An interesting comic with math content today:
Some math questions don’t have math answers. In math, as in language, you do lots of stuff merely because if you didn’t, you’d be considered a rube, or worse yet, you’d be misunderstood entirely. In English, you don’t say that you haved seeed this movie before – you say you had seen it. Though the general rules and logic seem to favor haved, you won’t say it in public, and if you do it isn’t usually celebrated. Same with math, where there are plenty of conventions that at first glance seem arbitrary if not plain wrong. You can see the logic of oneteen over eleven, but logic is not what is asked for here. Language exist in a community of speakers and listeners, and if a speaker unilaterally changes the language, she better be ready for the consequences. The job of creating a new and better language is rarely something that can be done unilaterally though sometimes it can seem that way: somebody creates a YouTube that millions download in the first few days, as if they had spent their entire life waiting for it.
When a linguist hears a young child say “My teacher holded the baby rabbits,” the linguist is not appalled at the child’s bad grammar. On the contrary, the linguist reasons that, clearly, the child has acquired the rule about verb + -ed for past tense. The child must have acquired the rule, says the linguist, because “holded” is not a word the child would have simply copied from the adult speakers.
Similarly, a mathematician would have reason to celebrate a child’s use of one-teen as evidence that a rule about the composition of numbers has been acquired. Hearing the child say “seventeen” is insufficient evidence of that, as “seventeen” might simply be a something heard and copied. A child who says “one-teen” is not copying, but applying a rule, a rule about splitting a number in ten and what’s left.
Non-standard and non-letter-perfect expressions of mathematical ideas by students are a great source of information about the level and kind of student thinking. Whether one likes or not, all mathematics learning starts at the place where the student is, and never at the place where the teacher is. Whether you think the student’s thinking is correct, wrong, poor, interesting or awesome – Jerry has no way of learning anything from a starting place other than the one in which he finds himself. Any kind of teaching Jerry that doesn’t make room for Jerry to start from wherever Jerry happens to be starting from is not aimed at Jerry’s learning.
Conventions are important, and you violate them at your peril. It is a convention that the digit that goes with the number of fingers on my left hand is written as “5” and not as “7”, and that the digit that goes with the number of days in a week is written as “7” and not as “5”. Those are conventions, age-old, entrenched, and pretty essential. Also just about universal.
Other conventions are more localized. For example, in some physics classes, a ratio is distinguished from a rate by saying a rate has two different units, like miles per gallon or pits per peach, whereas a ratio has no units (sometimes called a scalar, or dimensionless, or pure number). In a current math textbook, in contrast, ratio is used as the all-encompassing term, and rate is the special term used only when there are different units involved. In some math books, the term “counting number” is used for numbers 1, 2, 3, 4,… and in other math books the term counting number is used for numbers 0, 1, 2, 3, 4,… and if you wonder what the difference is, in one the number zero is explicitly included and in the other the number zero is specifically excluded. You see similar differences for the term “natural number”, and in some books “natural number” is synonymous for “counting numbers” and in other books it is not. These are the kind of differences that people come to blows over, or at the very least get into tirades about, all without any noticeable enlightening effects. People often treat such conventions, not as conventions, but as some kind of Gospel Truth, and the ensuing wars often have the flavor of religious wars. But you could do in Rome as the Romans do, and make a practice whenever you enter a mathematics conclave to ask what the local definition of “counting number” or “ratio” is and then politely and gracefully using that one.