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CRUME'S First Volume - What Will Mathematicians Think?

This question could be appended to Lynn Steen's closing list of "Twenty Questions about Research on Undergraduate Mathematics Education" - the latter henceforth referred to simply as RUME. Though this question will surely be asked and answered in a variety of ways, we wonder whether it's entirely appropriate. Should RUME be made accountable to, or judged by, mathematicians? Would a graph theorist, for example, stand in judgment of a colleague's work in several complex variables? To some extent, the practical answer is yes - this has been and will continue to be done in tenure decisions.

But wait, we are getting ahead of our story. Let us first introduce the protagonist - the first of a projected series of annual volumes titled Research in Collegiate Mathematics Education [CMBS Series, Issues in Mathematics Education, Vol. 4, 1994]. The volumes are a project of the AMS/MAA Committee for Research in Undergraduate Mathematics Education (CRUME), and managing editor Tom Dick has produced a slick-looking volume with only a few, mostly unobtrusive, typos. The editors, Ed Dubinsky, Alan Schoenfeld, and Jim Kaput, intend to serve two audiences - researchers in collegiate mathematics education and mathematicians who may be looking for applications to instruction or are just plain curious. In the preface, they warn that although authors were asked to avoid jargon when plain language would suffice, any field has its technical terms. And further, such research will not necessarily have immediate applications.

What is RUME?

The volume opens with an introductory piece by Schoenfeld giving a personal view of the field from one of its foremost practitioners. Like mathematics, RUME, he asserts is "a many splendored thing," ranging from large studies to those with very few subjects. The latter may be puzzling to those brought up on behaviorist psychology and traditional control group vs. experimental group studies. How can anyone learn anything about the teaching/learning of mathematics by looking at a single student? Yet this is precisely what Schoenfeld and colleagues did, spending a year-and-a-half analyzing seven hours of video tape of a single student working with graphing software. Surely a fool's errand. or perhaps not? The product was a 12O-page manuscript describing a particular "knowledge architecture. " Schoenfeld, et al, found some misconceptions were tied to complex knowledge structures and not easily fixed. Ideas tied to other ideas, whether right or wrong, were robust and resisted modification, while ideas not tied to others were fragile and easily forgotten. For this one needs a year-and-a-half? Surely every college mathematics teacher knows this instinctively. or do we typically say students are having trouble because ( 1 ) they haven't done their homework, (2) they don't have the background, or (3) the schools aren't doing their job? While such reasons often apply, apparently they did not for one ambitious high school student taking calculus during the summer at Berkeley.

A Variety of Research Methods and Topics

The volume contains eight research papers: three on functions, three on calculus, one on algebra and calculators, and one on preservice teachers. one thing that emerges from reading these papers - RUME can be very time consuming. A lot of data, whether interviews, questionnaires, or classroom observations, is needed to obtain even the most tentative findings.

Pat Thompson's research review and synthesis of students' understanding of function is an elaboration of his invited address to the 1993 Annual Joint AMS/ MAA Meeting. He focuses on representation, student cognition, and instructional obstacles, suggesting that the use of multiple representations, as currently construed, may not be well thought out. Tommy Dreyfus and Ted Eisenberg report a teaching experiment with top Israeli high school seniors on visualizing function transformations. While not an unmitigated success, due to the game nature of the Green Globs software used, they found a hierarchy from least to most difficult to visualize. Steve Monk and Ricardo Nemirovsky report the case of Dan, selected from interviews to find out how high school students understand functions arising from physical recording devices. Dan was observed while operating an Air Flow Device containing a bellows to push or pull air into or out of an air bag whose volume and flow rate were plotted. Throughout, Dan used the visual aspects of these graphs in an interpretive way, never literally like a picture, as naive students sometimes do. He moved toward abstraction, focusing on steepness and whether the graphs were spread apart. As he refined his understanding of the situation, these graphs and their uses became increasingly transparent for him.

In a 4-year evaluation of Duke's Project CALC, Jack Bookman and Charles Friedman found that, although initial reaction was quite negative because students had to give up deeply held beliefs that mathematics consists of facts, rules, and procedures, their attitudes improved over time and they eventually became better problem solvers than traditional calculus students. In a longitudinal study, Marty Bonsangue followed the academic performance of minority and non-minority engineering students at Cal Poly from Fall 1986 through Spring 1991.

In a naturalistic study, Sandra Frid compared technique-oriented, concepts first, and infinitesimal approaches to teaching calculus at one large university and two small colleges in Western Canada. Although the former used Stewart, Single Variable Calculus and the latter two developed their own textual materials, all presented calculus as highly structured with the ultimate goal being symbolic representations and justification. Across all three approaches, Frid found students who were Collectors - those believing mathematics to be a collection of formulas, rules, and procedures which they personally didn't understand but had to memorize, Technicians - those who saw calculus as more organized and logically structured and had confidence in their mastery ofthe rules and procedures, and Connectors - those who approached calculus by trying to understand it for themselves and make connections and had confidence in their ability to apply calculus to new problems. Even though all these students used symbols to solve "skill problems," few used symbolic representations during interviews to explain what the derivative is or why a function is not differentiable.

In a traditional study, Mary-Margaret Shoaf-Grubbs compared two classes of female elementary college algebra students, one using a graphing calculator and the other not. Both groups had the same instructor and lesson plans and examined the same number of graphs, but the calculator group finished significantly ahead in spacial visualization.

Rina Zazkis and Helen Khoury interviewed senior preservice elementary teachers regarding nonstandard place value tasks such as converting 12.34five to base ten. These students had all previously taken a "math for teachers" course which covered bases other than ten using manipulatives and were able to perform addition and subtraction of integers in different bases correctly. Errors such as interpreting O.23five as 23ten/lOOfive indicated these students' notions of place value were fragile and incomplete.

This is Question 18 of Lynn Steen's concluding short, yet provocative, list of twenty questions. One might just as well ask whether scientific research, circa 19OO, on whether malaria was caused by infected bed sheets, bad air, or mosquitoes is still valuable reading. Probably not, yet knowing the Anopheles mosquito is the vector remains important today because we act on it - spraying ponds, taking chloroquinine, looking for vaccines, etc. Informed use can take a long time - as late as WWII, pre- control infection rates amongst U.S. troops in the South Pacific could be 18OO per thousand (infections recur) per year. Like public health advances, reform in mathematics education will require public understanding and resolve, as well as knowledge. The question should not be, will this research still make good reading years later, but will there be lasting research results, facts or theories, on which we will act?

Steen's final question is, "Will it count for tenure?" The answer will determine how much RUME is produced and its quality. And that depends on whether mathematics departments are really interested in how students learn mathematics. Certainly many colleges and universities claim teaching as their most important mission. Does this merely mean small class size, being available for student consultation, preparing for class, having good eye contact and blackboard technique? Is the best we can do merely to strive to stimulate students of all abilities, without putting out an unreasonable amount of effort? On the other hand, could there be something to the idea of sequencing the material psychologically, as well as logically? Are there natural psychological entry points for certain topics such as function or limit? What's a true group task? Such questions are the stuff of RUME. Surely we could become more effective teachers if we knew the answers.

This article was taken from UME Trends Vol.6, No.6, Jan. '95, pgs. 22-23.

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