I’ve gone back and forth on this more times than I can count.

I think that pushing students to figure out large portions of the high school math curriculum for themselves, even with some structured guidance, doesn’t work very well in practice. Discovery can increase inequities as students who have strong background knowledge succeed and those who don’t struggle. It can exacerbate feelings of frustration toward math when students feel unsuccessful over and over again. It’s easy for a few students (or a lot of students) to fall through the cracks and miss key ideas. And asking students to figure things out for themselves takes a ton of time, and I think there are often better ways to spend my time. If discovery edges out time for students to practice and apply what they’ve learned, all that time they spent exploring becomes pretty worthless as what they discovered floats away without reinforcement.

However, I do value discovery in other ways. I think every student should experience mathematical discovery at least a few times a year, and some topics lend themselves to this well enough that my reservations can be put aside. I really love the Binomial Theorem, and once students have solid background with calculating combinations and multiplying polynomials, a structured exploration of the intersections between combinations, binomial expansions, and Pascal’s triangle can be a ton of fun. Figuring out new ideas is an essential part of the practice of mathematics, and that’s an experience I want all students to have.

I also think it’s often helpful to have students try to figure something out to see how well they apply their prior knowledge and identify where I might need to provide some extra support. I like to begin a unit on arithmetic series by telling students about how Gauss would finish his work early in elementary school. One day his teacher, to keep him busy, asked Gauss to add up all the numbers from 1 to 100. The teacher was amazed when Gauss found the sum in seconds. How did he do it? With this prompt, if some students can figure out how to sum an arithmetic series, awesome! I can try to spread those ideas through discussion and group work. More likely, I’ll lead some summarization and explicit connection and move into a practice or extension activity. I’ve also learned it’s important not to do this every day; if I’m always asking students to figure something out but I’ll explain it to them after 10 minutes whether they get it or not, they figure out the game and are much less likely to engage.

There’s some fun middle ground in a lot of situations as well. Recently I was beginning a unit on function transformations in Algebra II. I started by asking students to sketch rough graphs of a bunch of quadratics in vertex form — something they were pretty rusty on, but able to remind themselves of in small groups. After spending time playing with quadratics, we summarized the rules for the different types of transformations. This was a great transition into more abstract function transformations, making the connection between their prior knowledge and our next unit explicit. I was the one introducing new ideas, but students were still exploring to start the lesson and taking some ownership of their learning.

It’s easy to treat a discovery lesson as some big monolithic thing, but my choices depend on the content, students, and my broader goals and the time I have. Here are my core principles:

- Every student should experience mathematical discovery at least a few times a year to participate authentically in the practice of mathematics
- Beginning a lesson by asking students what they already know about a topic is a great way to get a sense of where they need support and to activate background knowledge
- Making concepts and connections explicit is an important practice to prevent exacerbating inequities that already exist
- Everything depends on context
- The central activity of math class should be students doing math; if students are spending all their time trying to figure out new things, they’re not spending enough time applying what they already know