LEARNING PROGRESSION IN FRACTION OPERATIONS— ZOOMING IN ON MULTIPLYING AND DIVIDING FRACTIONS
An important upper-grade standard is being able to solve multiplication and division problems in context. Using the area model for multiplication, one can illustrate what happens when one multiplies a fraction by a fraction. Using the area model, one can take % of 2/9 and show how the overlapping area is the product (see Section 8.7). Dividing fractions using the area model one to consider the measurement model in interpretation.
For example, У divided by 1/3 can be interpreted, “how many 1/3s can go into '/г?” Or “how many 1/3-cup servings can I get from У cup?” The first thing one might consider is that the serving size we want is 1/3 cup and then ask how many of that serving size can be found in a У cup. If we use the common-denominator method, we would have 3/6 divided by 2/6 and get 3/2 or 1 /. If we look at it as an area model, what we have is that 1/3 can fit into У, 1 У times (see figure below). This is a measurement division model, where we are measuring off the unit of 1/3 to see how many will fit into У (see Figure 8.4).
Apply and extend previous understandings of multiplication and division.
Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, for example, using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Providing contexts for division of fractions helps students to
interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent
Figure 8.4 Modeling division of fractions using the area model. Source: Authors.
the problem. For example, create a story context for (2/3) + (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) + (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) + (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? (CCSSM 6th grade, p. 42)
In the following lesson study vignette, we zoom in on a lesson on dividing fractions. Through this, we collected evidence of a common challenge with the concept of unitizing fractions.
Mathematics educators have emphasized a constructivist approach to teaching fractions instead of a procedure-focused approach (Bruner, 1986; Glasersfeld, 1990; Schoenfeld, 1985). The latter is based on the invert-and-multiply algorithm that is generally more efficient (Bergen, 1966; Capps, 1962; Krich, 1964) and leads to an algebraic thinking. However, this approach is often confusing for students and results in memorization of the procedure without actual understanding of the concept (Capps, 1962; Elashhab, 1978; McMeen, 1962; Siebert & Gaskin, 2006).
On the other hand, a constructivist approach provides students with an opportunity to build their understanding of the concept as a result of their own learning experience (Bruner, 1986; Glasersfeld, 1990; Schoenfeld, 1985). To reach this goal, students are encouraged to invent their own ways of approaching a problem by working in small groups or individually and discussing their thinking processes with each other (Flores & Priewe, 2013). Afterward, having conceptual knowledge, students are more likely to make sense of procedures when dividing fractions (Carpenter, 1986).
Mathematical ideas can be represented enactively (concrete representations), iconically (pictorial representations), and symbolically (written symbols) (Cramer & Wyberg, 2009). Manipulatives as concrete materials are used to help students build mental representations, which in turn prompts them to think abstractly (Cramer & Wyberg, 2009). Pictures work in similar way (Siebert & Gaskin, 2006). Concrete and pictorial representations are especially useful when teaching division of fractions. Students might use the measurement interpretation of division while solving the problem or repeated subtraction in order to make sense of the concept (Flores & Priewe, 2013).
Research on teaching division of fractions reveals a number of misconceptions students tend to have while learning the concept. The main misconception is a lack of understanding of what the whole is and how the whole as a candy is different from the whole as a serving (Flores & Priewe, 2013). As a result, students are confused about how to interpret the reminder (Flores & Priewe, 2013), which in turn causes inability to communicate when students try to discuss their work.