A Wall Street Journal opinion article drew considerable attention to Common Core (CC) math standards—particularly the sixth-grade standard for fractional division— in early August. In it, math professor (emerita) Marina Ratner criticized approaches she saw in her grandchild’s classroom, where students were expected to represent fractional problems with pictures. She says:
The teacher required that students draw pictures of everything: of 6÷ 8, of 4 ÷ 2/7, of 0.8 × 0.4, and so forth. In doing so, the teacher followed the instructions: ‘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 the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient . . .’
Ratner then asks: “Who would draw a picture to divide 2/3 by 3/4?”
I tend to agree. Requiring students to draw a picture to prove they “understand” the meaning of 2/3 ÷ 3/4 is likely to confuse more than it enlightens. Many others agree, including Barbara Oakley, a professor of engineering at Oakland University in Rochester, Michigan. She expressed her ideas in another Wall Street Journal opinion article called “How We Should be Teaching Math” published just this week. (Here is a link to a non-paywalled version.) In it, she states, “True mastery doesn’t mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it’s 25—it’s a single neural chunk that’s as easy to pull up as a ribbon. Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing.”
A More Sensible Approach to Fraction Division
As readers of this series know, my purpose is to describe how teachers can interpret and implement various CC math standards in a common-sense fashion. To do this requires not reading the standards as one would dissect and interpret the Bible. Here is the standard addressing fractional division, in its entirety:
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 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?
While the standard suggests that students “use visual fraction models,” that term can be interpreted in several ways. A sensible interpretation is presenting such models to students as a means of explanation. A pictorial representation can illustrate what it means to divide 7/8 by 3/16, for example, without having students actually draw it. As shown below in an example from the book “Arithmetic We Need; Grade 6,” textbooks have done this for ages.
Figure 1. (Brownell, et. al., 1955)
Having used the “Arithmetic We Need” series when I was in school, I relied on the ruler example as a way to interpret what a fractional division problem represents. Thus, 3/4 divided by 1/16 can be thought of as how many 1/16-inch intervals are contained in a 3/4 inch interval—or how many 1/16-pound units are contained in 3/4 of a pound. Also, the book asks students to check their work by multiplying the quotient (12, in this case) by the divisor of 1/16 to obtain 3/4. In this way, the relationship between multiplication and division is demonstrated as required by the CC standard quoted above.
Students also learn the difference between two types of division problems: those that ask “what part of” and those that ask “how many times.” Such distinction helps students understand why the answers to some problems are less than 1, and the answers to others are greater than 1—something that often confuses students who have worked with whole numbers up until now. Figures 2 and 3 illustrate how this works numerically. Figure 4 provides a pictorial representation for a “what part of” situation. This type of pictorial representation is what is mentioned in the CC standard: “use a visual fraction model to show the quotient.”
Figure 2 (Brownell, et. al., 1955)
Figure 3 (Brueckner, et. al, 1959)
Figure 4 (Brownell, et. al.; 1955)
In addition to requiring visual fraction models and equations to represent fractional division problems, the standard also requires that students explain how the fractional division algorithm works. How the student should explain it is not made clear, other than “use the relationship between multiplication and division.” Should the student draw visual models? Should he demonstrate why it works?
After students have learned the rule for multiplying fractions, this rule can be used to explain why 1 divided by any fraction is the reciprocal. For example, since 1 divided by 3/4 is a number which when multiplied by 3/4 equals 1, students can easily see that 3/4 multiplied by 4/3 equals 12/12, which equals 1. Students can then generalize that concept for any fraction, since the conceptual underpinning is now obvious.
We know from the principle of reciprocals that 1 ÷ 3/4 = 4/3. We also know that if we wanted to know what 2 ÷ 3/4 is, we could multiply 1÷ 3/4, (which is 4/3) by 2, to get 8/3. Extending this principle, if we want to know what 2/3 ÷ 3/4 is, we multiply 1 ÷ 3/4 by 2/3. The result is 2/3 × 4/3 = 8/9. This method provides a relationship between multiplication and division as the standard requires and provides an explanation that may be accessible for some sixth graders for why the “invert and multiply” rule works. It is not a new method of explanation; as shown in Figure 5, an arithmetic text from 1879 by Olney (1879) uses it. It has also been used in the Saxon Math program, and most recently has been used by the JUMP Math program that originates from Canada and which was started by Canadian mathematician John Mighton, who is a fellow of the Fields Institute (Mighton, et. al., 2013).
Figure 5. (Olney, 1879)
Not all students in sixth grade will understand why the invert and multiply algorithm works, let alone be able to explain it. From my experience, students in pre-algebra and algebra courses who have had some experience solving equations like 3x = 5 are in a better position to synthesize and apply prior knowledge to gain the understanding of why the algorithm works.
This is because the facility they have with algebraic representation enables a better view of what is happening mathematically than they have had previously. With the knowledge of reciprocals and that a fraction multiplied by its reciprocal equals 1, an equation like 2/3(x) = 4/5 can be an introduction to explaining the invert and multiply rule. Having taught pre-algebra students, my experience is that when presented with such a problem, their first response is to say “divide by 2/3 to solve for x”. The equation can be written as x = 4/5 ÷ 2/3. But they also now know that multiplying 2/3 by the reciprocal 3/2 equals 1. Multiplying both sides of the equation by 3/2 looks like: 3/2 × (2/3) x = 4/5 × 3/2 or x = 4/5 × 3/2. Since x also equals 4/5 ÷ 2/3, the relationship is complete and the two results can be shown to be equal: 4/5 ÷ 2/3 = 4/5 × 2/3.
Students’ knowledge of solving simple equations now enables a more efficient explanation of why the invert and multiply rule works that follows more naturally. Thus, I recommend presenting this explanation in seventh grade.
The Goal Is Genuine, Age-Appropriate Understanding
The standard I have discussed here illustrates the theme of “understanding” and “explanation” that pervades many of the CC math standards. Opinion is divided in the education community about how to teach understanding. It is certainly worthwhile to explain to students why the fractional division algorithm works. Even more important, however, is recognizing that a student who knows what problems fractional division can solve and can perform the procedure possesses some understanding. Marking students down on standardized tests who have enough understanding to solve problems but cannot do the things judged to indicate understanding is placing the cart before the horse. It is tantamount to saying, “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.”
Some students will have more understanding than others. Some may understand prior to learning the procedure, while, for others, procedural fluency tends to lead to understanding. But some students may also just not be interested in why the invert and multiply rule for fractional division works, for example. To some in the education community, lack of this understanding exemplifies what’s wrong with math education today. No. That’s not what is wrong with math education today. Students possessing a rote understanding of procedures they cannot perform is a far greater problem.
Brownell, William A., G.T. Buswell, I. Sauble. “Arithmetic We Need; Grade 6.” Ginn and Company. Boston, 1955.
Brueckner, Leo J., E.L. Merton, F.E. Grossnickle. “The New Understanding Numbers.” The John C. Winston Company. Philadelphia,1959.
Mighton, John, S. Sabourin, A. Klebanov, S. Rahbar, J. Loring. “JUMP Math 6.2; Assessment and Practice.” Toronto, Ontario, Canada. 2013.
Olney, Edward. “A Practical Arithmetic.” Sheldon & Company. New York, 1879.
Image by WoodleyWonderworks.