Introduction

In the last two decades, considerable efforts have been made to challenge lecturing as a predominant mode of instruction in university mathematics education (e.g. Laursen & Rasmussen, 2019; Rasmussen et al., 2017). Much of these efforts unfolded under the call for active learning and learner-centered pedagogies (CBMS, 2016). These umbrella terms embrace a broad range of instructional practices, many of which foreground students’ mathematical communication, group problem solving, and receiving and providing feedback from both instructors and peers (e.g. Ng et al., 2019). Research shows that institutionalization of these practices is not easy, partially due to institutional constraints (Johnson et al., 2018) and partially because instructors struggle to provide substantial opportunities for students to mathematize autonomously and agentively (e.g. Nardi & Barton, 2015; Paoletti et al., 2018). Notably, these findings also emerge from tutorials—instructional settings that are often purposefully set up for students to apply their mathematical knowledge from lectures to solve problems (e.g. Kontorovich & Ovadiya, 2023; Pinto, 2019).

The COVID-19 pandemic shifted higher education online, drawing attention to synchronous learning and instruction on digital communication platforms (e.g. Zoom, Teams). Spontaneous interactions, instructors’ observations of student reactions, students’ sharing their work, mathematical writing—these are just a few examples of practices whose online versions are radically different from in-person ones (e.g. McKenna & Green, 2002). Although learning and teaching worldwide mostly returned to physical classrooms, the online modality is unlikely to disappear soon (Major, 2020). This modality was used in distance education before the pandemic, and many universities capitalize on online affordances to pursue their internationalization agendas (Møgelvang et al., 2023). In the beginning of 2023, the University of Auckland oscillated between online and on-campus studies due to severe weather. Universities in Israel used online learning to ensure the safety of students and staff during terror attacks. These instances may illustrate a broader ongoing process, in which online becomes an educational format that can replace in-person teaching and learning when needed. Accordingly, understanding how various pedagogies operate online remains relevant to contemporary educational realities.

Research has acknowledged the affordances of online education (e.g. Møgelvang et al., 2023), while noting that teaching amidst a crisis differs from teaching in calmer periods (Youmans, 2020). Most research taps into these aspects drawing on students’ academic outcomes and reflective interviews (e.g. Møgelvang et al., 2023).

Our study revolves around field-initiated tutorials during the first pandemic year. The tutorials were aimed at supporting student involvement in online mathematical discussions. Studies show that students can be involved, engaged, and academically successful in learning implemented with online learning tools (Jones et al., 2022; Ng et al., 2019). Thus, we mobilize this case to understand what can facilitate and obstruct implementing familiar practices in an online setting. Specifically, our goal is to characterize instructional interactions as they unfolded on Zoom—a popular platform for online synchronous communication—and to explore how the online affordances shaped instructional interactions. Specifically, we aim to characterize different student-student and student-instructor interactions that took place and consider possible reasons behind them.

Tertiary Learning and Teaching Online

Mathematics education has been transformed by the availability of advanced technological tools (Engelbrecht et al., 2020). But technology alone is “not likely to bring any change, and intense pedagogical discussion should be undertaken” (Borba & Villarreal, 2005, p. 2). This perspective received a new meaning during the COVID pandemic. Although distance learning using various communication platforms (e.g. mail, radio, television, internet) is not new, the COVID-19 pandemic forced all educators to implement remote teaching in an all-encompassing manner (Kyriazis et al., 2023). Various methods of synchronous and asynchronous online teaching were used, with successful deployment dependent on many factors. These include students’ access to technological equipment, affective considerations and difficulties with individual support (Barlovits et al., 2021).

Various digital technologies have been incorporated in tertiary education (Cohen et al., 2022). Research shows that learning with technologies, including online learning, can facilitate students’ access to resources and reduce the gap between learning and extracurricular activities since various technologies allow for personalization and collaboration (Bardelle & Di Martino, 2012). Online learning allows for anonymous commenting, which can support hesitant students’ participation, and asynchronous viewing of videos, which allows students to rewatch and rewind (Seaton et al., 2022). Online tools can successfully support student engagement in learning. Ng and colleagues (2019) describe a course in which some tutorial groups were taught using non-traditional learning methods with online tools. Using Liljedahl’s (2016) “thinking classrooms” students collaboratively designed problems, solved them and evaluated other groups’ solutions with an online, interactive whiteboard. Ng and colleagues measured learning gains using qualitative methods and found that the students were more active, more engaged and more academically successful than their counterparts in the standard tutorials. The researchers proposed that this was due to the teaching methods implemented successfully in an innovative environment.

In the in-person context, non-lecture practices have been shown to benefit student learning, attitudes, success and persistence in mathematics and related fields (e.g. Griese & Kallweit, 2017; Lahdenperä et al., 2019; Laursen & Rasmussen, 2019). Mathematical discussions have been suggested as supporting meaningful tertiary student participation (Hershkowitz et al., 2014; Tabach et al., 2020). Small group discussions and whole class discussions can be used to support students’ mathematical activity of recreating mathematical ideas in a bottom-up manner (Stephan & Rasmussen, 2002).

Notwithstanding, modifying teaching and learning from traditional lecture-based practices is challenging and necessitates much preparation and support (Mesa et al., 2020). While considerable advancements have been made to pursue a more active learning agenda in university mathematics education (Gomez-Chacon et al., 2021) and incorporating online tools into courses has been shown to be productive (e.g. Jones et al., 2022; Kanwal, 2020), it remains unclear whether, and how, non-traditional learning can be pursued in an online environment.

Research maintains that online learning and teaching are not likely to disappear any time soon (Chan et al., 2021). Yet, there are many challenges with this format. Students can miss opportunities to raise questions due to anxiety of participating in an online environment (Radmehr & Goodchild, 2022), where the lack of a physical setting might inhibit the instructors’ ability to emotionally support students (Hopper, 2001). In online settings students access learning resources from multiple sources before, during and after the official instruction (Borba et al., 2017). This can modify students’ interactions with the content, with the course staff, and with their peers (Engelbrecht & Harding, 2005), potentially discouraging peer collaborations (Taranto & Arzarello, 2020). Additionally, there is some evidence that socioeconomic status, and the corresponding lack of resources, might inhibit university students’ learning through technology (McKenzie et al., 2014).

Although tertiary asynchronous online learning has been studied in various contexts (e.g. Geraniou & Crisan, 2019), its synchronous counterpart, where our interest lies, has been mostly studied via examining academic outcomes (e.g. Drijvers et al., 2021) and participants’ self-reports (e.g. Niu Voon et al., 2014). Overall, these studies show that the mathematical content can be shaped by teaching online (Krause & Martino, 2021) and student participation can also be affected (Taranto et al., 2020). Yet the interactions involved in online mathematical discussions remain less clear. This study uses qualitative methods to zoom-in to online synchronous tutorials on a popular communication platform to examine the instructor-student interactions during mathematical discussions, with an eye on how they can be shaped by the affordances of a particular online setting.

The Commognitive Framework

This study leverages the commognitive framework to explore instructor-student interaction as it occurs in an online communication space (Sfard, 2008). Commognition is a socio-cultural framework that has been acknowledged for its potential to explore communication and learning processes through the mathematical talk and actions of students and instructors, as well as their cooperation with each other (e.g. Nardi & Barton, 2015). To our knowledge, the framework has not yet been used to study instructional interactions in an online setting.

From the commognitive standpoint, learning is associated with one’s participation in a discourse that is renowned as mathematics. Visible markers of one’s participation involve using specific keywords (e.g. “equation”) and visual mediators (e.g. “\(\left|\text{x}\right|\)”), producing narratives generally endorsed as true (e.g. “a positive number less than 1 to a natural power is less than 1”) and performing conventional routines (e.g. proving). In practice, students’ learning is mostly occasioned by instructors who set task situations—circumstances that call for a certain kind of action (Lavie et al., 2019). Each action is viewed as a performance of a routine with two elements: a task, or a goal, that one pursues and a procedure as a sequence of steps (Lavie et al., 2019). We add that a task situation changes with each action that a student or an instructor takes, opening the space for revising the task and performing another procedure.

Mathematics learning often entails transitioning between incommensurable discourses – discourses that differ in their keywords and their use, characteristic routines, and underpinning rules (Sfard, 2008). In these terms, the commonly explored issues with the secondary-tertiary transition (Di Martino et al., 2023; Gueudet, 2008) can be construed as discrepancies between the discourses of secondary school and university mathematics (e.g. Sfard, 2014). Sfard (2008) maintains that students’ engagement in an incommensurable discourse requires proactive scaffolding of instructors as the discourse “oldtimers”.

This special student-instructor interaction is shaped by a learning-teaching agreement (LTA) which governs communication between students and an instructor. The LTA has been used to describe interactions in a broad sense (e.g. Nardi, 2014) and includes three components. Sfard (2008) contends that “to attain their respective goals, [students and instructors] need to be unanimous, if only tacitly, about […] the leading discourse, their own respective roles, and the nature of the expected change” (p. 283, our italics). The first component focuses on whose discourse is regarded as setting the standards for communication: the students’ or the instructor’s. For the second component, Sfard (2008) distinguishes between an instructor as a discourse leader and the students who are expected to “show confidence in the leader’s guidance and are genuinely willing to follow in the oldtimers’ discursive footsteps” (p. 284). The final component concerns agreement on “the final goals of the process of learning and as to the manner in which the learning is likely to occur” (Sfard, 2008, p. 285).

In addition to the roles of discourse leader and learner suggested by Sfard (2008), there are other roles involved in the interactions between group members. Research shows a rich palette of finer-grain roles that instructional interactions can entail (Strijbos et al., 2010). Studies about collaborative learning in small groups distinguish between scripted roles, which are responsibilities assigned by the instructor, and emergent roles, which are individuals’ contributions that emerge more spontaneously (Strijbos et al., 2010). Both these types of roles can be described through the duties and actions of the participants, aligning with the definition of roles in small groups as stated functions and/or responsibilities that guide individual behavior and regulate group interaction (Harre, 2012). Each role can be operationalized as a specific sort of action filled by a group member and can be stated as functions and responsibilities that guide the member’s activity (Harre, 2012).

The final component of the LTA is an umbrella construct that summons context-sensitive operationalizations. Pragmatically speaking, at the end of the semester, students are expected to become capable of generating written solutions to problems that abide by discourse-specific rules of undergraduate mathematics as these are sanctioned by the course instructors. The importance and necessity of using conventional mathematical terminology and notations is acknowledged by instructors and students in mathematics courses (Thoma & Iannone, 2022). We use these three components of the LTA to characterize instructional processes as they unfold online.

Method

The Online Tutorials

The study is contextualized in a first-year linear algebra course for mathematics and computer-science majors offered by a large university in Israel. The course is traditionally taught through lecturing (e.g. Malek & Movshovitz-Hadar, 2011). The tutorials, in which the instructor solves problems, are also taught in this manner. In 2020, the course was given online. In the second semester, six students volunteered to participate in an alternative tutorial format, which was advertised as promoting mathematical discussions and student interaction (see Nardi & Barton, 2015 for a similar course design). Before each tutorial, students were expected to engage with the course lectures (“live online” or recorded), watch tutorial videos from the previous semester, and work on homework problems. The tutorial sessions were held on the Zoom platform twice a week during a 13-week semester, with a total of 39 academic hours. Our data corpus consisted of video-recordings of the tutorial sessions as they featured on the instructor’s screen.

The tutorial sessions were led by Mia (pseudonym) — a PhD student in mathematics courses with over 20 years of tertiary teaching experience. Mia was awarded prizes for excellence in teaching and was consistently rated high in student feedback. Mia proposed converting her tutorial section to a student-centered one. The course staff, while giving official approval, were ambivalent and only vaguely aware of what the innovation involved. Mia received no further support nor input on her innovation. We return to this aspect of Mia-institution interaction when accounting for our findings.

The students had their cameras open, as per the institution’s policy in the pandemic period. Mia also asked the students to leave their microphones open, and the students mostly complied. Mia used an external camera to project the problems and her writing for the whole group to see (see Fig. 1 for an example). This mimicked the use of a physical classroom whiteboard by allowing Mia to easily write mathematical symbols, mark pieces of text and draw lines and arrows. The Zoom platform did not support the annotation option while using an external camera and the chat option was not used by the students. Thus, the distinctive feature of the online environment was the ease with which the instructor could write down mathematics for the whole class to see and the challenge of the students to do the same. Consistently, our analysis followed the students’ verbal discursive patterns; whereas we were able to pay symmetric attention to the instructor’s verbal patterns and to her writing.

Fig. 1
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Shared screen visible for students and translation

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Analysis

In the first phase of the analysis, we translated and transcribed the tutorial sessions, with special attention to the mathematical narratives that the participants generated. These are written and oral descriptions of mathematical objects, of processes on these objects and of relations between these (Sfard, 2008). The transcripts also included the written narratives which were available for everyone to see on the shared screen. The instances of student-instructor engagement and student-student engagement, with the intonations, were also marked. As all the students had their cameras and microphones open, they could contribute to the discussion at any moment, not very differently from in-person tutorials. Consistently, the resulting transcripts, at first glance, seemed to be capturing interactions that could have taken place in an in-person setting (we return to this point in the discussion). This allowed us to embark on the transcripts with a standard commognitive tool, which uses “things said and done by individual learners in direct interactions with others” (Sfard, 2008, p. 278).

The second phase was targeted at determining the unit of analysis. The extended transcripts were processed through abstraction (summarizing the material), searching for patterns and peer discussions about these patterns (Denzin & Lincoln, 2011). This overview of the transcripts revealed that the interactions were centered around problems—questions that students posed about the course material, difficulties with homework problems raised by students and problems that Mia prepared for each tutorial session. Thus, a discussion of a single problem became the main unit of our analysis.

In the next phase, we approached the transcript with the commognitive toolkit, paying special attention to the LTA construct. In the case of each problem, we examined the discourse leadership: Mia or the students. This was easy to identify based on who led the creation of mathematical narratives and advanced the problem solution. This is when we noted that sometimes the students communicated among themselves, while other instances consisted of exchanges between a single student and Mia. These observations gave rise to three initial categories: Mia talking to the class, exchanges between Mia and the class, and exchanges between students. We then turned to the remaining LTA components of the discursive roles and the nature of the expected change, to develop these categories systematically.

Discursive Roles

Informed by Harre’s (2012) approach to roles as duties, we initially used a deductive approach (Cresswell, 2013) and constructed an exhaustive list of activities that the participants conducted while working on a problem. For example, each discussion opened with setting the group problem, that is, locating a specific mathematical problem in the locus of the discussion. To solve the problem, the participants proposed and performed mathematical routines. A single routine was rarely sufficient to reach a solution, and thus a sequence of routines (or sub-routines) was needed. Not every proposed routine (and accordingly, a subtask and a sub-procedure) was taken by the group. Thus, someone had to determine which proposal would be followed and which one would be abandoned, setting the course of mathematical work. Additionally, the validity of an articulated mathematical narrative needed to be confirmed and recorded in writing by the scribe. We next used an inductive approach to derive general categories (Cresswell, 2013). The names of the roles and their descriptions were gradually developed through an iterative comparison between the participants’ work on different problems. In the process, initial categories were proposed, clarified, split, and merged until the development of a comprehensive set that was applicable to the whole data corpus.

Let us use a short example to illustrate the roles of the main task setter, mathematics assessor, and a routine proposer, setter, and performer. In one of the sessions a student set up the main problem, asking to determine which of four given sets constitute subspaces. Mia responded with “what do we need to prove it?” With this, Mia put forward the need for a proof (a subtask) and alluded that it can be constructed with some previously covered material (procedure). Three students proposed routines by suggesting proving that “there exists a zero”, “zero and linearity”, and “linear combinations.” Mia responded, saying that “the zero is the simplest one”, which determined the course of the group action. Then, another student performed the routine, reporting, “It looks like only one set doesn’t have it [zero]”. Finally, Mia asked, “Yes? You agree? Disagree?” calling the group to assess the provided answer.

The analysis of the final component, the course of the discursive change, displayed an additional duty. The group discussions tended to start with students sharing diverse and informal mathematical narratives before a call was made to change the modality of the discussion. Often, Mia was the one to call to “write things down”, switching the oral discussion to a written one. Occasionally, students asked “what would be considered an acceptable answer on an exam?”, implicitly calling to switch to mathematics which would be endorsed in assessments. In this way, the duty of calling for a discursive change was added to our categorization.

The Nature of the Expected Change

To reveal the nature of the discursive change, we considered candidates for the goals that the students and Mia could pursue in their tutorials in general and in the case of each problem in particular. This analysis attended to the characteristics of problem solutions (solution-narratives hereafter) that the group strived for. Having no access to the participants’ in-the-moment goals, we examined discursive cues for markers that these goals were achieved. This examination showed that in the discussions, students initially presented their ideas orally, through the medium of the microphones, and the final narratives were written by Mia on her cards for the whole group to see (see Fig. 1). The students and Mia rarely endorsed the written narratives as solutions explicitly. More often than not, Mia completed the writing with “Questions?” as if inviting students to seek further explanations. We saw this routine as Mia’s tacit endorsement of the mathematical validity of the written solution-narrative and as a signal that the goal of the problem discussion was achieved, at least in her view. Other discursive cues, such as saying, “that’s the solution” and transitioning to a new problem also signaled this. Typically, students endorsed the solution-narrative by not objecting to it.

Equipped with the presented adaptation of the LTA components, we applied them to the three initial candidates for the instructional interactions aiming to capture key similarities and differences between them. We systematically contrasted problem discussions within the same category to reveal roles and discursive changes characteristic to each of them. This is how we noticed that each interactional category offered certain roles to students while retaining others for the instructor. The goals of the interactions and nuanced differences between the expected discursive change also came to light in this way. The distinction between the types of interactions was occasionally not obvious and after the three categories were developed their application to the data revealed that the discussions switched between the types. The results of this analysis are presented next.

Findings

We present the findings in two stages. We first present three types of instructional interactions that occurred in this setting. Next, we focus on the dominance of one of these types.

Three Types of Instructional Interactions

The data analysis resulted in three types of instructional interactions, to which we refer as lecture-ish, instructor-oriented, and cross-student. We distinguish between them based on the discourse leadership, the duties offered to students and the nature of the expected change involved. Table 1 summarizes these and is followed by detailed descriptions and examples.

Table 1 The components of the LTA for the types of interactions
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Lecture-ish Interaction

The first type of interaction emerged from long sequences of Mia’s utterances, where she led the work on the problem. This interaction unfolded alongside Mia writing a solution-narrative, occasionally interrupting herself with questions for the students. Each question was tightly linked to the previous solution step and summoned a certain eloquent answer; for instance, “why is that true?” and “what do we get?”. We refer to this instructional pattern as lecture-ish due to its core similarity to a traditional lecture format (Fukawa-Connelly, 2012). Opposed to a “radical” format featuring a lecturer’s monologue, in the lecture-ish version, students were provided with the space and time to answer Mia’s questions. In most cases, she did not continue generating a solution-narrative until an expected answer came from the students. Notably, an expected response from a single student was sufficient for Mia to continue.

The following example of the lecture-ish interaction comes from the fifth tutorial on complex numbers. Mia talked for almost three minutes, without interruption, explaining how to use induction to prove that |zn|=|z|n for any complex z and how to write the proof formally. She continued as below.

5099 Mia: This (pointing on card to proof for n = 2) is the basis [of the induction]. Now we prove for n + 1. What is z to the power? We need to prove for n + 1.

5100 What is z to the power n + 1? (Writes: |zn+1|)

5101 So, when we want to prove an equality, we start with one side and go to the other side. So, we will start for… It doesn’t matter where we start, we will start there (writes “=”).

5102 Now we want to use the induction assumption,

5103 So we want… That there should be z to the power of n.

5104 (Pause of 8 s) OK?

5105 Roi: z times z to the power of n (Mia writes: = |z⋅zn|).

5106 Mia: What did you use to say there is equality? When we do a proof, and we are doing a formal proof, we need to justify each step. So why is this true?

In this excerpt, Mia sets the task of proving the claim for zn+1 [5100]. She suggests and sets the routine of using induction [5102] and of using the routine of decomposing zn+1 [5103]. She then waits for students to perform this routine, by pausing until Roi answered [5105], which allowed her to write this on the card in mathematical symbols. The final solution-narrative is presented in Fig. 2. It shows the claims, the justifications, the text introducing the proof and the conclusion.

Fig. 2
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- Cards at conclusion of lecture-ish interaction and translation

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The analysis of the changes that Mia and students pursued in the lecture-ish format, revealed that their interactions were targeted at the generation of a written solution-narrative that is consistent (i.e., contains no internal contradictions), well-substantiated (i.e., justifies each sub-narrative), formal (i.e., utilizes conventional terminology and notation), and comprehensive (i.e., considers all cases). As a scribe and a common endorser of students’ narratives, Mia did not allow her writing to deviate from the first two descriptors even once. Regarding the third descriptor, Mia often reformulated students’ narratives replacing deictic talk and colloquialisms with conventional keywords. For example, when a student said, “it will ruin it”, Mia wrote it as, “it will not fulfill the condition”. Similarly, when a student explained his claim by saying, “it is tiny”, Mia wrote this as, “the modulus is less than 1”. Overall, deviations from the last descriptor were rare and seemed unintentional. For example, in one discussion, the group problem concerned a claim about all natural numbers, but the written solution proved it for integers larger than 2. A student asked about n = 1, challenging the comprehensiveness of the recorded solution-narrative. This triggered Mia to reopen the discussion and amend the written solution.

Instructor-Oriented Interaction

We refer to the second type of interaction as instructor-oriented. There, students generated mathematical narratives and questions directing them to Mia as a mathematical authority. With such questions as “what do you mean?” and “why would that work?”, Mia turned the narratives into a short exchange with the student. The students’ open cameras and microphones facilitated this discursive ping-pong, and while Mia rarely used students’ names, it was clear to the question posers that Mia’s utterances were directed towards them.

The following excerpt exemplifies such an interaction. This comes from the thirteenth session on subspaces, where Mia set up the following problem: What is the greatest value of n such that there exist n subspaces Wi ⊂ ℝ2 × 3, 1 ≤ i ≤ n, where \(W_{1} \subsetneq \cdots \subsetneq W_{n}\)? After the students worked independently for almost three minutes, Mia started a discussion.

1301 Mia: Did someone find some kind of n? Did you find some series, not necessarily maximal, some n?

1302 Zohar: I think infinity.

1303 Mia: Infinity. Ok. So then what?

1304 Zohar: Should I justify? I don’t have a proof, I have an idea.

1305 Mia: Ok. So what’s the idea?

1306 Zohar: That I can always take… Ok, let’s say the trivial space—zero, and I can always add on to it a specific matrix, that is an element of ℝ2 × 3 and then umm build another set that includes the sub, those two matrices and then another one that doesn’t exist in it, and then another one, and there is infinite numbers in R so I can always add another one like that, and they will be included one in the other.

1307 Mia: Ok. So let’s start constructing [what you suggested]. What’s the smallest [subspace]?

1308 Zohar: The trivial one – zero, the zero set. And then one that has the zero and the identity matrix, for the sake of argument. [Mia writes: {\(\overrightarrow{0}\)} \(\subsetneq\) {\(\overrightarrow{0}\)]

The discussion continued and Fig. 3 presents what Zohar stated is an example of a chain of nested subspaces.

Fig. 3
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Card at conclusion of instructor-oriented interaction and translation

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Mia asked the group for a solution [1301] and Zohar authored the narrative that n could be infinity [1302]. Mia did not write this down, rather she instigated an interaction with Zohar that would ultimately result in a solution-narrative that could be assessed for mathematical correctness. After Mia prompted him, Zohar proposed a routine in [1306] by describing the construction of a set of subspaces. In [1307] Mia set this routine by asking Zohar to perform it, which he did in [1308]. Zohar’s performance of his suggested routine produced a mathematical narrative.

To summarize, the instructor-oriented type of instructional interaction is characterized by a dialog between the instructor and a single student. The student authors oral mathematical narratives, and the instructor instigates the generation of further narratives and writes mathematical symbols based on the student’s suggestions. When valid narratives were suggested, Mia instigated further discussion by asking for explanations or justifications. These episodes were initiated either by a student’s question or by a student suggesting an approach to a problem. Students volunteered to participate in these interactions, knowing that they would be required to answer Mia’s questions publicly. These episodes concluded with some mathematical notation written on the card that could be discussed and an oral narrative pertaining to these notations.

Cross-Student Interaction

The cross-student interaction took place when two or three students conversed with each other, leading the discussion about a problem, while Mia kept silent or confined her contribution to occasional clarifying questions. The exchanges consisted of several turns, where students acknowledged and built on each other’s narratives through rejecting, endorsing, extending, and elaborating.

We exemplify a cross-student interaction which changes to instructor-oriented from the twentieth tutorial about linear transformations. The discussion pertained to the following problem: Let T: ℤ54→ ℤ54 be a linear transformation such that T(1,2,3,4)=(0,0,0,0). For which n ∈ ℕ does there exist such a T for which dim Ker T = n. For these values, give an example of such a T and find a basis for and the dimension of Im T and Ker T.

2002 Mia: Think about the problem. <10 s pause>.

2003 Zohar: For four I think it happens.

2004 Roi: Not for more than four.

2005 Zohar: Not for more than four we can agree on. Because, then it’s actually the zero transformation.

2006 Leo: There aren’t more than four [dimensions].

2007 Zohar: Yes. If it is four, then it is the zero transformation.

2008 Mia: If n equals four?

2009 Zohar: Yes, then the dimension is….

2010 Mia: The dimension is four, so?

2011 Zohar: So, the dimension of the image is zero, which means it’s the zero transformation.

This cross-student interaction consists of five student talk turns, [2003]-[2207]. Students replied to each other and incorporated pieces of the other students’ narratives into their own narratives, suggesting that they were listening to each other. Indeed, in [2003] Zohar set the task to check the case of n = 4. Roi in [2004] aligned with this task and specified that n must be at most 4. Zohar continued setting the task and the procedure in [2005] to examine the transformation in this case. In [2006] Leo endorsed the mathematics, by agreeing to Roi’s routine and concluding that there is no point in checking for larger n. Notably, while appearing to confine the number of her interventions compared to the lecture-ish and instructor-oriented interactions, Mia still managed to change the course of students’ mathematical dialogue by writing “n = 4” (see Fig. 4), which set the procedure to check this case. (The discussion became instructor-oriented and ended in a lecture-ish interaction; see the next section for a detailed presentation of the general pattern.)

Fig. 4
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Card at conclusion of cross-student interaction and translation

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Generally, in the cross-student interactions Mia was often the one who set a broader group task, while students proposed, set and performed the procedures to solve it. These interactions ended when Mia changed them by intervening. This happened when sufficient narratives, or fragments of narratives, that could be used to further the discussion were generated.

To summarize, the three types of instructional interactions described above have different learning-teaching agreements in place, as can be seen by the variations in the three components of this agreement (see Table 1 again). In the lecture-ish and instructor-oriented interactions the instructor’s discourse, representing university mathematics, set the standards. In the cross-student interactions the instructor was a peripheral participant in students’ exchanges.

The roles of students and the instructor varied in the duties offered to students. In the lecture-ish interactions students were often offered to perform the routines. However, when they did not share the routine outcome in a few seconds, Mia implemented the routine herself. Students were also offered to set the problem. In the instructor-oriented pattern, students proposed the routine, positioning Mia as a mathematical endorser and a routine setter. Mia took these duties, sometimes implicitly, acquiescing to the offered routine. For example, by saying, “yes” and “ok” she explicitly set the routine a student proposed. She also encouraged the student to perform it by asking such questions as “and then?” or by not stopping the student who started to generate a solution-narrative. In the cross-student interaction students set the task, proposed and set the routine and endorsed the mathematics of each other. The instructor retained the duty of calling for a discursive change by transforming the interaction.

All Types of Instructional Interactions Lead to Lecture-ish

As mentioned beforehand, the work on most problems constituted a mixture of the interactional types presented in the previous section. Many of them started as lecture-ish and rarely deviated from this format until they were completed. Other problem discussions opened as instructor-oriented or included some cross-student interactions. A close analysis of these revealed that they transformed to lecture-ish at some point, typically after the instructor’s call to “write things down.” In this way, we concluded that all problem discussions throughout the semester converged to a lecture-ish interaction at some point.

Due to space limitations, we illustrate this convergence with a single example. The example comes from the sixth session, when the tutorial patterns can be considered as established. We chose this example since it includes all three types of interactions and there is an explicit call for a change in the discourse. Additionally, this instance illustrates how Mia’s questions and statements, that appear as building on student contributions at first glance, tacitly advances the solution-narrative to abide by the early-discussed characteristics— consistent, well-substantiated, formal, and comprehensive.

In the tutorial about complex numbers, Sarah asked Mia for help with the following problem: “\({z}^{200}=1+i\). True or false: The equation has a solution whose modulus is greater than 1.” After Mia read the question out loud, Roi opened the discussion.

628 Roi: I am, I am simply saying, if this to the power of two hundred… its modulus is the square root of two.

629 Mia: Let’s find the modulus of the root. What is the modulus of the root?

[Writes |zk| = ]

630 Roi: Then… it’s two to the power of one divided by four hundred [Mia writes 21/400].

631 Mia: And now is it a number that’s larger than one or smaller than one?

632 Leo: Larger than one.

633 Roi: It must be larger than one because if I put an exponent on it….

634 Zohar: Larger than 1. Since the minimum at infinity is one but it’s one, it’s larger than one.

635 Roi: Yes. Larger than one.

636 Zohar: If its smaller than one… If its smaller than one to the power of two hundred … to the power two hundred, then… then it will come out a tiny number.

637 Roi: That’s it! I am saying, I am saying, let’s say the modulus of the root was half […] and I put it to the power of two hundred it will never get to the square root of two.

638 Mia: (a) Let’s write it down. I am writing it. (b) Let’s give it a name like R [completes |zk| = 21/400 = R]. (c) And we know that R to the power of four hundred is equal to [Writes R400 = ] < pause > Two [Writes 2]. (d) And then if something to the power four hundred is larger than 1, what do we know about the exponent base?

639 Zohar: Larger than one.

640 Mia: Larger than one. Yes. Because a number smaller than one to any power will always be smaller than one. Thus, we know that R is larger than one. Ok?

This excerpt can be divided into three episodes, each corresponding to the presented instructional type. The exchange in [628–630] is the closest to an instructor-oriented pattern. In [628], we see Roi making a step toward finding the modulus of the equation’s roots. Mia’s utterance in [629] follows Roi’s. It involves the keywords of “root” and “modulus” that he mentioned and appears as if spelling out the task that Roi pursued for the rest of the group. Yet, Mia does not exactly “revoice” what Roi said. Her narrative replaces Roi’s deictic “this” and “it” with the term “root” and uses notation signaling there is more than one solution to the equation. Even more importantly, Roi’s narrative was concerned with the modulus to the power of 200, while Mia sets the subtask of determining the modulus of the root. Roi performs the new subtask in [630] to obtain \({2}^{\frac{1}{400}}\). Then, Mia responds with the next subtask—to estimate whether this number is larger or smaller than 1. In this way, we see the instructor breaking down the main task by setting up a chain of three smaller ones.

The last subtask sparked a cross-student interaction in [632–637]. There, three students maintain that the obtained number is larger than 1 and substantiate it using different procedures (routine setting and performing). This exchange showcases how students can not only share their narratives but also build on them. Indeed, just after Zohar notes that a number that is smaller than 1 to the power of 200 “will come out a tiny number” (see [636]), Roi picks up half as an example of the root modulus to argue that “it will never get to the square root of 2” (see [637]). This is a development compared to his rather approximate substantiation in [633], where he wanted to “put an exponent on it.”

The lecture-ish part of the interaction in [638–641] opens with Mia calling to “write it down” and claiming the role of a scribe. To recall, she was the ultimate scribe in the explored setting since she had a special video camera set up over her handwritten cards, which enabled the students to see Mia and her writing together. As in [628–630], Mia appears to operate with the mathematical narratives generated by the students. But a closer look reveals a more sophisticated discourse leadership in-play. In [638b], she denotes the root modulus, making the manipulation with it easier. Then, instead of estimating the modulus, as Zohar and Roi did beforehand, Mia raises \(R\) to the power of 400. Accordingly, Mia proposes a new estimation procedure and selects it for the group by executing it on the card. The students are given an opportunity to calculate \({R}^{400}\), but an answer does not emerge instantaneously, and Mia performs the calculation herself (see [638c]). In [638d-640], she returns to the task of estimating \(R\), while drawing attention to the base of the exponent. Zohar estimates correctly in [639] but provides no substantiation. Mia endorses Zohar’s response, but after not receiving the intended substantiation, she generates it herself. Notably, Mia a priori positioned the substantiation as something “we know”, which explains why it did not find its way into her written solution-narrative.

Summary and Discussion

This paper is part of a special issue on students’ digital experiences, with particular attention to how digital resources have met the expectations of the mathematics education research community before, during and after the pandemic crisis. We embarked on this topic with a focus on mathematical discussions in an online setting as they occurred during the pandemic to better understand how the affordances of communication platforms can shape the interactions between the students and the instructor.

In this study, we attempted to offer a fine-grain analysis of instructional interactions that unfolded in online tutorials in Linear Algebra in the first pandemic year. The tutorial brought together an experienced instructor, Mia, and a small group of six students who volunteered to partake in a format that was non-traditional to this mathematics department. The format was advertised as promoting mathematical discussions and student interactions around problems from the course. Throughout the semester, students engaged with the course content and worked on the assigned problems before partaking in the tutorial sessions. Hence, on the face of it, the necessary ingredients for the tutorials to deliver on its promises were in place.

In the sessions, students were engaged, generated narratives that were key to problem solutions, interacted with the instructor, and occasionally with each other. These are discernible shifts from the traditional lecture format that dominated the particular institutional setting. These shifts would not occur without the instructor opening the communication space for them. However, what we labelled lecture-ish instruction dominated the sessions, when cross-student interactions were rare, short, and did not go as far as they could have. The instruction became lecture-ish at some point in each problem discussion, and there, while drawing on students’ contributions, the instructor played the leading role in solution generation. Additionally, student contributions were endowed with lower epistemic status than the instructor’s contributions. Indeed, the call to “write down the solution” when only Mia had the technological ability to do so created a hierarchy between students’ oral contributions and Mia’s writing. While Mia maintained that she was merely going to “write it down”, the analysis shows that the change in the communication medium entailed a substantial advancement of the solution-narratives that the students did not reach on their own.

We maintain that a delicate intertwining between the LTA that was established in the tutorial and the affordances of the specific communication platform shaped student-instructor interactions to a significant extent. The shaping occurred through the set of duties, from the execution of which problem solutions emanated. Indeed, as a discourse oldtimer, the instructor could fulfill each of these duties more successfully, efficiently, and quickly than the students; we believe that all our participants were aware of this situation. Yet, we contend that the technological affordances predisposed the instructor to certain roles, while diverting the students from them. We elaborate on this thesis next.

The identified duties pertained to proposing, setting, and performing mathematical routines. Compared to performing, proposing and setting routines involves a higher degree of mathematical agency since they summon autonomous decision-making, substantiation, and consideration of what move to take next. In this sense, the duties afforded substantially different learning opportunities, making it inseparable from duties that the protagonists relinquished, took, and offered to each other. This is consistent with previous findings from in-person tutorial classrooms where lecturers favorably oriented towards learner-centered instruction still struggled with providing students with a high degree of mathematical agency (e.g. Kontorovich & Ovadiya, 2023; Ng et al., 2019).

The platform affordances here come into play: Mia was the only group member with technological tools to write conveniently and visibly, and so the students unanimously endorsed her as a group scribe. However, the analysis showed that in the switch from an oral to a written modality, she did not only scribe the previously shared narratives but also formalized them, broke down tasks into smaller ones, selected routines to pursue, and performed many of them herself. In this way, Mia’s call for switching to a written medium was a call for a change in the LTA that would restore her role as the leader of the mathematical discourse.

We contend that the institution also played a role in the described group dynamics. During the pandemic, the university equipped instructors with digital tools (e.g. tablets and digital writing pads) that were especially useful to write mathematics for all students to see. By that, the university predisposed the instructors to the role of scribe. Our findings illustrate that when instructors, who are expected and used to leading a mathematical discourse, are provided with advanced technological tools for mathematical writing, it may be challenging for them to scribe only. This makes us wonder whether allocating the same tools to students might have impacted the duties that they are given and those they take.

An additional factor that might have led to the predisposition for lecture-ish interactions pertains to how the described tutorials came about. Interactive mathematics instruction was foreign to Mia and she had minimal guidance in terms of pedagogical features and goals. For example, while inquiring into student thinking is an explicit goal of some student-centered approaches (e.g. Laursen & Rasmussen, 2019), Mia reflected that there was no particular pedagogy or a set of didactical principles to which she planned and aspired to adhere in the tutorials. Her students were also new to this type of teaching, as their other courses were taught in a lecturing manner. There are many documented projects, where groups of instructors, guided by researchers, developed and implemented non-traditional practices in the tertiary level (e.g. Gomez-Chacon et al., 2021). These organizational circumstances illuminate the critical role of a clear student-centered agenda, change leadership, and the community of like-minded peers. In contrast, the tutorials described in this study were initiated by Mia, who had minimal support from her colleagues and the mathematics department. The presented case demonstrates that as lecturing is so entrenched in university teaching (Johnson et al., 2018; Sfard, 2014), without holistic and systemic support learner-centered initiatives are likely to fold back to traditional practices, possibly with some ancillary features.

And let us not forget that the analyzed case unfolded in the pandemic. Teaching amidst a crisis requires more compassion, support, a predictable structure, and a deeper sense of community than in calmer periods (Youmans, 2020). Given that good intentions are not sufficient to transform teaching practices in regular times (Mesa et al., 2020), it is not surprising that Mia’s and her students attempts to teach and learn differently were especially challenging.

This study is in a particular setting and generalizations would be irresponsible. The presented findings are inseparable from the specificity of the online setting and communication platform on which the presented case unfolded. For instance, our students self-selected for the study, all of them had access to video cameras and microphones, and Mia insisted on them being kept open in all tutorials. Yet, we believe that with attention to online synchronous interactions, the presented case expands the current body of knowledge on online teaching and learning, which mostly draws on learning outcomes and reflective interviews (e.g. Møgelvang et al., 2023).

Overall, this study’s contribution to the existing literature is three-fold. Theoretically, we adapt Sfard’s (2008) construct of LTA to the context of university-level mathematics education. To our knowledge, this is the first study to utilize the construct for the analysis of student-teacher interactions. Empirically, we identified three types of instructional patterns in online tutorials. Research on tutorial teaching and learning is on the rise (e.g. Kontorovich & Ovadiya, 2023; Thoma & Iannone, 2022), and the identified patterns may be extendable beyond this setting. Practically, there are several points to consider when attempting to utilize non-lecture practices in an online setting: the technological resources that pave the way for some participants to fulfill certain roles by enabling them to mathematize in the ways that other group members cannot, the communication affordances of the platform that predispose the roles and duties of the interacting group members to some extent, and the critical role of holistic support that newcomers to non-traditional teaching, both the instructor and the students, may need.