Three Different Meanings of Mathematics

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 x^2+\sin x 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 (0,0) of f(x)=|x| 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 \delta-\epsilon 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.


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


14 Responses to “Three Different Meanings of Mathematics”

  1. Pinky Says:

    In my Ed classes we discuss IBL methods, and the benefits of it. As far as I can see, all three methods should be used. IBL is a great tool, but for some lessons it isn’t necessary or other methods just work better.
    I think a lot of students haven’t had practice learning in an IBL style, so there is a bit of a curve where they either don’t grasp the material right away (and think that is a bad thing) or don’t find value in being able to discover or understand what they plan to apply.

    • bretbenesh Says:

      Hi Pinky,

      I agree about the learning curve—so much so that it seems to me like (at the college level, anyway) it is difficult to be partially IBL. I have tried this before, and it seems like I get all of the pains of IBL and none of the benefits by using half measures.

      Of course, the problem is just as likely due to bad implementation than the fact that I wasn’t using IBL 100% of the time. So I am not ready to completely give up on partial IBL yet.

  2. TJ Says:

    This all sounds about right.

    And I wasn’t really ready to have that conversation, yet. It seemed rather farther down the priority list. 😉

  3. TJ Says:

    “all the pains of IBL and none of the benefits by using half measures.”

    This is my experience, too. And I have heard something similar from many in the community with more experience.

    • bretbenesh Says:

      I need to find a good time to go whole hog with IBL. I definitely want to do it when I get to our “intro to proofs” course, and I think that abstract algebra could be good for me, too.

  4. Andy "SuperFly" Rundquist Says:

    I really like these three definitions. I think it would help me with conversations with my students a lot. I had a lot of fun talking with my first year seminar students this week about mathematics after we read the Mathematician’s Lament. We played with some fun problems and talked about the different ways different approaches to math teaching would approach those problems.

    • bretbenesh Says:

      Thanks for having them read The Mathematician’s Lament. You are a kind-hearted physicist for doing so.

      I should have written in the main post that I think that most people think that the sole meaning of mathematics is “the application of existing mathematics.” That kind of thinking drives this mathematician crazy. Thanks for trying to help broaden your students’ minds.

  5. Aaron Says:

    I think there are a lot of ways to think about what “mathematics” is; I thought it was interesting that each of your three descriptions used the word “mathematics.”

    From my perspective, “mathematics” could be a collection of concepts and procedures; it could describe the activity of a community of practice (of the people we call “mathematicians”); it could be a way to categorize the activity of others; or it could be something else. And I think each of these perspectives has ramifications for the types of learning goals we create for our students.

    For example, if you view math as a collection of concepts and procedures then developing a particular habit of mind (in the sense that Al Cuoco uses the term) would probably not be a learning goal.

    • bretbenesh Says:

      Hi Aaron,

      Yeah, using “mathematics” to describe what “mathematics” means is a bit circular. My saving grace is that I said that these meanings apply with respect to teaching, so I am not making a case for what “mathematics” means in general. I guess I am assuming that the readers have a broader sense of what “mathematics” generally means, and I am relying on that sense to talk about what mathematics means with respect to teaching.

      Could you explain the difference between “the activity of a community of practice” and “categorize the activity of others?” Is the latter just like the former, only you do not consider yourself to be a part of that community?

      • Aaron Says:

        I think these broader notions of “mathematics” have ramifications for teaching. For example, if mathematics is “what mathematicians do” (including how they ask questions and interact with each other) then my job as a teacher is to create a particular type of community and enculturate my students in the “ways of being a mathematician.” That would lead to very different learning goals and teaching practices than if I thought my job was to “teach my students what a derivative is.”

      • bretbenesh Says:

        I agree that the broad definition of “mathematics” matters. That just wasn’t the purpose of my post.

        On the other hand, I think that if you ask mathematicians what mathematicians do, you will get a much narrow range of answers than if you ask them to describe the mathematics they teach (this is speculation—I haven’t actually asked).

  6. Joshua Bowman Says:

    I have some thoughts about IBL and doing mathematics, but it’s late, so I’m just going to link to a post I wrote a couple of years ago about what I think mathematics is and what it means to do mathematics:

    tl;dr version: Mathematics is a process—specifically a human process—and much of what we admire are the products of mathematics, not mathematics itself. To do mathematics requires asking and answering questions, and the depth of mathematics done depends on the depth of the question (which is not always evident beforehand).

    OK, here are some comments about IBL, too. I haven’t had the kind of conversation you describe, but I have seen the discomfort of (usually upper-level, math-or-related-major) students in an IBL setting when they realize they have to go through this process of constructing the results and the paths to get there. However, when I have students in lower-level classes (generally calculus, which I haven’t taught with a full IBL implementation) carry on similar kinds of discussion and debate to reach a conclusion, their reactions are almost entirely positive. They say they finally realize that math can make sense. So I wonder how much of the skeptical reaction is grounded in being invested in a familiar process that has worked for the interlocutor.

    • bretbenesh Says:

      Hi Joshua,

      I think that your mining metaphor is reasonable and well thought-out.

      Moreover, I speculate that a lot of mathematicians would agree with it. However, I also speculate that many (most?) professors do not explicitly work to convey this in their mathematics classrooms (I don’t do it enough, anyway. I hope I am not projecting here).

      Part of this second speculation is what you said about your upper-level IBL students: they are uncomfortable with it, likely because their previous mathematics teachers have not used your mining-type definition of mathematics operationally in the classroom. If they had, I am guessing that they would be more comfortable in an IBL class.

      So I admit that I totally dodged the “what is mathematics” question, partially because people like you have already given good answers. My point is that many mathematicians (including me) do not use this same answer when we teach—we use a different meaning of “mathematics” that may not fit your mining metaphor at all.

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