Math Talk That Builds Understanding: Explain, Justify and Validate

Teacher and students in a math class

As a group of professors, we often discuss ways to improve students’ engagement and mathematical understanding in our introductory mathematics courses. In 2012, we decided to re-design our courses using online technology. We chose to integrate MyLabsPlus version of MyMathLab, an online learning tool where students can practice mathematical concepts, access helpful resources, and get immediate feedback. As faculty members, we are able to generate timely student progress reports using assessment measures, which give us great insight into students’ mathematical reasoning and understanding.

We observed higher student achievement, but we wanted to take our Intermediate Algebra course to the next level. We wanted to further engage students in mathematical reasoning, as they develop their mathematical knowledge. According to Ball and Bass (2003) mathematical reasoning “comprises a set of practices and norms that are collective, not merely individual or idiosyncratic, and rooted in the discipline” (p. 29). Thus, we sought to further strengthen students’ mathematical reasoning by cultivating rich mathematical discourse. For example, during instruction, we often asked questions encouraging students to share their thinking, and give feedback to their peers. We frequently asked students to explain how they arrived at a solution. Based on our observations, students were more engaged with lessons that provided an opportunity to participate in rich mathematical discourse, which potentially had a positive impact on their overall course performance. We were seeing the results we wanted, and we knew we were making progress to further course improvement; thus, we employed the following strategy to cultivate students’ mathematical reasoning and to orchestrate rich mathematical discourse.

Stein and Smith (2011) acknowledged five practices to cultivate mathematical discourse: anticipate, monitor, select, sequence and connect. In planning lessons and the important constructs that we wanted students to conceptualize, we anticipated misconceptions and considered means that could address them. Next, we monitored the nature of students’ work. We asked students to explain, justify, prove, and discuss with their peers as they worked on mathematical tasks. This provided insights on what students understood and the variety of representations students used to communicate their mathematical ideas. Subsequently, we selected samples of students’ work to discuss with the class as a whole.

The selected work to be shared provided opportunities for students to engage in mathematical reasoning, and also provided examples of multiple ways to think about the mathematical problem. In some instances, we presented alternative perspectives to enhance the quality of the discourse. We considered a suitable means to sequence students’ ideas. Sometimes it was beneficial to discuss incorrect solutions first, and address misconceptions before having students share correct solutions; while other times, sharing various approaches used to arrive at the correct answer was more advantageous. Finally, we deemed it important to make connections across mathematical concepts, and to real world phenomena so that students can make sense of what was learned and consider the value of what was learned within the context of their reality. We also provided students an opportunity to compare and contrast various strategies, as well as to consider efficiency and accuracy of their responses.

In closing, we orchestrated rich mathematical discourse in Intermediate Algebra to increase opportunities for students to communicate mathematically, and provide evidence of mathematical reasoning. We encouraged our students to explain their thinking, justify the validity or lack thereof of their mathematical arguments, show how they arrived at the correct solutions and make connections between mathematical concepts as well as real world activities.

To learn more about how we redesigned our Intermediate Algebra course, read our full case study.



Ball, D., and Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics, pp 27-44. Reston, VA: National Council of Teachers of Mathematics.

Stein, M. K., & Smith, M. (2011). 5 Practices for orchestrating productive mathematics discussions. National Council of Teachers of Mathematics. Reston, VA.


About the Author
Ruthmae Sears, Ph.D.

Ruthmae Sears, Ph.D.

Dr. Ruthmae Sears is an assistant professor at the University of South Florida. She coordinates the Beginning and Intermediate Algebra Mathematics courses that incorporate the SMART lab in the teaching of mathematics. Additionally, she is the mathematics education doctoral program coordinator, the Tampa Bay representative for the Mathematics Teacher Education Partnership, and the secretary for the Florida Association of Mathematics Teacher Educators.  She has published and presented on curriculum issues, reasoning and proof, and technology.