### Final Responses

Aarnout Brombacher

#### A. Unsolved Problem(s)(August 15, 1997)

Steinhaus (solved August 13, 1997)

Until a few days ago, the Steinhaus problem remained unsolved. That is I was trying to develop a general solution for the ratio of the area of the inside triangle to the area of the original triangle for any value of k. However, I finally conquered this task on August 13, 1997 - as with many problems, once you finally find a solution the problem seems to have been less difficult that the hours spent on the problem would have suggested. In any case, I started the problem by developing a GSP model that illustrated aspects of the solution (but it gave no hint of how to find the general solution). Remembering a mention of the problem in the Mathematics Teacher of May 1996, I looked at that volume and found a formula credited to Steiner there - the reference for the citation, however, did not contain either the formula or the proof! After getting frustrated with a similar triangles approach, I tried placing the original triangle in a Cartesian plane with A(0, 0), B(b, 0) and C(c, d) and using coordinate geometry - while this approach would have worked - the algebra simply got too much! Finally, after throwing my previous similar triangle attempts away and starting afresh a solution revealed itself after only a short time!

Three Circles - though not one discussed in EMT 725; this problem remains a great challenge. (unsolved)

While I have not worked on this problem for a long time it remains one that teases me - I have been able to establish a construction for it and have even found a second construction for the solution. The solution to the actual question evades me - I have tried several approaches too long to detail here, some using the knowledge of the construction as the basis, others involving similar triangles, I even tried a linear regression type approach to at least determine a hint of the solution I was aiming at - nothing! I know that I will work on this problem for a long time to come......

Chicken McNuggets and Unit Fractions (unsolved) - no write-up to link to.

These two problems are unsolved not because of their complexity but rather because I have not spent enough time on them yet. I have found the "answers" for both problems - that is I know the solution and I am even satisfied that I know the "answers" to be correct what remains, however is to find some elegant algebraic (or other) explanation that demonstrates the validity of the "answers" and also generalizes the problem. With time I will resolve both of these I know.

#### B. Proposition Debate

Proposition: Problem solving can not be a central part of the mathematics curriculum in the secondary school because it takes too much time. There is too much other material to be covered.

I will present a counter argument to the proposition.

I believe that the very wording of the proposition demonstrates the erroneous foundation of the statement - the statement differentiates between problem solving and other material. If we view problem solving as another topic to be taught then the proposers of the debate may have a point - the curriculum is full enough and further topics would be tough to deal with. On the other hand, and this is my point, you cannot separate problem solving from the learning of mathematics, at least not in my mind.

In his paper Problem Solving in Context(s), Alan Schoenfeld (1988) makes the following remarks:

At its core, doing mathematics is fundamentally an act of sense-making, an act of taking things apart (mathematically) and seeing what makes them tick. ... Every theorem is, in essence, a statement of the following type; 'thing fit together in a particular way for the following reasons.' In short, mathematicians spend most of their time making sense of things.

...

For students to see mathematics as a sense-making activity, they have to internalize it as such. That is, they need to learn mathematics n classrooms which are microcosms of mathematical culture, classrooms in which the values of mathematics as sense-making are reflected in everyday practice. For Mathematics education, then, the issue is a cultural one: How can we create classroom environments which are microcosms of the right mathematical culture?

I read this paper in 1990 and these remarks changed my attitude about mathematics teaching and learning. I began to interpret mathematics as the act of solving problems - finding answers to questions and no longer saw it, as I had experienced it for much of my undergraduate work in the subject, as a meaningless collection of rules and tools. One might ask how it was possible to complete a whole degree (never mind a school career as a student) not really enjoying the subject. As I reflect, I think that I had a love for things and problems numerical, I liked puzzles and brain-teasers and enthused by my father's interest in these matters, I continued to have fun. Enjoying these conundrums etc. it seemed "natural" that I should pursue studies in mathematics - for there was, I was told, a link between these things that I enjoyed and mathematics. Well, not until reading the Schoenfeld paper did it strike me that the two were the same, it was just that I had never experienced the two in the same way.

After reading the paper and digesting its message my teaching changed dramatically. I structured my teaching around problem solving - that is I tried to teach mathematics as a tool for solving problems. Let me hasten to make the remark that I do not limit problems here (in the way that some discussions will) to "real-world" problems. No, I believe that we can structure much of the mathematics we teach as a response to some problem - we can teach the process of completing the square as a response to the problem of trying to find the roots of a quadratic. I hope that I am being fairly honest when I say that for the last six or seven years, as I have taught mathematics, I have taught it through problems.

Based both on conviction and experience, I would argue that problem solving should be the vehicle for teaching math, if we choose this attitude then time is not an issue, I go as far as to say that time is saved because students are much happier with the work they are doing, more mathematics, particularly more meaningful mathematics is learned and the experience of mathematics becomes very much more rewarding.

In closing, I would like to make two observations. While I love teaching, and teaching mathematics in particular, the discovery, on arriving at the University of Georgia (Athens) to pursue a masters degree in mathematics education, that I would be expected to do pure math classes was a very unhappy one - I think, retrospectively, that it was a good thing that I did not know this before arriving here (I'm serious when I say that - I'm not sure if I would have come). I have been fortunate though, I have been taught by two math lecturers - though very different in their approaches, they have both (one maybe more than the other) exemplified mathematics as a sense making activity. In addition, the experience of the three EMT classes (668, 669 and 725) in which we do a great deal of mathematics and where problems form the basis of our discussions has done a great deal to help me gain confidence in my capacity as a mathematician - a label that I am gaining comfort in wearing. I really wonder if I could have been a good mathematician, if my school and undergraduate experiences of the subject had been different? - if I had learnt mathematics as a sense-making, problem solving activity.

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