How do we want students to understand a fraction such as 3/4?
Yesterday I had the privilege of working with an elementary team that is starting lesson study for the first time. They decided their theme would be around developing persistence in their students through lessons based on teaching through problem solving, and their topic is fractions (5th grade). Fractions are a popular topic for lesson study at the elementary level because students frequently have difficulty with one or more aspects of fractions.
A common approach to introducing fractions in the U.S. is through a part-whole model. The fraction 3/4 would be represented by a geometric shape divided into 4 equal parts, with 3 of them shaded. There are many problems with this model (see Watanabe 2002), but the one I want to discuss here is the awkwardness of applying this model to fractions larger than 1, such as 5/4. “If the whole has only 4 parts,” students object, “how can there be 5 of them?”
The idea behind fractions is that if you want to express some amount using a unit, such as a meter, but that unit is too large, you can subdivide that unit into smaller but equal-sized units – e.g. divide a meter into 4 parts to get fourths of a meter. Now it is possible to express a quantity using this new, smaller unit, rather than the larger one. Just as with the larger one, we can string together 1, 2, 3, 4, 5, or any number of these new smaller units to describe a given quantity.
Thus the Common Core defines a fraction like 3/4 as 3 units of size 1/4, or 3 x 1/4. This is not an explanation of how to multiply a fraction by a whole number. The expression “3 x 1/4” means three (groups) of size 1/4, and “3/4” means the same thing.
Gunderson & Gunderson (1957) argue for and describe teaching students fractions initially using the fraction words for awhile – e.g. “3 fourths” – before teaching the notation. Some teams in Chicago that have done Lesson Study on fractions have tried this, with success.
In the Japanese textbook series we often use in Lesson Study work, fractions are introduced in the context of linear measurement, as fractions of a meter. This has at least four benefits compared, say, with pizzas. First, unlike pizzas, a meter is a fixed size. Second, linear measurement is easier for students than area. Third, fractions of linear measurement translate easily to fractions on a number line, an idea emphasized in CCSSM. And fourth, it is easier to accurately draw arbitrary fractional parts, especially if your notebook has a grid. (To draw fifths, say, draw five adjacent rectangles of the same size.)
Gunderson, Agnes, and Ethel Gunderson. “Fraction Concepts Held by Young Children.” Arithmetic Teacher 4 (October 1957): 168–73.