Today’s story starts with this TED video about teaching math. The video is short, you should go watch it. I happened across the thing accidentally one night and, unsurprisingly, I found myself annoyed and slightly angry. The video concerns itself with answering the following question about mathematics: “What will I use this for?”
“Here we go again”, I thought to myself, “the same old story about math being too hard and too terrible to inflict on our poor students, so we should just give up.” Now, I have ranted about this before in a more specific context, but this piece called out for some more general thoughts.
My immediate and strongest reaction to this talk is that fundamentally that the question is the wrong question. Students will always inevitably ask “When am I going to use this?” To me, the right answer to this question is “give it time and you will find out.” Of course, no one will actually be satisfied with this answer. So, I’ll fall back on Jordan Ellenberg’s version, from How not to be Wrong, a great book about the nature of mathematics.
“Mathematics is not just a sequence of computations to be carried out by rote until your patience or stamina runs out—although it might seem that way from what you’ve been taught in courses called mathematics. Those integrals are to mathematics as weight training and calisthenics are to soccer. If you want to play soccer—I mean, really play, at a competitive level—you’ve got to do a lot of boring, repetitive, apparently pointless drills. Do professional players ever use those drills? Well, you won’t see anybody on the field curling a weight or zigzagging between traffic cones. But you do see players using the strength, speed, insight, and flexibility they built up by doing those drills, week after tedious week. Learning those drills is part of learning soccer.”
He goes on to point out that mathematics is like soccer, to get really good at it you have to spend some time doing boring things. I think this is a good line of reasoning, and it would be enough for me, but it’s apparently not enough to motivate kids these days. The poor teacher in the TED talk could not get his kids to accept this answer. Even though he did his best, they pestered him with the same old question.
But it’s still the wrong question.
I tend to get testy about this question in the context of math and science education. This is true for two reasons:
It’s what I did in school, and I like it, and I get defensive when the subject is treated badly by the world (“Oh, math, I hated math”).
Because I’m defensive about it, I feel like the question is disproportionately applied to mathematics (and science, and other related areas).
For example, I might say to myself, no one would walk up to you and proudly proclaim that they were “no good” at speaking, or reading, or even writing. But math and science are fair game.
Then I found this piece about how people would like to just blow off Shakespeare because he is now completely irrelevant to our multi-ethnic lives due to the fact that he is a dead white guy.
Then I thought maybe I should be a bit less defensive, since apparently the English majors are getting it too.
But it’s still the wrong question.
At this point you are probably asking yourself, “well what’s the right question then?” And I would have to admit that I am not completely sure. I think if I knew the right question then I’d probably be working to completely overhaul mathematics education and in the process I’d be raising a new generation of super-minds who would then solve all the world’s problems.
While pondering this more, I found another TED video that was more to my liking. This one was called The Nature of Mathematics and it lived up to its title. I think the speaker does a good job of illustrating a small aspect of the true nature of mathematics (pattern matching, conjecture, proof, theorem) while not being patronizing or otherwise overbearing. Also along these lines was a third video that is much more overwrought and emotional. Here, the speaker makes an important point, which is part of the right answer: we teach math badly. But he seems a bit too torn up about it.
This point is also made by a longer piece that I have linked to before called A Mathematician’s Lament which you should go read now because it actually makes most of my point better than I can. Here’s the opening paragraph to get you started:
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made — all without the advice or participation of a single working musician or composer.
The piece then goes on to describe the nightmare that is the mathematics curriculum. What it makes me wonder is why I ever liked math, given what I apparently suffered through. If I had to guess it would be one experience I had in school and one outside of school.
The school experience was proving the Pythagorean theorem in geometry class. Now, the article above tears up geometry class as a crime against humanity, but the one that I took was actually pretty enjoyable and was the first hint that I got that mathematics was actually a creative activity rather than just a rote computational one. Reconstructing the proof of this theorem in my own words was a sort of revelation. What it told me was “you can figure out anything if you stare at it enough”.
The experience that I had on my own had to do with astrophysics. Astrophysics was my first dork hobby. I think the Voyager pictures did it to me. Anyway, one of the main ways that we have learned what we know about the universe is by taking a particular kind of measurement of starlight called a spectrograph. A spectrograph breaks the light down into the spectrum of colors. In the early 1800s a man named Joseph von Fraunhofer discovered that if you do this with a particular sort of optical instrument you get not only the pretty rainbow of colors, but also a pattern of light and dark lines in the spectrum. Later, it was discovered that these lines were a result of the light interacting with either atoms in the stars or atoms the space between the stars and us. When quantum mechanics was developed in the early 1900s, we had an exact way to map the patterns of lines to specific elements. This led to the relatively amazing discovery that the universe is made up the same elements that the Earth is. This is pretty neat.
But it gets better. Since light, in some ways, acts like a wave, it is subject to the Doppler shift. So if the light source is moving towards you, the light shifts blue (shorter wavelength) and if it is moving away from you it shifts red (longer wavelengths). In the 1920s Edwin Hubble used this fact to show that the Universe is expanding, leading to the Big Bang theory and all that.
That’s not the particular thing that jazzed me though. I was reading one of my astronomy text books and it described how you should use a spectroscope to detect and measure the orbits of binary stars. I remember the whole picture coming together in my head:
Find stars that are so dim you can’t even seem them with your eyeballs.
Point a telescope at them.
Take many pictures of their spectral lines. The spectrum tells you the mass and temperature of the stars, so you can know that there are two different bodies there.
Now measure the red and blue shifts over time.
From this, you know the relative velocities of the stars towards you and away from you. So if the stars are oriented correctly you can compute their mutual orbit, even if you can’t resolve the double star optically.
This. Blew. My. Mind. Open.
What this taught me was that if you know how a few conceptually simple rules (an actual understanding of the nature of spectral lines requires that you understand quantum mechanics and probably some relativity, so it’s not really that simple, but it’s simple to explain), and are pretty smart, you can put together clever solutions to really complicated problems about objects that are really far away from you. It also taught me that I had to know calculus, and at the time I was only 12 or 13 … so I waited impatiently to gain that knowledge. It did motivate me to pay attention to my math and physics.
So here is the question that we should be asking: “How do you give the students that head exploding experience”? I’m not saying that every kid will have this sort of epiphany over a math or science problem. But some of them will. Others will go crazy over music, or Shakespeare, or the history of land wars in South East Asia. Who knows. However these things work the educational system has to put the opportunity for discovery in front of the kids. Instead, we are paralyzed with irrelevant questions about future economic or practical utility. We need to teach people that the question to ask is not “What will I do with this”, but rather “Why are you showing this to me, and why might it be interesting?” and “How can I find out more?”.
But maybe that puts too much burden on the student. If that’s the case then the next question to ask is “Why do we teach math so badly?” The videos and the article that I quote above all paint a bleak picture. Rather than being taught as a vibrant and creative endeavor, math is presented as a linear slog from one set of rote computational methods to another. It’s really not until late in your college career that you get any inkling that there is more to it than that. Consider that my 14 year old self was happy to have figured out a lot of the basic mechanics of astrophysics on my own but for some reason stopped myself at a wall when I needed more math. Why, I wonder now, didn’t I think to try and work more of that out on my own? Or, I wonder even more, why didn’t I just ask my dad, who was a university math professor? The answer is that it never occurred to me. I figured I had to learn all the “pre-calculus” first.
Now, I’m not saying that we can teach Calculus to 14 year olds (although we could to some of them). What I am saying is that even the 14 year old me had already been brainwashed by math education into knowing that he had to follow the linear sequence to get to the stuff he was interested in. We teach math not as something you can, with guidance and practice, learn how to do yourself. We teach it as rules and definitions and “facts” handed down from on high. The musical analogies in the second Youtube video and A Mathematician’s Lament are apt. We teach math as if it were made up of all the rules and regulations of music, but where you don’t actually get to play any of the songs until you’ve been doing it for more than ten years.
I actually wonder what it would be like to teach math more like we teach music. Not to idealize it, but music education requires that you mix the practice of rote mechanics within the learning of more abstract and more creative structures. That is, you have to practice the boring stuff, but in return you see incremental progress through the actual performance of actual songs. It seems like this mix is what we would like for mathematics education as well, but what we get instead is too much of the former and not enough of the latter. Thus, everyone, even the math geeks, has to go through almost a decade of tedium before even seeing an inkling of the creative and artful side of the game. This is a shame, and for some reason we’ve never been able to figure out how to make it better. Every “new math”, or “common core” just makes it worse.
At the center of all this pain and angst is the question with which we started our story, and which never seems to go away: “What will I use this for?”. Ultimately I think this question just lazy. It’s lazy for the students because it focuses them only on short term and concrete goals and learning to become a human being requires that you learn to think about more than that. But even worse, it’s lazy on the part of teachers and the adults, who should have already learned that lesson, and are ignoring it anyway. I think if the only question we ever ask is “What will I use this for?” then we are doing ourselves and our kids a disservice. I think they deserve more. There is too much out there to be learned for us to bury the whole process in that question. If that’s the only thing we are going to try and answer, then maybe I will back pedal from my old position and just conclude that we should give up and not teach them anything.