Reasoning through Content

I love writing because as I write, I understand things a little better. I’ve written and changed my mind about why math might be worth learning more times than I can count. Each time, I get a little sharper in my thinking. Long time readers will probably find this repetitive, but I enjoyed writing it, and hopefully there’s a bit more insight than before.

Math class has lots of purposes. Learning math keeps future doors open, allows people to better understand the world around them, helps students to experience beauty and wonder, and empowers learners by showing them what they are capable of. But the most important argument to me is that math teaches reasoning. It’s not the only way to get there, but I think math class has enormous potential to teach students to notice and generalize patterns, shift representations and see a problem from a new perspective, balance attention to the big picture with attention to the details, connect ideas that seem different on the surface, and move flexibly between abstractions. These are skills that could serve students in the future within and beyond math classrooms.

But I hesitate to argue for reasoning as my primary goal in teaching math. Reasoning is the end goal, but it’s not the means of getting there. I don’t think that reasoning is a skill that one can practice; students can’t go to class, do some hard problems, and assume they are better at reasoning. I can’t confidently say that I’ve seen my students reasoning more effectively or consistently at the end of a year. All humans can reason, and all humans struggle to reason consistently. It’s a gradual learning process, and I’m lucky if I have a small part in that as a teacher.

More than my feeling that teaching reasoning directly doesn’t work, it’s also inequitable. Reasoning always happens in context, whether we choose the context of factors and multiples, interpreting statistics, or making decisions about finances. Students who already have knowledge of that context are likely to learn, and students who don’t, won’t. Pretending that reasoning is divorced from context advantages some students while leaving others floundering.

The Road to Reasoning Goes Through Content

Reasoning is a fuzzy goal, with a long and uncertain path to reach it. But content is what I know how to teach. I don’t have much confidence that I can teach a student to notice patterns and create generalizations in a given class. I do think I can do a decent job of teaching students to solve problems with quadratic functions. So I start with content. If I do my job and give each student a foundation in the content we’re learning, they all have a chance to apply their knowledge in new ways, to stretch their ways of thinking, and to reason. And I believe that, as students practice reasoning in more contexts, it becomes a little more likely that they will be able to apply those skills somewhere new. At first it’s small jumps — learning some things about quadratics and polynomials that can then be applied to rational functions. But, over time, I think math education sets students up with skills they can transfer outside of the math classroom. I don’t claim to know exactly how to get there, but that’s the goal.

This might seem like a dull perspective. We teach math to teach reasoning, but all I want to do is focus on content? But there are so many rich questions left unanswered. What content is most useful to teach? I don’t love what I’m asked to teach right now. Functions and algebraic manipulation are great, and should be in the curriculum, but what could we cut to make room for the math of gerrymandering, statistical analysis for social good, linear and exponential modeling in the world, and more? And once we’ve decided on the content, how can I teach that content in ways that help students reason? What are the teacher moves to link students’ experiences in math class to future decisions? What experiences will help students see math as a tool that empowers them in their lives? I don’t have any more answers at the moment, but I think these are great questions, and they are even better questions to ask about the breadth of content I teach each year.

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