### Fourth graders’ argumentation about the density of fractions between 0 and 1

PurposesEffective teaching; Homework activity; Lesson activity; Professional development activity; Reasoning; Representation
DescriptionResearch has shown that, through argumentation, even young children can build proof-like forms of argument (Maher & Martino, 1996a, 1996b; Styliandes, 2006; 2007; Ball, 1993; Ball et al., 2002; Maher, 2009; Maher & Martino, 1996; Yankelewitz, 2009; Yankelewitz, Mueller, & Maher, 2010). The purpose of this VMCAnalytic is to illustrate events of students involved in argumentation during which students are interacting with each other and a researcher.

The view of argumentation presented in this VMCAnalytic is consistent with Practice 3 of the Common Core Standards for Mathematical Practice that indicates that students should be able to build arguments, critique the arguments of others, and build justifications for their conclusions (CCSSMP 3, CCSS, 2010). The events show forms of argument that naturally occur in a student’s mathematical arguments.

In this VMCAnalytic, fourth grade students are arguing about whether or not infinitely many fractions can be placed between 0 and 1 on the number line. The events depicted in this VMCAnalytic were taken from a larger data set gathered as a result of a year-long research intervention involving fourth graders exploration of fractions. This research was supported by a grant from the National Science Foundation: MDR-9053597 directed by R. B. Davis, C. A. Maher.

The events depicted here occurred on November 3, 1993 and are taken from the third month of these sessions (Schmeelk, 2010). This session is the 14th session of 17 sessions of the fraction intervention (Yankelewitz, 2009). In the sessions leading up to the session from which the events in this VMCAnalytic are taken, students had explored various problems involving whole numbers and fractions. These tasks began with comparing relationships among the Cuisenaire rods generally and included finding the number names for the different Cuisenaire rods given that a particular rod or train of rods was given the number name 1. Students also used the rods to compare fractions and find equivalent fractions. The context of some of the problems involved sharing a candy bar equally. It is important to note that, other than this research intervention, students in this VMCAnalytic had no previous instruction on operations with fractions (Schmeelk, 2010).

The events in this VMCAnalytic are taken from the third of the four clips that capture the entire session on November 3rd. In Clips 1 and 2 of this session, students discussed number lines, the relationships among Cuisenaire rods, a ruler, a number line, the idea of infinitely many, and how they might plot positive and negative integers and fractions on a number line. If viewers are interested in a more detailed picture of how students developed these ideas, Clips 1 and 2 are available for viewing on the Video Mosaic Collaborative Repository at vidoemosaic.org (See “The infinite number line, Clip 1 of 4: Naming points on the number line” and “The infinite number line, Clip 2 of 4: Placing integers on the number line”).

In the third clip and just prior to the events in this VMCAnalytic, students worked on a task that involved placing unit fractions (1/2, 1/3, 1/4, and so on, through 1/10) on a segment of a number line labeled from 0 to 1. Each student was asked to construct his or her individual number line with points to indicate where the fractions should be. This VMCAnalytic shows students’ argumentation about whether or not infinitely many fractions can be placed between 0 and 1 on the number line. Claims, counter claims, warrants, justification, backing, qualifiers, and justifications are elements of argumentation that are evident in the student discourse. The clip, in its entirety, can be found at www.videomosaic.org by accessing the “The infinite number line, Clip 3 of 4: How many numbers between 0 and 1?” clip.

The events in this VMCAnalytic were prompted by Researcher Maher’s statement to the class that mathematicians claim that there are infinitely many fractions between 0 and 1. In the first event, Erik expresses doubt that this claim is true. In the second event, Alan supports the original claim made by mathematicians by making a claim of his own, namely that you can divide the distance on the number line between 0 and 1 into very small parts, even into zillionths.

In the following events, students debate this claim. Erik challenges this claim and makes a counter claim. Michael qualifies his original claim when he decides that Alan’s claim can be true. Alan supports his claim through the illustration of a very small quantity, a dust particle.

In Event 5, Andrew uses a warrant to further support Alan’s claim by introducing the idea of using a microscope as a tool. Andrew asserts that if you put a number line under a microscope, it would look like you had “enough room” to put smaller and smaller fractions between 0 and 1.

Erik challenges the idea that when you use a microscope you get more space on the number line. The argumentation then focuses on whether using a microscope gets you more space on the number line, or just enables you to see the space that is already on the number line. Several students make claims and provide support for their positions. In Events 8 and 9, Erik questions Alan to clarify his position about the claim that there is more space than the naked eye can see. Research Maher then summarizes the various students’ positions about this idea.

In Event 10, David provides a warrant to support the idea that the microscope enables you to see more on the number line without gaining more space. In Events 11 and 12, several students express their agreement or disagreement with the claims that have been made and give support for their positions.

In Events 13 – 15, Alan, at the overhead, draws a diagram of a magnified number line to provide backing for the claims that he has made. He argues that if you could magnify the very small space between the 0 and the 1/100 on the number line, it would become apparent that the resulting space could be partitioned into smaller and smaller fractions. As he explains his position, he refines his argument, including the claim, the justification, and the evidence. In Event 15, Alan modifies his claim as he asserts that if you could further magnify the spaces between the tick marks between 0 and 1/100, you could continue to partition those spaces up into little spaces.

In Event 17, other students state their agreement with Alan’s claim and give justifications for their positions. Meredith synthesizes the discussion when she says, “I think what he is trying to say is that if you look at it through the microscope then there is a lot of space but if you just look at it through the human eye then there isn’t very much space in there.”

The events in this VMCAnalytic were chosen purposefully to illustrate these fourth-grade students’ argumentation about plotting infinitely many fractions on a segment of a number line. Their exploration of fractions goes beyond what is represented here. If viewers are interested in a more detailed picture of how students developed these ideas, the complete set of clips from November 3rd are available for viewing on the Video Mosaic Collaborative Repository at vidoemosaic.org. These clips include Clips 1, 2, and 3 that were mentioned previously, as well as Clip 4: “The infinite number line, Clip 4 of 4: Placing fractions and mixed numbers on the number line”. Additionally, videos from the larger data set gathered throughout the year of this research intervention are also available videomosaic.org.

References:
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The elementary school journal, 373–397.

Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians, (Vol. III, pp. 907-920). Beijing: Higher Education Press.

Maher, C. Children’s Reasoning: Discovering the Idea of Mathematical Proof. In Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (Eds.). (2009). Teaching and learning proof across the grades: A K-16 perspective. Routledge.

Maher, C., & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. JRME, 27 (2), 194-214.

Maher, C. A., & Martino, A. M. (1996a). The development of the idea of mathematical proof: a 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.

Maher, C. A., & Martino, A. M. (1996b). Young children invent methods of proof: the gang of four. In: P. Nesher, L. P. Steffe, P. Cobb, B. Greer, & J. Golden (Eds.), Theories of mathematical learning (pp. 431–447). Mahwah, NJ: Lawrence Erlbaum Associates.

National Governors Association Center for Best Practices, C. o. (2010). Common Core State Standards for Mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C.

Schmeelk, S. E. (2010). Tracing students’ growing understanding of rational numbers (Doctoral dissertation, Rutgers University-Graduate School of Education).

Stylianides, A. J. (2006). The Notion of Proof in the Context of Elementary School Mathematics. Educational Studies in Mathematics, 65(1), 1–20.

Stylianides, A. J. (2007) Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321.

Yankelewitz, D. (2009). The Development of Mathematical Reasoning in Elementary School Students’ Exploration of Fraction Ideas (Doctoral dissertation, Rutgers, The State University of New Jersey).

Yankelewitz, D., Mueller, M., & Maher, C. A. (2010). A task that elicits reasoning: A dual analysis. The Journal of Mathematical Behavior, 29(2), 76-85.
Created on2014-12-22T20:05:48-0400
Published on2015-04-14T14:51:31-0400
Persistent URLhttps://doi.org/doi:10.7282/T39K4CZC