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new and old thoughts about teaching mathematical proofs

A long, long time ago, I used to teach math. One of the central questions in mathematical education at the college level is how to teach mathematical proofs. Sometimes, you had pessimistic conversations. People simply had “mathematical maturity” and there wasn’t much you could do about it. There is truth to this –  some people simply can’t grasp what a proof would entail.

Beyond this simple observation, there was remarkably little thinking about how to teach proofs. Of course, there are occasional books that try to break down the process of creating and writing proofs, such as How to Prove It. Still, I felt there was something missing in the conversation about proof teaching. This blog post is my modest contribution to the topic.

My hypothesis: An important barrier to teaching math proofs is that they combine two very, very hard skills and that most math teachers only focus on one of the those skills. Specifically, proofs entail (a) symbolic manipulation and (b) recipes that get you from A to B. Math teachers and books are actually pretty good at (a). For example, almost every text will teach you about the symbols – set theory; formal logic; deltas and epsilons; etc. What is almost completely overlooked is that students find it hard to glimmer the “recipes” that make up proofs and there is no theory, or set of instructional strategies, for helping students intuitively understand recipes. In practice, you simply take courses on various topics (numerical analysis or matrix theory) and you mimic the proofs that people give you. Not great, but better than nothing.

The old Dolciani high school text books had an interesting response to this issue. In the geometry text, the proofs would always have two parts: “analysis” (outline of the idea) and “proof” (traditional proof with all details). You also see this in advanced texts and journal articles. When a long, hard proof is coming up, the author will present an outline.

Here is my modest suggestion: When teaching proofs, always outline the proof as a flow chart. In other words, take the old notion of the proof outline (or “analysis” in Dolcian’s terms), make it visual, and then put it in front of all proofs that require more than a few sentences. By repeatedly visualizing proofs as chains, teachers will be forced to extract the recipe from the text in a way that more students can understand. They will also more easily identify common themes that appear in multiple visualizations of proofs. Also, pictures are easier to remember than dense, equation filled masses of text.

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Written by fabiorojas

October 2, 2015 at 12:01 am

One Response

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  1. I sniff Wittgenstein’s “to follow a rule”, here—have you heard of it? Alternatively, there is Michael Polanyi’s “tacit knowledge”; I’ve been meaning to look at how these two things might intersect. Furthermore, theoretical biologist Robert Rosen has a great section in Life Itself on how any given syntax can only partially grasp the entity that is the natural numbers, per Gödel. Perhaps we could think of it this way: while you’re teaching a syntax, you also want to somehow connect your student to the actual entity.

    One way to motivate the ‘recipe’ approach is to think of a forest with well-trod paths. Those are the ones most commonly used, the ones which can get you a lot of the places you’ll want to go, so it’s important to be able to recognize them. That doesn’t mean you’ll never veer off-path, of course. One might even introduce the idea that some kinds of math (and programming) attempt to characterize those paths and come up with a way to pick only the paths out, with a new syntax and/or rules of inference.

    I like the term “flow chart”; in a somewhat analogous way, there is a way to do functional programming such that the resultant “data flow diagrams” are very easy to comprehend, even for the [relatively] uninitiated. One is tempted to call this a more natural way of presenting the information, although I’m sure that is a very tendentious statement.

    Neat stuff! We need more people thinking along these lines. I want to believe that humans can learn to learn at least an order of magnitude faster than currently done. To get there will take a lot of work, which will challenge a lot of people’s conceptions of “how things ought to be [done]”. But hey, what else is new?

    Like

    labreuer

    October 5, 2015 at 11:20 pm


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