Building Algebraic Understanding

Letters in math?!?! As a middle school teacher students tend to panic when they first see variables creep their way into their math problems. While in elementary they have solved problems for unknown numbers using empty boxes or pictures, the concept of letters as variables is a major sticking point for students. As a teacher who teaches 6th through 8th grade students I see students from the initial exposure of variables through solving multi-step equations with variables on both sides and systems of equations. 6th graders “reason about and solve one-variable equations and inequalities” (Common Core State Standards Initiative, n.d., p. 41) while 8th graders “analyze and solve linear equations and pairs of simultaneous linear equations” (Common Core State Standards Initiative, n.d., p. 53). Throughout middle school students make huge gains in their understanding of algebraic expressions and equations. 

Although there are huge gains many students struggle with conceptualizing algebraic expressions. As a middle school teacher I have developed my Technological Pedagogical Knowledge by exploring the models and manipulatives that support a rational understanding of solving equations (Mishra & Koehler, 2006) . I have found that through the use of manipulatives students can better visualize what is actually happening when solving equations. 

When thinking about equations, balance scales are one of the major resources to teach these concepts. One program I have used is Hands On Equations. Hands on Equations uses the concept of a balance scale to teach students using “legal moves”. This game based approach using dice and pawns for students to manipulate the equations in real life. While the students can use this to picture my Knowledge of Teaching and Students (Hill & Ball, 2009) has led me to the realization that students struggle when they have to create the problems using the manipulatives as well as when moving from the more concrete to the abstract concept of solving the systems algebraically.

As a virtual teacher one of my biggest struggles is getting the manipulatives into the hands of students. While looking at digital manipulatives, I have found Polypad allows students to explore equations through virtual balances and algebra tiles. When working with students I have found success with using the balances online but the setup process is a little complicated for students. 

While the common manipulatives have gained their popularity for a reason, there are some non-traditional manipulatives that help students to grasp these concepts. In my classroom I often introduce the concept of multistep equations by wrapping and unwrapping a box. Throughout my demonstration we record the steps needed to wrap the box. Students then can make the connection that in order to figure out what the present is they need to work backwards to undo their steps. Additionally I have used easter eggs or other mystery boxes to demonstrate algebraic concepts. Students can use these manipulatives to visualize the equation and determine the steps to figure out what is in each container. 

While teaching algebraic concepts to students I have found that by starting with concrete models students can reason through algebraic equations and can visualize what steps they are actually taking. When thinking about math many of my students can relate to Jones saying “I never considered myself to be successful in mathematics; however, a survivor of mathematics” (Jones, 2021, p. 1). For many of my students their identity in math is often based on their experiences of struggle and failure. Although at times it takes longer to use manipulatives in math, doing so can help students see how the problem is solved.  When it is time to actually solve the equations on paper they can better visualize and reason through the problem which can help shape their mathematical identity as being a successful mathematician. Variables do not need to be a scary thing. By making the process more concrete for students we can hopefully help cure some of the fear when letters sneak their way into math. 

References:

Common Core State Standards Initiative (CCSSI). (n.d.). Mathematics standards. Mathematics Standards | Common Core State Standards Initiative. https://corestandards.org/wp-content/uploads/2023/09/ADA-Compliant-Math-Standards.pdf 

Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71.

Jones, D. A. (2021). A critical race examination of four young black men's participation in community college developmental/remedial mathematics. [Doctoral dissertation, Michigan State University]. ProQuest. https://www.proquest.com/docview/2543473935?parentSessionId=sff0SaEimfrSBi0Oa71k8UmkpSaADhenmry0F1wWSEQ=&accountid=12598&sourcetype=Dissertations%20&%20These

Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108(6), 1QA017-1054.