The Algebra 2 students wrestled with the concept of e in their very first lesson that included this irrational number. After an exploration that led students to notice that e is a constant (rather than a variable), the teacher asked the students why they thought e was represented as a letter rather than the number 2.7 (its approximate value). One student replied,“Because it’s irrational.” The teacher probed further: “What does irrational mean?” The student responded, “It’s not a real number.” 

A quick refresher for those of you who are far removed from Algebra 2: e is a real number! It is irrational (like 𝛑 and √2) meaning it cannot be expressed as a ratio (fraction)—hence ir-rational, or not rational. The only numbers that are considered unreal, or “imaginary,” are those that contain i, which is defined as the √-1. 

Hopefully we haven’t lost you. The point of this anecdote is to illustrate how important it is to let students talk, to ask probing questions, to give them the space to work out ideas and misconceptions. If the teacher hadn’t asked a follow-up question, she might never have found out that the student wasn’t secure in his understanding of rational and irrational numbers, as well as real and imaginary numbers. 

Talk in math class is invaluable. Discourse goes well beyond giving students the chance to talk to demonstrate their understanding. Indeed, the instructional shifts emphasized by the NYC DOE include moving away from “Discourse to Demonstrate Understanding” and towards “Discourse to Develop Understanding.” 

Research, summarized beautifully in A Discourse Primer for Science Teachers (2015), shows us that people process through speaking. We don’t (usually) spit out fully formed thoughts, but rather work through ideas as we speak. If we don’t give students the chance to talk, we’re cutting short their opportunity to process ideas. 

Discourse has several other benefits as well, each of which is summarized in the green column below:

Talk (in math class) helps students process ideas; it develops their language and social skills; it creates a classroom community where knowledge is shared and students’ ideas act as resources for other students. As demonstrated in the Algebra 2 classroom, it makes student thinking public, giving teachers and other students the opportunity to address misconceptions. 

A culture of talk also shows us who is silent. We’ve all experienced the student who does not want to talk or share, despite a classroom full of chattering students. A student’s silence leads us to question why they are silent: Do they feel unwelcome or unsafe in the class? Do they lack foundational skills to help them access the problem? Do they have something going on outside of the classroom that is interfering with their learning? Each of these questions then gives us an opportunity to further help the student feel like they belong in our math class and are capable of success (which are closely intertwined feelings). Imagine how much we would miss if we only allowed students to talk when called on to provide an answer! 

Think about a time one of your students said something that caused an “Aha” moment for you. How did you create the space for them to share? What did their words tell you about them, about what they understood or didn’t understand, about how you could help them feel successful and confident, inside or outside of the classroom? We’d love to hear your experiences with talk in the classroom.