Introduction

Mathematics teacher educators’ (MTEs) post-teaching or debriefing conversations with preservice teachers (PTs) include providing “guidance and feedback” (Lampert et al., 2013, p. 231) and “avoiding judgments” (Brown et al., 2020, p. 89), while focusing on the lessons’ mathematics. Debriefing conversations after an “educative experience” include a “purposive discussion” of experience (Lederman, 1992, p. 145) for reflection (Pearson & Smith, 1985) that produces learning. Yet, debriefing conversations can involve power and control issues that limit PTs’ reflection (Sundli, 2007). To interpret our interactive moves during debriefing conversations, we share findings from a self-study of our debriefing practices. We argue that MTEs use the voice of the discipline (D’Ambrosio, 2004) of mathematics teacher education to inform interactive moves during debriefing conversations.

Study Context

We consider ourselves “constructivist teachers” (Steffe & D’Ambrosio, 1995, p. 148) who create models (Ulrich et al., 2014) of PTs’ “ways of operating as they confront situations in mathematics teaching and learning” (Kastberg, 2014, p. 353). Such models are developed through our interactions with PTs learning about teaching. At the time of the study, we were four constructivist mathematics methods teachers at Purdue University teaching mathematics methods in an undergraduate teacher education program. We supported PTs’ mathematics teaching using “children’s mathematics” (i.e., mathematics children develop; Steffe, 1994, p. 132). Models of PTs using children’s mathematics and our cultures, knowledge, and experiences inform our teaching of mathematics teaching (Loughran, 2007). Sue-Ellen and Signe are white Americans and long-time collaborators, with experience teaching post-secondary mathematics. At the study’s beginning, Sue-Ellen had taught mathematics methods (10 years) and early childhood (5 years). Signe had taught mathematics (10 years) and mathematics methods (15 years). Mahtob is Iranian and had taught methods once and mathematics in Iran. Lizhen is Chinese with a linguistics background who had not taught methods. Sue-Ellen, Mahtob, and Lizhen were doctoral students and Signe was a mathematics education professor during data collection. As novice MTEs, Lizhen and Mahtob were attentive to Signe and her experience creating a power differential in the team.

MTEs play a significant role in PTs’ views of mathematics (Drake, 2006), related emotions, and mathematics teaching (Bibby, 2002; Jenßen et al., 2021). Purdue University required nine mathematics credits prior to a three-credit mathematics methods course in preparation for student teaching. Methods built from PTs’ mathematics experiences and included an elementary school practicum (early-field) with nine, four-hour sessions. During the practicum, PTs interviewed elementary students, conducted whole-class assessments, co-taught one mathematics lesson, and solo-taught another. PTs engaged in one debriefing conversation. Using pre-planned priming questions, like “Tell me about what you found exciting in your lesson,” and “Tell me about a student’s mathematical idea that you were curious about during the lesson” (critical-friend dialogue, Sept. 19, 2018), MTEs initiated PTs’ reflection on teaching and attended to PTs’ interpretation and use of children’s mathematics to inform teaching decisions.

PTs’ early-field teaching provides opportunities for reflection on teaching (Brown et al., 2020). MTEs can support PTs (Stanulis & Floden, 2009) through debriefing, yet there is unequal power and control (Sundli, 2007) in such interactions. Additionally, MTEs may struggle to engage PTs in wondering about teaching rather than noticing (Roller, 2019) and retelling events. Brown et al. (2020) illustrated how debriefing conversations emerge as PTs and MTEs share views of events and potential action. Martin and Russell (2018) described teacher educators’ stances in debriefing conversations, highlighting differences between transmission and interpretive stances. While debriefing conversations can promote PTs’ reflection on teaching (Graham & Young, 2000; Pearson & Smith, 1985), how MTEs experience such conversations is less clear. We use self-study methodology to document, interpret, and improve our debriefing conversations.

In debriefing conversations our interactive moves built on constructivist listening (Weissglass, 1990) to motivate PTs’ development of “authority of experience” (Munby & Russell, 1994, p. 92). Listening is the “main component of constructivist teaching” (D’Ambrosio, 2004, p. 136). MTEs create models of PTs’ mathematics teaching by listening to their descriptions of mathematics teaching. D’Ambrosio (2004) described three voices involved in listening during teacher-learner (i.e., MTE-PT in our study) interactions: voice of the discipline (VOD), voice of the learner, and the teacher’s inner voice. Due to space considerations, we focus on MTEs using their VOD of mathematics teacher education. VOD includes MTEs’ experiences, practices, and pedagogical knowledge of mathematics teacher education as well as “ways of thinking, strategies and understanding” (D’Ambrosio, 2004, p. 137) of mathematics teacher education. For example, an MTE’s experience teaching and learning mathematics, such as subtraction, informs how she observes and listens to PTs’ accounts of teaching subtraction.

Aims/Objectives

The goal of the self-study was to document, interpret, and improve our debriefing practice. The question guiding our inquiry is: How do MTEs use the voice of the discipline in debriefing PTs’ mathematics teaching?

Methods

Analysis of our debriefing conversations using the D’Ambrosio’s (2004) VOD is part of a project focused on improving debriefing practice. Data collection and analysis (2017 – present) were challenged by MTEs’ life events including COVID-19, completing PhDs, and beginning new positions. From 11 recordings of four MTEs’ debriefing conversations with different PTs (spring and fall 2018), each MTE selected one debriefing conversation.

Our self-initiated, improvement-aimed, interactive study utilized self-study methodology (LaBoskey, 2004). Two monthly critical-friend dialogues were instrumental as provocations for reframing and interpreting our debriefing experiences. Critical-friend dialogues involve “interchange of thought or talk” (Placier et al., 2005, p. 57) through cycles of critique and inquiry. Dialogues build knowledge by exploring divergent views and meanings of convergent views. We acknowledge the power differentials in the team. Signe led the critical-friend dialogues, while Sue-Ellen acted to mediate Signe’s dominant voice by interjecting questions and inviting Mahtob and Lizhen to contribute.

We applied three qualitative methods to two data sources (debriefing conversations and related critical-friend dialogues). First, we used D’Ambrosio’s (2004) VOD to analyze one debriefing conversation per participant during a critical-friend dialogue (Placier et al., 2005). Dialogues produced retrospective accounts of debriefing conversations and VOD uses. We discussed factors including time, place, relationships, and the VOD that informed our actions. Second, we separately analyzed the four debriefing transcripts using the VOD. We engaged in analytic dialogues (Guilfoyle et al., 2004; Placier et al., 2005) of our analysis resulting in three categories of VOD uses during debriefing conversations: bridging, exploring, and telling. Third, to address educative authenticity (Grant & Lincoln, 2021) of our interpretive inquiry, we used the categories to analyze our critical-friend dialogues. This step addresses what we learned and how we came to know about our debriefing practice. All four debriefing conversations contained evidence of categories of VOD uses, yet we used Mahtob, Lizhen, and Signe’s debriefing conversations in this report.

Outcomes

MTEs used the VOD in debriefing conversations in three ways: bridging, exploring, and telling. We represented these interactive moves in vignettes using excerpts from Mahtob’s, Lizhen’s, and Signe’s debriefing conversations and critical-friend dialogues.

Bridging: Mahtob’s Debriefing

For bridging, the MTE uses two ideas: (1) one from her VOD and (2) another from her hypothesis regarding the PT’s view of teaching and children’s mathematics. Using these two ideas, the MTE attempts to create an opportunity for a PT to reflect on teaching or children’s mathematics. Importantly, while the MTE may have these two ideas, her attempts to bridge may be unsuccessful in encouraging such reflection.

Mahtob’s debriefing conversation followed her observation of Allie’s lesson using base-ten blocks and procedures to solve two-digit subtraction problems. Throughout the course, Mahtob was challenged by Allie’s approach to teaching as telling and demonstrating procedures, since Mahtob viewed children’s mathematics as developing through challenges to their existing concepts. Allie modified her lesson plan to address Mahtob’s suggestion that third graders use base-ten blocks to model numbers and solve subtraction problems. In the excerpt, Allie noted that the child’s answers using the blocks and the procedure were different.

Mahtob: What was the most exciting or interesting part of your lesson?

Allie: I think when I was looking at the papers they [children] were turning in. Cause you suggested to do the algorithm and do the blocks and have them solve it differently, and a lot of the kids had a right answer for the block method and the wrong answer on the same page using the algorithm. So, it was really exciting.

Mahtob: I see. So, was that the exit ticket that you are talking about?

Allie: … Yea it was really exciting to see that they were really using that [block method].

Mahtob: Yea, cause I think so far that’s what you have been observing with them [children]. That they’re not...

Allie: [finishing Mahtob’s sentence] Getting carrying over [regrouping tens to ones in the standard algorithm]

Mahtob: Yea. 

(Mahtob-Allie debriefing, fall 2019)

In the debriefing conversation, Mahtob used her idea from her VOD regarding children’s interpretation of ones and tens place in two-digit numbers. After Allie described the children’s actions, Mahtob used her VOD in relation to Allie’s noticing the differences in children’s use of blocks and procedures. This effort to bridge created a dialogic space for Allie to reflect on the children’s use of blocks and the subtraction procedure. During the critical-friend dialogue, Signe and Lizhen affirmed Mahtob’s efforts to bridge using her pedagogical knowledge that mathematics teaching should use children’s mathematics. Signe described Mahtob’s agreement with Allie as “letting her talk” (critical-friend dialogue, June 4, 2021). Mahtob described the space as an opportunity for Allie to reflect on the children’s work and share new ideas about their thinking. Yet, Mahtob was concerned. “I don’t want to push too hard…. because I felt like when I’m doing that, she just stopped engaging….” Mahtob described trying to bridge to Allie’s interest in the children’s engagement with the blocks and her challenge to keep the children focused. “...I was trying to redirect her to talk about the challenges [in teaching] mathematics or children’s mathematics that she found.” Mahtob eventually realized that her bridging move would not result in Allie attending to the children’s mathematics:

So, during the debriefing, I wanted to point it [children’s mathematics] out. But I think she was excited about the lesson. I just didn’t want to bring it up into her face and say, ‘I don’t think so.’ I was hoping that when she analyzed her students’ work, she might actually get to that [children’s mathematics] (Mahtob’s critical-friend dialogue, June 4, 2021).

Mahtob’s responses to Signe’s and Lizhen’s perspectives on the VOD made explicit her difficulty eliciting Allie’s reflection on the children’s use of regrouping and blocks. Mahtob’s bridging move drew from children’s mathematics she saw to her hypothesis about Allie’s focus on a subtraction procedure. Although Mahtob elicited Allie’s description of the differences between the children’s use of the blocks and the procedure, Allie returned to the importance of teaching, regrouping, and managing the children’s behavior. Not wanting to risk Allie disengaging, but with her idea from the VOD still in mind, Mahtob hoped her questions would provide Allie an opportunity to connect her observations to children’s mathematics.

Exploring: Lizhen’s Debriefing

As constructivist teachers, one goal during the debriefing conversation was identifying ideas within the PTs’ description of teaching that related to our VOD. An MTE uses ideas from her VOD to hear and see PT’s teaching. In exploring, the MTE uses her ideas from the VOD to create questions or statements in her search for evidence of the PT’s view of teaching and children’s mathematics. In contrast with bridging, exploring requires only the MTE’s idea(s) from the VOD, to identify ideas from the PT’s description of teaching. Exploring can transition into bridging if the MTE hypothesizes a PT’s idea.

Lizhen’s debriefing conversation used ideas from her VOD involving children’s fraction thinking. Exploring was cognitively challenging for the MTEs, including Lizhen. Simultaneously holding her idea about children’s fraction thinking in mind while listening to Mandy’s description of teaching for evidence of children’s mathematics was a significant challenge. Lizhen’s debriefing conversation with Mandy and her teaching partner was informed by a folding activity with string used in the methods course. Their lesson included using paper folding to generate fractions. They used two sets of fractions (1/2, 1/4, 1/8 and 1/3, 1/6, 1/9) with their fifth graders. From class discussions, Lizhen anticipated that Mandy would use children’s mathematics in her teaching.

Lizhen: First, tell me what you’re excited about?

Mandy: I’m excited when they [children] wanted to move on. Like if they saw that we did a half, they wanted to go ahead and start doing the fourth. They just came up with different ideas, how to get there, especially when we started doing the third, the sixth, and the ninth. Because you could see it clicked, when they realized how to get from one [fraction] to the next, to the next. So, I thought that was really exciting to see.

Lizhen: Yeah, I remember you talked about the fourth in Tuesday’s class on campus. You said, ‘OK, if this is half then you fold it [string] one time; if it’s third then you fold it [string] twice.’ When students do half, fourth, or eighth, if they build on their previous problem, they just need to fold one more… But coming to one-third, one-sixth, and one-ninth, that will be different? What did you find about students’ thinking when they folded these three fractions?

Mandy: They did find the third. They recognized if you fold a third in half, you get a sixth. It was the ninth, a lot of them first said, ‘Oh you just take a sixth and fold it in half again.’ So, a lot of them were getting one twelfth, which was to be expected. But once we talked about the multiplication, when they fold it in half again, that six [parts] turns into twelve [parts on the paper strip].

Lizhen: Yeah, how did the students fold one ninth?

(Mandy’s teaching partner responded.)

Lizhen: Ah, did any student fold one ninth right away?

Mandy: I didn’t have anybody who did it right away.

Lizhen: So, they built on the previous problem?

Mandy: Yes, they just continued trying to do like the half, the fourth and the eighth. You fold it in half, then you fold it in half, you have fourth; then you fold it in half, you have eighth. So, they figured it was the same way [by folding the paper strip in half] with the third, the sixth, and the ninth.

(Lizhen-Mandy debriefing, fall 2019)

Lizhen’s ideas from her VOD were grounded in children’s fraction thinking. One idea was how using halving to create 1/2, 1/4, 1/8 might be overgeneralized for fraction sequences like the second set (1/3, 1/6, 1/9). Lizhen considered how the PTs conceptualized children’s fraction thinking. Lizhen’s idea from the VOD motivated her to explore the PTs’ view of children’s fraction thinking using questions. Mandy described children’s actions (i.e., paper folding) rather than their mathematics. She further described her demonstration of multiplication to show how halving a six-part paper strip creates twelve parts. Lizhen continued to explore Mandy’s experience for evidence of children’s fraction thinking. Mandy repeated her ideas about the children’s error without addressing the children’s thinking. Thus, Lizhen stopped exploring.

During the critical-friend dialogue, Signe identified the PT’s second fraction set as “unusual” (critical-friend dialogue, June 11, 2021), noting, as Lizhen had, that the halving strategy children developed for the first fraction set would be problematic when applied to the second set. Signe referenced Lizhen’s focus on fraction thinking from the VOD. Signe further described Lizhen’s questions and requests for information as exploring Mandy’s view of children’s fraction thinking in relation to creating one-ninth. Mandy’s reference to multiplication did not deter Lizhen from using her VOD to question PTs about children’s fraction thinking.

Telling: Signe’s Debriefing

MTEs may use telling after unsuccessful bridging or exploring attempts. Like bridging and exploring, telling begins with the MTE’s idea from the VOD. However, rather than attempting to use or identify hypotheses about PTs’ ideas, the MTE uses her ideas from her VOD to motivate PTs’ reflection.

Jason planned to teach a two-digit addition lesson to first graders using base-ten blocks and problems like 33 + 11. Signe’s written feedback on the lesson pointed out children’s potential confusion with double-digit numbers. “Jason, I encourage you to consider using different digits for the tens and ones position so you can see how the children are differentiating the digits” (Signe’s feedback to Jason, Nov. 2, 2018).

During Jason’s lesson, Signe observed that Jason had not addressed her feedback regarding 33 + 11. Signe attempted to center her thinking on Jason’s description of teaching and children’s mathematics, hoping Jason might address the problematic nature of numbers like 33 and 11. However, after several attempts to explore and then bridge, with time running out, Signe resorted to telling to motivate Jason’s reflection on the role of the digits.

Signe: So, one thing I wondered about is, what made you feel mathematically curious when the students were saying mathematics to you?

Jason: I would just say once again [Jason had already described his curiosity during the debriefing] …The difference between tens and ones. I mean, that’s what the whole lesson was about. That’s what I’m most curious about. If she just saw the number three, and was like, ‘alright, the tens place is three.’ That they’re actually thinking ‘that means thirty.’

Signe: Yeah, so that’s a guideline for you. So, if you’re like, ‘oh, ok, I’m curious about that,’ then that’s something that you REALLY need to find out about with the kids. Because that’ll really motivate you to focus.

One comment I had for you, building on from that. When they say three, and the number’s thirty-three, how will you know? So, I know you think hard about these things, so I really wondered about the choice of the numbers, because it’s thirty-three, both digits are the same. So, if the students report digits, then it’s not clear to me that you’ll know which one they mean. So maybe strategically selecting the number that you’re using so that both digits, the ones and the tens, aren’t the same digit.

Jason: Yeah, that’s smart.

(Signe-Jason debriefing, fall 2018)

In this excerpt, Jason wondered, as he had earlier in the debriefing, whether the child interpreted a three in the tens place as 30. Signe used her VOD to explore Jason’s description of teaching for his ideas about children’s place-value thinking. Signe also used her VOD to bridge Jason’s curiosity about children’s use of tens and ones and the classroom teacher’s use of a place-value chart and choral counting.

In our critical-friend dialogue, Signe reflected on her difficulty eliciting Jason’s reflection on the children’s challenge in using base-ten blocks. She described her difficulty eliciting Jason’s reflection on using children’s choral counting to support development of a unit of ten. Jason responded with an idea for teaching addition using a number line. Signe elicited Jason’s idea about children’s meaning for digits multiple times during the debriefing and described deciding to “share (her) intellectual and academic expertise” to “support him to see a way forward” from his idea (critical-friend dialogue, July 2, 2021). Signe’s idea about digits from her VOD did not help her create a situation that could motivate Jason’s reflection on children’s place-value thinking. So, she had to “just go to straight up giving advice” (critical-friend dialogue, July 2, 2021), i.e., telling. While Jason agreed that Signe’s suggestion was a “smart” choice, our critical-friend dialogue revealed that Signe questioned and was uncomfortable with her use of telling.

Discussion

We have focused on the MTE-PT debriefing, a form of post-teaching conversation, to investigate ways of supporting PTs’ reflection on their field teaching experience. D’Ambrosio’s (2004) VOD situates teachers’ acts of meaning during interactions with learners within the teacher. Applied to debriefing conversations, D’Ambrosio’s VOD situates MTEs’ acts of meaning during debriefing conversations within the MTE. From this view, we hypothesized that MTEs use the VOD to inform interactions during debriefing conversations. Our self-study produced three interactive moves informed by our VODs: bridging, exploring, and telling.

Defining the interactive moves of bridging, exploring, and telling allowed us to identify relationships among the moves, as MTEs worked to create reflective opportunities for PTs during debriefing conversations. For example, Mahtob’s bridging move to create reflective opportunities for Allie involved Mahtob’s use of her VOD regarding teaching to connect with Allie’s description of children’s subtraction procedures and base-ten blocks. Similarly, the goal of Lizhen’s VOD use was to explore Mandy’s construct of children’s factional thinking aimed at creating reflection opportunities for Mandy. When bridging and exploring feel unproductive, MTEs may use telling to create reflective opportunities. Signe used telling after her bridging efforts did not result in Jason’s reflection about the children’s mathematics.

Findings illustrate how MTEs’ actions in debriefing conversations move beyond MTEs’ knowledge categories to ways knowledge is used in teaching about teaching mathematics. While our descriptions of MTEs’ using the VOD may sound cognitive, interactive moves are also informed by emotions, places, and relationships with PTs. Our critical-friend dialogues unearthed evidence of this complexity. Space does not allow us to unpack ways such factors inform the VOD use . We assert that MTEs’ uses of the VOD occur within a flow of information that transforms MTEs’ understanding of the VOD and PTs’ conceptions in real time.

MTEs’ mathematics teacher education experiences and anticipation of programmatic and teaching demands PTs will face inform debriefing conversations. Debriefing as mentorship (Stanulis & Floden, 2009) has been identified as providing opportunities for PT learning (Brown et al., 2020; Roller, 2019), but can contain challenges informed by issues of power and control (Sundli, 2007). Our findings illustrate ways these issues play out as MTEs use the VOD as the primary source of insight regarding best practice for mathematics teaching. Sundli (2007) cautions teacher educators that endorsing particular behaviors and thinking may encourage PTs to replicate rather than analyze such behaviors. Brown et al. (2020) note there is “no ‘best’ model” of mathematics teaching, “a meta-rule for communication which promotes the development of personal conviction in each of us” (p. 106). Our findings further support these cautions. Although we intended to support PTs’ reflections on teaching, the cognitive demand (Roller, 2019) of wondering about observed events in PTs’ teaching challenged us. Our unique VODs, served as defacto “best” models of teaching Brown et al. (2020) cautioned about.

Findings illustrated that we used the VOD to explore and bridge to PTs’ experiences of mathematics teaching and learning, yet often moved to telling to motivate PTs’ reflection on teaching. Our use of telling drew from assumptions characteristic of the transmission and interpretive stances (Martin & Russell, 2018). We assumed that we should identify “teaching behaviors that should be improved,” characteristic of a “transmission stance,” and that “teacher candidates will always have uncertainties” the supervisor must identify and support with “relevant discussion,” characteristic of the interpretive stance (p. 337). Telling is informed by both stances, the events in the course, our relationships with PTs, and the preceding debriefing conversation with an eye on challenges we imagined the PTs would encounter. We experienced resonance with the description of Brown et al. (2020) that debriefing conversations take unexpected paths and should be engaged in with a focus on unpacking the teaching of the PT. Telling can encourage PTs to abandon their thinking and follow the MTE’s suggestions (Sundli, 2007). Yet, we experienced the cognitive demand of wondering aloud about PTs’ teaching (Roller, 2019), for example Lizhen’s challenge to maintain her thoughts while creating a model of Mandy’s.

Consciousness of MTEs’ VOD use in debriefing PTs’ teaching informs our efforts to improve debriefing practice. Despite intending debriefing conversations as reflective opportunities in PTs’ learning about teaching, we identified our use of the transmission stance. Improving our debriefing practice rests on maintaining an interpretive stance throughout the debriefing conversations by providing reflective opportunities PTs can capitalize on.

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