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Part II. Additional recommendations concerning:
A: Students taking general education or introductory collegiate courses in the mathematical sciences
General education and introductory courses enroll almost twice as many students as all other mathematics courses combined . They are especially challenging to teach because they serve students with varying preparation and abilities, many of whom have had negative experiences with mathematics. Perhaps most critical is the fact that these courses affect life-long perceptions of and attitudes toward mathematics for many students’and hence many future workers and citizens. For all these reasons these courses should be viewed as an important part of the instructional program in the mathematical sciences. This section concerns the student audience for these entry-level courses that carry college credit. An important resource for discussions about these courses is Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus, published by the American Mathematical Association of Two-Year Colleges and available in its entirety on the Internet.
Mathematical sciences departments should ensure that all students meeting general education or introductory requirements in the mathematical sciences are enrolled in courses designed to
General Introductory Courses
At Princeton University, the Math Alive course is designed for those who haven't had college mathematics but would like to understand some of the mathematical concepts behind important modern applications. It consists of largely independent 2-week units in cryptography, error correction and compression; probability and statistics; birth, growth, death and chaos; geometry and motion control; and voting and social choice. Each unit is divided into two parts. For each part students can download lecture notes in pdf or ps format. Each part has a problem set and a corresponding online lab. Links for the lecture notes, online labs, and problem sets through corresponding links are on the course website. Each problem set is posted on the web one week before it is due. Solutions to the problem are available on the web after the submission deadline. The syllabus describes class topics, lists due dates and posts problem sets and labs.
A widely-used text for a similar but somewhat less mathematically demanding course is For All Practical Purposes by COMAP. The companion website contains extensive resources for students and teachers.
At The University of Texas at Austin, the mathematics course developed for the liberal arts honors program is designed to present ’culturally significant and beautiful concepts with concomitant emphasis on potent strategies of discovery and exploration.â? The course presents infinity, the fourth dimension, geometric gems, topology, coincidence, chaos, fractals, and other topics. Each topic is intended to illustrate the process of starting with a simple observation and applying techniques of effective thinking that lead to the creation of new ideas, such as making errors and learning from them, breaking complicated questions into simple components, understanding simple things deeply, and finding the essence of an issue. One project directs students to take a non-mathematical issue they care about and apply the methods of analysis, not the mathematics itself, to analyze the issue or produce a creative work about it. The text used is The Heart of Mathematics: An invitation to effective thinking, by E. Burger, and M. Starbird.
The introductory course in Contemporary Mathematics at Virginia Commonwealth University is taken by more than 2000 students each year. It uses the text Excursions in Modern Mathematics Peter Tannenbaum and Robert Arnold and includes topics such as voting and fair division, applications of graph theory/networks, population growth, symmetry, and fractal geometry. In the context of these applications of mathematics, students strengthen their algebraic and graphing skills. The course serves as a prerequisite for the statistics course that is required in most humanities and social science degree programs. Students take three exams and four quizzes; write two papers; participate in making a group presentation; make a poster session presentation; turn in a dozen in-class/homework worksheets; and respond to weekly prompts in a Learning Log. Grades are based on tests (30%), quizzes (20%), presentations (20%), papers (20%), worksheets etc. (20%). Students who complete a learning log may drop the lowest 10% of the grades for their assignments. A detailed instructor's guide, discusses the use of writing-to-learn, group projects, independent study projects, poster sessions and other approaches that expect active student engagement.
Mount Holyoke College offers a variety of courses to incoming students not enrolling in a calculus sequence. The introductory ’explorationsâ? in algebra, number theory, geometry, and fractals and chaos offer a way for students to begin their study of mathematics. These courses emphasize mathematics as an art and as a way of seeing and understanding. The explorations presuppose neither special talent for nor prior strong interest in mathematics. They intend to awaken interest by demonstrating either the pervasiveness of mathematics in nature and its power as a tool that transcends disciplines or its qualities as an art that brings aesthetic pleasure to the participant. Another alternative for students is an interdisciplinary case-study course in quantitative reasoning.
Resources that can be used to introduce students to contemporary topics in general education courses include the AMS website What’s New in Mathematics and PLUS magazine, an Internet magazine from the United Kingdom, which aims to introduce readers to the beauty and the practical applications of mathematics. A number of articles in the MAA journal Math Horizons are also appropriate for a general undergraduate student audience.
Precalculus ’ New Approaches
Based on her workshop program in calculus, Nancy Baxter-Hastings has developed a workshop program in precalculus, designed to eliminate the distinction between classroom and laboratory work. Her text is Workshop Precalculus: Discovery With Graphing Calculators. The method alternates between three primary components: summary discussions, introductory remarks, and collaborative activities. Students are expected to learn by working in groups, discussing problems as a class, and writing individual reports. The book encourages students to do initial computations and symbolic manipulations by hand in order to understand how results are produced, turning to calculators once they understand a concept. The book offers the following suggestions for workshop instructors: take control of the course, keep the class roughly together, allow students to discover, promote collaborative learning among students, encourage students’ guessing and development of intuition, lecture when appropriate, have students do some work by hand, use technology as a tool, be proactive in approaching students and give them access to ’rightâ? answers, provide plenty of feedback, stress good writing, implore students to read well, and have fun!
Precalculus with Applications, based on Functioning in the Real World: A PreCalculus Experience by Sheldon Gordon et al., is taught at Farmingdale State University of New York. It is designed to prepare students for calculus as well as for quantitative courses in the natural and social sciences. The course introduces students to the fundamental families of functions using contextual, tabular, graphical, and algebraic representations. A common theme is the notion of fitting functions to real-world data. Each family of functions is introduced in context and the emphasis throughout is on realistic applications. Matrices and their use in solving systems of linear equations are also introduced, as are the notion of recursion and applications via models involving difference equations. The course requires three class tests, a series of three individual investigatory projects (which count as equivalent to two class tests), and a cumulative final exam.
The Precalculus Weblet consists of an online textbook, syllabi, homework, and exams developed by the members of The Washington State Board for Community College Education. It contains links for exploring precalculus concepts using current information on the Internet. All the material on the website is freely available for personal use.
In 1988 Moravian College replaced the traditional 2-term Precalculus-Calculus I sequence with a one-year course, Calculus I with Review. The course addressed the problem of making calculus accessible to students with weak algebra and problem-solving skills. The idea was that by providing conceptual background and discussing specific algebra techniques just prior to introducing a calculus topic, the students are motivated to understand the usefulness of the techniques and immediately apply them to calculus problems. The developers of the course believed that it is important to have good supplemental material, and to this end they prepared a text, A Companion to Calculus, which can be used with scheduled or informal tutoring sessions or as a supplement for individual study. Chapters are keyed to primary topics in any first calculus course, and the text approaches all concepts in four ways: descriptive (verbal and written), symbolic, numeric, and graphic. With assistance from the Fund for the Improvement of Post-Secondary Education (FIPSE), the Moravian project team mentored a number of other institutions in creating similar courses. In fact the idea of integrating precalculus material into calculus course became so successful that textbooks have been written specifically for such a course. Two of these are Calculus 1 With Precalculus: A One Year Course by Ron Larson, et al., and Integrated Calculus: Calculus with Precalculus and Algebra by Laura Taalman.
An example of a calculus course that integrates calculus and precalculus and places special emphasis on active learning is the Workshop Calculus Program. The textbooks Workshop Calculus: Guided Explorations with Review, vol. 1 and vol. 2, and Workshop Calculus with Graphing Calculators Guided Exploration with Review, vol. 1 and vol. 2, developed by Nancy Baxter-Hastings, Dickinson College, seeks to help students develop the confidence, understanding, and skills necessary for using calculus in the natural and social sciences and for continuing their study of mathematics. Lectures are replaced by an interactive teaching format that does not distinguish between classroom and laboratory work. Students are expected to learn by doing and by reflecting on what they have done, and the instructor is expected to respond to students as they learn.
Introductory Statistics ’ New Approaches
There have been two parallel and complementary strands in the statistics reform movement, commonly referred to as "data-driven" and "activity-based." Texts used by faculty who choose a data-driven, activity-based approach to introductory statistics include the Workshop Statistics series by A. Rossman et al., and Statistics in Action: Understanding a World of Data by A. Watkins et al. Texts that adopt a data-driven approach that can be supplemented with activities include Introduction to the Practice of Statistics by D. S. Moore et al., Statistics: A Guide to the Unknown by J. L. Devore et al., Statistics: The Exploration & Analysis of Data by J. L. Devore and R. Peck., Seeing Through Statistics by J. Utts, Statistical Ideas and Methods by J. M. Utts and R. F. Heckard, and Intro Stats by D. DeVeaux and P. Velleman. A bibliography of statistics textbook reviews is maintained by Ann Cannon, Cornell College.
ARTIST stands for Assessment Resource Tools for Improving Statistical Thinking. ’This website, with support from the National Science Foundation, provides a variety of assessment resources for teaching first courses in Statistics:
1. Assessment Builder: a collection of about 1100 items, in a variety of item formats, according to statistical topic and type of learning outcome assessed. This database can be used to generate files to be edited and manipulated by statistics instructors.
* Information guidelines, and examples of alternative assessments (such as projects, article critiques, and writing assignments)
* Copies of articles or direct links to articles on assessment in statistics. References and links for other related assessment resources. 3. Research Instruments: instruments that may be useful for research and evaluation projects that involve assessments of outcomes related to teaching and learning statistics.
4. Implementation issues: questions and answers on practical issues related to designing, administering, and evaluating assessments.
5. Presentations: copies of conference papers and presentations on the ARTIST project, and handouts from ARTIST workshops.
6. Events: information on past and upcoming ARTIST events.
7. Participation: ways to participate as a class tester for ARTIST materials.â?
The article ’An Activity-Based Statistics Courseâ? by M. Gnanadesikan et al. from the Journal of Statistics Education includes examples of types of activities that work well in various classroom settings along with comments from colleagues and students on their effectiveness. Another source for the activity-based approach is Teaching Statistics: A Bag of Tricks, by A. Gelman and D. Nolan. The software Fathom, developed with support from the National Science Foundation, allows users to type in their own data, to use the over 300 data files that come with Fathom, or to import data from text files or directly from the Internet.
Other useful sources on pedagogy are the two MAA publications Statistics for the 21st Century, edited by S. Gordon and F. Gordon, and Teaching Statistics: Resources for Undergraduate Instructors, edited by T.L. Moore. The latter resource contains seminal articles on statistics education and descriptions of products available for teaching statistics. An online resource, A Sampler of www Resources for Teaching Statistics, was compiled by R. Lock. A useful source on content is the MAA publication Perspectives on Contemporary Statistics, edited by D. Hoaglin and D. Moore. Additional online resources for teachers of statistics include the Journal of Statistics Education, the Data and Story Library, and the SIGMAA on Statictics Education.
College Algebra ’ New Approaches (see Part 2, Section A.2)
Engaging Students in Project Work
Laurie J. Burton, Western Oregon University, reported about a technique used in a general education mathematics survey course aimed at engaging students and improving their communication skills. She incorporated weekly projects by dividing the class into groups of four and requiring the groups to write summaries of their projects On a rotating basis, one student was responsible for the written submission, while the others served as editors. Over the semester each student was responsible for two weekly, typed ’write ups,â? each worth 12.5% of the course grade. In the written submissions the students were required to include an introduction, a statement of assumptions, a rewriting of each problem, a display of all steps the mathematics, and a clear sentence reporting he answer. Burton reported that ’The class started off slowly to say the least! I wrote an extensive set of directions for them, but clearly many of them didn't bother to read their packet! The first three weeks of projects were dismal. Eventually they all sort of clued in and by the end of the term students were turning in really nice projects. Clearly they had learned something. I was really impressed and happy as a teacher that the students were making such clear progress.â? Information about the nature and use of current projects for the course, Introduction to Contemporary Math, is available through a link on her website.
The course Mathematics in Action: Social and Industrial Problems, developed by Morteza Shafii-Mousavi and Paul Kochanowksi, Indiana University, is project based. See the description in Part 1, Section 3.
The University of South Carolina Spartanburg (USCS) developed Project-Based Instruction in Mathematics for the Liberal Arts. The website provides projects and resources for instructors and students who wish to teach and learn college mathematics or post-algebra high school mathematics via project-based instruction. In 1994 a group of faculty members at USCS began to develop and test an innovative pedagogy integrating technology and activity- or project-based instruction in mathematics for liberal arts majors. The group collected, modified, and wrote items for a packet of activities designed to form the core of material that would be used to supplement and eventually replace the textbook in the ’College Mathematicsâ? course. Subsequently, M.B. Ulmer wrote a booklet to lend structure to the use of the activities, which supplanted used of a regular textbook in many sections. Ulmer reports that success rates have risen dramatically for students who have gone through the program and that their subsequent performance in required statistics courses has also shown improvement.
Using the recent anthrax crisis as an example, NSF Director Rita Colwell observed, ’When we have little direct control over our fate, a firm understanding of probability can alleviate some of the stress.â? Colwell's remarks were made at a 2001 forum on quantitative literacy held at the National Research Council and jointly sponsored by the National Council on Education and the Disciplines, the Mathematical Sciences Education Board, and the Mathematical Association of America. The forum's white paper defined quantitative literacy (also called "numeracy") as the "quantitative reasoning capabilities required of citizens in today's information age." Relevant documents include Mathematics and Democracy: The Case for Quantitative Literacy (Steen, 2001) and Quantitative Literacy: Why Numeracy Matters for Schools and Colleges.
The Mathematical Association of America recently established a Special Interest Group on Quantitative Literacy (SIGMAA QL). Information about previous work of the CUPM subcommittee on Quantitative Literacy Requirements is maintained by Rick Gillman, Valparaiso University. The Quantitative Literacy webpage of MAA Online contains links for information a reports concerning quantitative literacy that were formerly located on the website of the National Council on Education and the Disciplines at the Woodrow Wilson National Fellowship Foundation.
In ’General Education Mathematics: New Approaches for a New Millennium,â? Jeffrey O. Bennett, University of Colorado at Boulder, and William L. Briggs, University of Colorado at Denver present some observations regarding the problems of developing appropriate mathematics curricula for non-science, engineering, mathematics (SEM) students, along with recommendations for their solution The authors state that the ways students need mathematics are for college, for career, and for life. When a committee at the University of Colorado examined what mathematics would be appropriate to meet these needs, four content areas emerged: logic, critical thinking, and problem solving; number sense and estimation; statistical interpretation and basic probability; and interpretation of graphs and models. Bennett and Briggs advocate a context-driven approach for instruction in these areas.
Developing Mathematical and Quantitative Literacy across the Curriculum
The University of Nevada, Reno, established a Mathematics Center with a focus on integrating mathematics across the curriculum. The main goal of the Center is to improve the quantitative and mathematical skills of all students, and to help them better appreciate the importance and utility of mathematics. The Center does this primarily by working with faculty in various disciplines to assist them in enhancing the quantitative and mathematical content of their courses, and then providing them and their students with the necessary support. The plan calls for influencing courses ranging from the natural and social sciences to English and the fine arts. It also calls for bringing applications from other disciplines into the elementary mathematics classes. The Core Curriculum is a high priority for the project.
Several institutions have been involved in an NSF program called Mathematics Across the Curriculum. Schools involved include Dartmouth University, RPI, Indiana University, and the United States Military Academy,
Macalester College has established an interdisciplinary program, Quantitative Methods for Public Policy, that involves many different departments in teaching quantitative literacy in the context of public policy analysis. This work is supported by a grant from the Department of Education's Fund for the Improvement of Post-Secondary Education.
Offering Choices to Satisfy a General Mathematics Requirement
Stetson University offers students a wide variety of mathematics courses to complete the mathematics requirement. Courses meeting the general mathematics requirement include Finite Mathematics, Mathematical Game Theory, Chaos and Fractals, In Search of Infinity, Great Ideas in Mathematics, Mathematics and Multiculturalism, Geometry, Introduction to Mathematical Modeling, and Cryptology, as well as the calculus courses.
Goucher College also offers a variety of courses to students completing their general mathematics requirement. Available courses include Topics in Contemporary Mathematics, Introduction to Statistics, Problem Solving and Mathematics-Algebra, Problem Solving and Mathematics-Geometry, Functions and Graphs, Discrete Mathematics, various levels of calculus courses and Linear Algebra.
The Math Lab Program at Francis Marion University is designed to give students access to mathematics across a wide range of entry-level courses and to make it possible for students to work at their own pace. The Math Lab features an individualized format that makes it possible for a student to complete the course in more or less time than the regular semester. However, to succeed in the Math Lab program students must have motivation and self-discipline. Francis Marion strongly recommends that students use the available resources including extra help sessions, extensive mini-lab hours, computer tutorials, videotapes, and instructor office hours. The Math Lab Program offers the introductory courses of College Algebra with Analytic Geometry I, College Algebra with Analytic Geometry II, College Trigonometry with Analytic Geometry II, and Calculus I. All but the first course satisfy the General Education Requirement. The first course does, however, earn credit toward graduation. Syllabi for all courses are located at the site indicated above.
Examples of Introductory Course Syllabi
The following are course syllabi sites where the instructor has not only indicated a schedule, but has also described the objectives, philosophy and environment of the course.
* Georgia Perimeter College: Descriptions and syllabi for 13 introductory courses from pre-algebra to calculus
* Richland Community College: Finite Mathematics
* Mary Washington College: Finite Mathematics with Applications
* Bowling Green State University: A Probability/Activity Approach for Teaching Introductory Statistics
Support for Faculty Teaching Developmental Mathematics
Faculty teaching developmental mathematics courses at various institutions can often feel isolated and may have little information on what is new in the field. A committee of the MAA has started a web page, a mailing list and other activities to support these instructors. Support is also available from AMATYC, which focuses considerable attention on developmental mathematics. See the section on developmental mathematics on their Electronic Proceedings pages.
A great deal of information about teaching an online beginning algebra course is available from Roseanne Hoffman, Montgomery County Community College.
Mathematical sciences departments at institutions with a college algebra requirement should
Refocusing College Algebra
Founded in 1996, the Historically Black College and University (HBCU) Consortium for College Algebra Reform developed the Contemporary College Algebra program. Its purpose is to refocus college algebra to address the quantitative proficiencies that students need for mathematics and other disciplines, society, and the workplace. Thus emphasis is placed on trying to empower students as problem solvers in the modeling sense rather than making them try to master lists of algebraic rules. To support the purpose, the course emphasizes developing communication skills, engaging students in small group activities/projects, using technology for doing mathematics, and trying to build student confidence. Discussions with faculty in different disciplines and with people in the workplace influenced the development of the program. In particular, the heavy emphasis placed on data as well as on graphical and numerical analysis reflects these discussions. Data analysis is used to generate the need for functions, which in turn leads to modeling situations in various disciplines using recursive sequences. Creators of the program believe that the ability to understand elementary data analysis, to extract functional relationships from data, and to model real-life situations mathematically is fundamental to the education of every student. The pedagogical environment is focused on student learning, which includes a strong emphasis on small-group in-class activities and out-of-class projects. Technology is used extensively as part of discovery activities. The program has expanded beyond the HBCU Consortium to include majority schools and tribal colleges.
A conference on College Algebra was sponsored by the HBCU College Algebra Reform Consortium in December 2002. Two articles that resulted from a workshop sponsored by the Consortium are ’College Algebraâ? by Arnold Packer, Johns Hopkins University, ’An Urgent Call to Improve Traditional College Algebra Programsâ? by Don Small, U.S. Military Academy, and ’Who Are the Students Who Take Precalculus?â? by Mercedes A. McGowen, William Rainey Harper College. Conference participants recommended the following as major characteristics of a college algebra program:
* Real-world problem based: a topic is introduced through a real-world problem and then the mathematics necessary to solve the problem is developed. Example problem: Schedule a multi-faceted process.
* Modeling (transforming a real-world problem into mathematics): - using power and exponential functions, systems of equations, graphing, and difference equations ’ primary emphasis is placed on creation of a model and interpretation of the results. Example: Model the stopping time versus speed data presented in a driver’s manual by plotting the data and fitting a curve to the plot. Interpret how well the resulting stopping time function models reality at small speeds. Revise the model, if necessary, to account for zero stopping time at zero speed. Use the (revised) function to predict stopping times for speeds not given by the data. Revise the model to account for different road surfaces.
* Emphasize communication skills: as needed in society as well as in academia ’ reading, writing, presenting, and listening. Example: Students learn how to read, understand, and critique news articles that include quantitative information and to make informed decisions based on the articles.
* Small group projects: involving inquiry and inference. Example: Analyze the soda preference of students by conducting a survey and comparing the results with data from the school’s dining hall or a local fast food restaurant.
* Appropriate use of technology to enhance conceptual understanding, visualization, and inquiry, as well as for computation. Example: ’What-ifâ? a model for paying off a credit card debt by changing the monthly payment, interest rate, size of debt, etc. Plot the results to visually compare the different scenarios.
* Use of hands-on activities rather than all-lecture format.
The Texas Southern Consortium for College Algebra Reform, part of Project Intermath, has two goals: (1) to develop a contemporary college algebra course that educates students for the future rather than training them for the past; and (2) to change the culture surrounding the college algebra program. The primary goal of its contemporary college algebra course is to empower students to become exploratory learners. Most of the topics in the course begin with the analysis of data. The course involves the use of small-group projects developed by interdisciplinary faculty teams, incorporates a strong technology component, emphasizes the development of students’ communication skills, and attempts to improve students' mathematical self-esteem and confidence in their problem-solving skills. The specific objectives of the goal of changing the college algebra culture are to energize faculty to develop modes of instruction that actively engage students in their learning, instill in faculty a sense of ownership and pride about teaching college algebra, encourage faculty in disciplines that require college algebra to develop a sense of involvement and responsibility for the college algebra program, and obtain administrative support for a reformed college algebra program.
Three faculty members at the University of Houston Downtown, William Waller, Linda Becerra, and Ongard Sirisaengtaksin, wrote a case study about the process of initiating change in their college algebra course. They write that the challenges they believed needed addressing in their previous course were student performance and student preparation, and that traditional methods were not effective in meeting these challenges. Their aims were to provide students with numerous opportunities to learn, lead students to learn fundamental concepts and skills through solving real-world problems, stimulate student interest and increase motivation (thereby improving retention), increase mathematical literacy, use diverse teaching strategies, and offer a technology-dependent curriculum.
’A Research Evaluation of a Reform College Algebra Courseâ? by Joan Cohen Jones, Eastern Michigan University and Andrew Balas, University of Wisconsin Eau Claire, describes how the authors, a mathematics educator and a mathematician, structured a college algebra course with the aim of empowering students by having them construct their own understanding through discussing concepts in small cooperative groups. In the course, students had to apply traditional algebra skills to problems in real-life situations. Research conducted by the authors indicated that the students improved in their attitudes toward mathematics and their confidence in their ability to solve problems, that students attributed their success less to the instructors and more to themselves and their peers, that successful groups bonded well, and that the groups served as a forum to explore and test ideas.
The College Algebra Reform Papers website at the State University of New York ’ Oswego contains articles by William Fox, Francis Marion University, Scott Herriott, Maharishi University of Management, and Laurie Hopkins, Columbia College, discussing the appropriateness of college algebra practices and offering suggestions for improvement. William Fox addresses the issue of integrating modeling and problem solving in developing new courses to replace the traditional college algebra course. Scott Herriott compares the traditional college algebra curriculum with more recent reform approaches and also discusses related issues of national and local educational policy. Laurie Hopkins focuses on the role of technology, and specifically the use of handheld computer algebra systems in the college algebra classroom. The website also includes two ’provacateurâ? responses to the articles.
The reform of college algebra has been the topic of discussions, paper sessions and panels at many mathematics meetings. The website for 1997 MAA Session on Courses Before Calculus contains a list of topics and speakers, and a number of articles on the subject are in the Newsletters of Mathematicians for Education Reform. The AMATYC publication Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus also contains a great deal of information about developments in college algebra courses.
College Algebra ’ New Approaches
Hamid Behmard’s college algebra course at Chemeketa Community College uses College Algebra and Trigonometry with Modeling and Visualization by Gary Rockswold. It covers polynomial, rational, exponential, logarithmic, and related piece-wise defined functions. The algebra of functions, complex numbers, sequential functions, and linear systems are also included. The course incorporates group activities and writing and the syllabus states: ’Upon successful completion of this course, students shall be able to:
1. Create mathematical models of abstract and real world situations using linear, quadratic, polynomial, rational, exponential, and logarithmic expressions.
2. Use inductive reasoning to develop mathematical conjectures involving these function models.
3. Use deductive reasoning to verify and apply mathematical arguments involving these models. (Distinguish between the uses of inductive and deductive reasoning.)
4. Represent these functions in graphical, tabular, symbolic and narrative form, and then use mathematical problem solving techniques to solve problems involving these functions.
5. Make mathematical connections to, and solve problems from other disciplines involving these functions.
6. Use oral and written skills to individually and collaboratively communicate about these function models.
7. Apply appropriate technology to enhance mathematical thinking and understanding, solve mathematical problems, and judge the reasonableness of their results.
After examining the student population in the college algebra course and consulting with departments that required that course, the Hiram College Department of Mathematics eliminated the course and replaced it with Mathematical Modeling in the Liberal Arts. In this course, students use data together with linear, quadratic, polynomial, exponential, and logarithmic functions to model naturally occurring phenomena in medicine, economics, business, ecology, and other disciplines. The course uses numerical, graphical, verbal, and symbolic modeling methods.
Bonnie Gold’s article ’Alternatives to the One-Size-Fits-All Precalculus/College Algebra Courseâ? describes Monmouth University’s mathematics department’s experience replacing a single college algebra course taken by almost all students by four courses designed for particular student populations: elementary education majors, biology majors, social science majors, and students who eventually go on to a standard calculus course. The three new courses were designed in consultation with faculty from the relevant departments. In addition, the course that prepares students for calculus no longer satisfies the general education mathematics requirement, whereas the other three courses ’ as well as a pre-existing quantitative reasoning and problem solving course ’ do satisfy the requirement. For an electronic copy of the article, contact Bonnie Gold.
At American University, Elementary Mathematical Models is a course at the level of college algebra or precalculus that uses simple discrete growth models to provide a context for the study of elementary real functions. The mathematical content has a large degree of overlap with traditional college algebra or precalculus courses and includes properties and applications of linear, polynomial, rational, exponential, and logarithmic functions. The course goals emphasize looking realistically at the methodology of applying mathematics through models, with consistent use of numerical, graphical, and symbolic methods over the entire course. The use of simple difference equation models throughout is intended to provide a unifying theme. The course begins with arithmetic growth and linear functions, and concludes with logistic growth models. A qualitative discussion of how chaos can arise in discrete logistic models is the climax of the course.
At Georgia College and State University a new college algebra course focuses on integrating technology in the form of graphing calculators and providing learning support: strategies for test taking, dealing with math anxieties, mastering mathematical concepts, and developing graphing calculator skills. The article College Algebra, Learning Support, and Technology: What is the Connection? by Margo Alexander briefly describes a study done to compare college algebra students who concurrently took a learning support course against those who did not have additional support.
The Mathematics Department at Missouri University re-examined the department’s college algebra course and replaced its traditional college algebra course with two new courses designed to make their study more interesting for calculus-bound students and more relevant for students not planning to take calculus: College Algebra for Calculus Bound Students and College Algebra for Non-Calculus Bound Students.
Paul Dirks, Miami-Dade Community College, developed a course entitled Contemporary College Algebra that incorporated group activities, a heavy use of technology, and outside-of-class group projects. He reported that he was guided by the description below (from a 2002 AMS-MAA-MER session on education reform):
Contemporary College Algebra, a data-driven modeling course, is an example of a reformed college algebra course that serves as a base course for a quantitative literacy program. The course focuses on problem solving in the modeling sense rather than the exercise sense. Communications (reading, writing, presenting), use of technology, small group interdisciplinary projects, analysis of real data sets, graphical analysis, and recursive sequence models are all strongly emphasized. The course is designed to prepare students to be mathematically literate in today's information society. The focus is on preparing students for the future rather than training them for the past.
Dirks stated that he was at first unsure about whether his students had the mathematical and communication skills required to succeed in this course, but after three semesters of teaching it, he reported that they have exceeded his expectations. He said that he has observed improved student engagement in critical thinking (outlining issues clearly, posing non-trivial questions, organizing their discoveries, and presenting results in a variety of forms), increased exercise of creativity and autodidactic activity (learning new mathematics and adapting old, learning and using new technologies, creatively presenting results); and a phenomenon best expressed by the statement, ’The whole is more than the sum of its partsâ? (group work pushing toward a better solution). Dirks stated that he has forever changed the way he teaches as a result of this experience, that even if this is not the ’final answer,â? he feels his teaching is moving in the right direction.
All of the content of Suzanne DorÃ©e’s Applied Algebra course at Augsburg College is presented in applied contexts: the examples, exercises, and the text narrative itself, and the topics were chosen in consultation with client disciplines. They are organized into three groups: linear models, exponential models, and polynomial models. The course is equivalent to intermediate algebra but does not presume that students have mastered introductory material. Concepts and skills are included only if needed in subsequent study or for everyday life. The applications are intended to be relevant and meaningful for both traditionally aged and adult learners and for students from a diversity of cultures, life experiences, and areas of interest. The locally produced text materials, sections of which are available on DorÃ©e’s website, have been used since 1997 by instructors who have employed a variety of pedagogical approaches. Slides from a talk about the course are also on her website.
At the University of Arkansas students can enroll in a special section of College Algebra taught in conjunction with the Mathematics Resource and Tutoring Center (MRTC). The course consists of in-class and MRTC activities plus computer work. The computer work consists of eight interactive modules where the student must demonstrate understanding of the concepts and techniques from the text by scoring 90% or above in order to move to the practice problems for the module. Once students complete all module practice problems correctly, they may take the associated test. This purpose of the course is to prepare students for higher-level mathematics courses. As a consequence, the course offers every student as many different opportunities to learn or re-learn fundamental algebraic material as possible.
Some currently available texts and text supplements that take novel approaches to college algebra:
* Functions and Change: A Modeling Approach to College Algebra by Bruce Crauder et al.
* College Algebra: A Contemporary Approach by David Dwyer and Mark Gruenwald
* Functioning in the Real World: A Precalculus Experience by Sheldon P. Gordon et al.
* Contemporary College Algebra - A Graphing Approach by Thomas W. Hungerford
* College Algebra through Modeling and Visualization by Gary K. Rockswold
* Modeling, Functions, and Graphs: Algebra for College Students by Katherine Yoshiwara et al.
* Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos by Dan Kalman
* Projects for Precalculus by Janet Andersen et al.
In designing general education and introductory courses, mathematical sciences departments should ensure that students taking subsequent courses, such as calculus, statistics, discrete mathematics, or mathematics for elementary school teachers, are appropriately prepared. In particular, departments should
College Algebra ’ New Approaches
Tim Warkentin and Mark Whisler, Cloud County Community College, wrote ’Questions about College Algebraâ? to describe their experience assessing alternative formats for their college algebra course. They conclude that ’The change with the greatest impact is likely to be the change in format that we instituted in the fall of 2002 in College Algebra. We are offering all of our daytime sections of College Algebra as classes that, along with its companion class, College Algebra Explorations, meet every day.â?
’A Research Evaluation of a Reform College Algebra Courseâ? was conducted by Joan Cohen Jones, Eastern Michigan University, and Andrew Balas, University of Wisconsin Eau Claire. The research indicated that ’that the students improved in their attitudes toward mathematics and their confidence in their ability to solve problems. They attributed their success less to the instructors and more to themselves and their peers. Successful groups bonded well, and the group served as a forum to explore and test ideas.â?
A proposed evaluation of a project ’Redesigning College Algebra Delivery from Direct Instruction to a Computer Environment by Brian Beaudrie, Northern Arizona University, discusses a number of the difficulties involved in making a transition from a classroom-based course environment to a web-based one and sets forth plans for how the result will be evaluated.
Precalculus ’ New Approaches
In ’Analysis of Effectiveness of Supplemental Instruction (SI) Sessions for College Algebra, Calculus, and Statistics,â? Sandra Burmeister, Patricia Ann Kenney, and Doris L. Nice explore data from 177 courses in mathematics for which SI support was given (1996). The SI sessions are based on theoretical notions of ’metacognitionâ? and aim to help students develop a cognitive monitoring system and make effective use of learning strategies. The data indicate that SI sessions promote student success. There were positive differences in grades for students who participated in SI sessions in college algebra, calculus, and statistics when compared with students who did not participate. Additionally, in 1994 Kenney and James Kallison reported on research studies on the effectiveness of SI in mathematics classes.
In ’Precalculus in Transition: A Preliminary Reportâ? by Trisha Bergthold and Ho Kuen Ng, San Jose State University, the authors discuss their initial investigation of low student achievement in our five-unit precalculus course. We investigated issues related to course content, student placement, and student success. As a result, we have streamlined the course content, we are planning to implement a required placement test, and we are planning a 1’2 week preparatory workshop for students whose knowledge and skills appear to be weak. Further study is ongoing.
Integrating Precalculus and Calculus
An evaluation of the Moravian College integrated calculus and precalculus course by the Fund for the Improvement of Post-Secondary Education examined student persistence rates, the performance of integrated-course students compared to students in the traditional sequence on a set of problems included in the final examinations of both courses, instructor attitudes, and student attitudes. It concluded: ’Uniformly, student persistence through the sequence was higher for the integrated course than for calculus preceded by precalculus. Integrated-sequence students performed at least as well on a set of common problems as the traditional-course students, and sometimes better. In general, both faculty and students liked the integrated sequence better.â?