Wednesday, July 29, 2015

Numeracy instruction

kw: book reviews, nonfiction, mathematics, mathematical thinking, instruction, learning methods

When I saw A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra) by Barbara Oakley, PhD, I was intrigued. The main title was in the form of a cutesy equation, which I supposed was the editor's conceit. Having read the book, I am not so sure.

Dr. Oakley knows what she is talking about, because she began as a math-phobic, but learned to love it. Throughout the book she has 1-page items by people with a similar background, who now are comfortable with mathematics, at one level or another.

The book is a breezily-written compendium of learning techniques and tips gathered into 18 chapters. It is not intended to be taken in wholesale. Different people learn in different ways, and in many cases, a single chapter, or even one point in a chapter, can unlock the math potential for someone. But I wonder…

At a certain level, to be human is to be mathematically adept at some level. Very young children, asked to choose one of two piles of coins, will pick the one that is spread out rather than a neat stack of the same number of coins. They equate spatial area with quantity, and don't realize that the two piles are equivalent. But, I suspect they have not yet learned to count, and it takes this further level of sophistication before they have the mental equipment to fairly evaluate the two piles.

I think that is analogous to an experience I had at about age 12. Someone had showed me a few Algebra equations. I saw something like 10x=5 and wondered, "How can that be?" I thought the "x" was supposed to represent a digit, like the stuff on the left would be a number from 100 to 109. So I thought something else had to be going on. This caused quite a delay in my getting the point when I began Algebra class later that year. But I think, a month or so into the school year, when it all began to "click", that my brain had simply grown up enough to have the right tools for doing algebra.

We all do a certain amount of calculation. Most of us can quickly evaluate the change we're given at the store (if we used cash). People who bowl soon learn to keep score without writing down their calculations. When we drive (without a GPS), and we see "Chicago, 95 miles", we check the odometer, note what it shows, and can then glance at it later and know how many miles we still have to go. Many times, we can even be told a "problem" like this:
John and Mary ride bicycles toward each other at 5 mph, from 1 mile apart. Their pet bird, which flies 20 mph, flies back and forth from one to the other until they meet. How far does the bird fly?
Most of us, by age 10 or so, can figure that John and Mary each ride half a mile, because they are going the same speed. The bird flies four times as fast as either of them, so its total flight is two miles. This kind of "figuring" is actually algebra, without the equations. In fact, it would take me longer to write down the equations for solving this using "traditional" algebra, than it did to write the two sentences above. What is funny is when someone tries to tackle the problem as a series of flights of decreasing length by the bird. The equations to get that to work are gnarly!

So every one of us has some amount of math built in. Standard equipment. But that "some amount" varies a great deal from person to person. Not everyone can learn algebra, no matter how it is taught nor how hard they try. But most can, and by "most" I mean "more than half but not a great deal more". Some kids who had no trouble with algebra never, ever get the point of Trigonometry. My senior year of high school, we got done with the ordinary curriculum for the year a few days early, so the teacher did an experiment. He got out a few copies of a basic text in Calculus and taught it to us. In just a couple of weeks, I learned enough Calc so that I pretty much breezed through the Calc 101 course the next year in college, which used that same textbook!

I was a working mathematician, at a certain level, for decades. But there are branches of math that have never made sense to me, and others that I can puzzle out with desperate levels of effort. I had to take Differential Equations three times to pass it. I'm still not comfortable with it, but the explanation of how to use it takes only two pages in my old CRC Handbook of Chemistry & Physics, and actually contains most of what one needs to do nearly any Diff Eq problem!

And so it goes. Each human brain has a certain mathematical limit. With luck, we might grow mathematically to our full potential, but it is time consuming. Most of us never need all that stuff. But we also grow into certain abilities over time. Just as the brain doesn't finish emotional maturation until about age 25, it must be true that certain math circuits only get set up at certain ages. It may be that the ten years between my second and my third try at learning Diff Eq made more difference in my ability, than the exposure I'd had during the first two attempts.

With all that in mind, I find no sense in delving into what Dr. Oakley has to say. The book is a fantastic resource. Someone who needs encouragement and help in how to learn math and science will do well to read the book quickly, then return to read over certain sections with more care: those sections that seemed to make the most sense the first time through. The first two chapters will be helpful to everyone. As for the others, while the author attempts to make them generally applicable, each will actually be best suited to people with a certain kind of mind, one way or another.

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