How I Try to Teach Problem Solving
Problem Solving
People like to say that the point of math class is to teach problem solving skills. That sounds nice. How do I teach problem solving skills, though?
There's no such thing as a generic problem solving skill. That’s the first thing I need to remember. Here's a good primer for the research on this topic. Humans aren't very good at transferring their knowledge from one context to a very different one. We can get good at problem solving in one area, but that doesn’t mean we’ll automatically be good at problem solving in a different context. Transfer does happen, of course; it's just hard to predict. The first part of teaching problem solving is to teach math. Lots of math skills could come in handy in lots of different situations. The best way to prepare for that is to learn lots of math. Then, it's important to solve lots of different types of problems and to learn when a certain concept or strategy is or isn't relevant. If I always give my students proportions problems with the same format and the same contexts, they will struggle to generalize those skills to new situations. If I give them lots of different types of proportions problems and we spend time thinking about how to know when proportional thinking is useful, it's much more likely that knowledge will transfer.
Another important piece of the research is that kids develop problem solving skills naturally. They apply those skills when they have some knowledge of a context, and when they are motivated to do so. I'm always amazed at the problem solving kids show in Minecraft, or in finding clever ways to circumvent my school's web filters and play Slope. So the goal isn't to teach kids how to be problem solvers; it's to help them apply that natural problem solving skill in more and more contexts.
Ok Where Is This Going?
The whole "we can't teach problem solving" is kindof a copout. Problem solving is what lots of people like best about math. I wrote a few weeks ago about how this year I have two extra periods a week with my students, in smaller groups than my regular classes. Some students get pulled for different services during this time so we can't move forward with the curriculum. I'm spending that time on a few different things, but one is problem solving. I just said above that I don't think you can teach problem solving in general. So what am I doing?
I really have two goals for these problem solving blocks. First, I want students to experience problem solving. Problem solving is an important part of the discipline of mathematics, and I want them to get a taste of what that's like. It's the experience I care about. Second, I think a key part of being a successful problem solver is a productive disposition. Some people see problems they haven't been shown how to solve and want to give up right away. My goal is to give students accessible, positive problem solving experiences so they develop a disposition to try things, even if they aren't sure what to do right away.
Here are six principles I use to design these blocks:
Accessibility. The whole point of this thing is that students find the problems approachable and doable. I'm not picking problems because I like them, or because a few kids might find them a good challenge. I'm picking problems that every student can access and engage with. That's not only the problems themselves; it's also about how I scaffold and introduce the problems to help make them feel doable. The first problem I used was to play a few rounds of "101 and You're Out." Then, once students got the hang of it, I gave them the six rolls at the start, and their job was to get the highest possible score using those rolls. Then we played some more, including using an 8-sided die and playing the highest possible score version with an 8-sided die. It might seem like a simple problem for 7th graders but that's fine — it led to some great thinking, it was accessible for everyone, and students enjoyed it.
Exploration, not insight. I read a few years ago and it has stuck with me ever since. In the universe of problem solving there are "move problems"where the solver can try different "moves" in different sequences, exploring different pathways until they reach a solution. Then there are "insight problems" where the solution comes from having a flash of insight, rather than gradual progress. One problem we've used is the 8 queens puzzle, asking how to place 8 chess queens on a chessboard so no queen can "see" any other queen. But we started by trying to put 4 queens on a 4x4 board, then 5 queens on a 5x5 board, and so on, building up slowly toward tougher and tougher problems. There are great insights to be had in this problem — but you can make a ton of progress by just trying different approaches.
Distribute success. I'm teaching a class of students — these classes are smaller than my regular classes, but still top out at 17 kids. My goal is for as many of those kids as possible to experience success during the problem. It can't be structured in a way where one kid figures something out and spoils the problem for everyone else; everyone should have a chance to experience success. We used this square cakes problem from Play With Your Math, and I put the number 1 - 40 up on the board. Whenever someone found a possible cake I circled the number and put their solution up. Some students really enjoyed finding very large numbers, so I wrote them on the board too. This way every kid feels like they have something to contribute, rather than focusing on whoever figures something out first.
Variety. When people think of problem solving in math class, they often think of problems like those on the Play With Your Math website, things that "feel" mathematical. But there are lots of ways to practice problem solving. This year we will solve plenty of problems like those, but we will also play games, make math art, code, and more. I'm working on lessons that practice using AI, or lessons with an intro to a personal finance topic and then a chance to research and explore things like which credit card is the best deal. There are lots of ways to apply problem solving in the world, and students should get to see a broad slice of that.
Reflection. Every problem solving block ends with a short reflection, either on what they learned from the problem or how they felt about the problem. This type of reflection is valuable metacognition for students, and also helps me to see which problems feel successful and which don't.
Silliness. Ok so maybe this is just the way I like to teach, but I try to mark our problem solving blocks with a bit of distinctive silliness. I shared a room with a teacher once who did something called "Fierce Friday." I forget what it was, it had something to do with reading, but to get students excited about it he would show pictures of fierce things. The day of the week of these blocks rotates because of our schedule, so I'm calling them Fierce Feats of Problem Solving. Here are a few fierce pictures.
At the start of each problem I remind students to get fierce and put up the photo and we laugh a little bit about it. And whenever we're about to try something hard — maybe I pause, share a few strategies, and push students to solve a tougher version of the problem — I remind them to be fierce and put the picture back up. It's one of those things that has to be authentic to the teacher, but it's been a ton of fun and has also helped me to narrate some of the perseverance I'm hoping to help students build.
One Final Note
It’s too soon to tell if this whole plan is working. I’ve taught five of these Fierce Feats of Problem Solving blocks so far. But one thing that has felt good so far is that each time I do one, I see different students engaged and excited about the problem. Maybe one student isn’t very enthusiastic about the square cakes problem but loves playing Nim the next week. Maybe another student finds Nim frustrating, but is excited to do some math with Sierpinski triangles and see how many triangles they can squeeze into one. Problem solving doesn’t have to be a hit every single day. But if each kid can find a few problems that they really enjoy digging into, I can help them understand and appreciate an important facet of mathematics.