“Math punk” Tom Henderson has written a brilliant essay that I have co-opted and edited here.
In a nutshell, what is the problem with math education in the US? It’s that students are mostly trying to minimize feeling stupid rather than trying to maximize their ability to solve problems.
This manifests as “show me The Steps.”
Many students want a sequence of steps that they can perform that will give them an answer. This is not unreasonable; they know that their performance on exams, and therefore their performance on the All-Important Grade Point Average, is largely determined by being able to Do The Steps. So they want to know the formulas, so that they can float them on top of their short-term memory, ace the exam, and then skim them off.
For their entire mathematical careers, math has been a sequence of Steps, and if they get them wrong, they get red pen, bad grades, No No No Look What You Did. Plus, bonus, there is no apparent relevance of these algorithms other than To Get The Answer.
But that’s crap. “The Steps” aren’t math, and what’s more, The Steps aren’t generally useful in life. What’s useful is the ability to deconstruct thorny problems and figure out a way to tackle each of the pieces.
The Steps are seeing the sorts of symbols that count as “right”, and trying to replicate that dance of steps. It turns out that the easiest thing in the world is to look at a student’s work, and tell the difference between “Knows what’s going on, made mistakes and dozed off” vs. “Can memorize steps, has no idea what’s going on.”
Now, a better way to explain mathematics sort of looks crazy at first. It’s handwaving. It’s referring to certain groupings of symbols as “alphabet soup” and writing it down as a wild scribble with one or two symbols around it. It looks nothing like standard “math class” from the outside.
That’s because the better way avoids showing The Steps and instead shows enough of The Idea that the student can reconstruct what the steps MUST be.
And that brings us to a better way to learn mathematics: you get a fear-free zone, you check your ego at the door, you try a bunch of things that will wind up not working, you ask a pile of dumb questions, and before long, you figure out some crazy way to get the problem solved. And only then do you realize that your crazy, lame-brained, that-can’t-possibly-work solution is in point of fact the official method nine times out of ten. Because math is, at its core, just a collection of “the best way we could figure it out” stories, organized semi-sensibly, and with a specialized vocabulary and language on top.
So, what’s wrong with math education in the US? What’s wrong is: whatever it is that makes students uninterested in learning any more math than is required to minimize feeling stupid. My solution? Provide that safe space; find the genius in every question; and provide interesting problems to solve.
Ultimately, that’s all it takes to get students to say “Oh. That’s not nearly as hard as I thought it was going to be.”
Brian Greene makes the point about science, and it holds just as true for math: “We rob science education of life when we focus solely on results and seek to train students to solve problems and recite facts without a commensurate emphasis on transporting them out beyond the stars.”
Interested in communicating with a faraway friend without allowing anyone to eavesdrop on you? Of course you are; this problem affects a middle-schooler’s daily life, and yet it is also the basis for modern commerce: communicating your credit card number over the Net without allowing a thief to eavesdrop is a non-negotiable requirement for our economy. How is this problem solved? First, you try for a while. Then let’s talk about prime numbers and see what they can do for us.
Interested in taking a rocket to the moon? Well, if you want to drive a spaceship, you probably ought to understand how gravity works differently from your intuition when you’re far from Earth. Let’s talk about ellipses, and while we’re at it, let’s predict the next approach of Halley’s comet. (And if you’re a high-school sophomore or so, we can talk about inverse-square laws while we’re at it.)
What if we picked up a signal for an alien civilization on our radio telescopes tomorrow? Do you think they’d speak English? Not a chance. Let’s think about how we would communicate without a shared language, a shared culture, even a shared planet? Well, there’s one thing we have in common with any race advanced enough to send us a signal like that: they know math. Let’s think about how that might work…
One of the biggest rarely-asked questions in math is simply “why should I care?”
I love answering that one.
Well, I really mean to answer “what’s wrong with using the results of students’ standardized tests to evaluate teacher performance?”
On its surface, nothing. Indeed, student performance should be the primary metric of teacher quality. When I worked for Bodsat Prep, I was gauged by that metric myself: what matters is the point gain.
The main problem, though, is that the standardized tests don’t test everything that we want our students to know. In the same way that the standardized test at the DMV doesn’t tell you whether you’re a good driver (though it does catch some really bad drivers), standardized testing for schools doesn’t tell you whether you’re on track to be a productive member of society (though it does catch a number of people who really aren’t).
Also, if a standardized test samples say a random 10% of what one should know, then how well one does on such a test is, statistically, a good indicator of how much of the target material one knows. But if it’s a non-random 10%, then you can count on prepared students to know that 10% very well. And possibly nothing else.
The thing I can’t get out of my head when I read this NYT article on kids paying other kids to take the SAT for them is simply this:
All they charged was $3600? Cripes, that’s a bargain at ten times the price.
Don’t get me wrong: you shouldn’t cheat, both because it’s wrong (which should be enough reason) and tactically too risky (in case the first argument wasn’t enough).
But I mean, come on, let’s do the math: a one-percent increase in salary over your life is easily a five-digit number even if you’re kind of a slacker. Two significantly different SAT scores mean admission to schools of two significantly different calibers. And I doubt the salary increase we’re talking about here is just 1%.
For those of you who are fans of the Drake equation, which uses best-guesses to try to figure out whether there’s intelligent life out there, I challenge you to apply this reasoning to SAT prep.
In fact, you might even try to create an analogous equation governing this stuff, like I just did. (I hope you have more luck than I did; if so, please let me know.) But, equations aside, it’s not really that hard to think about.
To figure out what a higher SAT score is worth, just do the following steps:
First, get a lifetime earnings calculator. (Google it; there are many.)
Then, use it to estimate the student’s lifetime earnings, given that he or she attends the best school to which he or she can gain admission given the initial SAT scores.
Then, take the average (expected) gain in SAT scores given a particular preparation method.
Then, use the calculator to estimate the student’s lifetime earnings, given that he or she attends the best school to which he or she can gain admission given the final (expected) SAT scores.
The difference between the two lifetime earnings is the value of the higher SAT score.
And now that I’ve said all that out loud, I’m starting to realize that “only four-digit” prices for SAT prep only make sense for providers who can offer only single-percentage-point gains with a high variance, as delineated in this article in the Wall Street Journal regarding the average benefit of SAT prep.
Now these days, I am no longer doing SAT prep, having left Bodsat Prep in 2016. However, the work I do preparing students for the AMC competitions (as well as the Math 2 and Physics SAT Subject Tests) still seems to be governed by this math.
Since we don’t often see prices like this, the conclusion I come to is: almost no one is delivering reliable results. (Or the people who are aren’t also good at pricing.) Interesting.