Imagine entering a second-grade classroom in which Ms. Candace, a preservice teacher (PST), has launched a number talk activity (Parrish, 2010). She is leading a whole-class discussion for the first time with 22 second-grade students, as her peers (10 PSTs), a teacher educator (TE), and a cooperating classroom teacher carefully observe how students engage with the mathematics. Ms. Candace has prepared for this moment of leading a class discussion by engaging in a series of training activities based on the cycle of enactment and investigation (McDonald, Kazemi, & Kavanagh, 2013) in the context of an elementary school-embedded mathematics methods course.
Described here are illustrative examples of work with PSTs starting with Ms. Candace, the preparation activities and learning cycle, and connections for TEs and coaches as they work with teachers to lead mathematics discussions. The experiences and claims are drawn from a larger study that investigated the preparation of PSTs in a school-embedded mathematics methods course. I discuss how teacher discourse practices can be developed through planning, rehearsing, and teaching number talks, highlighting specific talk moves that were worked on initially, as PSTs who were new to leading discourse often took up these practices. I also unpack complex situations that took place during number talks, such as attending to student errors, and highlight a common dilemma that both novice and experienced teachers face of knowing when ‘to tell’ during discussions. This paper expounds upon rehearsals in mathematics teacher education through the learning cycle, a practice-based approach to teacher preparation that engages PSTs in the everyday work of teaching (AMTE, 2017).
Ms. Candace’s Number Talk
Ms. Candace first shows the class a picture of a dot image (see Figure 1) and asks the class of second-grade students, “How many dots do you see?” She pauses for a few minutes as students begin to put their fists to their chests and lift up their fingers indicating the number of unique strategies they have come up with to the find the total.
Figure 1: Dot Image in Ms. Candace’s Number Talk Discussion
After recording student responses to the total number of dots for the class to see, Ms. Candace asks Markus to discuss his strategy for how he came up with an answer of 14 total dots (see Figure 2).
Figure 2: Markus’ Strategy in Ms. Candace’s Number Talk Discussion
Markus: If you add 4 and 4, that’s 8.
Ms. Candace: Where did you get the 4 and 4 from?
Markus: On top and the bottom.
Ms. Candace: This one (circling the top row). And where did you get the other one?
Markus: On the bottom.
Ms. Candace: Right here (circling the bottom row)?
Markus: Yes.
Ms. Candace: Okay, so what did you do with the 4’s?
Markus: I added them together to make 8. Then I moved one down to make another 4 (See Figure 2).
Ms. Candace: You moved an imaginary one down? (points at the empty space in second row) This one?
Markus: No.
Ms. Candace: Can you come up and show me?
Markus: (Comes up to the board, circles the second dot in the second row and draws an arrow down)
Ms. Candace: Okay, I see (Ms. Candace points to the image again, so all students can see).
Markus: And then I got 12.
Ms. Candace: How did you get 12?
Markus: By adding the 3 rows of 4
Ms. Candace: Okay, so you did 4 plus 4 plus 4 equals 12?
Markus: Yes, I added the fours together and then I added the two.
In this interaction, Ms. Candace aims to position Markus as competent and capable by restating his ideas and affirming his methods, and supports him as he makes his thinking visible for the class (NCTM, 2014). Ms. Candace uses multiple discourse or talk moves, which are “strategic ways of asking questions and inviting participation in classroom conversations” (Chapin, O’Connor, & Anderson, 2013, p.11) as Markus reasons through his process for finding the solution. This interaction exemplifies just one of many that PSTs enrolled in the mathematics methods course had with second-grade students as they learned how to lead number talk discussions within a learning cycle to support student mathematical ideas and reasoning.
Attending to Student Thinking within the Cycle
There is clear evidence that noticing, attending to, and responding to student thinking is essential to ambitious teaching and key to student learning (Kazemi, Franke, & Lampert, 2009). As PSTs and inservice teachers learn to lead discussions, it can be challenging to go beyond “show and tell” to facilitate discourse that intentionally supports students as they develop methods for solving problems, explain their thinking, connect strategies, and justify solutions (NCTM, 2014). Leading whole-class discussions involves teaching practices such as eliciting and interpreting student thinking, probing and building upon their reasoning, orienting student ideas toward each other, and making student representations visible (Boerst, Sleep, Ball, & Bass, 2011; Teaching Works, 2015). One approach that has garnered much interest in teacher preparation and also professional development settings is the use of a cycle of enactment and investigation for teachers to learn to enact practices and to develop knowledge for ambitious teaching. Within a learning cycle, teachers first observe and make sense of the talk moves involved in a number talk through watching videos or modeling from a TE or instructional coach. Together, they examine the practice and decompose it into smaller chunks. They then approximate practice or rehearse the activity as a TE or instructional coach provides support. In a rehearsal, the teacher leads a discussion with their peers, eliciting and responding to their thinking. The TE can pause the enactment by asking the teacher to revise his or her actions and try out a different talk move. The TE presses the teacher to think deeply about his or her role in attending to student thinking. The goal is for the teacher to become more adept at certain practices such as eliciting and responding to student thinking, and to continue to build their teaching skills.
To illustrate the learning cycle, I specifically focus on PSTs in a mathematics methods course, in which the cycle began as the TE modeled a number talk while PSTs discussed ways to flexibly move dots between two ten-frames to find the total number of dots (Figure 3). The TE used talk moves throughout the discussion in an attempt to simulate a classroom environment. Following the activity, the PSTs were asked to reflect on the experience of being a “student” during the number talk and also to reflect upon the different talk moves that were made to support their thinking. The goal of modeling was to help PSTs ‘see’ how a teacher carefully elicits and interprets student thinking, responds flexibly, and makes student thinking visible.
Figure 3: Ten Frames used in TE’s Number Talk
After seeing multiple representations of number talks, the PSTs were each given an opportunity to plan and rehearse a unique number talk. They completed a number talk protocol (adapted from Teacher Education by Design, https://tedd.org/) to explicitly define goals for the activity, anticipate student ideas, and identify talk moves that support student discourse.
In the rehearsals, the TE acted as a coach as the PSTs in the class acted as ‘students’ to approximate a classroom setting. The TE paused the enactment by asking PSTs to revise their actions and try out different talk moves with the aim of thinking deeply about their role in attending to student thinking. Additionally, the rehearsing PST was able to call a “teacher time-out” to think aloud and consider different instructional moves given the content and practice goals of the discussion (Gibbons, Kazemi, Hintz, & Hartmann, 2017). Following each rehearsal, the class engaged in a debrief in which the PSTs responded to the following two prompts: What went well for you? and What are you wondering? After this individual reflection, the whole-class debriefed together about how to support students to engage in discourse moves. Some of the debriefs included discussions about how to best support student discourse, student’s mathematical thinking, what PSTs learned about students as learners, and how the rehearsal process supported them in developing ambitious teaching practices.
Trajectory of Using Talk Moves in Enactments
The PSTs used four main talk moves during the whole-class discussions to promote student mathematics discourse, specifically elicit, ask for reasoning, revoice/repeat, and orient student thinking to each other or the content (see Table 1). The PSTs followed a consistent pattern in leading the discussion by first eliciting student solution strategies, revoicing or repeating their ideas, and then pressing students for reasoning. While it was less common, some of the PSTs worked on orienting students’ thinking to each other where they would connect student ideas.
Table 1: Examples of Talk Moves PSTs Used during Number Talks to Support Student Reasoning
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Elicit student ideas
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Press for reasoning:
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Revoice/repeat:
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Orient students to each other’s thinking or the content |
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The types of talk moves used encouraged student participation and allowed them to construct and reinforce their understanding of mathematical ideas. Students capitalized upon the opportunities to participate as they conversed with each other about strategies, and built a more complete understanding of the mathematical idea being discussed. As the PSTs became more familiar with number talks, the hope was that they would begin to layer in goals for what they wanted students to do. This learning cycle focused on PSTs interacting first with the one student in the discussion who provided the strategy, and then pressing that student for their reasoning. PSTs were encouraged to work on allowing contributions from multiple students to construct ideas, including clarifying and restating each other’s thinking, connecting strategies, and sequencing the methods to build upon each other.
Next Step, Addressing Student Errors
During the number talks, the PSTs were able to effectively elicit and press students for reasoning, and occasionally orient students to each other’s thinking, but they found it challenging to address student errors. As is the case with many teachers learning to lead discussions, making the appropriate in the moment decision, especially in response to student error, takes time to develop (Bray, 2011). The teacher is faced with many options that include: addressing (or not addressing) the error; correcting the error; guiding the student to realize the error; or bringing in other students to the discussion (Santagata, 2005).
When students made an error, some PSTs would continue to elicit and press the student for their reasoning with the intention that the student would realize the misconception. Another approach PSTs took was to elicit additional strategies from the class with the intention that a new strategy would address the previous misconception. For instance, in the following vignette, PST Ms. Anna leads a discussion on 31 + 25, in which Gerry explains how he added the tens place first and then the ones place, a method he had worked on in class.
Gerry: I added the tens first.
Ms. Anna: Could you tell me which tens?
Gerry: The two and the three, which is 50.
Ms. Anna: So you did tens, two plus three?
Gerry: Equals fifty. I added the 5 and the 1 which equals six, so now I know it is 61.
Ms. Anna: So you are saying that the tens is like, there is a zero right after it (points to 2 and 3), which is like 50, right? And then you did one plus five, and you got 6. And the total answer is?
Gerry: I think it is 61.
Ms. Anna: Okay.
In this interaction, Ms. Anna tries to address Gerry’s error by pressing him for his reasoning, but he restates his partial conception of adding two digit numbers. Ms. Anna turns to the class to see if someone else could restate Gerry’s strategy and revise the error. Another student offers a strategy by adding doubles, 30 + 30, and then subtracts four to come up with 56. At this point, Ms. Anna has elicited two unique student strategies, one that resulted in a total of 56 and another that resulted in a total of 61, yet she is unclear how to proceed with the two different solutions.
This interaction is one that all teachers can relate to and one that takes careful navigation, but also allows a great opportunity for learning. While both students took unique and solid approaches to reach their total, there is dissonance in the classroom with the final answer. It is in this tension between answers that teachers may be afraid to engage in or feel they must resolve the problem immediately. However, it is also a space where students are motivated to learn. Should the teacher ‘tell’ the student what to do and address the error, or allow for more discussion about the methods or ways of arriving at a solution? What a teacher does next may depend on the goal of the discussion, or the teacher’s comfort level with leading a number talk.
Teacher Learning
In this work, the number talk activity was the entry point for PSTs into leading whole-class discussions. The number talk format allowed elementary students to see themselves as capable learners who had the tools to be successful and were integral to the learning process. Rehearsals within the cycle of enactment and investigation allowed for PSTs to work on key discourse practices in a less complex setting, specifically the mathematics methods classroom, and then confidently enact their rehearsed talk moves in the elementary classroom as they encouraged students to formulate ideas and offer diverse explanations. Learning cycles such as this one may provide PSTs the opportunities they need to engage in and think deeply about ambitious teaching practices and potentially strengthen the connection between university and fieldwork experiences. TEs play an essential role as they develop pedagogies for PSTs to make sense of the interplay between the teaching practice, student thinking, and the mathematics content. Rehearsals may be considered an integral component for preparing PSTs as they enact teaching practices in their field-based experiences. It is through this iterative learning process of investigation, practice, and reflection, that PSTs develop proficiency in teaching.
As the use of rehearsals becomes a more common way to support teacher learning, in addition to applicability with PSTs, it may be considered as a strategy to be used with inservice teachers and instructional coaches in professional learning communities to help support the development of discourse practices. Following a similar structure, instructional coaches could facilitate professional learning communities in which they read about, watch video, and decompose different discourse moves, rehearse with colleagues, and then enact number talks with students, while receiving coaching throughout the process. Yet, another approach would be one in which inservice teachers observe each other enacting similar number talks in their classrooms, and then debrief the shared experience.
The use of rehearsals and enactments of number talks within the learning cycle was found to be a powerful form of preparation to support teachers in learning to lead discussions. TEs, instructional coaches, and teacher leaders could develop communities of practice in courses and schools centered on the cycle of enactment and investigation with the ultimate aim of supporting students as they engage in rich mathematical discourse.
References
Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. http://amte.net/standards
Boerst, T. A., Sleep, L., Ball, D. L., & Bass, H. (2011). Preparing teachers to lead mathematics discussions. Teachers College Record, 113(12), 2844–2877.
Bray, W. S. (2011). A collective case study of the influence of teachers’ beliefs and knowledge on error handling practices during class discussion of mathematics. Journal for Research in Mathematics Education, 42(1), 2–38.
Chapin, S. H., O’Connor, C., & Anderson, N. C. (2013). Classroom discussions in math: A teacher’s guide for using talk moves to support the Common Core and more, Grades K–6 (3rd ed.). Sausalito, CA: Math Solutions Publications.
Gibbons, L., Kazemi, E., Hintz, A., & Hartmann, E. (2017). Teacher time out: Educators learning together in and through practice. Journal of Mathematics Education Leadership, 18(2), 27-46.
Kazemi, E., Lampert, M., & Franke, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In R. Hunter, B. Bicknell & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia. (Vol. 1, pp. 12-30). Palmerston North, NZ: MERGA.
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National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies, grades K-5. Math Solutions.
Santagata, R. (2005). Practices and beliefs in mistake handling activities: A video study of Italian and US mathematics lessons. Teaching and Teacher Education, 21(5), 491–508.
Teacher Education by Design (TEDD). (2014). https://tedd.org/. University of Washington.
TeachingWorks. (2015). Leading a group discussion. Ann Arbor, MI: University of Michigan.