This question was intended to see if students could identify appropriate problem solving techniques in context, that is, as applied to a specific and concrete problem. The problems here were chosen because about half the different methods explicitly discussed in class are represented. Before summarizing the student responses I will briefly discuss the most efficacious approaches.

i. This is a good example of a problem approached by doing an investigation - namely take different values for n, say between 1 and 5 or 6, recording the results and looking for a pattern. If a pattern is found, then this becomes a conjecture and one would then proceed to prove that this is the general result by mathematical induction.

ii. This is a classic example of a pigeonhole principle problem. We make 15 "boxes" which are pairs of opposite points where the 30 people are sitting. Since there are 16 men, two must go into the same box and then they are sitting opposite each other as required.

iii.This is an example of problem that is best attacked using the invariance principle: find something that remains the same under all of the transformations. In this case label the vertices in some order by 1`,2, ..., 20 and let the value of vertex i be w_i. Let W_e be the sum of the vertices with even index and W_o the sum of vertices with odd indices. Note that under the transformation one even vertex and one odd vertex are each incremented by 1. This means that no matter how many transformations are performed the difference W_e - W_0 always remains the same as its initial value. Therefore if this number is not a mulitple of 3 at the outset then it can never be the case that at sometime all the values are divisible by 3 (for in that case W_e - W_0 is divisible by 3).

iv. This is another pigeonhole principle problem and similar to a homework problem (all points in the plane are colored red and blue; prove that there exists a rectangle all of whose vertices are the same color) and also a final problem in which there are 4 colors instead of 2. In fact, the final problem and this one are actually exactly the same problem.

v. This problem is somewhat ambigious since it does not actually state what it means for two squares to be neighbors. For the purpose of this problem two are neighbors if they share at least a common vertex. This is an example of an extremal problem. The maximal distance from the square with 1 and the square with 64 is seven. Since we have to increase by a total of 64 -1 = 63 in at most seven steps, at least one of those steps has to be 9 or more.

SUMMARY OF STUDENT RESPONSES.

I. This was the problems the students had the most success with. Eleven students worked on this part of the interview, one student declined because he/she was not informed prior to the interview that actually working on problems while being observed was part of the process. Of the eleven who did participate in this question every one mentioned the use of inducation and all but one said they would figure out some values for small choices of n and look for a pattern and this would be what they would do the induction on. When asked what feature of the problem suggested this several students mentioned that it had an integer parameter it was a statement about natural numbers.

ii. This was the second most successful problem for the eleven students. Six said with confidence that they would apply the pigeonhole principle and the features they mentioned that alerted them to this were: the unequal numbers of men and women, the requirement of showing that two have some characteristic, more men than women. Other responses did include elements that could lead to a solutuion: "This problem is like the problem about kids in chairs. I would start trying to disprove it and look for the reasons why there have to be two men opposite." "I would draw a diagram. I would start assuming it wasn't true and each time I put a man down there would be a woman opposite and then I would run out of women." "I would set up a correspondence." "I would draw a picture. The first and sixteenth places go together. I would look at different cases. There are more men than women. Not pigeonhole. Yes. No. Maybe." "I would make it easier by using smaller numbers and look for a pattern."

iii. This problem had about equal success as compared with the previous one. Six students mentioned the method of invariants and had the idea that it was like a problem considered in class. Among these comments were: "I would work backwards and see what needs to happen to make all divisible by three. This is like the cube problem." "This is an invariant problem. We are changing the intergers at the vertices and need to see what changes and what is the same." "An invariants problem. Performing sequences of moves, transformations/operations we need to figure out what stays the same." "This is invariants. We did a similar problem." "I would make the problem smaller. It is an invariants problem, similar to the cube problem." "Invariants. Look at the integers from 1 to 20." Other comments included: "I would make the problem smaller. I think it is like the checkerboard problems." "It is an extremal problem." "I remember problems of this sort. Use congruence, reduce to a smaller gon." "I recall the cube problem." "I don't know."

iv. Only three students gave fairly close answers though a few other may have been on-track and could have come upon an appropriate method with more time. The four closest responses were: "This is a pigeonhole problem when formulated differently in terms of colors." "Pigeonhole, prove that there exists a rectangle given all possibilities." "Pigeonhole. Reduce to a line segment case." Other "close" responses included: "Remember a problem similar to this. It is similar to the triangle problem. I would start drawing.""I remember a similar problem. How many combinations of five points. Will repeat. Then get a rectangle." "I remember something similar." "This is like a coloring problem." Still other responses that did not suggest closeness to the appropriate method were the following: "Solve by contradiction." "Is it a graph theory problem?" "I have no clear idea." "This problem is difficult to understand. I understand the given but not what is to be found."

v. Only one students recognized this as an extremal problem. The other responses were all over the place. Several students suggested pigeonhole, another mentioned coloriing, and still others would have tried to investigate the problem by "making it smaller" presumably by using a smaller board. This is not surprising because of the two to three hundred problems totally submitted by the students, including the final, a very very small number were of this type. Students found this the most bewildering and most difficult to master.

In all, I found the results of this exercise quite satisfying: after more than half a quarter in which nearly all the students once again found themselves primarily in situations where they were asked to work on routine exercises they had retained several problem solving methods in context and recognized features of a problem that suggested one method over another.