So, to begin with, let me introduce this sequence of blog posts tagged [RUME 2016]. I was recently awarded a travel grant to the ICME conference coming up in Germany. Part of the requirements for this award is that I'm supposed to help disseminate stuff I learn at that conference, so I decided to practice / build readership of this blog by blogging about RUME talks I found particularly fun and interesting. I've got basically three blog post topics (which may span multiple posts) in my brain, with a fourth that may come later.

The first talk I'd like to blog about is Rina Zaskis's talk “On symbols, reciprocals, and inverse functions.” (Igor' Kontorovich is a coauthor on this talk but could not make it because Auckland is v. far away.) This talk immediately grabbed my attention because I wrote a paper some time ago about students' construal of similarity between functions and linear transformations, and in particular, the ways they think about the inverses of each of these things. More on this later (in a forthcoming post or two).

Dr Zazkis gave students a scripting task, in which she asked students (pre-service secondary teachers) to extend this imaginary (but completely plausible!) interaction:

T: So today we will continue our exploration of how to find an inverse function for a given function.

S: So you said yesterday that $ f^{-1}$ stands for an inverse function.

T: This is correct.

S: But we learned that this power (-1) means 1 over, that is, $ 5^{-1} = \frac{1}{5}$, right?

T: Right.

S: So is this the same symbol, or what?

T: …

(I think this is such a brilliant task.)

So here's three ways you could sensibly answer this question:

  1. The group theory approach: In every group, every element $ g$ has an inverse $ g^{-1}$ such that $ g * g^{-1} = e$, the identity element. So then these are totally the same thing; the only difference is what's the group and what's the operation (the group of rational numbers under multiplication vs the group of all functions under composition). Unfortunately, it's probably not the best idea to drop some group theory on some high-school students, so we should probably explore other approaches.
  2. The context-dependent approach: The common symbol $ \Box^{-1}$ means different things depending on what's in the box (e.g., a function vs. a number). This smacks of rule-based thinking and obscures the legitimate connection between inverses, so I don't like it twice over.
  3. The middle-ground approach: The common symbol $ \Box^{-1}$ means slightly different things depending on what's in the box, but there is a relation between these slightly different meanings. I'm calling this the middle-ground approach because it seems to bring out this relationship without invoking all the machinery of group theory. This is probably how I would choose to answer the question, should it come up; I'd probably talk in some amount of detail about how we could consider both things instances of some more generic idea of inverses. I think we'd all pretty much agree that this is a better way of explaining the relationship, even though it may be difficult right now to articulate why.

But Spencer, you're thinking, the title of this blog post is something about linguistics, and this is just a bunch of math. You're right; now it's time to borrow some linguistics words to give better descriptions to ways #2 and #3 above.

The words Dr Zazkis chose to borrow were homonymy and polysemy. These are good words that do exactly the work that we'd like them to do. Here are some definitions I synthesized from various google results:

Homonymy: the relation between words with identical forms and sounds but different and unrelated meanings. Example: “river bank” vs. “savings bank” vs. “bank shot” vs. “bank of interview questions.” (Yikes!)

Polysemy: the relation between words with identical forms and sounds but multiple, contiguous meanings. These meanings emanate from a central origin, and they form a network such that understanding any one meaning contributes to understanding any other meaning. (The wiki page on polysemy is v interesting.)

It's probably clear where this is going now: way #2 above is understanding the different $ \Box^{-1}$s as homonymous, and way #3 is understanding them as polysemous. This fits so super well: we could certainly call $ 5^{-1}$ “five-inverse” just like we call $ f^{-1}$ “f-inverse,” and there is absolutely a central origin for all these words (i.e., the group-theoretic construct of inverses).

What I really like about this, pedagogically speaking, is the network-y bits of the definition of polysemy I gave above: understanding one kind of inverse will help you understand another. Calling both $ 5^{-1}$ and $ f^{-1}$ “inverses” helps us recognize and talk about both the similarities (in both cases, the one thing “undoes” the other thing) and the differences (the operations are different) between the two cases. What's more, I think this is precisely the way in which way #3 feels better than way #2. Look how much mileage we got by borrowing some ideas from linguistics!

I've got two more post ideas lined up that build on this idea of borrowing interesting things from linguistics. The first borrows the idea of metonymy to talk about inverses of linear transformations; the second borrows the idea of backward transfer (from people who study second-language learning).

I'll close this post with this lovely quote Rina Zazkis presented, from Henri Poincaré:

Mathematics is the art of giving the same name to different things.