From Arithmetic to Algebra: Variables, Word Problems, Fractions and the Rules

The Wrong Path Taken by Past Curricula

There are myriad reasons why my students continue to have difficulties representing phrases symbolically. By sixth grade, which is a benchmark year, most students in the U.S. are required to translate, express and evaluate expressions, not to mention create, justify and apply this information to solving equations. Without careful attention given to the cultivation of this series of skills, students will not be successful. My students do not have a solid method to use when they problem solve. They also have a limited knowledge of the English language. These two characteristics present in my students exacerbate misconceptions and do not allow them easy access points into the curriculum.

Below, I have outlined the progression of the major concepts in this area that I have gathered from the textbooks and curriculum maps that I use. They typically address the Common Core State Standards quite nicely as individual skills and are taught in numerical order. There is a lack in how they are presented coherently together. The progression starts with writing and evaluating expressions with exponents. Students here are already on shaky ground, with limited background knowledge on the meaning and the manipulation of exponents. Simple introduction of exponents at this juncture may not be appropriate.

Figure 1. Progression of 6^th Grade Standards

Progressing to the next main sections in the diagram, one can see that students are expected to do many things at once. They are writing expressions with variables with very few tools to help them. They are asked to work with variables without really understanding their role in expressions. This is especially problematic for ELL students. In my opinion, most curricula do not include a large variety of terms used, nor enough examples that will lead to the development of fluency. When rote understanding of terms is assessed, students are left with voids in the concepts underlying them. ² Continuing this progression, students then are asked to create their own versions of algebraic expressions, with at most partial ideas of the roles that variables play. They then are supposed to evaluate these expressions, which requires them to bring to bear a whole new series of skills and background knowledge.

By seventh grade, my students need to be working with ‘higher-level’ problem sets. However, I see myself as a seventh-grade math teacher going back to sixth grade content to help fill in their gaps of learning. The reason why my students continuously struggle is fleshed out above. Maybe the issue is a not thoughtful sixth-grade teacher with a very superficial exposure to material involving expressions, or maybe it is uneven attendance. Following the model above gives students a fragile foundation to build on and has them unprepared to engage in solving equations.