*By Dawn Woods, RME Elementary Mathematics Coordinator*

Many students find fraction concepts difficult to understand yet the understanding of fractions is essential for learning algebra and advanced mathematics (National Mathematics Advisory Panel, 2008). As an elementary mathematics educator, I noticed that many of my students struggled with fraction concepts across the curriculum. I wondered how I could help my students experience and understand fraction concepts conceptually so they could succeed not only in my classroom but also in advanced mathematics.

My search for answers began with research. I discovered that to understand fractions means to recognize the multiple meanings and interpretations of fractions. Furthermore, I needed to explicitly present these different constructs in a contextual way to build understanding. Mathematics educators generally agree that there are five main fraction constructs and that they are developmental in nature.

**The first construct presents fractions as parts of wholes or parts of sets.**Research suggests that this construct is an effective starting point for building fractions (Cramer & Whitney, 2010). However, it is important to realize that the part-whole relationship goes way beyond the shading of a region. For example, it could be part of a group of animals such as (¼ of the animals are dogs), or be part of a length, (we ran 1 ½ miles) (Van De Walle, Karp, Bay-Williams, 2013).

Researchers such as Cramer, Wyberg, and Leavitt suggest that the fraction circle manipulative is a powerful concrete representation since it helps to build understanding of the part-whole relationship as wells as the meaning of the relative size of fractions (2008). Here, they use fraction circle models to help build mental images that aid in the ability to judge relative sizes of fractions. It is also important to remember that the fractional parts do not need to be identical in shape and size, but must be equivalent in some other attribute such as area, volume, or number (Chapin & Johnson, 2006). However, it is important to teach beyond this first construct to include other fraction representations and models.

**The second construct presents fractions as measures.**Measurement (Van De Walle, Karp, Bay- Williams, 2013) involves identifying a length and then uses that length to determine a length of an object. The number line plays an important role in this construct by partitioning units into as many subunits that one is willing to create (Chapin & Johnson, 2006). For example, in the fraction ¾, you can use the unit fraction ¼ as the selected length and then measure to show that it takes three of those to reach ¾ (Van De Walle, Karp, Bay-Williams, 2013). Research suggests that students who develop an initial understanding of rational numbers as measures, develop ideas of unit, partitioning, order, addition and subtraction (Cramer & Whitney, 2010) while using the number line as a model. Essentially, this powerful construct illustrates that there are an infinite number of rational numbers on the number line as it focuses on how much rather than parts of a whole.

**Fractions can also result from dividing two numbers.**This construct is often called the quotient meaning, since the quotient is the answer to a division problem (Chapin & Johnson, 2006). Think about the number of cookies each person receives when 15 cookies are shared between 3 people. This problem is not a part-whole scenario (Van De Walle, Karp, Bay-Williams, 2013) but it still means that each person will receive one-third of the cookies expressed as

^{15}⁄

_{3},

^{5}⁄

_{1}, or 5. Connecting division to fractions enables students to feel comfortable with seeing division expressed in multiple ways such as 16 ÷ 3,

^{16}⁄

_{3}, and 5

^{1}⁄

_{3}and is important for continued success in advanced mathematics.

**The fourth construct presents fractions as operators.**In this construct, a fraction is a number that acts (or operates) on another number to stretch or shrink the magnitude of the number (Chapin & Johnson, 2006). For an example, a model of a car may be 1/16 the size of the original or a cell maybe magnified under a microscope to 400 times the actual size demonstrating a multiplicative relationship between the quantities. This construct takes fractions beyond representation to a place where students know how to use fractions to solve problems across the curriculum.

**The fifth and final construct characterizes fractions as the ratio or comparison of two quantities.**A ratio such as 1/3 can mean that the probably of an event is one in three (Van De Walle, Karp, Bay-Williams, 2013). Or a ratio can also represent part-whole relationships such as 11 children at the park compared to the total number of 18 people. We could write this part-to-whole relationship as the fraction

^{11}⁄

_{18}. However, it is important that realize that all fractions are ratios but not all ratios are fractions (Chapin & Johnson, 2006). Part-to-part comparisons such as the number of children to the number of people at the park, 11:18, is not a fraction because this comparison does not name a rational number but presents a comparison of two numbers.

Fraction understanding, although a challenge to students, is a critical mathematics concept. For students to really understand fractions, they need to experience fractions across all five constructs in meaningful ways that build conceptual understanding. This conceptual understanding, in turn, provides students with mental representations that enable students to connect meaning to fractions across a variety of contexts.

Chapin, S.H., & Johnson, A. (2006). Math matters, 2nd edition. Sausalito: Math Solutions Publications.

Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D.V. Lambdin & F.K. Lester, Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 11-22). Reston, VA: NCTM

Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School 13(8), 490-496.

National Mathematics Advisory Panel (2008). The final report of the national mathematics advisory panel. Jessup, MD: Education Publications Center. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

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