The Algebraic Nature of Fractions: Developing Relational Thinking ...

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Relational Thinking and Fractions 28 We have further argued that Relational Thinking is a critical precursor – perhaps the most critical – to learning algebra with understanding, because if children understand the arithmetic that they learn, then they are better prepared to solve problems and generate new ideas in the domain of algebra. However, Relational Thinking is almost entirely neglected in typical U.S. classrooms with the unfortunate result that children experience all types of learning difficulties as they move beyond arithmetic into learning algebra. Some proposed solutions focus on a renewed emphasis on prerequisite skills (e.g., U.S. Department of Education), while others emphasize the use of concrete materials and models (e.g., Lesh, Post, & Behr, 1987). We have presented an alternative view of how to address these difficulties that centers on cultivating children’s implicit use of fundamental properties of the real-number system to solve arithmetic problems, to better align the concepts and skills learned in arithmetic and algebra. At the heart of this view is the reciprocal relationship between arithmetic and algebra as it is revealed in children’s reasoning about quantity.
REFERENCES Relational Thinking and Fractions 29 Ambrose, R., Baek, J-M. & Carpenter, T. P. (2003). Children's invention of multiplication and division algorithms. In A. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructive adaptive expertise (305-336). Mahwah, NJ: Erlbaum. Baek, J. (2008). Developing algebraic thinking through explorations in multiplication. In C. E. Greenes & R. Rubenstein (Eds.), Seventieth Yearbook: Algebra and Algebraic Thinking in School Algebra (141-154). Reson, VA: National Council of Teachers of Mathematics. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.) (1999): How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. Brinker, L. (1997). Using structured representations to solve fraction problems: A discussion of seven students' strategies. Paper presented at the annual meeting of the American Educational Research Association Annual Meeting, March, Chicago, IL. Carpenter, T. P., Franke, M. L., Jacobs, V., Fennema, E. & Empson, S. B. (1998). A longitudinal study of invention and understanding in children's use of multidigit addition and subtraction procedures. Journal for Research in Mathematics Education, 29(1), 3-20. Carpenter, T.P., Franke, M.L., & Levi, L. (2003): Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann. Carpenter, T.P., Franke, M.L., Levi, L. & Zeringue, J. (2005). Algebra in elementary school: Developing relational thinking. ZDM, 37(1), 53-59.
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The Algebraic Nature of Fractions: Developing Relational Thinking ...

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