DescriptionThe events depicted in this analytic were taken from a larger data set gathered as a result of a year-long research intervention involving fourth graders’ exploration of fractions. The events depicted here occurred on September 29, 1993 and are taken from the first month of the intervention during the fifth session. (For a detailed analysis of the student reasoning during this intervention, see Maher & Yankelewitz, 2017; Yankelewitz, 2009).
The students’ investigations in this VMCAnalytic build on their explorations in the prior sessions, during which they constructed Cuisenaire rod models to support ideas about basic relationships among whole numbers and fractions. During these sessions, students assigned number names to individual rods and also to trains constructed of more than one rod based on a specific rod, or train of rods, given the number name 1 and assigned as the “unit." During the sessions prior to this one, students compared 1/2 and 1/3 and developed Cuisenaire rod models to support their claim that 1/2 was greater than 1/3 (Steencken, 2001; Steencken & Maher, 2003; Yankelewitz, 2009; Maher & Yankelewitz, 2017). During Session 3, Researcher Carolyn Maher contextualized the idea of the unit by describing an actual situation in which she used chocolate candy bars as a metaphor for thinking about fractions. Students agreed that 1/2 of a larger candy bar (unit) was bigger than 1/2 of a smaller candy bar (unit) and so they should keep the same "candy bar" or the same unit when assigning number names to rods during their Cuisenaire rod explorations (Alston & Van Ness, 2017). In Session 5, depicted here, students revisit the comparison of 1/2 and 1/3 and are asked to quantify the difference. The argumentation described here supports solutions to the task, "Which is greater 1/2 or 1/3, and by how much?" It is important to note that, other than this research intervention, students in this analytic had not had any formal instruction on operations with fractions.
Researchers have found that given certain learning environments, students, even young children, can develop convincing arguments (Krummheuer, 1995; Weber, Maher, Powell, & Lee, 2008; Conner et al., 2014; Whitenack & Yackel, 2002; Ball, 1993; Ball et al., 2002; Stylianides, 2006; 2007; Zack, 1997; 1999; Maher & Martino, 1996; Maher, 2009). The original purpose of this analytic was to access viewers understanding of argumentation. The current purpose is to outline the argumentation in each event. For a detailed analysis of the argumentation in each event, including students’ claims, data, warrants, and backing and accompanying diagrams that show the elements and structure of the argumentation in these events, see Van Ness (2017). For a more detailed discussion of students’ exploration of the unit, see VMCAnalytic Comparing Models and Justifying the Choice of Unit, (http://dx.doi.org/doi:10.7282/T3XW4MQT) (Van Ness & Alston, 2015) and Switching the Unit (Alston & Van Ness, 2017), Justifying the Choice of the Unit (Van Ness & Alston, 2017) and Establishing the Importance of the Unit (Van Ness & Alston, 2017). For a more detailed discussion of the events that come after the ones presented here, see the VMCAnalytic Comparing 1/2 and 1/3: Confusion about the Unit, (http://dx.doi.org/doi:10.7282/T3ZW1NS9) (Van Ness & Alston, 2015) and Switching the unit (Alston & Van Ness, 2017).
References
Alston, A. S., & Van Ness, C. K. (2017). Switching the Unit. In C. A. Maher & D. Yankelewitz (Eds.). Children’s Reasoning While Building Fraction Ideas (pp. 65-81). Rotterdam, The Netherlands: Sense Publishers.
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.
Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. Educational Studies in Mathematics, 86(3), 401-429.
Krummheuer, Götz (1995). The ethnography of argumentation. In Cobb, Paul, Bauersfeld, Heinrich (Eds). (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Studies in mathematical thinking and learning series., (pp. 229-269). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc, xi.
Maher, C. A. (2009). Children’s reasoning: Discovering the idea of mathematical proof. Teaching and learning proof across the K-16 curriculum, 120-132.
Maher, C. A., & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. JRME, 27 (2), 194-214.
Maher, C. A., & Yankelewitz, D. (Eds.). (2017). Children’s Reasoning While Building Fraction Ideas. Springer.
Steencken, E. (2001). Tracing the growth in understanding of fraction ideas: A fourth grade case study. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.
Steencken, E. P., & Maher, C. A. (2003). Tracing fourth graders’ learning of fractions: early episodes from a year-long teaching experiment. The Journal of Mathematical Behavior, 22(2), 113-132.
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.
Van Ness, C. K. (2017). Creating and Using RUAnalytics for Preservice Teachers’ Studying of Argumentation. Doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.
Van Ness, C. K. & Alston, A. S. (2015). Comparing Models and Justifying the Choice of Unit. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3XW4MQT
Van Ness, C. K. & Alston, A. S. (2015). Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions. [video]. Retrieved from http://dx.doi.org/doi:10.7282/T3V40X3R
Van Ness, C. K., & Alston, A. S. (2017). Establishing the Importance of the Unit. In C. A. Maher & D. Yankelewitz (Eds.). Children’s Reasoning While Building Fraction Ideas (pp. 49-64). Rotterdam, The Netherlands: Sense Publishers.
Van Ness, C. K., & Alston, A. S. (2017). Justifying the Choice of the Unit. In C. A. Maher & D. Yankelewitz (Eds.). Children’s Reasoning While Building Fraction Ideas (pp. 83-94). Rotterdam, The Netherlands: Sense Publishers.
Weber, K., Maher, C., Powell, A., & Lee, H. S. (2008). Learning opportunities from group discussions: Warrants become the objects of debate. Educational Studies in Mathematics, 68(3), 247-261.
Whitenack, J., & Yackel, E. (2002). Making mathematical arguments in the primary grades: The importance of explaining and justifying ideas. Teaching Children Mathematics, 8(9), 524.
Yankelewitz, D. (2009). The Development of Mathematical Reasoning in Elementary School Students’ Exploration of Fraction Ideas (Doctoral dissertation, Rutgers, he State University of New Jersey).
Zack, V. (1997). You have to prove us wrong”: Proof at the elementary school level. In Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 291-298). Lahti: University of Helsinki.
Zack, V. (1999). Everyday and mathematical language in children’s argumentation about proof. Educational Review, 51(2), 129-146.
GenreHomework activity, Lesson activity, Professional development activity, Reasoning, Representation