A long time ago, John Conway posted a very thoughtful response to a posting of mine about mathematical habits of mind. Below is a belated response from my project. I would ver much like to continue this discussion. Thoughtful criticism from intelligent people is the best way to improve the work we're doing.
`` However, there is something about Michelle's paper that worries me TREMENDOUSLY. I'm sure it's mostly accidental, but it doesn't seem to mention what I regard as the most important thing of all about the mathematical experience, from both the practical and theoretical standpoints.
This is the habit of PRECISE thinking about PRECISELY worded problems. This is the most important thing to teach, and should precede ANY kind of thinking about FUZZILY worded problems, in my view (and I'm speaking here as a teacher, rather than as a professional mathematician).''
He is right (of course); his worry is due to an accidental slip (or, more precisely, to a lack of a precise description of what we mean) in the paper.
Here's an annotated version of the paragraph in question:
``There is another way to think about it, and it involves turning the priorities around. Much more important than specific mathematical results are the habits of mind used by the people who create those results, and we envision a curriculum that elevates the methods by which mathematics is created, the techniques used by researchers, to a status equal to that enjoyed by the results of that research.
[This includes (1) precise thinking about precise problems and (2) precise thinking abut not-so precise problems. It also includes using (3) heuristics and intuition to come up with plausible conjectures. Examples from some curriculum materials we are developing: (1) Students are given the explicit task of cutting a rectangle up to form another rectangle on a different base. (2) Students are asked to define what they mean by ``best'' if the want to find the best spot for an airport that will serve three cities. (3) Students are asked to imagine a line segment, starting at one base of a trapezoid, moving up parallel to that base (connecting points on the non-parallel sides), stopping at the top base, and are then asked to use this thought experiment to conjecture a general formula for the median of a trapezoid.]
The goal is not to train large numbers of high school students to be university mathematicians, but rather to allow high school students to become comfortable with ill-posed and fuzzy problems, to see the benefit of systematizing and abstraction, and to look for and develop new ways of describing situations. ''
[ We should have emphasized precision when we wrote ``to see the benefit of systematizing and abstraction, and to look for and develop new ways of describing situations;'' it was certainly on our minds.]
Right after that,we say:
``While it *is* necessary to infuse courses and curricula with modern content, what's even more important is to give students the tools they'll need to use, understand, and even make mathematics that doesn't yet exist.''
A curriculum organized around habits of mind tries to close the gap between what the users and makers of mathematics *do* and what they *say*. Such a curriculum lets students in on the process of creating, inventing, conjecturing, and experimenting; it lets them experience what goes on behind the study door *before* new results are polished and presented. It is a curriculum that encourages false starts, calculations, experiments, and special cases. Students develop the habit of reducing things to lemmas for which they have no proofs, suspending work on these lemmas and on other details until they see if assuming the lemmas will help. It helps students look for logical and heuristic connections between new ideas and old ones. A habits of mind curriculum is devoted to giving students a genuine research experience.
[There are two main thrusts of the paper: (1) to call for a mathematics experience based on the *interplay* between deduction and experiment that is so crucial to doing mathematics (that's the integration of the logical (precise) and heuristic (less-precise) ways of thinking) and (2) to concentrate on the way mathematics is *developed* as opposed to the way it is presented. For example, students are introduced to proof as a method for communication *and* as a technique for discovery. The communication of an argument is a rather precise activity, and we show students a few ways to do this (two-column, paragraph, as well as some presentation techniques from Russia, China, and Israel). The *search* for an argument is a more fuzzy activity, which makes it harder to teach, but we spend a great deal of time developing some general principles for navigating through the morass of detail that you face when trying to establish a conjecture.]
So, the point is: we certainly agree that precision and precise thinking are central to mathematics, and, if that didn't come across in the paper, it's due more to shoddy writing than to shoddy intent. The paper will be appearing soon in JMB. Maybe some letters to the editor published in subsequent issues can clear this up.
-- Michelle Manes Education Development Center firstname.lastname@example.org