If students only solve a narrow range of problems they will not be able to apply their knowledge in new contexts in the future. If my students solve a broad range of problems they are more likely to develop deep knowledge. All my clever demonstrations, cute explanations, and bad jokes matter a lot less than I like to think. What matters is students thinking about mathematical ideas, and the best way to get students thinking is to have them solve problems.

I’ve spent some time over the last few weeks trying to expand my knowledge of problems. Mostly, this has involved finding interesting problems in my textbook and working through them. I use the Larson, Hostetler & Edwards Precalculus textbook sporadically but most of my curriculum is a homebrew. While I enjoy the freedom to teach what I want to teach, I also run the risk of teaching topics in narrow ways based on my knowledge and biases.

There’s a lot of drudgery in textbooks, and some are better than others. I didn’t work through every problem. But flipping to the last page of each section led me to a surprising variety of interesting problems. I found plenty of challenges and new perspectives and saved lots of problems to use in my curriculum. I think textbooks can get an unfair reputation. For the most part they are resources of examples and problems. Examples and problems are the backbone of any math curriculum. Teaching straight from the textbook can be incredibly uninspiring if I just parrot the examples and assign 1-33 odd, but if I insist on inventing everything myself I’m missing the opportunity to save myself effort and expand my knowledge of many topics.

The largest danger of textbooks, in my opinion, is the structure of the text and not the problems themselves. There’s an implied pedagogy in typical textbook design. Start with examples, then assign students some repetitive practice. Fast students might get to some harder problems at the end, but probably not. I want to use some of the new and challenging problems I’m finding in my examples. Rather than only teaching the basics and hoping a few students can figure out hard problems, I can raise expectations by making challenging and unusual problems an explicit part of my teaching, providing students with support, leading discussions, sharing perspectives, and summarizing takeaways. The drudgery of textbooks to me is in the repetition of paint-by-numbers mathematics. I can enrich my curriculum by incorporating variety. It’s all there, it just needs to be structured in the right way.