David M. Bressoud April, 2005
Recommendation 2: Develop mathematical thinking and communication skills
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Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind. More specifically, these activities should be designed to advance and measure students’ progress in learning to
All of us want our students to learn to think mathematically. I know of no instructor whose only goal is to have students memorize formulae. Most students want to understand and so be able to use the mathematics they are learning. The obstacle is not intent. The obstacle lies in the real cognitive barriers that exist throughout the undergraduate curriculum. We who teach mathematics ignore these barriers at our peril.
Bill Thurston put it well when he said:
“… in classrooms, … we go through the motions of saying for the record what we think the students ‘ought’ to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material ‘covered’ in the course, and then grading the homework and tests on a scale that requires little understanding. We assume that the problem is with the students rather than with communication: that the students either just don’t have what it takes, or else just don’t care.
Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs.” 
There is a substantial body of work on mathematical reasoning and problem solving, led by researchers such as Dubinsky, Schoenfeld, and Tall. One of the best and most accessible introductions to this field is the address to the International Congress of Mathematicians given by David Tall in 1994, “Understanding the Process of Advanced Mathematical Thinking.”  Amid quotes from Hadamard, Halmos, and Thurston, Tall describes many of the cognitive obstacles that students face and suggests some means to overcome them.
There is no magic bridge course that will turn neophytes into mathematicians. That is why the emphasis in this CUPM recommendation is on every course. In every course, the instructor must be aware of the various strengths and weaknesses with which students begin, of the difficulties they are likely to encounter, and of techniques that can help them develop the ability to think mathematically. Every course, if it is teaching meaningful mathematics, will introduce new concepts and new ways of relating mathematical ideas. Every course should stretch and strengthen students’ ability to think mathematically. This is not to say that there is no place for a bridge course or a course that focuses on problem-solving. Such courses can be very useful for helping students break out of the template approach to learning mathematics. But these courses alone are not sufficient.
It is not easy to recast a mathematics class so that it helps students achieve and requires them to demonstrate understanding, but we can draw upon a wealth of experience and resources to supplement our efforts. Under Recommendation 2, the CUPM Illustrative Resources includes information and references for
This last activity, writing mathematics, is one of the most effective tools I have found for forcing students to think about mathematics and for helping me to understand how they think about it. I would no longer consider teaching a math course in which I did not, at some point, require students to write about how they solved a challenging problem or found their own proof of a theorem. The literature on the use of writing in teaching mathematics is vast. If you are not yet using writing in your classes, the CUPM Illustrative Resources can help you find an approach that will work for you.
 Alan Schoenfeld. Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. Chapt. 15, pp 334–370, in Handbook for Research on Mathematics Teaching and Learning (D. Grouws, Ed.). New York: MacMillan, 1992. Also available at http://gse.berkeley.edu/faculty/ahschoenfeld/Schoenfeld_MathThinking.pdf
 David Tall. Understanding the processes of advanced mathematical thinking. L’Enseignement Mathématique, vol. 42 (1996), pp 395–415. Also available at www.warwick.ac.uk/staff/David.Tall/themes/amt.html
 William P. Thurston. On proof and progress in mathematics. Bulletin of the AMS, vol. 30 (1994), pp 161–177.
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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 firstname.lastname@example.org.|