But first, look at this cat:
Such beautiful creatures. But they'll burn you if you're not careful. Anyway,
What do I mean by “problem solving”?
For me it means that students are regularly asked to make progress on questions that they have never been told how to answer. This isn’t an air-tight definition, but it will do for now.
There are a lot of supposed benefits of a problem-solving approach to learning math. Here are a few that come to mind:
- It’s truer to the work that mathematicians do
- It’s more fun for students
- It develops habits of mind that are transferable
I think I agree with the first two ideas, and I’m skeptical of the third, but that's all sort of beside the point. My core responsibility in the classroom is to teach these kids a bunch of skills and concepts in a way that compares favorably to the way they’re learning them next door. If the most effective way to teach is lecturing and drilling, I will teach that way, even if it’s boring and unlike the way that mathematicians work.
The good news is that fun, truth and effective learning coincide in this case. I think.
I want my students to solve difficult problems in class because I believe it’s the most effective way for them to learn and remember the content. Here are my pedagogical assumptions:
- Difficult tasks help organize knowledge: When a person is faced with a difficult task, they search their memory for a way to accomplish the task. They think about the tools that they have and how well they fit the task at hand. This search reinforces a person’s understanding of their tools and how they are used. In math, the tools are the sorts of things we want kids to know: procedures, skills, concepts and habits of mind.
- Organization takes the form of connections between topics: It's a pretty solid result that novices organize knowledge by topic, and experts organize them by their underlying structure. A difficult problem doesn’t cue students into the tools that they’ll need to use, and so anything might be relevant. As students attempt difficult problems they need to start organizing what they know into more useful clusters than “first semester” or “lines stuff.” Instead, when presented with an equation they’ll start thinking about what tools they have for solving equations. When presented with a challenging proof they’ll need to think about other problems that they’ve proven in the past and decide which ones are relevant for the current puzzle.
- Students will fail often: Some studies have shown that knowledge sticks better after a person has taken a difficult test and failed. This makes a certain amount of sense – the brain is most attentive when we know we’re missing something. The right answers come in a problem-solving class, but they will always follow failure.
- Different approaches invite justification: It’s helpful for learning to have different approaches to discuss. Multiple approaches create the need for explanation, and explanation and justification also help students organize and remember their mathematical knowledge. When solving a good problem, students will almost never have just one approach. The teacher can skillfully select multiple approaches to bring to the fore.
- The mind remembers stories very well: "If you want to make something memorable, you first have to make it meaningful." But how do you make it meaningful? Stories that connect with the rest of the things that you know can do this. As Dan Meyer has put it, good math stories come in three acts. First comes the hook, where the problem is posed. Then comes the development, where students struggle with the math and run into trouble. Then comes the resolution, which we might talk about as a whole group, or students might discover on their own. The daily story telling comes easily: “Today we tried to do this, and we ran into trouble. Then we discovered X, and then we were able to solve the problem. But what about Y? See you guys tomorrow.”
Coming up in this series I'll point to things that I was doing wrong last year and how I think that I can fix them.