This student says that a proof is "something that clearly demonstrates that something has to be true." So a proof is all about guaranteeing truth. In other words, it's a form of justification.
Here are the rest of the class' takes on proof:
- "A statement that solves a mathematical (mostly Geometric) problem."
- "Take an example of a problem, explain it to show how whatever you are trying to prove must be true."
- "A mix of words and equation or pictures that convincingly show some problem or statement to be true."
- "A proof is a way to guarantee that something is true for all situations."
- "A proof is something that is uncontestible."
- "Solid evidence."
- "Undeniable evidence that can't be proven otherwise."
And then there's one kid who wrote the following:
- "A proof is a statement of why things work the way that they do, and is also backed up by evidence."
Do you see how that's different? All the other kids were talking about proof as justification, but this kid here is talking about proof as explanation.
I think that proof as justification is more concrete. Students get the hang of that whenever there's a controversy or doubt, and any good problem can create a good deal of controversy or doubt. It's relatively easy for me to provoke a need for a justifying proof with kids of almost any age.
But Michael Serra told us in the comments that we ought to "focus on proof as a means of 'explaining why' rather than a way of convincing someone the given conjecture is true." And that makes perfect sense. If you only see proof as a justification, then proofs of things that you already believe to be true would seem to be strange, unnatural exercises. There's very little reason to value multiple proofs if the only purpose of proof is to justify a claim.
These kids, for the most part, see proof as a means to justification, and it's making it difficult to motivate all sorts of things in class. For instance, last week we were looking at proofs without words of the Pythagorean Theorem. We did an easy one on Monday. On Tuesday we did a harder one. A kid said, "I don't see the point of this. Wasn't yesterday's proof easier? Shouldn't we just be looking for the simplest proof of something?"
He's working with proof as justification. I need to take him to proof as explanation. I don't know how. Ideas?