Understanding Abstractions

This doesn’t feel true about mathematics. Much of the math I teach I would enjoy going down a similar rabbit hole with students, though it hopefully wouldn’t take as long.

But this comic also made me think about calculus. There are plenty of gaps in my calculus understanding — I’m not sure I could prove the product rule without some significant help, for instance. I’ve worked through proofs of Lagrange error before but I’m a long way from really understanding how that whole thing works. Not to mention the Fundamental Theorem of Calculus, which I can use pretty fluently yet don’t particularly understand why it’s true.

Maybe this is a reminder to deepen my own content knowledge. At the same time, my instinct is that there are times when it’s appropriate for a tool to remain an abstraction. I would like to verify that abstractions work — for instance, use Desmos to verify that a few product rule applications do, in fact, produce appropriate derivatives. I wonder if I could come up with criteria for when an abstraction is far more useful than understanding why that abstraction is mathematically correct.

3 thoughts on “Understanding Abstractions

  1. Michael Paul Goldenberg

    I believe every math teacher (and learner, which includes teachers) draws the line on how many layers deep s/he’s willing to go in a given pass with a particular topic. The person in the cartoon is either infinitely patient/indulgent or has never heard of resources like Wikipedia. But then, it’s a comic.

    I don’t generally struggle with drawing such lines in class (and offering suggestions about where students can look for more). The problem is drawing them for myself as a learner. At what number of levels do I come to grips with the fact that I either have too many holes in my knowledge to pursue the original topic/issue/problem successfully or that I’ve lost the thread entirely and really don’t have much of a clue why I’m still following link after link.

    I can’t speak for anyone else; I just know how frequently I’ve chased my own tail without feeling like I got anywhere close to what I’d hoped to understand. It’s okay: I am well aware of the vastness of the territory, the limitations I bring to the table, and the general unlikelihood of finding just the right guide to get me through the labyrinths. I wish I had a firmer foundation, more aptitude and intuition, better mathematical maturity and overall chops. Still, it’s fun to play around in various branches of the mathematical tree. If we inspire students to see that as fun, not torment, we can feel pretty damned good about our teaching.

    1. dkane47 Post author

      I would describe it as using a tool without understanding why that tool works (and perhaps without sufficient knowledge to figure out why that tool works). But that’s just trying to capture how it’s being used in the comic.


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