Unitizing Area Numerically and Algebraically; The Gap Between Arithmetic and Algebra
Creators: name: Weitzer, Courtney
Publisher: name: No Publisher Supplied
Year: 2014

description: In this analytic, Researcher Carolyn Maher employs a very useful and effective strategy to facilitate Stephanie’s learning.  The idea of “reasoning down” involves analyzing by cycling back to previous experiences or knowledge (Driscoll 1999).  Through questioning, Researcher Maher discovers that Stephanie’s confusion with the meaning of the square of (a+b) is due to her difficulty with the meaning of a^2.  A possible reason for this struggle lies in the leap to algebra, a means of applying an understanding of numbers and numeric operations by reasoning abstractly, from arithmetic, a concrete skill.  This transition to algebra is “mysterious and intimidating” as numerical computation is “replaced by a new set of experiences, characterized by their reliance on letter symbols, not numbers, and by a set of arcane rules for manipulating these symbols” (Driscoll, Ch. 2, 1999).  Stephanie has trouble with reasonably fathoming what a variable is because it is not just one known value. Since (a+b)^2 = a^2 + 2ab + b^2 for all (infinitely many) real values of ’a’ and ’b’, then by definition, it will be true no matter what specific values we choose. Stephanie therefore builds representations based on the arithmetic that she already knows in order to construct further knowledge (Davis 1984). Her concrete examples of how area is generated show her smooth transition back to arithmetic, in order to construct a generalization.  She is also "[making] sense of problems" in order to "persevere in solving them" (CCSS-Mathematical Practice #1, 2011). Having an available visualization as a tool for thinking about mathematics is important for algebraic reasoning as it helps us to "model with mathematics" (CCSS-Mathematical Practice #2, 2011). A part of this reasoning is the ability to focus on the structure of the mathematics by recognizing and analyzing in order to ultimately generalize a pattern: which in essence, is the core of mathematical reasoning, according to Hahkioniemi, Fosnot and Driscoll.  Algebra seeks to generalize beyond specific instances as it consists of “structuring [grounded] in the progression of strategies, the development of big ideas, and the emergence of modeling” (Fosnot 2010).  This is a beautiful segue into proof and higher mathematics as it involves the ability to "reason abstractly and quantitatively" (CCSS-Mathematical Practice #4 2011). Stephanie’s inability to concretely represent the variable ‘a’ in a visual manner is evidence of her conceptual struggle with the idea of an object that varies and can represent infinitely many values.  Stephanie has trouble conceptualizing the variable ’a’ as the representation of a number because she literally cannot visualize ’a’ as taking up a certain amount of space in her mind, or even visually on the paper. She finds it extremely difficult to see ’a’ as 1-unit ‘a’ times or to unitize ‘a’, when ’a’ is seemingly a letter and not a number.  Rather, she must take a higher level of understanding by conceptualizing ’a’ as a generalization of the behavior in which all real numbers would generate the unit squares.Common Core State Standards Initiative. Common Core State Standards For Mathematics - Mathematical Practices: Preparing America’s Students for College &amp; Career. Compiled by Metamorphosis Teaching Learning Communities: © 2011.Davis, R. (1984). “Learning Mathematics: the Cognitive Science Approach to Mathematics Education.” Ablex. New Jersey.Driscoll, Mark. "Fostering Algebraic Thinking". A Guide For Teachers Grades 6-10. Heinemann: Copyright © 1999 by Educational Development Center, Inc.Fosnot, Catherine Twomey; Jacob, Bill. “Young Mathematicians at Work” – Constructing Algebra. Heinemann: Copyright © 2010. NCTM: Reston, VA.Francisco, J., Hahkioniemi, M. “Insights into Students’ Algebraic Reasoning”. PME 30, Prague, 2006, Vol. 3, 105-112.