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
but 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 –32is 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.