As a teacher I encounter all of the typical kinds of students. There’s one kind of student I routinely encounter, usually in a freshman calculus course, that really boils my blood: the failing student who “has always been good at math.”
Oh it’s so annoying! And it’s even worse to hear because the stuff we teach in calculus isn’t really math either. The irony is so thick in the air when a student says it I’m surprised I don’t cough. Invariably, they never actually understood the “math” they were always so good at. I don’t get angry because they don’t understand the material (indeed, that’s the point of the class!). I get mad because they have absorbed a message about what constitutes mathematics that is, at best, misguided.
Of course, the problem is deeper than a handful of students who accidentally say ironically stupid things. The problem is that American high school students are taught something named “math” for four years which is not even close to math.
As I often try to do, I’d like to give an analogy for what I mean. Imagine that in high school the music courses required you to become really good at reading and manipulating music on paper. In order to pass the first music course, you need to be able to draw the treble and bass clefs in exactly the right position, and know how to draw the whole, half, quarter, and eighth notes with the right orientations. You also need to know which positions in the sheet correspond to which musical notes, and they give you little mnemonics to remember this, such as “Elephants Go Bowling Down Freeways.” Note that in all this time you never sing a note, nor do you pick up an instrument. Indeed, it’s deemed very advanced to hum a tune out loud or to compose a song.
The second class is Music 2 - Transposition & Key Changes. You’re expected to be able to take a piece of sheet music and transpose it into different keys, and modify sharps, flats, and natural signs appropriately. The third class is Meter and Pitch, and you go on learning the theory of music without ever listening to or playing any songs.
Would you say a student who excels in this practice (but can’t play an instrument) is good at music? No way! Would you say they have a mind (or an ear) for the subject? Certainly not! (Fair disclaimer, this analogy is borrowed from a much better writer, Paul Lockhart).
But this is precisely what we subject mathematics high school students to. We force them to memorize the precise details of solving algebraic equations and polynomial inequalities. We require them to be adept at simplifying fractions involving radicals, drawing accurate sketches of ellipses and hyperbolae from their equations, and writing out the steps of a geometric proof in such pedantic detail as would make a grown mathematician cry.
I fear my rant may disguise my true intentions: the problem is not the content. Geometry and calculus and algebra are very fine subjects of mathematics. The problem is that they’re taught in a way that strips out all the math and leaves a vapid husk of an education. In fact, so few non-mathematical people understand how bad it is that the only way to explain it is to use analogies like the music dystopia above.
So when a freshman college student tells me that he was always good at math, it translates to “I was very good at following obscure steps to manipulate mysterious symbols, without any real understanding of what I was doing.” Just as the pitiful music student from before, there is no evidence that the student was ever any good at mathematics. That’s not to say he can’t be good, but that he has just been misled his whole life.
But this raises the obvious question: if math isn’t about all of this drudgery, then what is it?
The simplest way to say it is that mathematics (especially at the elementary and secondary level) is about recognizing and reasoning about patterns. These patterns can come from anywhere: shapes, numbers, relationships at a party, physical systems, tournaments, card games, knotted rope, doodles with colorful pens, literally anything!
All my bravado aside, the point is that solving algebra problems and doing two-column geometry proofs is not mathematics. Generating your own questions about whatever interests you and trying to answer them is. Big names in mathematics education seem to focus so much on technology or test scores that they neglect to ask their students to think about something. I know that teaching mathematics can be done better than this.
And I have actually done it! If you’re interested into what goes into such a lesson, you should read my post on teaching graph theory to high school students at my blog Math ∩ Programming. It consolidates a number of my experiences with students aging 13 - 18, which includes roughly twenty hour-long sessions to date.
This subject (graph theory) is closer to mathematics, and much better at promoting critical thinking and problem solving, than anything taught in American high schools. The reason it is a nice topic for such a lecture is that teachers have no preconceptions about the “right” way to teach it, and students have never heard of it before. I’d love to do the same thing with calculus and algebra and geometry, but teaching an entirely new (and equally accessible) subject has far fewer barriers. I encourage you to read it and think about the principles of critical thinking and problem solving at work there. These are the keys to doing mathematics, and there’s no good reason why we can’t provide that in high school.