An insightful remark by one of my Twitter friends today sparked some thoughts about mathematics education — specifically, my thoughts about the new fads in math education that seem to crop up all the time in schools across the U.S. like ever-more creative varieties of fungus. As someone who loudly and often condemns many of these new “methods” for teaching math, it turned out I had far too many thoughts for Twitter.
I’ve worked for many, many years as a private math tutor, at varying degrees of full/part time (right now I don’t do it for my main job, but maintain a few students because it’s fun and I like teaching and I like the company I work for). My students have crossed a wide variety of middle and high schools, with a wide variety of curriculum tracks (from off-beat “hippie”-ish private schools to standard state curricula to college, and including a range of clients from students for whom math was not their forte to students who were brilliant and wanted more enrichment to students who were smart but lazy). Pretty much the only selection bias my students have had in common is that they and their families tend to be in a financial bracket to hire me and care enough about academics to do so, which has slanted the demographic I work with to be much more likely to attend schools that purportedly each really try to have a good math curriculum.
And so many of them . . . well, don’t.
Education is a hard thing. I realize that. It’s really, really hard to figure out the optimal way of teaching a lot of disparate students with a lot of disparate skill levels some knowledge they don’t necessarily all want to learn. Teachers are often underfunded, with too many students, and constrained by administrations or standardized tests that work against them. I’m willing to cut teachers a lot of slack for not batting 1000 for all students at all times. But what really gets my goat when it comes to mathematics education is what I call “fad math.”
Fad math is my students whose school thought the best way to teach them geometry was to put them in small groups and say, “figure it out” with no additional guidance (this is based on the textbook, by the way, which does not teach at all). Fad math is my students whose curriculum jumps from topic to topic with no discernible connective tissue, and then assigns a mountain of problems . . . maybe three each on each of the wildly different topics. Fad math is the private school that decided real-world math was more important than algebra and calculus, so taught taxes and mortgage amortization instead. I could go on. (And on.)
I get the motivations behind trying out these fads. Mathematics teaching in this country is (in many ways correctly) perceived as broken, and people are looking for the magic formula, the Holy Grail, the thing that will work. For example, in the first example above, I get that the idea was probably originally that more “figuring out” should happen in math teaching and less rote memorization; I support that in principle, but when we’ve reached a point at which any actual teaching has disappeared and students are basically being required to re-derive modern mathematics from scratch (with the end result that most of them just learn nothing), we’ve gone way, way, way, way too far. The second curriculum I mentioned comes from the fear that students lose material when they study one topic in bulk rather than repeating the skills over a period of time, which, again, is a problem worth addressing — but gutting students’ ability to gain an in-depth understanding of any material is not the way to do that. And while I applaud schools looking to teach students about real-world math like taxes and mortgages — again, a good idea! — lacking a more traditional math base completely derailed students who moved or transferred high schools and also snapped off the math foundation needed for any students who wanted to go on to a STEM field in college, including fields like pre-med and economics.
Fad math, in my view, seems more about people being proud of a shiny new idea — “this idea will work for pumping math knowledge into kids’ heads!” rather than being concerned with teaching, which is, in my opinion, where people should be concerned. My best math teachers have never used any gimmicks, ever. But they were really, really, really good at explaining things in a way that made sense.
(And that’s the kind of teacher I try to be, too: one who explains things in a way that makes the lightbulb go off and the student say, “Oh! So then that’s why this happens!” It’s amazing to be able to help someone reach that place.)
For what it’s worth, here are the main problems I see with math teaching in this country from working with my students:
- Math taught as a “how to” instead of a “why.” If all you’re doing is memorizing that this number should go here and that one should go there when you see a certain symbol, then that’s . . . . well, almost useless. If all students are doing is following a flow chart by rote memorization, there’s no mathematical understanding going on. Students have to know why a thing makes sense in order for it to, well, make sense. Teachers need to teach the why instead of just giving a recipe for the how.
- No connection between mathematical ideas. Math is ridiculously interconnected. Every topic is related to every other. And when you help a student relate a new topic back to an old one, it enhances understanding of the old one while giving the student an intuitive basis for the new one. Trying to learn math concepts in isolation is a ridiculous proposition, and yet, that’s what students are so often asked to do.
- A horrifying number of high school math teachers don’t seem to have a deep understanding of the concepts they’re teaching. I can’t count the number of times my students have come to me confused about explanations their teachers gave them — explanations that were off-base, muddled, or just plain wrong. The cynic in me bets that this is because the teachers learned math by learning “how” as well, so when students ask the “why,” the teachers might genuinely not know. (This is not, of course, true of all math teachers, and is perhaps not even true of most math teachers — I don’t know — but that it is true of a noticeable number concerns me.)
- A smaller number (but even more horrifying in its existence) of math teachers either just don’t care or are actively derogatory towards students. This includes everything from being hostile toward giving extra help to sexism toward female students.
- Math teachers also have roadblocks thrown in their path from every conceivable corner. Class sizes are too large, stripping away teachers’ abilities to tailor explanations to individual needs or to give a struggling student the extra help that might make the difference. Standardized tests and state-imposed math standards often do more harm than good, as they pressure teachers into hammering the “how” into students hard enough that they’ll get the right answers without ever addressing the “why.” Teachers are underpaid, overworked, and often struggle against turgidly bureaucratic administrations. And in math specifically, the sort of “fad math” I’ve referred to here is often forced on teachers from the outside, hobbling their ability to actually teach.
I don’t mean to sound hostile toward teachers. Like I said, I’ve had some brilliant math teachers in my time — in fact, I don’t think I’ve ever had a single math teacher who was bad at what she or he did. And god, look, that’s probably a huge part of the reason I fell in love with math and went into it: because of my teachers. Two of my high school teachers in particular (one in my high school and one in a summer program) are probably directly responsible for me going to MIT and majoring in math.
But this just reinforces my point: Good teachers are so freakin’ important. Not a single one of my math teachers ever used a gimmick or a fad or some krazy new-fangled new idea for gettin’ math into the heads of them dumb-dumb math-hatin’ students. They used blackboards, white boards, or transparencies. And — and now that I’m thinking about it, this was true without exception — they pretty much spent the entirety of each class period writing and talking and explaining things. And it worked. I mean, I know it might seem like, okay, this is me talking, and I’m smart and good at math so it worked for me — but no, it worked in general. I still use visualizations taught to me by my geometry teacher 17 years ago with my students and they find them incredibly helpful. My calculus teacher was exceedingly proud of the fact that every single one of her AP students would consistently get 4’s and 5’s on the AP exams. Every single one, in a public school. (A good public school, but still.) And she didn’t teach to the test; she just taught well (and was incredibly beloved by students, not just me).
Yeah, I was very lucky. But I’d like all students to be able to be that lucky. To be as well-taught and inspired as I was. To feel that they’re not just passing tests, but that they truly understand what they’re learning.
There’s a lot that is hard about education reform. A lot. But for Pete’s sake, one thing we can do is stop it with the fad math. Stop dropping shiny new assembly line algorithms across school curricula in the hope that they’ll press out perfect little cubes of students who know how to factor properly. You can’t teach math by plugging a student into a flow chart.
You teach math by teaching it. There are many, many excellent ways and methods of teaching, of course, and I’m not saying discussing those isn’t valuable — I could probably write a book on all the different ways I’ve discovered to explain calculus. But so many of these math fads stop valuing teaching entirely. And that makes our education system, one in which there are already so many things to fix, just that much more broken.
- There’s this commercial for an online tutoring service that drives me bonkers. It’s meant to show a good math tutor. The student calls up the tutor and says, “How do you find the area of a triangle?” The tutor says something like, “Well, [Student’s Name], the area of a triangle is one-half base times height! So you take the base, and multiply it by one-half and by the height!” and she writes A = 1/2 bh. And the two of them smile at each other like this is just peaches. And I scream every time I see this commercial, because teaching a kid to memorize a formula, that’s not teaching math. In fact, area of a triangle is one of the easiest things to explain — A = bh is quite intuitive for a rectangle (and if not can easily be demonstrated via a visualization of 1×1 boxes), and then you can show the area of a triangle as being half the area of a rectangle by drawing a rectangle and slicing it down the diagonal to make two triangles, so for a triangle A = 1/2 bh. (Slightly more rigorously, you can teach area of a parallelogram in between those, but “triangle as half a rectangle” is actually easier for most students when intuiting the reason for the formula, and the other connections can be drawn later.) In any case, this commercial is everything I hate about bad math education in one thirty-second soundbite.↵
- In most cases it’s pretty easy for me to tell when it’s just the student who’s confused versus when the teacher was actually confusing.↵