This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.
Here’s my entry.
I’m not exactly sure how to say the thing that I want to say, because the thing I want to say isn’t quite true, and can easily be misunderstood. But the thing I want to say is that problem solving is overrated in math teaching.
A story may help explain what I mean. I’ve been planning a conference for math educators, and I work on recruiting people to lead sessions. In the middle of each day, we have a block of time for doing some math together. But the thing is, it can be a tricky session to lead. Some of our participants have advanced degrees in mathematics; others teach younger students and haven’t used algebra in years. How do you make sure you’re doing math that’s meaningful for everybody?
This gets especially hard if the session is framed as the attempt to solve a math problem. The problem is one of aim — will the problem be productive for everyone in the room? What if someone immediately sees the problem for what it is, and makes quick work of it? As Benjamin Dickman put it, “if the plan can be brought down with Oh I’ve Seen This Already…then to me it’s a risky move.”
(Sure, you can tell people what to do if they’ve seen it already, but doesn’t that feel like a consolation prize?)
It’s the same problem that I had to grapple with at math camp during the summer. One of my jobs at camp was to teach the counselor class. Counselors ranged from 16 to 20 years old. The class involved giving counselors time to grapple with some of the challenging math that the camp asked kids to do battle with.
Here is a typical problem from camp:
What is the sum of all the digits of the numbers from 1 to 1000, i.e. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + … + 9 + 9 + 9 + 1 + 0 + 0 + 0?
That’s tricky. But it was significantly trickier for some of the youngest counselors I taught, who were themselves high school students, and significantly less so for the seasoned math major who is headed straight toward a PhD program in math.
What to do?
In this case, I started class by emphasizing that the goal was not to solve the problem, but to find math to understand.
I put a representation — 10 through 15, with the digits all splayed out like 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 — up on the board, and asked the class what they saw and noticed.
I presented the original problem. Then I asked each counselor to come up with a question that was related, but not the same, as the original problem.
I asked everyone to pursue one of these questions, or maybe the original problem.
Explicitly framing class as a search for understanding, calling on students to notice whatever catches their attention, focusing attention on new representations, asking students to think about a problem as living in the context of a network of similar problems…these are all aspects of math that I emphasize, at the expense of solving a specific problem.
Getting a bunch of counselors who signed up to work at a math camp to work on math is not exactly teaching on the hardest difficulty setting. I mean, damn, if every group of kids that I worked with could spend a half hour essentially poking around mathematically after the set up I left these counselors…my job would be so much easier.
I’m not saying that my counselor class should be anyone’s model.
But I am trying to say that I felt like I was subverting problem solving. Because when you make problem solving the goal, you create a lot of challenges for yourself. What happens if someone finishes first? If the goal is solving the problem, why should the teacher ever help? or why should they refuse to help? How do you make sure everyone feels valued if they don’t solve the problem? How do you keep people from feeling that your extension is a consolation prize?
I’m still not quite sure what to say about problem solving, but maybe it’s like this: a lot of things get much harder once you tell your class that the goal is to solve a problem. So most of the time, I don’t.
I think some people think of math as essentially focused on problems, but this is not how mathematicians necessarily describe their work.
For example, while Timothy Gowers (in Two Cultures of Mathematics) does describe himself as oriented towards problem solving, he sees this as the minority position within mathematics, where the dominant orientation is towards theoretical understanding. Michael Atiyah is his chosen representative of this dominant orientation. Atiyah describes himself as unmotivated by problems:
I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.
Talking, learning, discussing math: these are distinctly mathematical activities. It’s not all about problem solving — it’s not even primarily about problems.
There is a line of thought that argues that math education must be focused on problems and problem solving, because that’s what mathematicians do. But mathematicians do so many things beyond solving a problem.
More than anything else — Atiyah says this — mathematicians are interested in learning about math. They learn from colleagues, who explain things to them. They learn from classes and seminars. They ask questions. They teach, and learn from teaching. They learn new ways to represent ideas. They notice new things, learn to see new things as important. And, yes, they work on problems.
There’s another thing too. If you’re oriented towards problem solving, you’re oriented towards the achievement of the few. If mathematical success is about solving valuable problems, that’s going to be inherently inequitable. Your history of math becomes the history of individual achievement, mostly represented by men from privileged cultures. It’s the same inequities that show up in the classroom around problem solving — some have a head start, some finish first — but played out over an entire field.
And it’s the sort of thing that can get inside your head. Am I worthy of working on a problem? Will I ever contribute by solving something of value? (Or even: Has my culture ever contributed anything of value? Have people like me ever made important contributions?)
These worries make perfect sense in a world full of unsolved puzzles waiting to be solved. But, if we’re oriented towards understanding instead of problem solving, the whole situation gets turned on its head. Achievement is understanding, and there is more than one way to spread understanding. It’s not just about being first — in fact, it’s not about that at all.
These two ways of seeing achievement in math can clash. For example, when a graduate student worried online if he could meaningfully contribute to math, Bill Thurston answered that he was thinking of “contribution” too narrowly:
Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining — they depend very heavily on the community of mathematicians.
If we focus too much on solving a problem as a goal, then we end up focusing on the “solving.” And then we can end up paying too much attention on who did the solving, instead of looking at the way clarity was spread through some mathematical community.
(There are ways of combating this slide, but it’s hard work!)
If our culture and classrooms are focused on understanding, equity has a better shot.
So much for teacher conferences, summer camp, and mathematical culture. But the last piece of this sort-of-critique of problem solving comes from my regular classroom work during the school year.
I have definitely found that, when students are starting off learning something new, solving a specific problem has limited use. The drive to answer the question can narrow attention in unhelpful ways, focusing students on whatever it is that will help them most solve the problem — even if that’s not a new and interesting feature.
So I try to hold off on problem solving, at first, and I very rarely frame that as the goal of a class session.
I remind students: the goal is spreading understanding.
I ask students to notice new things.
I ask students to study examples of student work, both correct and incorrect. Because learning from existing mathematical work is mathematical. It’s only second-rate if your goal is problem solving.
I ask students to write their own problems, and to see problems as existing within genres of problems. (If Monday was about finding the volume of prisms, very likely my first ask on Tuesday will be for students to come up with a prism to sketch and to find its volume. These sorts of tasks are surprisingly challenging! Remembering stuff is hard, but the work is deeply mathematical.)
I ask students to compare representations.
And if all these things aren’t exactly in conflict with problem solving, that’s also not our goal. We’re aiming for understanding, and problem solving happens along the way.