TEACHING STRATEGIES, USING REPRESENTATIONS, AND OVERCOMING COMMON MISCONCEPTION
The difficulty lies in the interpretation of the meaning of fractions, which requires explanation through connections with other mathematical knowledge, various representations, and/or real-world contexts. Often students do not realize that they have been exposed to fractions in their everyday activities long before they were formally instructed in school. Such knowledge may be thought of as constructed knowledge, and currently, there is a big gap between this constructed knowledge that students bring to the classroom and the instructed knowledge that the teachers try to deliver. Students, who are mathematically proficient, make the connections between the two bodies of knowledge and students that do not understand may never make the connection. Modeling the mathematics is one way to achieve these connections.
Figure 7.3 What is the whole? Visualizing a unit.
One of the key ideas in fraction is the notion of unitizing (Barnett-Clarke, Fisher, Marks, & Ross, 2011) state, “The concept of Unit is fundamental to the interpretation of rational numbers.” (p. 19) Each fraction depends on some unit of measure. One activity that helps students become flexible with their ability to unitize is to use a familiar manipulative like the pattern blocks and change the unit (see Figure 7.3).
According to Lamon, “unitizing is a natural process” (p. 105) that can be thought of “chunking” a specific size of something. For example, 24 eggs can be 2 (dozen) where the dozen is chunked as a unit or it could be 4 (6 packs). One group of teachers wanted to see how students make sense of unitizing fractions. The following fractions were presented to students in grades 5-7.
Students had many different approaches. One question they kept at the forefront of their thinking was finding out “What is the whole?” In the first example, one of the groups changed the fraction 1 1/3 to 4/3 and said if we know that 4/3 = 12 stars and we know 2/3 is half of 12 then we can find out that 2/3 is 6 and one third is 3.9 starts is one
Figure 7.4 Students reasoning through What is a whole? Source: Authors.
whole. Students also used the unit fraction to find the whole. For example, students stated that 5/6 = 15 and 15/5 = 3 there for the unit fraction 1/6 = 3 and 6/6 would be 3 x 6 = 18. And 1 У2 would be 18 + 9 = 27 (see Figure 7.4).