Problem Solving and the Common Core

Context

I am a passionate instructor and I want my students to “get” math. As a student, I did well in math because I could memorize algorithms and apply them where necessary. This got me as far as my required math course in college and I had no interest in pursuing math any further. It was when I started teaching math, however, and began to understand the connections and the math behind the algorithms I had previously memorized that I discovered I truly enjoyed math and wanted to better understand it. As a result, it is forefront in my mind when planning curriculum that the goal is not strictly related to math fluency and the use of algorithms but rather an in depth conceptual understanding and application of mathematical principals, patterns and properties.

Equations and Expressions is one of the five domains making up the common core math standards in grades 6-8. In sixth grade students begin to reason with numerical expressions. Progressing into seventh grade, students are expected to continue to develop their understanding of generating equivalent expressions and to apply their knowledge of writing and solving equations to solve real life problems. In eighth grade, students begin to connect linear equations with their graphical representation on the coordinate plane. Additionally, students are expected to give examples of linear equations with one solution, no solution or an infinite number of solutions. Finally, students in eighth grade are expected to apply their knowledge of linear equations to solving pairs of simultaneous linear equations. In order for students to be successful with the eighth grade content, it is key that they have a solid foundation in algebraic reasoning and extensive experience generating and manipulating equations. Without this base, students will have difficulty progressing towards more complex algebraic reasoning.¹

This unit’s emphasis, therefore, is on the continued development of a solid foundation in algebraic reasoning. In general, this strong understanding is key to a student’s continued success in the academic and real world. Mastery of Algebra is often viewed as a gatekeeper and key predictor of success in higher-level mathematics in high school and beyond. As many studies have shown algebraic reasoning is essential to further mathematical study, “because it pushes students’ understanding of mathematics beyond the result of specific calculations and the procedural applications of formulas”.² The development of these skills creates more flexible thinking, preparing the learner for more abstract mathematical concepts.