A Mathematician’s Lament

“Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics… Mathematics is the purest of the arts, as well as the most misunderstood.” – Paul Lockhart, A Mathematician’s Lament

I’ve been on a “humanist” math philosophy reading kick lately. I’ve gotten myself partway through two of Reuben Hersh’s books and many of his articles and found many interesting TED talks supporting the view that teaching, regardless of the subject, has changed in many ways and we need to adapt.

Lockhart’s article is one of the better pieces I’ve read on what is apparently a fairly subversive view of mathematics and mathematics education. I sat reading it, completely absorbed, getting more and more fired up with each page. The article addresses ideas which have been circulating in my mind a lot lately; one semester into my teaching career, and I’m already seriously throwing into question 80% of what I thought I knew about math education.

I am a brand new teacher – graduated with a bachelor’s in pure math less than two years ago. I never went to education school, which Lockhart refers to as “a complete crock”. He argues that “teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.”

I can see this pretty clearly in my own experience this year. I teach geometry, which Lockhart calls the “instrument of the Devil” and precalculus, which he refers to as “a senseless bouillabaisse of disconnected topics” (is it bad if that quote makes me hungry?). Just halfway through my first year of teaching each of those courses, I already see frustration in the students with the traditional way of teaching. They shut off completely during geometric “proofs”, which I have now almost completely abandoned after realizing that not only did the students not understand them, but apparently lost the ability to explain simple concepts in their own words because they were so preoccupied with having to write a “geometric two column proof”. Precalculus has lots of review, and many of the students are completely shut off by having the same material they’ve seen before just lectured back to them in about the same depth.

It’s too easy for me to “be a passive conduit of some publisher’s “materials” and to follow the shampoo-bottle instruction “lecture, test, repeat” than to think deeply and thoughtfully about the meaning of one’s subject and how best to convey that meaning directly and honestly to one’s students,” as Lockhart describes it. So easy, and so boring, that in less than one semester, I am already itching to get further away from stand-and-deliver-teaching and the mind-numbing textbooks we use.

“Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply that we as a culture don’t know what mathematics is. The impression we are given is of something very cold and highly technical, that no one could possibly understand— a self fulfilling prophesy if there ever was one.”

“All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake— it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”

Many teachers are afraid to let real math happen. They want to stand and lecture and prevent the kids from trying anything new or ever making any mistakes. It seems they believe math is a set of absolute truths and they think they have been appointed by the gods to stand and deliver these truths to the congregation. Remember the lessons of medieval churches; congregations can’t read the holy word themselves and make sense of it, so we need the holy priests to deliver the word unto them. Looks like we need a real Reformation for math.

I could go on. But I think this is good enough for a first post. I’ll leave you with one last line from Lockhart (who really is a smart cookie, by the way – I laughed, got angry, got inspired, and felt very stimulated by the article).

“It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them!”

If my ramblings have piqued your interest in the article (which is a little long at 25 pages, but truly excellent to read), you can find it online here.