developing multiplicative thinking from additive reasoning · 2018. 1. 1. · or imagine partition...

102 September2013•teaching children mathematics | Vol. 20, No. 2 www.nctm.org

A s students progress through elementary school, they encounter mathematics concepts that shift from additive to multiplicative situations (NCTM 2000). When they encounter fraction problems that require multiplicative thinking, they tend to incorrectly extend additive properties from whole numbers (Post et al. 1985). As a result, topics such as fraction equivalence are difficult for students to understand even after formal instruction (Kamii and Clark 1995; Ni 2001). Fraction equivalence is

“generally viewed … as the ability to call the same number by different names, the ability to ignore or imagine partition lines, and/or a manifestation of flexible thought” (Kamii and Clark 1995). When generating equivalent fractions, students typically multiply or divide the numerator and denominator by the same factor. By building on additive strategies, such as unit and composite unit relationships, teachers can give students a foundation for developing an understanding of these multiplicative strategies on their own.

DevelopingMultiplicative

Thinking from

AdditiveReasoning

Using the context of restaurants and ratios to find equivalent fractions can push students’

strategies forward.

By Jennifer M. Tobias and Janet B. Andreasen

Copyright © 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

www.nctm.org Vol. 20, No. 2 | teaching children mathematics • September 2013 103

Fractions versus ratiosFractions are commonly defined as describing how many equal-size pieces are selected of the total number of equal-size pieces in a whole (Lamon 2005). For example, 3/4 means to take 3 of 4 equal pieces.

Ratios are defined as representing part-whole or part-part situations where the quantities are somehow related (Marshall 1993; Van de Walle, Karp, and Bay-Williams 2010). They are “consid-ered [to be] a comparative index rather than a

number” (Behr et al. 1983). For example, a ratio of 3/4 means that 2 quantities are compared, such as 3 boys to 4 girls. Although multiplica-tive concepts are initially difficult for students to comprehend, a—

mathematics curriculum must not wait … to advance multiplicative concepts, such as ratio and proportion. These principles must be introduced early when considering addi-tive situations. (Post et al. 1993)ra

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Additive strategiesAdditive strategies for equivalent fractions require adding or subtracting equivalent ratios. First an initial ratio is needed. Students can then use this relationship to generate other equivalent fractions. For example, consider the following problem:

For every 12 customers in a restaurant, 4 are children. How many customers are there if 20 are children?

The initial relationship given is that 4 of every 12 customers are children, which can be rep-resented as 4/12. By using 4/12, continually adding (see fig. 1) could determine that when 60 customers are present, 20 will be children.

As figure 1 shows, 4/12 is treated as a ratio in the sense that it represents 4 children of every 12 customers. Although placed in a part-whole context, 4/12 is not viewed as a quantity but rather as a relationship between customers in a restaurant. When viewing fractions as a part-whole ratio, the additive strategy looks similar to what students will do when solv-ing fraction addition situations. However, the equation represents an equivalence situation with respect to the fact that the amounts being

added generate an equivalent form of that relationship. This does not generate an answer that is the result of combining fractional amounts. For example, when adding 4/12 and 4/12 to find an equivalent ratio, the result is 8/24. When adding fractions, 4/12 plus 4/12 results in the solution of 8/12.

Using subtraction to find an equivalent rela-tionship is another form of an additive strat-egy. If an extension question were presented to the Restaurant Customers problem (finding how many are children if there are 36 custom-ers), students could use their solution of 20/60 and the relationship of 4/12 from the previous question to work down to 36 customers (see fig. 2).

Although the student’s work looks similar to fraction subtraction, the equation represents a situation showing how 20/60 and 4/12 are used to generate 12/36. As a result, all three of these ratios are equivalent, which can be illustrated by the fact that each has three times as many customers as children.

Incorrect additive strategiesWhen finding equivalent fractions, teachers often use the common saying, “Whatever you do to the top, you do to the bottom.” This statement has no meaning for students and can result in the use of incorrect additive strat-egies (see fig. 3). Moreover, students may see the relationship between 12 and 60 as being a change of 48. They may then think that the relationship between total customers and chil-dren is also a change of 48. This results in an answer that 4/12 is equivalent to 52/60. With respect to students’ thinking, they may see this as a result of doing “the same thing to the top and bottom” and think that 52/60 is equivalent to 4/12.

Similarly, students may also use an incor-rect additive relationship within the same fraction (see fig. 4). When adding the same

Using fraction equivalence to develop students’ multiplicative reasoning1. Give students a context for the numbers in a problem.2. Choose numbers carefully so that students can find a relationship

among them.3. Present students with multiple problems types so that they must

consider alternative strategies to the ones they already know.

subtraction may be used as another form of an additive strategy.

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find 20/60.

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number to both quantities (see fig. 3 and fig. 4), the ratios are not equivalent because that initial relationship is no longer preserved. For example, when 48 children are added (see fig. 3), the 48 additional customers would include only the extra children. Similarly, in figure 4, having a difference of 8 customers in both 4/12 and 52/60 would imply that every child—after the first 4—came in by himself or herself. Because each additional child would add 1 customer, the difference of 8 would mean that 52 of 60 customers are children. With the original relationship of 4 children in every group of 12 customers, this would mean that for every 4 children added, an additional 8 adults (12 customers total) are also needed.

Unit relationshipsUsing a unit relationship to generate an equiv-alent fraction requires the use of a unit frac-tion, which either may be given in a problem or may need to be derived. For example,

For every 30 customers in a restaurant, 6 were children. At this same rate, how many children would there be if there were 40 customers?

The initial relationship given is 6/30. Unlike the previous example, 6/30 cannot be continu-ally added to get the solution of 8/40. However, finding a unit relationship of 1/5 can be used to generate 8/40 (see fig. 5).

incorrect adding strategies can result from common sayings that have no meaning for students, such as, “When finding equivalent fractions, whatever you do to the top, you do to the bottom.”

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+ 48

Children 4 52

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+ 48

a difference of 8 customers in both 4/12 and 52/60 implies that every child after the first 4 came in alone. each additional child adds 1 customer, so a difference of 8 means that 52 of 60 customers are children. the original relationship of 4:12 means that for every 4 children added, an additional 8 adults (12 customers total) are added.

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Children 4 52 + 8

Customers 12 60

exploring the restaurant Customers problem offers an opportunity to employ additive strategies.

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Finding a unit relationship and using an additive strategy with the unit, students can start to formulate multiplicative ideas of dividing 6 and 30 each by 6 to find the unit 1/5.

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students may not need to go all the way down to the unit relationship of 1/5, instead using the composite unit relationship 2/10.F

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By using multiplicative reasoning, a student can see that the 5 in this case represents the number of groups of 4 and 12 that are in the 20 and 60, respectively.F

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When finding the unit relationship of 1/5 from 6/30, students may see the relationship of 1/5 as being 1 child for every 5 customers:

I was thinking if you could divide it so that there’s 6 groups so that you know there’s going to be 1 kid in each group because there’s 6 groups. And then you would wind up with 5 people in each group. And then 1 of those 5 people is the kids.

Or students could find the relationship of 1/5 as a result of using groups, but in the sense that 1 group of customers of 5 groups of customers is a group of children.

One group out of those 5 groups was kids, so that’s 1/5 of those. 1/5 is kids. So 1/5 is the whole fraction that you get. So 1/5 of all customers was kids.

In either case, students can start to formu-late multiplicative ideas of dividing 6 and 30 each by 6 to find the unit 1/5. By understand-ing 1/5 either in terms of individual customers or in terms of groups of customers, students can develop an understanding that the rela-tionship 1/5 means that 1/5 of the customers are children. This understanding can then be used to interpret 6/30 and derive the answer 8/40 in terms of the 1/5 relationship.

Composite unitsComposite unit strategies are similar to the additive and unit strategies already described. Rather than using a unit relationship, students

may use a composite of a unit. Using the same example of finding how many of 40 customers are children, students may not need to go all the way down to the unit relationship of 1/5. Instead, students can use the relationship 2/10 (see fig. 6).

By understanding 6/30 as comprising 3 equivalent groups of 2/10, students can then use 2/10 to find the solution of 8/40. This alle-viates the need to go down to the unit of 1/5, and it provides another way for students to flexibly think about the 6:30 relationship.

Multiplicative reasoningWhen students can think flexibly in terms of additive strategies, using either a given relationship or deriving a unit or composite unit, teachers can guide students to develop multiplicative reasoning. Multiplicative rela-tionships with equivalent fractions are similar to multiplicative relationships with whole


Teaching equivalenceTeachers who want to develop students’ under-standing of multiplicative reasoning with fraction equivalence must present problems that guide students toward that thinking. Giving students a context for the numbers in a problem allows them to better understand how the num-bers relate to one another. From here, students can use informal methods for solving problems, which teachers can develop into more formal, or multiplicative, ways of thinking.

When presenting problems to students, choose numbers carefully so that the children can find a relationship, just as with the first Restaurant Customers problem, 4/12 = x/60. Because 60 is a multiple of 12, this fact can be used to find the solution of 20. Although every child may not see this relationship, and some may use the incorrect strategies described, the topic can spark classroom dialogue.

numbers in that they can be derived from repeated addition. As discussed with the first Restaurant Customers problem, the relation-ship of 4/12 can be used additively to find the solution of 20/60. Additive strategies can be cumbersome in situations where the numbers are so large that it would be inefficient to use them. By using multiplication, the number of groups can be taken into account directly and represented as the factor that gets multiplied by both the numerator and denominator (see fig. 7). Rather than directly telling students that they can multiply the numerator and denominator by the same number, teachers can start with additive strategies and move students toward finding more efficient strate-gies. This can lead students to discovering and developing multiplicative relationships on their own and tying them to whole-number understandings.

the restaurant Customers problem uses the context of children in a restaurant to guide students’ informed strategies, preparing them for multiplicative thinking.

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When students discuss different strategies, they have the opportunity to distinguish between methods that do and do not work.

Third, present students with different types of problems. If the only type of problems that they see are those where numbers are mul-tiples of one another, students may not under-stand problems that do not have that relation-ship. We saw this with the second example problem of 6/30 = x/40. Because 6/30 cannot be continually added to get to 8/40, students are forced into other ways of thinking with similar situations. As a result, other methods, such as devising a unit of 1/5 or a composite unit of 2/10, were used to fi nd the solution of 8 (see fi g. 8). Multiplicative reasoning may also include the use of division. Dividing the numerator and denominator of a fraction by the same number is also a common strategy for fi nding equivalent fractions and is typically used when simplifying.

DiscussionTo understand multiplicative relationships, students must understand that multiplication is a form of repeated addition and that division is a form of repeated subtraction. Knowing that elementary school-age children initially prefer additive approaches when solving problems requiring multiplicative reasoning, teachers can guide students to develop multiplicative strate-gies by using their additive thinking as a founda-tion for understanding. By putting equivalent fractions in the context of missing-value ratio

problems, children can start to develop pro-portional reasoning strategies—which they will need to understand in middle school—while developing an understanding of equivalent frac-tion relationships.

Understanding relationships among equiva-lent fractions is important for students to be successful with such higher-level fraction topics as addition and subtraction, and such higher-level mathematics as algebraic and proportional reasoning. By using the additive understandings that students already know, teachers can extend multiplicative reasoning to fractions.

Common Core Connections

3.NF.A.3A3.NF.A.3B

REFERENCESBehr, Merlyn J., richard lesh, thomas r. Post,

and edward a. silver. 1983. “rational-Number Concepts.” in Acquisition of Mathematics Concepts and Processes, edited by richard lesh and Marsha landau, pp. 91–125. Orlando, Florida: academic Press.

Kamii, Constance, and Faye B. Clark. 1995. “equivalent Fractions: their diffi culty and educational implications.” Journal of Math-ematical Behavior 14 (4): 365–78.

lamon, susan J. 2005. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers. 2nd ed. Mahwah, NJ: lawrence erlbaum associates.

Marshall, sandra P. 1993. “assessment of rational Number Understanding: a schema-Based approach.” in Rational Numbers: An Integration of Research, edited by thomas P. Carpenter, elizabeth Fennema, and thomas a. romberg, pp. 261–88. Hillsdale, NJ: law-rence erlbaum associates.

National Council of teachers of Mathematics (NCtM). 2000. Principles and Standards for School Mathematics. reston, Va: NCtM.

Ni, Yujing. 2001. “semantic domains of rational Numbers and the acquisition of Fraction equivalence.” Contemporary Educational Psychology 26 (3): 400–17.

Multiplicative relationships with unit and composite fractions may also include the use of division.

FIg

Ur

e 8

÷6 × 8

Children 6 1 8

Customers 30 5 40

÷6 × 8

÷3 × 4

Children 6 2 8

Customers 30 10 40

÷3 × 4


Post, thomas r., Kathleen a. Cramer, Merlyn Behr, richard lesh, and Guershon Harel. 1993. “Curriculum implications of research on the learning, teaching, and assessing of rational Number Concepts.” in Rational Numbers: An Integration of Research, edited by thomas P. Carpenter, elizabeth Fennema, and thomas a. romberg, pp. 327–57. Hills-dale, NJ: lawrence erlbaum associates.

Post, thomas r., ipke Wachsmuth, richard lesh, and Merlyn J. Behr. 1985. “Order and equiva-lence of rational Numbers: a Cognitive analysis.” Journal for Research in Mathemat-ics Education 16 (1):18–36.

Van de Walle, John a., Karen s. Karp, and Jennifer M. Bay-Williams. 2010. Elementary and Middle School Mathematics: Teaching Developmentally. 7th ed. Boston: allyn and Bacon.

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INSPIRING TEACHERS. ENGAGING STUDENTS. BUILDING THE FUTURE.

Jennifer M. Tobias, jtobias@ilstu

.edu, is an assistant professor of

mathematics education at illinois

state University in Normal. she is

interested in how students develop

an understanding of fraction

concepts and operations and the

preparation of teachers. Janet B.

Andreasen, Janet.Andreasen@

ucf.edu, is an instructor of

mathematics education and the coordinator of

secondary education at the University of Central

Florida in Orlando. she is interested in mathematical

knowledge for teaching, using technology in

teaching mathematics and in training teachers,

as well as meeting the needs of all students in

mathematics classrooms.

developing multiplicative thinking from additive reasoning · 2018. 1. 1. · or imagine partition...

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