Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus

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Each year, over 1,000,000 students take college-level courses below calculus such as precalculus, college algebra and others that fulfill general education requirements. Most college algebra courses, and certainly all precalculus courses, were originally intended to prepare students for calculus. Most are still offered in this spirit, even though only a small percentage of students have any intention of taking calculus. This volume examines how the courses below calculus might be refocused to provide better mathematical experiences for all students. This initiative involves a greater emphasis on conceptual understanding with a de-emphasizing on rote manipulation. It encourages the use of realistic applications, math modeling and data analysis that reflect the ways mathematics is used in other disciplines. It promotes the use of active learning approaches, including group work, exploratory activities and projects. It emphasizes communication skills: reading, writing, presenting and listening. It endorses the appropriate use of technology to enhance conceptual understanding, visualization, and to enable students to tackle real-world problems. The 49 papers in this volume seek to focus attention on the problems and needs of the courses and to provide guidance to the mathematics community. 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.
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Product Details

  • ISBN-13: 9780883851791
  • Publisher: Mathematical Association of America
  • Publication date: 2/1/2006
  • Series: Maa Notes Ser.
  • Pages: 396
  • Product dimensions: 8.40 (w) x 11.00 (h) x 0.90 (d)

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The Conference: Rethinking the Preparation for Calculus Editor's note: The papers presented at this conference form the basis for this volume. This article, which was written shortly after the conference was held in October 2001, gives the rationale for the conference. For a more up-to-date view on some of the issues discussed here, please see the article, "Where Do We Go from Here: Creating a National Initiative to Refocus the Courses below Calculus," by Sheldon Gordon, later in this volume. Rationale for the conferenceDuring the last decade, calculus renewal efforts occurred at all levels of post-secondary institutions as outgrowths of the Tulane Conference in 1987 and the subsequent national conference on Calculus 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 to rise. The calculus renewal movement continues to have a significant impact on undergraduate mathematics education. Instructors have experimented with alternative teaching methods that included the use of technology, collaborative learning, and out-of-class projects. These methods were integrated into new curricula with an increased emphasis on conceptual understanding. Today, all new editions of calculus texts, even so-called traditional ones, incorporate significant themes and problems developed as part of the calculus renewal movement. One major, although unanticipated, outcome of the calculus renewal effort is the development of comparable efforts to revise college algebra and developmental mathematics offerings. There have also been several efforts to rethink precalculus courses, most notably those by Baxter Hastings, Connelly et al, and Gordon, et al. However since the publication of the volume, Preparing for a New Calculus [2], the mathematics community has paid insufficient attention to courses that bridge precalculus courses with calculus courses. The need to address this issue is essential since, as Lynn Steen points out, "Clearly precalculus (and its alter ego college algebra) is the single most common mathematics course in undergraduate education." The enrollment data in the fall of 2000 supports Steen's statement. In particular, the precalculus enrollment then was twice the enrollment of calculus I at all types of institutions and four times the enrollment of calculus I at two-year schools [12]. It is now time to renew a national dialogue on these issues. The Rethinking the Preparation for Calculus project focused on precalculus courses that are not terminal---that is, those that are requirements for some type of calculus. All of us are aware, in general, that only a small percentage of students who take precalculus courses ever go on to take calculus and that many of them who do are not particularly well-prepared for calculus and never complete the course. But this has never been carefully documented. In a comprehensive study done at the University of Lincoln at Nebraska, Steve Dunbar found just how small that percentage actually is [13]. As a result, large numbers of students lose the opportunity to pursue mathematics or mathematics based disciplines. Mathematics instructors can readily identify with the observation noted in [3]: Students who were substantially underprepared reported more conceptual problems and feelings of being overwhelmed in the early stages of their major.... Not only did most of these students abandon their ambition to continue with a S.M.E. (Science, Mathematics, and Engineering) major, they also suffered emotional damage by attempting what proved an impossible task. Several colleges and universities, for example, University of Michigan, University of Texas at El Paso, and SUNY Farmingdale have recognized the need for rethinking the precalculus curriculum and have implemented completely different approaches. Furthermore, given the importance that two and four-year schools attach to the development of articulation agreements among two and four-year schools, it is essential that there should be some serious discussion on the topic in conjunction with any discussion of changing precalculus courses.Overview of the conferenceAt the 2001 joint meetings in New Orleans, Jack Narayan discussed the idea of having an NSF-funded conference focusing on the precalculus curriculum and was encouraged to develop a proposal. Shortly thereafter a steering committee was formed consisting of Jack Narayan (chair), Steve Dunbar, Sheldon Gordon, Nancy Baxter Hastings, Christopher Hirsch, and Jo Ann Lutz. The committee proposed to organize a special invited conference to bring together mathematicians with a deep interest in this topic. The purpose of the conference was to rethink the preparation for calculus, with the following considerations: 1) Students are having different mathematical experiences in high school. The routine use of graphing calculators is standard, there is a greater emphasis on group work and collaborative learning, and there is a growing emphasis on conceptual understanding and realistic problem solving, not just skill development.
2) Calculus in college is placing different expectations on students, particularly an emphasis on conceptual understanding and the use of technology.
3) New technologies provide a wider selection of tools for both the teaching and learning of mathematics.
4) College algebra courses are in the process of changing. The expected outcomes of the conference included:

· articulating some principles for changing precalculus offerings
· providing guidance to the mathematics community
· developing a cohesive effort among those individuals who have done groundbreaking work in this area to make a larger impact on the mathematics community
· focusing attention on problems and needs in the precalculus area that will lead to new funding programs/opportunities from NSF and other funding agencies
· publication of the conference proceedings as a volume in the MAA Notes series The proposal for the conference was funded by the National Science Foundation (DUE 0136162) and the Calculus Consortium for Higher Education (CCHE). In the fall of 2001, fifty-five mathematics educators participated in the invited conference in Arlington, VA to rethink the preparation for calculus. The major themes for the conference included: · Transition from high school
· Changes in college algebra
· Precalculus reform projects
· Technology
· Implementation issues
· Research in student learning
· Influencing the mathematics community Invited position papers for each theme were presented and discussed. Participants were then encouraged to help identify challenges and make recommendations. This MAA Notes volume represents the contributions of many of the participants. The discussions were based on a series of basic principles about precalculus courses that are stated and addressed by Nancy Baxter Hastings and Sheldon Gordon (see [14], [15]):
· Precalculus courses serve two distinct student populations: the overwhelming majority for whom precalculus is a terminal course and the relatively small minority for whom it is a gateway to higher mathematics. The needs of both populations should be met.
· Precalculus courses need to prepare students for calculus both conceptually and algebraically. It is not enough just to emphasize the development of manipulative skills; students need help to learn how to understand and apply the basic calculus concepts. Very few students have the ability to develop those conceptual connections on their own.
· Calculus is no longer the first mathematics course that is considered a prerequisite for courses in other quantitative disciplines. Precalculus and college algebra are now prerequisites for (non-calculus-based) courses in many fields. The mathematical needs of those fields are often not satisfied by standard, algebra skills-oriented precalculus/college algebra courses.
· Students need to see an emphasis on mathematical modeling to learn how mathematics is connected to the real world. The basic mathematical concepts and methods should be developed in contexts to help the students transfer their learning outside the mathematics classroom.
· Precalculus courses should help students learn to use modern technology wisely and appropriately. Moreover, current research into the learning process has much to tell us about how students acquire fundamental precalculus (mathematical) concepts. Only a small minority of students learn mathematics the way their professors did. The original intent for the conference was to focus exclusively on precalculus courses that are intended as the immediate precursors to calculus. However, the discussions at the conference almost immediately demonstrated that it was impossible to separate such precalculus courses from all precursor courses, especially college algebra. As a consequence, most of the comments in this article, as well as the current volume in its entirety, reflect this broader vision. The principal recommendation from the conference was to collect extensive data from many different types of institutions to identify which students take precalculus (and college algebra) courses and why they take these courses. In addition, it was recommended that data be collected concerning the success rates in these courses, which successor courses the students actually take, and how they do in successor courses. The conference participants felt that such data is critical for convincing the mathematics community at large that precalculus courses need to change, as well as to acquaint potential funding agencies of the magnitude and implications of the problem. Moreover, the participants felt that any efforts to rethink precalculus should involve high school mathematics teachers and faculty in client disciplines....
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Table of Contents

Preface vii

Introduction 1
1 The Conference: Rethinking the Preparation for Calculus, Jack Narayan and Darren Narayan 3
2 Twenty Questions about Precalculus, Lynn Arthur Steen 8
Background 13
3 Who are the Students Who Take Precalculus?, Mercedes A. McGowen 15
4 Enrollment Flow to and from Courses Below Calculus, Steven R. Dunbar 28
5 What Have We Learned from Calculus Reform? The Road to Conceptual Understanding, Deborah Hughes Hallett 43
6 Calculus and Introductory College Mathematics: Current Trends and Future Directions, Susan L. Ganter 46

Theme 1. New Visions for Introductory Collegiate Mathematics 55
7 Refocusing Precalculus: Challenges and Questions, Nancy Baxter Hastings 57
8 Preparing Students for Calculus in the Twenty-First Century, Sheldon P. Gordon 64
9 Preparing for Calculus and Preparing for Life, Bernard L. Madison 78
10 College Algebra: A Course in Crisis, Don Small 83
11 Changes in College Algebra, Scott R. Herriott 90
12 One Approach to Quantitative Literacy: Understanding our Quantitative World, Janet Andersen 101

Theme 2. The Transition from High School to College 109
13 High School Overview and the Transition to College, Zalman Usiskin 111
14 Precalculus Reform: A High School Perspective, Daniel J. Teague 121
15 The Influence of Current Efforts to Improve School Mathematics on the Preparation for Calculus, Eric Robinson and John Maceli 129

Theme 3. The Needs of Other Disciplines 151
16 Fundamental Mathematics: Voices of the Partner Disciplines, William Barker and Susan L. Ganter 153
17 Skills versus Concepts at West Point, Rich West 160
18 Integrating Data Analysis into Precalculus Courses, Allan J. Rossman 169

Theme 4. Student Learning and Research 179
19 Assessing What Students Learn: Reform versus Traditional Precalculus and Follow-up Calculus, Florence S. Gordon 181
20 Student Voices and the Transition from Reform High School Mathematics to College Mathematics, Rebecca Walker 193

Theme 5. Implementation 211
21 Some Political and Practical Issues in Implementing Reform, Robert E. Megginson 213
22 Implementing Curricular Change in Precalculus: A Dean's Perspective, Judy E. Ackerman 219
23 The Need to Rethink Placement in Mathematics, Sheldon P. Gordon 224
24 Changing Technology Implies Changing Pedagogy, Lawrence C. Moore and David A. Smith 229
25 Preparing for Calculus and Beyond: Some Curriculum Design Issues, Al Cuoco 235
26 Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course, Bonnie Gold 249

Theme 6. Influencing the Mathematics Community 255
27 Launching a Precalculus Reform Movement: Influencing the Mathematics Community, Bernard L. Madison 257
28 Mathematics Programs for the Rest-of-Us, Naomi D. Fisher and Bonnie Saunders 265
29 Where Do We Go From Here? Creating a National Initiative to Refocus the Courses below Calculus, {\it Sheldon P. Gordon 274
Ideas and Projects that Work: Part 1 283
30 College Precalculus Can Be a Barrier to Calculus: Integration of Precalculus with Calculus Can Achieve Success, Doris Schattschneider 285
31 College Algebra Reform through Interdisciplinary Applications, William P. Fox 295
32 Elementary Math Models: College Algebra Topics and a Liberal Arts Approach, Dan Kalman 304
33 The Case for Labs in Precalculus, Brigitte Lahme, Jerry Morris, and Elias Toubassi 310
34 The Fifth Rule: Direct Experience of Mathematics, Gary Simundza 320

Ideas and Projects that Work: Part 2 329
35 Mathematics in Action: Empowering Students with Introductory and Intermediate College Mathematics, Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick, and Sylvia Svitak 333
36 Precalculus: Concepts in Context, Marsha Davis 337
37 Rethinking College Algebra, Benny Evans 341
38 From The Bottom Up, Sol Garfunkel 345
39 The Functioning in the Real World Project, Florence S. Gordon and Sheldon P. Gordon 348
40 The Importance of a Story Line: Functions as Models of Change, Deborah Hughes Hallett 352
41 Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus, Nancy Baxter Hastings 355
42 Maricopa Mathematics, Alan Jacobs} 360
43 College Algebra/Quantitative Reasoning at the University of Massachusetts, Boston, Linda Almgren Kime 364
44 Developmental Algebra: The First Mathematics Course for Many College Students, Mercedes A. McGowen 369
45 Workshop Precalculus: Functions, Data, and Models, Allan J. Rossman 376
46 Contemporary College Algebra, Don Small 380
47 Precalculus: A Study of Functions and Their Applications, Todd Swanson 386
48 Success and Failures of a Precalculus Reform Project, David M. Wells and Lynn Tilson 390
49 The Earth Math Projects, Nancy Zumoff and Christopher Schaufele 393

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