One mistake we make in the school system is we emphasize understanding. But if you don’t build those neural circuits with practice, it’ll all slip away. You can understand out the wazoo, but it’ll just disappear if you’re not practicing with it.
-Barb Oakley, source
I stumbled across the above quote in a recent interview in the Wall Street Journal, and it struck me as a useful way to think about my teaching.
When I first started teaching, I spent most of my planning time thinking about how I wanted to introduce new topics to my students. I was always looking for clever ways of explaining ideas and interesting new perspectives and hooks relating content to prior knowledge or student interests. I designed inquiry lessons carefully leading students to the big mathematical ideas I wanted them to grapple with.
Now, I spend much more of my time thinking about practice. Not that how I introduce a topic is irrelevant, just less important than what students actually do with the knowledge they’ve gained. I think about how to space that practice and interleave different topics, how to build toward more rigorous applications, how to ensure students engage with a topic in multiple contexts and use multiple representations over time. I work to create collaborative structures that will support students in doing challenging math while still providing individual accountability. I design sequences of activities that move between whiteboarding, technological manipulatives, and pencil-and-paper to keep students engaged for a full class.
The core principle of my teaching is that students are active in their learning. Students learn math by doing math. Practice can have a negative connotation among teachers, and research suggests repetitive practice on low-level tasks is ineffective for learning. But focused, purposeful practice that pushes students outside their comfort zone, is designed to move toward meaningful goals, and involves useful feedback is absolutely necessary for deep, durable learning.
There’s a constant balance here. John Sweller’s Cognitive Load Theory suggests that if the demands of problem solving are too great, students may not retain what we want them to learn even if they are successful in solving the problem. I am partial to Ben Blum-Smith’s summary: “any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees”. At the same time, Robert Bjork’s work on desirable difficulties suggests that if students don’t experience any difficulties in the learning process, what they learn is unlikely to be retained in long term memory or transfer to new contexts. Meaningful learning is hard; if it feels easy it’s likely a missed opportunity.
I’m uninterested in arguing about whether discovery or direct instruction is better. From my perspective, those terms have been overused and caricatured to become meaningless pejoratives. As Dan Willingham says, memory is the residue of thought. What are students thinking about? What does that thinking look like? Those are the key questions I’m interested in, and I think they lead conversations past surface features to the substance that has a real influence on learning.
So students learn math by doing math, and my job is to constantly monitor what that experience is like for students. To what extent are they challenged and thinking deeply about mathematics? To what extent are they overwhelmed and struggling to connect the dots? If I can find a balance between these two poles while keeping students doing substantive math that builds toward ambitious goals, it’s a good day for me.