Honoring All Parts of the Mathematical Process

I went to a sectional MAA meeting 1.5 weeks ago, and I saw a fantastic talk by Aaron Wangberg. He basically said that there are about four parts to doing mathematics:

  1. Experimenting and developing intuition
  2. Making definitions
  3. Making conjectures
  4. Proving and problem solving

(These four may not be exactly the four that Aaron had, but they are close enough for this post). Aaron went on to say that we spend almost all of our time in the four area—proving and problem solving—in a typical math course. He thinks that we should work on all four aspects with students.

(A quick aside: I have always felt a bit uneasy about Moore Method-style notes like those found on jiblm.org They are great, but Aaron gave me a way to put my finger on one of the things that I don’t like about them: they definitely do not provide space to make definitions.)

In many ways, this is not new. However, he put things in such a way that organized my thinking to the point where I think that I can start doing this (which I am sure is not being captured in this blog post). I am newly inspired, and I am trying to figure out how to start doing this justice in my courses.

The other motivator for me is my success in my capstone class this semester. Briefly, I am giving them open problems to work on. They are doing amazingly well. They are going through all four of the parts of doing mathematics listed above, and it is awesome to watch. I am so happy with this course (and so are the students, based on the feedback I have gotten, which is from nearly everyone in the course).

I understand that these are senior math majors, so I would need to scaffold things a lot more for, say, my probability and statistics course (next semester) or courses for elementary education majors (next year). However, I am going to plan on doing this.

Here is my one barrier: I know that this is going to require me to grade very differently. I know that my grading system (and courses in general) have gotten too rigid, so I need to figure out how to grade in a way that (1) I feel good about and (2) allows for stuff like “developing intuition” to count.

Let me know if you have ideas!

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12 Responses to “Honoring All Parts of the Mathematical Process”

  1. Min Ro Says:

    It reminds of a story an upperclassman told during graduate school. He was talking with advisor and mentioned the definition of something, and his advisor replied, “No, that’s the wrong definition. Go find a better definition.” The student replied, “But this is the definition in the book.” And the advisor insisted, “No, it’s the wrong definition. There has to be a better one.”

    Especially in my undergraduate years, definitions seemed to be carved in stone and the use of intuition was bad. That latter one makes a bit of sense when relying too much on it, but I feel like I had to unlearn a lot of accidental lessons.

    • bretbenesh Says:

      I think that I had the same experience, and I think that this is true in general (just look at what happened when astronomers changed the definition of “planet” so that Pluto was no longer a planet).

      You make me imagine a class where we might have an evolving definition—that would be awesome!

      • Min Ro Says:

        One accidental example of evolving definition (through multiple classes) would be “compact spaces.” First as “closed and compact” for subspaces of Euclidean space, then “sequentially compact” (every sequence has a convergent subsequence) for metric spaces, and then finally the open cover defintion for general topological spaces.

        Beyond this example, I can’t think of a good example for an evolving definition. I was thinking about the historical evolution from groups of permutations to the abstract definition. But I don’t how well that would work.

      • bretbenesh Says:

        I am seeing that in my capstone class. They are trying to solve a nim-type game, and they are coming up with definitions akin to P-position and N-position, but they are refining them as they go.

        I think that this would actually work even better if the students generate the definition, since they do not know the “right” definition. For instance, they might initially define “irrational number” to be one with a non-repeating decimal expansion. However, they may change their definition once you ask them to prove that the square root of two is irrational.

      • Min Ro Says:

        Oh, I see now! That sounds quite interesting!

  2. gasstationwithoutpumps Says:

    I got through a MS in math without ever really doing anything but problem solving and proofs (and memorizing definitions). Computer science lead to much more use of parts 1, 2, and 3 than mathematics ever did for me.

  3. TJ Says:

    I don’t want to hijack this conversation, because I basically just agree with you. As usual.

    But here I go:

    First, for transparency: I recognize that I am not a disinterested observer.
    Also, I have not read every single set of notes on JIBLM.

    I can tell you that at least one set of notes on JIBLM is designed exactly to get students involved in making both conjectures and definitions. It’s not that a set of notes in that style can’t do it, it’s just that it isn’t always obvious how they do it. A good instructor still matters.

    • bretbenesh Says:

      Dear TJ,

      First, thank you for using “disinterested” in the way it was meant to be used.

      Second, thanks for pushing back on me. I am certain that you are correct. I was writing, perhaps a bit carelessly, in response to the Moore Method-type notes I have mostly seen, which starts as “A group is a set combined with a binary operation that satisfies the following axioms …” as Definition 1.

      Certainly, if this is a problem with the types of notes found on jiblm.org, it is a problem of history and culture, not format.

  4. Lessons Learned from Fall 2019 | Solvable by Radicals Says:

    […] an open research question, and they went to town. I got to see them successfully get through all parts of the mathematical process, and it was a joy to see them succeed. I am thrilled I get to teach this course again this semester […]

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