered using the researcher's class diary (DIARY), and notes based on videotapes made of 1 class of each section (VIDEOS). Appropriate choices of analysis used included the analysis of covariance, the median test, chi-square contingency tables, and qualitative methods to investigate changes in understanding function and views of mathematics. Results. No significant differences were found between the results achieved in the two sections of precalculus. Six questions from the final exam were found to be statistically significant by using the median tests. Changes were observed on perspectives on the role function plays outside of the classroom for the reform group. Conclusions. Findings of this study indicate that the role of calculus needs to be better defined if precalculus is to prepare a student for calculus. Qualitative results suggest that reform precalculus students are able to extend the intensive application orientation to a context outside of the classroom, furthering their understanding of the impact of mathematics on their lives. Further research is suggested to explore the connections between learning styles and the reform or traditional approaches in mathematics. 320. {pri:learn} J. Price. Learning mathematics through writing: some guidelines. College Mathematics Journal, 20(5):393-401, November 1989. The author describes his use of writing assignments in a course in elementary number theory. The bulk of the article consists of a handout, given to the students, called `Guidelines for Written Homework.' 321. {obr:aninv} T. E. O'brien Pride. An investigation of student difficulties with two dimensions, two- body systems, and relativity in introductory mechanics. PhD thesis, University of Washington, 1997. This dissertation reports on an investigation of student understanding in introductory mechanics. The topics include two-dimensional kinematics, the impulse-momentum and work-energy theorems, momentum conservation in two-body systems, relative motion, and simultaneity. The investigation, which extended over more than four years, involved primarily students enrolled in introductory calculus-based physics, but also students studying physics at more advanced levels. Conceptual and reasoning difficulties were identified through individual demonstration interviews and descriptive studies that were conducted throughout the period of instruction. The findings were used to guide the design and modification of a tutorial curriculum to address specific student difficulties. Ongoing research provided a continuous assessment of the effect on student learning. 322. {pri:speak} D. Primm. Speaking Mathematically: Communication in Mathematics Classrooms. Routledge, London, 1987. 323. {pru:cogni} E. Prus-Wisniowska. Cognitive, Metacognitive, and Social Aspects of Mathematical Proof with Respect to Calculus. PhD thesis, Syracuse University, 1995. The author conducted a study in the form of a naturalistic teaching experiment involving eight participants. Data was mainly collected through problem solving interviews with the participants. Cognitive, metacognitive, and social aspects of students' calculus proofs were examined. 324. {pug:using} D. Pugalee. Using Journal Writing to Characterize Mathematical Problem Solving. PhD thesis, The University of North Carolina at Chapel Hill, 1995. (This annotation is quoted from the dissertation abstract.) `This study explored the use of journal writing as a method for explaining the problem solving behaviors of an introductory high school algebra class. Since the study of problem solving has largely depended upon the analysis of protocols from `think-aloud' sessions, the study included a comparison of data from this method with the journal writing method. This comparison supported the legitimacy of using journal writing when compared to a `think-aloud' method in the study of problem solving processes. Data from twenty students engaged in solving non-routine mathematics problems through journal writing activities was analyzed in a qualitative framework to explain those processes. The data demonstrated the value of using journal writing in providing a description of those processes.' 325. {pug:conne} D. Pugalee. Connecting writing to the mathematics curriculum. Mathematics Teacher, 90(?):308-310, April 1997. Citing relevant literature, the author argues for the use of writing in the mathematics classroom. He gives two examples of writing assignments from his own classes, and analyzes them in terms of their benefit to student learning and the insights they provide in terms of assessment. 326. {putnam-91} The William L. Putnam Exam. Mathematics Magazine, 64:143, 1991. 327. {qui:writi} R. Quinn and M. Wilson. Writing in the mathematics classroom: teacher beliefs and practices. The Clearing House, 71(?):14-20, September/October 1997. The article includes a review of the literature concerning writing in K-12 mathematics education. The authors conducted a survey of K-12 teachers. The results revealed that although the teachers tended to have favorable attitudes toward the use of writing in mathematics education, few actually used writing to any significant extent in their classes. Reasons cited were poor student writing ability, lack of classroom time, and lack of teacher time. 328. {rab:teach} A. F. rabb liu. Teaching methods and student understanding in calculus. PhD thesis, The University of Arizona, 1997. This study is a comparative case study of what three college calculus teachers did in their classrooms and what their students understood about the concept of derivatives. The teachers were solicited on the basis of peer, supervisor and student recommendations as being good teachers; several volunteer student subjects were selected from each class. Using a naturalistic participant-observer paradigm, the data were collected primarily via extensive classroom observations and in-depth interviews with the teachers and students. Examination of written work, such as student exams, was employed for additional confirmation of hypotheses generated in the field. This study contributes to the bodies of knowledge on pedagogy, effective teaching, classroom dynamics, student understanding and teacher beliefs. The results should be of interest to teachers, teacher educators, mathematics text authors and people interested in how students learn and think about mathematics at the collegiate level. The study of these three classrooms reveals that there is a variety of effective teaching models for undergraduate calculus classrooms. There were, however, important commonalties among these models, the examination of which leads to some characterization of effective teaching practices. These teachers kept the focus on what their students were learning, rather than on covering material. In three different ways, these teachers each gave their students the opportunity to interact with the mathematics before the lesson ended. All three teachers displayed a willingness to grow and learn as teachers. Calculus students do not always learn what their teachers think they have taught. The students in this study displayed a variety of mistaken ideas about the concept of derivative and about other mathematical topics. For example, many students had trouble distinguishing between properties of the function and properties of the derivative. Some students believed that the derivative at a point was a line, rather than the numerical value associated with the slope of a line. Students and teachers disagreed about the correct definition of the derivative, with students attributing little importance to the idea of limits. 329. {raf:calcu} J. S. Rafael. Calculus reform from a constructivist perspective. PhD thesis, University of Calgary (Canada), 1997. The calculus reform movement has grown to large proportions. More than 500 mathematics departments in the United States are currently implementing some level of calculus reform, affecting approximately 1/3 of all students enrolled in calculus. (Ganter, 1997). Educators involved in the movement are looking for ways to improve undergraduate calculus instruction. With the traditional way calculus is taught, memorization and repeated practice of template problems is often the route to a passing grade. Reform activists are looking for ways to help students achieve higher levels of conceptual understanding. In the following chapters, I will present a history of calculus reform; a philosophy of mathematics leading to a constructivist framework for mathematics education; a more precise definition of what reform is and how it fits the constructivist perspective; a summary of current efforts and research; and some ideas for the future based on the constructivist ideas. 330. {ram:theef} C. L. Ramey. The effect of project-based learning on the achievement and attitudes of calculus i students: A case study. PhD thesis, University of Missouri - Kansas City, 1997. This study examined reasons highly motivated students are willing to accept changes in the traditional approaches to the teaching of mathematics. The subjects were high school students enrolled in a traditional calculus I course or in a project-based calculus I course block with AP Physics. Since final exam scores showed no difference in achievement between the two groups of students, the researcher conducted student interviews to determine reasons some students are willing to risk enrolling in the project-based calculus I rather than the traditional calculus I course. Classroom teachers were also interviewed and students were observed in the classroom. The researcher found that the project-based students were willing to accept changes in the approach to teaching mathematics. Project-based students liked the challenge of the class, enjoyed the project work and the hands-on applications of mathematics, as well as, the extensive use of technology, as opposed to a traditional approach to teaching calculus. Students were able to find real-world applications in learning calculus and expressed a perception of improved problem solving and critical thinking skills. The projects provided students with external motivation and developed skills for working in a cooperative group setting. Further research needs to address why females are less likely to risk the project-based class, how after-school work affects the student's choice of classes, how the education level of the parent(s) affects the student's choice of classes, and how mathematics educators can increase use of project-based learning activities in classrooms for all students-not only high-achieving students. 331. {ras:refor} C. L. Rasmussen. Reform in differential equations: A case study of students' understandings and difficulties. Paper presented at the Annual meeting of the American Educational Research Association. To be added later. 332. {ras:quali} C. L. Rasmussen. Qualitative and numerical methods for analyzing differential equations: A case study of students' understandings and difficulties. PhD thesis, University of Maryland College Park, 1997. Prompted by advances in technology and the calculus reform movement, recent reform efforts in the second-year course in differential equations are decreasing the traditional emphasis on analytic techniques for finding exact solutions to differential equations and increasing the emphasis on qualitative and numerical methods for analyzing solutions. To date, however, there is little research on the effect of these reform efforts on students' understandings. The purpose of this study was to examine students' understandings of and difficulties with qualitative and numerical methods and to identify the factors that help shape these understandings and difficulties in one such effort at reform. The particular effort investigated used a traditional textbook and a Mathematica supplement to introduce qualitative and numerical methods for analyzing differential equations. A case study approach was used, focusing on six students in one section of an introductory course in differential equations for scientists and engineers. Data collected included four task-based interviews with each student, classroom observations, instructor interviews, document analysis, and an end-of-the-semester questionnaire. Analysis of data from both an individual cognitive perspective and a sociocultural perspective revealed a gap between the intended and the achieved curriculum. Students had difficulty making important conceptual, symbolic, graphical, and contextual connections and some qualitative and numerical methods appeared to be learned in isolation from other aspects of the problem. From the individual cognitive perspective, the following obstacles were found to influence the development of students' understandings: the action-process-object dilemma, the tendency to overgeneralize, interference from informal or intuitive notions, and the complexity involved with graphical interpretations. From the sociocultural perspective, students' understandings were influenced by instruction that did not seek out students' explanations, classroom interactions that implicitly established procedure and rule-based mathematical justifications, a curriculum that tended to break the subject up into small, skill-based pieces, and the use of technology that was disconnected from the learning process. 333. {rezn-mtps-94} B. Reznick. Some thoughts on writing for the Putnam. In A. H. Schoenfeld, editor, Mathematical Thinking and Problem Solving, pages 19-29. Lawrence Erlbaum, Hillsdale, NJ, 1994. 334. {ric:theef} K. A. Rich. The effect of dynamic linked multiple representations on students' conceptions of and communication of functions and derivatives. PhD thesis, State University of New York at Buffalo, 1995. This study explored the effects of multiple representations and dynamically linked multiple representations on the learning and retention of derivative concepts in high school calculus classes. Three regularly scheduled classes in a suburban high school, all taught by the same instructor, were assigned to receive instruction as follows: (1) Control (N = 20); (2) Multiple Representations (N = 21); (3) Dynamic Linked Multiple Representations (N = 18). No significant differences existed among these groups prior to instruction as measured by the SAT-I Math component, SAT-I Verbal component, and NY State Course III Regents mathematics scores. Immediately following one week of instruction on the concepts of derivatives and their uses a posttest was given. This was followed one week later by a surprise retention test - a parallel form of the first posttest - and by a structured interview about the calculus concepts learned and mental processes students were using. The results showed that all groups learned the material to a high level and on some comparisons there were no significant treatment differences. A common trend in the data which produced some significant differences was that instruction with multiple representation tended to show larger effects at the one-week retention posttest than at the immediate posttest. Such results were least apparent on total test score and most apparent when cognitive processes were categorized and scored for variety and appropriateness of use. The largest differences occurred in comparisons between the control group and the two multiple representation groups; the smallest differences occurred between the latter two groups. Despite using more representations to solve problems, the increases due to multiple representation instruction were not achieved at the expense of increased instructional or testing time. These findings support those of previous research on the effectiveness of multiple representations and dynamic linked multiple representations for mathematics instruction. In addition, this study demonstrates the effective use of these methods for calculus instruction and instruction with young adults. In conclusion, the fields of multiple representations and dynamic linked multiple representations are relatively young but promising areas of research. 335. {ric:thero} G. W. Richgels. The role of student's beliefs about mathematics in the learning of the mathematical definition of limit. PhD thesis, The University of Wisconsin - Madison, 1993. Researchers have found the student concept of limit to be very stable. They have found it difficult to effect a change from the non-mathematical to the mathematical concept of limit. Researchers have hypothesized that student beliefs about mathematics influence and prevent students from learning the mathematical concept of limit. Literature on conceptual change and mathematical problem solving indicate the importance of beliefs. The history of mathematics has also been influenced by beliefs about mathematics. This study is an attempt to identify the role that beliefs play in the learning of the mathematical concept of limit. Twenty-three high school students volunteered to take part in the study. During their high school calculus course, these students participated in three videotaped assessments, a videotaped instructional unit, and completed a geometry knowledge questionnaire. First students responded to written questionnaires; their ranked interview responses were used to arrive at a composite belief score for 18 beliefs that were being studied. No relationships between beliefs and the limit concept were found in examining the students composite belief scores. Next, the students component item responses were examined. From this analysis three component items were identified as being related to the learning of the concept of limit. These component items were related to two beliefs about mathematics. This study determined that: (1) there is a connection between a student's beliefs about determination of the validity of a mathematical technique and the formal limit concept; (2) there is a connection between the formal limit concept and student beliefs about the concept of limit; and (3) that recitation of the formal limit definition should not be the sole criterion for determining whether or not the formal concept of limit has been learned. 336. {rid:intro} L. H. Riddle. Introducing the derivative through the iteration of linear functions. Mathematics Teacher, (5):377-381, 1994. Discusses the use of computer graphics to introduce the concepts of derivative and tangent line using the iteration of functions, especially linear functions. 337. {riv:astru} B. S. Rives. A structural model of factors relating to success in Calculus, College Algebra and Developmental Mathematics. PhD thesis, University of Houston, 1992. The purpose of this study was to examine selected affective and cognitive variables related to student success in college mathematics, especially calculus, college algebra and developmental mathematics. The exogenous variables of gender and locus of control, and the endogenous variables of mathematics preparation, mathematics attitude, and length of time since the student's last mathematics course were posited to have direct and indirect relations to student success in mathematics. Data were collected from 1550 college mathematics students at three campuses of two universities-a 35,000 student campus and an 8,700 student downtown campus of a large urban four-campus system, and a 1,500 student campus of a regional four-campus university system. Three instruments were used to collect data: Rotter Locus of Control Survey (two-point forced choice), Aiken-Dreger Revised Mathematics Attitude Scale, and Rives Demographic and Math Preparation Scale. LISREL 7 was used to estimate the path coefficients and test the goodness-of-fit of five models: a measurement model, a fully specified hypothesized model, and three competing models (null model, modified model, and an elaborated model). The posited model was a very good fit [$(/chi/sp2$] = 28.03, 15 df; GFI = 0.995; AGFI = 0.986; RMR = 0.076). Three non-significant paths were eliminated to produce a more efficient model in which all paths were significant. The findings of the project indicated women had significantly less mathematics preparation, less positive mathematics attitude, and more time since their last mathematics course; however, when all other factors were equal, they tended to be more successful in the course. Students with internal locus of control had a more positive attitude toward mathematics than students with external locus of control. Mathematics preparation was directly related to success as well as indirectly related to success through length of time since the last mathematics course. Mathematics attitude was directly and positively related to success. 338. {rod:acomp} C. D. Roddick. A comparison study of students from two calculus sequences on their achievement in calculus-dependent courses. PhD thesis, The Ohio State University, 1997. This study compared traditional calculus students and Calculus & Mathematica students on their achievement in courses which require calculus as a prerequisite. The first part of the study involved quantitative analyses of grades in calculus-dependent courses. The second part of the study was a qualitative analysis of six students in an introductory engineering: mechanics course. Task-based interviews were conducted over a ten-week period to investigate conceptual and procedural understanding of calculus and ability to apply that knowledge of calculus to engineering mechanics problems. The theoretical framework was based on the theory that active learning with a conceptual focus can enhance the decontextualization of knowledge, which can promote transfer. Results from the quantitative analysis show a significant difference in a differential equations course favoring traditional students, and a significant difference in the first of a calculus-based physics sequence favoring the Calculus & Mathematica students. Other significant differences, favoring the top third of Calculus & Mathematica students, were found in the introductory physics courses and an engineering mechanics course. Results from the qualitative analysis showed that Calculus & Mathematica students were more likely to approach problems from a conceptual viewpoint of calculus knowledge whereas traditional students were more likely to approach problems procedurally. Students who had generalized and abstracted their understanding of the derivative and integral were better able to apply their calculus knowledge to engineering mechanics problems. 339. {roe:there} T. L. Roepke. The relationship between mathematics anxiety and students' perceptions of mathematics and mathematics instruction. PhD thesis, The University of Toledo, 1988. The purpose of this study was to examine relationships between the way a student views the subject of mathematics in a formal or informal manner and the student's level of mathematics anxiety. The study also compared the student's view of what mathematics instruction ought to be as well as the view of mathematics instruction received with the level of mathematics anxiety. The study was conducted with 134 subjects. Of these students, 78 were enrolled in Calculus I and 54 were enrolled in Mathematics for Elementary Teachers. The subjects were students at either a large public university or a small private university in the midwest. The students were asked to complete four instruments. The Beliefs About Mathematics and the Beliefs About Mathematics Instruction Scales measure the extent to which students view mathematics and its instruction as formal or informal. The Tuckman Teacher Feedback Scale measures how a student views the mathematics instruction received in terms of certain characteristics of the mathematics teacher. The Mathematics Anxiety Rating Scale measures the student's level of mathematics anxiety. Correlation as well as regression analyses were conducted on the above data to test several hypotheses. A weak correlation indicated that those students who viewed mathematics more formally tend to have higher levels of mathematics anxiety. No such relationship existed between the student's view of mathematics instruction as formal or informal and levels of mathematics anxiety. Negligible correlations were found between characteristics of former mathematics teachers and the levels of mathematics anxiety. No correlation existed between grades received in mathematics courses and levels of mathematics anxiety. Analyses were conducted to compare the results of the instruments for the two groups of students. The Calculus students tended to view both mathematics and its instruction more informally than did the Mathematics for Elementary Teachers students. The Mathematics for Elementary Teachers students tended to be more mathematics anxious than did the Calculus students. The groups did not differ on grades received in mathematics or on the way they viewed their former mathematics teachers. This study could have implications for how mathematics is taught and viewed in the classroom. Those students who viewed mathematics as a formal system consisting of rules and procedures to be followed tended to exhibit higher levels of mathematics anxiety. 340. {rog:criti} G. M. Rogers. Critical thinking and intellectual development: A study of the effects of an integrated first year curriculum in science, engineering and mathematics. PhD thesis, Indiana State University, 1991. The purpose of this study is to assess the outcomes related to intellectual development and critical thinking of an innovative freshman year curriculum after its first year of implementation at Rose-Hulman Institute of Technology. The curriculum is a broad-based, integrated curriculum in which the materials traditionally offered in a first-year engineering curriculum (physics, calculus, chemistry, computer science, graphics, engineering statics, and engineering design courses) are unified in a three course, twelve credits per quarter, sequence. The primary goals of the curriculum are to infuse concepts, improve techniques, and cultivate effective search strategies. Students were pre-tested at the beginning of the academic year using the Watson-Glaser Critical Thinking Appraisal (WGCTA) to assess critical thinking skills and the Cognitive Complexity Index of the Learning Environment Preferences (LEP) to assess intellectual development. A comparison group was selected using a cluster analysis technique which clustered students in the traditional curriculum with students participating in the experimental program based on demographic data and pre-test scores. At the end of the academic year, both groups were post-tested using the WGCTA and the LEP. T-tests were utilized to analyze the data. It was found that after the first year there was no significant difference on measures of intellectual development and critical thinking between students in the experimental program and those in the comparison group who were in the traditional freshman curriculum. However, because of the complexity of the variables examined, the fact that the students in the experimental program were highly selected, and the two groups tested were very homogeneous, it is recommended that further study be done to examine the long-term effects of the program. 341. {roj:enhan} M. E. Rojas. Enhancing the learning of probability through developing students' skills in reading and writing. PhD thesis, Columbia University Teachers College, 1992. The author compared two randomly selected sections of a community college first course in probability. One class received special materials designed to enhance their reading, writing, and communication skills. She found that students using the auxiliary materials had higher mean scores on exams than the other students, and she also noted improvement in their attitudes toward writing. The author suggests further research to determine whether an environment that stimulates writing increases student achievement in statistics. 342. {roo:theef} M. Rooney. The effects of brain hemisphere dominance on mathematical achievement in calculus at the college level. PhD thesis, University of Arkansas, 1991. This study examined the effect of student and teacher brain dominance upon course grades and final examination scores in nine sections of Calculus I (Calculus and Analytic Geometry) at the University of Arkansas. Torrance's Style of Learning and Thinking (SOLAT) was used to assess the brain hemisphere preferences of 324 students and 8 instructors. Based on the numerical results of this instrument, each subject was placed into one of three categories: (1) Left brain dominant, (2) Right brain dominant, or (3) Whole brain dominant. An initial comparison of the students' ACT mathematics scores showed no significant differences in mathematical ability among the three brain dominant groups. Yet, left dominant students received statistically higher course grade means than right dominant students. There were no significant differences in final examination scores among the three groups. The interaction of left dominant students with left dominant teachers produced a significantly higher course grade mean than any other student-teacher interaction. The study further examined the withdrawal and failure rates among students of different brain dominant groups. Right dominant subjects received three times as many F or W grades as left dominant subjects. Among instructors of different brain hemisphere preferences, however, there were no significant differences in percentages of withdrawals and failures. In addition, the study examined the relationships between student sex and student brain dominance, as well as those between student sex and final course grades in Calculus I. Females were more likely to be left brained than males, while males were more likely to be whole brained. No significant difference in course grade means existed between the sexes, with males and females being equally likely to receive a grade of A, B, C, D, F, or W. 343. {ros:using} B. Rose. Using Expressive Writing to Support the Learning of Mathematics. PhD thesis, The University of Rochester, 1989. (This annotation is quoted from the dissertation abstract.) `This study explores, through both conceptual and empirical components, the role of exploratory or personal writing to support the learning of mathematics. ...A review of the literature on expressive writing and writing in mathematics and a conceptual analysis based on a preliminary study are combined to identify (a) complementary ways in which expressive writing can be employed in mathematics instruction, and (b) a theoretical framework of potential benefits for the classroom along the three dimensions of student as writer, teacher as reader, and the student-teacher interaction. The setting for the empirical component was a calculus course for business majors at a small private college. As the author taught this course, she used autobiographical narratives, in-class focused writing, and spontaneous dialog journals and monitored them carefully. Qualitative research methods were used. ...' 344. {row:alter} E. R. Rowley. Alternative assessments of meaningful learning of calculus content: A development and validation of item pools. PhD thesis, Utah State University, 1996. Research indicates that classroom teachers lack the training to produce and analyze valid measurement instruments. Furthermore, college calculus teachers are faced with measuring student achievement relative to new goals coming from a major curricular reform movement. As a result, there is a need for valid measurement instruments to be developed for measuring student achievement in calculus. The purpose of this project was to develop effective measurement items relevant to differential calculus of a single variable. The items were organized into computerized item pools. A booklet was written which provides information and instructions necessary for researchers and teachers to access the item pools and construct valid measurement instruments. 345. {sal:motiv} U.A. Saleh. Motivational orientation and understanding of limits and continuity of first year calculus students (motivation). PhD thesis, The University of Oklahoma, 1994. This study examines first year calculus students' understanding of limits and continuity and the effects of motivational orientation and gender in those understandings using Skemp's and Fless's models of understanding. The study also examines student views about the causes of success in calculus using Weiner's attribution theory to determine what factors are responsible for success. Finally students' perceived ability is compared with their performance on a test of understanding of limits and continuity to see whether or not the two variables are positively correlated. The results show that motivational orientation significantly affects understanding of limits and continuity. Task-oriented students performed better than ego-oriented students. However, the results show that gender does not significantly affect understanding. In other words, there is no significant difference in understanding between males and females. Finally, the results show a positive correlation between perceived ability and understanding. This means the more confidence students have in their ability to do well, the better they perform. 346. {san:teach} R. A. Sanchez. Teachers' and students' mathematical thinking in a calculus classroom: The concept of limit. PhD thesis, The Florida State University, 1996. The mathematical topic that is the focus of this dissertation is the concept of limit as it was introduced in the first semester at University level. The purpose of this study was to investigate one teacher's thought processes when teaching the concept of limit as it occurs in a classroom setting during the instruction in an introductory calculus course, and selected students' cognition and understanding of the concept of limit in this context. I attempted to describe how students developed the idea of limit and how students understood the concept of limit and the factors (students' background knowledge, motivations, social interaction, teacher) which might influence or interfere with that understanding, as well as how understanding of the concept of limit changed with the instruction. Also, I tried to determine what was the teacher mathematical thinking, her teaching strategies, and the role that the concept of limit plays within calculus. In undertaking this study qualitative research methodology was employed. This study used different methods of data collection: (a) classroom observations, (b) interviews, and (c) copies of materials used in class, written and graphic materials related to the concept of limit. The focus was on one calculus classroom in a Venezuelan University over a period of three months. I found that students' learning was characterized by following the routines established by the teacher' examples and doing the worksheet problems. The teaching of the concept of limit consisted of exposing students to formal definition ( [$\epsilon -\delta$] definition), rules, and procedures with mandates to memorize and practice. The students were unable to express a formal definition of limit after the unit on limits. 347. {santos-rcme-98} M. Santos-Trigo. On the implementation of mathematical problem solving instruction: Qualities of some learning activities. CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education III, 7:71-80, 1998. 348. {sas:authe} C. M. Sasse. Authentic learning: What makes a classroom and its tasks authentic? PhD thesis, University of Missouri - Columbia, 1997. Many educators and researchers have identified constructivist pedagogies such as cognitive apprenticeships, case problems, and simulations as vehicles for authentic learning in classroom settings. This study examined two college classrooms, an introductory calculus course using lab problems and a graduate management course using a simulation, to describe how constructivist concepts like situated cognition, collaborative discourse, teacher scaffolding, and student ownership are manifested in the classroom setting. Data from direct observations and participant interviews were collected for each classroom, and qualitative, interpretative analyses were conducted on the combined data. While both classrooms included authentic problems, collaborative structures, and teacher scaffolding, the findings suggest that merely including these elements is not sufficient to creating an authentic learning environment. Specifically, the lab problems situated in realistic contexts and collaborative structures did not result in authentic learning for students as typical school norms dictated student engagement. The management simulation was, however, apparently successful in transcending the school context despite the use of conventional classroom activities and strong direction by the teacher. 349. {heri-99} L. J. Sax, A. W. Astin, W. S. Korn, and K. Mahoney. The American Freshman: National norms for fall 1999. Higher Education Research Institute, UCLA, 1999. 350. {sch:amode} M. Schiro. A Model and Study of the Role of Communication in the Mathematics Learning Process. PhD thesis, Boston College, 1994. (This annotation is quoted from the dissertation abstract.) `An important topic in mathematics education research currently and in the recent past relates mathematics and communication. ...The ability to communicate mathematically is seen as crucial in fostering the development of competent problem-solvers capable of inquiry and the conveyance of thought. ...The related areas of communication including reading, writing, speaking, and listening both across the curriculum and in mathematics were reviewed. These seemingly discrete components were unified in a model of the role of communication in the mathematics learning process. This model was built upon the child as a communicative being at the core of the learning process that involves the communication of cognitive and affective elements through various channels to a communicative audience. A segment of the model was examined by investigating the relationship between increased communication through journal writing and sharing and mathematics achievement and attitude. The first part of the study included a pretest, instruction in standard cubic units, and a posttest in six fifth grade classrooms. Instruction in three experimental classes differed from that in control classes by being accompanied by journal writing and sharing. Two classes participated in the second part of the study that followed the same procedures for instruction in common fractions. ...' 351. {sch:onlea} M. Schneider. On learning the rate of instantaneous change=a propos de l'apprentissage du taux de variation instantane. Educational Studies in Mathematics, (4):317-350, 1992. Divided into two parts, this article analyzes why some pupils feel reserve about instantaneous velocities and instantaneous flows. The second part relates reactions of pupils facing a problem that implicates the instantaneous rate of change. Describes some characteristics of this problem that enables the authors to explain its instructional impact. 352. {schoen-85} A. H. Schoenfeld. Mathematical Problem Solving. Academic Press, Orlando, FL, 1985. 353. {schoen-cog-89} A. H. Schoenfeld. Teaching mathematical thinking and problem solving. In L. B. Reznick and B. L. Klopfer, editors, Toward the thinking curriculum: Current cognitive research, pages 83-103. American Society for Curriculum Development, Washington, D.C., 1989. 354. {schoen-hrmtl-92} A. H. Schoenfeld. Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws, editor, Handbook of Research on Mathematics Teaching and Learning, pages 334-370. Macmillan, New York, NY, 1992. 355. {schoen-rcme-98} A. H. Schoenfeld. Reflections on a course in mathematical problem solving. CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education III, 7:81-113, 1998. 356. {schoen-ume-95} Alan H. Schoenfeld. A brief biography of calculus reform. UME Trends, 6(6):3-5, 1995. Reflects on the history of the calculus reform movement and the development of course implementation projects, including use of technology in instruction. 357. {sch-pet:calcu} M. Schoonover and J. C. Peterson. Calculus reform: Experiences of a two-year college collaborating with universities. In ?, volume ?, pages ???-???, Boston, US, 1993. Annual Meeting of the American Mathematical Association of Two Year Colleges. As part of an effort to reform calculus instruction practices, Chattanooga State Technical Community College (CSTCC), in Tennessee, participates in two consortia. One consortium, the Chattanooga Calculus Consortium (CCC), is headed by the University of Tennessee at Chattanooga and involves one other four-year institution and three Chattanooga-area high schools. The CCC was funded by the National Science Foundation (NSF) in 1991 to adapt, implement, and evaluate calculus reform materials being developed at St. Olaf College; inform secondary schools about and involve them in calculus reform movements; prepare computer laboratory manuals; and prepare additional problem sets. The second consortium, the Western Appalachia Calculus Consortium (WACCO), is directed by the University of Kentucky and involves eight other higher-education institutions in three states. Funded by the NSF in 1992, WACCO was designed to share the identification, acquisition, and evaluation of reform approaches and materials through twice annual progress meetings; an Internet-accessible archive of materials; a journal; and workshops with high school educators. Additionally, CSTCC has been working to revise elementary differential equations to move students away from rote memorization to a meaningful understanding of differential equations. Participation in the consortia has benefited CSTCC in the following ways: theoretical and technological assistance has been provided with calculus reform and the college has been allowed input into the reform. The college has shared reform and technology with area high school teachers. 358. {sch:calcu} C. S. Schrock. Calculus and computing: An exploratory study to examine the effectiveness of using a computer algebra system to develop increased conceptual understanding in a first-semester calculus course. PhD thesis, Kansas State University, 1989. The purpose of the study was to investigate the effects of a computer algebra system on students' comprehension and computational skills in a Calculus I classroom. The effect of the CAS on students' attitude toward mathematics and computers was also considered. Three Calculus I classes at a small midwestern university were involved in the study. Two classes acted as control groups and received traditional instruction emphasizing computation. The third class, the experimental group, received instructions emphasizing conceptual understanding. The Maple CAS was used in this class for demonstrations and to help students perform the calculations thus de-emphasizing computation. This system performed symbolic as well as algebraic manipulations. A survey was used to examine student attitudes and expectations before and after the course. A midterm examination, was administered to all classes to test conceptual understanding, and a final examination, was administered to test computational ability. Students' attitude and confidence were positively affected by the use of Maple. In addition, the experimental group showed significantly higher conceptual understanding than the control groups while showing no significant loss of computational ability. 359. {sch:doesw} A. W. Schurle. Does writing help students learn about differential equations? Primus, 1(2):129-136, 1991. Reports on the differences in two sections of a college course on differential equations. Results on a common examination showed that writing assignments substituted for traditional homework assignments did not improve test performance. However, survey results indicated that students felt the writing assignments, in lieu of additional homework, improved their comprehension. 360. {sch-92:rev} R. Schwartz. Revitalizing liberal arts mathematics. Mathematics and computer education, (26(3)):272-277, 1992. Suggests instructional modifications that utilize liberal arts students interests and strengths to stimulate their interest in mathematics. Suggestions recommend that teachers (1) relate mathematics to current critical issues; (2) make use of projects and open-ended problems; (3) utilize collaborative learning; and (4) provide opportunities for written and oral communication. 361. {sel-sel:error} A. Selden and J. Selden. Errors and misconceptions in college level theorem proving. In Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics, volume 3, pages ?-?, Cornell University, 1987. The authors document, analyze and classify seventeen errors exhibited by students in abstract algebra classes. Suggestions for curriculum improvement are proposed based on the results. 362. {selden-rcme-00} A. Selden, J. Selden, S. Hauk, and A. Mason. Why can't calculus students access their knowledge to solve nonroutine problems? CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education IV, pages 128-153, 2000. Abstract from the paper: In two previous studies we investigated the nonroutine problem solving abilities of students just finishing their first year of a traditionally taught calculus sequence. This paper reports on a similar study, using the same nonroutine first-year differential calculus problems, with students who had completed one and one-half years of traditional calculus and were in the midst of an ordinary differential equations course. More than half of these students were unable to solve even one problem and more than a third made no substantial progress toward any solution. A routine test of associated algebra and calculus skills indicated that many of the students were familiar with the key calculus concepts for solving these nonroutine problems; nonetheless, students often used sophisticated algebraic methods rather than calculus in approaching the nonroutine problems. We suggest a possible explanation. These students may have had too few tentative solution starts in their problem situation images to help prime recall of the associated factual knowledge. We also discuss the importance of this for teaching. 363. {selden-jmb-89} J. Selden, A. Mason, and A. Selden. Can average calculus students solve nonroutine problems? Journal of Mathematical Behavior, 8:45-50, 1989. 364. {sel-sel:unpac} J. Selden and A. Selden. Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29:123-151, 1995. (This annotation is quoted from the paper abstract.) `This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a `bridge course' designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them.' 365. {selden-maa33-94} J. Selden, A. Selden, and A. Mason. Even good calculus students can't solve nonroutine problems. In Research Issues in Undergraduate Mathematics Learning, volume 33, pages 19-26. MAA Notes, 1994. 366. {sev-som:integ} A. Sevilla and K. Somers. Integrating precalculus review with the first course in calculus. Primus, 3(1):35-41, 1993. Describes a course designed by Moravian College, Pennsylvania, to integrate precalculus topics as needed into a first calculus course. The textbook developed for the course covers the concepts of functions, Cartesian coordinates, limits, continuity, infinity, and the derivative. Examples are discussed. 367. {sfard-esm-91} A. Sfard. On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22:1-36, 1991. 368. {MR1625407} A. Sfard. The many faces of mathematics: do mathematicians and researchers in mathematics education speak about the same thing? In Mathematics education as a research domain: a search for identity, Book 1, 2, pages 491-511. Kluwer Acad. Publ., Dordrecht, 1998. This item will not be reviewed individually. 369. {sfard-flm-98} A. Sfard, P. Nesher, L. Streefland, P. Cobb, and J. Mason. Learning mathematics through conversation: Is it as good as they say? For the Learning of Mathematics, 18:41-51, 1998. 370. {she:writi} R. Shepard. Writing for conceptual development in mathematics. Journal of Mathematical Behavior, 12:287-293, 1993. (This annotation is quoted from the paper abstract.) `Using writing to learn in mathematics is discussed from a conceptual development perspective. Shuell's (1990) phases of learning are described as a general model for conceptual development and subsequently matched with Britton's (Britton, Burgess, Martin, McLeod, & Rossen, 1975) categories for transactional-informative writing. It is proposed that both research and pedagogical initiatives may be improved by considering writing assignments according to the type of thinking and level of understanding required. Sample math assignments for each writing category are also provided.' 371. {she:stude} T. Sherman. Student research papers in freshman calculus-modelling with differential equations. Primus, 3(1):19-34, 1993. To build understanding of, and the ability to use, first-year science and engineering calculus, students were introduced to differential equations and asked to carry out research on differential equation modeling projects. Describes a model for the research paper; appendices provide a copy of the student handout and five examples of student research topics. 372. {shi-gal:thean} M. Shield and P. Galbraith. The analysis of student expository writing in mathematics. Educational Studies in Mathematics, 36:29-52, 1998. (This annotation is quoted from the paper abstract.) `The use of writing as a learning activity in mathematics has been the subject of many publications. However, little evidence has been presented to support the claims that writing enhances learning in mathematics. One difficulty in such research has been the lack of a detailed method for analysing the writing products of students. In the present study, a scheme for coding the parts of written mathematical presentations was developed. At the grade 8 level at which the study was conducted, a limited style of expository writing was found to predominate. The writing was shown to closely resemble the style of the typical mathematics textbook used by the students.' 373. {shul-er-86} L. S. Shulman. Those who understand: Knowledge growth in teaching. Educational Researcher, pages 4-14, February 1986. 374. {MR99a:00011} A. Sierpinska and J. Kilpatrick, editors. Mathematics education as a research domain: a search for identity. Book 1, 2. Kluwer Academic Publishers, Dordrecht, 1998. An ICMI study. The 34 papers in this collection include the following: Part I. The ICMI Study Conference {report}. Part II. Mathematics education as a research discipline: Josette Adda, A glance over the evolution of research in mathematics education (49-56); Norma C. Presmeg, Balancing complex human worlds: mathematics education as an emergent discipline in its own right (57-70); Paul Ernest, A postmodern perspective on research in mathematics education (71-85); Erich Ch. Wittmann, Mathematics education as a "design science" (87-103); Roberta Mura, What is mathematics education? A survey of mathematics educators in Canada (105-116); Gunnar Gjone, Programs for the education of researchers in mathematics education (117-127). Part III. Goals, orientations, and results of research in mathematics education (7 papers). Part IV. Different research paradigms in mathematics education (10 papers). Part V. Evaluation of research in mathematics education (4 papers). Part VI. Mathematics education and mathematics: Anna Sfard, A mathematician's view of research in mathematics education: an interview with Shimshon A. Amitsur (445-458); Ronald Brown, What should be the output of mathematical education? (459-476); Michele Artigue, Research in mathematics education through the eyes of mathematicians (477-489); Anna Sfard, The many faces of mathematics: do mathematicians and researchers in mathematics education speak about the same thing? (491-511); Heinz Steinbring, Epistemological constraints of mathematical knowledge in social learning settings (513-526); Anna Sierpinska and Jeremy Kilpatrick, Continuing the search (527-548). 375. {MR98h:00013} A. Sierpinska and S. Lerman. Epistemologies of mathematics and of mathematics education. In International handbook of mathematics education, Part 1, 2, pages 827-876. Kluwer Acad. Publ., Dordrecht, 1996. One gets the impression that the authors themselves do not take epistemology of mathematics seriously. Epistemology is treated as an area about which everybody who likes it may utter an opinion even when his or her knowledge about that area is totally unreflected and conventional. The authors of this chapter from the International handbook of mathematics education do not seem to have considered the possibility that a serious and careful epistemological study may produce new and interesting or useful results. They claim that such an attitude is common among mathematicians (p. 833). Already in their introduction to the topic of epistemology, by stating that "the answers to the question of origins of knowledge have been traditionally put into two categories: apriorism and empiricism", they seem totally ignorant of the fact that Kant's Critique of pure reason, with which epistemology in the modern sense began, was devoted to just overcoming that false dichotomy. Consequently, they do not pay any serious attention to the most important and most elaborated epistemology of mathematics of recent times, which had a great influence on mathematics education as well, namely Piaget's genetic epistemology. Piaget was a Kantian, and he tried to further elaborate and dynamize Kantian epistemology. Let me just mention one central issue with respect to this. Kant's greatest innovation with respect to mathematics was to see that mathematics as a science of the possibility of things, as it used to be called, could not be based on the "possibile logicum" alone, but demanded a new notion of possibility and impossibility. Piaget, who was totally aware of this, devoted at least three entire books and hundreds of experiments to this notion of possibility, none of which is included in the bibliography (7 pages). It seems obvious that the vision of possibility is most central and profound to a person's mental health and activity and thus deserves a lot of concern from mathematics educators. The authors themselves make a short remark that "proofs of impossibility are a trademark of a mathematical frame of mind" (p. 832), but neither elaborate on this nor draw any consequences from it. The chapter has two parts: 1. Epistemologies of mathematics (15 pages), 2. Epistemologies in mathematics education (24 pages). The first part seems very confused and confusing. The second part does not deal with epistemology of mathematics at all, but rather presents a variety of epistemological views circulating in the mathematics education community of today and which the authors seem to be particularly fond of. This part, however, is well informed and well written and is certainly useful to student teachers who want to know what is most popular among certain circles in the community of today (Piaget seems out of fashion, as does any serious interest in the nature of mathematical knowledge). What worries me is the lack of continuity and reflection, which keeps mathematics education from making more progress than it actually does in becoming an area of serious academic study. For instance, not even the report New trends in mathematics education IV, which emerged from the Karlsruhe Congress and which was published by UNESCO in 1979, is mentioned. 376. {sim:beyon} M. Simon. Beyond inductive and deductive reasoning: the search for a sense of knowing. Educational Studies in Mathematics, 30:197-210, 1996. (This annotation is quoted from the paper abstract.) `Examination of data from several mathematics education research projects has led the author to postulate a form of mathematical reasoning that learners engage in spontaneously and that is not inherently inductive or deductive. Transformational reasoning is generated through the learner's inquiry into how a mathematical system works. This sense of `how it works' may lead to a sense of understanding that may not be provided by inductive and deductive reasoning.' 377. {sim:teach} L. M. Simonsen. Teachers' perceptions of the concept of limit, the role of limits, and the teaching of limits in Advanced Placement Calculus. PhD thesis, Oregon State University, 1995. The main goal of the study was to investigate high school advanced placement calculus teachers' subject matter and pedagogical perceptions by examining the following questions: What are the teachers' perceptions of the concept of limit, the role of limits, and the teaching of limits in calculus? Additionally, the sampling technique used shed some light on the question: Are these teachers' perceptions associated with their participation in a calculus reform project focused on staff development? A multi-case study approach involving detailed examination of six teachers (three had participated in a calculus reform project and three had not participated in any calculus reform project) was used. The data collected and analyzed included questionnaires, interviews, observational fieldnotes, videotapes of classroom instruction, journals, and written instructional documents. Upon completion of the data collection and analysis, detailed teacher profiles were created with respect to the questions above. The results of this study were then generated by searching for similarities and differences across the entire sample as well as comparing and contrasting the group of project teachers and the independent teachers. The teachers in this study perceived calculus as a linearly ordered set of topics in which the concept of limit formed the backbone for appreciating and understanding all other calculus topics. The teachers felt the intuitive understanding of limits was essential to further understanding of calculus. Nevertheless, little classtime was devoted to developing an intuitive understanding. Furthermore, little emphasis was given to drawing connections between limits and subsequent calculus topics. The independent teachers devoted considerable time to discussing formal epsilon-delta definition and arguments. The complex relationship between teachers' perceptions and classroom practice appeared to be affected by the significant influence of the teachers' goals of preparing students for the advanced placement exam and college calculus and the authority given to the calculus textbook. Differences between the group of independent teachers and the group of project teachers were found related to the following factors: (a) commitment to the textbook, (b) planning, (c) use of multiple representations, (d) attitude toward graphing technology, (e) classroom atmosphere, (f) examinations, (g) appropriate level of mathematical rigor needed for teaching calculus, and (h) the stability of perceptions. These factors, however, were not fully attributed to participation in the calculus reform project. 378. {tall-simp-ictcm-98} A. Simpson and D. Tall. Computers and the link between intuition and formalism. In Proceedings of the Tenth Annual International Conference on Technology in Collegiate Mathematics, pages 417-421. Addison-Wesley Longman, 1998. Abstract from paper: We consider the wide spectrum of meanings which individuals can give to a figure. It may be conceived as being passive, merely being associated with a given concept or organisational, allowing the individual to represent several pieces of relevant information compactly in a single diagram. Alternatively, it may be generative, in the sense that a learner uses it to guide their thinking. Such generative imagery may be conceptually generative suggesting intuitive insights into mathematical relationships. It may also be formally generative, linking to the formal arguments required to develop a coherent formal theory. In this paper we consider how visual computer software may be used in generative ways, with particular reference to how different kinds of visual magnification can be used to offer generative ideas about continuity and differentiability. 379. {sko:anana} P.R. Skoner. An analysis of the relationship between the study of Calculus in high school and achievement in first-year undergraduate Calculus. PhD thesis, Indiana University of Pennsylvania, 1992. The purpose of this study was to determine whether students who studied calculus in high school achieved at a higher level in undergraduate calculus than those who had no previous exposure to calculus. Other differences between students in the two groups were also investigated through the use of a questionnaire to provide direction to curriculum coordinators in grades K-12 who sequence exposures in mathematics for students who will later study calculus as an undergraduate. Grades in undergraduate calculus were used to measure achievement. Seven background variables related to undergraduate calculus achievement were investigated: SAT verbal and mathematics scores; high school class rank, overall grades, and mathematics grades; gender; and undergraduate instructor. Survey questions dealt with attitudes toward mathematics, attribution of successes and failures in mathematics, perceived areas of weakness in mathematics background, and the perceived advantage of studying calculus in high school. Data from 87 students were used. The effect of the incidence of studying calculus in high school was measured by analysis of covariance using SAT mathematics scores and high school class rank as covariates. Chi-square tests were used to test the significance of survey responses. Students who studied calculus in high school achieved significantly higher course grades in first semester undergraduate calculus than their counterparts. However, in second semester, the achievement level of the students in the two groups did not differ significantly. Increases in students' self-reported attitudes were found for both groups through various grade levels. The students who studied calculus in high school reported the most positive attitudes, although the differences between those attitudes were only significant in grades 4 through 12. Students in both groups attributed successes and failures mostly to teachers and felt that it was an advantage to study calculus in high school. Based on the findings, high school students should study calculus even if it results in some repetition at the undergraduate level. However, mastery in basic skills beginning at the elementary level should be demonstrated before students are advanced to higher levels of mathematics. Psychological factors should also be considered in placement and advising situations. 380. {sla-90:abil} R. Slavin. Ability grouping, cooperative learning and the gifted. point-counterpoint- cooperative learning. Journal for the Education of the Gifted, (14):3-8, 1990. The article discusses how cooperative learning (emphasizing group goals and individual accountability), the limited use of acceleration by extremely able learners, and differentiation within classes can reduce tracking and separate enrichment programs while meeting the needs of gifted students in the regular classroom. 381. {sla-90:learn} R. Slavin. Learning together. American School Board Journal, (177):22-23, 1990. In cooperative learning, students are typically assigned to heterogeneous teams. The Johns Hopkins University Elementary School Program uses four principal methods involving student team learning, including Student Teams Achievement Divisions, Teams Games Tournament, Team Accelerated Instruction Mathematics, and Cooperative Integrated Reading and Composition. 382. {sla-90:res} R. Slavin. Research on cooperative learning: Consensus and controversy. Educational Leadership, (47):52-54, 1990. Four literature reviews found that cooperative learning methods using group rewards and individual accountability consistently increase student achievement more than control methods in elementary and secondary classrooms. More research is needed to gauge cooperative learning's effectiveness at senior high and college levels and for instilling higher order concepts. 383. {sla-91:syn} R. Slavin. Synthesis of research of cooperative learning. Educational Leadership, (48):71-82, 1991. For enhancing student achievement, the most successful cooperative learning approaches have incorporated two key elements: group goals and individual accountability. Positive effects have been consistently found on outcomes such as self-esteem, intergroup relations, acceptance of academically handicapped students, attitudes toward school, and ability to work cooperatively. 384. {sma:compu} D. Small and et al. Computer algebra systems in undergraduate instruction. College Mathematics Journal, (5):423-433, 1986. Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. 385. {smi-con-91:und} E. Smith and J. Confrey. Understanding collaborative learning: Small group work on contextual problems using a multi-representational software tool. In ??, volume ??, page ??, ??, 1991. American Educational Research Association. The interactions of three high school juniors (two female and one male) working together on a series of contextual mathematics problems using a multi representational software tool were studied. Focus was on determining how a constructivist model of learning, based on an individual problematic-action-reflection model, can be extended to offer explanatory power for small-group collaborative learning. This extension is constructed by adopting several concepts from the socio- historic or Vygotskian school, including the zone of proximal development, cultural tools, proleptic talk, and appropriation. The subjects worked together during a 10-week secondary mathematics course that focused on problem solving with Function Prove. Although constructivist and socio-historic approaches to cognition have, at times, been interpreted as offering opposing viewpoints, it is suggested that there is a potential complementary, particularly in the area of collaborative peer learning, since researchers in neither area have as yet offered a strong explanatory model for how students jointly construct mathematical knowledge. 386. {smi:probl} P. Smith. Problem-solving through Writing: A Course for Preservice Teachers of Secondary School Mathmatics. PhD thesis, George Mason University, 1996. (This annotation is quoted from the dissertation abstract.) `The purpose of this developmental dissertation was to create an undergraduate, college-level course in teaching problem-solving through writing, to provide mathematics pre-service secondary school teachers with the knowledge and strategies that will better prepare them to respond appropriately to the challenges of the contemporary mathematics classroom at the high school level. ...The goal of the course was to improve the teaching and learning of problem-solving in the mathematics classroom through the use of writing strategies. The lessons centered around method objectives and content objectives in behavioral form. The unique features of the course included: (1) the use of writing strategies such as collaborative learning, peer editing, presentations, projects, and journal writing to teach problem-solving in the mathematics classroom; (2) field experiences to observe other teachers; and (3) the use of the writing process in problem-solving. /dots' 387. {sno:aninv} K. Snook. An investigation of first year calculus students' understanding of the derivative. In ? The Association of Research in Undergraduate Mathematics Education, 1997. Researchers have suggested that many students pass calculus without a basic understanding of some of the fundamental concepts required for further study in their disciplines. Research reports indicate that students are procedurally proficient in completing problems that are similar to those in their textbooks, yet they demonstrate limited understanding of the calculus concepts and processes involved. This paper describes a study designed to investigate first year calculus students' understanding of the derivative. The focus of the study was threefold: (1) to identify levels of understanding of the derivative indicated by subjects' written performances on derivative problems, (2) to identify levels of understanding of the derivative indicated by subjects' verbal performances on derivative problems during talk-aloud interviews, and (3) to investigate the relationship between these two indicators of subjects' understanding levels. A ten problem Derivative Test was designed and administered to 225 college calculus students. Results of this written Derivative Test were used to identify subjects' levels of understanding of the derivative as presented in ``freshmen'' or ``elementary'' calculus. Five subjects participated in two individual interviews in which they re-solved the ten Derivative Test problems in a ``talk-aloud'' manner. Interview data were analyzed using mapping techniques. The levels of understanding indicated by subjects' verbal performances during the interviews were then determined using the developed Combined Model of Understanding. Synthesis of the analyses of the two performances determined the existence of relationships between the levels of understanding of the derivative indicated by subjects' written and verbal performances. Each problem on the Derivative Test was categorized by type of derivative interpretation; geometric, physical, algorithmic, or relational, and by manner of presentation; traditional or nontraditional. Relationships between subjects' performances depended upon the type and presentation of derivative problems. It was concluded that the levels of understanding determined by subjects' written performances are comparable to the levels of understanding determined by their verbal performances within some, but not all, problem categories. This paper overviews the developed Combined Model of Understanding and the mapping procedures used in the analysis of collected qualitative data. The paper then describes the results of the study, the implications of these results, and recommendations for further research. 388. {sno:there} K. Snook. The relationship between students' written displays and their conceptions: A study of first year calculus students' understanding of the derivative. volume ?, pages ???-???, ?, 1997. International Group for the Psychology of Mathematics Education, North American Chapter (PME-NA). This research report summarizes the results of a study designed to investigate first year calculus students' understanding of the derivative. The focus of the study was threefold: (1) to identify levels of subjects' displayed understanding of the derivative, (2) to characterize subjects' conceptions of the derivative and the levels of understanding of the derivative these conceptions indicate, and (3) to investigate the relationship between subjects' displayed understanding of the derivative and the understanding of the derivative indicated by their conceptions. 389. {sno:towar} K. Snook. Toward accurately assessing students' understanding in calculus. volume ?, pages ???-??? The Association of Research in Undergraduate Mathematics Education, 1998. Accurately assessing students' understanding of mathematical concepts is a difficult task. In 1997, during a study of first year calculus students' understanding of the derivative, I developed a model to assess students' understanding of mathematical concepts. The model uses Concept Development, Problem Approach, Knowledge Use, Connections, Reasoning, Procedural Problem Solving, and Conceptual Problem Solving as components to determine Low, Intermediate, or High levels of understanding. I conducted the original study at a site where calculus instructors taught in a traditional and lecture format. I used both quantitative and qualitative analysis. I found that students in these traditional calculus classes exhibited higher levels of understanding of problems categorized as Traditional and Algorithmic than problems categorized as Non-Traditional and Relational both on a written exam instrument and during interviews. At the completion of my study I questioned how variations in classroom activities and assessment instruments would impact on students' levels of understanding. An implication of my study was that results of assessments that use only problems categorized as Traditional and Algorithmic or only problems categorized as Nontraditional and Relational may not be indicative of a student's level of understanding. To obtain a more comprehensive assessment of students' understanding, a variety of problem types and presentations should be used. Additionally, information about student understanding must come from a variety of sources. In the actual study, a talk-aloud problem solving interview allowed students more opportunity to reveal their depth of understanding than did the written instrument, but this technique is difficult as an assessment tool in the classroom. I felt that if students were exposed to many different types of problems both in class and on tests, and if they had opportunities to discuss and verbalize solutions, then it would be possible for instructors to ascertain from these activities an accurate picture of a student's level of understanding. I have extended this research into my practice as a teacher and supervisor in the freshmen calculus course at the United States Military Academy (USMA). At USMA the calculus course has incorporated interdisciplinary applications for several years. Small classes are taught in an interactive manner with a focus on applications of the mathematics under study. I have emphasized expanded use of applications and varied assessment of students' understanding. Applied classroom problems, computer mini-projects and major projects are assessed components of the course (some graded and some ungraded). The course uses a ``reform'' calculus text, and calculus instructors develop supplemental applied problems and projects used in the course. In-class activities and discussions provide instructors with a close equivalent to ``talk-aloud problem solving'' in order to assess students' levels of understanding. Outside of class projects offer students an opportunity to use mathematical modeling and scientific computing to solve an applied problem and analyze their solution. Students submit a written report and instructors assess the quality of students' analyses, as well as their mathematical solutions. Assessment of these types of projects is fruitful in determining students' level of understanding. Exams consist of both procedural and conceptual problems. Problems are designed with possible inferences about students' understanding in mind. I have been able to use the results of these assessment activities along with my previously developed model of understanding to determine students' levels of understanding of mathematical concepts. In my final paper and talk I plan to share samples of the in-class and out of class activities and link students' work to levels of understanding on the model of understanding. I also plan to discuss the development of the various types of assessment activities as they evolved in an effort to ensure they would provide accurate information about students' understanding. 390. {sno-thesis:aninv} K. G. Snook. An investigation of first-year calculus students' understanding of the derivative. PhD thesis, Boston University, 1997. This study was designed to investigate first year calculus students' understanding of the derivative. The focus of the study was threefold: (1) to identify levels of understanding of the derivative indicated by subjects' written performances on derivative problems, (2) to identify levels of understanding of the derivative indicated by subjects' verbal performances on derivative problems during talk-aloud interviews, and (3) to investigate the relationship between these two indicators of subjects' understanding levels. A ten problem Derivative Test was designed and administered to 225 college calculus students. Results of this written Derivative Test were used to identify subjects' levels of understanding of the derivative as presented in 'freshmen' or 'elementary' calculus. Five subjects participated in two individual interviews in which they re-solved the ten Derivative Test problems in a 'talk-aloud' manner. Interview data were analyzed using mapping techniques. The levels of understanding indicated by subjects' verbal performances during the interviews were then determined using the developed Combined Model of Understanding. Synthesis of the analyses of the two performances determined the existence of relationships between the levels of understanding of the derivative indicated by subjects' written and verbal performances. Each problem on the Derivative Test was categorized by type of derivative interpretation; geometric, physical, algorithmic, or relational, and by manner of presentation; traditional or nontraditional. Relationships between subjects' performances depended upon the type and presentation of derivative problems. It was concluded that the levels of understanding determined by subjects' written performances are comparable to the levels of understanding determined by their verbal performances within some, but not all, problem categories. The study recommends further research to investigate and evaluate the value of the Combined Model of Understanding as a framework for describing students' understanding of the derivative and other mathematical concepts. The study also recommends that investigations of relationships between students' written and verbal performances be conducted at sites where calculus reform materials or pedagogy are used and students are exposed to a wider variety of types and presentations of problems. Research aimed at designing written instruments which accurately assess students' understanding across a variety of problems is also recommended. 391. {sol:actos} M. E. Solis. Actos visuales y analiticos en el entendimiento de las ecuaciones diferenciales lineales. PhD thesis, Univercity of Mexico, Mexico, 1996. To be added later. 392. {sol:learn} A. Solow. Learning by discovery and weekly problems: Two methods of calculus reform. Primus, 1(2):183-197, 1991. Discusses and provides sample lessons of learning by discovery and weekly problem sets, which are presented as alternative methods for teaching college calculus. Both approaches stress conceptual understanding and guide the students to explore the ideas of calculus in small groups in a computer laboratory setting. 393. {sot:techn} H. Soto-Johnson. Technological vs. traditional approach in conceptual understanding of series. PhD thesis, University of Northern Colorado, 1996. The primary purpose of this quantitative/qualitative study was to study the effects of three calculus teaching methods on the conceptual understanding of series. The teaching methods were (a) Project CALC, (b) Revised Illinois Project, and (c) the traditional calculus course. A secondary purpose was to research students' attitudes towards calculus and the use of technology in the calculus classroom based on the three teaching methods. Both of the Calculus Reform projects, Project CALC and the Revised Illinois project, incorporated Mathematica labs into the classroom. An ANOVA ([$\alpha$] =.05) was used to test mean differences in the post-test and post-attitude survey. There was not a statistical difference in students' conceptual understanding of infinite series or students' attitudes towards calculus among the three teaching methods. The students from the traditional course had significantly better computation skills than the Project CALC students. The students from the traditional course had a significantly lower attitude towards the use of technology in learning calculus. All interviewed students thought calculus was applicable but only the Project CALC students felt they had been exposed to real world application problems. All students thought technology was an important tool in learning calculus but the Revised Illinois project students did not think it was used effectively. The Project CALC students and instructors asked more conceptual questions during class than the other two teaching methods. These students also had a better 'team' relationship with other students in the classroom and the instructor. Overall the Project CALC students had a better understanding of the importance of series of functions. They understood the concept of derivative and could explain how it is almost the inverse of integration. They were able to relate differentiation, integration, and series to one another. The students from the Revised Illinois project and the traditional course did a better job of explaining that the rate at which a series decreases determines whether a series converges. 394. {spe-wal:catwa} B. Speiser and C. Walter. Catwalk: First-semester calculus. Journal of Mathematical Behavior, (2):135-152, 1994. Describes the use of time-lapse photographs of a running cat as a model to investigate the concepts of function and derivative in a college calculus course. Discusses student difficulties and implications for teachers. 395. {spe-wal:const} B. Speiser and C. Walter. Constructing the derivative in first semester calculus. In Proceedings of the 16 th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, volume ?, pages ???-???, Baton Rouge, Louisiana, 1994. International Group for the Psychology of Mathematics Education. North American Chapter. To be added later. 396. {stac-macg-pme-97} K. Stacey and M. MacGregor. Multi-referents and shifting meanings of unknowns in students' use of algebra. In E. Pehkonen, editor, Proceedings of the 21st International Conference of the International Group for the Psychology of Mathematics Education, volume 4, pages 190-198, Gummerus, Finland, 1997. 397. {MAA3} Lynn-Arthur Steen, editor. Undergraduate Mathematics Education in the People's Republic of China. Report of a 1983 North American Delegation. MAA Notes Number 3. Mathematical Association of America, Washington, DC, 1984. Presents the observations and opinions of a delegation of North American mathematics educators who visited the People's Republic of China at the invitation of the Chinese Mathematical Society to discuss with Chinese mathematics teachers issues of mutual interest about university mathematics education. The delegation visited six cities and met with research mathematicians, university teachers, officials of the Ministry of Education and middle school teachers. The report addresses the structure of education in China, undergraduate curriculum, applications of mathematics, computer science, special programs, administration and personnel, and mathematics examinations. 398. {cog-conflict-aera-88} Leslie P. Steffe, editor. The Role of Inconsistent Ideas in Learning Mathematics. University of Georgia, Athens, Department of Mathematics Education, 1988. Collection of papers presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 5-9, 1988); ERIC Acc. No. ED293725. Four research papers and a discussion piece on the role of inconsistent ideas in learning mathematics: 1. ``Cognitive Conflict in Procedure Applications" (Merlyn Behr and Guershon Harel); 2. ``Inconsistencies in Preservice Elementary Teachers' Beliefs and about Multiplication and Division" (Anna O. Graeber and Dina Tirosh); 3. ``The Relationship of Students' Definitions and Example Choices in Geometry" (Patricia S. Wilson); 4. ``Inconsistencies in the Learning of Calculus and Analysis" (David Tall); 5. Discussion paper: ``Cognitive Conflict in Learning Mathematics" (Leslie P. Steffe). 399. {ste-gra:theef} J. Sterling and M. W. Gray. The effect of simulation software on students' attitudes and understanding in introductory statistics. Journal of Computers in Mathematics and Science Teaching, (10):51-56, 1991. The authors investigated students' statistical understanding as a result of their interaction with simulation software. Two sections of an introductory course were used in the study; one section served as the control group and the other served as treatment group. The researchers used examination questions to measure student achievement in each group and found that the experimental class scored significantly higher than the control group on exam questions about concepts covered by the software. 400. {ste:using} A. Sterrett, editor. Using Writing to Teach Mathematics, volume 16 of MAA Notes. The Mathematical Association of America, 1992. The Table of Contests follows: Mathematicians Write, Mathematics Students Should, Too, Ann K. Stehney Writing for Educational Objectives in a Calculus Course, Sandra Z. Keith Writing in Mathematics: A Plethora of Possiblilites, Timothy Sipka A Reply to Questions from Mathematics Colleagues on Writing Across the Curriculum, Emelie Kenney Writing in Mathematics at Swarthmore: PDCs, Stephen B. Maurer A Writing Program and its Lesson for Mathematicians, Ann K. Stehney Writing in the Math Classroom; Math in the Writing Class (Or How I Spent My Summer Vacation), Thomas W. Rishel A Writing Intensive Mathematics Course, Arthur T. White Writing, Teaching, and Learning in Mathematics: One Set of Experiences, Richard J. Maher Technical Writing for Mathematics Projects, J. Douglas Faires and Charles A. Nelson But This is Not an English Class, Andr/'e Michelle Lubecke You Can and Should Get Your Students to Write in Sentences, Melvin Henriksen Three R's for Mathematics Papers--'Riting, Refereeing, and Rewriting, Thomas Q. Sibley Attempting Mathematics in a Meaningless Language, Martha B. Burton Using Expressive Writing to Support Mathematics Instruction: Benefits for the Student, Teacher, and Classroom, Barbara Rose Rewriting our Stories of Mathematics, Linda Brandau Writing in Mathematics: A Vehicle for Development and Empowerment, Dorothy Buerk Two Perspectives on a Writing Intensive Course in Operations Research, Mary Margaret Hart McDonald and Coreen Mett A Writing Fellows Program Meets an Abstract Algebra Calss: The Instructor's and the Fellow's Perspective, John O. Kiltinen and Lisa M. Mansfield Writing Abstracts as a Means of Review, David G. Hartz Journals and Essay Examinations in Undergraduate Mathematics, Gary L. Britton Weekly Journal Entries--An Effective Tool for Teaching Mathematics, Louis A. Talman Writing Assignments and Course Content, Joanne E. Snow Library and Writing Assignments in an Introductory Calculus Class, John R. Stoughton Teaching Mathematics Within the Writing Curriculum, David T. Burkam Writing About Proof, Keith Hirst Using Writing to Improve Student Learning of Statistics, Robert W. Hayden Integrating Writing into the History of Mathematics, Dorothy Goldberg Writing to Learn and Communicate Mathematics: An Assignment in Abstract Algebra, Anne E. Brown Writing in a Non-Euclidean Geometry Course, Richard S. Millman The Essay as a Cognitive Map, James V. Rauff 401. {ste:facto} C. S. Steward. Factors influencing enrollment in mathematics courses by college freshmen. PhD thesis, Texas A&M University-Commerce, 1997. Purpose of the study. The major purpose of this study was to identify specific elements of mathematics programs which had a positive or negative influence on the enrollment of college freshmen in mathematics classes. A secondary purpose of this study was to produce a register of potential approaches in the development of mathematics programs on the basis of positive and negative incidents recorded. Procedure. The Critical Incident Technique was utilized in this study. Responses were classified as either upper group or lower group according to standardized college entrance scores. There were 183 responses from the lower group and 168 from the upper group. Findings. The findings established that teacher personality and traits is the single most influential element on enrollment in mathematics classes. Students consider teachers that are thorough and encouraging as effective and those that are strict or lacking control in the classroom as ineffective. High school course selection is also important with more positive experiences occurring in Algebra I, Algebra II, and Trig/Pre-Calculus than in Geometry. Conclusion. Based upon the findings, the following conclusions are advanced: (1) Teacher personality and traits is the single most important factor influencing enrollment in mathematics classes. (2) The applicability, grade received in a course, and the opportunity to peer tutor, are more important to high school students than to their elementary counterparts. (3) School transfer within the school year is remembered by elementary students as being significant more often than by high school students. 402. {tall-stew-mis-79} Ian Stewart and David Tall. Calculations and canonical elements: Part i. Mathematics in School, 8(4):2-4, 1979. ERIC Acc. No. EJ213334. The author argues that the idea of canonical elements provides a coherent relationship between equivalence relations, the basis of modern approaches to too many mathematical topics, and the traditional aspect of computation. Examples of equivalent relations, canonical elements, and their calculations are given. 403. {str:thein} M. L. Stribling. The influence of mathematics background and continuity of coursework on success in college calculus. PhD thesis, University of Kentucky, 1995.