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doi:10.7282/T3251N0P

Working with fractions is consistently a challenge for both younger and older students. Specifically, understanding the operation of division may be harder for younger students, who could be asked to imagine splitting a piece of something into yet smaller pieces. Unfortunately, the idea of division is often introduced to students as a rule of multiplying the initial value (numerator) by the reciprocal of the second value (denominator) in the number sentence, without understanding. In the article, Rules without Reason: Allowing Students to Rethink Previous Conceptions, Mueller, Yankelewitz and Maher (2010) discuss the importance of students making connections to real world problems in building their understanding, as well as in creating models of situations to gain relational understanding of a concept. According to Skemp (1976) relational understanding of mathematics “consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point.”(13). In this analytic, a group of fourth-grade students represent situations with Cuisenaire® rods that enable the modeling of division by fractions. The students use their knowledge of building models with rods and creating multiple representations to reason about the connections between the rod models and number sentences involving fraction division.
The videos for this analytic come from the video collection stored at the Robert B. Davis Institute for Learning (RBDIL). The video data are from a 1993 session and involves a class of twenty-five 4th grade students in a suburban/rural school in New Jersey. The goal of the intervention was to promote the development of mathematical ideas that eventually lead to various forms of reasoning, which Maher and Yankelewitz (2017) report are basic goals of learning mathematics in general. The students are using Cuisenaire® rods to make sense of numerical relationships between and among these rods. Prior to the episode described in this analytic, students have had experience giving rods number names based on defining a unit rod (a rod with number name, one). They then gave names for the other rods in the set, setting up conditions for comparison of fractions.
The first three events are from one of three videos on Division with Fractions. The next two events are from two of three videos on Division with Fractions. The last event comes from three of the third of this video series.
This analytic focuses on the arguments the students make in regards to dividing by unit fractions and the use of rods and mathematical symbols to represent claims that are made verbally. Through questioning from Researcher Martino, collaboration with peers, and manipulation of rods, the students create division number sentences involving fractions to mirror the models created with their rods. The rods are used by showing how many of a certain color can “fit” or be lined up against the rod defined as the unit (whole/one) in a given task. Although some of the arguments may be incomplete in these early stages, the tasks included in this analytic invite the students to engage in and combine ideas from the Standards for Mathematical Practice as proposed by the NCTM (2000). Through these fraction tasks, students are observed to make sense of the problem, use rods to build a mathematical model, construct the beginning stages of an argument, discuss and refine potential arguments with peers, and eventually write a number sentence.
The first task asks the students to answer the question, “If a red orange train has the number name one, what would you call the white rod?” The students are also asked to model this situation with appropriate rods. Using a model of a red orange train with twelve white rods lined against it, Brian and Danielle conclude that the white rod should be called one twelfth. The students use direct reasoning by making a claim based on the given example. To take the concept a step further, the students are asked to re-write a given question using numbers instead of colors of rods and answer the question they have created. The question is, “How many whites are in a red orange train?” Through this task, a classmate (Erik) recalls the idea of equivalent fractions to support the work of his classmates. The class concludes that there are twelve, one twelfths in one or similarly there are twelve, one twelfths in twelve twelfths.
Researcher Martino probes the students to move further and write the given question/answer set as a number sentence. Through collaborative work, Danielle creates a general claim to describe the situation. She states, “It’s one divided by the fraction and then just the plain number.” Danielle may be considering the numeral in the denominator of the fraction to represent a whole number when she is using the phrase “plain number.” While her argument is incomplete, it is important to note that Danielle is using data of a few examples (one divided by one sixth equals six and one divided by one eighth equals eight) to make a general claim. Meredith comes to the same conclusion as Danielle, and Brian writes his idea as a number sentence. Finally, Michael arrives at the same conclusion as Meredith and explains his solution using a definition of division in his own words as well as the representation of the rods to model his claim. While the division of fractions can be a difficult task to understand in a relational way, through collaboration with peers, constructive questioning from the researcher and the use of appropriate mathematical tools, students at the 4th grade level were able to create number sentences describing the division of fractions in this analytic.

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