Problem Solving and Creativity

I try not to put too much stock in the endless hole of people talking about Silicon Valley and startups and the tech world on the internet. But this quote caught my eye:

You need to know the things that you need to know to solve the problem. And you need to not believe things that will get in the way of solving the problem.

Sourced here, which credits the quote to Scott Klemmer, though I can’t find the original anywhere. He was talking about design and what research says about creativity, but I think it applies well to problem solving in math. Leads me to the question: is there anything else to problem solving?

12 thoughts on “Problem Solving and Creativity

  1. David Butler

    I think it’s not so much that you need to know the things you need to know, but you need to be prepared to learn new things in order to solve the problem. And it’s not so much that you need to believe that nothing will get in the way, but that you are capable of surmounting whatever does get in the way.

    Reply
    1. dkane47 Post author

      Interesting. I’m curious how often in problem solving situations students are effectively able to learn new things that they need to solve the problem. I don’t dispute that it happens, but I think significant learning makes the problem solving process much more difficult.

      Reply
      1. David Butler

        Oh yes having to learn something at the same time will impede problem-solving. As teachers we can give students problems that we know they know the stuff to solve. But in life and work, often you don’t know what you need to and you actually do have to stop and learn something and then come back to the problem when you know more. To succeed in these situations you have to be willing to learn. Indeed, with our students, they’ll usually come to learn a new connection that they didn’t have through their problem-solving. Not being ready to learn and expecting to know all the necessary things already can stop them from solving the problem too.

        Reply
  2. banderson02

    Not totally sure what you are looking for here Dylan, but I would say that you also need to be able to see connections between things that might not be apparent.

    For students, there is a wide range of “things I need to know” to solve problems. Those “things” vary dependent on what you are doing of course, but it also varies dependent on the experiences and knowledge of that student. Math is beautiful because we can approach it from multiple angles with multiple solution paths. We also bring to the table a very different skill set. This quote pushes me more towards the feeling of “I do, We do, You do.” This is drastically different that what I do with students which is: what can you do, where do you need to go, how are you going to get there?

    For me, finding new connections/relationships is one of the great things about mathematics, and it should be explored and believed in.

    Reply
    1. dkane47 Post author

      I would argue understanding connections and relationships to be a critical way of knowing things. Knowing things means knowing them in a way that facilitates transfer and helps students see their use in a variety of contexts. That’s definitely not apparent in the original quote. But are the connections that you’re talking about part of the problem solving process, and there are things you need to know to do that? Or is making connections its own skill?

      Reply
  3. Michael Pershan

    is there anything else to problem solving?

    Yes.

    Now, it’s open to discussion whether this other stuff is teachable. (Our scorched-earth traditionalist friends say ‘no,’ though the Wiliams and Willinghams of the world seem to say ‘just a bit and very quickly.’)

    Reply
    1. dkane47 Post author

      And plenty more say “yes, and that’s what math class should be focused on”. I think it depends in part on how we define “knowing things”. Which is unhelpfully vague to begin with.

      Reply
  4. Benjamin Leis

    That reminds me of the numerous “problem solving” strategy posts which usually boil down to understand what you’re being asked, get organized and don’t give up. Nothing’s wrong with advice but it doesn’t really help that much when confronted with a true problem where you need to find a solution that you don’t already know.

    Or to put things in the context of the quote, in a true math problem you don’t initially know all the things you need to know otherwise it would just be another exercise.

    It seems to me that the key to problem solving is to engage in lots of interesting problems where you figure things out and that sharpens intuition over time.

    Reply
    1. dkane47 Post author

      I think I agree with your final point on sharpening your intuition, but I think problem solving can be a problem where you initially know everything, but the challenge is figuring out how those bits of knowledge fit together. I think what I would call problem solving can happen at multiple levels, and the essential knowledge that students don’t have is knowledge of which strategy to use.

      Reply
  5. Michael Pershan

    Reading some of the other comments here reminds me how confusing I find most talk of problem solving and knowledge. Yes, let’s say, problem solving mostly comes down to knowledge. But — Dylan I think you said this — what kind of knowledge?

    Our traditionalist friends often talk as if really what it comes down to is a difference in basic skills and factual knowledge. Which, ok, yes, this stuff often helps a lot.

    But knowledge has a structure, and more abstract knowledge is functionally equivalent to a lot of mathematical strategies. I think of Schoenfeld’s take on Polya, which is that his strategies aren’t specific enough. Schoenfeld’s strategies are, e.g. When dealing with problems with an integer parameter, it may be of use to examine values of n = 1, 2, 3, … in sequence, in search of a pattern.. (Source)

    Now, nobody is going to go around claiming that this is a 21st century problem solving skill, but this is NOT just a fact or a basic skill. Knowledge has structure. It’s not just a vast field of facts — so we shouldn’t talk as if it is.

    The other thing that our traditionalist friends sometimes say is that there is basically no transfer of learning between contexts. This is, basically, wrong. If transfer doesn’t happen at some level then you just haven’t learned a skill, you’ve learned an example.

    I find Perkins’ distinction between the high and low roads to transfer helpful in this context. (Source) Far transfer does happen sometimes, though (as far as I understand things) in inconsistent and unpredictable ways. Near transfer, at some level, ought to happen for students. Directly teaching things like Schoenfeld’s strategies — abstract, but not too abstract, knowledge — should help here.

    As far as teaching for far transfer goes, I think of it in a cost/benefit sort of way. You don’t want to make far transfer your only goal in teaching because it happens so mercurially. On the other hand, if it’s fun, doesn’t get in the way of other goals, and is balanced with lots of knowledgey teaching…why not? Might as well take a Hail Mary when there’s 5 seconds left on the clock.

    Reply
    1. dkane47 Post author

      I need to read that Perkins piece.

      “teaching for far transfer” — I like the idea, but what does that look like?

      Reply
      1. Michael Pershan

        I don’t really take teaching for far transfer so seriously in my classes, but I think this is what people are trying to do when they teach kids mathematical habits of mind, strategies or principles that would be relevant across all of math.

        Reply

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