Sunday, October 10, 2010

A Number Approaching Infinity

I stopped taking math courses after my sophomore year in college, so I'm aware that there is a lot about math I don't know, and even more that I don't know I don't know. By the time I hit those final years of formal math education, my main strategy was to sit next to the prettiest girl in class, under the assumption that I'd need help from my peers to survive, and being a hormonally driven teen, figured I'd have the most fun with an attractive study mate.

That's how I met my friend Renae, the girl who got me through calculus, and who has since made a career serving as chief financial officer for several highly profitably enterprises. Renae was, in fact, the best math teacher I'd had since elementary school because she was smart, helpful, and I had fun being with her. 

Out of sheer stubbornness, I continued to register for math courses that I didn't need for my degree: refusing to be defeated by the cliche, "Math is hard." That said, I saw the beginning of the end coming when I rearranged my entire schedule to follow Renae into the "Calculus for Business Majors" course series, rather than stick with the straight-ahead, pure mathematics of hard core calculus, where a typical day in the classroom involved a TA with a poor command of English, solving problems on the blackboard. I might not have been willing to admit that math was hard, but that action of following Renae convinced me that math without smart, pretty girls, was no fun. I later tried a discrete mathematics course on my own. It hurt my head and I dropped the course after 3 mind-numbing sessions.

I can't speak for college level math education, but as a preschool teacher, I know that there is no reason that math shouldn't be fun, even without smart, pretty girls helping you. All it is, after all, is a process of learning increasingly complex and wonderful ways to do things that give us great pleasure as human animals: patterning, classifying, and sorting. At bottom, when we boil it down, that's the entirety of math -- patterning, classifying, and sorting -- which is ultimately the foundation of analytical thinking.

We enter the world as mathematicians, exploring all the ways we can order our world, craving an understanding of the logic of things.

We repeat our mathematical inquiries over and over. 

This boy spent 10 minutes repeating the pattern of putting a small, pink detergent bottle lid into the mailbox, closing the mailbox, opening it, removing the pink lid, then putting it back inside, repeating the pattern over and over.

Tom Hunter wrote a brilliant, simple song, which he later turned into a children's book, entitled Build It Up and Knock It Down:

Build it up
And knock it down
And build it up again.
Knock it down
And build it up
And knock it down again.

Subsequent verses echo the same circular, two-step pattern, so familiar to the natural play of young children. Turn it on and turn it off and turn it on again. Pick it up and put it down and pick it up again. Put it in and take it out and put it in again. It might drive us crazy as adults, it might seem to us like they're stuck, but really the children are simply testing their formula, practicing it until it's second nature: A-B-A-B-A-B . . .

Young children, in the course of their play, go on to discover increasingly complex patterns all around them.

And then using those discoveries to do important things like take turns in a board game . . .

Or engage in a meaningful process of many steps.

Play itself is impossible without the ability to think logically. That's why we're driven to mathematical play. These are things we really must know in order to satisfy our curiosities. There is no greater motivator than the prospect of discovery. They're all ponies -- it's a discovery! Let's put them together, and to make sure we understand this, let's put blocks around them.

And even within this classification of ponies, it gets more complex as we notice that some of them have "warm" manes and tails, while others have "cool" manes and tails. It's another discovery! Let's group them within their group, two at this end and three at the other.

When we explore a shape . . .

. . . or put blocks in a box . . .

. . . or only the cars . . .

. . . when we line things up . . .

. . . or create one-to-one correspondence . . .

. . . we are mathematicians discovering for ourselves the classifications and patterns of the world, and using them for our own pleasure be it great beauty, great truth, or just horsing around with friends.

Math is not hard, but at some point, for many of us, it stops being about discovery, which makes it no fun. 

The opposite of play isn't work, it's rote. ~Edward Hallowell

I don't know if we, as adults, need to know more math than we already know. Maybe higher math turns us off, not because its hard, but because its boring to those of us with a particular learning style. Maybe this kind of complex patterning, classifying, and sorting can really only live on paper and doesn't have a place in the rest of the world where we can use all of our intelligences to play with it. Maybe at that level it's just a playground for a certain type of human brain. And that would be okay with me, although I hope I'm wrong.

But I don't know. These are undoubtedly the musings of an idiot because of all the things I admittedly don't know I don't know.

Maybe we already typically know enough math to live our gratifying lives. And still I'm certain that the number of exciting mathematical discoveries that I could potentially make is a number approaching infinity.

(Update: It occurred to me that you might want to click on over to Jenny's magnificent piece at Let The Children Play! about teaching literacy in a preschool with a play-based, progressive curriculum. Whereas I use a lot of words here to write about about math, Jenny uses the far superior medium of photos to illustrate what literacy education should look like in preschool. I sometimes think of Jenny as my blogging Renae!)

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Deborah said...

Hi Tom,
I think that too often, our own lack of understanding of what math is or what kinds of activities actually are a mathematical process are not understood. This is a great post of learning how to identify the mathematical processes that are happening all all around us in the preschool classroom.

Juliet Robertson said...

High 5 Maths!

What I'd like to add to your post is the vital role of adults who can express an interest, use a variety of mathematical terms and allow children to make mathematical explorations.

My hero is a retired educator called Peter Patilla who helped me see maths in a completely different light in school eg that one can teach 3-12 yrs olds in a group all at once. He always advised me to let children make up their own rules for games. Snakes and ladders is never boring now...!

Scott said...

Great post, Tom. My wife is a high school math teacher and she would agree with these observations about math and young children. One of her new passions is helping early childhood teachers realize what foundational math looks like (and it has nothing to do with symbolic numerals!).