Helping Students Make Good Mathematical Arguments

How do you teach someone how to make a good mathematical argument? Here’s my theory:

  1. Make sure they understand the stuff being argued about
  2. Show them what a good argument looks like
  3. Help them make good arguments on their own

I think I did a better job of this in this year’s geometry class that I have in the past, so I’ll share what I did.

First, I made sure that students understood what was being argued about. This is a plea for teachers not to teach mathematical concepts or skills through proof — a problem I see happening more at higher levels of math instruction. There’s simply too much that is new for students in a proof. It’s not the time to make sure students also understand every else.

In my class this year this happened both at the level of the course and that of the lesson. I started the unit with lots of time working with the sorts of diagrams that would feature later in proofs. But I also did this within a lesson. If I wanted to show students an example of a proof, I would make sure to launch the lesson with an activity that just involved analyzing that diagram.

Here is a diagram, I’d say. I just want you to tell me as many pairs congruent sides or angles in it as you can, and tell me how you know:

Then, after making a nice list of all the congruencies in the diagram, I’d preview the structure of the proof (thanks, Catrambone):

I would then model writing a proof along these lines using the congruencies the class found at the start of the lesson.

There’s a bit more that needs to happen at this stage to make sure students have a chance to really think about this. There are a few things that I’ve tried at this stage — asking students to turn and explain each step of the proof; removing a step from the proof and asking students to explain what it needs to be; asking everyone to take notes from memory with some or all of my proof erased. I’m still working on getting this stage of my little routine down.

Following this, I shared a new diagram and asked students to once again find congruencies. When I did this in class I used this new diagram and this is what students observed:

They came up with those questions as well. I asked the class to prove that the inner triangle was equilateral, using the structure that they had studied in that first proof.

So, the first stage is making sure students know what the proof is talking about. The above is how I’ve done a better job of that this year.

The last step is trickier than it may seem. Yes, it’s “obvious” that helping students make good argument is important for learning how to do exactly that. But the emphasis I’d put is on help and (especially) good. Because in my experience, there’s strong gravity that pulls students towards arguments that are intuitively good but not up to the rigorous standards of mathematical proof.

The issue is basically this: everyone knows a lot about shapes. In fact this is not an issue — this is fantastic. I love hearing people’s ideas about shapes. But if you don’t provide the right kind of help, you’ll confuse and anger your students. The reason is because your job, as a mathematics teacher, is to teach a very particular form of mathematical argument. And students will have many wonderful ideas about shapes all their own. In past years I fell into a tricky spot where my students were baffled why their own extremely reasonable arguments for why a pair of triangles must be the same size and shape aren’t acceptable as proofs.

In case you haven’t heard these things live and in person, the human eye is attracted to symmetry and change. If you ask a group of people why a pair of lengths are the same, they will be drawn to arguments that incorporate those sorts of features. “Those lengths have to be the same because the diagram is symmetric. If it weren’t symmetric then when you moved that angle over there, that other angle would have to change also. And then the whole diagram would be broken.”

What’s the issue here? Nothing, really. It’s beautiful. It’s just not rigorous enough to pass muster in a geometry class. And while some people might be tempted to push on students — how do you KNOW that it’s symmetrical? how do you KNOW that angle will change? — my experience is that this only angers teens, who grow disgruntled that their own reasoning has been invited to the party (finally!) and then rejected as unacceptable.

Far better, I now believe, to help students make GOOD arguments, where “GOOD” is defined as “whatever is acceptable from the point of view of the course.”

In practice, this means that I have started to include a lot of visual choices for students as they are asked to form justifications. Here is a snap shot from a problem set I assigned in class:

There are now choices. Your argument will look like one of these four choices — always. Learn to use them, and they’ll become part of your language of justification. (Thanks, self-explanation literature.)

In short, students’ choices are constrained. To put it one way, especially in the early stages of learning, students are allowed to choose a wrong argument but they are not allowed to make a bad argument. (Again by bad I don’t mean “bad,” just “not ideal within the context of this course.”)

I think I’ll be working on improving my teaching of proof until the day they drag me feet first out of a classroom, but these two ideas feel like progress to me.

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