M3N5(b) - Meaning of Fractions

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.
b. Understand the fraction a/b represents a equal sized parts of a whole that is divided into b equal sized parts.

Teaching and learning of fractions continue to be a major challenge for both teachers and students. Many children (and teachers) think of fractions as parts of a whole [M3N5(a)]. However, M3N5(b) suggests that we look at fractions in a slightly different way as well. For example, according to M3N5(b), the fraction 2/3 means there are 2 pieces of 1/3. So, why is it important that children understand fractions in this manner?

In "Elementary School Teaching Guide for the Japanese Course of Study: Arithmetic (Grades 1 – 6)," the authors suggest that there are 5 meanings of fractions. For example, the fraction 2/3 may mean:

1. two parts of a whole that is partitioned into three equal parts,
2. representation of measured quantities such as 2/3 liter or 2/3 meter,
3. two times of the unit obtained by partitioning 1 into 3 equal parts,
4. quotient fraction (2 ÷ 3), and
5. A is 2/3 of B – if we consider B as 1 (a unit), then the relative size of A is 2/3.

Thus, M3N5(b) corresponds to the third meaning above. So, why is it not sufficient to think of 2/3 as 2 out of three equal parts (of a whole)? What advantages are there to think of 2/3 as two 1/3’s?

The most important reason for going beyond the part-whole view of fractions is that we want students to understand fractions as numbers. The part-whole interpretation of fractions is more about relationships, and it does not necessarily signify a quantity/number. When someone makes 6 out of 8 free throw attempts, the fraction 6/8 doesn’t signify a number. In fact, if he makes 8 of 10 attempts in the next game, we can say he was successful at 14/18 of attempts in those two games combined. This combination is NOT addition of numbers 6/8 and 8/10, in that case, we have to find a common denominator to find the sum. Rather, 6/8 and 8/10 are both ratios. The part-whole interpretation will signify a number if the whole we are considering is the number 1.

The part-whole interpretation is important, and may be a prerequisite, before students can consider 2/3 as 2 pieces of 1/3’s. For this interpretation to be truly useful, students must first understand 1/3 as a number – it is a number such that if you have 3 of them together, you will make the number 1. In other words, 1/3 is a number that is equal to the number in a group when 1 is divided into three equal sized numbers – 1 out of 3 of the number 1.

There are many places in the elementary school curriculum the interpretation of a/b as a copies of 1/b’s. For example, if students’ view of fractions is limited to the part-whole interpretation, they will have a hard time making sense of an improper fraction. After all, what does 4 out of 3 mean? On the other hand, if you consider 4/3 as 4 pieces of 1/3-units, then there is nothing different about 2/3 and 4/3. Or, consider the simple addition/subtraction of fractions with like denominators. For example, 3/5 + 4/5 means putting together 3 pieces of 1/5’s and 4 pieces of 1/5’s, giving us 7 pieces of 1/5’s all together, or 7/5. This reasoning is, in principle, the same as thinking of 30 + 40 as adding 3 tens and 4 tens, thus 7 tens.

The importance of looking at non-unit fractions as collections of unit fractions is not a Japanese idea. Thompson and Saldanha indicated in their chapter on fractions in the Research Companion to the Principles and Standards for School Mathematics, that this is a very important view of fractions. Unfortunately, they also note that this idea is rarely seen in US mathematics textbooks. As we begin implementing the GPS, therefore, it is important for us to remember this perhaps unfamiliar way of looking at fractions.

By the way, the fourth meaning is discussed in M5N4(a). The fifth meaning of fraction, i.e., fractions as ratios, are not treated until Grade 6 when students are introduced to the idea of ratios (makes sense, doesn’t it?). Moreover, in the Japanese elementary textbooks, the idea of a fraction of a set (or discrete model of fractions) does not appear until Grade 6 because they believe that the meaning of fractions in that context is much closer to the ratio meaning of fraction. [In fact, the part-whole meaning of fractions is very close to the ratio meaning of fractions.] This is an interesting contrast to GPS M2N4(a) and something Georgia educators must think about carefully.

I will be discussing models of fractions in another post.