The following is the first of what I hope will be three posts about thought-provoking books about math teaching, each accompanied by a treasured insight I got out of reading it. All three books (Bob and Ellen Kaplan’s Out of the Labyrinth; Paul Lockhart’s A Mathematician’s Lament; Catherine Twomey Fosnot and Maarten Dolk’s Young Mathematicians at Work) have a lot going on. If you read (or have read) any of them, I’m confident you’ll have (or have had) all kinds of thoughts totally unrelated to the ones I’m about to share. But the thoughts I am highlighting have all felt very exciting to me and that’s why I’m sharing them. It was like something clicked into place and answered a question that had been loitering inchoately in my mind.
Out of the Labyrinth: Setting Mathematics Free
by Robert and Ellen Kaplan
I’ve written about my appreciation for this book elsewhere. So without further ado -
Nugget: Mathematics is a vital interplay between the general/abstract and the specific/concrete. Without generality and abstraction, mathematics lacks power and grandeur. But without specifics, mathematics lacks life.
From Out of the Labyrinth:
“The spirits of Hilbert and Ramanujan lean over our efforts: the one ever lifting us up toward the form of the whole, the other dipping down again and again to catch at the invigorating singular…. This stirred soup is a spiral nebula, exceptional in each of its stars.” (p.157)
(To explicate the cultural reference: Ramanujan and Hilbert were both early twentieth century mathematicians. They represent opposite poles of the spectrum from general to particular. Hilbert was the grand theorist, among other projects attempting to find a formal system that could unify the entire edifice of mathematics. Ramanujan was a delighter in the details. In a famous anecdote, when his friend G. H. Hardy remarked that he had arrived in a taxi whose number was 1729 and that this seemed a dull number, Ramanujan exclaimed, “No Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”)
Everybody talks about the need to make math concrete for students, but what clicked into place for me when I read Out of the Labyrinth is that mathematicians at every level have literally the same need. What counts as concrete is different at different levels but this is the only difference. A problem is not compelling if you can’t feel it, if you can’t turn it and prod it in your mind. It’s not that at some level of mathematical development you cease to need the problems to be real. It’s that as you deal with and get to know a mathematical object, at any level of abstraction, through its interactions with things that are already real for you, it becomes real. Then questions about it can become interesting.
Students learning algebra for the first time often experience equations containing variables as a sort of hazy, insubstantial thing. The methods for manipulating the equations seem arbitrary, and there is nothing satisfying about solving them because they barely seem to exist in the first place. I had the good fortune to encounter variables for the very first time in a context that made them extremely real, so I never had this experience. But I remember that haziness, that alienating lack of substance, from my first encounter with the definition of a group. “What is this pointless object that kicks all the substance out of the rich terrain of Number and leaves a hollow shell?” I remember thinking.
This is funny to me now since my subsequent mathematical education has made groups just as concrete as numbers. I now experience the symmetric groups on 3 and 4 elements as familiar little toys. The point is that this was a process. Groups did not begin to feel real to me until I saw how they explained and unified things I could already touch: numbers, the symmetry of shapes, motions in space. A long, slow process of solving problems, cultivating techniques, and most importantly investigating interactions and connections between things I already understood, made concrete objects out of what had first to me literally felt like thin air.
The philosophical lesson I draw from this is that a powerful way to look at teaching mathematics is that it is a process of cultivating this sense of concreteness in more and more abstract objects. The process begins in the concrete details of a problem – that’s what gives it life. Then it pulls upward, beginning to perceive the ethereal layer above the details that tie them together with other details. Then you try to hold both layers at the same time, and perceive their relation, until the concreteness of the details seep into the ethereal layer and it gains fleshy substance. Then you can move higher still. (First number, then variable, then function, then operator, for example, over the course of a decade.)
More concretely, a math lesson has to start from a problem that feels real and tangible for the students; but this doesn’t have to mean “real-life”. It just means that they need to be able to hold the problem, touch it, manipulate it in the mind. This depends on their mathematical development. If it’s a calculus class and the precalculus class did its job, the graph of y=sin x is a real, tangible thing. So a provocative question about the graph of y = sin x is more compelling than a routine question about a car’s speed.
Conversely, a problem that does not feel concrete lacks life, and concreteness grows slowly. We should beware of problems which require us to teach something before the problem can be posed. The lesson must begin with an interesting question about a tangible reality. The reality can be pysical, social, or purely mathematical. It just has to be real.