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Does anyone else think that it's hard to teach fractions because fractions are hard? The tenor of these conversations is always about how difficult it is to te

Does anyone else think that it's hard to teach fractions because fractions are hard? The tenor of these conversations is always about how difficult it is to teach fractions, with an implicit judgement that this is bad because fractions should be easy. The thing is, you construct fractions by generating the equivalence class of ordered pairs (Z_1, Z_2) where Z_2 =/= 0. I actually find elementary set theory easier to grasp than the formal construction of arithmetic over the rational numbers. A lot of people would respond that we don't teach children the formal construction -- fine, but a lot of the behavior of the rationals comes from that formal construction. I recall a class in my freshman year of college that required differential equations to do system analysis. The only problem was -- differential equations was next quarter. I think the rote algorithm they gave us to calculate simple differential equations was one of the most frustrating experiences of my academic career. I tend to think that if you teach students things as though they can't understand it, you'll often find that they can't understand it.

vanderZwan 13y ago
> A lot of people would respond that we don't teach children the formal construction Arguably, the way we teach fractions already follows the natural progression of formally defining numbers pretty much: You start with addition (which is somewhat hardwired into our natural number sense), learn that you can also reverse that to get subtraction. Then the big leap of symbolic faith[1] is subtracting a big number from a smaller one to get negative numbers. After this there's the idea of counting repeated additions and giving that a shortcut, resulting in multiplication, and the reveral of that gives division. The big mental jump then is dividing a small number by a bigger number, giving fractions. [1] You have to remember that mathematics is basically formalising our limited number intuition into language, so we can manipulate the symbols as we would manipulate real world objects, greatly increasing our capacity to reason about these ideas. "Doing with Images makes Symbols," to quote Alan Kay.
JoshTriplett 13y ago
Zero apples. One apple. Two apples. Half an apple. One and a half apples. 1 inch. 1.5 inches. 2 inches. Fractions are not conceptually hard.

vanderZwan 13y ago

> A lot of people would respond that we don't teach children the formal construction Arguably, the way we teach fractions already follows the natural progression of formally defining numbers pretty much: You start with addition (which is somewhat hardwired into our natural number sense), learn that you can also reverse that to get subtraction. Then the big leap of symbolic faith[1] is subtracting a big number from a smaller one to get negative numbers. After this there's the idea of counting repeated additions and giving that a shortcut, resulting in multiplication, and the reveral of that gives division. The big mental jump then is dividing a small number by a bigger number, giving fractions. [1] You have to remember that mathematics is basically formalising our limited number intuition into language, so we can manipulate the symbols as we would manipulate real world objects, greatly increasing our capacity to reason about these ideas. "Doing with Images makes Symbols," to quote Alan Kay.

JoshTriplett 13y ago

Zero apples. One apple. Two apples. Half an apple. One and a half apples. 1 inch. 1.5 inches. 2 inches. Fractions are not conceptually hard.