## Wednesday, October 19, 2011

### What makes a good problem?

Let's start things off with a short taxonomy of problems:
1. Valuable problems are problems that worth doing. You'll learn something from engaging and eventually solving them.
2. Hooky problems are problems that keep you up at night. They aren't boring -- they're what you do when other things are boring. In short, they're fun.
3. Hard problems are difficult to solve.

There are clearly more types of problems out there, but this is what I need to start with.

As a teacher, I want to assign valuable problems to help my students learn. The problem is that a lot of valuable problems are hard, and people sometimes get frustrated with hard things. When people don't get frustrated by hard problems, it's often because the problems are hooky. So I have an interest in understanding what sort of things teenagers find hooky.

Anyway, I hope to continue thinking about this in a series of posts. Here's the first marker of a hooky problem that I have to share.

Here's a question that is, you know, it's just fine. It's not bad.

Here's one that I worked on for more time than I care to admit:

Here's one thing that makes the second problem hookier: it provides its own feedback. You know if you've found a rule, or if you haven't had a rule. You know how to check your answer. It's self-checking. That gives you the opportunity to try things and fail and retrench while sitting secluded in your room or your desk.

In contrast, the top problem doesn't offer much help. If you know how to find inverses and domains and range, then you'll feel confident. If not, you're sunk.

To be clear, whether a problem is self-checking is neither inherent to the problem nor absolute. If a high schooler has been taught to confirm domain and range with graphs on the graphing calculator, then the top problem might be self-checking in a way that approaches the bottom problem. Whether a problem offers its own feedback depends on the base of knowledge someone brings to the problem.

But the bottom problem is still hookier, and that's because the base of knowledge it requires is exceedingly low for anyone attempting the problem. All it requires is for the solver to be able to feel comfortable working with fractions.

Anyway, I need to continue this so that I can become a better problem writer. I hope to refine my thoughts on playful and deceptive problems so that I can try to understand how these relate to hooky problems.

Brilliant readers: what makes a problem hooky?

1. Can 1/n always be written as the sum of two unit fractions, or just sometimes? I'm guessing always, but I'm trying to figure out how I want to reword your problem statement.

I see so many students in my beginning algebra classes struggling with fractions, and this problem may be a big help. I'll pose it to them. (I've seen it before but hadn't thought of it as a pedagogical tool.)

2. Maybe people won't know how to answer that, but can offer you examples of problems that hooked them.

3. I'll have to think about your question, but thanks very much for the idea. This is what I have been missing when I think about how I teach my students. I know that it is these kinds of problems that grabbed me and made me love math, but I didn't make the connection that I needed to offer students the same opportunity. I want them to be as fascinated as I am by the subject as I am, but what I see in their faces is boredom. I haven't been offering them hooky problems!

4. Sue: For any 1/n there's always at least a few unit fractions that will work. For example, 1/10 = 1/20 + 1/20. More generally, 1/n = 1/2n + 1/2n. (If I understand the problem correctly, there's precisely one other pair of (a,b) that will ALWAYS work, no matter what n is.)

Also, my next posts will be an attempt to catalog some more hooky problems and to explicit redesign problems to be hookier. Short answer is that this is something that I'm still struggling with.

Suugaku: What sort of problems have you loved?