Exploring and developing a framework for analysing whole-class discussions in mathematics

ABSTRACT

Leading whole-class discussions in mathematics instruction in a productive way is identified as a core practice. However, productive discussions are rarely seen in classrooms. Thus, we need more knowledge about how different teacher actions in response to students’ comments can affect the students’ learning. If research is to be able to produce such knowledge, detailed analytical frameworks and concepts are required. Thus, this study sought to propose concepts and refine definitions of existing categories of teacher actions to help develop Drageset’s framework ([2014]. Redirecting, progressing, and focusing actions – a framework for describing how teachers use students’ comments to work with mathematics. Educational Studies in Mathematics, 85(2), 281–304.). This was done by analysing 35 mathematical whole-class discussions, generated from multiple-choice tasks supported by classroom response systems, in secondary schools in Sweden and the UK. The results suggest five new categories of teacher actions and enrichments of two existing categories in Drageset’s framework.

KEYWORDS:

• Whole-class discussions
• teacher actions
• mathematics

Introduction

There is a growing body of literature that recognises the importance of classroom discussions in high-quality mathematics instruction for all learners (e.g. Anthony & Walshaw, 2009; Jacobs & Spangler, 2017; Schoenfeld & Kilpatrick, 2008). While these classroom discussions can be conducted in a small-group or whole-class setting, recent research has shown the importance of conducting whole-class discussions (Webb et al., 2019; Wester, 2021). In fact, researchers (Jacobs & Spangler, 2017) argue that leading whole-class discussions is a core practice in mathematics instruction. Furthermore, whole-class discussions have been shown to have the potential to improve students’ learning and generate affective outcomes, including a positive attitude toward mathematics (Jacobs & Spangler, 2017).

However, productive mathematical whole-class discussions, in which students can improve their learning,¹ are rarely seen in classrooms (Park et al., 2017) despite years of attempts by professional development programmes to achieve these discussions. It has been suggested that this absence of productive whole-class discussions is caused by challenges teachers face (Staples, 2007). One major challenge for teachers concerns how to lead these discussions productively by building on students’ contributions. Research has shown that leading discussions requires a high level of teacher knowledge and skill (O’Connor et al., 2017; Walshaw & Anthony, 2008). Therefore, there is a need for a better understanding of how different teacher actions in response to students’ contributions can influence student learning.

In mathematics education research, theories from other fields have been imported and adapted to characterise and understand various aspects of mathematical whole-class discussions (Herbel-Eisenmann et al., 2017; Ryve, 2011). For instance, positioning theory has been used to explore and understand agency, authority, and identities (Herbel-Eisenmann et al., 2017), and discursive psychology approaches have been helpful in exploring how psychological matters are dealt with in discussions (Ingram, 2018). Further, building on sociocultural theory, the concepts of exploratory talk (e.g. Mercer, 2008; Mercer & Howe, 2012), dialogic teaching (e.g. Alexander, 2018), and dialogic teaching and learning (Hennessy et al., 2016) have been developed and examined. Other examples of sociocultural research focus on the evolution of different practices such as Hufferd-Ackles et al.’s (2004) math-talk community. In addition, analytical approaches building on sociolinguistics have been used to identify useful teacher actions in classroom discussions (e.g. Michaels & O’Connor, 2015). In this area, conversation analysis has provided methodological tools and concepts to offer in-depth insights into the learning process (Ingram, 2018), for example detailed exemplifications of productive contexts and practices (e.g. Ingram et al., 2019) and patterns in discussions such as Initiation–Response–Evaluation (IRE) (Koole, 2012).

One category of concepts and frameworks describes a whole classroom practice or culture, such as funnelling and focusing (Wood, 1998), Wood et al.’s (2006) descriptions of four different classroom cultures, and Hufferd-Ackles et al.’s (2004) descriptions of the quality levels of four practices within whole-class discussions. In contrast to this, frameworks which can capture details, such as single utterances, in the discussions are also needed in order to develop the knowledge of how teachers lead of whole-class discussions influence students’ learning (Drageset, 2014; Jacobs & Spangler, 2017). Frameworks such as Advancing Children’s Thinking (ACT) by Fraivillig et al. (1999) and Extending Student Thinking (EST) by Cengiz et al. (2011) have been developed to meet this need. Further, the work on Accountable Talk (e.g. Michaels et al., 2008), a research-based pedagogy for improving the quality of classroom discussions, includes a set of teacher moves for productive discussions that have been used as an analytical framework (e.g. Heyd-Metzuyanim, 2019; Michaels & O’Connor, 2015). However, there is still a need for more detailed frameworks for analysing teacher (and student) actions in these whole-class discussions (Jacobs & Spangler, 2017).

A more recently developed and detailed framework in the context of mathematics instruction has been presented by Drageset (2014, 2015b). His first version of the framework in 2014 comprises 13 concepts of teacher actions in response to a student’s mathematical contribution. Furthermore, Drageset’s 13 categories of teacher actions are grouped into three superordinate categories: redirecting, progressing and focusing actions. One important difference between Drageset’s framework and the above-mentioned detailed frameworks is that the former includes both productive actions and actions that can be unproductive when it comes to achieving a productive whole-class discussion. Drageset’s framework has been shown to be useful by other researchers (Gustafsson, 2023; Klemp, 2020; Kooloos et al., 2020; Solomon et al., 2018). However, the framework was developed based on data from the specific context of Norway, with five teachers teaching the topic of fractions for a week. Moreover, the interaction patterns in all Drageset’s analysed lessons could be categorised as IRE patterns. Drageset states that this might have influenced the emergence of the categories, and points out that other patterns of turn-taking may result in additional categories. Thus, there is a need to explore the potential of the framework in other contexts and practices (Drageset, 2015a).

In this study, I propose concepts and refine definitions of existing concepts of teacher actions in Drageset’s framework, which can help researchers conduct detailed analysis of whole-class discussions; such detailed analysis may help in the development of a deeper understanding of how teachers lead whole-class discussions in mathematics instruction. I approach this by investigating the potential of Drageset’s framework to conceptualise teachers’ actions when leading whole-class discussions generated from multiple-choice (MC) tasks supported by classroom response system (CRS) technology. These whole-class discussions were conducted by twelve teachers in mathematics classrooms in secondary schools (students aged 13–17 years) in Sweden and the UK. The research questions guiding this paper are:

• Which categories of teacher actions can be found that do not clearly belong to the existing ones in Drageset’s framework?

• What role could these categories of teacher actions play in mathematical whole-class discussions?

Teacher actions in whole-class discussions

Researchers have offered different descriptions of what characterises productive whole-class discussions. For example, Stein et al. (2008) considered productive whole-class discussions to occur when teachers “effectively guide whole-class discussions of student-generated work toward important and worthwhile disciplinary ideas” (p. 319). Drawing on a more detailed description from Grossman et al. (2014), I understand productive whole-class discussions to entail the teacher and all the students working together, using all the participants’ thinking and knowledge as a resource to improve the students’ learning toward a specific mathematical goal. In these discussions, students should have opportunities to practise and apply mathematical reasoning and communication skills.

To achieve productive whole-class discussions, teachers can use various teaching moves to support different instructional goals and student actions (Jacobs & Spangler, 2017; Michaels & O’Connor, 2015). Drawing on Jacobs and Spangler (2017), I define teaching moves as “actions that teachers take that observers can see or hear, such as asking questions, providing a representation, or modifying a task” (p. 778). A crucial instructional goal concerns eliciting students’ thinking. Teachers can address this goal by inviting and encouraging students to share their thinking (da Ponte & Quaresma, 2016; Fraivillig et al., 1999; Michaels & O’Connor, 2015), using wait time (Chapin et al., 2009), or asking for elaboration and clarification (Michaels & O’Connor, 2015). Teachers should also engage students with their peers’ mathematical thinking (Franke et al., 2015; Jacobs & Spangler, 2017; Webb et al., 2014) to improve their learning (Webb et al., 2014). Some examples of teaching moves regarding this goal are: encouraging students to add something more to peers’ contributions; restating or explaining peers’ solutions; comparing their thinking with that of others; and commenting, evaluating or developing peers’ ideas (Cengiz et al., 2011; Chapin et al., 2009; Franke et al., 2015; Herbel-Eisenmann et al., 2013; Jacobs & Spangler, 2017). Teachers must also support students’ mathematical thinking. To address this goal, they can: draw attention to specific ideas; control the pace of the discussion; or ask students to repeat, revoice, clarify or summarise a lengthy discussion, or to make connections (Jacobs & Spangler, 2017; Michaels & O’Connor, 2015). Finally, in productive discussions, teachers should support students in deepening their reasoning and learning by using challenging moves, including pressing them for connections, elaborations and justifications (Brodie, 2010; da Ponte & Quaresma, 2016; Jacobs & Spangler, 2017). These challenging moves have been shown to predict the quality of students’ participation in whole-class discussions (Matsumura et al., 2008).

The redirecting, progressing and focusing framework

In earlier research Drageset (2014) presented the redirecting, progressing and focusing framework, which focused on single-teacher action during a teacher-led discussion. This framework highlights essential teaching elements rather than an entire practice, aiming to give a detailed description of how teachers lead discussions in response to student contributions related to mathematical tasks and content (Drageset, 2014).

Drageset’s framework (2014) emerged from an analysis of mathematics lessons conducted by five teachers in primary schools (students aged 10-13) in Norway who taught the topic of fractions. Drageset points out that all analysed data refers to situations that can be described as involving an IRE pattern. The teachers managed all turns, and chose who was allowed to speak and when. Various themes emerged during Drageset’s analysis of single teacher comments as a part of the discussion, resulting in 13 categories of teacher actions (presented in Table 1). He argues that these 13 concepts can help show that there is room for many variations in IRE patterns.

Table 1. Drageset’s analytical framework (2014) with teacher actions.

Redirecting actions	Progressing actions	Focusing actions
Put aside	Demonstration	Requests for student input
Advising a new strategy	Simplification	Enlighten detail
Correcting question	Closed progress details	Justification
	Open progress initiatives	Apply to similar problems
		Request assessment from other students
		Point out
		Notice
		Recap

Categories of teacher actions

As seen in Table 1, Drageset’s framework (2014) 13 concepts for describing and analysing teacher actions are divided into three superordinate categories: redirecting, progressing and focusing. In addition, the category of focusing actions is divided into two subcategories, referring to either actions that request student input or those in which teachers point out important details themselves.

Table 1 shows three redirecting actions that redirect students’ ideas by either putting aside their contributions by ignoring or rejecting them, explicitly advising them on a new strategy, or issuing correcting questions that reject their ideas (Drageset, 2014).

The four progressing actions support teachers in moving the process forward. Demonstrations describe actions in which the teachers solve tasks themselves and do the intellectual work (Drageset, 2014). Simplifications refer to actions in which the teachers reduce the task’s difficulty level by giving hints or changing the task (Drageset, 2014). Closed progress details entail actions in which the teachers move the solution process forward one step at a time (Drageset, 2014). In open progress initiatives, teachers ask questions to which there are multiple acceptable student responses (Drageset, 2014).

Redirecting and progressing actions are common elements in the practice of funnelling, in which the teacher dominates the discussion and does most of the intellectual work. In funnelling, the students’ thinking is limited to finding the “correct” response the teacher is looking for (Drageset, 2014). In isolation funnelling does not generate a productive mathematical discussion, but redirecting actions might be helpful in keeping the students on track, and progressing actions might be helpful in accelerating the process when necessary (Drageset, 2014). If redirecting and progressing actions are wisely applied in specific situations, they can play a pivotal role in high-quality discussions (Drageset, 2014).

Teacher actions that focus on important mathematical details by requesting student input or pointing out something important themselves are called focusing actions (Drageset, 2014). These actions have the potential to elicit and advance students’ learning and move the interaction away from “show and tell” (Drageset, 2014), and are therefore an essential part of a productive mathematical whole-class discussion.

Focusing actions that request student input describe actions requiring students to enlighten important details, perform a justification of why a response or choice of method is correct, apply the knowledge to solve a similar task or assess a contribution from another student (Drageset, 2014). Moreover, teachers can point out important details by either recapping a student’s contribution by repeating and summarising, or getting the students to notice something of mathematical importance (Drageset, 2014).

Earlier developments of Drageset’s framework (2014)

Drageset (2015a) supplemented the framework by developing a similar categorisation of the five student actions: (1) explanations, (2) student initiatives, (3) partial answers, (4) teacher-led responses, and (5) unexplained answers. These concepts can help describe different types of student contributions in discussions, according to Drageset (2015b). Further, together with the concepts for teacher action, it may be possible to characterise and describe qualities in discussions on a turn-by-turn basis (Drageset, 2015b). He mentions that it would also be possible to study and describe relations and patterns between several types of student and teacher actions.

In a Dutch study conducted in upper secondary school mathematics classrooms, Kooloos et al. (2020) adjusted Drageset’s framework to better fit their context. Firstly, they divided Drageset’s teacher actions into two superordinate categories of Henning et al.’s (2012) concepts of divergent student-guided actions and convergent teacher-guided actions to better match their theoretical framework’s critical characteristics of classroom discourse. Secondly, they added a third superordinate category of encouraging actions. There are two categories of such actions, confirmations and encouragements, that entail actions that invite students to continue talking and explaining their thinking. Kooloos et al. point out that confirmations can be performed through either an evaluative confirming statement or a questioning manner like “yes?”. Further, they added a fourth superordinate category called regulative actions, which include the action rules of classroom discourse. This action involves the teacher articulating the rules and norms of communication. Moreover, they removed some of Drageset’s actions that they did not find in their data and added the action of reformulation, which includes the actions in Drageset’s (2014) category point out. According to Kooloos et al., reformulations also include correcting and modelling students’ mathematical language and facilitating communication. Finally, they added the category of external directed, which entails actions in which the teacher asks a specific student a question; and external general, in which the teacher asks the students in general.

In an intervention study in the UK, Solomon et al. (2021) implemented a realistic mathematics education (RME) approach to instruction with low-attaining students. In this approach, teachers were to achieve productive mathematical whole-class discussions. Here, Drageset’s framework was used as an analytical tool to explore how the teachers used students’ contributions in discussions. However, to investigate how intellectual authority was distributed between teacher and students, Solomon et al. added the location of authority to Drageset’s framework. Teacher actions that they classified as teacher as authority were all redirecting actions; the progressing actions demonstration, simplification and closed progress details; and the focusing actions recap and notice. In addition, the progressing action open progress initiatives was classified as some student authority, while the focusing actions in which the teacher requests student input were classified as student as authority.

In a study on early mathematics, Klemp (2020) found that many teacher comments did not fit any of Drageset’s categories and thus added a new category, classroom management actions. However, Drageset (2014) does not claim to cover this type of action as his framework focuses on how teachers respond to students’ mathematical contributions.

Methodology

To explore and develop Drageset’s framework (2014), data was used from an intervention study that characterises whole-class discussions to explore the potential of implementing MC tasks supported by CRS. A brief description of the study is presented in the following sections; for more details, see Gustafsson (2023).

Intervention

The intervention aimed to help teachers achieve productive mathematical whole-class discussions by implementing MC tasks supported by CRS. MC tasks consist of a question or instruction followed by two to five options tagged with the letters A, B, C … These options contain no, one, or several correct choices and no, one, or several incorrect choices, called distractors. A web-based CRS (sometimes called clickers or personal, classroom, instant or audience response system) was used to support the implementation of these MC tasks. The CRS involved a teacher’s computer connected to a projector screen, one computer for every student, and a web-based CRS application. When using CRS teachers can launch tasks and then collect and monitor students’ responses in real time, to finally compile them in a bar chart. Then, the teacher can instantly analyse the results and decide whether and how to conduct peer and whole-class discussions.

Three types of MC tasks, each constructed with support from Gustafsson and Ryve’s (2022) framework, were implemented. Figure 1 shows an example of a task type focusing on students’ conceptual understanding. The task type, called Odd one out, contains multiple defendable answers (Gustafsson & Ryve, 2022). In Figure 2 an example of a task type focusing on procedural understanding is presented, called Evaluating solutions (Gustafsson & Ryve, 2022). Figure 3 presents an example of a task type called Trolling for misconceptions or mistakes (Gustafsson & Ryve, 2022).

Figure 1. Task type odd one out (Gustafsson, 2023).

Figure 2. Task type evaluating solutions (Gustafsson, 2023).

Figure 3. Task type trolling for misconceptions or mistakes (Gustafsson, 2023).

Teachers’ implementation of the tasks was supported with a simple teaching guide. The guide contained the aim of the task, why certain choices could be correct, and comments on common mistakes or misconceptions that could be reasons for incorrect choices. Further, all teachers were instructed to use the following activity structure: (1) launching the task and letting all students think on their own, (2) students responding by themselves, (3) students discussing in pairs, (4) visualising a summary of students’ responses in a chart, and finally (5) teachers leading a whole-class discussion. However, the teachers received no support in how to lead the whole-class discussions.

Participants, context and data

To explore and help develop Drageset’s framework (2014), teacher-led whole-class discussions that originated from the research project described above were analysed and characterised. This data was chosen because it involved a different context to that of Drageset’s data.

A total of twelve teachers were recruited, nine secondary school teachers from Sweden and three from the UK (with students’ ages ranging from 13 to 17 years). All the teachers had experience of teaching mathematics supported by CRS technology. Teachers were recruited from two countries in order to achieve a triangulation of data sources and sites to strengthen the study’s credibility by reducing the local factor effect (Shenton, 2004).

The data sources comprised 35 whole-class discussions originating from 35 implemented CRS tasks in MC format conducted by the twelve teachers. Gustafsson and Ryve’s (2022) framework for constructing CRS tasks in MC format was applied to construct appropriate tasks. Eleven teachers conducted three whole-class discussions, while one teacher from the UK ran out of time and implemented only two tasks. On average, students worked individually for 2.5 min before submitting a response. Then peer discussions were conducted for an average of about 2.5 min before the whole-class discussions were conducted, lasting an average of about 7.5 min. Further, the lessons were recorded with a fixed camera positioned at the front of the class focusing on almost all of the students, and a second camera positioned at the back, focusing on the teacher and the board. The second camera was used to follow the teacher. Neither of the cameras’ positions blocked the students’ or the teachers’ view (Kimura et al., 2018). An additional microphone was attached to the teacher to guarantee good sound quality in the recording of all the teacher’s talk.

Data analysis

The data analysis and coding process was conducted in an iterative process. First, the author watched and transcribed all 35 whole-class discussions with timings in Nvivo 12. As nine teachers came from Sweden, some of the presented transcriptions below have been translated by the author. Pauses longer than one second and inaudible parts were noted. Identified student speakers were recorded as numbers in order, and student speakers who were not identified were recorded as “S”. Then Drageset’s framework (2014) was applied, and all the teacher’s actions were coded according to the framework’s 13 categories of teacher actions. It is important to note that in a teacher-led whole-class discussion, each turn depends on the previous turn (Linell, 1998). Thus, the whole-class discussion can be viewed as a construction generated by the jointly coordinated actions of different actors (Linell, 1998). Therefore, I chose to analyse and categorise the teachers’ actions turn-by-turn, but not entirely isolated from the students’ actions. Consequently, these teacher actions were analysed as part of the discussion, whereby each action could be seen as a response to the previous student’s comment.

After the first draft of coding, difficulties were discussed and resolved with two colleagues with extensive experience in classroom research. An example of an analytical difficulty was when teachers asked students to repeat their contributions, because this action might have different goals. However, when we looked at teachers’ actions as part of the discussion in relation to students’ contributions, we came to a decision: if it was clear that a student’s response was hearable and incorrect it was coded as a correcting question, because the goal might have been to let the student correct their response; if it was clear that the student’s response was hearable and correct it was coded as enlighten detail, because the goal might have been to place the focus on an important mathematical detail. Once again, difficulties were discussed with the same two colleagues, and the author re-analysed the data. For the relative frequency of each original category in the data, see Gustafsson (2023).

As described above, the author noted some teacher actions that were difficult to categorise. Transcripts with these actions were noted. Further, to be able to contribute to Drageset’s framework (2014, 2015b), I chose to try to replicate his described analytical method when analysing the selected transcripts. First, the teacher actions were analysed one at a time as part of the discussion. Each teacher action was seen as a response to the previous student action, and the next student action as an effect of the previous teacher action. Then similar actions were brought together to become a preliminary category. Second, all of the developed preliminary categories were compared with each other and merged if possible. Third, Drageset’s three superordinate categories were used as an analytical tool to further categorise the suggested newly developed categories as redirecting, progressing or focusing actions. Then, the categories were compared with the existing subcategories in Drageset’s framework to be either rejected or suggested as a new category or as a specific example of an existing one.

Results and analysis

The data analysis resulted in suggestions for five new categories of teacher actions and enrichments and refinements of two existing categories in Drageset’s framework (2014, 2015b). These suggestions are presented in Table 2.

Table 2. A suggested development of Drageset’s analytical framework (2014, 2015b).

Redirecting actions	Progressing actions	Focusing actions
Put aside	Demonstration	Requests for student input
Advising a new strategy	Explanation	Enlighten detail
Correcting question	Simplification	Justification
	Closed progress details	Apply to similar problems
	Open progress initiatives	Request assessment from other students
	Encouragement	Add on
		Connection
		Point out
		Notice
		Recap
		Challenge students’ ideas

Note: New categories are shown in bold, and enriched categories are underlined.

A short presentation of each teacher action and the contexts of the discussions are outlined and then illustrated in an excerpt from a transcript. Following this is a brief analysis of the transcript. Finally, a further analysis of each action is presented, including connections to existing categories and what role they might play in whole-class discussions.

Progressing actions

In this section, suggestions are presented for two new categories of progressing actions that emerged in the analysis.

Explanation

On many occasions, the teachers performed explanations of reasons, concepts or methods in response to a student’s mathematical contribution. For example, they sometimes performs complete explanations of the logic behind a correct or incorrect choice in an MC task.

Example 1

In this first example, students were to discuss and explain different correct and incorrect displayed solutions for calculating the numeric value of the algebraic expression 3a + 1 if a = 5. The example illustrates a teacher’s explanation of the reason behind an incorrect choice (C) in an MC task displaying the incorrect solution 3 + 1 = 4:

S2: Where’s the number five?

T: She’s lost it and completely ignored the variable with the value of five there. Let’s move on to the next task.

In Example 1, the student takes the initiative and asks a question about a mathematical detail in a fictitious incorrect solution. The teacher responds and performs an explanation of why the answer is incorrect to meet the student’s needs, and finishes the discussion by explicitly pointing out that they need to progress to the next task.

Example 2

In Example 2, involving a task focused on explaining the differences and similarities between the concepts of percentage and percentage units, the teacher and the students discuss the truthfulness of the statement ‘A political party increased their support from 17% to 20% in an election. Thus, they increased their support by 3%’. The teacher struggles to get a good explanation from the students, and decides to give them an easier example:

T: If we take a simple example: If something increases in weight from 10 kg to 12 kg, how many kilos has it increased by?

S: 2 kilos

T: By how much per cent has it increased? From 10?

S: 20 per cent

T: All things measured in percentages must have the same rights. So, the percentage change must be calculated as the change divided by the primary value, right? But the change in number is what it is. That’s why you need some way to describe that the number is changing, but not an increase of three per cent. Do you understand? So, then percentage units have been introduced. Plus, three percentage units ((the teacher writes + 3 p.u. on the board)); I’ll abbreviate it. So, when the number changes by something, you say percentage units. But how much per cent was the change?

In Example 2, the teacher simplifies the original task by giving a clear example, then asks a funnelling question and finishes by trying to explain percentage units. Thereafter, the teacher redirects the focus to the original task to see if the students can apply the knowledge in this context. It may be that teacher performs the explanation himself to make some progress in the discussion, as the students have not come up with any correct explanation at the beginning of the discussion.

Example 3

In certain cases, these explanations provide explanations of methods. In Example 3, the teacher and the students discuss different fictitious correct and incorrect student solutions for simplifying the task 14x – 3(x + 5). The following sequence is near the end of the discussion:

S: It’s 11x decreasing by 15 and that’s the final answer.

T: Excellent! What do we call these things here?

(pointing at x-terms)

These are what?

S: Factors!

T: They’re like terms. That’s why we can combine them. We combine the 14x and the 3x because they’re alike; they talk about the same thing. They’re talking about x.

In Example 3, a student gives a correct answer and the teacher confirms it. The teacher then performs a short answer question, emphasised with a gesture to place the focus on the x-terms. The student gives an incorrect short answer, and the teacher responds with a correct answer and continues, explaining why x-terms can be added. As this action places the focus on an important aspect of the task, it can be seen as a focusing action of notice. However, the teacher chooses to explain why the solution is correct, thus performing a major part of the task the students are supposed to discuss.

General comments

The above transcripts show instances in which teachers choose to explain concepts, methods or reasons for incorrect or correct answers displayed in the MC tasks. I argue that in these situations this is not a notice action, because it is not about stopping the progress and pointing out an important element in the students’ comments (Drageset, 2014). In my data, when a task’s focus was on getting students to explain and argue, it seems more likely that the teachers occasionally chose to perform explanations to make progress and to support the students’ thinking in the discussion when they were struggling to provide explanations. Therefore, I argue that these explanations belong to the superordinate category of progressing actions. One could also argue that these explanations belong to the demonstration category, but Drageset’s (2014, 2015b) descriptions and examples show that demonstrations involve solving steps in procedural or problem-solving tasks rather than explaining reasons, methods or concepts in tasks focused on reasoning.

Encouragement

On many occasions, in the analysed data, teachers may confirm students’ comments by saying “yes”, “okay” etc., sometimes adding a gesture to strengthen the confirmation to encourage them to continue talking.

Example 4

In the following example, the teacher and the students discuss a lottery that says that every third ticket is a winner, in a Trolling for misconceptions or mistakes task.

S: They kind of … you know this type of lottery … and they actually didn’t … 

T: Yes. (nods affirmatively at the student)

S: And they could apparently be in order.

T: All right.

S: But someone could only buy one ticket and get a winning ticket, and if other customers buy three tickets they might not get the winning ticket in the third; they could get it in the second.

In Example 4 the student tentatively begins to talk, and the teacher responds by nodding and saying “yes” to encourage them to continue. The student continues talking, and the teacher again confirms and encourages them by saying “all right”. The student continues talking and develops the explanation.

Example 5

The following example shows a teacher explicitly encouraging students to continue by saying “continue”, “elaborate further” etc. Below, a student tries to describe the value of milli in an Odd one out task.

S: So, it’s not the same thing. Milli is less than one. No, I don’t know.

T: Yes, elaborate further!

S: Milli is one-thousandth.

In this example, the teacher responds with a confirmation to an incomplete explanation and then explicitly encourages the student by saying, “elaborate further”. The student continues talking and contributes a correct response.

General comments

Teacher comments such as “yes” or “okay”, and utterances like “move on” can function as actions that encourage students to move on and continue talking and explaining. Thus, these actions can help to move the progress forward and support students in making their thinking public. While one could argue that an explicit encouraging action is a kind of open progress action, I suggest that encouragement is different, because open progress initiative is an action that initiates a progress (Drageset, 2014) rather than encouraging an ongoing progress.

Focusing actions

In this section suggestions are presented for new categories of focusing actions, which emerged in the analysis.

Add on

Regarding focusing actions, an action was noted in the data when the teacher asks the students to add on to another student’s contribution. The example below is based on a discussion about an Evaluating solutions task concerning distance, speed and time:

S: I thought … more than 20 km in an hour, so the answer must be under one hour. And 0.8 was the only option that was under an hour, so you find out already then. Then I also thought that 0.8 is equal to 4/5. And 1/5 of 25 km is 5 km so then it must be 0.8. I don’t really know.

T: You’re doing great, Anna. Can anyone help Anna? How should she write this? She reasons so well. How does she put this in writing?

The teacher confirms that Anna’s reasoning is promising. After this, the teacher invites other students to contribute and express this reasoning with mathematical symbols.

General comments

In this action, the teacher seems to focus on the student’s reasoning and make it more straightforward and accessible to all the students. This is done by asking other students to engage in the reasoning and help develop a peer’s contribution by adding on a written symbolic representation of the reasoning.

Connection

In certain situations in the analysed data, the teacher stops the progress by asking a question that forces the students to make a connection between different fictitious solution methods displayed in the Evaluating solutions tasks. Here, the students discuss different correct and incorrect methods for a task concerning percentage:

S1: Disa (CHOICE D) has multiplied 15 per cent by three.

T: Which one of the other solutions shows a thought like Disa, but in a different way?

S2: Christer (CHOICE C)

The teacher responds to a partial student explanation by asking a question about similarities to other fictitious solution methods displayed in the choices of the MC task.

General comments

In these teacher actions, the teacher explicitly asks questions about connections and forces the students to make mathematical connections between representations or methods. One could argue that these actions belong to Drageset’s (2014) category enlighten detail. However, Drageset describes enlighten detail as an action in which teachers ask a student to explain what something means or how something happens, which is necessary in order to allow the other students to follow their line of thought. However, in the examples above, the teacher focuses on mathematical connections rather than the details of something that could help the students follow the line of thought. Thus, it seems that this teacher action is focused on requesting students’ input in order to make a connection, with the aim of enhancing their mathematical learning.

Challenge students’ ideas

On some occasions, in the analysed data, one of the teachers challenge students’ ideas by making statements and pretending to be against a student’s idea. In this example, the teacher and the students discuss and compare three javelin sporting results from three different competitions in an Odd one out task. Which of the athletes should be removed from the national team?

S: Yes, if you add them all up, then Eric has the shortest result.

T: So, that’s why Eric should be removed?

S: Yes

T: But he’s thrown the farthest of them all?

S: He only threw well once. He got lucky.

The teacher follows up on a student’s explanation by asking a confirmative question. The student responds, and thereafter the teacher challenges them by making a provocative statement and pretending to be against their idea. The student then tries to sharpen their explanation.

General comments

In these teacher actions, the teacher pretends to be against students’ ideas to challenge these ideas. This action seems to force students to sharpen their arguments and further explain their thinking. Research (Jacobs & Spangler, 2017) has stressed the importance of pressing students for justifications, and Drageset (2014) has a category of justification in which the teacher asks students to justify their method or answer. Thus, one could argue that challenging students’ ideas can be categorised as a justification action because it can be viewed as an indirect press for justification. However, then the definition and examples of the category of justification need to be supplemented to include these provocative and challenging statements. Meanwhile, a difference between challenging by pretending to disagree and justifications is that in the former action the teacher points out important information that argues against students’ ideas and can rather be categorised as a pointing out action. Thus, I suggest that this is a separate category of pointing out actions.

Enrichments of existing categories

In this section, suggestions are presented for enrichments of existing categories of teacher actions.

Justification

In certain situations, as can be seen in the data, the teachers follows up on a student’s response by explicitly asking them to justify why an answer, choice or method is incorrect in the implemented MC task. Below is an example from a discussion on the task type Trolling for misconceptions or mistakes:

S: I thought that … Choice B can’t be correct either.

T: Why can’t Choice B be correct?

S: I think that if someone were to choose B, then they think that one hour is.

100 min, but it’s not.

Choice B in this MC task elicits the common misconception that one hour equals 100 min. The student suggests that Choice B cannot be correct, and the teacher presses them to explain why. Finally, the student gives a correct suggestion for why the answer in Choice B is incorrect.

General comments

The intention behind this action is likely to force students to perform justifications and explain the reasons behind common misconceptions or mistakes in these MC tasks. Drageset’s (2014) description of the category justifications focuses only on asking students to explain why a method is correct. I argue that these types of questions, which explicitly ask students to explain why something is incorrect, also belong to the category of justification. The above example can help make this category richer and clearer. Thus, I suggest that Drageset’s description of the category justifications should be refined to entail students being asked to justify why an answer or method is correct or incorrect.

Request assessment from other students

Sometimes the teacher asks students if they understand the other students’ explanations and can follow the line of thought. In this example, the students discuss and compare lines of equations in a Trolling for misconceptions or mistakes task:

S: While one line increases by three continuously, the other also increases by two the whole time. So, they won’t be parallel. They go away from each other (SHOWING WITH HANDS).

T: I understand. Did you follow that, Amy?

The student tries to explain why two lines cannot be parallel. The teacher confirms the explanation and then asks another student if she has followed the line of thought.

General comments

This action might also be a teacher action that aims to request assessment from other students or check if the students are paying attention. However,Drageset’s (2014) description of the category request assessment from other students offers no clear examples or descriptions of occasions on which the teacher asks students if they understood other students’ contributions or could follow the line of thought. Thus, I argue that this example could help enrich this category and make it clearer.

Discussion

This study sought to suggest new concepts and refine definitions of existing concepts of teacher actions in Drageset’s framework (2014, 2015b) – actions that can be helpful in analysing, describing in detail and building a profound understanding of whole-class discussions. This was done by investigating the potential of Drageset’s framework to capture teachers’ actions during 35 whole-class discussions generated from three types of MC tasks supported by CRS. Overall, the findings showed that the elaborated data contained teacher actions that did not clearly fit Drageset’s (2014) categories. The results suggest a developed framework with five new categories of teacher actions – two progressing and three focusing ones – and enrichments of and refinements to two existing categories of focusing actions.

Concerning progressing actions, the analysis revealed cases in which teachers conducted explanations of reasons, concepts or methods. I suggest that teachers occasionally chose to conduct explanations to control the pace of the discussion and make progress. This action can support students’ mathematical thinking (Jacobs & Spangler, 2017) when they are struggling with explanations. Although the teacher does all the mathematical thinking in this action, this can sometimes be necessary if the goal of the discussion is to be reached within a reasonable time (Drageset, 2014). Further, I suggest that the implemented tasks, and the fact that they were designed to generate discussion and get students to explain and argue (Gustafsson & Ryve, 2022), can partly explain why teachers performed explanations.

Another progressing action that the analysis revealed involved teachers encouraging students to continue talking by offering confirmations or explicitly encouraging them. This action, noted in previous research, can function to elicit students’ thinking (da Ponte & Quaresma, 2016; Fraivillig et al., 1999; Michaels & O’Connor, 2015). Michaels & O’Connor label these explicit encouragements as the “say more” family. Further, the indirect encouraging actions, with comments like “yes” or “okay”, have also been identified as acknowledgement tokens in IRE patterns (Huq & Amir, 2015). A strongly delivered acknowledgement token by the teacher has been shown to serve as a confirmation of a student’s contribution, and a passively delivered acknowledgement token may serve to maintain the teacher’s listening (Huq & Amir, 2015). In this study, the results contain examples in which these may have served as confirmations that encouraged students to continue talking. However, to achieve productive discussions and use students’ thinking as a resource (Grossman et al., 2014), teachers must first elicit their thinking (Jacobs & Spangler, 2017). Thus, encouragement actions can play an essential role in whole-class discussions, and these actions have also been suggested as developments of Drageset’s framework (2014) by Kooloos et al. (2020).

Regarding focusing actions, the analysis revealed actions not captured by Drageset’s framework (2014), in which teachers asked students to add on to another student’s contribution, facilitated connections, and challenged a student’s ideas by pretending to disagree. The importance of the asking students to add on to another student’s contribution has been noted in earlier research and the label add on is used in Accountable Talk (Franke et al., 2015). However, this action facilitates the critical aspect of productive discussions to engage students with their peers’ mathematical thinking (Franke et al., 2015; Jacobs & Spangler, 2017; Webb et al., 2014). Further, the analysis also revealed examples in which teachers asked students if they understood other students’ explanations or could follow the line of thought. This action also engages students with their peers’ mathematical thinking and may aim to either evaluate their peers’ contribution or support them in listening to one another (Michaels & O’Connor, 2015). Thus, I suggest that this specific teacher action can enrich the existing category request assessment from other students.

The findings show teacher actions that encourage students to make connections. Research has stressed the importance of helping students make connections between concepts or methods (Jacobs & Spangler, 2017; Stein et al., 2008). In fact, Stein et al. (2008) suggested that making the connection between students’ contributions is a key practice in whole-class discussions on problem-solving tasks. The analysis in this study revealed how teachers used the fictitious solution methods displayed in the implemented task type Evaluating solutions to force students to make connections between solution methods. Previous research has shown that tasks designed with fictitious solution methods can help teachers achieve productive whole-class discussions (Evans & Dawson, 2017). According to Evans and Dawson, the pre-knowledge of well-designed student solution methods supports teachers’ planning and decreases the need for improvisation when leading whole-class discussions.

The findings present the teacher action challenge students’ ideas by pretending to disagree, in which the teacher makes statements and pretends to be against the students’ ideas. The analysis showed that this action may force students to sharpen their arguments, explanations or justifications. However, I suggest that this action may sometimes function in the opposite way if the teacher has the main authority in the classroom. In such cases, students might be afraid to continue contributing by explaining their thought process. Further, this action might be similar in name to “agree/disagree” in Accountable Talk. However, agree/disagree is described as an action that aims to elicit, link and compare students’ contributions and thinking (Michaels et al., 2013). In addition, the findings also contain examples in which teachers pressed students for justifications of why something was incorrect, which can enrich Drageset’s (2014) existing category of justifications. Moreover, I argue that both challenge students’ ideas by pretending to disagree and justifications of why something is incorrect can support students in deepening their reasoning and learning, which is a critical aspect of productive whole-class discussion (Jacobs & Spangler, 2017). The design of the Odd one out tasks may have partly influenced these teacher actions. In these MC tasks, the choices were often constructed with multiple defendable answers. All or several choices can be correct, depending on how you argue. Thus, the teacher can challenge a student’s idea by pretending to disagree in order to facilitate their argumentation without saying it is mathematically incorrect.

Researchers should have access to tools that allow detailed analyses and descriptions of how teachers use students’ contributions in teacher-led whole-class discussions (Drageset, 2014). However, in a literature review, Jacobs and Spangler (2017) identified a need for more detailed frameworks for analysing whole-class discussions; the results of this study can add to this area of research. This is done by suggesting additional categories of teacher actions, and the enrichment of existing ones, in Drageset’s (2014, 2015b) framework. This developed framework can help researchers explore in depth how teachers’ actions during whole-class discussions influence students’ learning. This could involve, for example, exploring teacher and student actions on a turn-by-turn basis around specific teacher actions, such as challenge students’ ideas, something that more holistic frameworks cannot support (Jacobs & Spangler, 2017).

However, these suggestions emerged from data in the context of mathematics instruction with MC tasks and CRS; while some of these actions have been identified in previous studies and some connections have been discussed above, this context likely influenced the results of this study. Therefore, the developed framework needs to be tested in other contexts in further research.

I also hope that this developed framework can help teachers plan, analyse and develop their practice of leading mathematical whole-class discussions, both in the context of MC tasks and in other contexts. However, in the context of mathematics instruction supported by CRS and MC tasks, research on whole-class discussions has received little attention (Gustafsson, 2023). This is the case despite the importance of conducting both peer and whole-class discussions if one hopes to improve students’ learning (Green & Longman, 2012; Webb et al., 2019; Wester, 2021). Thus, the results of this study can help researchers explore this critical practice of leading whole-class discussions generated from MC tasks supported by CRS, and further enhance the knowledge in this research area, by enabling a more profound understanding of how teachers’ actions can influence students’ opportunities to learn.

Ethical declaration

The study contains video-recordings of classroom discussions between teachers and their students and is, therefore, needed to be considered in relation to the Swedish law on ethical review of research involving humans (SFS2003:460). The video-recordings were not classified as sensitive by the Head of the Advisory Ethics Committee at Mälardalens University. In addition, the confidentiality claims and the data use requirement of the study are fulfilling the recommendations from the Swedish Research Council.

Acknowledgements

I would like to thank all the teachers and students for their voluntary participation in the implementation of the MC tasks. I also want to thank Andreas Ryve and Per Sund for their great support during the research and writing process.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For a more detailed description and definition of productive whole-class discussions, see the section “Teacher actions in whole-class discussions”.