by Viswanath Sankaran, Mathematics
Teaching Effectiveness Award Essay, 2002
Most freshmen view calculus as a set of rules to be memorized and applied mechanically to solve problems. While this attitude works by and large for their first differential calculus course (Math 1A), Integral Calculus (Math 1B) poses a new challenge. Here, most problems involve a crucial “guessing” step (called the substitution) that transform them into more amenable problems. An “insightful” guess leads to the solution, a “wrong” guess can get one stuck.
So the question is: How can a teacher communicate this insight? In lectures, the instructor gives a number of examples in which he/she provides this main step and the students carry out the rest of the solution. With practice, one hopes, the students will acquire this skill. But in reality, students just memorize what substitutions to use for what problems and are often clueless when faced with an unfamiliar problem on the exam or elsewhere. The thought process by which experts arrive at the correct “guess” is mysterious to the student. Teaching students to think, first demands we demystify this thought process.
In discussion sections, I addressed this by first giving the whole section a problem and asking them to work on it for 3 minutes. At the end of this time, I would solicit suggestions for the crucial first step. I typically got 2-3 different suggestions. I would divide up the board space and write each suggestion on a different part of the board. The key thing was to treat each suggestion with equal respect and betray no knowledge of which ones were “good” and which ones “bad.” I would also ask the suggesters why they came up with the idea they did and usually their responses would convince the class that these ideas were at least plausible and definitely worth trying. I would then start out on one of the threads, with me acting as scribe and the class telling me the next step all the way. We often hit a dead-end and the class would be unable to think of a way to proceed. I would then point out where exactly the difficulty lay and once they were convinced that this method would not work and why, we’d go on to the next thread. Many times, the method would get us tantalizingly close to the solution and I would often say “We’re almost there. We just need to take care of this little extra term here. Can someone see how to modify our initial guess so that this term disappears?” and usually someone would! If none of the threads led us to the solution, I would lead the class, by an analysis of what failed, to the correct guess.
The above process typifies what goes on in the expert’s head during problem solving. This approach taught them that there is nothing wrong with having “wrong” ideas. It’s a natural part of everyone’s thought process. They realize that its not just them alone — their classmates come up with wrong ideas as well. One just doesn’t hit upon the right idea straight away. It taught them to critically analyze why an idea fails and to use that wisdom in finding the right idea. It built mathematical insight!
The fact that I respected all their suggestions as valid starting points for the problem largely removed their fear and hesitancy in expressing half-baked ideas. A testimony to the whole approach was the fact that as the semester progressed, most students were able to anticipate troubles few steps down the line without actually doing the problem. Their problem solving skills sharpened, as evidenced from the responses I progressively got to my questions. They also lost all fear of speaking out — and most importantly many of them told me that they had had a lot of fun doing math that semester!