Learning to Read

Yes, this is math class, but especially in a problems-based curriculum we are teaching reading at the same time, right?

I recently gave two classes the following problem*: “How many triangles are there whose three vertices are points on this 3×3 square grid?”

Once they put their heads together, kids are able to make nice progress on this problem.  It’s intended to teach the habit of “taking things apart” (most kids tend to say, “break it down”), and comes with a suggestion to count the number of triangles you can make within a 2×2 grid, then a 3×2 grid, then the full 3×3 grid.  Not all kids take this suggestion (and good for them), but most wind up categorizing triangles by type somehow and then counting how many there are of each type of triangle.

Looking back, I think about the little things that tripped up the kids, all of which had to do with implicit assumptions about what the problem was asking.  Some kids wanted to know if the vertices of the triangles had to be on the dots in the grid, or if they could be in between.  Other kids assumed that the triangles had to be right triangles, or else asked if it was okay to use non-right triangles; these kids thought that there was real ambiguity in the question.  I want too much to be helpful, and for the students to be able to get on with their math, so in this case I answered their questions directly rather than doing what (I suppose) I should do: telling them to read the question again.  At times I have done a better job guiding students back toward the question.  I resolve to be so good again!

Other questions that the kids had about the wording of the problem struck me as more legitimate.  One question that came up was, “if two triangles are the same shape, should we count them as different triangles?”  Here I think the kids have a case that the question is ambiguous.  After all, we are just beginning to study graph theory, in which we’re about to call several pairs of isomorphic graphs that look nothing alike “the same.”  I’ve encountered many problems over the course of teaching that, while not ambiguous to someone “in the know,” can actually be read a few different ways by an attentive student.  I suppose that teaching students the conventions  of mathematical language is also “teaching them to read.”

*this problem appears in our textbook, inspired by a problem on a Math Counts competition.