I hear students complain often (though I don’t teach math) that in their math courses, they do fine on the homework, fine on the quizzes–even fine on their midterms sometimes–(in calculus) but they fail the final, so they fail the course. The students complain that they get problems on the final that they “have never even seen before.” Isn’t that the nature of real life problems that we can’t solve right away–they look brand new?
I was talking to my brother who is a professor of math and insists on doing group work in his calculus courses. Here are some strategies he employs to get students to take responsibility for applying solutions he’s already explained in lecture to new problems rather than depending on him to show them how ONCE AGAIN–leaving them still clueless.
1) He NEVER goes over the homework in lab or re-works problems from his previous day’s lecture. Instead, he gives problems in lab that demand they apply the concepts from the previous day’s lecture. Inevitably, and repeatedly, though only for a few weeks, students come to lab never having looked at their lecture notes. Then, they say they can’t do the problems because they’ve never seen that kind of problem before. He makes them get out their lecture notes to see WHERE (not if) material that will help with the solution is located. The first time, he says, he does give them hints to help them find the material, but after that he just makes them work as a team to find the material (by then it should be pretty obvious that it’s on the previous day’s lecture notes.) Then, he sets them to work figuring out how to apply the concepts. Soon they get over complaining that he’s giving them problems they’ve “never seen before.” In other words, they are held accountable for attending lecture–taking reasonable notes–and sharing with others in the effort to figure out problems for which he’s provided reasonable background for finding solutions. Caveat: He says that if he makes up a problem that lends itself to a formulaic solution, it is no good because the bright light in each group will set about showing everyone else HOW to use the formula. This means the instructor must design the problems with care and a clear comprehension of the concept s/he expects the students to apply (which needs to be fundamental, but not obvious). The idea here is that to grow we need to face a reasonable degree of perplexity. The challenge: this is WORK for the instructor.
2) The second “rule” he employs says that students should always ask him intelligent questions when they get stuck on a TEST problem. The principle behind this is that he knows that when they’re stuck, have tried to move on and are getting no where fast, it’s because they’ve made an error earlier on that is making progress impossible. He knows that if they can fix the place where the bungling took place, they will be able to solve the problem if they really know the concepts. So he helps them find the place where the blooper is, preferably by getting them to ask questions, but sometimes by just telling them to look back, and back, and again back until they locate the mistake. If they’re clueless, he’s able to see it right away, and so can the students, so they realize they need to work on the concepts more.
3) Finally, he always does the exam himself during the exam period and posts the solutions directly after. The students are then able to go check the solutions immediately and then go and ask him questions about what they did if appropriate. It serves as a great learning moment, but also gives him a chance to recognize if he wrote a problem that, indeed, could not be solved–then he informs them of the fact on the solution board. And sometimes the students, by asking him questions, either during the exam or during the after-exam conference, help him see an error HE made in his rush to solve the problem or to see an alternative route to the solution. This, I believe, is brilliant teaching–no wonder math education is one of the most popular majors at U of Mich. Flint. I believe it achieves the goal of encouraging autonomy (and interdependence). And it clearly gives the students the kind of confidence you are concerned about here. We can all think about the principles he’s applying here and find ways to structure our learning environments to make similar things happen. It takes work and courage (and autonomy on the part of the professors, because many dept. colleagues don’t believe his method can work for them–not enough time to COVER MATERIAL!)
Hope this serves as a bit of an inspiration to someone. It did to me!
–Roberta Gilman, Faculty, Linguistics, Univ. of California, Santa Barbara, CA