I claim that this video is evidence against a common approach of teaching proof in geometry class. A weird claim, right? But I stand by it! Here's why.

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Many teachers of proof and reasoning think that their students have under-developed logical reasoning skills. The way that story goes, students struggle with proof because they aren't sufficiently logical yet.

To address the deficient logical skills of their students, many teachers of geometry assign students work with

*non-geometric logical thinking*. You'll find these exercises in many geometry texts.
The theory behind these exercises is that we can improve students'

*mathematical logical thinking*by giving them exercises in more concrete, practical,*everyday logical thinking*. Since logical thinking is one skill, you can work it out in an easier context and then apply it to a more difficult, abstract context.
But the video above shows that

**logical reasoning is not one skill**. These young kids are perfectly able to reason logically about merds and bangas, even though you can see that logical thinking come and go in the above video.
When their logical reasoning falls apart, it's for two reasons. First, there are times when the kids aren't sure whether they're supposed to use their logical reasoning or their practical knowledge to answer the question. ("This is math, not nature class.") Second, when kids/people have practical knowledge about something they tend to use it rather than reason about it. ("They do

*not*bounce when they fall.")
Applying these lessons to math, when our kids struggle with reasoning in geometry it's likewise for one of two reasons:

- They don't yet really understand what's expected of them. The expectations that mathematicians have aren't always natural or easy to pick up. Kids don't always get the expectations.
- Kids, like adults, use their practical knowledge about the world to guide their reasoning. If kids aren't reasoning well, it's often because the practical knowledge that they're leaning on isn't strong enough yet. (It's like the van Hieles said, sort of.)

But it's

*not*because kids can't reason logically, because young kids can reason well about merds, bangas and all other sorts of things.**Questions!**

- But clearly
*some*people end up with super-duper logical reasoning skills that they can apply in any and all contexts. Where do those skills come from? - Why do kids get better at logical reasoning about make-believe as they get older? Is it because they get socialized to the idea of reasoning about make-believe?
- Take this too far, and you end up saying that if you just teach kids a bunch of facts they'll end up with the ability to write beautiful proofs. That's clearly not true. Informal reasoning about geometry is clearly important for developing your ability to reason more carefully about geometry. But why?
- Say that you're convinced that practice with
*non-mathematical logical reasoning*doesn't help much with proving stuff in geometry. What could you replace that unit with that would help?

I think the conflict you see between "pure" logical reasoning and leaning on practical knowledge is a big problem, and doesn't necessarily go away as people get older and more mature.

ReplyDeleteIf I had to narrow it down to a single skill, I think it would be the ability to draw conclusions from uncertain (or even false) statements (i.e. being able to reason while suspending disbelief). It's what differentiates real science (rigorous hypothesis-testing) vs. science driven by confirmation bias. It's also very useful when trying to write airtight proofs (see your most recent post on the mathmistakes blog): having the ability to think "if premise A is false, then my proof falls apart" naturally leads to "ok, well now I need to show premise A is true" rather than just taking for granted certain "obvious" premises.

What I'm wondering now is whether I overstated things. Maybe those non-mathematical logic puzzles are good for teaching kids that mathematicians value the thinking through of false premises.

DeleteI've been mulling this a lot over the summer (odd, because I probably will go another year without teaching geometry), and I think you're right. My kids get logic, they can follow the rules and identify the thinking of logic by if...then or by venn diagrams with little problem. But they don't get what is expected in proofs.

ReplyDeleteSo why did we get it back in the day? I'm not sure most of us did. I remember back when I took geometry a zillion years ago that I was shockingly good at proofs (at an otherwise dismal year of understanding) while everyone else struggled.

So if even top kids had some struggles with proofs , maybe we're just seeing what proofs are like when dealing with much lower ability kids. Maybe the switch to diagrams and proving facts isn't as axiomatic as we think.

I'm in the middle of these thoughts right now and would love a great answer to your question number 4 (if there even is one). No, there has to be a better way.

ReplyDeleteMy tentative answer is: informal argument-writing experience.

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