The development of students’ conceptual knowledge through modelling fractions.

PurposesLesson activity; Student collaboration; Student engagement; Student model building; Reasoning; Representation
DescriptionIn the 21st century, there is a significant change in mathematics education. There is a focus on the conceptual teaching of mathematics that has replaced the procedural teaching methodology. Currently, teachers employ the use of mathematical activities and tasks to help students discover new ideas on their own. Research has shown that students in a conceptually oriented mathematics class outperform students in a procedural oriented mathematics class on tests and on measures of attitude towards mathematics. (see, for example, Boaler, 1998; Cain, 2002; Fuson et al, 2000; Masden and Lanier, 1992; Schwartz, 2008).
In this Analytic, there is attention on analyzing critically how students modeled fractions using Cuisenaire rods. As a result, they discovered, and expressed symbolic representation of fraction models. They assigned different number names to different rods while observing the ground rule of identifying a whole unit that was given the number name, one.
An objective of teaching mathematics is to help students model mathematical situations as they endeavor to solve routine and non-routine problems. This discovery-based learning leads to the acquisition of conceptual knowledge. Application of certain rote rules for writing symbols on paper in very specific ways is often meaningless to the student. Rather, according to Davis (1990), students should build up powerful symbol systems in their minds that allow the acquisition of creating and using mental representations to model mathematical situations.
Prior to the episodes described in this analytic, Researcher Carolyn Maher demonstrated an actual situation (Candy bar metaphor) where she used chocolate candy bars to help students contextualize and understand the concept of fair sharing when comparing fractions. In this scenario, students had the opportunity to discover that half of a smaller candy bar can be smaller than the corresponding half of a larger candy bar. The students, then, all agreed that one should use the same candy bar, or refer to the same whole unit when comparing fractions.
In this narrative, we see students use previously acquired experiences (ground rule for specifying the unit) from the Candy Bar metaphor demonstrated by Researcher Maher, to build new ideas as they model fractions problems. In conceptual learning, students link and assimilate new and existing schemas resulting to a better understanding of mathematics. Davis (1992) writes, “Students are determined to understand, and they create their own ways of understanding. What they learn thereafter is built upon this foundation of previously built-up understanding” (p 226).
The videos used in this analytic were recorded on October 4, 1993, in a fourth-grade class with Researcher, Carolyn Maher and classroom teacher, Joan Phillips. In this session, Researcher Maher posed the following problem to students: Which is larger, one-half or two-thirds and by how much? She invited the students to use Cuisenaire rods to build as many models as they could to represent their solutions.
Students use Cuisenaire rods to build models to represent the fractions one half and two thirds and identify the rod that was given the number name, one, and then find the difference. Researcher Maher observed how the students came up with several models, and asked them to explain how they built them, as well as what their conclusion was, and how it compared to the models built by the rest of their peers. One of the outstanding features of this relational learning is that students provided arguments through their explanations for every mathematical action taken in a problem-solving process.
There are six events in this analytic. Students are actively building fraction models, comparing them, and coming up with various arguments in an endeavor to support and justify their solutions. They built four different models to support their claim that two thirds is greater than one half by one sixth. They model the problem situation, construct, and justify their conclusions providing evidence using Cuisenaire rod models, indicating a high level of understanding of the problem solution.

Erik and Alan’s model

In clip 1, students are actively involved in building various models. Researcher Maher asked students to work in groups and, use Cuisenaire rods to illustrate and compare the difference of one half and two thirds. Erik and Alan are representing their ideas using the rods. They used four rods of different colors; dark green, light green, red, and white. Erik gave the dark-green rod the number name number one, the green rod, number name one half, the red rod, number name, one third, and the white rod, number name, one sixth. Erik explains to Researcher Maher that, two red rods (that’s two thirds since one red rod is one third) are longer in length than a light green rod (one half) by a length equal to a white rod (one sixth). They convince Researcher Maher that two thirds is greater than one half by one sixth (a number name given to a white rod). Researcher Maher asks them to find out if they can use another model to represent the problem.

Danielle’s model

In clip 2, Danielle came up with a different model than Erik and Alan by using a train of orange and brown rods (a whole unit), two blue rods, three green rods and eighteen white rods. She then compares the length of one blue rod (given number name one half) and two green rods (given number name, two thirds). She concludes that two green rods are longer than the blue rod by three white rods (each with number name, one sixth), and three out of eighteen is equal to one sixth. This means that two thirds is greater than one half by one sixth. According to Maher & Weber (2010), students may have different representations of the problem solution since there are often multiple ways of expressing and representing ideas in the context of a collaborative and supportive environment. Children, naturally, have wonderful ideas, and they are empowered when they can express them.

How much is two thirds bigger?

Researcher Alston challenges Danielle to come up with another model. She builds a model using the dark green rod to represent a whole unit, two light green rods each to represent one half, three red rods each representing one third, and six white rods each representing one sixth. She compares two red rods (two thirds) and one light green rod (one half), and realizes that a train of two red rods (train rod number name, two thirds) is larger than a light green rod by one white rod (number name, one sixth), determining that two thirds is greater than one half by one sixth (the length of a white rod).

Are one sixth, two twelfths and three eighteenths different?

Researcher Alston asks Danielle and Gregory if they can use a different train apart from orange and a brown rod. They decide to use orange and a red rod to make a train (whole unit). In this model, they also use green rods, purple rods, and white ones. A green rod given number name, one half. A purple rod given number name, one third. A white rod with a number name, one twelfth. Therefore, two purple rod has a number name, two thirds. They compare a green rod (one half) with a train of two purple rods (two thirds). They realize that two purple rods (two thirds) is longer than a green rod (one half) by two white rods. Danielle says to researcher Alston that two thirds is bigger than one half by two twelfths. However, she argues that one sixth, three eighteenth and two twelfths are the same but different. She says they are different because they have different number of white rods, and same because they all represent the difference between two thirds and one half.

Caitlin’s model

In clip 5, Researcher Maher asks Caitlin to use the model she had built and clarify which is bigger, one half or two thirds. Caitlin made a model consisting of a train of a red and orange rod and gave it the number name one. She gave each of two green rods the number name one half and each of three purple rods, the number name, one third. When the researcher asked Caitlin to point out the rod with the number name one, she indicated a train of an orange and a red rod. This shows understanding for defining a whole unit. She pointed out that the number name for a dark-green rod is one half and that of a purple rod is one third. Researcher Maher asks for a rod with number name, two thirds. Caitlin picks two purple rods, combines them to make a train of two purple rods. She says to Researcher Maher that the number name for a dark-green rod is one half. Researcher Maher puts two purple rods along the dark-green rod and asks, “Can you tell me which is bigger?” Caitlin asserts that, two thirds is bigger than one half, although she did not illustrate by how much. Researcher Maher asks Caitlin to find out how many red rods bigger is two thirds than one half, and leaves her to work on it.

Researcher Maher asks Brian and Michael what they are doing because they seem busy. They reply that they have built five different models to represent the problem. Researcher Maher asks Brian and Michael, “Hey gentlemen, you are so busy, what are you doing? How many models did you make?” Michael responds,” five” and the students count all the models they made to confirm. The two boys assert that the difference between two thirds and one half is one sixth. Researcher Maher asks, “Did the relationship hold?” Michael responds, “Yes! the difference is one sixth.” Researcher Maher asks if they expected that to happen. Brian and Michael reply, “Yes!” The researcher then asks if they can convince everybody what has happened. They agree and Researcher Maher gives them a piece of paper to record their models.

Jackie explains her models

In clip 6, Researcher Maher asks Jackie to explain her models. She has constructed two models. She gives a train of red and orange rod the number name, one; gives the purple rod, a number name, one third; the dark-green rod, a number name, one half; and a red rod, a number name, one sixth. Researcher Maher asks what rod has number name, one third. Jackie responds by showing a purple rod while pointing at it with a finger. Since a purple rod has a number name, one third, Jackie asserts that, two purple rods joined together to make a train, has a number name, two thirds. She tells researcher Maher that a dark-green rod has a number name, one half, and concludes that two thirds is bigger than one half. Researcher Maher asks,” By how much? Jackie responds, “one seventh.” Researcher Maher asks, “Why? Do you have a rod with number name one seventh?” Researcher Maher then counts the red rods and finds six. Jackie corrects herself. She interjects and says that a red rod represents one sixth of the whole unit.
Jackie made another model consisting of a dark-green rod (number name, one) and two light-green rods. She gives a light-green rod, a number name, one-half. She has a train of three red rods and she gives one red rod, a number name, one third. The researcher asks her why a red rod has different number names in the two models. She asserts that number names of a red rod are different because the two models have different whole units.


1. Davis, R. B. (1990). Chapter 7: Discovery learning and constructivism. Journal for Research in Mathematics Education. Monograph, 4, 93-210.

2. Davis, R. B. (1992). Understanding ‘Understanding.’ Journal of Mathematical Behavior 11, 225-241.

3. Van Ness, C. K., & Alston, A. S. (2017). Establishing the Importance of the Unit. In Children’s Reasoning While Building Fraction Ideas (pp. 49-64). Sense Publisher

4. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

5. Schwartz, J. E. (2008). Elementary mathematics pedagogical content knowledge: Powerful ideas for teachers. Prentice Hall.

6. Maher, C. A., & Weber, K. (2010). Representation systems and constructing conceptual understanding. Special Issue of the Mediterranean Journal for Research in Mathematics Education, 9(1), 91-106.
Created on2016-11-25T10:28:59-0400
Published on2017-10-18T08:54:19-0400
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