Talk, talk, talk. The importance of math talk continues to be emphasized because we know it is important to have students talk about mathematics if we want them to develop meaning for mathematical ideas. But how do we engage students in math talk when instruction has moved mostly online as a result of the COVID-19 pandemic?
It is important to expand our ideas concerning math talk to consider several ways in which teachers and students can exchange ideas about mathematics, regardless of where students are when they are engaging and learning.
Although talking is important for learning, there are also several other ways in which teachers and students exchange mathematical ideas—for example, through writing, videos, forums, etc. Exchange is the main concept here, meaning that, through whatever media, students need to engage in sharing their ideas as well as in understanding, questioning, and appreciating the ideas of others. It is the exercises of constructing and presenting one’s viable argument, following and understanding another’s argument, and critiquing the arguments and reasoning of others that matter—whether in-person or online, whether synchronous or asynchronous, whether spoken or through other forms of communication. Using these different ways of communicating and conceptualizing classroom discourse from a broad perspective helps us consider how to engage students in mathematical exchanges with their classmates and with us, as teachers. From this broad perspective of math talk, the idea of different types of classroom discourse that exist in face-to-face classrooms continues to be important in alternative learning environments.
It is all about how students and teachers are questioning, listening, explaining, and using different modes of engagement to communicate with one another about mathematics and for what purposes.
The Math Discourse Matrix from the book Activating Math Talk: 11 Purposeful Techniques for Your Elementary Students (Sztajn, Heck, Malzahn, 2021) summarizes four different types of discourse in which students and teachers can engage: correcting, eliciting, probing, and responsive. These types of discourse have different purposes and continue to play a role in distance and hybrid learning as much as in face-to-face learning. Let’s examine what each type means and how it may look in different learning environments. The focus is on the key characteristics of each discourse type and its enactment, whether in-person, remote, or hybrid.
Correcting discourse is about fostering accuracy, automaticity, and speed, focusing mostly on the recall of facts and procedures, which have their place in mathematics. When engaging in correcting discourse, the teacher is listening (or looking) for a right answer and the students are listening (or waiting) for the teacher’s approval of their answers. There is little student-to-student interaction, mostly because the teacher is positioned as the “knower” who determines the correctness of mathematical ideas. In online environments, correcting discourse continues to focus on students providing a right answer, with very little attention to how they got there. Mathematical exchanges are brief, with little opportunity for conceptual development.
The main idea in eliciting discourse is to increase participation and student sharing, asking several students to present their mathematical thinking and opening up the conversation beyond accuracy to consider students’ solution strategies as well. In this type of discourse, the teacher uses more open tasks and makes sure students are comfortable. When learning face-to-face or remotely, the goal is to make the exchange non-threatening, with students’ ideas often remaining unchallenged. Students can use talk, writing, or recordings, and can share synchronously or asynchronously. Tools such as electronic boards, pictures, or the use of small groups (e.g., break out rooms) can facilitate this type of discourse and make sure most students are sharing and comfortable. Overall, the key aspect of eliciting discourse is that there are several different opportunities for students to share and all ideas are welcomed and valued.
When teachers start asking or inviting key mathematical questions about the solutions students share, they move into probing discourse. This type of discourse goes into greater depth and critique, showing appreciation for students’ mathematical justifications and strategic competence. In this case, the instruction engages students in constructing and presenting their mathematical arguments, which, again, can happen through talk, writing, videos, audio recordings, electronic boards, or other communication venues. Expectations for students involve detailed explanations of their mathematical thinking (through probing questions) and connections among their ideas and key mathematical concepts (through pressing questions). Although we tend to think of this type of interaction as synchronous, it can happen in a longer and asynchronous fashion (e.g., facilitated discussion boards, online forums), extending exchanges about several mathematical solutions over longer periods of time.
In responsive discourse there is a shift in terms of whose responsibility it is to initiate and engage in the mathematical conversation. This means the teacher is no longer positioned as the only “knower” in the classroom; instead, students are also responsible for asking questions of each other, probing and pressing for mathematical ideas and exchanges, making connections among different students’ mathematical arguments, and helping each other draw conclusions and understand the math. In this type of discourse, students take initiative to make thinking and justification available for discussion. These collective, content-rich, and goal-focused math discussions can happen in environments where students have opportunities to see, discuss, and connect different solutions, in a variety of formats and using several communication tools beyond straight discussion in synchronous classrooms, such as interactive slides, visible online polling, or interactive video platforms.
In any learning environment, a strong focus on correct answers and procedures (what & how), participation (who), strategies and justifications (why), or releasing and sharing responsibility for learning (authority and connections) helps identify the types of math discourse in which your students have the opportunity to engage. And whereas correcting discourse is important for developing accuracy, and eliciting discourse is important for engaging many students with mathematics, it is probing and responsive discourse that help students develop conceptual understanding as well as fluency, flexibility, and strategic competence. Thus, in these new remote and hybrid environments, we must continue to use discourse-promoting tasks and focus on the types of questions, explanations, justifications, and modes of communication students use to present and critique mathematical arguments and justifications, draw conclusions, and make connections.