Breaking down understanding: Pythagorean theorem example

What does it mean to understand? In learning and teaching, often we are worried about whether what’s been learned involves true understanding or is just facts and skills that can’t be “applied” or “transferred”.

In Understanding by Design, Wiggins & McTighe present many examples of lack of understanding. One example was a question about the distance between two points on a grid. Let’s say (2,8) and (6,5). Students were assumed to know the Pythagorean theorem: a^2 + b^2 = c^2. They could solve the problem by finding the number of units between the points on the x-axis: |2-6|=4, finding the number of units between the points of the y-axis: |8-5|=3, and then applying Pythagorean theorem: \sqrt{4^2+3^2} = 5. But most students could not solve the problem! Their takeaway is that students knew but didn’t understand Pythagorean theorem.

I challenge these students even know the Pythagorean theorem. Is the Pythagorean theorem “a^2 + b^2 = c^2“, as we stated, and as most of the students could probably tell you? No!! The statement “a^2 + b^2 = c^2” by itself means nothing at all. Let’s do better: “the Pythagorean theorem states that for any right triangle with legs of length a and b, and hypotenuse of length c, the relation a^2 + b^2 = c^2 is true”. Ok, there is a statement that a mathematician and maybe even a logician would be happy with, but students may need some prodding to get to if they knew it at all.

We aren’t done because our audience is students rather than mathematicians. Mathematicians are a slightly crazy type of person who is happy with this statement. If students realized what we’ve just done, they would be appalled–and for good reason. Why? We said “any right triangle…”. That is an infinite class of things. If we made a similar statement like “any New Yorker is rude” or “any Vikings team will not win the Super Bowl”, that would be called ignorant and awful. But the mathematician is comfortable because they have extreme confidence that they can spot any right triangle in any context and say a few definitely true things about it. Now, most of our students are probably able to see a right triangle and say, yes, that is a right triangle, no, that is not a right triangle. But they may not have a fluent perceptual skill of running out in the wild and eagerly seeing right triangles like mathematicians (remember: slightly crazy).

And yet being able to detect right triangles like a boss still isn’t enough! As the student, when sitting down in front of this problem, we enter a very strange place: a grid. We can we do in grid world? We could draw a smiley face with the two points as eyes, color in squares, or make mazes. Creating a right triangle with legs parallel to the axes and hypotenuse that is the line between points (2,8) and (6,5) is just one of countless possibilities. That is legitimately considered an invention when it’s not a practiced skill, and a vast majority students who have never even encountered the idea of creating shapes to support their geometric reasoning are not going to invent it on the spot.  “Find the distance” might spark me to draw a straight between them (“‘the shortest distance between two points…’ wait, do I want the shortest distance?”). But the creation of that triangle, then the detection of the right triangle (since we may have drawn the correct lines without necessarily thinking “triangle”), then the application of Pythagorean are all steps needed to solve the problem.

So boiling down the issue here to a lack of understanding of Pythagorean theorem is, if not wrong, totally unhelpful. Nor is it helpful to say that the students are “failing to apply their knowledge”, or the student just needs to “learn more transferable knowledge”. All those sound like the responses of an obnoxious politician.

There are times when recognizing and pointing to a lack of understanding is a useful communication. There is some pattern to the student’s actions where people who know better can agree they don’t have “understanding”, even if we don’t have a perfect description of what that entails. Call this the gestalt perspective on understanding.

The philosophy here is that we can attempt to break down a lack of understanding into knowledge and skills that are missing. Call this the reductionist perspective. In this case I choose to consider the problem like some kind of environment where the student can perceive things and take actions while applying some of their existing beliefs about the world. I’m not sure it’s an accurate look inside the mind of a student, but I think it helps bridge to ideas like perceptual learning and affordance that we otherwise might not recognize.