The fascinating thing about mathematics is that everyone perceives it differently. My construction worker dad sees every mathematical computation as a concrete, real-life , unquestionable number. As a linguist, I see language as more specific and less abstract than math. I can make anything equal two. I can make two apply to anything. I can figure its differentials; I can manipulate it into a fraction, a circle, an imaginary number, even coffee. There is nothing concrete in how I deal with numbers.
As a tutor, someone who delves into the mind of another human being to figure out how it ticks and then teaches it what it wants to know, I mull over this revelation with an intensity reserved for how one finagles extra value into a dollar.
It's not the first slap-in-the-face I've gotten about the instructional aspect of mathematical concepts: I've recently taken on a student with a variation of autism who treats math as a series of processes, no end result necessary, not purpose driven, just a certain kind of machine that does a certain task. Perplexed, it took me two weeks to figure out how to teach her.
And then there's my own children, one in preschool and one in her second year. The younger one counts infinitely, knows implicitly the position of the fingers necessary to display numbers one through ten, and has a spacial recognition that puts me and my cognition in the dark ages. The older one is the opposite. Just in the current school year we've gotten her to count to twenty consecutively, though she reads at a nine-year-old's level. Neither think about math the way I do, nor do they follow any traditional math teaching method contained in my math books.
At a standstill with the home-schooled math student, I started looking at math not as a subject to be accomplished but as a region to be understood. What is math? It's not simply a series of figures with operation signs and the strategies to manipulate them. It's not necessarily purpose-driven, but it doesn't have to exist in an etherial realm, either. And it's far from being rote memorization. And it's certainly not as arbitrary as I might imagine.
Math is a perception. Like an artist who works in 3D but can't draw more than a stick figure, math is a process of interpretation. Logical and Spatial intelligence, on the IQ tests, but really, a way of integrating the world to one's imagination, and then communicating it.
For my beginning math student where words already exist in her psyche, teaching with language is the best option. Use units, use story problems. Find the perimeter kinds of problems. Write math equations instead of drawing them. Label graph paper with ones column and tens column and hundreds column. Speak of multiple digit numbers as a "six" in the ones place, a "four" in the tens place, and "add the ones column and any result with a digit in the tens place needs to be carried to the tens column." It's the language that will get her through the basics into the higher math. No pages of numbers and operations symbols are going to be sufficient. Rulers with numbers and units, polygons with units, and labeled graph paper for working the figures--these are the tools.
These are the tools, and I am the teacher. I pack my toolbox with the linguistic versions of all the mathematical concepts suitable for a "first grader," and I pull them out and carefully order them in story form. I line up the concept to the student, and walk away, because once I've done that, they have all they need to know. That toolbox isn't for me.
No, it is for me to do my job, this job to find out what math can possibly have to do with coffee. It is for me to find the psychology behind both what needs to be learned and the individual learner, and show that learner how those two things are the same.
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