Defining “Cat” Without the Cat

One of the hardest parts of learning math is the vocabulary. I mean, “isda”? “Ibon”? “Punung-kahoy”? What is this, Tagalog?

Wait, sorry, that’s Tagalog.

Still, mathematical words can feel like a foreign tongue. And they’re much harder to acquire than terms in Tagalog or English. To see why, consider how you might learn a new word. A word like, say, “cat.”

First, you might just pick up its meaning from context.

Or, second, someone can just tell you the meaning.

But there’s one situation where learning a definition demands extra effort: when you’ve got no prior experience with the object being defined. In this case, learning the definition is not simply a matter of affixing a name to something you already know.

You need a more elaborate introduction.

It’s this third scenario where mathematical words tend to belong. They name ideas that are invisible, abstract, and yet highly precise. The “derivative,” say, is something exotic and ethereal—even more so than cats.

Which brings me to my point: our mathematical culture gets this exactly backwards.

We tend to define a new term in the abstract, draping it in high-minded language like purple garments—all while nobody has any idea who’s under the robes. A much better method: explore motivating examples, and then give definitions.

How do you define “cat” without a cat? The fact is, you can’t.

19 thoughts on “Defining “Cat” Without the Cat

  1. I wholeheartedly agree with you. I learned all those vocabularies by brute force. Memorizing all of the definitions and just did well in the algebraic procedures to pass the classes. As a math teacher now, I won’t do that to my students. Using patterns and visual and allow students to describe concepts in their own words (that thingy, this little weird shape, etc) sometimes lead to better engagement.

    (btw, my own education got to the worst when I took abstract algebra in college, for the same reason as you have described…)

    1. Yes, abstract algebra is the classic case of trying to define ethereal, invisible objects. A certain level of abstraction is as inevitable as the name suggests. But no reason to make things worse with bad pedagogy!

  2. I found this all through my Engineering education as well. Differential equations was taught in the abstract, then two years later you take circuit theory. I had almost completely forgotten DiffEq because I had no context in which to retain it. It would have been much easier to learn the math with the motivating example.

    We also learned a lot of electronic theory (i.e. theoretical model of a transistor) and very little practical design using transistors. I found it very strange how much of STEM education seemed to try to force you to learn abstract concepts in isolation rather than teaching with examples.

    1. Yeah, that’s an interesting puzzle.

      I think people are drawn to the efficiency and power of abstractions. One statement covering many cases. But of course it doesn’t matter how grand and all-inclusive the teacher’s statements are; it matters what the students internalize!

      1. A colleague always told me, “Math dictionaries are great for people who already know math.” That is, the statements are concise and precise… and absolutely unintelligible if you don’t have a frame of reference.

        I love this essay, Ben. A colleague once told me that students can’t discover definitions. Then I showed her examples and counterexamples of ratios and asked if she could use that to create a definition of ratio, and she was like, “Oh.” Thanks for sharing!

  3. Ha, that lovely furry little creature we all love 🙂 Food for thought. Happy to see you back! I thought you had gone for a while. Got your book for Christmas! Happy New Year to you!

    1. Glad to be back! I’ve been working on book #3, but that’s almost done, so I’m hoping to be blogging a little more. Thanks for reading, and happy new year!

  4. Actually, it turns out to be impossible to define mathematical terms except to a tine 0.00000001% minority of mutants born with mathematical minds. And, amusingly enough, you can prove this mathematically.
    Consider: every mathematical term typically requires at least 2 more mathematical terms to define it. As, for example, a fiber bundle. Oh, what’s a fiber bundle? It’s a space that is locally a product space, but globally may have a different topological structure. Great. What’s a product space, and what’s a topological structure?
    A product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. What’s the Cartesian product of a family of topological spaces? And what’s a natural topology?
    You can see where this is going.
    At each step of the definition, the number of terms you need to define increase by 2^N. A typical definition in mathematics typically has around 16 levels, which means you need to define 2^16 terms. That’s 65,536 terms.
    Now, the average non-mutant person will take at least a month (often more) to come to grips with each definition. This means that no human (except the tiny 0.00000001% minority of mutants born with mathematical minds) can possibly live long enough to understand any serious mathematical term.
    Short version?
    Math is impossible. Humans can’t do it. Only mutants can handle math. You can give examples until you’re blue in the face, but it won’t help.
    As Nobel laureate Chen-Ning Yang put it: “There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence.”
    If a Nobel laureate can’t deal with math, what chance do you have?

  5. The derivative is something exotic and ethereal, except in the art world where it is banal.

    I quite like the math words that have separate meaning in the vernacular.

    Rational is one of my favorites. Something that can be described by a simple ratio is able to be understood and reasonable. Integer is another as indivisibility implies integrity.

  6. Agreed. I end up creating handouts that explain things in normal language (I find outrageous examples help) and have cartoons for visual. I sometimes wonder what curriculum writers are thinking. Kids like to have fun. Why make math as un-fun as possible with boring examples?

  7. Examples of abstruse math words that only acquired real meaning for me decades after being introduced: elliptical, parabolic, and hyperbolic, as applied to partial differential equations. Now, I understand them as how boundary and initial conditions move through the domain. Sadly, I’m still very bad at distilling this insight to others.

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