LEARNING NEW CONCEPTS

Ten of the eleven students interviewed said that they had learned new concepts in Math 30 and that the opportunity to participate in conversation principally facilitated by the problem solving groups was an important factor, thought my no means the only important influence, for example, some students mentioned the value of the mini-lectures. In at least one instance a student indicated that they learned new concepts exclusively from the lectures but the discussion fostered a deeper understanding. Among the areas of mathematics or specific topics mentioned by students where they learned new concepts were: graph theory, geometry, the Fibonacci numbers, mathematical induction, binomial expansion and Pascal's triangle.

LEARNING NEW TECHNIQUES

There was, even a stronger affirmation that new techniques were learned with every student responding positively. One student remarked that this was only to a certain extent and the only new method that was understood stronger was the pigeonhole principle. Five other students also mentioned the pigeonhole principle. Other methods also mentioned included: the extrema principle, coloring (two students) and invariants. Other things that students mentioned in responding to this question is that they learned to look at problems differently, that the techniques were more ingrained in their mathematical processes for having used them and that they learned to simplify problems.

LEARNING NEW STRATEGIES

On the question of learning new strategies the answers ranged from a simple yes (two students) and yet another who said, "Yes, some heuristics." to more specific and detailed answers. One student said they had picked up the strategy of relaxing conditions. Another said, "Definitely. (Before) I didn't have a strategy to do a proof." Others cited were: generalize, specialize, simply. "These were used a lot" one student offered. Another student opinoned: "I definitley learned new strategies. By talking with more knowledgeable students (in groups) they shared their approaches and from this I learned new strategies." Yet another student said this: "Yeah (I learned new strategies) by doing lots of problems, a diversity of problems."

REINFORCE KNOWLEDGE/TECHNIQUES/STRATEGIES

An overwhelming majority of the students thought that knowledge, both of concepts and techniques which they brought to the course was strengthened over the course of the quarter. Some students responded with a simple yes, or said it occured "a little bit." In most other cases there was more detail provided and the answers were stronger. For example, one students said, "I took Math 100 in Fall 1999. In Math 100 we did some set theory, pigeonhole principle and induction. They were reinforced in Math 30, in partcular the pigeonhole princial." "Yes, for example, I became better at building a proof." "Quite a bit. It helped me to conceptualize processes and to do computation. The course helped me to apply these methods." "Think it did. I had more opportunity to practice induction as well as general skils of problem solving." "I think it definitely reinforced previous knowledge. It forced me to go back and review, for example, probability." "Induction was already second nature to me but pigeonhole was reinforced in class." "I didn't come with much knowledge, tecniques or strategies. I learned more about how to approch problems then any specifics."