Robert Lewis, a professor at Fordham University, has published this essay entitled “Mathematics: The Most Misunderstood Subject”. The source of the general public’s misunderstandings of math, he writes, is:

…the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand “Quick, what’s the quadratic formula?” Or, “Hurry, I need to know the derivative of 3x^2 – 6x +1.” There are no such employers.

Prof. Lewis goes on to describe some ways in which this central misconception is worked out in our schools and in everyday thinking. The analogy between mathematics instruction and building construction, in which he compares current high school mathematics instruction to a building project where the scaffolding is constructed and then abandoned because we think the job is done, is pretty compelling. The whole essay is well worth reading.

I do think that it’s a bit too easy to lay the blame for the current state of mathematics instruction at the feet of American high schools, as Lewis does multiple times. Even if high schools do have flawed models of math instruction, certainly they are not alone in this. How many universities, even elite institutions like Fordham, have math classes or even entire curricula predicated on teaching math as rote mechanics? And what about the elementary math curricula? Pointing the finger at high schools is the natural thing to do for college professors, because we are getting students fresh from that venue and can see the flaws in their understanding, but let us not develop tunnel vision and think that fixing the high schools fixes everything. Laying blame on the right party is not what solves the problem.

Lewis brings up the point that we should be aiming for “genuine understanding of authentic mathematics” to students and not something superficial, and on that I think most people can agree. But what *is* this “authentic mathematics”, and how are we supposed to know if somebody “genuinely understands” it? What does it look like? Can it be systematized into a curriculum? Or does genuine understanding of mathematics — of anything — resist classification and institutionalization? Without a further discussion on the basic terms, I’m afraid arguments like Lewis’, no matter how important and well-constructed, are stuck in neutral.

Again coming back to higher education’s role in all this, we profs have work to do as well. If you asked most college professors questions like *What is authentic mathematics?*, the responses would probably come out as a laundry list of courses that students should pass. *Authentic mathematics consists of three semesters of calculus, linear algebra, geometry, etc*. And the proposed solution for getting students to genuinely understand mathematics would be to prescribe a series of courses to pass. There is a fundamentally mechanical way of conceiving of university-level mathematics education in which a lot of us in higher ed are stuck. Until we open ourselves up to serious thinking about how students learn (*not* just how we should teach) and ideas for creative change in curricula and instruction that conform to how students learn, the prospects for students don’t look much different than they looked 15 years ago.

I fear that when you look for the greatest common divisor of all the items on that laundry list of “authentic mathematics”, you might find rote mechanics and a list of required courses. It would be a shame to settle for that.

I’ve long thought that if the goal is to give students a feel for “authentic mathematics,” however you define that term, calculus is a poor choice for a required mathematics course. It’s so, so easy for calculus to devolve into rote mechanics. Why not make, say, graph theory the standard required course? Its applications are just as engaging as calculus, and it lends itself to learning “authentic mathematics” (logical thinking, creative problem solving, etc.) much better than calculus does.

I was a math major at MIT — loved everything about math and took all the math related classes I was allowed as both undergraduate and graduate student. Now, many years later I find I don’t remember much at all of what I learned. In fact, I recently started reviewing and re-learning from the beginning with algebra, geometry and trig. My goal is to work my way back through all or most of the math I once knew and understood. Does my loss of math ability mean I never had a good understanding of math?