Supporting Students' Mathematical Arguments with Algebra Readiness Lessons

by Angela Knotts

As mathematics teachers continue working to integrate the Common Core State Standards into their classroom practice, one of the most common questions is how to teach the Standards for Mathematical Practice (MPs), and how to teach them with content standards.

Let’s consider MP3: Construct viable arguments and critique the arguments of others. Students should be able to put together logical explanations for why something is or isn't true, and be able to explain whether someone else's argument makes mathematical sense, and why.

One set of WestEd resources that can support middle school students in engaging with this and other MPs is the Math Pathways and Pitfalls Algebra Readiness lessons. MPP-AR lessons are designed to prioritize students' developing deep conceptual understanding of the math involved, rather than simply practicing techniques for finding answers. For example, a lesson might focus on a common error such as rewriting 30 = x + 7 as 30 + 7 = x. Instead of emphasizing procedures for manipulating equations, the lesson might invite students to investigate whether these equations are equivalent or not, and how students might check. 

Or students might consider a situation in which a fictional student has represented the statement "There are six times as many leopards as giraffes" with the equation 6L = G. To help students understand this error, instead of giving them rules for how to translate word problems into equations, the teacher might ask students to come up with a reason for why the equation does or doesn’t make sense as a representation of the statement.

When students have opportunities to think through the logic of whether or not something makes sense — beyond just identifying whether or not a correct procedure was followed — they are developing the ability to reason, whether abstractly (as in manipulating symbols) or quantitatively (as in translating a word problem into an equation). Developing an ability to reason is central to the expectations of MP3 and other Standards for Mathematical Practice. 

In the MPP-AR lessons, students are given opportunities to think independently and to discuss the mathematics with a partner and as a whole class. The teacher can facilitate discussions to support students' sense-making and can scaffold the mathematical arguments that they share with their peers. 

For some video examples of classrooms working on MP3, visit this Raising the Bar video collection: Planning Instruction With Higher-Order Questioning.

Interested in learning more about our services? The WestEd/SVMI Mathematics Network offers a variety of sustained mathematics learning events for teachers, leaders, coaches, and administrators. These learning events include professional development and leadership development opportunities, as well as networking and planning sessions for member school districts. 

This material is based upon work supported by the National Science Foundation under Grant No. (DRL-1314416, PI Davenport). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Research Associate with Science, Technology, Engineering, & Mathematics (STEM) at WestEd