The Leap from Patterns to Formulas

In this article by Darin Beigie, the author presents a compelling argument for the use of geometric counting activities as a conduit to introducing the concept algebraic expressions. Algebra is a challenging mental leap as students must depart from the relative 'fingers-and-toes' security of numbers for the abstract world of variables. Geometric patterns, he argues, make numeric patterns more discernible. Mr. Beigie describes four lessons he uses with his seventh grade pre-algebra students. The first uses cubes and prisms; the second, polyhedrons; the third, regular polygons; and the fourth, perimeter and area patterns of polygons.  Beginning with cubes and prisms, students count and record such dimensions as edge lengths, perimeter of a face, total length of edges, area of face, and total area. Working with cubes of various sizes, students create a sequence of numbers. From this sequence, students can detect patterns which can then be readily translated into an algebraic expression. He repeats this with the other geometric shapes, eliminating such mind-blocking issues as order-of-operations. In addition, his use of verbal prompts guides learners to recognize numeric patterns they might not have detected with pencil and paper alone.

It is an absolute miracle that I did as well in mathematics as I did. It frustrates me, somewhat, that there existed other means of instruction that might have attenuated, if not eliminated, much of my frustration. Ever since taking ETE 107, my ability to help my children understand mathematics has improved dramatically. Until ETE 108, I never realized how geometry could support the study of algebra. I always believed that I did not have the strongest active working memory. I have never done well at manipulating letters or numbers in my head. But since being shown how to work with numbers visuo-spatially, my understanding of math and my mathematical self-concept has improved. When I am a teacher I will use manipulatives as often as is possible. While I may never teach math at the middle school level, this article still speaks to the benefits of hands on manipulatives in the learning of mathematical concepts. Students deserve to be exposed to multiple representations of a numeric expression, as many as necessary for them to develop a deep understanding. By doing so, their ability to move into the abstract will be bolstered.

Belgie, D. (2011). The leap from patterns to formulas. Mathematics Teaching in the Middle School 16 (6), 328-335.