What Does Mathematical Discourse Sound Like?
Almost like a great song, top-notch mathematical discourse is something you know when you hear. Yet, to cultivate quality, you and your students need a vision—or an “ausion” (a word I just made up to describe an ideal auditory vision)— of what you’re going after. As described by Jeremy Kilpatrick and colleagues,
Mathematics classrooms are more likely to be places in which mathematical proficiency develops when they are communities of learners and not collections of isolated individuals. Research on creating classrooms that function as communities of learners has identified several important features of these classrooms: ideas and methods are valued, students have autonomy in choosing and sharing solution methods, mistakes are valued as sites for learning for everyone, and the authority for correctness lies in logic and the structure of the subject, not the teacher. (Kilpatrick, Swafford, and Findell 2001, 425)
Three attributes stand out in any classroom where I hear the hum of powerful student discourse: Conversations are
- student led; and
- rich with academic and domain-specific vocabulary.
Let’s listen in on each.
Discourse is Respectful
“I would respectfully disagree with Shanae because I think the square root of 136 would have to be between 11 and 12. What do you all think?”
Students take responsibility for their own learning and for supporting the learning of others. They focus their conversations on ideas, not personalities, and are supportive and collegial in discussion. Learners give and receive feedback, relishing opportunities to grow their thinking. Rather than saying “What?” and “I don’t get it!” they may use phrases such as
- I am wondering . . . ;
- Could you explain . . .; and
- I agree / disagree with ____ because . . . .
These conversation patterns take time to develop and require explicit teaching of academic vocabulary. When such discourse patterns are in place, they evolve from what author Peter Johnston calls the dynamic-learning belief system: one where students see themselves and their classmates as works in progress, capable of revising their thinking and growing their understanding in light of new evidence (Johnston 2004). This respect for self and others is cultivated by conscientious teacher talk targeting students’ sense of agency and flexibility.
Discourse Is Student Led
“That does not make sense to me. Can you say more?”
“Sure, my idea is that since 11 squared is 121 and 12 squared is 144, and 136 is in between . . .”
Learners lead the conversation, posing questions of one another that probe for thinking: “How?” “Why?” They listen and respond to the inquiries of their peers by explaining drawings, defending solutions, justifying answers, or expanding thinking. Learners take responsibility to ask questions when they are lost, and know that it is their job to explain effectively in order to allay the confusion of their peers.
The teacher facilitates patiently from the periphery, stepping in only as needed to propel the conversation forward. Posing purposeful questions, one of the NCTM’s Mathematical Teaching Practices, offers us an opportunity to “assess and advance students’ reasoning and sense making about important mathematical ideas and relationships” (NCTM 2014).
Discourse Is Rich with Academic and Domain-Specific Vocabulary
“So, you are explaining that because 11 to the exponent 2 is 121, and 12 squared is 144, and 136 is in between those two perfect squares, its square root needs to be between their square roots, between 11 and 12?”
Students engaged in high-level math discourse speak in complete sentences, using academic vocabulary as well as domain-specific mathematical terminology to describe their thinking. They avoid general pronouns, such as it or they, and instead refer to “the triangle on the left,” and “my table group, Estella and Marco.” Regular interaction with and use of Tier 2 and Tier 3 terms scaffolds students’ understanding, and for this reason, vigilant teachers encourage and promote precision of language during students discourse.
We can cultivate students’ fluency with academic and domain-specific language with our relentless and compassionate insistence on accuracy of speech. To this end, we can be willing to stop a student mid-conversation, and ask him to rephrase: “That? What is the ‘that’?” Though this habit can temporarily divert the conversation, it signals to all listening the value of precision.
As we insist, students will test us, but if we hold true to high standards and offer support within our math workshops, learners’ language will rise up to meet our highest expectations.