I think the main reason that difficulty with fractions and rational numbers is so pervasive is because students are not taught to think in terms of multiplicative relationships, which begins with multiplication. Students are taught to compute a product; they learn procedures to manipulate numbers. They don’t learn the concept of scaling, of working with two or more different unit sizes, of comparing the unit sizes as ratios thinking along the lines that there is x of this for every one of that, and becoming comfortable switching their point of reference regarding the unit size—i.e. what is one whole or 100%. Conceptually, they are stuck on additive relationships, which require the same unit size since you can’t directly add or subtract quantities that are based on different scales. They are only comfortable with the scale as the standard unit 1.

Alison Blank has two posts on the meaning of multiplication and concludes that it is a change of unit or a change of scale. Some of her reader comments relate multiplication to proportionality. Multiplication explicitly or implicitly always involves a ratio. In rate problems and unit conversion problems the ratio is explicit, such as miles/hour. In set problems, such as 3 six-packs of juice contains how many cans, the ratio is implicit, 6 cans/1 pack. In area problems, such as the area of a 5″ x 9″ rectangle, the ratio is implicit: 5 sq. in. /1 row, or 9 sq. in. /1 column. The structure of multiplication, the relationship among the numbers, is proportionality. In the juice example, the proportion is based on the ratio of 6 cans for every six-pack, or 6 to 1. The counts must follow that same ratio: 18 cans in 3 six-packs, or 18 to 3, which is equal to 6 to 1. Stated as a proportion, 6 cans is to 1 six-pack as 18 cans is to 3 six-packs.

Too often we ignore conceptual meaning in favor of computing an answer. Repeated addition is a computational strategy, not a meaning of multiplication. If you believe that repeated addition is a meaning of multiplication, then your logic would mean that continuous counting and counting up are conceptual meanings of addition. We need to focus on conceptual meaning and create activities that embody the concept so students develop an intuitive understanding of the concept before we formalize it. It seems to me that Mr. K is doing that with fractions on this lesson.