Using Manipulatives to Enhance the Teaching of Fraction Division
Karen Zwanch and Steven DeShong
Fractions are notoriously difficult for students to learn and apply, and division problems involving fractions tend to be challenging (Barlow & Drake, 2008). In particular, the meaning of fraction division tasks as either partitive (fair sharing) or quotative (measurement) adds an inherent complexity. In partitive tasks, a whole is divided into a number of groups and the quotient represents the size of the group. In comparison, quotative tasks require the division of a whole into groups of a certain size and the quotient represents the number of groups that result. This distinction adds a level of complexity to students’ understanding of fraction division, as the meaning of division influences the interpretation of the task and its solution (Fischbein, Deri, Nello & Marino, 1985). The learning of fractions, however, can be supported using manipulatives (Hackenberg, Norton, & Wright, 2016). Furthermore, although manipulatives tend to be used less in math classes by students in higher grade levels, the continued use of manipulatives is beneficial to students’ mathematical learning (Uribe-Florez & Wilkins, 2017). The purpose of this article is to demonstrate how one manipulative was used in a sixth-grade math classroom to support students’ understanding of fraction division, and more specifically, how this lesson was structured to incorporate teaching practices for “strengthening the teaching and learning of mathematics” (NCTM, 2014, p. 9).
This lesson was taught by Mr. DeShong, a preservice secondary mathematics teacher, during his middle school field experience. During this time, he was teaching 118 students who were grouped by ability into either sixth-grade math or advanced sixth-grade math classes. The manipulative—color square tiles—was chosen because of its flexibility in allowing students to create meaning and model mathematics in a variety of ways 1. The primary task used to explore fraction division was: You have 4-2/3 cups of hot chocolate powder. Each serving requires 2/3 cups hot chocolate powder. How many servings can you make? (Empson & Levi, 2011). Students were given approximately 20 minutes to work through the problem and were not given any instruction on how to use the color square tiles. Before reading about how students solved the task, try it yourself, and challenge yourself to find two strategies. Solving the task will help you engage with the students’ models and solutions as they are presented in the article, and understand how the lesson was structured around NCTM’s (2014) mathematics teaching practices.
|Figure 1a. Most common arrangement of the color square tiles to model 4-2/3 cups.||Figure 1b. Most common solution using the color square tiles to model 7 servings.||Figure 1c. Unique arrangement of the color square tiles to model 4-2/3 cups.|
|Figure 1d. Unique solution using the color square tiles to model 7 servings.||Figure 1e. A common solution error, in which students leave "leftover" hot chocolate powder (green tiles represent leftover powder).|
Figure 1. Each image shows how sixth-grade students modeled and solved the hot chocolate task using color square tiles.
Students’ models of the hot chocolate problem demonstrate the use and connection of mathematics practices (NCTM, 2014) by connecting their manipulative models to fraction division. The most common way that students used the color square tiles to model this fraction division problem was to create four groups of three tiles each, representing the four cups of hot chocolate powder. Students then commonly added an additional two tiles to their model to represent the additional 2/3 cups of hot chocolate powder. The ways that students arranged the tiles varied; the most common arrangement is shown in Figure 1a.
To construct any of the models in Figure 1, students needed to determine how many tiles should be used to represent one cup. To the teacher, it may seem obvious that students can use three tiles to represent one cup because the goal is to determine how many groups of 2/3 cups can be made; by using three tiles to represent one cup, two tiles can be used to represent 2/3 cups. Furthermore, it may be obvious to the teacher that the task is quotative in nature because students are measuring out 2/3 cups of hot chocolate powder at a time, and the quotient represents the number of groups, or cups of hot chocolate, that can be made. These determinations were cognitively demanding for students, however, as they had to coordinate the goal of modeling 4-2/3 cups of hot chocolate mix with the goal of measuring out groups of 2/3. Some groups initially tried to use one tile to represent one cup of hot chocolate mix. Some groups resolved this difficulty independently when they realized they could not represent 2/3 cups if one tile represented one cup. Other groups were prompted by Mr. DeShong to think about how they could use the tiles to “see 4-2/3 cups of hot chocolate.” Some students then tried using two tiles to represent a cup before finally deciding on three tiles to represent a cup.
“Building up” the model using one, then two, then three tiles to represent a cup was a strategy that allowed students to physically test their model. Initially, some students may require teacher probing to build a model with an appropriate number of tiles, such as that demonstrated by Mr. DeShong when he asked students if they could “see” the 4-2/3 cups in their model. The repeated opportunity to use manipulatives, however, can lead students to see that the denominator is important in deciding how many tiles to use. This can prompt discussions about using manipulatives to find common denominators to determine, for example, how many servings of hot chocolate could be made from 4-2/3 cups of powder if a serving requires 1/6 cups of powder. Discussions can also support a link between the manipulatives and the development of fraction division algorithms. In particular, quotative fraction division tasks, such as the hot chocolate task, can be used to formulate the subtractive algorithm for division. These discussions support students in building procedural fluency that is rooted in conceptual understanding, which is an effective mathematics teaching practice (NCTM, 2014).
Examining the students’ models for representing this problem demonstrate the strength of the color square tiles to students; students were able to flexibly decide how to arrange the tiles, and they could use the different colors to further support their reasoning and justification. For example, many students arranged the tiles as shown in Figure 1a. In this organization, students explained that each row of three tiles represents one cup of hot chocolate powder and that the final row of two tiles represents 2/3 cups of hot chocolate powder. To some students, the colors of the tiles they selected did not mean anything in terms of the model, but to other students, each cup of hot chocolate was represented by a different color. This allowed them to visually keep track of each cup of hot chocolate mix. Using different colored tiles for each cup also allowed some students to communicate and justify their solution strategies to others. By supporting students’ communication, the color square tiles allowed the students to describe and justify “their mathematical understanding and reasoning with ... representations” (NCTM, 2014, p. 29), which demonstrates the use and connection of mathematical representations (NCTM, 2014).
Students used the different colors of the tiles and the flexibility of separating and rearranging the tiles to solve the hot chocolate task. Their solutions are evidence of the success of the lesson in implementing a task that promotes reasoning and problem solving (NCTM, 2014). Each of the students’ solutions represent the quotative meaning of division, in which 4-2/3 cups of hot chocolate powder is divided into groups of a predetermined amount, 2/3 cups, and the solution, 7 cups of hot chocolate, represents the number of groups that can be formed. The most common solution is shown in Figure 1b. Many of these students rearranged the 14 tiles into groups of two tiles each to determine that 4-2/3 cups of hot chocolate powder can be used to make 7 servings of hot chocolate. Other students did not rearrange the color square tiles, but instead placed a second layer of color square tiles on top of the existing arrangement in groups of two. For example, two green tiles represented one serving, two red tiles represented a second serving, etc. The color coordination and regrouping of tiles allowed students flexibility in their thinking and helped all students to engage with the problem.
One group arranged their tiles in a unique way (Figure 1c). Although this arrangement may not initially seem distinct from the model in Figure 1a, the student explained his model in a unique way. In this model, the red tiles in the center represent the 2/3 cups of hot chocolate powder, and the groups of three tiles surrounding it represent the four cups of hot chocolate powder. This allowed the student to more easily see both the groups of three tiles representing one cup of powder each and the group of two tiles representing one serving of hot chocolate (Figure 1d). Each of these solutions demonstrate the students’ engagement in problem solving throughout the hot chocolate task.
All of the models and solutions shown in Figure 1 are examples of direct modeling, which is a representation of the problem that reflects its quantitative structure (Kent, Empson, & Nielsen, 2015). Although direct modeling can also be accomplished with drawings, the color square tiles and other manipulatives have advantages. First, students can more flexibly arrange manipulatives to model 4-2/3 cups using a shape or color combination that aligns with their thinking (Figure 2 shows possible arrangements using different manipulatives). Second, students who do not yet anticipate the need to partition the whole into thirds to facilitate the direct modeling can build up their representation to determine how many tiles can be used to model one cup. Additionally, students can use the color and movement of manipulatives to facilitate the communication of their thinking to others.
Although the color square tiles were made available to all students, not all students chose to use them. Some groups used their existing fraction knowledge to solve the hot chocolate problem with either repeated subtraction (beginning with 4-2/3 and subtracting 2/3, seven times) or repeated addition (beginning with 2/3 and adding 2/3 seven times, resulting in 4-2/3; see Kent, Empson, & Nielsen, 2015 for a discussion). Solving the task using these methods is equally valid, and mathematically equivalent, in comparison to modeling the problem with color square tiles. As problems become more complex, however, using manipulatives may facilitate the thinking of students who initially choose not to use them. For example, consider a similar hot chocolate task, in which there are initially 5 cups of hot chocolate powder. To measure out servings of 2/3 cups now results in 7 servings of hot chocolate, with a remainder of 1/3 cup of hot chocolate powder. In terms of servings of hot chocolate, however, the remainder is 1/2 of a serving. This distinction between 1/3 cups of hot chocolate powder and 1/2 of a serving of hot chocolate can be difficult, and using manipulatives will help students interpret the remainder. This type of extension of the task in future lessons is another example of how procedural fluency can be built from conceptual understanding (NCTM, 2014).
Figure 2. Other manipulatives that can flexibly be used by students to model the hot chocolate task. Each can support students reasoning similarly to the color square tiles.
What About the Leftovers?
Even with the support of manipulatives, one mistake emerged in a few students’ solutions to the hot chocolate problem. This mistake is shared for two purposes. First, to illuminate a common error that teachers might encounter in implementing the hot chocolate problem. Second, this mistake is shared as an example of Mr. DeShong’s use of questions that supported a productive struggle (NCTM, 2014). There were multiple opportunities for the support of productive struggle throughout this lesson, but this brief exchange is offered as an exemplar.
The mistake stemmed from students being unsure of what to do with “leftover” hot chocolate powder. These students arranged their tiles in the same way as many others (e.g., Figure 1a), but when trying to determine how many servings of hot chocolate could be made, they believed the solution to be five servings; the five servings are represented in Figure 1e by the five groups of two yellow tiles. Students making this mistake commonly explained that they had grouped two color square tiles from each “cup” of hot chocolate powder, and that the third tile was “leftover” or “extra.” The leftover hot chocolate powder is represented in Figure 1e by the four green tiles that are left ungrouped. Mr. DeShong encouraged the students who had leftover hot chocolate powder to use all of the hot chocolate powder or asked them if he could have the leftover powder to make some hot chocolate for himself. By questioning students rather than correcting them, they were afforded the opportunity to continue making progress on the task. Simple questions, like those asked by Mr. DeShong, were sufficient to help students see the need to use all of the hot chocolate powder, and to propel the groups’ discussions closer to a solution, thereby typifying the support of productive struggle (NCTM, 2014).
This problem has been shown in previous research (Kent, Empson, & Nielsen, 2015) to help students construct a deeper understanding of fractions as quantities and of fraction division. The use of manipulatives, however, can make the hot chocolate task and other similar fraction tasks accessible to students at varying levels of mathematical thinking. Manipulatives allow students to represent problems in non-standard ways that enhance traditional fractional representations, allow students to build up their models if they are not initially sure how to represent each cup of hot chocolate powder, and allow students to use color to reinforce their own understanding and to communicate their understanding to others. In addition to facilitating helpful models of fraction division, using manipulatives to model this type of problem can also help students make meaning from the results of fraction division, particularly as it relates to the subtractive algorithm for division or to interpreting remainders in fraction division problems. Manipulatives may additionally help students overcome the common misconception that division makes smaller (Graeber & Campbell, 1993) by introducing students to meaningful situations in which dividing by a number between zero and one results in a larger quotient. What these advantages have in common is that they all reinforce meaningful interpretations of traditionally difficult mathematical concepts for elementary and middle grades students by supporting the connection of mathematical representations (NCTM, 2014).
In this lesson, Mr. DeShong used the mathematical goal of exploring fraction division to inform his selection of the hot chocolate task and the use of color square tiles to support the students’ exploration. The students’ models, solutions, and mistakes provide an exemplar for the use of manipulatives in middle grades mathematics classes, and particularly in helping students learn fraction division. During the lesson, students were encouraged to use and connect mathematical representations, to reason and problem solve, and to engage in a productive struggle. Each of these qualities is a demonstration of one of NCTM’s (2014) mathematical teaching practices, which led to the lesson’s overall success. In reflecting upon the lesson, there are several points that could be expanded upon in future lessons to further students’ learning of fraction division. For example, using manipulatives to illustrate the importance of common denominators in fraction division, modeling partitive division with manipulatives, using manipulatives to support the interpretation of the remainder in fraction division tasks, and generating algorithms for fraction division. Such mathematical goals could be equally well supported in a middle grades math class by the use of manipulatives and by the mathematics teaching practices shown in this lesson.
1Other manipulatives could also be flexibly used on this task, but Mr. DeShong chose to offer only one manipulative to the students to streamline their solutions and to ease the comparison between solutions. The possibility of modeling this task using other manipulatives is included with students’ solutions.
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Frank W. Cox High School