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

Research has shown that, through argumentation, even young children can build proof-like forms of argument (Ball, 1993; Ball et al., 2002; Maher & Martino, 1996a, 1996b; Styliandes, 2006; 2007; Maher, 2009; 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 and other researchers. 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 2 and Interview 4. In the first four interview sessions, Stephanie explores ideas about expanding a binomial to the second power. During Interview 1 Stephanie explored the distributive property and provided a convincing argument justifying why a(x+y) = ax + ay. She also used properties and algebraic ideas to explore the meaning of (x+y)(x+y). During Interview 2, prior to Event 1 of this VMCAnalytic, Stephanie used definitions and properties to explore the meaning of (a+b)^2. Viewers interested in Stephanie’s argumentation during this algebraic exploration can access the VMCAnalytic, “Eighth Grader Stephanie’s Argumentation about Meaning for the square of a Binomial using Algebraic Reasoning” found on the videomosaic.org website. In the events of this VMCAnalytic, Stephanie is seen making a number of conjectures and claims about the meaning of (a+b) raised to the second power. She tests her conjectures and claims, produces counterexamples, poses counterclaims, refutes, modifies, and refines her conjectures, claims, and arguments. In this VMCAnalytic, conjectures are viewed as questions; claims are viewed as statements made with more confidence. While it not always clear whether Stephanie is posing a conjecture or a claim, judgments have been made distinguishing them based on factors such as inflection of Stephanie’s voice indicating confidence or lack of it and the context during which the statement is made. The focus of this VMCAnalytic, however, is not to determine whether it is a claim or conjecture being posed, but rather to make visible the argumentation that results during the investigation of the validity of a particular idea. Events 1 through 7 are taken from Interview 2 which took place on January 29, 1996. Event 8 is taken from Interview 4 which occurred about three weeks after Interview 2. In the time that elapsed between Event 7 and Event 8, Stephanie continued to explore these ideas, as indicated in the original clips (See “Early algebra ideas about binomial expansion, Stephanie’s interview two of seven” clips 5 and 6, and “Early algebra ideas about binomial expansion, Stephanie’s interview three of seven” clips 1 – 7) found on the videomosiac.org website. Just prior to Event 1, Stephanie explored (a+b)^2 algebraically and determined that (a+b)^2 = (a+b)(a+b). This exploration can be accessed through viewing “Early algebra ideas about binomial expansion, Stephanie’s interview two of seven: Clip 1” on the videomosaic.org website. In Event 1 of this VMCAnalytic, Stephanie considers the binomial expansion of (a+b)^2 as an area problem. Stephanie expresses confusion about this idea, so Researcher Maher invites Stephanie to review some basic ideas about the meaning of area. In Events 2 and 3, Stephanie conjectures about how she might represent a drawing of a square with a side length of (a+b) units. She questions her first representation and modifies it. By the end of Event 3, Stephanie has drawn a square with side lengths (a+b) units and partitioned it into four sections. In Event 4, Stephanie computes the areas of each of the partitions in the larger square. These representations can be considered conjectures and claims about what a square with side lengths (a+b) units looks like. In Event 5, Stephanie claims that (a+b)(a+b) = (a+b)^2. In Event 6, she claims that the area of the large square, i.e., (a+b)^2 is equal to the sum of the areas of the partitions. She writes the sum of the areas of the partitions as aa + ab + bb + ab, and then uses definitions and properties to simplify the expression to get a^2 + 2ab + b^2. She then claims that (a+b)^2 = a^2 + 2ab + b^2. In Event 7 she tests her claim by substituting numbers for a and b and verifies that the two expressions have the same value. Additionally, she states that in order to be sure that her equivalence is true, it must be true for all numbers, and she cannot possibly test all numbers because there are too many, suggesting a limitation of her argument. In Event 8, which is taken from Interview 4 about three weeks after Interview 2, Stephanie uses the geometric model to refine her argument that (a+b)^2 = a^2 + 2ab + b^2 presented in Events 1 through 7. In this event, Stephanie spontaneously draws a square with sides of length (a+b) to represent (a+b)^2 and partitions the square into four sections. She finds the area of the square by summing the areas these sections. She writes the areas of the sections as an addition expression that she then simplifies. The sum is represented as the simplified expression a^2 + 2ab + b^2. Stephanie displays confidence in the representation of her geometric model as she declares that “a plus b squared equals … a squared, ab, …, b squared, ab, and it would be a squared plus 2ab plus b squared.” Prior to the geometric exploration presented in this VMCAnalytic, Stephanie also explored raising a binomial to the second power using algebraic ideas. If viewers are interested in a more detailed picture of how Stephanie used argumentation to explore these ideas algebraically, they can study the VMCAnalytic, “Eighth Grader Stephanie’s Argumentation about Meaning for the Square of a Binomial using Algebraic Reasoning” 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 geometrically. 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