Amplifying Learning in the Mathematics Classroom

The importance of mathematics teaching and learning in our schools is hard to overstate.  From computational skills to inductive reasoning, mathematics is everywhere in our lives.  If only there were a playbook for implementing what works best in mathematics teaching and learning. There is now!

As we set out to capture the latest research in effective teaching in mathematics, we found that a playbook was the best way to translate the research into practice. This is a Playbook, which, then, contains a collection of tactics and methods used by a team to accomplish a common goal and get things done (Merriam-Webster, 2023). In the case of mathematics, the learning intention is to understand what works best in mathematics teaching and learning, so that we can implement these ideas, approaches, strategies, and interventions into our classrooms. Therefore, each play is designed to support our thinking and decision-making around mathematics teaching and learning in our classrooms. When athletic coaches and their teams use playbooks to get things done (e.g., score a goal in soccer, score a run in a cricket match, score a touchdown in American football), they select the plays that best fit the current context or situation. So, what’s the play needed in your classroom?

Going over every option is well beyond the scope of this blog. However, we want to highlight one play that you can put into action tomorrow: Math Talks.

Why Discourse Matters

For many of us, mathematics is commonly associated with simply solving a problem, getting the right answer, and being able to do it again with the next problem. So, what is the value in talking about the mathematics? To answer this question, we need to go back in time and visit the thinking of a legend in child development: Lev Vygotsky. Vygotsky (1976) described discourse as a platform on which students communicate their mathematical ideas with teachers and peers to share their understanding, clarify misperceptions, and evaluate ideas. The National Council of Teachers of Mathematics (NCTM, 2014) went further and stated that in discourse-rich classrooms, students work in pairs, small groups, and as a whole class to share ideas and clarify understandings, construct convincing arguments regarding why and how things work, develop a language for expressing mathematical ideas, and see things from other perspectives. This allows learners to do more than just go through the motions of mathematics, instead developing conceptual knowledge of mathematics and transferring that knowledge to new contexts.

Henningsen and Stein (1997) noted that teachers are charged with pressing students to provide meaningful explanations to help support higher level mathematical thinking and reasoning. It is not enough for students to think like mathematicians; they need to communicate like them as well. A classroom that provides meaningful discourse offers many opportunities for speaking, reading, listening, and writing (Walter, 2018). Students must process information, share ideas, listen, and respond to others, and collaborate on a consensus, which often requires clarification questions, summarization of key concepts, connection of relevant ideas, and analysis of conversations (Walter, 2018). If students cannot access and participate in discourse, their opportunities to learn mathematics may be diminished (Banse et al., 2016).

How can we assess what students know or are able to do without discourse?

More importantly, how can students assess their own learning if they don’t use the language of mathematics to reflect and refine their thinking?

Understanding Math Talks

Math Talks are respectful but engaged conversations in which students can clarify their thinking and learn from others through talk (Chapin et al., 2009). They are daily, rigorous teaching and learning experiences for all students, where meaningful mathematical discussions construct knowledge and support mathematical learning of all participants (Hufferd-Ackles et al., 2004). When students engage in Math Talks, they:

• increase their mathematical knowledge and understanding because they are held responsible for justifying their reasoning (Rawding & Wills, 2012),

• have a voice and use strategies for overcoming difficulties through discussion (Gresham & Shannon, 2017),

• concentrate on reasoning and making sense of mathematics (Gresham & Shannon, 2017),

• clarify their thinking and learn from others through talk (Chapin et al., 2009),

• develop their metacognition and discuss, debate, and reevaluate situations in a respectful manner (Walter, 2019), and
• discover misunderstandings, deepen meaning, boost memory, and develop language and social skills (Chapin et al., 2009).

Elements of Math Talks

Hufferd-Ackles et al. (2004) studied math-talk learning communities and identified four distinct but interrelated components:

1. Questioning: This looks closely at who is the questioner in classroom interactions. Is the teacher exclusively asking questions, or do students share this role alongside the teacher?  In our own classrooms, we should monitor this balance of questioning to ensure we are allowing learners to take ownership of their mathematics learning by asking the questions.
2. Explaining Mathematical Thinking: Although this connects with questioning, it focuses on students’ ability to explain their mathematical thinking fully and fluidly. Did they contribute in a significant way, such that the teacher and other students can assess, question, or build on their ideas?
3. Source of Mathematical Ideas: This component compares the teacher’s level of involvement with the students. Are students expected to mimic what the teacher models, or does the teacher elicit students’ ideas such that their involvement directs the learning?
4. Responsibility for Learning: The final component concentrates on students’ role in their learning journey. Are they passive listeners, or are they engaged and involved in the classroom discourse as co-learners and co-teachers?

Before moving forward in this Corwin Connect blog, evaluate the Math Talks in your classroom. Are these four components evident in the discourse? If not, how could you modify your next Math Talk to ensure these components are present? How will you build the capacity in your learners to engage in these four components?

So how are we going to put this into practice?

Strategies for Effective Math Talks

A student may not enter our classroom as an expert in mathematical discourse, but we can help them become one before they leave. Deciding which aspects of productive discourse to teach will depend on students’ previous experiences with classroom discourse,  students’ ages, and current classroom dynamics (Walter, 2018). To help us think through these decisions, Zwiers et al. (2014) suggested that teachers should be able to answer these questions:

• What do I want to see and hear?
• How can I teach students the best things to say next in conversation?
• How can I teach cohesion in taking turns and whole conversation?
• How do I teach listening?
• Which conversation skills can I teach?

However, sharing our expectations and the relevancy of mathematics discourse, we should involve our learners. Wagganer (2015) suggested the following five strategies that encourage meaningful Math Talk:

1. Discuss why math talk is important.
2. Teach students how to listen and respond.
3. Introduce sentence stems.
4. Contrast explanation versus justification.
5. Give an example.

When students understand why Math Talks are important, they properly engage in meaningful mathematics discussions (Wagganer, 2015). Invite students to share their reasons for why Math Talks are important and how they individually and collectively benefit from them. They will often acknowledge how working together, like in small groups, helps them to learn to work autonomously to make sense of mathematics. It also influences what gets worked on during the concluding discussion because the teacher often uses this time to monitor and guide students’ thinking around key mathematical ideas and uses their observations to determine which strategies to highlight during the whole-class discussion (Smith et al., 2009).

This is just one aspect of mathematics teaching and learning that will amplify our impact on students. To move from guessing to gaining, we must always keep these big ideas in mind:

1. Mathematics teaching must integrate all aspects of mathematics learning: procedural knowledge, conceptual understanding, and the application of concepts and thinking.
2. The expectations for mathematics learning must be clearly shared with and communicated to our learners.
3. Our mathematics classrooms must be full of rigorous mathematical tasks that align with those expectations.
4. We must make thinking visible in our classrooms so that we can see learning happening.

Read more about what works best in mathematics teaching and learning in our new playbook: The Mathematics Playbook: Implementing What Works Best in the Classroom.

With that, let’s get to talking about mathematics!

References

Banse, H. W., Palacios, N. A., Merritt, E. G., & Rimm-Kaufman, S. E. (2016). 5 strategies for scaffolding math discourse with ELLs. Teaching Children Mathematics, 23(2), 100–108. https://doi.org/10.5951/teacchilmath.23.2.0100

Chapin, S. H., O’Connor, M. C., & Anderson, N. C. (2013). Talk moves: A teacher’s guide for using classroom discussions in math, grades K-6 (3rd ed.). Math Solutions.Barbieri, C. A., Miller-Cotto, D., & Clerjuste, S.

Gresham, G., & Shannon, T. (2017). Building mathematics discourse in students. Teaching Children Mathematics, 23(6), 360–366. https://doi.org/10.5951/ teacchilmath.23.6.0360

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28, 524–549.

Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math talk learning community. Journal for Research in Mathematics Education, 35, 81–116.

Merriam-Webster. (2023). Playbook. https://www.merriamwebster.com/dictionary/playbook

National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring mathematical success for all. NCTM. National Council of Teachers

Rawding, M. R., & Wills, T. (2012). Discourse: Simple moves that work. Mathematics Teaching in the Middle School, 18, 46–51.

Vygotsky, L. (1978). Mind in society: The development of higher psychological processes (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Harvard University Press.

Walter, H. A. (2018). Beyond turn and talk: Creating discourse. Teaching Children Mathematics, 25(3), 180–185. https://doi.org/10.5951/teacchilmath.25.3.0180

Zwiers, J., O’Hara, S., & Pritchard, R. (2014). Common core standards in diverse classrooms: Essential practices for developing academic language and disciplinary literacy. Stenhouse.