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Saturday, April 14, 2007

Who Can Blame Them

Mathematics is not arithmetic, algorithms, or formula memorization. It is a science- the science of patterns. We hairless apes are extremely good at it, every last one of us. It is a terrible commentary on our culture that so many people think they are bad at math, because all humans are exceedingly good at math. Our minds have been crafted by millions of years of evolution to recognize patterns. We are so good at it that sometimes we see patterns that aren't even there, like say the visage of the Virgin Mary in a Pringle.

Imagine a toddler out for a walk with Mommy. They encounter in their travels a slobbery Golden Retriever puppy. Mommy, in her most nurturing of voices, accommodates her child's inquiring mind by christening the bounding amalgamation of fur and drool with the moniker of doggy. After extricating themselves from this cute little scene, they proceed with their journey. A few blocks away, they are greeted by the franticly bellowing bark of a Great Dane, whose towering form threatens the structural integrity of its chain-link enclosure. The toddler points at the dog, and with sage-like wisdom proudly proclaims doggy. Mommy is filled with pride. "My child is a genius," she thinks, as she begins mapping out the path from pre-school to Nobel Prize. But the daydream is soon interrupted and all hopes dashed. The object of the Dane's ire turns out to be a mischievous orange tabby that has intruded into his domain. To Mommy's chagrin, her little prodigy raises her chubby little finger toward the fleeting feline and with as much confidence as the diaper-clad can muster, announces yet a third doggy. Mommy is crest-fallen. It looks like she can treat herself to that designer handbag she has been eying, because her little one is clearly not college bound.

The preceding anecdote replays itself in some form or another all over the world. The mind of a child is forming new connections at a rate that adults can only superficially comprehend. Sometimes those patterns are correct but situationally inappropriate. This does not make them wrong. The pattern denoted as "furry, four-footed creature with a tail" is correctly applied to doggy. Of course, it can be applied to many other animals. It will take time, practice, and myriad "wrong" assessments before the nuances of kitty, horsey, and other cuddly quadrupeds are established. Such is the nature of language. It is sometimes messy and confusing, despite our best efforts, and our keen pattern-sense and mathematical reasoning will be both a blessing and a curse.

Consider that in English, trough, bough, dough, and enough do not rhyme, while beau, go, and sew do. That's enough to send anyone screaming for the hills, yet few native English speakers will claim to be bad at it.

The inherent order of mathematics is much heralded by its practitioners. It is neat,clean, and makes sense- most of the time. Unfortunately, math is a human construct, just like language. Often times, our beloved mathematics is guilty of the same confusion and inconsistencies as we saw above. Consider the notation f(x). For many, it immediately brings back horrible memories of their first taste of algebra and the day they discovered they were bad at math. They were cruising along through arithmetic. They learned how to do long division and divide fractions. They even went with the flow as the symbols started changing. The symbol for multiplication went from x to a dot to an asterisk, then to parentheses as in 3(4). It was confusing, but they had hung in there. Now they were being taught to use letters sometimes to represent numbers. Now that was pushing it, but still they pushed on. Enter the dreaded f(x). It looks familiar and friendly, and when asked to find f(4), the students find a suitable pattern from their arsenal and proudly answer 4f. I'm sorry says the teacher, the answer is kitty.

For many, this was the point where math went from fun to completely mind-boggling. What these unfortunates fail to understand, largely because they have not been properly told, is that the problem is not that they are bad at math, but that they are so good. This is one of many confusing notations employed in such an allegedly orderly field. Others include

  • sinn and tann
Where sin2x = (sin x)2 and tan2x = (tan x)2

sin–1x = arcsin(x) and tan–1x = arctan(x).

  • The square root symbol Öb , which actually refers to "the non-negative square root of b" but is often lazily used to mean both the positive and negative roots by math teachers.

  • Confusion abounds in order of operations: What is –32 ? Many think it is (–3)2, and so they arrive at an answer of 9. But that is wrong. The convention among mathematicians is to perform the exponentiation before the minus sign, and so –32 is correctly interpreted as –(32), which yields –9.
  • Or 3/5x, which depending on whether you use the BODMAS (bracketed operations, division, multiplication, addition, subtraction) or My Dear Aunt Sally (multiplication, division, addition, subtraction) will yield (3/5)x or 3/(5x).

These are just a few examples. I'm sure there are many more. The important thing is to appreciate how difficult and often times contradictory math notation can be, lest young math enthusiasts become older math phobics, simply because their highly adapted pattern recognition instinct was confused by faulty patterns.


Andy said...

This is a wonderful post, Tony. One of the things that tripped me up in high school math courses was the inconsistent notation of symbols, formulas, etc from year to year, sometimes even in the same textbook. One of the things I found liberating about computer programming in college is most leading programming languages are extremely consistent with symbols (at least inside a particular language), primarily because the program that compiles your source code cannot tolerate inconsistencies (e.g. the program will not be able to be rendered down to machine code and the compiler will throw errors). Also, if I need to do mathematical operations in my programming, I can explicitly specify in my code the exact order of operations as long as I use parentheses appropriately. And, if I want to create new symbols to perform more advanced operations (e.g. by defining a new operator or overloading an existing one), I have to be extremely explicit as to what data types (operands) the operation can work upon.

Personally, I think there should be more use of computers and programming in math courses, so that students can treat certain mathematical techniques as machines ... primarily so they can "look under the hood" to see what's going on as the computer performs the operations. Anything that helps students form a more precise internal visualization of what is actually going on becomes a great aid.

Phil said...

Math is the recognition of a _sequence_ of patterns. Choose the first operation, now choose the second based on what came from the first, repeat until you think you are done. Art doesn't work like that. Draw a picture of an elephant, draw a picture of a friend - this utilizes a much different form of pattern recognition. Sports don't work like that. See the ball, react to it based on past experience and current situation. I don't think we math recognizers see how hard the patterns we recognize are for others to see until we start thinking about how the patterns that others see are difficult for us.