David M. Bressoud January, 2006
B.2: Develop mathematical thinking and communication
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Courses that primarily serve students in partner disciplines should incorporate activities designed to advance students’ progress
There are three incidents from my years at Penn State, three times when a faculty member complained about what students did not know, that stand out for me. The first was when a biology professor whose students, all of whom had completed the two-semester sequence in calculus, insisted that they had never encountered the concept of exponential growth. The second was when a professor in electrical engineering, teaching students who had taken our differential equations class, complained to me that they could not “read” a differential equation, turning it into a verbal explanation of the relationships among the variables. The third was when a mathematics professor teaching junior/senior-level numerical analysis expressed frustration that his students were stymied by a request to explain how to find a linear approximation to a differentiable function in a neighborhood of a specified point.
I know that Penn State is not unique, that most of us have heard similar complaints. I know that those who taught calculus to these students are convinced that they covered these topics, that there should be no excuse for these students not to know them. And I know that the students were not lazy or inattentive, that they earned good grades in the calculus classes they took. But these students never learned how to transfer their skills and understandings from the mathematics classroom to other disciplines. It is precisely this ability to transfer that partner disciplines need their students to learn from us.
This point is made repeatedly in the Curriculum Foundations Guide . The chemists wrote that chemistry students “must learn how ‘to listen to the equations.’” The physicists wrote that “the most important factor is that students gain enough active understanding that they are able to think through and solve a wide variety of problems involving the fundamental concepts in a wide variety of contexts.” The electrical engineers wrote that mathematics courses “should emphasize concepts and problem-solving skills more than emphasizing repetitive mechanics of solving routine problems … There is often a disconnect between the knowledge that students gain in mathematics courses and their ability to apply such knowledge in engineering situations.”
Teaching for transference is hard, but we have learned a lot in the past twenty years about what it takes to accomplish this. The lessons we have learned are summarized in the three bullet points in this recommendation. Students learn how to apply mathematical insights to problems outside of mathematics by practice doing it. This is what modeling is all about. Modeling is seldom done well, especially when attempted within the context of a course with a prescribed set of skills and concepts that must be mastered. Real modeling is messy and time-consuming. If we want students to learn how to transfer these skills and concepts, we cannot teach the mathematical topic and then seek applications. We must start by giving students a problem that does not quite fit into any of the categories they know how to handle. We then help them to expand their mathematical knowledge so that they can find a solution to this problem. We need to constantly challenge them to explain and clarify their thinking as they work through a problem, both for their own benefit and to assist the other students who are trying to learn how to attack an unfamiliar problem. And we must insist on clear written and oral explanations. As we know, nothing clarifies an uncertain concept in our own minds so much as the need to explain it to someone else.
There is a fourth incident from my years at Penn State that stands in sharp contrast to the previous three. In my last year there, I taught calculus from David Smith and Lang Moore’s Project CALC: Calculus as a Laboratory Course . It is a modeling course. Every calculus topic is introduced as part of a solution to a modeling problem with which the students must grapple. At the time, such an approach was totally unfamiliar to my students. Early in the semester, they found it frustrating and agonizingly slow. They wanted me to “teach” them the mathematics. But by the end of the first semester, they expressed and demonstrated a confidence in using mathematics that they had never felt before. Most of those students continued into the second semester. They set the pace, and they moved much faster than I had ever seen a calculus class move through and master that material. Midway through that second semester, one of my students came to thank me. She had just taken an engineering exam on which her mind had gone blank. She could not remember one of the critical formulas that she needed. And then, she said, she remembered the skills she had learned in my class. Rather than relying on a memorized formula, she translated what she knew about the physical relationships into mathematics and solved the problem.
That is the goal we really seek for our students. She had learned how to use
mathematics. May all our students be given this opportunity.
 David Smith and Lawrence Moore, Project CALC: Calculus As a Laboratory Course, www.math.duke.edu/education/proj_calc/
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|David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at email@example.com.|