These standards present a challenge to many elementary school teachers who have been using traditional teaching methods that focus on procedures and algorithms. These teachers now need to acquire the content knowledge and the instructional strategies to enable their students to make sense of these fractions standards.
That students have difficulty with fractions sense is well known. For example, in a 2009 National Assessment of Educational Progress (NAEP) study only 25% of 4th-graders could select from among 5/8, 1/6, 2/2, and 1/5 the fraction closest to ½. Evidently, these students’ teachers had not been able to explain successfully some basic notions related to fractions.
Yet it is the elementary classroom teacher who is expected to correct this situation. It is through them, according to the Common Core math standards, that students are to develop “expertise” in the “conceptual understanding” of mathematics. The standards define this expertise as “comprehension of mathematical concepts, operations, and relations.” In other words, besides being able to perform calculations with fractions, students must comprehend what they are doing, to have what is known as “fractions sense.” The acquisition of fractions sense cannot happen if teachers themselves do not have this conceptual understanding and the means to communicate it.
To illustrate: At our Developing Fractions Sense workshops, when we ask elementary teachers to explain what is meant by the fraction ½, often we receive this response: “It is one part out of two.” There is no reason to believe that these teachers would tell their students anything different. And so we see how the problem begins: The concept of fraction has not been well defined for students. Without this basic building block, how can students possibly be expected to be successful with fractions?
What is the correct way to define the fraction of ½, or more generally the fraction 1/b? The standards say in 3.NF.1: “Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts.” Notice the important words that were omitted in the teacher response: “whole,” “partitioned,” “equal.”
Even if a teacher reads this definition, there is no reason to assume that the teacher will immediately understand the necessity for each of these important words. A school math leader or coach may need to provide a fuller explanation. But let’s assume that the teacher does understand the definition. The teacher still needs to develop an instructional strategy to communicate this understanding to students. Simply asking students to write down and repeat the definition will not do. The teacher needs to illustrate the definition in various ways for students to grasp its full meaning and be able to operationalize it in practice.
I would suggest that the most effective way to begin teaching fractions is with fraction blocks (also known as pattern blocks). The teacher can clearly illustrate partitioning a whole—represented by the yellow block—into 2 equal portions using the red blocks, 3 equal portions using the blue blocks, or 6 equal portions using the green blocks (see Figure 1). This process allows students to understand almost at once the meaning of 1/4, 1/3, and 1/6.
Figure 1: Fraction blocks clearly illustrate partitioning a whole into 2, 3, and 6 equal parts.
Next the teacher can illustrate the second part of 3.NF.1, namely, “Understand a fraction a/b as the quantity formed by a parts of size 1/b [referring to the same whole].” For example, now that the student recognizes a green block as having the value of 1/6, the teacher can have the student count green blocks: one-sixth, two-sixths, three-sixths, and thereby learn that the first word in five-sixths, for example, gives the number of equal blocks and the second word in five-sixths gives the size of each block. The teacher can then use rectangular-grid and number-line models to reinforce the meaning of 1/b and a/b, at all times emphasizing the notion of “whole,” “partition,” and “equal parts.”
The critical importance of knowing what constitutes the whole in a fractions problem is illustrated by the following question: “Mrs. Jones wishes to donate two cakes to the birthday party. Each cake is cut into 6 equal slices. At the end of the party, Mrs. Jones notices that 3 slices from the first cake and 4 slices from the second cake were eaten. What fraction of Mrs. Jones’ cakes was eaten at the party?” (Before reading further, please provide your own response.) At our workshops we find that almost inevitably teachers respond with the answer of 7/6 or 1 1/6 obtained by adding 3/6 to 4/6. The instructor will ask the workshop participants, “Is 1 1/6 more or less than a whole?” The participants will reply that it is more than a whole. The instructor will then ask, “Was all the cake that Mrs. Jones donated eaten?” They will respond that it wasn’t. They will then realize that the answer cannot be more than a whole; it has to be less than 1.
The essence of this problem is recognizing what constitutes the whole. The question asks for the fraction of “the cakes” consumed. Note the plural: cakes. Both cakes together form the whole in this instance. We can represent the two unsliced cakes with two of the yellow blocks. However, each cake was divided into six equal parts, and so we need to use the green blocks, which are sixths. Seven slices were eaten out of the total of 12 slices. Thus it is easy to see that the fraction of the cakes consumed is 7/12.
Fraction blocks are invaluable for helping students make sense of fractions. For example, by covering a red block with three green blocks the teacher can help students see that 1/2 ÷ 3 = 1/6 and 1/2 ÷ 1/6 = 3. The teacher can ask for a related multiplication equation from the same illustration and help the students, if needed, to formulate that 3 x 1/6 = 1/2.
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Figure 2: Fraction block illustration of 1/2 ÷ 3 = 1/6 and 1/2 ÷ 1/6 = 3.
The fraction blocks also are helpful in working with more advanced concepts. For example, if a teacher wishes to illustrate the meaning of 3/4 x 8/6, the teacher can take 8 of the green blocks, partition them into four equal parts, and then take three of those parts to get the answer 6/6 = 1.
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Figure 3: Fraction block illustration of 3/4 x 8/6 = 6/6.
The goal of this discussion is to point up the need for elementary teachers to have both a conceptual understanding of the fractions standards and the instructional strategies for delivering those standards in a way that will make sense to students. Often elementary teachers do not have any special strengths in mathematics knowledge or the pedagogy of teaching mathematics. Consequently it is essential that they be provided with resources, including staff development, that specifically target the fractions standards. It would be advisable for a math specialist at each elementary school to be available to conduct demonstration lessons, to provide math content instruction, and to coach teachers regarding instructional strategies.
Dr. Henry Borenson is the inventor of Hands-On Equations, a system he designed to enable young children to learn algebra. He is also the President of Borenson and Associates Inc. His company provides the Making Algebra Child’s Play workshop for teachers of grades 3 to 8. Their newest workshop is the Developing Fractions Sense workshop for teachers of grades 3 and 4. Visit www.borenson.com for additional information.