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.


Clickers for the mind

I had taken for granted that feedback is a critical part of learning: it’s information that we use to adjust our performance and incrementally get better. However, Dan Meyer gave an excellent example of when feedback goes wrong. When working through an algebraic equation in a computer program, the student writes a step and the equation turns red: the equation is wrong. They flip a sign and then it’s green: the equations is correct! And yet the student doesn’t understand anything.

I finally have a better framework for thinking about feedback: reinforcement learning, the theory behind animal training, particularly via a technique called clicker training.

In clicker training, animal trainers help an animal associate the “click!” of a clicker (just a small object that makes a click when pressed by the trainer) with positive reinforcement like a small treat for a dog or a fish for a killer whale. The purpose of the clicker is that the trainer can time the click exactly to when the animal performs a correct step. The positive reinforcement is a very powerful way to instill the behavior in the animal, and it works from household pets to performance animals.

Back to our algebra software: the program turns green, click!, positively reinforcing the student’s step. The problem is that we’re reinforcing the wrong action: “keep flipping signs until it’s right”. How Children Fail is a whole book of these kind of training failures in the classroom setting. The author explores how students he’s observed fall into patterns of trying to get to right answer, whether that is saying “I don’t know” or probing for the right answer like our student in the computer program. Anything but doing the hard work of understanding and working out the real problem!

It’s like reinforcing the dog for dragging every item in the house to your lap because those happened to include her fetch stick. Instead, we can break down the actions into a chain of tiny parts, and reinforce these one at a time. The principles of reinforcement learning, which I’ve been reading about (after getting a dog of my own) in the book Don’t Shoot the Dog, tell us how to do this kind of training. Here’s one example from the book:

We were watching a horse being trainer to bow, or kneel on one knee, by a traditional method involving two men and a lot of ropes and whips; the horse under this method is repeatedly forced onto one knee until it learns to go down voluntarily.

I said it didn’t have to be done that way and asserted that I could train a horse to bow without ever touching the animal. (One possibility: Put a red spot on the wall; use food and a marker signal to shape the horse to touch its knee to the spot; then lower the spot gradually to the floor so that to touch it correctly and earn a reinforcer the horse has to kneel.)

This act of shaping is a subtle art. Even training my puppy to sit wasn’t a straightforward procedure — I almost wanted to reach for the ropes and whips after twenty minutes without her ever getting in the right position. In animal training, trainers understand that verbal communication is starting from scratch. The dog has no idea what “sit” means when we start. The math student likewise has no idea what the concept of “equality across the sign” means. (Actually the task is even harder because we don’t know whether the observed behavior of flipping the sign comes from understanding the mistake or from trying all the possibilities. Meanwhile a sit is a sit.)

I believe that successful teaching practices are those that use effective shaping. It applies no matter what perspective you bring to education. Discovery learning shapes using affordances in the learning environment as I’ve talked about before with Portal. Explicit instruction shapes through worked examples that slowly build on previous understanding and have clear points of failure when applying misconceptions like the sign of a unit.

The link to animal training and its behavioral history has been quite surprising to me. Behavioralism has been a relegated branch of psychology, particularly by the cognitive science training that I had. The example of applied behavioralism that I see is in the design of addictive but meaningless games like Cow Clicker, where you are reinforced for clicking an invisible cow. But there’s no question that human minds respond to the same effects and they can be used for good. (Another amusing application: relationships.) There is still more to explain in terms of when concepts are understood versus when we’re just grinding through procedures, but this is where I’m at for now.