Week two of Your Daily Wu begins thusly…
Wu on fractions
[T]he publishers mistakenly believe that intuitive arguments and analogies suffice. Thus, fractions are simultaneously (and incomprehensibly) parts of a whole, a division, and a ratio. (p. 4)
what i think he’s saying
The only reasonable explanation of the meaning of a fraction is the one provided in undergraduate modern algebra: A fraction is an equivalence class of ordered pairs (a,b) of integers, with b not equal to zero, and with the rule that (a,b) is equivalent to (c,d) if and only if ad=bc.
But wait…that really *is* what he’s saying
[A] fraction m/n (for whole numbers m and n, n≠0) is just a symbol consisting of an ordered pair of whole numbers with m preceding n. (p. 10)
Or is he?
This is from “Teaching Fractions According to the Common Core Standards”, available on Wu’s homepage.
The shift of emphasis from multiple models of a fraction in the initial stage to an almost exclusive model of a fraction as a point on the number line can be done gradually and gracefully beginning somewhere in grade 4. This shift is implicit in the Common Core Standards. Once a fraction is firmly established as a number, then more sophisticated interpretations of a fraction (which, in a mathematical context, simply mean “theorems”) begin to emerge. Foremost among them is the division interpretation: we must explain, logically, to students in grade 5 and grade 6 that m, in addition to being the totality of m parts n when the whole is partitioned into n equal parts, m/n is also the number obtained when “m is divided by n”, where the last phrase must be carefully explained with the help of the number line.
So are those multiple interpretations of a fraction (1) incomprehensible and foolish (the former adjective is explicitly Wu’s in “Phoenix Rising”; the second is implicit), or (2) where we start with students before we formalize? And then do we then return to them at some point? I’m confused.