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.

Hi Honey, I’m Home

I just got back last night from a week-plus adventure that took me and the BF through: 2 different states (Oregon & California, in case you were wondering), 4 National Parks/Monuments, n>4 National Forests, 5 campgrounds, 1 Bed & Breakfast (for my birthday–isn’t my BF great?), and a grand total of 10 campfires, 7 of which used no more than one match.  Yes, I lit a fire with zero matches. I am that amazing.

I find it somewhat bizarre that, while surrounded by some of the incredible natural beauty our country has to offer, my mind has a tendency to drift to topics related to school/work. Why is that? Though, in all honesty, my mind has a tendency to bounce around seemingly randomly in a wide range of topics, only some of which happen to be school-related.

One thing that did make sense, in my hiking-trail musings, was the thought about one super-cool aspect of my new job. I will be helping out with the outdoor program. It’s written in my contract and everything. Now, I’m not exactly sure what this means, not exactly sure who else I’ll be working with, or how often we’ll go on trips, or where we’ll go, how involved the trips will be, etc. All I know is that I get to go on fun trips and get paid for it. Yippee!

I feel like I learn so much when I’m out camping and hiking. I learn about geology, biology, ecology, history, not to mention all the things I learn about what I’m capable of doing both physically and mentally. At home, I could care less about identifying different types of trees, but when I’m out hiking I want to know what I’m seeing. It’s like learning anything: when you’re immersed in the context the information means something to you and so you crave it more. I’m excited about getting to share these types of experiences with students. As well as getting to know students outside of the typical classroom environment. I know that this is going to be a great way to connect with students.

And maybe I’ll even get to go back and do some of those hikes that were closed because of snow. :)

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This is totally what I meant! Imagine the bikini-clad writer lying on the towel wearing hiking boots and a baseball cap and you’ll get a good idea of my affliction.