Depth Vs Breadth and the difficulty of getting students to redraft work

One of the primary objectives of the course was to influence students beliefs about what it means to do and learn mathematics, especially that problem solving is an ongoing process and that we are never really done; even when a solution to a problem has been obtained this can be the impetus for seeking alternative ways to solve the problem or else as the springboard for extensions and generalizations. This objective was perhaps the least achieved in the course. Though students were encouraged to redraft their work and resubmit problems and were often given suggestions for continuing or generalizing a problem in only one instance did a student do so. This occurred whether the comments were short, of medium length or quite detailed and whether directed towards one of the less experienced students, a student in transition from lower division mathematics or a relatively mature mathematics student. I have the impression that despite my direct and indirect efforts to inculcate this process view of solving problems the students preferred a sense of closure when working on a problem even if it was not completely or correctly solved so that when they wrote it up they did so at that point when they thought they had put enough effort into it and wanted to move on to another problem. I also think the students were driven by a personal need to feel productive as measured by the quantity of problems they worked on rather than the overall depth of treatment they gave to a particular problem. Moreover, during my introductory remarks about the course I told students that the main criteria for their evaluation would be their effort (not necessarily the number of correct problems the submitted) but this may have been misinterpreted and contributed to this phenomenon.

The difficulty of getting students to take risks and to venture into new terrain

A related observation is that I think a majority of students did not realize their full potential for growth during the quarter because they preferred to stick to problems with which they were familiar and were unwilling to take the risk of working on new types of problems. As a consequence these students learned fewer techniques and methods than the others. Both the group that preferred the comfort of the familiar and the smaller cohort of risk-taking students contained inexperienced and mature members so this was not the deciding factor. Moreover, while the latter cohort appeared to contain exclusively students who were very confident in their mathematical abilities, some confident students were also in the group that did not venture out very much. I think this particular group of students, confident but hesitant to take risks, was very much influenced by expectations formed in previous mathematics courses where the problems students are asked to do are almost always familiar and where the success of a student is determined by the quantity of such problems answered correctly on homework, quizzes and examinations and only indirectly accounts for the student's overall intellectual growth.

Students did acquire new content knowledge or strenthened extant knowledge

It does appear that a majority of the students did learn some new mathematical concepts and were even more successful at learning some new methods of problem solving and that previous knowledge and techniques were strengthened. First of all when contrasting the student responses to familiarity with terms for concepts and techniques used in course between the initial questionnaire and the one filled out at the end of the quarter there was an increase in the number of terms that the students indicated that they were certain/thought that they understood and a decrease in the number of items that they were sure that they did not understand. The greatest increase was from concepts drawn from elementary number theory and elementary combinatorial analysis and in particular problem solving methods. As is to be expected what material was learned for one student may have merely been strengthened for another depending on the backgrounds of these students. So, all of the experienced students had previously taken the department's transition course, Introduction to Proof, and were familiar with mathematical induction. For these students the multiple opportunities to use induction and to make the choice of when to use it strengthened their application and there was marked improvement in their exposition using this technique in the problems that they submitted. Additionally, this group was very successful at investigating problems involving a numerical parameter in order to determine what is to be proved and then to apply mathematical induction to its eventual demonstration. In fact, every student became adept at such investigations, in recognizing patterns in the form of well known sequences and in formulating a conjecture. During the interview process every student could identify this as a problem solving technique and could identify this as an appropriate approach to one in a series of problems.

Nearly all of the more advanced students had previously had a course in elementary number theory. I think many of these students began the class familiar with, but had not mastered, concepts such as congruence modulo an integer as indicated by responses on the original questionnaire and early problem solving. They could recall some theorems and properties but initially did not know how to apply this knowledge to problems. However, they did display growth and a better understanding of these concepts later in the quarter. One student among the experienced cohort of students and one of the "middle level" students also picked up some of the concepts from graph theory which were presented and these students were successful at applying this knowledge to problems. Also, a number of students became more adept at "counting smart" from working on enumeration problems from elementary combinatorics. During the interview process one student mentioned that as a consequence of working on the problems he/she finally understood Pascal's triangle.

For the students who had not taken the department's proof course this was their first introduction to mathematical induction. Each of these students appeared to be able to recognize when to use induction and had instances in which they successfully applied this method of proof. Elementary number theory was also new to these students, in particular, properties of remainders and congruence modulo an integer N. Some of these students were able to learn this well enough to use it in solving problems. Overall I do think new concepts can be taught through problem solving but I will need to do much more planning to fully achieve this objective, in particular, more care must be given to just which concepts to introduce and how to get students to spiral back to them when they have significant freedom to choose the problems on which they work.

Students acquired new tactics and strategies for solving problems

It appears that the greatest success was in learning problem solving techniques and methods that are specific to mathematics as contrasted with more general approaches or heuristics though all students did demonstrate the use of some heuristics. First of all, it was quite common for students while orienting themselves to a problem to try to represent it in different ways. This appeared to come naturally as part of the process of putting the problem into words, symbols and diagrams that had personal meaning to the student but some students went beyond this and made an effort to translate problems into more mathematical forms so as to exploit particular resources or techniques. Second, a number of students did, at times, try to work backwards on problems or divide a problem into pieces (establish sub-goals) whose solutions would, altogether, be sufficient to provide a solution. Students often tried to get at a problem by working on a simpler problem and they also sometimes tried to relate a problem to a similar problem.

A Caveat

This latter technique can be powerful but it can also easily lead students astray. In doing so the students made such judgements based on surface features. For example, there were a number of problems where each point of the plane is assigned a label (e.g., colored red or blue) and one must prove that certain configurations arise (e.g. there is a rectangle with all vertices the same color, an equilateral triangle with all vertices the same color). Typically students treated all these as similar problems and tried to apply one result to the proof of another. While these problems shared some similar methods I know of no way to use the knowledge of one solution to obtain the other. Another example of the pitfalls of this approach is described below.

Every technique that was taught was learned by some students and some techniques were learned by every student. The technique that was best mastered was the aforementioned approach to problems with an integer parameter: do some investigation, find a pattern by comparing to known sequences, use induction. The interviews with students demonstrated that every student could recognize such a problem and describe the method. Comparable in its mastery was the pigeonhole principle. Every student could identify this as a technique learned, could remember a problem they solved using this method and nearly all could identify a problem during the interview which could be solved with the pigeonhole principle. Several students could also identify problems which suggested the use of the invariance principle and also recall this as a technique used in the course. Far less successful was the extremal principle. Very few students solved problems during the quarter with this method, fewer still identified this as a method they had learned, could identify a problem which made use of this technique, even when they may have used it on an assigned problem.

In saying that students learned these techniques it is necessary to add the caveat that I think their decision process in choosing a particular method often recapitulated strategies from other mathematics classes; they would look for specific clues that indicate whether it is an invariance problem, a pigeonhole problem or an induction problem, etc. Many of the students became fairly competent at this for the particular methods they mastered. However, students often appeared to lock onto some feature of a problem and only after lots of contrary evidence would they switch their plan of attack. For example, during the interview a problem was given where students were asked to find all solutions in integers to the equation a^2 + b^2 = 3(c^2 + d^2). One student believed that a similar problem was solved in class, namely, "Find an infinite number of solutions in integers to the equation x^2 + y^2 + z^2 = 3xyz." Superficially these are quite similar. Both ask for solutions in integers, both involve a single equation which contains quadratics in multiple variables. However, the first equation has no non-trivial solutions (a solution other than a = b = c = d = 0) while the other has, as it requests, infinitely many solutions. Only after making many, many substitutions did this student begin to suspect that there were no (non-trivial) solutions and therefore a different line of attack would be necessary.