Sunday, February 24, 2013

The difference between given and found problems

There are all sorts of subtle ideas that lurk behind the veil of explicit communication. Every once in a while, it's worth taking stock of what those subtle ideas are to make sure that we're communicating wisely.

Every day, I give my students problems to solve. When I do that, I'm also telling them:

  • "This is a problem that you can make progress on."
  • "This is a problem that it is worth your time and effort."
  • "This problem is new and not the same as what you've studied before."
Because I teach my lessons in units, there's an additional implication present:
  • "This problem is connected to what we studied yesterday."
Are these things that I want to be communicating to my students? I don't know. (Those wiser than me will hopefully chime in on this.) But I do have a few observations to make:
  1. All of these implications make the problem easier to solve. My assurance that the problem is solvable and that it's worthy of their time helps them take the leap into the problem with confidence. The implication that this question has something to do with what we have been studying significantly narrows the available tools and techniques to choose.
  2. This is fundamentally different from the way problems are outside of the classroom. When I find a question that I want to make progress on, I have no assurance that it has a good or interesting solution or that I'll be able to do it. I have no idea what tools I'm going to need, and making those decisions is part of the difficulty of the problem.
What do we do about this? What can we do about this? We can try to present problems to students that are more akin to how they're found in the world, but the mere fact that we're offering them invests the problem with all of the implications detailed above. If we want to eliminate the suggestion that a problem is connected to the previous day's ideas we could eliminate units and integrate our topics more densely. 

Other than that, to really help students solve problems as they're found out there, we need to create more opportunities for them to find problems on their own. 

Not like I do that, but hey, aspirations, right?


  1. Lovely ideas. But I wonder if it's possible to give them problems you aren't guaranteeing to be solvable.

    I've only once solved a problem I had never seen a solution for. That was Spot It, and I knew it had to have a solution, because the game exists. (I wanted to know how they managed to put the cards together so every pair of cards has exactly one match out of 8 pictures.)

    Have you ever solved a problem you thought interesting, that didn't come from a book, and that you didn't know you could solve?

  2. "Other than that, to really help students solve problems as they're found out there, we need to create more opportunities for them to find problems on their own. "

    They would have to give a crap about finding the answers, and while that might be the case in your school, it's not the case for any but the top, say, 30% of students in higher math classes.

    1. At some point you and I should swap emails to nail down each of our respective student populations.

      In any event, no, I don't think that my kids would give a crap, but I also don't see why that's a prerequisite. Kids do all sorts of stuff in school that they don't actually give a crap about. I mean, kids read books, hard books, and come up with arguments about aspects of the books.

      Sometimes those arguments are given ("Explain why Atticus Finch bla bla bla") and sometimes the assignment is for students to find an argument ("Write a 3-5 page essay on what you think is the central theme of To Kill a Mockingbird.")

      In math that could look something like, "Read the newspaper this weekend and come up with a list of interesting math questions." Then maybe those questions are swapped around to other kids.

      To be clear, this post is not about imagining some sort of Utopian, progressive atmosphere where kids are interested by what they study and they pursue their own ideas. I'm just pointing out that the act of giving a student a problem makes that problem easier. I'm worried that we're going too easy on the kids.

  3. I spent 40 minutes in a 2 hour workshop a couple of weeks talking about this issue and brain-storming with teachers ways we could use resources to find these problems.

    I suggested that both students and teachers need to be problem finders, not just problem solvers. The problems themselves could be real world, abstract, borrowed, stolen, or whatever. I'm not sure that we need to know whether or not the problem has a solution, but it shouldn't be obvious what it is, and there shouldn't be an easy algorithm to find it. People re-invent the wheel all the time.

  4. What I mean to say is, I love it when students come to me with problems that I don't know whether or not are solvable (and neither do they) but I'm happy with them coming to me with absolutely any problem at all that they want to solve for themselves.

  5. I just have to troll this Michael, I'm sorry...

    A list of problems...

    You're welcome

  6. Investigating interesting, student-generated questions would be a great way to motivate and guide instruction. But I think students need some basic content knowledge and skills before they can recognize the really interesting questions. And developing a skill requires guided practice through contrived, teacher-developed questions/investigations. A point guard can't be creative and flexible in a game situation until he/she's run a lot of dribbling, passing, and shooting drills.

    1. Agreed. I think my ideal classroom would have a nice variety of given and found problems.

  7. I know this isn't quite what you meant, but... math contests. True, they're not "found" problems in terms of being something students will stumble upon later in life. But they ARE:
    - Problems not necessarily connected to what you're studying at the moment (but that are usually still tailored for the grade level)
    - Problems that students may not be able to make progress on, or that they may make progress on only to have to backtrack
    - Problems for which you can't immediately provide a solution. (This can energize sometimes. Once gave a trig identity sheet I'd found noting 'I can help with all except #7, haven't figured that one out yet'. So where did half the class start?)

    All of which seems to be exactly what you're looking for. And seems to give them some of the skills they'll need to tackle their own "found" problems. Furthermore, recent research indicates repeated math contests may close the gender gap:

    So yeah.

  8. First, I love your Khan/Japan video. Second, I'm sure you know of this guy (he has a TED talk online), but just in case. He's a big "teach the kids to solve problems where the tools and parts aren't handed to them" guy.