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Marilyn Burns has said, on more than one occasion, "If you're not stuck, it's not a problem. Why are you doing it?"
Many people believe math is learned by working through formulas with good explanations of why they work. That's not my perspective. Learning by being told is too much of a spectator sport to be very useful for most learners. Real learning starts with a good problem. The short version of what makes a problem good: it rocks your math world, at least a little. The longer version says that good problems are three things.
They are meaningful. They are discussable. And they are challenging, in the Goldilocks sense. Not too hard, not too easy.
Seymour Papert, the computer scientist who invented the LOGO programming language for children, gave us two of the three. Papert's word for making an idea meaningful was constructionism, which he defined against its opposite, instructionism. Instructionism is the belief that the way to improve learning is to improve the delivery, the telling. To do this you need to teach it more clearly, break it into smaller steps, and/or explain it better.
Papert thought this was backwards. Better learning does not come from better telling. It comes from better building. A child builds a sandcastle, a LEGO tower, a program, a theory, and in making the thing she makes the understanding. "Here, let me show you, now do it a hundred times" is instructionism written as a lesson plan.
Papert also gave us a great expression, hard fun. Children do not avoid hard things. They avoid pointless hard things. The "hard" is the challenge. The "fun" is the meaning. Research backs this up. When elementary students hit an impasse in a math problem, they don't feel one flat emotion. They feel several things all at once. They can feel confusion, frustration, curiosity, or energized tension. Students who felt curious and frustrated were more likely to change strategies, reread the problem, and lean into the impasse "(the 'stuckness')" rather than step away. If a problem is just hard, frustration collapses into shutdown. If a problem is hard and meaningful, frustration plus curiosity becomes fuel. And when the stuckness resolves, students feel pride, satisfaction, sometimes outright joy. Over time those experiences accumulate into a taste for hard fun.
Good problems are discussable. Lev Vygotsky, a Russian psychologist and researcher made the claim that thinking is social as well as private. When students make their private thinking public, an idea that originated in one head becomes part of a larger discourse. It gets held up and turned around, by the thinker and by others. Speech does not always report a thought that is already finished. Discussion is sometimes where the thought gets made, refined, or enlarged.
You can see this happen. Two children disagree about an answer, and in defending hers, one of them says something she did not know she thought until the words were already out. The talk did not describe the thinking. The talk was the thinking, finding itself.
But discussion can be fragile. The researcher Noreen Webb spent a career watching math talk in real classrooms. She found that discussion is important, but that the most important element for growing math is engaging with someone else's idea. That does not happen just because you seat four children together. It has to be built on purpose, and it has to be built on a problem rich enough to produce something worth saying. You cannot orchestrate a discussion about a worksheet.
I was once asked to do a demonstration lesson at a high school. I generally don't like demonstrations. My reasoning has always been, "How does you watching me ski help you to ski?" But the teachers at this school did not believe that choosing a good problem mattered. So, I agreed.
They set me up with a pre-algebra class they considered "problematic." It had a reputation for being bored and disruptive. A quarter of the students were on IEPs. I chose an area problem I've used from grade 7 all the way into graduate school. There is nothing special about it on the surface. But students think the solution is easy, and then find out it's not.
I put the problem up and asked what they noticed and wondered. One student said he thought he did this problem in fourth grade. They seemed tired, maybe a little bored, but not resistant. I put them in small groups at the whiteboard. As they worked, I moved through the room asking questions, and excitement grew. One of the watching teachers noticed something: "Even the students who are not contributing much are still listening. That's more than they usually do in class."
We ran out of time before anyone finished. Then, as the bell was about to ring, the most extraordinary thing happened. Two groups approached their regular teacher and asked if she planned to leave their work up. She told them she'd need the board for her next class. They asked if they could take a picture of their work so they could finish it.
"I've never had a student ask to take a picture of their work before," she said.
The lesson worked because the students saw a problem worth doing. It was meaningful. We could tell, because it engaged them. It was discussable. There was talk about it from the moment we started. And it was challenging without being overwhelming. They kept working without giving up.
The discussable problem, the meaningful problem, and the challenging problem are not three problems. They are one problem, seen from three sides. If we're honest, most of the time school math is for the teacher, or the school, or the curriculum. For problems to be meaningful, there has to be meaning for the students.
All Learners Network is committed to supporting pedagogy so that all students can access quality math instruction. We do this through our online platform, free resources, events, and embedded professional development. Learn more about how we work with schools and districts here.