A teacher may look at two disparate problems and see what they have in common — maybe a math teacher sees the two problems below as two connected examples of problems where it is useful to work backwards. But I’m skeptical. I think that students who are still novices mathematically lack the knowledge structure to make this connection. Instead, students need to encounter problems that make connections at a smaller level, and build knowledge systematically — deep content knowledge first, and strategies coming alongside and after as student knowledge develops. (“Non-Routine Problems – A Change in Opinion”, Dylan Kane)

I share Dylan’s skepticism that “work backwards” can really become a useful problem solving strategy for students. But, wait, what about the testimony of teachers who* *report that their students have learned and use these strategies?

I think that this could be explained by our (usually) only teaching groups of students for one year at a time. Say that I’m teaching Algebra 1, and I encourage my students to work backwards. Naturally, my examples of “work backwards” are all Algebra 1 examples: solving equations in one variable, mostly. And, yes, with time our students do use “work backwards” to as a strategy for solving equations. Maybe they learn another opportunity or two to use something that’s recognizable as “working backwards.”

The thing is that we rarely see what happens next year in their next course. Do we think that our students will just know how to use this strategy to devise proofs? Do we think that, after they learn how to work backwards to discover proofs, they’ll connect this to the strategy they learned last year in Algebra 1?

The research that Dylan cites suggests that students won’t make this connection. Let me support that research with a skeptical question: how do you *actually *know that your students are learning strategies that mean anything outside of your classroom?

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A very legitimate challenge that partially attacks the value of math as a high stakes subject (it teaches problem solving! it teaches logical thinking!) Studying expertise shows that there is, at some point, an issue of specificity: to be an expert in topic A involves specific skills and knowledge that do not translate to expertise in other domains. Or: if you want to be a tennis professional, focus on your tennis serve, not your golf putting.

That said, perhaps the primary thing we should be teaching are the meta skills and math class should just be applied problem solving (et al)? I would still claim it is a valuable sandbox for a large collection of extremely valuable skills (or, dare I call them, habits of mind), but that should be a more explicit objective.

What are the ideas that you want your students to take from your class into the rest of their studies (and their lives)? Caution: you might not have any real control over what sticks!

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