A supplement to a news article.
Students' belief that a larger number has more factors is outlined as a particular example of "the more of A, the more of B" intuitive rule. Robustness of this belief among pre-service elementary school teachers is discussed by demonstrating students' tendencies to perceive conflicting evidence as an exception to the rule. In the presentation I will argue the importance of instructor's awareness of students' potential misconceptions. I will suggest pedagogical approaches that confront students' popular beliefs and attempt to deepen students' understanding of the relationship between natural numbers and their factors.
The results of a study that investigated the nature of the processes of knowledge construction, organization, and reconstruction by undergraduate students enrolled in a developmental algebra course are described. Students' concept maps were used to document the processes by which the most successful and least successful students constructed, organized, and reconstructed their knowledge and to provide evidence of how students integrated new concepts and skills into their existing conceptual frameworks. Profiles of the students characterized as most successful and least successful were developed based on analysis and interpretation of triangulated data. Successful students organized the bits and pieces of new knowledge into basic cognitive structures that remained relatively stable over time. Each student's basic structure gradually increased in complexity and richness, with new knowledge assimilated into or added onto the existing structure. Students who were least successful constructed cognitive structures which were subsequently replaced by new, differently organized structures lacking interiority and essential linkages to other related concepts and procedures.
This study investigated the mathematical behaviors and beliefs of mathematics graduate students. These students' mathematical behaviors were observed as they completed complex mathematical tasks, and their beliefs were assessed using the Views About Mathematics Survey (VAMS). The students in this study exhibited and reported exceptionally strong persistence, high mathematical confidence, and an expectation of sorting out information on their own when attempting complex mathematical tasks. When observing these students during problem solving, they were at times ineffective in monitoring their problem solving attempts and frequently attempted to classify problems as a familiar type practiced or learned in class. Although not always correct in formulating their responses, the solution approaches offered by these students always appeared to have a logical foundation. The combined qualitative and quantitative analyses revealed numerous complexities in applying one's mathematical knowledge in unfamiliar settings and new insights into the interaction of cognitive and affective variables during problem solving.
The purpose of this research is to examine how undergraduates' structural/conceptual orientation of variables influences their use of them in formulas and equations in interpreting algebraic situations. The goal of our study is to acquire deeper insights into undergraduates' overall understanding of variables. The subjects are pre-service KB8 teachers at a mid-western university who have chosen to specialize in mathematics. Data consist of the responses to problems given early in the semester and accompanying interviews. Although the algebraic expressions used in these problems were relatively simple, such as n + 2, their use in an unfamiliar context provided a basis to investigate to what degree students were able to deal with these expressions as conceptual entities in their own right. We found that students struggled in their efforts to make sense of the problem when not able to consider the algebraic expressions as conceptual entities.
In his research, Rasmussen (1998) proposes the "function as solution dilemma" to describe "students' difficulties with the function concept and [their difficulties with] the leap from unknown as number [in algebraic equations] to unknown as function [in differential equations]." Our research seeks to expand current knowledge of students' concept images of a differential equation and its solutions. Although many of the students in our study certainly had difficulties with the function concept, we did not find any student who did not ever consider the solutions to differential equations as functions. We describe a structure for characterizing the many understandings students do exhibit about solutions of DEs in a variety of contexts.
This ongoing, small-scale investigation, conducted at several universities, examines views of proof held by students that we characterize as 'deductive theorem-proof' and 'inductive conjecture-verification.' The undergraduates mathematics majors in our study had been required to write mathematical proofs in some prior courses. Participants, working in groups of three or four, were asked to prove three mathematical statements. Computer software, such as Geometer's Sketchpad, was available. Group work sessions were videotaped. Later, individuals viewed segments of the group video. We asked about their recollections, explanations, interpretations, and reflections of group activities during the problem session and in previous mathematics courses that used collaborative group work. Some groups, despite prior study of rigorous proof, chose a 'conjecture-verification' approach using the computer as their main tool; others adopted a formal 'theorem-proof' strategy and did not use the computer, or used it sparingly to increase their confidence in the truth of their conjectures.
The purpose of this study was to determine the impact of writing assignments on students' understanding of the limit concept. The study involved two sections of second-semester calculus (n = 37, 34). The treatment group completed six writing tasks focusing on the concept of limit, replacing some of the problem sets that the instructor would have normally assigned. The control group did not write but handed in problem sets more frequently. The investigator used the writing assignments, as well as interview transcripts from a subset (n = 5) of the treatment group to assess cognitive growth. This qualitative assessment was based on action-process-object-schema (APOS) theory. Quantitative analysis was performed on three computational problems common to the two sections' final examination. The students that performed the writing tasks seemed to increase their conceptual understanding of the limit concept. Furthermore, differences in two of the three final examination problem scores significantly favored the treatment students.
This study focuses on the effect of homework assignments incorporating reflective tasks on student understanding of reform calculus. The Fall 1997 study used quantitative (n = 25, 18) and qualitative (n = 7) techniques. Assignments included comparing and contrasting ideas, developing concept maps relating course material, and writing strategies regarding specified tasks. Examination scores were analyzed using analysis of covariance with pretest achievement score as covariate. "Think aloud" problem sessions were conducted with selected students and analyzed by category of thought, providing detail unavailable otherwise. No significant differences were determined on examinations. Inspection of regression lines of examination scores revealed an interaction between treatment and precalculus achievement. An increasing number of students appeared to benefit as semester progressed, which was supported in "think aloud" problem sessions. Students began to exhibit previously absent categories of reflective thinking and to use more repetition and variety in their categories by semester end.
This talk proposes to give an example of a study designed on a recent extension of APOS Theory. Clark et al. (1997) proposed the use of Piaget and Garcia's triad mechanism to describe the development of a schema, specifically with respect to students' understanding of the chain rule. My dissertation study was designed to collect data to test this theory. My results are positive in the sense that the triad description fits with my data. I present more detailed descriptions of the intra-, inter-, and trans- levels of the development of the chain rule schema than were given in Clark et al. Also, general descriptions of how the triad mechanism influences the design of research studies are presented. Specifically, the need to get observations of a subject relating to situations that deal with not only the depth of the concept, but also the breadth of the concept.
In two previous studies we investigated the non-routine problem solving abilities of students just finishing their first year of a traditionally taught calculus sequence. The two groups studied were those who had a C in first term calculus and those who had either an A or B. In this paper we report on a similar study, using the same non-routine first year calculus problems, with students who had completed the traditional calculus sequence and were in the midst of an ordinary differential equations (DE) course. More than half of the DE students were unable to solve even one problem and more than a third made no substantial progress toward any solution. The routine test of associated algebra and calculus skills indicated that most of the students were familiar with the key concepts for solving the non-routine problems but many were unable to access or capitalize on that knowledge.
This study used task-based interviews to explore first-semester calculus students' methods of recognizing the same local function behavior in graphs, tables and formulas. The purpose was to establish a context in which to later evaluate their developing understanding of limits. Ten students were given formulas of 5 functions, all undefined at x = 3, but with different behavior nearby. From among 8 graphs and 8 tables, students matched a graph and a table to each function, providing explanation for one function. A 4-item questionnaire asked about preferences for graphs, tables, or function formulas. These students relied on ordered pairs in matching graphs and tables to formulas, used only 2 pairs of the graph-table-formula triad, and seldom compared functions to one another. Hence these students did not immediately associate a function with its local behavior.
In this paper, students' cognitive construction of a schema were examined via the Action-Process-Object-Schema [APOS] theoretical perspective. Data consisted of these extensive interviews with students who had completed at least two semesters of calculus. The complexity and the non-routine nature of the problem that students attempted to solve required them to rely on everything they had learned on graphing functions in calculus. In order to cogently describe the student responses, we examined two important schema the students were using. It naturally followed that the interaction of these two schema was critical. Therefore, there was a two-dimensionality in what we called their overall "Calculus Graphing Schema". The calculus graphing problem studied required students to integrate the properties of the graph of the function with each other, as well as across contiguous intervals. The triad of schema development - intra, inter, and trans- aptly describe the data with respect to two dimensions. One dimension is the "Property Schema" and the other is the "Interval Schema". Additionally, a number of problems were demonstrated by students consistently throughout and these problems are discussed in some detail.
This study explored first-semester calculus students' developing understanding of limits, relative to their function knowledge and graphing calculator use. The purpose was to identify factors influencing their understanding of limits, and hence begin to build a model of the development of the limit concept. Over 4 task-based interviews, 10 students progressed from examining local function behavior to conjecturing the existence or non-existence of specified limits, based on tables or graphs, in increasingly difficult situations. Written and oral responses suggest that these students could read a table or graph to make a limit conjecture, but had difficulty determining whether a table or graph reflected the function's true local behavior, and did not consider this unless directed to do so. This partially stemmed from an inability to accept limitations of the graphing calculator, and partially from weaknesses in their function knowledge.
Preservice teachers should have an explicit understanding of each of the concepts they will teach. Our goal is to articulate an explicit understanding of the composition of functions and provide ways to determine whether a preservice teacher has gained an explicit understanding. We use Vinner's (1983) framework to define an explicit understanding of a concept as an agreement between a person's concept image of the concept and the community's accepted definition of the concept. Two things are vital for an explicit understanding of composition of functions: the ability to move between object and process perspective (Sfard, 1991), and certain content knowledge such as the modern definitions of domain, range, univalance, function, and composition of functions. We have developed a set of performance indicators by which a preservice teacher can demonstrate explicit understanding of the composition of functions, along with assessment strategies which provide ample opportunities to exhibit these indicators.
This poster will describe research work that investigates the effects of instruction using the Cooperative Learning Method as opposed to instruction using the Traditional Learning Method on students' understanding of the numerical experience associated with the global behavior of polynomial functions. Two classes are compared using three instructional methodologies. The effectiveness of these methods is compared based on quantitative measures of student performances, adjusted for differences in student preparation in college algebra.
In two previous studies we investigated the non-routine problem solving abilities of students just finishing their first year of a traditionally taught calculus sequence. The two groups studied were those who had a C in first term calculus and those who had either an A or B. In this paper we report on a similar study, using the same non-routine first year calculus problems, with students who had completed the traditional calculus sequence and were in the midst of an ordinary differential equations (DE) course. More than half of the DE students were unable to solve even one problem and more than a third made no substantial progress toward any solution. The routine test of associated algebra and calculus skills indicated that most of the students were familiar with the key concepts for solving the non-routine problems but many were unable to access or capitalize on that knowledge.
Indicators of Quality Undergraduate Mathematics Education is a NSF-funded project designed to produce a national set of statistical indicators for monitoring the condition of the first two years of undergraduate mathematics education. The indicators under development stem from investigation of five levels: department/institution, curriculum, faculty, classroom, and students. Sub-investigation sites include mathematics departments at a comprehensive state university, a community college, and a large research university. Each of the sites is conducting its own data collection, analysis and reporting, as well as taking part in the cross-institutional data collection activities. Employing mixed methodologies, data collection includes institutional databases, departmental profiles, faculty and student questionnaires, classroom observations, focus groups, and analysis of both texts and assignments. Quantitative data are examined in the light of information provided by the various aspects of the case studies (focus groups, textbook analysis) in order to find those measures that succeed in helping to differentiate between qualitatively different instructional situations.
Following up on survey research which found a positive correlation (r = .67, n = 97, p = .001) between how interesting and how surprising students found true statistical statements, the author conducted what turned about to be a year-long case study using students selected via purposeful "typical case" sampling to explore the various roles surprise plays and its relation to motivation. Examples of these roles and interpretation will be provided for discussion.
This presentation discusses the impact Calculus 1 students' conceptualization of functions, specifically composite functions, has on the students' interpretations of and facility with the chain rule. The primary questions to be addressed are: (a) What are the definitions that these students would be willing to accept and what definition do they consider the best?; (b) What facility do the participants reveal when presented with composite functions in a variety of settings?; (c) How do the participants conceptualize the composition of two functions?; (d) What facility do the participants reveal when presented with problems necessitating the use of the chain rule in a variety of settings?; (e) How do the participants conceptualize the chain rule?; and (f) What trends can be drawn from the responses with respect to conceptualization of and facility with composite functions and how do these influence both conceptualization of and facility with the chain rule?
Current reform efforts in undergraduate mathematics education ask instructors to engage students in the classroom with content that is meaningful. This study followed an instructor through her first semester of attempting to implement reform in a college algebra class for "at-risk" students. Data analyzed included classroom observations and videotapes, and a journal the course instructor kept. Results from these data confirmed that the instructor struggled with (a) balancing "content coverage" and student involvement and (b) learning to use challenging content to drive student collaboration. Recommendations included increasing the types of and availability of instructional development and other support opportunities.
College students enrolled in either a remedial algebra course or a non-remedial precalculus course completed the Composite Math Anxiety Scale in order to provide a mathematics anxiety score. The students also rated a list of ten coping strategies with regard to frequency of use and helpfulness. Mathematics faculty and counselors rated the same strategies, but only in terms of helpfulness. A multi variate analysis of variance (MANOVA) was performed on the student data. The three independent variables were mathematics anxiety, gender, and course level. The dependent variables were the ten coping strategies. Low mathematics anxiety students valued the majority of coping strategies more than high anxiety students. Mathematics students, faculty, and counselors agreed in rating the helpfulness of most strategies. Completing homework assignments on time, informing your instructor if you don't understand the course material, setting aside extra study time before exams, and asking questions in class received the highest ratings.
This study was designed to investigate the effect of explicit instruction in expert-like retrieval on students' spontaneous retrieval and successful application of an analogue. Eleven students in the second year of a four-year teacher education program participated in the study. The students averaged 21 years of age with five enrolled as mathematics majors and six enrolled as middle-years specialists. All students were enrolled in a mathematics education class that explored the teaching of mathematics from a problem-solving perspective. The students were given instruction that served as an analogue for two tasks that were presented one and two weeks later. The two similar tasks were administered a week apart with 91% of the students retrieving the analogue and successful solution rates of 73% and 55% respectively. When the students were given another, unrelated task by another instructor, the students' use of retrieval was now only about 27% with a 0% success rate. The results attest to the potential influence of context on retrieval. About three weeks later 100% retrieved an analogue with 45% retrieving the analogue associated with the unrelated task. The success rate was again 0% perhaps indicating the importance of a general strategy (retrieval) and the close partnership necessary with specialized knowledge.
Theories of college students' intellectual development state that students begin with narrow, black-and-white, uncomplicated views of the world but slowly develop more complex, contextual, shades-of-gray views (Baxter Magolda, 1992; Belenky et al., 1986; Perry, 1970). I present the results of two qualitative studies which adapted these theories on "ways of knowing" into descriptions of "ways of knowing mathematics" that help explain students' intellectual understandings about mathematics. By increasing our understanding of students' beliefs about mathematical knowledge ("What am I studying?"), students' habits in learning mathematics ("How should I study it?"), and students' practices in checking mathematical truth ("How do I verify it?"), we can improve our classroom practices in mathematics teaching and enrich our theoretical perspectives on mathematics learning.