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doi:10.7282/t3-za33-j671

This analytic presents a task-based interview with Stephanie, an eighth- grade student, about her understanding of the meaning of the square of a binomial. Stephanie uses both algebraic and geometric reasoning to explain her reasoning and support her arguments. Van Ness, (2017) writes, “The literature on argumentation supports the notion that argumentation and reasoning are inextricably linked, and it is difficult to discuss argumentation without also talking about reasoning”.
The interview begins with Researcher Maher asking Stephanie the meaning of the expression (a + b) ^2. Stephanie conjectures that (a + b) ^2 = a^2 + b^2. Researcher Maher asks Stephanie to test her conjecture by substituting numerical values for a and b. Stephanie begins by writing the expression a^2 + b^2 as aa + bb. Prompted by Researcher Maher, Stephanie assigns numerical values for the variables “a” and “b”, and correctly substitutes into the expressions (a + b) ^2 and a^2 + b^2, discovering that her original conjecture does not hold. Mcintosh, Reys and Reys (1992), makes the connections between student’s number sense and their ability to make connections to mathematical properties, which they claim is a key understanding in being able to reason algebraically. From Stephanie’s substitution of specific numbers to test her conjecture, we see an understanding of number and number relationships. Further probing by Researcher Maher prompts Stephanie to expand the expression (a + b) ^2 using a geometric area model. Using recursive reasoning, she first uses an area model to represents a^2 which aides in representation of (a + b) ^2 as an area model. Finally, using the area model of (a + b) 2, she proves that (a + b) ^2 = a^2 + 2ab + b^2. From her working, it is evident that, examining a variety of representation, it is possible to build a deeper understanding of a mathematical idea (Davis & Maher, 1997, Maher & Davis, 1990).




REFERENCES
Davis, R.B., & Maher, C.A. (1990). The nature of mathematics. What do we do when we “do mathematics”? [Monograph]. Journal for Research in Mathematics Education, 4, 65-78.

Davis, R.B., & Maher, C.A. (1997). How students think: The role of representations. In L.D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and image (pp. 93 – 115). Hillsdale, NJ: Lawrence Associates.

Maher, C, Powell, A & Uptegrove, E. (Eds). (2010). Combinatorics and Reasoning Representing, Justifying and Building Isomorphism. New York, NY: Springer.

Mcintosh, A., Rey, B.J., Rey, R.E (1992). A proposed framework for examining basic number sense. For learning of Mathematics 12, 3, 2-9.

Van Ness, C. (2017). Creating and Using VMCAnalytics for Pre-service Teachers’ Studying of Argumentation. Unpublished doctoral dissertation, Rutgers University, NJ.

Whitenack, J., & Yackel, E. (2002). Making mathematical arguments in the primary grades: The importance of explaining and justifying ideas. Teaching Children Mathematics, 8(9), 525.

Student elaboration

Student model building

Reasoning

Representation

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