I “overheard” an exchange on social media that can be summarized like this:

Person A: I teach mathematics using an IBL-style.

Person B: I could never learn mathematics that way, even though I am good at mathematics.

I spent a lot of time thinking about this exchange, and I have found it helping me immediately in several ways. I guess this means that I might ramble a lot in this post. [Edit 20 minutes after first posting: this is probably not new to most people, and I have had similar thoughts before. But this was a bit of an epiphany for me for some reason I cannot explain.]

First, while Person A and Person B are both talking about “mathematics,” I think that they mean two different things. In fact, I think that there are (at least) three meanings for the term “mathematics” with respect to teaching.

The first meaning is what I call “application of existing mathematics.” [Edit 20 minutes after original post: mathematics that is often described as "procedural" belongs in here, although I suspect there might be more]. This comes in two flavors: the application to mathematics, and the application to outside fields. In a stereotypical “traditional” mathematics classroom, this is what is mostly meant by “mathematics.” For example: in a calculus class, finding the derivative of is an application of several existing bits of mathematics (the Power Rule, the Sum Rule, etc) to a mathematical problem to get the answer. And almost any sort of word problem fits this description.

The second meaning is what I call “understanding existing mathematics;” I think a lot of people would say this is about understanding concepts. In a Peer Instruction class (at least, in a PI class that operates in a similar way to how I do PI), this is what is mostly meant by “mathematics.” For example: in a calculus class, asking students how many tangent lines can be drawn at the point of might be an example of that. To answer this, students need to understand the existing notion of tangent line to do this. Another example would be getting students to understand the definition.

The third meaning is what I call “creating new mathematics,” or “doing mathematics” (when I say “new mathematics,” I mean that it is genuinely new to the student, not new to the entire community of mathematicians). I imagine that this is mostly meant by “mathematics” in a good IBL classroom. Students need to engage in the actual process of how mathematics is done by mathematicians, which includes dead ends and wrong answers (but also includes successes).

[Disclaimer: I am not trying to put a value judgment on these three meanings, although I am probably failing given that I am using the term "do mathematics" for one particular meaning. But I do happen to think that all three are extremely important. I also am probably talking in absolutes more than I should; please insert your own nuance.]

So it seems to me like that conversation actually was:

Person A: I teach *students how to create new (to them) mathematics* using an IBL-style.

Person B: I could never learn how to *apply existing mathematics* that way, even though I am good at *applying existing mathematics*.

I am guessing that Person A *does* teach students how to apply existing mathematics, but that it is secondary (or tertiary) to teaching students how to create/do mathematics.

Questions:

- Do any seasoned IBL instructors want to comment on the accuracy of my claims?
- Am I missing any other meanings?
- Anything else?