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A Fresh Start for Collegiate Mathematics

A Fresh Start for Collegiate Mathematics

Edited by Nancy Baxter Hastings
Florence S. Gordon
Sheldon P. Gordon
Jack Narayan
Series: MAA Notes
Volume: 69
Copyright Date: 2006
Edition: 1
Pages: 409
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  • Book Info
    A Fresh Start for Collegiate Mathematics
    Book Description:

    Each year, over 1,000,000 students take college-level courses below calculus such as pre-calculus, college algebra and others that fulfill general education requirements. Most college algebra courses, and certainly all pre-calculus courses, were originally intended to prepare students for calculus. Most are still offered in this spirit, even though only a small percentage of students has any intention of taking calculus. The MAA'S goal has been to refocus the courses below calculus to provide better mathematical experiences for all students. This initiative involves a greater emphasis on conceptual understanding with a de-emphasis on rote manipulation. The use of realistic applications, math modeling and data analysis that reflect the way mathematics is used in other disciplines is encouraged, along with active learning approaches (including group work, exploratory activities and projects). The initiative emphasizes communication skills: reading, writing, presenting and listening. The appropriate use of technology to enhance conceptual understanding, visualization, and inquiry enables students to tackle real-world problems. The 49 papers in this volume discuss various aspects of this issue. The major themes include: new visions for introductory collegiate mathematics, transition from high school to college, needs of other disciplines, research on student learning, implementation issues, and ideas and projects that work.

    eISBN: 978-1-61444-302-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-viii)
  3. Table of Contents
    (pp. ix-xii)
  4. Introduction

    • 1 The Conference: Rethinking the Preparation for Calculus
      (pp. 3-7)
      Jack Narayan and Darren Narayan

      During the last decade, calculus renewal efforts occured at all levels of post-secondary institutions as outgrowths of the Tulane Conference in 1987 and the subsequent national conference onCalculus for A New Century, hosted by the National Academy of Sciences. An MAA special report,Assessing Calculus Reform Efforts[1], estimated that “at least 150,000 students or 32% of all calculus enrollments in the spring of 1994 were in reform courses.” Since 1994, several reform calculus texts have been among the highest selling nationally, and the number of institutions utilizing one or more aspects of reform in their calculus courses continues...

    • 2 Twenty Questions about Precalculus
      (pp. 8-12)
      Lynn Arthur Steen

      Approximately fifteen years ago a workshop similar to this one took place at Tulane University where a merry band of reformers sought to make calculus lean and lively. I had the opportunity to address that workshop with a list of twenty questions for calculus reformers. Thus I thought it appropriate to take a similar approach to this current workshop, to help launch your work by asking twenty questions about precalculus. (For comparison, I reproduce in Appendix A the questions that I put before the calculus reformers at Tulane. There you will find not 20 but 28 questions, the extra eight...

  5. Background

    • [Introduction]
      (pp. 13-14)

      It is one thing toclaimwe have a problem and another thing toshowwe have a problem. Based on their own experiences and observations, participants at the conference,Rethinking the Preparation for Calculus, felt that the number of students who continue on to calculus from precalculus was small and their success rate was low. However, they realized that in order to convince colleagues, administrators, state legislators, funding agencies, and book publishers that changes need to be made, they needed data that clearly indicate the extent of the problem. They also realized that much could be learned from the...

    • 3 Who are the Students Who Take Precalculus?
      (pp. 15-27)
      Mercedes A. McGowen

      This paper examines the questions: Who are the undergraduate students who enroll in precalculus courses? What courses do students take after completing a precalculus course? These questions are addressed by an analysis of enrollment in mathematics courses at two- and four-year colleges and at universities from 1980 to 2000, followed by a demographic profile of students at a Midwestern two-year college whose mathematics course enrollment closely parallels enrollment at two-year colleges nationally [1].

      As we begin the task of rethinking the precalculus curriculum, we first need to examine our assumptions about precalculus. Do we, as well as members of our...

    • 4 Enrollment Flow to and from Courses Below Calculus
      (pp. 28-42)
      Steven R. Dunbar

      In rethinking the courses below calculus, two questions naturally arise:

      What mathematics courses (calculus and otherwise) will students take after completing courses supposedly intending to prepare them for calculus?

      What mathematics courses have students studied before calculus and calculus preparation courses, and how recent is their knowledge?

      This report attempts to partially answer these two questions based on a study that tracked the actual enrollment of students in precalculus, calculus, and non-calculus based courses over 20 successive semesters, from fall 1992 to spring 2002. This is a study of the enrollment flow of students on a microscopic level, in that...

    • 5 What Have We Learned from Calculus Reform? The Road to Conceptual Understanding
      (pp. 43-45)
      Deborah Hughes Hallett

      In order to evaluate the impact of calculus reform, we first need to recall its goals. Although different people may phrase it differently, everyone involved would agree that they were trying to improve the teaching of calculus. Some would say they wanted more student involvement; others would say they wanted to take advantage of technology; others would say they wanted to emphasize problem solving and modeling. Most would agree that they wanted to improve conceptual understanding. What has been the impact of this effort?

      The teaching of calculus came under scrutiny in the 1980s for several reasons. One was concern...

    • 6 Calculus and Introductory College Mathematics: Current Trends and Future Directions
      (pp. 46-54)
      Susan L. Ganter

      For better or worse, the ideas of calculus reform are deeply embedded in the conversations of the mathematics community. Technology, cooperative learning, student projects, applications—the elements of this movement—have become a part of the vocabulary in mathematics departments across the country. Efforts to change the nature of the calculus course at the undergraduate and secondary levels have sparked discussion and controversy in ways as diverse as the actual changes. Such interactions range from “coffee pot conversations” to university curriculum committee agendas to special sessions on undergraduate education at regional and national conferences. But what is the significance of...

  6. Theme 1. New Visions for Introductory Collegiate Mathematics

    • [Introduction]
      (pp. 55-56)

      Courses below calculus face a number of challenges. They need to serve two populations: students who plan to continue their study of mathematics and students who do not. They need to meet the mathematical needs of today’s population of students (which is different from when many of us were in school). They need to prepare students who do continue their study of mathematics to take a calculus course where they are expected to think and understand and not just do computations. The six papers in this section describe some of the major changes that are taking place in the courses...

    • 7 Refocusing Precalculus: Challenges and Questions
      (pp. 57-63)
      Nancy Baxter Hastings

      With this paper, I hope to initiate a dialogue about some of the challenges confronting refocusing precalculus. The paper articulates some of the forces for change in introductory mathematics courses at the collegiate level and how these concerns are being addressed. It describes the distinguishing features of alternative instructional materials and the pedagogical changes that the new materials have fostered in the teaching and learning environment. Many of these changes have been implemented in calculus; in fact, even the so-called traditional books reflect many of these new directions. Precalculus, however, lags behind.

      Introductory collegiate mathematics is in the midst of...

    • 8 Preparing Students for Calculus in the Twenty-First Century
      (pp. 64-77)
      Sheldon P. Gordon

      Major changes have taken place in the mathematical education of students over the last decade. These changes have come about for a variety of reasons, including:

      (1) the changing demographics of the students taking college-level mathematics,

      (2) the growth of technology and what it can provide for the teaching and learning of mathematics,

      (3) the changing mathematical needs among the people who use mathematics.

      All of these factors have major implications for what we teach, and how we teach it, both at the precalculus level and in all other mathematics offerings.

      We first look at the changes in the student...

    • 9 Preparing for Calculus and Preparing for Life
      (pp. 78-82)
      Bernard L. Madison

      Every teacher has been confronted with the question, “Where will I use this?” Very often in school and introductory college mathematics, the answer is, “In calculus.” In fact, preparation for calculus has for several decades been a primary aim of grades 9–12 mathematics. Often this preparation extends to grades 13 or 14. Consequently, calculus has taken on extreme importance and extreme responsibility in US mathematics education. I will argue here that this importance and responsibility are inflated and serve neither calculus nor general education well. Neither students nor the general public see the value that calculus delivers and very...

    • 10 College Algebra: A Course in Crisis
      (pp. 83-89)
      Don Small

      This paper presents a case for transforming traditional college algebra from a failed program attempting to prepare students for calculus to one that enables students to address the needs of society, the workplace, and the quantitative aspects of disciplines. Characteristics of improved college algebra programs are described as well as the symbiotic relationship between a transformed college algebra and quantitative literacy.

      Traditional college algebra courses arenot working[3]. That was the strong consensus of the participants in the Conference to Improve College Algebra, held at the U. S. Military Academy, February 7–10, 2002. This conclusion was based on...

    • 11 Changes in College Algebra
      (pp. 90-100)
      Scott R. Herriott

      The high rate of students’ failure in the college algebra course could be a problem originating in admissions policies, placement activities, curriculum design, or instruction. This paper focuses on the curriculum as a partial solution to this problem, in light of the needs of the types of students who tend to enroll in the course.

      The traditional college algebra curriculum seems to assume that the course is a preparation for calculus. However, surveys at many institutions have shown that only a minority of college algebra students go on to take calculus of any kind, and only a small fraction of...

    • 12 One Approach to Quantitative Literacy: Understanding our Quantitative World
      (pp. 101-108)
      Janet Andersen

      The calculus reform movement sparked numerous conversations about pedagogy and curricula on campuses throughout the country. However, it is only recently that the national conversation has spread to courses below calculus. These include courses that potentially lead to calculus, such as college algebra and precalculus, as well as courses that students take primarily to fulfill general education requirements. Part of the confusion is that many courses serve one purpose in theory yet serve another purpose in practice. For example, many topics in college algebra and precalculus textbooks are included because they are necessary background for standard topics in differential or...

  7. Theme 2. The Transition from High School to College

    • [Introduction]
      (pp. 109-110)

      At the collegiate level, precalculus is frequently categorized as a “remedial course” and too often it is a terminal mathematics course for students who enter college not prepared to take calculus. On the other hand, at the high school level, precalculus seeks to prepare the best and the brightest to study calculus. In both cases, enrollments are at record levels. Not only are the audiences different, at the high school level, as a result of implementation of theCurriculum and Evaluation Standards, published by the National Council of Teachers of Mathematics (NCTM), the way precalculus is taught and what is...

    • 13 High School Overview and the Transition to College
      (pp. 111-120)
      Zalman Usiskin

      I have been asked to present a “high school overview” with some comments about the transition from high school to college. I am interpreting the first part of this mission to mean an overview of the mathematics curriculum in schools, instruction in classrooms, and performance among students today in the United States. Six years ago at the conference “Preparing for a New Calculus” I concentrated mainly on the curricular changes moving into grades 7–12 [13] and could only begin to speculate on the implications of those changes. Now we have some rather consistent information about changes that have occurred...

    • 14 Precalculus Reform: A High School Perspective
      (pp. 121-128)
      Daniel J. Teague

      In many secondary schools, precalculus reform has preceded calculus reform. Consequently, those of us in the high schools have considerable experience with the capabilities of successful precalculus students in a reformed curriculum and their performance the next year in calculus. From a personal perspective, two important issues are brought forward for consideration.

      Successful students in a reformed precalculus curriculum will have new attitudes and abilities that can be utilized successfully in the teaching of calculus.

      Reformed precalculus curricula can play an important role in preparing students for the rigor of theoretical mathematics.

      Both of these issues have implications in the...

    • 15 The Influence of Current Efforts to Improve School Mathematics on the Preparation for Calculus
      (pp. 129-150)
      Eric Robinson and John Maceli

      In this paper, we will describe some of the changes in K–12 education that affect the mathematical preparation of students entering colleges and universities and who pursue a study of mathematics that includes calculus. Although ideas and research can be traced back further, we will take the year 1989 as a starting point, when two significant publications appeared that served as catalysts for many individuals engaged in efforts to improve school mathematics education. They are:Everybody Counts: A Report to the Nation on the Future of Mathematics Education[14], published by the Mathematical Sciences Education Board (MSEB), theCurriculum...

  8. Theme 3. The Needs of Other Disciplines

    • [Introduction]
      (pp. 151-152)

      As Deborah Hughes Hallett observes in her paper, “What Have We Learned from the Calculus Reform Movement?” (which appears earlier in this volume): “In the long run, the largest impact of calculus reform is likely to be the creation of a community of mathematicians who innovate and reflect on their teaching—and who do so in collaboration with faculty in other disciplines and across institutional boundaries.” One of the challenges confronting refocusing the courses below calculus is for mathematicians to understand and to respond to the needs of partner disciplines. The three papers in this section address this challenge. Bill...

    • 16 Fundamental Mathematics: Voices of the Partner Disciplines
      (pp. 153-159)
      William Barker and Susan L. Ganter

      Given the impact of mathematics instruction on the sciences and quantitative social sciences—especially instruction during the first two years—there is a need for significant input from these partner disciplines when revising the undergraduate mathematics curriculum. The committee Curriculum Renewal Across the First Two Years (CRAFTY), a subcommittee of the MAA Committee on the Undergraduate Program in Mathematics (CUPM), has gathered such input through theCurriculum Foundations Project. The primary component of the Curriculum Foundations Project has been a series of eleven disciplinary workshops held across the country from November 1999 to February 2001 (see Figure 1).

      Each Curriculum...

    • 17 Skills versus Concepts at West Point
      (pp. 160-168)
      Rich West

      In 1990 West Point adopted a bold, new curriculum that changed the mathematics courses that all students must take during the first two years. The senior leadership at West Point has always struggled with the issue of emphasis: skills or concepts. With the new curriculum, we were torn between teaching students to understand and use the tools of mathematics in solving real-world problems as opposed to mastering the skills (usually precalculus) that they had already learned. After many discussions with our partner disciplines, we came up with a compromise: the partners would concur on the needed skills, and the mathematics...

    • 18 Integrating Data Analysis into Precalculus Courses
      (pp. 169-178)
      Allan J. Rossman

      The statistics education reform movement of the past fifteen years has emphasized genuine data, conceptual understanding, and active learning (see [3] and [9] for overviews). These features have also been hallmarks of calculus reform efforts, although the use of genuine data is naturally more prevalent in statistics courses than in calculus, for statistics has data analysis at its very core. Yet many calculus reform projects emphasize applications, which in turn often involve genuine data.

      The articles in this volume reveal that these features, including the use of real data, are prominent in the call for rethinking precalculus courses as well....

  9. Theme 4. Student Learning and Research

    • [Introduction]
      (pp. 179-180)

      One line of questioning that was repeatedly voiced at the conferenceRethinking the Preparation for Calculusrevolved around determining what works and what doesn't work. Participants asked: “How can we determine whether or not new pedagogical approaches or new curricular materials are effective? How can we measure the impact of a new approach on student learning?” In order to convince colleagues to adopt new pedagogies and instructional materials, we not only need data to show that there is a problem, we also need to measure the effectiveness of alternative approaches. In this section, Florence Gordon looks at the results of...

    • 19 Assessing What Students Learn: Reform versus Traditional Precalculus and Follow-up Calculus
      (pp. 181-192)
      Florence S. Gordon

      For the last decade, the reform of mathematics education at the college level has been accompanied by the so-called “math wars.” One dimension of this struggle has involved calls to prove that non-traditional courses are at least as effective as the courses they are designed to replace. From one point of view, this is certainly a reasonable request, especially as all of higher education faces pressures for accountability. From another point of view, I am not aware that anyone has ever been asked to prove anything about the effectiveness of the traditional courses. Most studies looking at the impact of...

    • 20 Student Voices and the Transition from Reform High School Mathematics to College Mathematics
      (pp. 193-210)
      Rebecca Walker

      A variety of questions arise while trying to rethink college precalculus. How will students react to a reform precalculus experience? Can a reformed precalculus experience help students develop a broader and more realistic perspective of mathematics? How successful will students be in calculus if they have a different type of preparation? Can a different learning environment promote deeper mathematical understanding? Is it possible to create a classroom environment where students expect the mathematics to make sense and where they will struggle with complex problems?

      None of these questions is easy to answer, but it is possible to begin to answer...

  10. Theme 5. Implementation

    • [Introduction]
      (pp. 211-212)

      In the six papers in this section, we learn from the experience of others who have implemented changes at their institutions, who have developed new curricular materials and designed new courses, or who are utilizing emerging technologies. We also revisit the impact of education reform on the transition from high school to college and the appropriate placement of students. In particular, Robert Megginson offers some suggestions for successfully implementing a new curriculum from the faculty standpoint, while Judy Ackerman offers suggestions from an administrator’s standpoint. In support of Zalman Usiskin’s earlier claims about the importance of placement, Sheldon Gordon argues...

    • 21 Some Political and Practical Issues in Implementing Reform
      (pp. 213-218)
      Robert E. Megginson

      The University of Michigan at Ann Arbor initiated some major curricular revisions in its precalculus and introductory calculus course in the early 1990s with which the author has been closely involved as a member of the Michigan mathematics faculty. A number of political and practical issues had to be addressed to help assure the success of the efforts. The purpose of this paper is to describe some of those issues and provide suggestions for dealing with them when they arise in other implementations.

      In the fall semester of 2001, about 3000 students enrolled in the three courses considered to be...

    • 22 Implementing Curricular Change in Precalculus: A Dean’s Perspective
      (pp. 219-223)
      Judy E. Ackerman

      Mathematics departments have not been overly enthusiastic about rethinking precalculus courses despite changes in calculus, changes in K-12 mathematics that have resulted from the NCTM Standards, and an increased emphasis on accountability. In four-year colleges and universities, some faculty equate precalculus with precollege mathematics or at best as the one mathematics course that students take to meet their graduation requirement. However, in the two-year colleges, precalculus often serves as a true pathway to calculus and to majors that require a significant amount of mathematics.

      For many years, calculus reform was the rallying point for mathematics faculty around the country. It...

    • 23 The Need to Rethink Placement in Mathematics
      (pp. 224-228)
      Sheldon P. Gordon

      Several years ago, Richard Riley, secretary of education in the Clinton administration, challenged the mathematics community to address the problems of articulation in mathematics education between high schools and two- and four-year colleges. Riley called for this national initiative, through the National Research Council, because of the growing breakdown in the once smooth transition between high school and college mathematics, as well as the differences between mathematical experiences in different colleges when students transfer from one institution to another.

      In large measure, many of the problems with mathematical transitions are due to the rapidly growing reform movements in mathematics education,...

    • 24 Changing Technology Implies Changing Pedagogy
      (pp. 229-234)
      Lawrence C. Moore and David A. Smith

      Sam looked up from the stack of orders on his desk and glanced at his watch. 3:30, time to work on his project with Andrew. He pushed the orders to one side and turned to his computer. No picture this time, but Andrew’s voice came through with sounds of students playing frisbee in the background.

      Sam, 28, was a non-traditional student, fitting his course work around his work schedule. Andrew was a traditional first-year student. The two had been partners now for four weeks—though they had never met in person.

      The background rock music ceased, and Sam heard Andrew’s...

    • 25 Preparing for Calculus and Beyond: Some Curriculum Design Issues
      (pp. 235-248)
      Al Cuoco

      This paper outlines an alternative to the topic-driven design principle that is the basis for most precalculus courses, arguing that the real power of mathematics lies in the methods used to produce results as much as in the results themselves. It describes a fourth-year high school course that adopts this design, with examples and student work.

      Curriculum design in US precollege mathematics is largely topic driven; a course is defined by the topics it treats. The major criteria for including a topic in any particular course include:

      does it review and deepen important ideas from previous courses?

      is it a...

    • 26 Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course
      (pp. 249-254)
      Bonnie Gold

      How do we want our future legislators, our future news reporters, our country’s future parents to feel about mathematics? Do we want them to believe it is a collection of rituals, requiring special skills only achievable by a few and of no practical value? Or would we prefer that they see mathematics as a way of describing many aspects of the world, central to many issues that will affect their lives, and a subject in which they can achieve whatever level of proficiency they need?

      If the last mathematics course students take is a traditional college algebra or precalculus course,...

  11. Theme 6. Influencing the Mathematics Community

    • [Introduction]
      (pp. 255-256)

      Changes are taking place. New materials are being developed. New pedagogies are being implemented. Data are being collected at schools here and there. Conferences are being held where colleagues meet to discuss the issues. Tools to assess student learning are being developed and new programs are being evaluated. Conversations are taking place between members of the mathematics community and colleagues in partner disciplines and between mathematicians who teach at the collegiate level and at the high school level. However, the biggest challenge confronting refocusing the courses below calculus is to launch a national initiative. This is a complex and immense...

    • 27 Launching a Precalculus Reform Movement: Influencing the Mathematics Community
      (pp. 257-264)
      Bernard L. Madison

      In 1982 I was chair of a mathematical sciences department that included statistics and computer science and a PhD program in mathematics. Resources other than booming student enrollments were scarce. In an act of frustration I wrote a letter to the presidents of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA) asking for help in addressing the problems facing my department and many other departments. That letter opened the door to my involvement, over the next decade, in several interconnected efforts where influencing the mathematics community was a central and critical focus, including the Mathematical Sciences...

    • 28 Mathematics Programs for the Rest-of-Us
      (pp. 265-273)
      Naomi D. Fisher and Bonnie Saunders

      Beginning in 2001, the American Mathematical Society (AMS) and the Mathematicians and Education Reform (MER) Forum began a three-year project, funded by the National Science Foundation, to help mathematics/mathematical science departments strengthen their undergraduate programs. EntitledExcellence in Undergraduate Mathematics: Confronting Diverse Student Interests, the project focuses on the different groups of students in mathematics and their particular mathematical needs. The goal is to develop curricula and instruction that is valuable to these different student populations. The project seeks to build a network of departments that are committed to (1) reviewing and assessing how well their undergraduate program is working...

    • 29 Where Do We Go From Here? Creating a National Initiative to Refocus the Courses below Calculus
      (pp. 274-282)
      Sheldon P. Gordon

      In the four-month period following theRethinking the Preparation for Calculusconference in October 2001, there were three other invited conferences regarding the undergraduate mathematics curriculum.

      1. CRAFTY’sCurriculum Foundations Summary Workshop(November 2001), organized by Bill Barker and Susan Ganter and supported by the NSF and the Calculus Consortium for Higher Education [1]. CRAFTY (the MAA committee on Curriculum Renewal Across the First Two Years) had previously organized a series of 11 workshops in which leading educators from 17 different quantitative disciplines came together to discuss and inform the mathematics community of the mathematical needs of their students today. The...

  12. Ideas and Projects that Work:: Part 1

    • [Introduction]
      (pp. 283-284)

      Just as our students say, “Show me an example,” participants at the conference,Rethinking the Preparation for Calculus, asked for examples of ideas and projects that work. The authors of the five papers in this section discuss some of the larger issues connected with curricular materials that they have developed. The next section, “Ideas and Projects: Part II,” contains fifteen short, descriptive papers for particular projects.

      In this section, Doris Schattschneider describes an alternative one-year college calculus course that integrates a review of precalculus concepts to help students, who have taken precalculus but are not prepared for calculus, succeed in...

    • 30 College Precalculus Can Be a Barrier to Calculus: Integration of Precalculus with Calculus Can Achieve Success
      (pp. 285-294)
      Doris Schattschneider

      The articles collected in this volume make one fact very clear: precalculus means many things to many people, and serves many different functions. One of the primary functions of a precalculus course (whether in high school or in a college or university setting) is to serve as a preparation for the first course in calculus. It is this “preparing for calculus” course, specifically in the college or university setting, that we assert is failing in its stated purpose. The dismal percentage of those who complete precalculus and continue on to complete the calculus I course makes it clear that something...

    • 31 College Algebra Reform through Interdisciplinary Applications
      (pp. 295-303)
      William P. Fox

      Traditional college algebra has been taught at Francis Marion University since before the school’s establishment as a four-year institution in 1970. The majority of students performed poorly in these courses. We initially tried an experiment using applications and projects to motivate the college algebra. We integrated real-world problems in the form of projects, applications, and activities to motivate students to better understand the principles of algebra. Performance improved and feedback from most of the students was positive. Based on the overall positive experience, two new freshmen algebra courses were added using modeling and problem-solving as their framework.

      In this paper,...

    • 32 Elementary Math Models: College Algebra Topics and a Liberal Arts Approach
      (pp. 304-309)
      Dan Kalman

      It would be difficult to overstate the importance of algebra in mathematics. If mathematics is the language of science, then algebra provides the alphabet, vocabulary, and syntax. In particular, the traditional college algebra course covers the elementary functions of analysis: linear and quadratic functions; polynomials and rational functions; roots, exponentials, and logs. These functions are inescapable in the most elementary applications of mathematics to other subjects. In a word, they are ubiquitous all over the place.

      It is tempting, therefore, to prescribe algebra as the minimal quantitative component of a higher education. Unfortunately, for many students who study algebra in...

    • 33 The Case for Labs in Precalculus
      (pp. 310-319)
      Brigitte Lahme, Jerry Morris and Elias Toubassi

      The goal of this position paper is to make the case for the use of labs in a precalculus course. The observations made in this paper are based on our experience with the precalculus program at the University of Arizona, where supplementary lab assignments are integrated into the precalculus curriculum. While these lab assignments frequently involve the use of technology, they are not computer labs in the traditional sense; rather, they are multi-step, real-life problems that students explore in a group setting.

      First, we set the context for the precalculus course at the University of Arizona. All students intending to...

    • 34 The Fifth Rule: Direct Experience of Mathematics
      (pp. 320-328)
      Gary Simundza

      In recent years, the use of multiple representations has become an important part of the teaching of mathematics at all levels. This paper describes how a laboratory approach to precalculus instruction can allow students to achieve mathematical insights using direct sensory experience of quantitative phenomena. Such experience can complement and inform the graphical-numerical-analytical-verbal models students use in problem-solving.

      Since the beginning of the calculus reform movement, the Rule of Three has become an essential feature of mathematics education. Facilitated by the availability of technological aids to constructing graphs and tables, the use of multiple approaches in describing mathematical functions and...

  13. Ideas and Projects that Work:: Part 2

    • [Introduction]
      (pp. 329-332)

      This volume focuses on challenges that need to be met and changes that need to be made. Repeatedly authors have stressed the importance of emphasizing conceptual understanding , instead of rote manipulations; the importance of focusing on situations where mathematics is used in the real world, instead of on rinky-dink word problems; the importance of utilizing the methods of data analysis, instead of relegating it to the last chapter (if it's there at all); and the importance of fostering an active learning environment, where students ask what-if type questions, make connections, explore mathematical ideas, and work collaboratively instead of sitting...

    • 35 Mathematics in Action: Empowering Students with Introductory and Intermediate College Mathematics
      (pp. 333-336)
      Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick and Sylvia Svitak

      The Mathematics in Action project aims to empower college mathematics students with a real-world mathematical literacy that will provide a solid foundation for future study in mathematics and other disciplines. The project was developed by the Consortium for Foundation Mathematics, a team of fourteen SUNY and CUNY faculty, with support from the National Science Foundation (DUE 9455638), and is based on the AMATYCCrossroadsStandards.

      The project’s goal to empower students mathematically focuses on developing desired student outcomes in five main areas: number sense, symbolic sense, a general function sense, a thorough linear function sense, and a sense of nonlinear...

    • 36 Precalculus: Concepts in Context
      (pp. 337-340)
      Marsha Davis

      Precalculus: Concepts in Context[1] was developed in response to calculus reform and to the authors’ general dissatisfaction with results in traditional precalculus courses. Believing that traditional precalculus instruction failed to prepare students for reform calculus (and we would argue, for traditional calculus as well), Judy Moran, Mary Murphy and I set out to reform precalculus with the publication of a laboratory manual,Precalculus in Context: Functioning in the Real World[2]. The lab manual was designed to supplement a standard precalculus course by providing opportunities for students to work collaboratively on lengthy, context-based problems. However, we soon discovered that...

    • 37 Rethinking College Algebra
      (pp. 341-344)
      Benny Evans

      There are many difficulties with traditional college algebra. Oklahoma State University mirrors many other campuses in that virtually every student on campus must take some mathematics course, and for the vast majority that course is college algebra. By and large the students don’t like the course, and they perform miserably. Success rates are embarrassingly low; perhaps the lowest of any course on campus. Such courses quickly draw the attention of the upper administration. Even worse, for most of the students, this is the last mathematics course they will ever see, and it shapes their perception of what mathematics is. Far...

    • 38 From The Bottom Up
      (pp. 345-347)
      Sol Garfunkel

      In 1989 the National Council of Teachers of Mathematics (NCTM) published a rather remarkable document, known to all now as the NCTMStandards. TheStandardsrecognized and announced that we have done a rather poor job of teaching mathematics at the K–12 level. It set out a rather bold agenda for new curricula and pedagogy built around a number of principles focused upon improving the mathematics education of all students. To a surprising extent this document went relatively unnoticed at the undergraduate level. That state of affairs changed rather dramatically in 1997–8, when new curricula built to embody...

    • 39 The Functioning in the Real World Project
      (pp. 348-351)
      Florence S. Gordon and Sheldon P. Gordon

      The calculus reform movement of the last decade or more has led to major changes that include an emphasis on geometric and numerical ideas as a balance to symbolic manipulations, student projects, realistic applications via mathematical modeling, the use of technology, and a more active learning environment. These efforts were intended to transform calculus into apump, not a filter.

      But if we are to change calculus, we must also consider how we “fill the tank”; that is,

      how do we increase the numbers of students who proceed on to calculus?

      how do we improve the mathematical experience of both...

    • 40 The Importance of a Story Line: Functions as Models of Change
      (pp. 352-354)
      Deborah Hughes Hallett

      If students are to remember what they learn, the courses they take must tell a coherent story. This story provides a framework onto which they can hang their newly acquired knowledge. Without such a framework, teachers find themselves having to repeat material. Precalculus courses often run the risk of not being memorable because they are defined as the skills needed in calculus rather than telling a coherent story. Thus, the first decision in designing a new precalculus course is to choose the story it will tell.

      The central theme we chose for our precalculus course is how functions can be...

    • 41 Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus
      (pp. 355-359)
      Nancy Baxter Hastings

      The Workshop Mathematics program [2, 3, 5] broadens student access to university-level mathematics by providing multiple entry points into the discipline. Courses in the program—Workshop Statistics, Workshop Precalculus, andWorkshop Calculus—seek to enable students, who might otherwise “fall through the cracks,” to develop the skills and understanding necessary to use mathematics in other disciplines and to continue their study of mathematics. Like other reform courses, workshop courses seek to encourage students to read, write and talk about mathematical ideas and develop confidence in their abilities to think about and do mathematics. They seek to promote student learning through...

    • 42 Maricopa Mathematics
      (pp. 360-363)
      Alan Jacobs

      Faculty in the Maricopa Community Colleges began the project, The Maricopa Mathematics Consortium (NSF grants: DUE9352897 and DUE9602386), in 1993, at the convergence of two significant reform movements: Calculus reform and the implementation of the 1989 NCTMStandards. We believed that the mathematics curriculum before calculus would need to change, not only to prepare students for a reformed calculus course, but also because our entering students will have had a different preparation in their school mathematics. We decided to reconstruct the entire curriculum below calculus and to write appropriate course materials. Since several college algebra/precalculus reform projects were already underway,...

    • 43 College Algebra/Quantitative Reasoning at the University of Massachusetts, Boston
      (pp. 364-368)
      Linda Almgren Kime

      For many years, the head of Continuing Education at the University of Massachusetts, Boston had been receiving complaints from industry that their employees couldn’t think quantitatively. She called together a handful of faculty from the Mathematics Department, Graduate School of Education, and Academic Support, and asked us to a design a course she could market to these companies. Having an academic bias, we took this as an opportunity to revise our traditional college algebra course—with the thought that later some of the modules could be used to serve the corporate world.

      The discontent (at least for some) with the...

    • 44 Developmental Algebra: The First Mathematics Course for Many College Students
      (pp. 369-375)
      Mercedes A. McGowen

      Since 1980, increasing numbers of students are repeating their high school mathematics courses as undergraduates and enrollment in the developmental courses has continued to grow. In Fall 2000, more than three million students were enrolled in undergraduate mathematics courses taught in departments of mathematics. Thirty-one percent of these students (981,000) were enrolled in remedial mathematics courses (arithmetic, algebra I, algebra II). Of these students, 763,000 were at two-year colleges (57% of the total two-year math enrollment) [1].

      The large number of students who enroll in remedial courses suggests that the traditional emphasis on showing students how to use a rule...

    • 45 Workshop Precalculus: Functions, Data, and Models
      (pp. 376-379)
      Allan J. Rossman

      The Workshop Mathematics program [1], developed at Dickinson College, leads students to discover, explore, and apply fundamental concepts of introductory mathematics and statistics courses. Active learning is the distinguishing feature of the “workshop” pedagogical approach, which replaces lectures with activities through which students interact with each other, with technology, and with the instructor. This program has been extended from its beginnings in Calculus [2] and Statistics [3] to the precalculus level, supported by a grant from the National Science Foundation (#9952483). In this project, Nancy Baxter Hastings and I are developing materials for a course that integrates ideas of data...

    • 46 Contemporary College Algebra
      (pp. 380-385)
      Don Small

      Contemporary College Algebrais designed to educate students for the future rather than to train them for the past. The course, developed in collaboration with faculty in several disciplines as well as with people in the workplace, provides a strong base for quantitative literacy programs.

      The primary goal of the course is to empower students to become exploratory learners, not to master a list of algebraic rules. Some of the means that are used to establish an exploratory environment for the students include:

      Queries for engaging students in questioning and exploring the material being presented

      Exercises that explicitly ask students...

    • 47 Precalculus: A Study of Functions and Their Applications
      (pp. 386-389)
      Todd Swanson

      In response to the calculus reform movement in the mid-1990s, we wanted to find a way to prepare students for a calculus course that was more conceptual, contained more real-life applications, and required students to view functions in multiple representations. With the aid of an NSF grant (DUE-9354741) we set out to write projects that would have students solve interesting, real-life problems that involved multiple representations as well as multiple topics from a precalculus course. These 26 projects were eventually published under the title,Projects for Precalculus[1] and could be used as a supplement to any precalculus course.


    • 48 Success and Failures of a Precalculus Reform Project
      (pp. 390-392)
      David M. Wells and Lynn Tilson

      During the years from 1988 through 1996, the authors developed a set of materials ([1], [2], and [3]) for college algebra and precalculus. Our initial discussions about teaching and writing occurred at Penn State-New Kensington, a regional campus of Penn State University. The campus has an enrollment of about 1000 students, most of whom are freshmen and sophomores. The precalculus course is populated primarily by students who plan to complete degrees in engineering, technology, or science, either at New Kensington or at Penn State’s University Park campus. College algebra is often taken as a terminal course or as a prerequisite...

    • 49 The Earth Math Projects
      (pp. 393-396)
      Nancy Zumoff and Christopher Schaufele

      Since 1991, with support from the National Science Foundation (NSF) and the U.S. Department of Education’s Fund for Improvement of Secondary Education (FIPSE), the authors have developed unique materials for use in mathematics courses ranging from algebra through calculus. These projects have resulted in three books,Earth Algebra(college algebra) [1],Earth Angles(precalculus) [2], andEarth Studies(applied calculus) [3], that all have applications to environmental issues that affect students’ lives. They are designed to generate more interest in the use of mathematics as a tool to analyze real situations. Using mathematics to study real problems that are interesting...