I sometimes think that the very precision of mathematics that gives it much of its power can lead to problems when it comes to mathematics education in the lower school grades. I was reminded of this recently when both sides of a debate about mathematics standards emailed me about a particular issue.
The point of contention was a statement that pupils should be aware of the relationship between perimeter and area of plane figures. From a strictly mathematical perspective, there is of course no relation between perimeter (meaning the length of the perimeter) and the area (that is, the numerical figure we ascribe as a measure of an enclosed plane region). A given perimeter length can enclose a whole range of different areas, going down as close to zero as you choose and up to a maximum determined by the kind of figure you wish to construct (rectangles, rectilinear, ellipses, etc.) As a reader of MAA Online, you know that, so do I, and so did the two antagonists who wrote to me.
But then, we all learned about lengths and areas many years ago, and the concepts are familiar to us. Things seem very different to someone learning mathematics for the first time. Recently I had occasion to talk with some researchers who had worked with several classes of young learners using a neat little computer tool that allowed them to use a given length of "perimeter-wire" in order to enclose rectilinear fields (of the farming variety) of several given target areas. Except for the case when the target area could be achieved with a rectangle, the students found it immensely difficult, and in many cases impossible without assistance. This known difficulty was, of course, the point of the activity!
Using the tool, the students were able to experiment with the way different rectilinear configurations of the same overall length produced enclosed regions with different numerical areas. In a variant, given a target area, they had to select the perimeter length from a given menu. How would you describe the activity? I would say they were investigating the relationship between perimeter and area. Indeed, I suspect they came out of the exercise having recognized that perimeter (both shape and length) and area (both shape and numerical measure) are related. Change one factor and the others change too.
So which of my two correspondents was correct? The answer was they both were. At issue was the perspective from which they were approaching the issue: formal mathematical as understood by a mathematician, or cognitive-conceptual in the elementary or middle school classroom.
In writing state standards, which is what occasioned the debate into which I ever-so-briefly injected my two-cents worth, it is surely important to find terminology that captures both perspectives. As I have argued previously in this column, while it is educationally crucial that we understand the way new material will appear to someone learning it for the first time, and design our instruction accordingly, we should not proceed in a fashion that leads them to adopt incorrect concepts or form false mental models, that must later be undone. (Not least because the evidence is clear that in many cases no amount of subsequent "corrective instruction" can eliminate a first-acquired, false notion.)
Those of us in mathematics need to be aware that our love of precision, so important within the discipline, can cause problems if taken blindly into the school classroom. Definitions matter. Words matter. The words we use as mathematicians have other meanings too. Knowing how to bridge the two cultures and their two linguistic usages is part of being a good mathematics teacher.
Devlin's Angle is updated at the beginning of each month. Find more columns here.