more from Wu

from Allison:

Going forward from what Catherine posted, Wu argues that most of what teachers know about fractions, and teach to their students, is wrong in the sense that it is unsupported by what the students have previously been taught.

Wu makes clear that you must be ruthless with your self when teaching. You must not allow yourself to use any concept you've not explicitly given your students. And almost all textbooks violate this rule in almost every lesson on fractions.

Nearly all textbooks avoid defining a fraction. Since they aren't defined, they don't define how you operate on fractions in terms of any real definition. Beyond that, when they do introduce operations on fractions, the books don't define operations in terms of what students know either. They teach rules without ever teaching where the rules come from. Their "explanations" invoke properties no one taught the students, but are just asserted. That means logically, their explanations are wrong, as you can't prove a statement by assuming the statement true.

For example, how does one justify multiplication of fractions? (I'll answer how later, but for now, discuss amongst yourselves.)

Yes, you can learn to use the rule properly without understanding where it is from, but you'll only go so far, and more, you will mistrust your teachers and mistrust math. Math will seem to be an endless series of magical statements with no relationship, and sooner or later, you'll run out of time/space/effort for keeping track of them. Properly understood, math is coherent, because every new step you take follows from the ones before.