My Former Students Are Grown-*ss Folks

When I started out, veteran teachers at my school said to me, “you won’t really understand what you’re contributing until your students grow up and come back as adults.” I didn’t really understand what they were saying because it sounded unfathomably distant in the future.

But I am beginning to find out.

It’s just starting to sink in that the kids I taught in 2001-2, who were then high school freshmen, are now about 1 year older than I was when I taught them. The ones I taught in 2002-3 are 1 year younger.

BANANAS.

Example: I just had an email exchange with one of my 2002-3 students, who is now involved in math education (!) working for the Young People’s Project. This is the second former student I know of to get involved in math education. *Proud.* I would go so far as to say, *kvelling.*

Elementary Mathematics from an Advanced Standpoint

Another of the many reasons I’m in grad school. I benefit as a teacher from understanding the content I teach in way more depth than I teach it. (I think everybody does, but it’s easiest to talk about myself.)

This does a number of things for me. The simplest is that it makes the content more exciting to me. Something that previously seemed routine can become pregnant with significance if I know where it’s going, and there’s a corresponding twinkle that shows up in my eye the whole time my students are dealing with it. A second benefit is that it gives me both tools and inspiration to find more different ways of explaining things. A third is that it helps me see (and therefore develop lessons aimed at) connections between different ideas.

So, this post is a catalogue of some insights that I’ve had about K-12 math that I’ve been led to by PhD study. The title of the post is a reference to Felix Klein’s classic books of the same name. The catalogue is mostly for my own benefit, and I don’t have all that much time, so I’m going to try to suppress the impulse to fully explain some of the more esoteric vocabulary, but I never want to write something here that requires expert knowledge to avoid being useless, so I’ll try to be both clear and pithy. (Wish me luck.)

Elementary level: Multiplication is function composition.

I’m developing the opinion that it’s important for especially middle and high school teachers to have this language. The upshot is that in addition to the usual models of multiplication as (a) repeated addition and (b) arrays and area of rectangles (and, if you’re lucky, (c) double number lines), multiplication is also the net effect of doing two things in a row, such as stretching (and possibly reversing) a number line.

The big thing I want to say here is that understanding this is key to understanding multiplication of signed numbers. I would go so far as to wager that anybody who feels they know intuitively why $-\cdot -=+$ understands it on some level, consciously or not.

When somebody asks me why a negative times a negative is a positive, I have often had the inclination to answer with, “well, what’s the opposite of the opposite of something?” (I have seen many teachers use metaphors with the same upshot.) The problem is that if you understand multiplication only as repeated addition and as the the area of rectangles, I’ve changed the subject with this answer. It is a complete nonsequitur. It’s probably clear why it has to do with negatives but why does it have to do with multiplication?

On the other hand, if on any level you realize that one meaning of $2\times 3$ is “double then triple”, then it’s natural for $(-2)\times(-3)$ to mean “double and oppositize, then triple and oppositize.” But for this you had to be able to see multiplication as “do something then do something else.”

Algebra I and Algebra II: Substitution is calculation inside a coordinate ring.

I just realized this today, and that’s what inspired this blog post. So far, I’m not sure the benefit of this one to my teaching beyond the twinkle it will bring to my eye, though perhaps that will become clear later. It’s certainly helping me understand something about algebraic geometry. The basic idea is this: say you’re finding the intersections of some graphs like $y=3x+5$ and $2x+y=30$ . You’re like, “alright, substitute using the fact that $y=3x+5$ . $2x+(3x+5)=30$ , so $5x+5=30$ …” and you solve that to find $x=5$ , for an intersection point of $(5,20)$ . A way to look at what you’re doing when you make the substitution $y=3x+5$ is that you’re working in a special algebraic system determined by the line $y=3x+5$ , in particular the (tautological) fact that on this line, $y$ is exactly three $x$ plus five. In this system, polynomials in $x$ or $y$ alone work the usual way, but polynomials in $x$ and $y$ both can often be simplified using the relation $y=3x+5$ connecting $x$ and $y$ . This algebraic system is called “the coordinate ring of the line $y=3x+5$ .”

I can’t tell if it will even seem that I’ve said anything at all here. The point, for me, is just a sublte shift in perspective. I imagine myself sitting on the line $y=3x+5$ ; then this line determines an algebraic system (the coordinate ring) which, as long as I’m on the line, is the right system; and when I substitute $3x+5$ for $y$ , what I’m doing is using the rules of that system.

Calculus: The chain rule is the functoriality of the derivative.

“Functoriality” is a word from category theory which I will avoid defining. The point is really about the chain rule. The main ways the derivative is presented in a first-year calculus class are as speed, or rate of change, on the one hand (like, you’re always thinking of the independent variable as time, whatever it really is), and the slope of the tangent line of a graph, on the other. There is a third way to look at it, which I learned from differential geometry. If you look at a function as a mapping from the real line to itself, then the derivative describes the factor by which it stretches small intervals. For example, $f(x)=x^2$ has a derivative of $6$ at $x=3$ . What this is saying is that very small intervals around $x=3$ get mapped to intervals that are about 6 times as long. (To illustrate: the interval $[3,3.01]$ gets mapped to $[9,9.0601]$ , about 6 times as long.)

Seen in this way, the strange formula $[f(g(x))]'=f'(g(x))\cdot g'(x)$ for the chain rule becomes the only sensible way it could be. The function $f(g(x))$ is the net effect of doing $g$ to $x$ and then doing $f$ to the answer $g(x)$ . If I want to know how much this function stretches intervals, well, when $g$ is applied to $x$ they are stretched by a factor of $g'(x)$ . Then when $f$ is applied to $g(x)$ they are stretched by a factor of $f'(g(x))$ . (Note it is clear why you evaluate $f'$ at $g(x)$ : that is the number to which $f$ got applied.) So you stretched first by a factor of $g'(x)$ and then by a factor of $f'(g(x))$ ; net effect, $f'(g(x))\cdot g'(x)$ , just like the formula says.

(As an aside, for the sake of being thematic, note the role here of the fact that the multiplication comes from the composition of the two stretches – multiplication is function composition. When I say “the derivative is functorial” what I really mean is that it turns composition of functions into composition of stretches.)

Calculus: The intermediate value theorem is $f(\text{connected})=\text{connected}$ . The extreme value theorem is $f(\text{compact})=\text{compact}$ .

This is a good example of what I was talking about at the beginning about the twinkle in my eye, and connections between ideas. When I used to teach AP calculus, the extreme value theorem and the intermediate value theorem were things I had trouble connecting to the rest of the curriculum. They were these miscellaneous, intuitively obvious factoids about continuous functions that were stuck into the course in awkward places. They both had the same clunky hypothesis, “if $f$ is a function that is continuous on a closed interval $[a,b]$ …” I didn’t do much with them, because I didn’t care about them.

I started to see a bigger picture about three years ago, in a course for calculus teachers taught by the irrepressible Larry Zimmerman. He referred to that clunky hypothesis as something to the effect of “a lilting refrain.” I was also left with the image of a golden thread weaving through the fabric of calculus but I’m not sure if he said that. The point is, he made a big deal about that hypothesis, making me notice how thematic it is.

Last year when I taught a course on algebra and analysis, having benefited from this education, I made these theorems important goals of the course. But something further clicked into place this fall, when I started to need to draw on point-set topology knowledge as I studied differential geometry. Two fundamental concepts in topology are compactness and connectedness. They have technical definitions for which you can follow the links. Intuitively, connectedness is what it sounds like (all one piece), and compactness means (very loosely) that a set “ends, and reaches everywhere it heads toward.” (A closed interval is compact. The whole real line is not compact because it doesn’t end. An open interval is not compact because it wants to include its endpoints but it doesn’t. A professor of mine described compactness as, “everything that should happen [in the set] does happen.”)

Two basic theorems of point-set topology are that under a continuous mapping, the image of any connected set is connected and the image of any compact set is compact. These theorems are very general: they are true in the setting of any map between any two topological spaces. (They could be multidimensional, curved or twisted, or even more exotic…) What I realized is that the intermediate value theorem is just the theorem about connectedness specialized to the real line, and the extreme value theorem is just the theorem about compactness. What is a compact, connected subset of $\mathbb{R}$ ? It is precisely a closed interval. Under a continuous function, the image must therefore be compact and connected. Therefore, it must attain a maximum and minimum, because if not, the image either “doesn’t end” or “doesn’t reach its ends,” either of which would make it noncompact. And, for any two values hit by the image, it must hit every value between them; any missing value would disconnect it. So, “if $f$ is a function that is continuous on a closed interval $[a,b]$ …”

P.s. Japheth Wood Is My Dude

One reason why.

Dispatches from the Learning Lab: Yup, Time Pressure Sucks

Continuing the series I began here and here, about snippets of new-feeling insight about the learning process coming from my new role on the student side of the desk…

This one is funny, because I knew it, I mean I knew it in my bones, from a decade working with students; but yet it’s totally different to learn it from the student side. I’m a little late to the blogosphere with this insight; I’ve been thinking about it since December, because it kind of freaked me out. Even though, like I keep saying, I already knew it.

Learning math under time pressure sucks. It sucks.

It sucks so much that I ACTUALLY STOPPED LIKING MATH for about 5 days in December.

I didn’t know this was possible, and I don’t think anyone who’s ever worked closely with me in a mathematical context (neither my students, colleagues, or teachers) will really believe it. But it’s true. It was utterly, completely unfun. There was too much of it and too little time. It was like stuffing a really delicious meal down your throat too quickly to chew, or running up the Grand Canyon so fast you puke. Beautiful ideas were everywhere around me and I was pushing them in, or pushing past them, so hard I couldn’t enjoy them; instead they turned my stomach, and I had the feeling that the ones I pushed past in a hurry were gone forever, and the ones I shoved in weren’t going to stay down.

I had some independent study projects to work on during winter break, and what was incredible was the way the day after my last final exam, math suddenly became delicious again. Engaging on my own time and on my own terms, that familiar sense of wonder was back instantly. All I had to do was not be required to understand any specific thing by any specific date, and I was a delighted, voracious learner again.

Now part of the significance of this story for me is just the personal challenge: most of the grad students I know are stressed out, and I entered grad school with the intention of not being like them in this respect. I was confident that, having handled adult responsibilities for a decade (including the motherf*cking classroom, thank you), I would be able to engage grad school without allowing it to stress me out too much. So the point of this part of the story is just, “okay Grad Program, I see you, I won’t take you for granted, you are capable of stressing me out if I let you.” And then regroup, figure out how to adjust my approach, and see how the new approach plays out in the spring semester.

But the part of the story I want to highlight is the opposite part, the policy implication. Look, I frickin love math. If you’ve ever read this blog before, you know this. I love it so much that most of my close friends sort of don’t feel that they understand me completely. So if piling on too much of it too quickly, with some big tests bearing down, gets me to dislike math, if only for 5 days, then the last decade of public education policy initiatives – i.e. more math, higher stakes – is nothing if not a recipe for EVERYONE TO HATE IT.

And, not learn it. Instead, disgorge it like a meal they didn’t know was delicious because it was shoved down their throat too fast.

In short. The idea of strict, ambitious, tested benchmarks in math to which all students are subject is crazy. It’s CRAZY. The more required math there is, and the stricter the timeline, the crazier. I mean, I already knew this ish was crazy, I’ve been saying this for years, but in light of my recent experience I’m beside myself. If you actually care about math, if you have ever had the profound pleasure of watching a child or an adult think for herself in a numerical, spatial or otherwise abstract or structural context, you know this but I have to say it: the test pressure is killing the thing you love. Its only function is to murder something beautiful.

If you teach, but especially if you are a school leader, and especially if you are involved in policy, I beg you: defend the space in which students can learn at their own pace. Fight for that space.