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A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus

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Nancy Baxter Hastings, editor
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Notes 69
[Reviewed by
James Poinsett
, on

Imagine a driver’s education course being taught by a professional racecar driver, who assumes he is preparing his students for a future career in NASCAR. Throughout the course, he repeatedly uses NASCAR jargon often technical language and emphasizes the driving techniques that are practiced by and primarily familiar only to people in the sport. After grading the final exam for the course, he’s shocked to find out that only three of his thirty students prove to be highly proficient, and that fifteen to twenty of the thirty students have failed the course. Upon further investigation, he discovers that the top three students were planning to pursue a NASCAR career and that the fifteen or twenty students who failed were only taking the course as a requirement to receiving their driver’s licenses. The ten or so students in between proceeded to the next driver’s education course, but in the end only one of the thirty original students made it through all three introductory courses to enroll in the advanced driving courses that lead to official racecar licensure. It would not be surprising if the teacher should conclude that driver’s education must be reformed to meet the needs of the larger population of students.

The growing problem of high failure rates in the most widely offered collegiate math courses across the nation, namely those under the broad title of precalculus, has led to attempts to reform these courses so as to serve the population of students who actually take them. As Mercedes McGowen defines it in this collection of 49 articles authored by mathematicians and math educators in colleges across the country, “precalculus refers to those courses intended to prepare students to take calculus: college algebra, trigonometry, the combined algebra/trigonometry course or a precalculus/elementary functions course.” This definition of precalculus is adopted by most authors in the text and in this review.

As in our story about driver’s education, most precalculus students have no interest in pursuing future courses in mathematics. They take the course only because it is a requirement for their degree. Many of these students fail. Only a tiny percentage of precalculus students ultimately take the three course calculus sequence, yet the courses are often designed to prepare these students to enroll in the calculus sequence. The authors of this collection describe this problem in further detail and offer suggestions for reform.

Some of the articles collected in this book are based on formal research design; others are of a more anecdotal or general nature. The predominant themes are the following:

  • lessen the traditional amount of time performing algebraic manipulations;
  • decrease time spent executing algorithms simply for the sake of calculation;
  • restrict the topics covered to the most essential;
  • decrease the amount of time spent lecturing;
  • deemphasize rote skills and memorization of formulas.

Current practices should give way to a reformed curriculum and pedagogy that replaces these traditional practices with more productive ones. Specifically,

  • embed the mathematics in real life situations that are drawn from the other disciplines that mathematics departments serve;
  • explore fewer topics in greater depth;
  • emphasize communication of mathematics through discussion and writing assignments;
  • utilize group assignments and projects to enhance communication in the language of mathematics;
  • use technology to enhance conceptual understanding of the mathematics;
  • give greater priority to data analysis than in traditional precalculus courses of the past;
  • emphasize verbal, symbolic, graphical, and written representations of mathematical concepts and objects;
  • focus much attention on the process of constructing mathematical models before finding solutions to these models.

As shown by the results from the articles describing research projects, the implementation of these suggestions improves student interest; highlights the relevance of mathematics in everyday life; and creates a quantitatively more literate general population.

In their article on ‘The Influence of Current Efforts to Improve School Mathematics on the Preparation for Calculus,’ Robinson and Maceli say that, “courses that precede or parallel calculus courses and stress mathematical modeling, the use of technology, and data analysis are important for today’s students — both non-math majors and math majors.” In “Fundamental Mathematics: Voices of the Partner Disciplines,” Barker and Ganter claim that many mathematicians believe that client disciplines of mathematics (those that use the content and techniques of mathematics to further understand their own area of inquiry) seek students with algorithmic proficiency. Barker and Ganter assert that this expectation is largely false: client disciplines primarily want students to have a conceptual understanding of mathematical topics. This leads to the common suggestion amongst many of the authors that there must be an initiated and sustained dialogue between mathematics departments and client departments to help guide the development of reform curriculum.

Arguing for the relevance of mathematics outside of a classroom, Rossman writes:

Data analysis can play an important role in enhancing students’ learning experiences in precalculus. Genuine data often provide motivation and interest for students, and they reveal that concepts of precalculus do have application to analyzing data from a variety of disciplines as well as from everyday life.

Describing a lab-based precalculus course where lab activities are worked on in groups, Lahme, Morris, and Toubassi say:

The labs provide students with the opportunity to use multiple strategies to solve problems including the use of formulas, graphs, tables, and verbal descriptions. The labs also provide a good opportunity to use technology such as graphing calculators, Excel, and the web. The ability to use different strategies and tools makes students more competent problem solvers.

Being good problem solvers, being fluent with the use of technology, and working in groups are all important pedagogical objectives of the precalculus reform movement. This results in curricular and pedagogical approaches that differ significantly from the traditional practices in precalculus courses.

These common threads may inspire many precalculus teachers to revise their pedagogy and curriculum. Some readers, however, will be disappointed by the lack of representation of opponents of the proposed reform. Several of the authors attempt to give voice to these opponents, but only in order to argue and justify their own points. A future edition should include a dialogue amongst proponents and opponents of precalculus reform to provide the reader with a larger perspective on the issue.

For example, an opponent may read the book and point out the lack of convincing evidence that such reformed courses will adequately prepare students for the symbolic and abstract rigor of upper level undergraduate courses. Offering the space to rebut the claim of Robinson and Maceli that non-math majors and math majors both benefit from these types of courses would greatly enhance the quality of this already informative and important collection of articles.

Since most of the articles are not in response to, or building on other articles in the collection, there is much repetition in the framing of the problem with traditional precalculus courses as well as the suggestions to improve them. Because of this, I recommend reading articles 2, 4, 5, 8, 9, 12, 16, 18, 20, 21, 25, 26, 29, 30, 31, 34, 39, 40, 45, 48 to get a rather comprehensive sense of the larger collection.

This is a must-read book for mathematicians who are not well-versed in educational research and practice, and also for anyone new to the profession. Even if some readers do not agree with the recommendations, knowledge of the reform movement will be valuable in provoking opponents to articulate the reasons why they believe in their pedagogical practices rather than those suggested in the book. Secondary math teachers should also read this text to see the roughly similar approaches mathematicians take to combat the problem of unsuccessful transfer of knowledge between teacher and student. In fact, an implication of this book is that partnerships between colleges and high schools should be established or enhanced for the precalculus reform come to fruition, with the possibility of higher achieving students than ever before.

James Poinsett is in his first year (05/06) of doctoral study in mathematics education at the University of Pennsylvania. Prior to starting at the university, he taught for two years at a New Jersey public middle school while completing a master’s degree in mathematics at Rutgers University. Currently, as a graduate fellow of the MetroMath program, he is part of a research team exploring the ways that high school kids use mathematics outside of school and how those practices can positively influence the teaching and learning of mathematics in school. He can be reached for further discussion at

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