For example, if you drew an equation, you could then answer the question, “What does the equation represent? What are its implications?” Your answer, or “the answer” might just be that it represents a problem. You could then answer the question by looking at the equation and saying, “It’s a quadratic. So if the solution is 2/3, then it must be 1. And if you look at the answer as a mathematical constant, then your answer is therefore 1.” Or, you might look at the equation as part of a larger problem. A “problem” might be the building of a large, complex structure. It might be a problem for the students, to learn about the mathematics involved. It might be a problem for the student to understand how to solve the problem, whether that means drawing the equation, writing down the answer, or some combination of these things.
In fact, the problem might be so big, complex, and important that it merits all the attention it can get.
This is exactly what happened in one of my classes, a “Problem Solving and Calculus for Adults” class (P.E. 603). Students would take the same problem in their second and fourth year of college, and each year they would find a different solution. This might be something as simple as figuring out who wrote “Alice is the smartest of the three mice,” or what the answer to the equation meant, or looking at a graph of how the solution developed. In any event, it was always interesting to see what problems the students came up with. (Most were problems to do with geometry and geometry for adults.)
These students also had their problems solved by my own students for two reasons. First, the class’s goal was to teach math, to develop mathematical skills (including understanding what a problem is), and to learn to think like a mathematician (also understanding what a theorem is). I wanted to find “the good” and “the useful” answers for these students to help them learn more about math and its applications outside my class. Second, students needed a solution and this meant they needed an exact “recovery” answer. I wanted the class to be able to “come back” and repeat a problem to see what had changed since the time the student had “stayed awake” trying different alternatives.
As this experiment goes on, my students will continue to learn and develop new problems that will challenge them to learn more about their mathematical development. They
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