Pre-service teachers enter their education courses with their personal experiences as the basis for what their future classrooms might look like (Brown et al., 2021). For many, teaching and learning math involves algorithms, textbooks, paper and pencil, and getting to that one right answer. To overcome these mathematical and pedagogical assumptions, I have my future teachers participate in number talks.
Number talks have been part of the elementary classroom since the 1990s. Created by Kathy Richardson and Ruth Parker, number talks “engage students in meaningful mathematical discourse and sense-making as well as transform the culture of the classroom to one of inquiry and curiosity” (Math Perspectives, 2022). During a typical number talk, students are presented with a math equation, problem, or scenario that the teacher feels the students can access. They are given time to mentally work to solve the problem in any way that makes sense to them. When students feel they have an answer, they give a small thumbs-up on their chest. This modest sign allows the teacher to get a sense of how the class is progressing without distracting others from continuing their thinking. When most students are ready, the teacher calls on a few students to share their answers. After these answers have been recorded, the teacher asks for volunteers to share how they came to their answer. This is the heart of number talks.
As the learner describes their mental process, the teacher records the math for all members of the class to see and prompts with questions to better understand the student’s mathematical reasoning. This process is as much for the learner as it is for the other members of the class. For the teacher, providing access to such learning experiences can offer insight into students’ mathematical conceptions (Callahan, 2011). For the learner, this justification process is an opportunity to develop their mathematical agency—their perception of themselves as doers of math (Bieda & Staples, 2020). For members of the class, hearing and seeing other’s strategies can prompt their own reasoning and expand their own mathematical learning (McGlynn-Stewart, 2010). It is less important that the class comes to a consensus on the “correct” answer or the “right” strategy. The justification process provided by number talks is the end as much as the means.
During a recent number talk with future teachers, I began by showing the following domino (see Figure 1) and then asked how many dots there were. After providing wait time and collecting answers, I asked the pre-service teachers how they had found the total.
Mel (pseudonym): Well, there’s three rows of 3, so 3, 6, 9—
Shelby (pseudonym): And then you had to add on the other 6 on the other side.
Author: And how did you add that on?
Shelby: Uh—I had to count with 1-1 correspondence, so I did 10, 11, 12 for the top three.
Author: So you counted-all for the top?
Shelby: Yeah, and then I saw the bottom altogether and I subitized it as three and went up to 15.
Author: You didn’t count the bottom three to know there were three?
Shelby: Right, I just went from 12 to 15.
Author: Cool, did anyone get their answer a different way?
Mel: I did 9, and then 12 [made a loop in the air with their finger], 15 [made another loop in the air with their finger].
Author: So you saw the top row as a whole and just jumped up 3, then did the same with the bottom row?
Mel: Yeah, that’s it.
Author: Did anyone solve it differently?
Ellis (pseudonym): I saw it as 9 plus 6.
Author: Oh, so you didn’t count them at all?
Ellis: Yeah, and I think about, whenever I add anything plus 9, I look at the next number and it’s one less [made a swiping motion from the 6-dot side to the 9-dot side] so 5.
Author: So you took one from the 6—
Ellis: Yeah, then it would be 15.
Author: How?
Ellis: Um, my brain just does that. I don’t think about it.
Author: Okay, so you took one from the 6 and put it with the 9, which did what?
Ellis: Made it 10.
Author: And then what?
Ellis: Plus the other 5 is 15.
For me, as the instructor, implementing this number talk revealed how my students were applying the early number concepts we had been discussing in class. Shelby, for example, described her counting through 1-1 correspondence (acknowledging each single dot) and her grouping through subitized arrangements (using the memorized pattern of three dots in a row as a single group). After completing the number talk, we debriefed the mathematical reasonings they had presented. I was able to call attention to the transition from the early number concepts that they had verbalized into additive reasoning, specifically Shelby and Mel’s use of counting-on and Ellis’s use of decomposition.
For the learner, sharing in this number talk fostered their reflection on both their mathematical and pedagogical content knowledge. Mathematically, the number talk provided an opportunity for the learner to justify their numerical reasoning and begin verbalizing how they operated with different quantities. Pedagogically, having the opportunity to focus mathematical learning on verbal justification over written, “correct” answers, exposes the learner to a different way of considering their own, future instruction.
For future teachers as a whole, participating in number talks can encourage their awareness of others’ mathematical reasoning. This practice can support their attention to their future students’ mathematical conceptions, allowing them to consider how to promote their students’ learning in a more systematic way. As a mathematical tool, number talks can support future teachers’ numerical reasoning as they reflect on and verbalize their own mathematical processes. As a pedagogical tool, number talks can encourage future teachers to notice their students’ reasoning and promote justification as a critical aspect of the mathematics classroom. For examples of number talk questions visit http://www.sfusdmath.org/sample-math-talks.html and https://www.kentuckymathematics.org/vr_nbrtalks.php.
Amy Smith, PhD, is an assistant professor in the Department of Education at Stetson University, specializing in elementary mathematics education.
References
Bieda, K.N., & Staples, M. (2020). Justification as an Equity Practice. Mathematics Teacher: Learning and Teaching PK–12, 113, 102-108.
Brown, C. P., Barry, D. P., Ku, D. H., & Puckett, K. (2021). Teach as I say, not as I do: How preservice teachers made sense of the mismatch between how they were expected to teach and how they were taught in their professional training program. The Teacher Educator, 56(3), 250-269.
Callahan, K. M. (2011). Listening responsively. Teaching children mathematics, 18(5), 296-305.
Kentucky Center for Mathematics. (n.d.). Number Talks. https://www.kentuckymathematics.org/vr_nbrtalks.php
Math Perspectives Teacher Development Center. (2022). Number Talks. http://mathperspectives.com/number-talks/
McGlynn-Stewart, M. (2010). Listening to students, listening to myself: Addressing pre-service teachers’ fears of mathematics and teaching mathematics. Studying Teacher Education, 6(2), 175-186.
San Francisco Unified School District Math Department. (n.d.). Sample Math Talks. http://www.sfusdmath.org/sample-math-talks.html