Practice-based teacher education has been suggested as one approach to help preservice teachers (PSTs) learn how to enact high-leverage teaching practices such as facilitating a whole class discussion (Association of Mathematics Teacher Educators, 2017; McDonald et al., 2013). McDonald et al. (2013) suggested a learning cycle that includes four parts: 1) introduce the practice; 2) prepare for and rehearse the practice; 3) implement the practice with students; and 4) analyze the enactment to inform future cycles. The rehearsal within the learning cycle provides space for the PSTs to practice enacting the instructional practice while receiving feedback from the mathematics teacher educator (MTE) (Kazemi et al., 2016). Due to program structure, course content focus, resources available, and the PSTs' needs, the rehearsal's enactment can look very different (Kazemi et al., 2016). The purpose of this article is to describe three rehearsals, each led by a MTE, that focused on facilitating a whole class discussion and the similarities and differences as well as the affordances and limitations we noticed across the vignettes.

Vignettes of Rehearsals in Teacher Education

Rehearsal with Peers

Author 2 currently teaches mathematics content courses for elementary and middle school PSTs at a Hispanic-serving institution in the United States's southern region. She regularly engages her PSTs in face-to-face rehearsals with their peers. To prepare for the rehearsal focused on whole class discussions, PSTs first read about and discussed math talk moves (Stein, 2007), then watched teaching clips and identified the math talk moves used by teachers in those videos (e.g., Huinker & Bill, 2017). The instructor then modeled how to incorporate several talk moves during task implementation. Next, the PSTs selected and/or adapted a cognitively demanding mathematical task aligned with specific content and process standards. Following this, PSTs engaged their peers with the task while incorporating three talk moves (e.g., revoice/rephrase, ask open-ended questions, press for reasoning). PSTs then wrote a reflection about areas of success and growth and modifications for the next rehearsal. For example, one PST wrote in their reflection, “I think my questions were mostly open ended but I did not do a very good job pressing for reasoning. I guess I was not too sure what kinds of follow-up questions I could ask to bring out students’ reasonings. This is something I need to work on. I feel this is the hardest area.”

Rehearsal using Teacher Moments

Author 5 currently teaches middle school and high school mathematics methods at a predominantly white institution located in the northeast region of the United States. He uses digital clinical simulations (Teacher Moments) as a form of practice that allows PSTs to navigate the delicate balance of authority and equity as they become attuned to how their decision-making when facilitating a whole class discussion can influence the way students access and engage with mathematics. For this simulation, PSTs logged on and engaged with a scenario that challenged PSTs to interact with four virtual student groups solving the same mathematical task. Each PST was presented with student work showcasing different approaches to solving a task (i.e., the group got stuck, the group used guess and check to solve correctly, the group used an algorithm, and the group had a (mis)conception). PSTs were then faced with the decision of selecting a group to start the whole-group mathematical discussion while explaining their choice.

In terms of authority, instructional moves influence who holds speaking rights to share their thinking, how what is shared is valued in the mathematics learning space, and who is granted access to engage with tasks that require a high level of cognitive demand (Langer-Osuna, 2018). PSTs used the simulation to reflect on how their decisions about instructional moves shift power toward perpetuating or disrupting a culture of exclusion within their own classrooms (Battey et al., 2016). Recognizing and reflecting on how their choice of instructional moves relate to power can support PSTs’ understanding of how their decisions may position particular ways of doing and learning about mathematics as more valued. For example, after the simulation experience, a PST reflected on how a teacher’s perception may perpetuate a culture of exclusion in a classroom or put the labor of including students’ thinking on other students, saying, “It’s probably not a great thing to do, but if a teacher is perceiving [something negative about a student], there’s always going to be someone who can play clean up, or other kids respond to that student differently. Maybe that is what is implied here [in the simulation.]”

Rehearsal using Virtual Simulations

Author 3 engaged her elementary PSTs enrolled in a 3rd-6th grade focused mathematics methods course in a rehearsal to practice leading a whole class discussion using Mursion, a software that provides virtual training simulations for teachers and educators (www.mursion.com). First, PSTs read “Unwrapping Students’ Ideas about Fractions” (Lewis et al., 2015) to provide an example of an inquiry-based lesson. PSTs then solved the initial problem posed in the article, “6 children are sharing 8 small sandwiches. They are sharing so that each child gets the same amount. How many sandwiches will 1 child get?” (p. 159) and anticipated how students would solve the problem. Next, they used the student work samples provided in the article to record assessing and advancing questions (NCTM, 2014). Lastly, PSTs worked in small groups to select and sequence 2-3 student work samples and outlined a hypothetical discussion.

In the next phase of the learning cycle, PSTs were given five pieces of student work on a similar equal-partitioning task to help them prepare to lead a discussion. They then facilitated a 15-minute discussion based on the student work through Mursion. During the discussion the PSTs interacted with five avatar students who were controlled by a simulation operator. The MTE joined each PST in their virtual lesson and provided feedback on the questioning used and the alignment of questioning and prompts to goals. PSTs completed the learning cycle with a reflection assignment in which they provided personal analysis of their discussion.

One common theme throughout the PSTs’ reflections included successful use of talk moves to engage students; highlighted as one PST wrote, “I think I did well at prompting students and having them restate and add onto their peers’ strategies by having students repeat each of the three strategies I went over.” On the other hand, many PSTs commented on their desire to tell less and prompt connections from their students. One PST shared, “I feel that I connected between strategies well. I was able to connect each student’s strategy to the next… I want to improve on my use of the applying reasoning talk moves… so that students were able to see/create the connections themselves rather than me explaining or facilitating the connection.” The experience with avatar students seemed to provide an opportunity for PSTs to examine the different types of prompts used and the depth those moves provided.

Similarities and Differences

Taken together, these vignettes demonstrate how rehearsals can be embedded into courses ranging from mathematics methods to mathematics content courses, as well as across grade bands. They also illustrate how rehearsals can vary in the amount of technology and resources required. Rehearsals can just as effectively work with or without technology such as working with peers or low-tech (e.g., Teacher Moments) or high-tech live simulations with avatars controlled by an actor in real-time (e.g., Mursion).

Despite the variation in enacting rehearsals, three key similarities surface from these vignettes: the importance of having PSTs analyze and reflect on the enactment, centering rehearsals around an instructional practice such as facilitating math discussions, embedding rehearsals into a course through a series of scaffolded readings and activities. While all MTEs included a reflection, the reflections focused on a different aspect: (1) student needs, (2) self-reflection focused on implicit biases, and (3) analysis of the questions asked. Such variation suggests that the focus of the reflection is an important consideration in designing rehearsals.  

Finally, the three vignettes highlight a similar goal but different ways of integrating an equity lens into rehearsals. Namely, the first vignette shows how rehearsals can guide PSTs toward equitable mathematics teaching through differentiated tasks to meet the needs of diverse learners. The second vignette shows how an MTE can guide PSTs to analyze equity in terms of power dynamics and building cultures of inclusion. The third vignette focuses on equity in terms of facilitating meaningful discussions that elicit and make use of students' mathematical thinking.

Affordances and Limitations

One affordance with a rehearsal is that it can be adapted to meet the needs of the PSTs and the goals of the MTE. For example, Author 2’s rehearsal highlighted talk moves and differentiation, Author 5’s rehearsal focused on power dynamics, and Author 3’s emphasis was content and pedagogy. Further, the teaching practices, while all related to facilitating a whole class discussion, varied: Author 2’s on talk moves, Author 5’s on power dynamics, and Author 3’s with the 5 Practices. We also noticed that peer rehearsals like Author 2’s can be easier and likely more affordable to implement; however, they are less predictable in how peers act and respond when participating in the rehearsal. Virtual approaches such as Author 5’s or Author 3’s can be more controlled, but MTEs and their PSTs need access to such resources. Further, the highly trained interactors in digital resources, such as Mursion, can help to ensure that the precise pedagogical challenge is raised, but there is less flexibility in the task, whereas less trained interactors, such as peers, in face-to-face rehearsals allows for more personalization but at the risk of bias and inconsistency.

Conclusion

The similarities/differences and affordances/limitations identified across these three vignettes help to demonstrate the need for many different types of approximations and theoretical grounding to support mathematics teacher educators in choosing different approaches. Further research is needed to understand how to sequence different kinds of simulations and the affordances or limitations of different sequencings. For example, Mikeska et al. (2022) begin to take up the question of how different types of simulations might be sequenced together for optimal impact. Additionally further study is needed to understand how to coordinate virtual and in-person rehearsal experiences to maximize PST learning. Our work here demonstrates some of the ways MTEs have interpreted and enacted rehearsals in their own courses.

References

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National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring  Mathematical Success for All. Reston,VA.

Stein, C. A. (2007). Let's talk: Promoting mathematical discourse in the classroom. The Mathematics Teacher, 101(4), 285-289. https://doi.org/10.5951/MT.101.4.0285