Conceptually Correct.
August 19th, 2005This morning I met with Sullivan, a rising fourth grader and son to one of the staff members of Leverett. His mother is worried that Sullivan’s understanding of fractions isn’t strong enough. It’s not suprising. Fractions are hard. I didn’t learn fractions until I was already in the fourth grade. And here he is, the summer before — well, he’s practically a third grader still. And it’d be surprising if a third grader fully understood fourth grade concepts. In fact, it’d be unnatural, preternatural, some-natural but not plain old vanilla natural.
Anyway, I agreed to meet with him for an hour today in the library at 11 am. By 11:30 he could add and subtract fractions with the same denominator symbolically and give a geometric explanation as to what those symbols actually mean. It was interesting to watch him. Every once in a while, he inverted numerator and denominator, which formally is just as good as the convention everyone else uses. Sullivan would get the same answers that we would, just upside down. This was great! I knew he understood what was going on. He just knew it upside-down. But that’s completely unimportant. The orientation is human invention, an artifact of the notation. But rather than let this habbit go much further, I suggested he comply with convention [for his good. I'm not sure his teachers would appreciate his throrough mastery for what it was if it were in an unfamiliar form].
Eventually I asked him what “three out of four plus two out of four” is. He very quickly told me that it was “five out of four” without much thought. He had applied the rules, and what the consequence of his answer hadn’t quite struck him. Then the follow-up, what does it mean to have five things out of four? He thought a while, and lo! Sullivan developed a theory of mixed numbers for me, guided by the principle that four things out of four things is one whole thing itself. By 11:53 am — I checked — he could convert improper fractions to mixed numbers based on an algorithm he had developed himself. Again, there was a little trouble with notation. He always got the integer part correct. And he always found the correct numerator, though sometimes he put it in the denominator, he something came up with an incorrect denominator, which he very consistently put on the numerator when he made this mistake. He very naturally, I think, carried the numerator from the improper fraction to the numerator of the fractional part of his mixed number. Again, I believe this is a weakness of the notation — not Sullivan. Because of the information he supplied, and correctly 19 times out of 20 — 8+6=14, not 12 — it was clear that he understands the rules.
More to that. For fractions like, 10/2 he gave an answer of “5 and 0/2.” Most impressive, at least to me, was that he did this instinctively. His inclusion of 0/2 shows me that he carried the entire algorithm to its conclusion each time, which means he was using an algorithm, which means that Sullivan is a metacognitve genius. I was very excited to explain all this to his mother, but we discussed radiation poisoning with Sullivan instead.
The algorithmic, symbolic nature of arithmetic makes it a very unnatural, abstract, and terrible introduction to math; mostly because it isn’t math. It’s computer science.
But Sullivan triumphed all that. “He’s a good man, and thorough.”
