Friday, July 4, 2014

Scaffolding Proof Writing

I'm reading a lot of papers on proof and geometry class right now, and I came across one that offers some sensible, manageable scaffolds for standardish proof problems. The title is Moving Toward More Authentic Proof Practices in Geometry, and it's an interesting read.

Here's the first two problems they offer in the paper:

In Problem 1, the "Given" and "Prove" are missing. In Problem 2, the diagram is missing. They seem to have made a big list of things that can happen in geometry proofs and designed problems by excluding some combination of these things. It's the job of kids to provide those missing things.

In Problem 3 the missing thing is the "Given" and "Prove."
In Problem 4 it's the actual given statements.
In Problem 5 it's the theorems.
In Problem 6 it's the auxiliary line.

The last three problems are a bit different. In Problem 7 they explicitly ask kids to make conjectures -- something I know that I should be more systematic about than I have been in class. In Problem 8 they ask kids to find mistakes in a given proof. Problem 9 is another "missing information" problem, where this time they left out the theorem but gave you the entire proof.

This past year I really didn't push my classes to do much written proving -- though "how do we know this?" was practically a mantra in my teaching -- but I was often disappointed by my students' ability to write down logical arguments when I did ask them to explain their thinking on paper. I think that these scaffolds could be an important part of what I do next year.


  1. I've had the most success with the approach in problem #5, where we are trying to get students into the habit of thinking to themselves "IF this THEN what?". Another scaffold is spending sufficient time teaching logic to students before they start proving things. I know my logic unit is currently falling short of getting students ready for proof.

    1. I do like problem #5. But I want to disagree with this: Another scaffold is spending sufficient time teaching logic to students before they start proving things.

      The idea that the problems that kids have with proof comes from problems with logic is problematic. What do we mean when we say that kids don't know logic? Kids make logical inferences all the time, do this.

      Here's what is true: we often use our practical knowledge in place of deductive reasoning, especially when the contexts are unfamiliar. In particular, researchers regularly are able to provoke children as young as 4-5 to reason logically by providing them with a fairy tale context. (link!) The problem that kids have in geometry is that they have all this practical knowledge about triangles, squares and parallelograms that they come to a proof with, and they end up using their practical knowledge.

      You're then faced with a choice: try to get kids to give up using their practical reasoning, or help them improve the practical reasoning they bring to proofs.

      Anyway, I've been thinking a lot about this lately. Thoughts? Response?

  2. I definitely scaffold things this way. I often lead in with Algebraic proof to get used to some of the reasoning.

    Then I show them other methods, including paragraph proofs and "flow" or flowchart proofs. I recently saw someone suggest it might be good to do the flowcharting first, then move to two column/paragraph proofs. I like that idea too.