I was working with a trigonometry student on trigonometric identities. He was expecting his teacher to announce a test or a quiz soon, and he was still uncomfortable with the topic. “I just don’t feel like I know the steps,” he told me. I’m afraid he found my response unsatisfactory: “There aren’t any steps. You just have to play with it. You’ll have to come to terms with that.”
When I’m tutoring and a student pulls out a question, I usually know the steps. However, occasionally I have to say, “I don’t immediately see how to do this one. Give me a minute and let me play with it.” I have come to realize that letting them watch me solve a problem while I think out loud (formally known as “modeling problem-solving behavior”) is actually more educational than teaching them steps. They would not agree. They want a recipe, dang it, and any delay in getting there is just wasting time.
I recently learned that in Russia as early as 1st grade (or rather the Russian equivalent of 1st grade) the schools distinguish between “problems” and “exercises.” An exercise is a question that one answers using an algorithm. The question was usually assigned specifically to allow the student to practice a particular algorithm. A problem, on the other hand, is a question that one does not initially know how to approach. What may be a problem for some students may be an exercise for older, more experienced students.
I have decided to begin distinguishing between problems and exercises when I talk about them. Old habits die hard and I imagine I will need to correct myself often at first. However, the trouble with schooling is that there are too many exercises and not enough problems. I think the first step towards addressing this issue is to point out the difference.