What Makes a Good Problem?

This may be the single most important question a teacher—in any subject—learns to ask him/herself. I am, for obvious reasons, more familiar with talking about how this pertains to math problems. Perhaps then, I am mistaken when I say that I believe there is a special aspect to finding good math problems that is not present in other disciplines. [Teachers of other subjects, please jump in and pound on me if I’m misrepresenting this—I can take it.]

What makes a good math problem? Well, one great description of great problems can be found here.[1] I’m sure we all have an idea in our heads of a Great Problem. Personally, I feel like looking at a specific problem and explaining what I like about it. So, go back to Avery’s blog and check out this cool paper problem.

Why I like it:

This problem is highly accessible. I would guess that most students haven’t seen it before. It also has very little language obscuring entry into the problem, essential when working with ELL’s. One thing I spent oodles of time on in my first year of teaching was rewriting problems to take out confusing language so my ELL’s (and the rest of my students who struggled with reading) weren’t getting stuck on the problem statements.  Note that this is not a “minimally defined problem” as per Avery’s definition of such. The problem is stated: How can you make this object with only a piece of paper and scissors? But there is not some wacky trying to be “real-life” confusing context that has been plastered over the surface (I HATE that!). There is no one “right” way to go about solving this problem, i.e. no standard algorithm. You can discuss different approaches and therefore have a rich mathematical discussion about it. It plays to the strengths of spatial reasoners, who often don’t get a lot of opportunities in the classroom to utilize their abilities. In other words, it gives students another way to be mathematically smart. I’m all about looking for ways for different students to showcase their talents—and ways for different talents to be used by students.

Update. Something I just thought of: I think one of the major things that characterizes a good problem is the ability to have a conversation about it either after you’ve solved it, or while you are in the process of solving it. Conversations should be mathematical in nature, of course. “Wow! That problem was really hard,” wasn’t the kind of thing I had in mind.

Why I think it is hard(er) to find problems like these in math:

Tradition: The vast majority of math problems you can find in textbooks and on the web are old school, boring, skill-based, practice, drill-and-kill type problems. Yeack! Which is not to say that you can’t find good problems. There are lots of really great problems you can find on the interwebs too, and even some pretty gosh-darned good textbooks as well.

Time Limits: There is so much to cover. And so little time. And, really, who has the energy to go searching high and low for super-cool problems, then figuring out how to fit them into your curriculum, then spending extra time on them—because they DO take extra time—in class, then figuring out some way of assessing them…whew, I’m getting tired just thinking about it. I guess I’ll just assign 1-29 odd again.

Territory: It’s unfamiliar. I didn’t learn math this way. I hated math when I was a kid. There’s probably not a connection there. That’s crazy-talk. Not to mention the frothing-at-the-mouth math traditionalists out there who will doggedly defend to the death the “back to basics” math territory they so strongly believe in.[2]

Testing: The big sticky one. Will it be on the state test? Testing problem solving is hard. Testing skills is “easy”. Though, looking at some of the latest findings about test validity, begs the question of whether we even know how to test skills effectively. Or at least in a way that will predict future success. Which is one of the things I think a good test should be able to do.

Trickiness: Teaching these kinds of problems is much harder than lecturing. It involves a totally different skill-set. For one thing the teacher has to let go his/her white-knuckle grip on the reins. There is no point in giving students a mathematically rich problem if you then go about explaining to them how to solve it. This means you have to take a step back. You have to allow some space and time for students to come up with ideas, some of which might not work *gasp* (I know all of my ideas work. All the time. Don’t yours?). You then have to make sure that all of your students are getting to access the materials/the mathematics. Are Catherine and Carol hogging all of the centimeter cubes and as a result Jim can’t do his work? That’s a problem. Fix it. Have Darryl and Ben misread the problem and are working on something different than the rest of the class? That might be a problem; but it might also be a great opportunity to learn something new. Are Kate and Dan getting heated, arguing over whose solution is correct? That’s definitely not a problem. They’re not only talking about the math, they’re emotionally invested in the math! I live for those moments. Is all of this happening at the same time? You betcha. That’s why they call it Complex Instruction.

[1] Disclaimer: I do have a slight bias towards this blogger, as we happen to be close, personal friends. Doesn’t mean it isn’t a damn good list. But maybe it explains the fact that I’m basically giving you a virtual tour of his blog in this post.

[2] Oooh. Are my personal leanings showing? Sheesh. Anyone who thinks there’s only one right way to teach/learn math needs a reality check. I think reading teachers have similar battles over phonics versus whole language. And I would guess that most would argue that you need both. Just like most math teachers I know say you need both procedural practice and creative problem-solving to learn math effectively.


5 thoughts on “What Makes a Good Problem?

    • Aw shucks, Jason. *blushing*

      I’m glad that you found the post interesting. If you (or anyone else) have more items to add to my list, put them on the comments! Those were the ones I came up with at 11:30 at night when I wrote the post. I’m sure there are plenty more that can be added.

      And they don’t even have to start with the letter T.

  1. Your comment on my blog helped me remember I meant to comment back here. Back story: At or around the time of this post I was having dinner with Sam Shah and he mentioned how PCMI had these amazing problem sets where each one perfectly scaffolded the next one. (He blogged it but I’m too lazy to find the link)

    So I think one of the hard parts about finding and using great problems in math is that math teachers (certainly standards/curriculum writers take a large part of the blame, but not all) don’t or can’t create a good narrative out of the somewhat tenuously connected things that are taught. There is little story. I have this picture in my head right now of the scene in Usual Suspects where the detective sees the bulletin board and all the parts come together. A good problem would have that feeling. Like a whole bunch of things you’ve learned is coming together and becomes necessary and related – and the skills or knowledge or whatever that you don’t have, but need, announces itself.

    It’s past midnight so I’m losing my train of thought here but I think the moral is that a great problem is only great within a well crafted story and that great problems in class are rare because the story is so difficult to create.

    I have reached the point where the word “story” is starting to sound funny in my head and isn’t making sense so I’m going to stop now.

    • Don’t tell Avery, but I think I’ve just fallen a little bit in love with you for this comment. 🙂

      Seriously, I think the word story is a great one–I am, after all, a writer. My writing group would give you sh*t about your lapse into backstory though. And I think that’s another reason why creating the narrative of a problem, or set of problems, is so amazingly hard: if you have to introduce and explain the whole procedure/process before students can work on a problem, there is no unfolding of the “what happens next” question. If books began with a summary of what the protagonist’s challenge would be, and explained what steps s/he would be taking to meet that challenge and resolve whatever the issue was NO ONE would read them. It is that desire to see what is coming up around the corner that pulls you forward through a narrative. And, you’re right, that’s damn hard to create in the context of a math class where you have to teach so many seemingly unrelated pieces of content.

  2. Pingback: A Critical Language for Problem Design «Research in Practice Research in Practice

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