Sunday, October 13, 2013

Explanations and Justifications

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?


  1. I never thought of it like that, but I agree: understanding multiple proofs allows for a deeper understanding of the theorem, and that fits better with seeing proofs as explanations rather than justification.

    Another reason for multiple proofs might be the side-effect axioms and lemmas that pop out of different proofs, though that fits in with the "deeper explanation" theory.

    I also just always thought that other proofs were cool to see and recognize. Do you think that the students understood the Pythagorean Theorem on a deeper level because of Tuesday's proof? Or is this one of those handy opportunities to sharpen their algebra/geometry/logic skills? Or both?

  2. I'm still not sure that I see the difference you are trying to express. To justify a statement is to explain why you can be sure of the statement, at least in my dictionary---not just why you believe it, but how you can be certain that you haven't made a mistake. And the more reasons/explanations/justifications you can find, the better you will understand whatever topic you are studying. That is why Polya taught us to "Look back."

    Are you limiting justification here to mean only "establishing a truth about which there was originally doubt"? If so, I agree that meaning is too limiting for math.

    Of course, there is also a weaker meaning of justification: explaining why you believe something---not in the sense of proving it, but just giving a reason for one's belief. That seemed to be how you used the word the other day, but it doesn't seem to mesh with what you are saying in this post.

    1. This is such a good comment. I see two related questions that you're forcefully asking.
      1. Let's say that to justify X is to give a reason for believing it. Is there really anything called "mathematical explanation" of X that isn't just giving a reason to believe X? When we seek an explanation, aren't we just seeking a reason to believe?
      2. Why can't justification motivate the need for multiple proofs or proofs of "obvious" statements?

      I don't want to attempt an answer now, but I did want to highlight these two questions as being very important ones for us to grapple with. Anyone else?

  3. Hmm, a few weeks ago I walked my calc students through a four-page handout to prove that the derivative of sine is cosine. First we saw from graphs that it was likely true, then we waded through these proofs that established that it must be true. Did they help us understand better? I don't think so. But I do think my students got some sense of what proving means. I kept offering to leave parts out (and I had told them they wouldn't be tested), but they kept wanting to get it all. I told them they were courageous.

  4. I'm solidly in the "proof as explaining" camp. "Explaining why" became a central theme of my teaching once I started to read the work of Sybilla Beckmann, for example:

    As one of the commenters in another thread pointed out, students often see no need for "proof as convincing" because they are often easily "convinced", even though we, the mathematics teachers looking over their shoulders, wouldn't be ourselves (and know they shouldn't be).

    In fact, I think it may do more harm than good to push the issue of "proof as convincing". I mean, who could fault students for dismissing the idea of proof if the goal was always to "convince" them of things that they regarded as "obviously true".

    A favorite example of mine on this score has to do with adding two odd numbers. How many of us have seen students become convinced that the sum of two odd numbers is always even after trying just a handful of cases? My students usually needed no additional convincing and would brush off my attempts to point out the potential dangers of extrapolating as they had.

    But, once we're convinced: Can we make a convincing argument for *why* two odd numbers always sum to an even number? Students explore that question much more productively.

    Plus, "explaining why?" seems to invite a variety of responses. After we've heard one explanation, a logical follow-up question is, "Can someone explain this a different way?" The Pythagorean Theorem seems like fertile ground for such a discussion, since there are so many creative ways out there of explaining it.

    1. Our students need to experience plenty of "patterns that fail", so they can appreciate the necessity to justify even "obvious" statements.
      Patterns that fail at 360blog
      big list of patterns that fail

  5. I think we can't help but be influenced by the plain-language meaning of the word "proof" which leans heavily toward the uncontestible, undeniable, solid evidence, justification way of viewing things.

    It is the same with trying to get students to understand "an argument" -- it is hard to get over their plain-language preconceptions around the word.

    I suggest you say what it is you really mean to your students, get them used to that idea, then try to convince them that that can be called a "proof".

    1. Excellent point about the language that we're using. I think the language is further evidence that "proof as explanation" is an abstraction, a development out of "proof as justification." The language itself points to an initial desire for justification.

  6. Michael

    Have you seen this?

  7. A prove always explains why something is true. If it doesn't it is usually wrong or at least badly written.

    I think that the problem with proofs is that kids are not used to this kind of science. Do the history or the geography teachers "prove" anything? No, they teach facts, don't they? So why does the math teacher actually insist on proving that strange relation between the sides of a rectangular triangle?

  8. Reading your question for the reader, I am reminded of why I have students present all of their solution methods to problems that I pose in class: to help develop the connections that students have between mathematical ideas. The larger the number of methods we see, the more connections we make, the more solid each of the concepts used becomes. Additionally, in the future then, when solving a problem, if the initial concept one uses becomes inefficient or ineffective, one can switch to using a related concept. This can not be done without having developed a vast interconnected web of mathematical concepts and ideas.

    A quick example of this that comes to mind are some of the Number Talks that Jo Boaler advocated in her online course this past summer. The more models we have seen for doing something, the more flexibility we have in the future. This helps us ensure efficiency and accuracy.