As happens with amazing frequency, Christopher Danielson said something interesting today on Twitter.
Dear children’s publishing industry: It is NOT a property of rectangles that they have “two long sides and two short sides”. #ThatIsAll
— Christopher (@Trianglemancsd) July 29, 2013
And, as also happens with impressive regularity, Max Ray chimed in with something that led to an interesting conversation — which, in the end, culminated in my assertion that not everything that is mathematically true is pedagogically useful. I would go further and say that a truth’s usefulness is a function of the cognitive level at which it becomes both comprehensible and important — but not before.
By way of an example, Cal Armstrong took a shot at me (c.f. the Storify link above) for my #TMC13 assertion that it is completely defensible to say that a triangle (plus its interior) is a cone. Because he is Canadian, I think he will find the following sentiment particularly agreeable: we’re both right. A triangle both is, and is not, a cone, depending on the context. It might be helpful to think of it as Schrödinger’s Coneangle: an object that exists as the superposition of two states (cone and triangle), collapsing into a particular state only when we make a measurement. In this case, the “measurement” is actually made by our audience.
When I am speaking to an audience of relative mathematical maturity, I can (ahem…correctly) say that cone-ness is a very broadly defined property: given any topological space, X, we can build a cone over X by forming the quotient space
with the equivalence relation ~ defined as follows:
If we take X to be the unit interval with the standard topology, we get a perfectly respectable Euclidean triangle (and its interior). Intuitively, you can think of taking the Cartesian product of the interval with itself, which gives you a filled-in unit square, and then identifying one of the edges with a single point. Boom, coneangle. Which, like Sharknado, poses no logical problems.
Of course, it is a problem when you’re talking to a middle school geometry student. In that situation, saying that a triangle is a cone is both supremely unhelpful and ultimately dishonest. What we really mean is that, in the particular domain of 3-dimensional Euclidean geometry, when we have a circle (disk) in a plane and one non-coplanar point, we can make this thing called a cone by taking all the line segments between the point and the base. But to that student, in that phase of mathematical life, the particular domain is the only domain, and so we rightly omit the details. In an eighth-grade geometry class, there is absolutely no good reason to introduce anything else.
We do this all the time as math teachers. “Here, kid, is something that you can wrap your head around. It will serve you quite well for a while. Eventually we’re going to admit that we weren’t telling you the whole story — maybe we were even lying a little bit — but we’ll refine the picture when you’re ready. Promise.”
Which brings me back to Danielson’s tweet. From a mathematical point of view, there are all kinds of problems with saying that a rectangle has “two long sides and two short sides” (so many that I won’t even attempt to name them). But how bad is this lie? Better yet, how bad is the spirit of this lie? I think it depends on the audience. I’m not sure it’s so very wrong to draw a sharp (albeit technically imaginary) distinction for young children between squares and rectangles that are not squares. It doesn’t seem all that different to me, on a fundamental level, from saying that cones are 3-dimensional solids. Or that you can’t take the square root of a negative number. Or that the sum of the interior angles of a quadrilateral is 360 degrees. None of those statements is strictly true, but the truths are so very inconvenient for learners grappling with the concepts that we actually care about at the time. It’s not currently important that they grasp the complete picture. And it’s probably not feasible for them to do so, anyway.
Teaching mathematics is an iterative process, a feedback loop. New information is encountered, reconciled with existing knowledge, and ultimately assimilated into a more complete understanding. Today you will “know” that squares and rectangles are different. Later, when you’re ready to think about angle measure and congruence, you will learn that they are sometimes the same. Today you will “know” that a times b can only be 0 if either a or b is zero. And tomorrow you will learn about the ring of integers modulo 6.
I will tell you the truth, children. But maybe not today.