I've put together an ever evolving list of characteristics of rich math problems. Anything you'd add, subtract, or edit? These are in no particular order...

1. The problem should be accessible. It should minimize vocabulary and notation, have multiple entry points, and include ways to collect data of some sort. It should have multiple methods that promote different learning styles and celebrate different ways of being smart.

2. The problem should be deep. It should be rich enough to spend hours, days, weeks, months, or years working on variations, generalizations, and extensions. It should lead to and connect as many different aspects of mathematics as possible, as this can then be the motivation for developing procedures, vocabulary, notation, and mathematical concepts.

3. The problem should be able to scale sideways so that students can explore related ideas in different contexts to reinforce concepts.

4. The problem should be captivating. This does not mean that it has to be a “real world” problem and it really shouldn't be a contrived real world problem (please explain to me how in the world I would know the number of total animals and feet I had on my farm, but not the number of cows and chickens). It might mean that it leads to a surprising result. It might mean that it feels like a puzzle waiting to be solved.

5. The problem should be mathematical. Progress should be able to be made by using problem solving heuristics and resources. The language of mathematics should benefit the student in solving the problem.

Two problems:

ReplyDelete1. How many coplanar points does it take so that there is precisely one unique conic section that contains all those points? (I've found two elegant ways of finding the answer for a circle; I suspect it depends upon the placement of the points for others.)

2. What is the radius of the biggest circle one can "drop" into a concave-up parabola (with directrix-focus distance of 2, let's say) such that the circle will touch the bottom of the parabola? (This is not a great problem by any means, but fun and pleasing.)

@Benedict: Thanks for the contributions! Could you talk more about the characteristics that make these problems good/interesting to you? Imagine I were going to the great repository of math problems to find you a new problem to work on, but I wanted to make sure you'd be interested.

ReplyDeleteThis is one of the best definitions of what a problem-solving approach to mathematics should be about. It fits right in with a lot of things going on with the math circles movement here in the Bay Area, such as http://mathteacherscircle.org . We're definitely going to link to this post from our discussion of what a math teachers' circle is!

ReplyDeleteI wonder if you'd like to join us for one of our workshops this summer? I think you'd get some good problems from us and we'd get some great ideas from you on how to improve our pedagogy.

We'd be happy to have you as a visitor in the mornings of the week of 6/28 when we host teams from around the country that are interested in the math circles approach to professional development, or, even better, any time during the week of 7/6 at our workshop for local middle school teachers.

You can reach me at Joshua dot Zucker at gmail.

I have been talking with teachers about this very topic. The idea of learning through problem solving, rather than learning for problem solving is one of my main themes. Here is what I suggest makes a good problem. It's quite similar to yours, and is based on the work of Marilyn Burns, Marian Small, John Van de Walle, Dan Meyer, and me.

ReplyDeleteA good problem is:

Given at the beginning of the learning, not at the end.

Non-routine. Students can not answer it immediately.

Allows every student an entry into it.

Is compelling enough that students are engaged in the problem.

Is interesting enough that students are compelled to persevere.

Fosters discussion and debate.

Invites multiple methods of solution.

I'm wondering here about California's adoption of the national Common Core State Standards. The eight Standards of Mathematical Practice get to a lot of the habits of mind NCTM created and you talk about.

ReplyDelete1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

The language isn't as classroom friendly as the lists above but the concepts are similar.

Are these national standards playing into your work/planning/ thinking at all?

@CA NYC: Welcome! Based on your name, I guess I'm NYC CA. Both the NCTM and Common Core Standards talk about process and habits and I am sure that their list and my list both come from some of the same sources. My issues with the NCTM and common core habits have always been that 1, they don't talk about ways to teach these things (and unlike fractions, teachers have little experience learning these things explicitly and there are very few existing materials/lessons for teaching habits, nevermind the fact that I've yet to see a district pacing guide that builds this time in) and 2, every standardized test I have ever seen has devalued these processes by saying that they are "inherently assessed in the content questions." I find this a total cop out.

ReplyDeleteI'm trying to give a more detailed description of what it means to, for example, "Make sense of a problem". Hence the long and detailed list.

Then I'm building in class time to explicitly teach these processes/habits (which I fortunately have the luxury to do).

Then I'm assessing kids specifically on these processes/habits (first quiz will be tomorrow...I'm interested to see how things go).

Then I will attempt to get kids to use these habits effectively on problems where I haven't told them to use a specific habit of mind.