Last week I posted a short video from a tutoring session I had with a kid. We were solving equations, and he had some interesting ideas, and it was nice to have those ideas and his mental workings become explicit.

Here’s another chunk of that video:

Comment on whatever you like, but here are some prompts:

  1. Help me understand his thinking. How did he devise his test for whether his solution is correct?
  2. What does this say about what he thinks about 2/0?

Or jump in with whatever you like in the comments.

  • First, I think I often say things along the lines of “I don’t know if this is right” or even “I don’t know why that doesn’t match the answer in the text, your logic looks great”. (I say ‘think’ because it’s possible I don’t do it as often as I believe.) So good on you for leaning towards how can we check.

    In terms of the test, he didn’t go back far enough, which can be a problem with even strong students. I recall on one occasion a student took the extra time to check their work and show how they were correct… except they used the first line they had written, which turned out to already be different from the printed problem. He also seems to think that the last step before a solution is division… which, yeah, it is most times. 2x=8? Divide. -5x=20? Divide. Dude must like algorithms.

    Of course, here it wasn’t a division, and since we don’t see that part of the video, I don’t know if he tried to divide earlier here too (so he’s forgotten that it wasn’t needed) or if he just feels like he missed a division, so doing it after the fact will somehow verify things. He probably also knows 0/2 is zero, and thus… is hoping that maybe going the other way will spawn a negative? I’m guessing there.

    Must say, having seen the previous video, the invalid answer now seems to be from him changing his ‘+4’ (which seemed correct to me there) to ‘-4’ (which is no longer correct), and I wonder why. Must have something to do with what looks like the removal of ‘x’ at the same time. He seems to have a desire to completely remove things from one side of the equation… which again, is fine, if tedious. Okay, that’s enough of my babble.

  • LSquared

    That looked like a complete guess to me. I can imagine the student thinking “I have no idea how people check if something’s right or wrong–they must be able to look at a problem and just know if they are right or wrong. I’ll do the first thing that comes into my head–sometimes you divide at that second to last step. I bet that’s what I was supposed to do” sort of thing.

    At this point I’d try to point the student’s attention back to the original problem. I might ask how to prove using the original problem that 1 is not the right answer, or I might even hint by asking if the student remembers how to substitute or plug in a number into an equation. (I’ve learned not to dwell on the wrong path for too long, it seems to make it harder for the student to un-learn it)

    I love your question/comment (I don’t know if it’s right)–it really gets you to see what the student knows and doesn’t know.

  • “How would you check?” is a great question — so great that I think we all ought to be asking it at the beginning. “What will the answer look like?” where I want them to say “it should be x = something”, and then I can say “OK, how about x = 10, does that work?” and then they can tell me what’s going on there before we’re all caught up in hunting for procedures.

    In a classroom, it’s great when two kids get different answers so that you can encourage them to try to convince each other while you fade into the background for a while. In a tutoring situation, they need to sell you on the correctness of their answer.

    I wonder what I’d ask next after this 2/0 craziness? I probably would avoid the whole issue of division by 0 not making sense, and instead go after something like “Why is x = -2 an answer? Why not stop at one of these earlier steps?” in the hopes that we could eventually come to the idea that the value of x should actually make the equation true.

    Maybe even more deeply, the analogy with weights from the previous part might need more reinforcement, that an equals sign is a statement that two expressions have the same value, and that we can find out if statements are true or false.

  • You ask, “How do you check? Do you know whether it’s right? Do you know whether it’s wrong?”.

    I find myself tutoring other teachers’ kids, and when we get to this point, I usually ask, “What the hell was that all about? Why did we just do that? What was the point of it all?” And I usually get a shrug for a response.

    This idea of checking doesn’t seem like it should come into play unless we really understand the purpose of what we’re doing. I would argue that this student didn’t know the purpose. By understanding the point of “finding x”, I believe the student is more inclined to check his or her answer without the teacher’s prompt.

    I think it’s often useful to ask ourselves of any answer, what does this mean? To the teacher, it’s obvious. To the student, it’s not.

  • Could we contextualize the original problem? Could we come up with a story for the original equation to see if -2 makes sense?