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doi:10.7282/t3-e2ak-3812

In this interview session Stephanie is asked by Researcher Maher about what (a+b) squared means. She makes a claim that it is the same as (a squared) + (b squared). She carries out a test of testing her claim by substituting a and b with numbers and establishes that the two are not equivalent. This sets the stage for exploring the meaning of (a+b) squared.
Sometimes teachers will ignore a wrong answer or correct a student who offers an incorrect answer. We see in this interview Researcher Maher making use of an incorrect conjecture as an opportunity for the student to discover, by reasoning, the correct answer. Starting from a correct response by the student for the meaning of "a squared" as “a multiplied by a”, Researcher Maher challenges Stephanie to think about a square of side 3 units. Stephanie immediately responds that 3 times 3 is 9. To begin an exploration, the researcher challenges Stephanie to consider a square of side 3 units, with each side made up of smaller one units. The representation enables Stephanie in a transition from a specific situation (3-sided square) to a general situation (n-sided square). Considering an n-sided square is abstract; however, considering a 3-sided square enabled her to visualize what a square of side a units could mean. According to Davis (1992), a central part of problem solving is building in one’s mind a mental representation of the problem situation. Once Stephanie recognizes the aspect of dimensions constituting units, she can understand how to represent a general situation, the abstract square.
Stephanie recognizes that (a+b) squared means (a+b) multiply by (a+b), and (a+b) cubed will be (a+b) multiplied as a factor, three times. According to Maher, Powell & Uptegrove (2010) Students use representations that they build to make sense of and attribute meaning to the mathematics that they are doing. Using the concept of 9 square units represented by the square diagram, Stephanie represents (a+b) squared on a diagram, she draws a square of side (a+b). Just like many mathematicians would do, she does not break (a+b) into two, part a and part b until she is challenged by Researcher Maher to partition a side into a and b units. In her effort to show the two parts of each side of the square, Stephanie partitions the square into four smaller regions constituting the different areas that make up the entire square.
Stephanie begins by finding the area of all the four parts as “a squared, ab, ab and b squared”. When prompted to give the area of the square of side (a+b) squared, she says it is the same as the sum of the areas of the four parts. She finally finds that (a+b) squared is equal to “a squared+2ab+b squared”. To justify that her product is correct, she substitutes the earlier values 2 and 3 for a and b respectively, and the result turns out to be the same as her calculation at the beginning of the session. After Stephanie had justified that her solution was correct, Researcher Maher proceeds giving Stephanie an opportunity to explore a deeper meaning of area as composed of parts. During this interview there was questioning not just by the Researcher Maher but also by the Stephanie. Failing to follow up an answer with "Why" by a teacher is a failed opportunity (Leinwand, 2009). Stephanie is asked to justify her responses, and this helps her eventually to develop conceptual understanding of the binomial expansion. Although Stephanie, previously, had not been binomial expansion, using geometry she is able to explore the meaning of terms in the binomial expansion. This points to the powerful aspect of multiple presentations of ideas.



References
Davis, R. B. (1992). Understanding "Understanding". Journal of Mathematical Behavoiur II.
Leinwand, S. (2009). Accessible Mathematics. Portsmouth: Heinmann.
Maher, C. A., Powell, A. B., & Uptegrove, E. B. (2010). Combinatorics and Reasoning;Representing,Justifying and Building Isomorphisms. New York: Springer.
Uptegrove, E. B. (2007). Dissertation on To Symbols From Meaning: Students’ Investigations in Counting. New Brunswick, New Jersey: Unpublished.

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