Eighth Grader Stephanie’s Argumentation about Meaning for the Square of a Binomial using Algebraic Reasoning

Research has shown that, through argumentation, even young children can build proof-like forms of argument (Ball, 1993; Ball et al., 2002; Maher, 2009; Maher & Martino, 1996a, 1996b; Styliandes, 2006; 2007; Yankelewitz, 2009; Yankelewitz, Mueller, & Maher, 2010). The purpose of this VMCAnalytic is to illustrate events of a student involved in argumentation. 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. The data from which this VMCAnalytic was made were taken from a larger data set collected in a longitudinal study designed to investigate students’ mathematical reasoning (Aboelnaga, 2011; Maher, 2005). This research was supported by a grant from the National Science Foundation: REC-9814846 directed by C. A. Maher. In this data set, Stephanie, an eighth-grade student enrolled in a traditional eighth-grade algebra class in a private parochial school, participated in a series of seven task-based interview sessions conducted by Researcher Carolyn Maher. The research team had investigated Stephanie’s mathematical reasoning since she was in first grade. At this point in her schooling, Stephanie had changed schools and began expressing dislike for math, indicating that she wanted to make sense of the symbol manipulation rituals that were new to her. This resulted in a teaching experiment where the goal was for Stephanie to explore and make sense of the meaning behind the rules and symbols (Aboelnaga, 2011). The data used in this VMCAnalytic come from two of the seven after-school interview sessions: Interview 1 and Interview 2. In the first four interview sessions, Stephanie explores ideas about expanding a binomial to the second power. During Interview 1, prior to Event 1 of this VMCAnalytic, Stephanie explored the distributive property and provided a convincing argument justifying why a(x+y) = ax + ay, which she restates and supports in Event 3 of this VMCAnalytic. The events of this VMCAnalytic show Stephanie’s exploration of the meanings of (x+y)(x+y) and (a+b)^2. In the events of the VMCAnalytic, Stephanie is seen making a number of conjectures and claims about the meaning of these expressions. She tests her conjectures and claims, produces counterexamples, poses counterclaims, refutes, modifies, and refines her conjectures, claims, and arguments. For the purposes of this VMCAnalytic, conjectures are posed as questions and claims are statements made with more confidence and evidence. While the distinctions between conjecture and claim are not always clear, in this VMCAnalytic, judgments are made distinguishing them based on factors such as the inflection of Stephanie’s voice, the context surrounding the statement, and evidence derived from making use of algebraic properties. In viewing this VMCAnalytic, however, it is not critical to distinguish whether a claim or conjecture is being posed, but rather to make visible the statements that evoke argumentation during the investigation of an idea. In this VMCAnalytic, Events 1 through 9 are taken from the first interview between Stephanie and Researcher Carolyn Maher. This interview took place on November 8, 1995, the fall of Stephanie’s eighth-grade year. In these events, Stephanie explores (x+y)(x+y). In Event 1, Stephanie makes two conjectures: 1. (x+y)(x+y) might be equal to x^2 * y^2; 2. (x+y)(x+y) might be equal to x^2 + y^2. Although Stephanie makes the first conjecture, she quickly abandons it after she makes her second conjecture and does not consider this other conjecture for the rest of the exploration. The purpose of Events 2 through 9 in this VMCAnalytic is to trace how Stephanie investigates the reasonableness of this second conjecture: (x+y)(x+y) = x^2 + y^2. In Event 1, Stephanie’s statement that (x+y)(x+y) might be equal to x^2 + y^2 is a conjecture rather than a claim because she expresses uncertainty about it, as evidenced by her posing it as a question. In Event 2 she refutes this conjecture using a counterexample. She rejects her original conjecture, stating that (x+y)(x+y) does not equal x^2 + y^2. In Event 9, making use of the distributive and additive properties, Stephanie poses another claim, that (x+y)^2 = x^2 + 2xy + y^2. She provides data for this claim by testing its validity by substituting values for the variables and evaluating the expression for x = 2 and y = 3. Stephanie uses of the definition of squaring a binomial and the addition and distributive properties as warrants for this claim. In the following five events (Events 10 through Event 14), Stephanie explores the meaning of (a+b)^2. These events are taken from Interview 2. Interview 2 took place about two months after Interview 1 on January 29, 1996. In these events, Stephanie first conjectures that (a+b)^2 equals a^2 + b^2, which is a similar conjecture to the one she makes in Event 1 that (x+y)(x+y) = x^2 + y^2, and, as with the previous conjecture, she refutes this conjecture by using a counterexample. In Events 13 and 14, Stephanie claims that (a+b)^2 = (a+b)(a+b) and makes use of the definition of squaring a binomial and raising a binomial to an exponential power as warrants to back this claim. In these events, she connects her previous exploration of (x+y)(x+y) with her present exploration of (a+b)^2 when she claims and justifies that (a+b)^2 = (a+b)(a+b). From the exploration of raising a binomial to the second power using properties of algebra, Stephanie continues her exploration using a geometric representation, that is, an area model (see “Eighth Grader Stephanie’s Argumentation about Meaning for the Square of a Binomial using Geometric Reasoning”). If viewers are interested in a more detailed picture of how Stephanie used geometric argumentation to explore these ideas, they can study this VMCAnalytic, which can be accessed on the videomosaic.org website. The events in this VMCAnalytic were chosen purposefully to illustrate Stephanie’s argumentation about raising a binomial to the second power algebraically. Her exploration of these ideas goes beyond what is represented here. If viewers are interested in a more detailed picture of how Stephanie developed these ideas, the complete set of clips for Interviews 1, 2, 3, and 4 are available for viewing on the Video Mosaic Collaborative Repository at vidoemosaic.org (See “Early algebra ideas about binomial expansion, Stephanie’s interview one of seven,” “Early algebra ideas about binomial expansion, Stephanie’s interview two of seven,” “Early algebra ideas about binomial expansion, Stephanie’s interview three of seven,” “Early algebra ideas about binomial expansion, Stephanie’s interview four of seven”). References: Aboelnaga, E. Y. (2011). A case study: the development of Stephanie’s algebraic reasoning (Doctoral dissertation, Rutgers University-Graduate School of Education). 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. 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. 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