What Are Your Problems?
In his article "The Two Cultures of Mathematics," Timothy Gowers argues that there are two types of mathematcians, the theory builders and the problem solvers.
He offers a small test to determine your type or what class you belong to, based on your reaction to two statements:
- The point of solving problems is to understand mathematics better.
- The point of understanding mathematics better is to become better able to solve problems.
From this same perspective, Jeremy Alm and David Andrews argue that unfortunately, most mathematics books, especially at the undergraduate level, seem to have been written by theory builders for theory builders.
That is, in their characterization (and this should sound too familiar), math texts (and thus too many classroom lessons) are "organized linearly, so that an understanding of the (n+1)st chapter, notation and all, depends on understanding the previous n chapters. The techniques and tools are developed before any discussion of the problems one can solve with them...."
In contrast (and for the other type of developing mathematicians), Alm and Andrews suggest that math texts should be organized around a set of problems: "Finding a solution to each problem would be the goal for the students, around which the necessary concepts and mathematical machinery would be built...."
A good idea...oddly, an idea that many math educators claim is the way math should be taught and experiences, yet we all seem to fall into line with the linearity of mathematics. But, does this linearity acutally exist, or is it imposed from the theory-side of our mathematical understanding?
And, if such a problem-solving focused text could be written for secondary schools, what problems would you choose? That is, given the totality of secondary mathematics, if students master the techniques and understand the concepts involved, what important problems should they be able to solve?
On the generic level, Alm and Andrews offer these criteria for the focus problems:
It would be interesting to sit down with each of you to discuss the list of problems you would suggest. The entire process seems daunting, but it not only could be fun....but is perhaps necessary.
- Comprehensible, possibly involving concrete examples
- Uncontrived...not "make-work"
- Compelling to the teacher, allowing his/her enthisiasm to rub off on students
- "Big"...involving multiple ideas and techniques
- Possibly needing multiple weeks to explore before solutions appear
- Small in number, but well chosen
Source: Adapted from J. Alms & D. Andrews' "Solutions in Search of Problems," Notices of the AMS, August 2012, pp. 963-964