Can we see more examples in class?
Why does this question bug me so much? It’s not because examples are bad. On the contrary, the research shows (and this is surely backed up by experience) that studying worked examples can be a highly effective strategy for learning a concept. So I ought to be happy to hear it, right?
When people ask this question because they want to study an example, I’m happy. But studying an example and seeing an example are two radically different things. Studying an example means making conscious efforts to examine the example in depth: isolating the main idea or strategy, actively trying out modifications to the objects involved, making connections to previous examples and mathematical results, and – very importantly – using worked examples to create new examples of one’s own. This is a strategy expert learners use, not only to become experts but to maintain and extend expertise.
But more often than not, when someone asks to “see” an example, especially in week 11 when it’s becoming crunch time in the semester, they want just that – to see and not to do. Underneath this request is often an assumption that if someone is exposed to enough examples then the understanding of the concept will form spontaneously (or at least with minimal work). Somewhere I’ve written about this before — people tend to think that knowledge is like a virus, in the sense that if we are just exposed to it enough then we’ll “get” it.
But there’s no reason to believe this and every reason to disbelieve. For example (irony!), this study that recently came across my radar [PDF] showed that in a sample of students preparing for a standardized physics exam, exposure to nearly 1000 routine exercises led to no improvements in conceptual understanding. We might find some resonance of the “virus” model of knowledge in our own experiences among the many moment when, after laboring for a long time on some concept, we suddenly understand it after seeing or hearing something that triggers the “light bulb”. But in reality, what really happened was that we prepared the way for that lightbulb moment to happen — by hard work, and by doing the kinds of things with examples I mentioned above. That’s a long way from just seeing a bunch of examples, as much as possible en route to some imaginary tipping point that will never come just by “seeing”.
The issue here is not whether examples ought to be studied. It’s who is doing what when it comes to those examples. For students, throughout most of their school math experience, the teacher is the “who” and lecturing is the “what”. So they ask to “see more examples”. Of course at the beginning of a semester, students aren’t experts, and showing them examples is important. But what I also have to do is (1) teach students how to study examples and (2) set and adhere to an exit strategy for giving examples. My job is not to give more and more examples. Instead it’s to say: Rather than give you more examples, let me instead give you the tools to create and verify your own examples. And then, at some point in the semester, formally withdraw from the role of chief example-giver and turn that responsibility over to the students.
I think that point comes around mid-semester for a freshman mathematics course, around week 4 (in a 14-week term) for a sophomore-level course, and about week 2 for an upper-level course. Certainly by week 11, every student needs to be taking a DIY approach to examples. Otherwise, while I try to be patient, I have the same kind of reaction to being asked to show more examples as I would if my 4-year old son asked me to spoon-feed him at dinner: No thanks. I know that would make you more comfortable, but comfort isn’t what you need. What you need is not to need me for this.