V.6 #1 - Mathematics - Toward a Deeper Understanding: Strategies to Foster Multiple Meanings of Frac
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Knowledge of fractions is such an integral part of the elementary school mathematics curriculum as it provides a foundation for learning measurement, ratio and proportion, percents, probability, and algebra in secondary school mathematics (Clarke, Roche, & Mitchell, 2008; Son, 2011). However, many students have difficulty developing conceptual understanding of fractions. Jones (2012) and Van de Walle, Karp, and Bay-Williams (2010) detail several reasons why fractions are difficult to understand:
Fraction symbolism is very different from whole number symbolism. In fractional notation, the numerator and denominator are separated by a horizontal line but represent one quantity and not separate values.
It is difficult to compare the size of fractions. Students can compare whole numbers easily on a number line once they have learned to count meaningfully, but comparing fractions on a number line, for example, requires complex understanding.
The rules for operations with fractions are different than for whole numbers. For example, students can add 1+2 to get 3 but cannot add 1/2 + 2/3 to get 3/5.
There are more rules for operations with fractions than for whole numbers and the rules appear contradictory. For example, why can’t you add 1/2 and 2/3 to get 3/5 but you can multiply 1/2 and 2/3 to get 3/6? In the latter, it’s okay to multiply numerators and denominators straight across.
Students mistakenly believe that parts of the whole do not have to be equal.
Meanings of Fractions
In order to develop a deeper understanding of fractions, it is important to understand all the possible concepts that fractions can represent. In the United States, the most common meaning, and the one taught first, is part-whole. However, there are four meanings that students should understand: part-whole, measure, quotient, and ratio (Jones, 2012). Many who research fraction understanding believe students should understand fractions more deeply with greater emphasis across all meanings of fractions (Siebert & Gaskin, 2006). This article describes the four meanings and strategies teachers can employ to help students gain greater understanding of fractions.
Part-whole. One meaning of fractions is part-whole where a fraction represents a quantity that is partitioned or divided into equal pieces. Imagine that Figure 1 is a pizza which is divided into eight equal slices. Two equal slices eaten (part) out of eight equal slices (the whole) is represented by 2/8.
Figure 1. Part-whole meaning of fractions.
Measure. Fractions can also be represented as a comparison of lengths. A length is identified and then is used as the unit for comparison. In Figure 2, the fraction 2/8 can be measured relative to the length of the whole or can be considered as two copies of the length 1/8.
Figure 2. Fraction represented as a measure. Quotient. A fraction can represent the division of two numbers. If I had two cookies and eight people equally sharing the cookies, each person would receive 2 ÷ 8 or 2/8 of a cookie. Ratio. A fraction can compare two sets or measurements. Ratios can be part-part or part-whole. Let’s say there are two boys for every eight girls in a third grade class, the ratio of boys (part) to girls (part) would be 2:8 which can be represented as 2/8. Let’s say that in a fourth grade class two boys have striped shirts (part) out of a class of eight students (whole), the ratio can be written as 2/8. Models for Understanding Fractions
There are several models to assist students in understanding the meanings of fractions. Different models provide different opportunities to learn; when instructing students about fractions, encourage them to use models for as long as necessary. Eventually, they may be able to use mental models of fractions instead of concrete or pictorial models. Three types of models discussed in this article are: area/region model, length/measurement model, and set model.
Area/region model. Area or region models help students to visualize part of a whole. Some commonly used manipulatives to assist students with this model are:
Circular fraction pieces (Figure 3). These “fraction circles” allow students to see the size of parts relative to the whole. Students can stack the pieces on top of one another to compare sizes of fractions. They can also be used to teach equivalent fractions and basic addition and subtraction of fractions.
Figure 3. Circular fraction pieces.
Paper folding. Paper is a great tool for students to begin to explore part-whole relationships. One benefit of using paper is that it is generally readily available in schools unlike some of the more “fancy” manipulatives. Have students fold pieces of paper to create different fractions (e.g., halves, thirds, fourths, eights). Pattern blocks. Extend this familiar manipulative to fractions. For example, have students explore how many green triangles cover one yellow hexagon. If they find out that it takes six green triangles to cover one yellow hexagon then one triangle represents 1/6 of the hexagon. Food. Common snacks such as apples, graham crackers, and cookies can be used to showcase parts of a whole. Food can also be a nice reward for demonstrating understanding of area model of fractions. Length/measurement model. With the length or measurement models, lengths or measurements are compared instead of areas. Two common manipulatives to assist students with this model are:
Cuisenaire rods. Cuisenaire rods (Figure 4) are a collection of rectangular rods in 10 lengths and 10 colors, each color corresponding to a different length. These manipulatives are versatile because any one of the 10 Cuisenaire rods can be designated as the whole or unit. For example, in relation to the orange rod (whole) the white rod represents 1/10 of the length. However, in relation to the red rod (whole), the white rod is 1/2 of the length. These manipulatives are also helpful for beginning addition and subtraction of fractions.
Figure 4. Cuisenaire rods. Note. Google Image from www.mathcats.com. Fraction strips (Figure 5). Students can create their own strips with whatever fractions a teacher wants to showcase or compare. Have students create and color code strips to highlight different fractions. Once created, students can save them in folders or plastic bags for future reference. Fraction strips are also a great tool for beginning concepts of equivalent fractions much like the fraction circles but in a linear format.
Figure 5. Fraction strips. Set model. In this model, the whole is a set of objects and subsets of the whole make up fractional parts. This model is more complicated and should not be taught until students develop understanding of the area/region and length/measurement models (Jones, 2012). In Figure 6, let’s say I have eight cookies (the set) and the two shaded circles represent chocolate chip cookies; the fraction of chocolate chip cookies is 2/8 of the total set. Encourage students to break up the set into the amount of equal groups (or parts) shown by the denominator and then take/shade in (depending on the problem’s context) the amount of those equal groups as represented by the numerator. For example, in Figure 7, if students are asked to find 2/8 of a set of 16 cookies they could split the set into eight equal groups and take two groups which, in this case, would yield four cookies. Counters in two colors on opposite sides are frequently used for set models but other tangibles such as unifix cubes, coins, cookies or whatever is readily available in a classroom can be used.
Importance of Understanding Multiple Meanings of Fractions
This article showcased four meanings of fractions and three models which can be used to assist students with grasping these meanings. Students must be able to explore fractions across models. A teacher will not be able to adequately assess student understanding unless students model fractions in different contexts. As an assessment tool after all models have been taught, I have students complete a Fraction Models Chart (Figure 8) to show they understand fractions in multiple meanings. I either give students a fraction or have them choose their own (or both). Students then draw pictures and write accompanying sentences to express the fraction across different models. Both the pictorial and written formats allow me to greater assess their conceptual understanding of the meanings of fractions.
Figure 8. Fraction models chart. Having students explore various meanings of fractions will hopefully alleviate some of the confusion surrounding fractions and avoid the often dreaded sigh or moan from students when faced with fraction tasks. Armed with a deeper understanding of fractions should set students on more positive journeys to work with ratios, proportions, percents, probability, and algebra in more advanced mathematics work. References Clarke, D. M., Roche, A., & Mitchell, A. (2008). Ten practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School, 13(7), 373-380.
Jones, J. C. (2012). Visualizing elementary and middle school mathematics methods. Hoboken, NJ: John Wiley & Sons, Inc.
Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
Son, J. W. (2011). A global look at math instruction. Teaching Children Mathematics, 17(6), 360-370.
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Allyn & Bacon.
Joan Gujarati, Ed.D., is an Assistant Professor in the Department of Curriculum and Instruction at Manhattanville College in Purchase, New York where she teaches the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujarati’s research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at Joan.Gujarati@mville.edu