Teacher Follow-up on Learners’ Initial Response to Teacher Questions

Abstract

Participating in classroom mathematics conversations, particularly engaging with the opinions of others, can improve learners’ mathematical comprehension. Teachers can use a variety of invitation and follow-up moves to encourage student engagement. The study aimed to explore teacher follow-up on learners’ initial responses to teacher questions and building on learner ideas in Grade 9 mathematics classrooms. A qualitative approach and descriptive case study design were utilised. Data were collected from three schools in the same circuit in South Africa through classroom lesson observations, face-to-face teacher interviews and researchers’ field notes. Three Grade 9 mathematics teachers, one from each school, were purposively selected, and the selected teachers’ already established Grade 9 classes, with a maximum of 30 learners each (owing to the COVID-19 health protocol), were automatically involved. Cognitively guided instruction, a learner-centred approach that uses social constructivism as a research paradigm, guided this study. The exploration was supplemented by Brodie’s conceptual framework on teacher follow-up moves. The findings revealed that the three teachers followed up on learner contributions and used different follow-up moves, namely insert, elicit, press, maintain and confirm. The study showed hybrid teacher moves. Although more teacher moves were skewed towards elicit and insert, which are closer to traditional practices, there is an indication of moving towards the press, maintain and confirm, which are reform-oriented moves. In addition to these moves identified by Brodie, acknowledging or praising the learner for responding was another move that emerged in this study.

Keywords:

• cognitively guided instruction
• learner-centred
• learner talk
• follow-up
• responsive teaching

Introduction

Given the persistent trend of very low learner achievement in mathematics in South Africa, an analysis of Grade 9 Annual National Assessment (ANA) results revealed that one of the challenges facing learners was that they were being taught mathematics without gaining understanding (Department of Basic Education, 2014). Low learner performance draws attention to the nature of mathematics teachers’ pedagogy in classrooms (Abdulhamid, 2016), like the absence of teacher follow-up practices on learners’ initial response to teacher questions and lack of building on learner ideas. Although much seems to have been done internationally on responsive teaching, little has been investigated in South Africa (Abdulhamid & Venkat, 2018; Moodliar & Abdulhamid, 2021), including teachers’ follow-up on learner responses in mathematics, particularly in Limpopo where the study was conducted. In this regard, Abdulhamid and Venkat (2018) have suggested that responsive teaching be considered in classroom interaction to improve teachers’ awareness of the need to give suitable follow-up to learners’ responses that increase possible ways of mathematics learning.

At all levels of learning, learners are exposed to questions because teacher–learner interaction positively affects classroom mathematics learning. Teacher questioning is an important tool for effective teaching, which Boyd (2015) regards as arguably the most powerful device in talk moves. For learner thinking to be incorporated into the learning process, teachers first need to elicit such information from learners. In Principles to Action, the National Council of Teachers of Mathematics (2014) notes that one of its eight mathematics teaching practices includes the eliciting and use of evidence from learners’ thinking. In South Africa, the National Curriculum Statement (Department of Basic Education, 2011) aims to produce learners who can use critical and creative thinking when solving problems. Ideally, questioning has several benefits that can be summarised as follows (Nappi, 2017; Wash & Sattes, 2016):

• questioning elicits learner activity and engagement;

• questioning provides opportunities for peer learning;

• questioning switches the learner’s role from listener and receiver to the speaker and producer of mathematical ideas;

• questioning and answering spreads the energy production in the classroom from the teacher to the whole class;

• questioning is a mode of formative assessment for the teacher to gauge and guide learner understanding; and

• questioning is the ideal strategy for immediate feedback.

Without a doubt, questioning is a very useful tool in teaching and learning, hence worth implementing in mathematics classrooms. In addition, Mercer (2012) asserts that if we can transform the quality of classroom talk we can, in turn, transform the quality of education. Therefore, focusing on teachers’ follow-up to learner responses in mathematics in South Africa is important, given the ongoing evidence of poor learning outcomes in mathematics, and largely traditional pedagogic forms in which the primary source of information is the teacher and learners are mere recipients. However, Brodie (2010) argues that although there is a shift towards more reform-oriented moves, teachers are developing hybrid pedagogies and responding to reforms in different ways. In the same vein, Staples (2007) asserts that collaborative inquiry in mathematics classrooms entails teachers being responsive to learners by allowing space for learner contribution and tracking their thinking. In such inquiry, both parties—the teacher and learners—are involved in the mathematical conversation as the teacher follows up on the learner’s idea. This article is guided by one research question: how do teachers follow up on learners’ initial explanations and build on learner ideas after questioning?

We begin this paper with a theoretical framework on Cognitive Guided Instructions on how it is described and related to the learner-centred approach. In the literature on teacher follows-up moves, when teachers respond to their learners’ first answer to the teachers’ questions, learner talk has been reviewed in detail. The data sources and methodological approaches used for this development are detailed prior to the central sections of this paper, where we present and illustrate the dimensions, codes and categories on teacher moves emanating from the three teacher lessons.

Theoretical Framework: Cognitively Guided Instruction

Learning, in the mathematics classroom, is greatly influenced by teacher–learner interaction and classroom talk. However, Brodie (2010) has indicated that empirical evidence on mathematics classroom talk reveals that there is a limitation in developing learners’ deeper mathematical thinking. In this modern world, teachers are expected to use a learner-centred approach instead of utilising traditional mathematics classrooms where, according to Dahal et al. (2019), the teacher does most of the talking as the learner listens. Instead, learner-centred approaches encourage learner participation in classroom discourse (Jones, 2019). On the other hand, learner-centred instructional approaches encourage learners to develop their abilities, resulting in the production of higher positive achievement among mathematics learners (Fallace, 2015). The teacher becomes a facilitator by scaffolding instead of being the only provider of information to learners (Machaba, 2017). This implements one of the aims of the South African curriculum which emphasises an active and critical approach to learning rather than rote learning (Department of Basic Education, 2011). In learner-centred approaches, both teacher and learner are involved. For example, when scaffolding, the teacher takes the role of supporting the learner, in this case in the form of follow-up to learner responses, and the learners get engaged as they respond to teacher moves. However, in support of Staple’s (2007) study, learner centredness and teacher centredness are not described as a dichotomy or polar opposite of each other but are viewed as existing on a continuum. In her study, Staple revealed that a teacher who used the traditional approach was able to probe and engage learners well. In addition, Darsih’s (2018) perspective is that a teacher becomes a creator of a climate conducive to learning, enhancing learners’ self-worth and sense of capability as they connect mathematical ideas.

This study was guided by cognitively guided instruction (CGI), one of the learner-centred approaches that can be applied in mathematics classrooms. Brodie (2010) asserts the need for teachers to adopt CGI to curb the problem of lack of learners’ involvement in mathematics classroom discourse. The learners’ thought processes become explicit when learner have to justify their thinking in response to teacher follow-up questions. In addition, Carpenter et al. (2015) view CGI as an approach that encourages the teacher to listen to learners’ views of mathematical thinking and the teacher uses this knowledge as the basis of instruction, hence, taking learners from one level of understanding to another in responsive teaching. This suggests that reform and traditional approaches are not a dichotomy. Instead, they exist in a continuum. Cognitively guided instruction, according to Carpenter et al. (2015), is an example of learner-centred instruction based on the assumption that learners have intuitive mathematical knowledge that can serve as a basis for more advanced knowledge. Consequently, CGI, a theoretical framework that promotes learner involvement, was suitable for this study.

Review of Related Literature

Teacher Follow-up

If teachers need to incorporate learners’ thinking into classroom activities, they should elicit this thinking by posing questions (Setiawati, 2017). One of the categories of teacher questioning techniques, according to a report on teacher questioning strategies in mathematics classroom discourse by McCarthy et al. (2016), is probing and follow-up. A question is described as a follow-up question when it follows a learner’s response and is directly connected to the learner’s response or broad idea (Dong et al., 2018). This is explained further by Nappi (2017) as a way of including a series of questions to explore an idea with the expectation of gaining answers and insights, in this case, getting insights from learners. Alternatively, follow-up is described as ‘teacher moves that respond to learner contributions and thus engage with them in some way’ (Brodie, 2010: 7). Therefore, follow-up emerges when teachers respond to their learners’ first answer to the teachers’ questions, thus creating a reciprocal dialogue between teacher and learner(s). The immediate decisions on learners’ responses that are established in the teacher’s succeeding teaching move in classroom discourse constitute responsive teaching, which Abdulhamid and Venkat (2018) assert is vital for suitable teacher follow-up to learner responses. Mhlolo et al. (2012) have also suggested the need for teachers to be assisted in strengthening the effectiveness of their pedagogical practices. Hence, this study on teacher follow-up will add to existing empirical evidence valuable to teacher educators and teachers on responsive teaching in the classroom.

Franke et al. (2009) identified teacher questions used to follow up on learners’ contributions once the teacher has asked learners to explain their thinking. They established leading (scaffolding) questions and probing questions. Leading questions guide learners towards particular explanations or answers, while probing questions mainly focus on attaining an explanation of the learners’ thinking. Furthermore, Brodie (2010), in her study on learners’ mathematical thinking, classified teacher follow-up moves into five distinct subcategories, namely insert, elicit, press, maintain and confirm.

Learner Talk

Learners must talk as they respond to teacher questions (National Council of Teachers of Mathematics, 2018). According to Franke et al. (2009), learner talk is one of the main components that promote learner acquisition of knowledge. Consequently, learners are allowed to justify or elaborate on their thinking in response to the teacher’s questions, including follow-up questions. While learners respond to teacher questions, teachers encourage mathematical discussion to elicit learner thinking (Dahal et al., 2019). On the other hand, Franke et al. (1998) have suggested that listening to learner talk assists teachers and classmates in following up on learners’ mathematical thinking. In concurrence with these research findings, Chin (2007) has proposed that when learners describe, explain and justify their thinking, other learners are assisted in internalising principles, constructing specific rules and, in the process, they become aware of the gaps in their understanding. Hence, learner talk becomes critical in the process of teaching and learning as learners get involved when they respond to teacher questions.

Methodology

This qualitative descriptive case study was an exploration of how teachers follow up on initial learners’ responses to teacher questions and build on learner ideas during mathematics classroom discourse. Descriptive case design was employed because the inquiry examined a phenomenon in Grade 9 mathematics teachers in a certain context and using multiple data collection methods and findings (Yin, 2018). The case study consisted of three rural public secondary schools from same circuit. It is essential to find out how teachers follow up on learner thinking when eliciting questions that are used in classroom talk.

Research Participants and Settings

The participants in this study were three Grade 9 mathematics teachers from three coeducational secondary schools in a single circuit in a rural community in Limpopo Province, South Africa. The circuit has six coeducational public schools with an enrolment of about 400 Grade 9 learners and 11 Grade 9 mathematics teachers. The study was conducted in Limpopo Province because the lack of mathematics education research in Limpopo has meant that there are no accounts of teachers’ follow-up to learner responses in mathematics. It can be argued that the lack of research knowledge from rural contexts in South Africa results in deficit assumptions regarding the quality of mathematics teaching and teachers in those contexts, particularly when the dearth of empirical evidence relating to mathematics teaching in Limpopo schools and classrooms is seriously considered.

The target population for this study was Grade 9 mathematics teachers. A sample of three Grade 9 teachers was purposively selected from the different secondary schools in the circuit, using the following criteria: participants had to be from Tshinane Circuit, they had to be Grade 9 mathematics teachers and they had to be willing to participate in the study. The selected teachers’ already established Grade 9 classes were automatically involved in this study. The Grade 9 class was selected because it is the exit grade from General Education Training (GET) to Further Education Training (FET) and the Grade 9 ANA results revealed that learners have challenges in mathematical thinking. Although the South African National Curriculum Assessment Policy Statement (Department of Basic Education, 2011) emphasises teaching and learning mathematics to develop deep mathematical thinking, the Grade 9 ANA results show that learners are facing challenges in developing deep mathematical thinking.

Owing to the COVID-19 protocol, each class from each school had a maximum of 30 learners, in compliance with national health regulations. Table 1 displays teaching information on the teacher participants.

Table 1. Teaching information

Participant	Number of years of experience in teaching	Learner race/social economic status	Years teaching mathematics in Grade 9 Class	Qualifications	Gender
Teacher A	16	Black/Low	11	Honours degree in teaching	Female
Teacher B	7	Black/Low	4	Tertiary diploma in teaching	Female
Teacher C	13	Black/Mid	8	Tertiary degree in teaching	Male

As indicated in Table 1, the participants’ information on mathematical tertiary training was obtained to gauge how qualified they were to teach mathematics. All teachers, although trained to varying levels, were qualified to teach Grade 9 mathematics. However, they had a different number of years of teaching experience.

Data Sources

Classroom lesson observations, teacher face-to-face interviews and field notes were the primary data collection methods used in this qualitative research case study. These data collection methods were utilised to gather much-needed data and help answer the research question on how teachers follow up on learners’ responses to initial teacher questions and build on learner ideas as they engage with their learners in mathematics conversations. The data were collected during August 2021.

Classroom Lesson Observations, Interviews and Field Notes

Each teacher was observed and audio-recorded in three consecutive Grade 9 mathematics lessons. The reason we chose three lessons is that we wanted to see some kind of consistency in the teaching. We would not have been able to determine the trend in teachers’ teaching practices by observing only one lesson per teacher. When lessons were in progress, information on lesson date and time, the number of learners in a class, gestures from participants and what was written on the chalkboard were recorded as field notes because such occurrences could not be captured in the audiotapes. The teachers were teaching their already existing classes at different time slots and on different days. Owing to the COVID-19 pandemic, the Grade 9 learners were coming to school on certain days of the week on a rotational basis, that is, two or three days per week. Consequently, these three lesson observations per teacher were done within two weeks. The main purpose of the observation was to establish how teachers follow up on learner thinking as learners respond to teacher questions.

After lesson observation, each teacher was engaged in an audio-recorded one-on-one and face-to-face interview. Audio-recording was used instead of video-recording to increase the confidence of participants through concealing their physical identity, unlike in video-recording. The teacher interviews were used to support and augment the data obtained from lesson observation. Each teacher interview session lasted about 20–30 minutes. These face-to-face interview interactions allowed teachers to provide their opinions and experiences, thus boosting the depth of the required data (Tracy, 2020). Therefore, utilising these three techniques in this study was useful in avoiding biases in the information or collecting shallow data from the participants that could have resulted from total dependence on the use of one instrument. Another reason for using three data sources was for triangulation to enhance the validity and credibility of the study.

Procedure for Data Analysis

Table 2 displays the five teachers’ follow-up moves and their specific detailed descriptions that informed the data analysis of this study.

Table 2. Subcategories of follow-up

Insert	The teacher adds something in response to the learner’s contribution. She can elaborate on it, correct it, answer a question, suggest something, make a link, etc.
Elicit	While following up on a contribution, the teacher tries to get something from the learner. She elicits something else to work on the learner’s idea. Elicit moves can sometimes narrow the contributions in the same way as funnelling.
Press	The teacher pushes or probes the learner for more on their idea to clarify, justify or explain more clearly. The teacher does this by asking the learner to explain more, by asking why the learner thinks they are correct, or by asking a specific question that relates to the learner’s idea and pushes for something more.
Maintain	The teacher maintains the contribution in the public realm for further consideration. She can repeat the idea, ask others for comments, or merely indicate that the learner should continue talking.
Confirm	The teacher confirms that the learner has been heard correctly. There should be some evidence that the teacher is not sure what has been heard from the learner, otherwise, it could be press.

Adapted from Brodie (2010).

In Brodie’s (2010) view, traditional practices promote learners to learn through memorisation, repetition and lack critical thinking. Elicit and insert moves are traditionally oriented because they bring more of teacher’s knowledge into the conversation. In contrast, reform practices encourage learners to think critically and to justify their thinking. The press move is reform oriented. However, maintain and confirm moves are neutral since they do not bring major output in the interactions. Therefore, the five moves work in a continuum.

According to Creswell (2014), organising data is necessary when preparing for coding. The audio-recordings for the classroom lesson observations and teacher interviews were transcribed into textual data. These recordings were listened to closely and played repeatedly to check for accuracy and to get a sense of the collected data. Our coding unit is the teacher turn, or teacher moves if the teacher made more than one move in a turn. The crucial category in our data is ‘follow-up’, which describes teacher actions that reply to learners’ inputs and thereby engage with them in some way. Categories on teacher follow-up moves were identified and classified according to a set of pre-determined codes informed by the literature, as shown in Table 2. The five sub-categories are not hierarchical. On each copy of transcribed data from lesson observations, a column was created where a code was assigned for each teacher move. Therefore, this study used deductive thematic analysis.

Portions from lesson observation transcripts were extracted from transcribed data to examine how teachers followed up on learner thinking. These extracts from lesson observations were generated indicating episodes that illustrated different teacher questions and learner responses. In addition, teachers’ views embedded in responses on tracking learner thinking were extracted from the transcribed interview data. Table 3 outlines the subtopics observed and class enrolment.

Table 3. Subtopics observed and class enrolment

Participant	Subtopics observed	Class	Number of learners
Teacher A	1. Factorisation of algebraic expressions 2. Factorisation of trinomials 3. Difference between two squares	9A	30
Teacher B	1. Properties and definitions of triangles 2. Classification of quadrilaterals based on their sides and angles 3. Properties of diagonals of quadrilaterals	9B	30
Teacher C	1. Flow diagrams 2. Describing the relationships in diagrams 3. Calculating the value of the independent variable when given the dependent variable	9C	30

Results and Discussion

Schools in South Africa are ranked according to quantiles: quantile 1 being the poorest and quantile 5 being the least poor. Pseudonyms were implemented to protect the identity of participants and maintain confidentiality. During the Covid-19 lockdown, learners were attending school on a rotational basis, resulting in teachers being observed teaching different topics. Follow-up moves on learners’ initial responses to teacher questions were, therefore, coded according to those varying topics.

Follow-up moves emerged when teachers responded to their learners’ first answer to the teacher’s question. This created a reciprocal dialogue between the teacher and learner(s). Different follow-up moves across the three sequential lessons for each teacher are summarised in Tables 4–6. In addition, one excerpt for each teacher’s predominant follow-up moves has been included as an example of teacher moves coded according to Brodie’s (2010) framework.

Table 4. Teacher A follow-up moves

Teacher moves	Lesson 1	Lesson 2	Lesson 3	Total
Insert	1	1	2	4
Elicit	8	5	6	19
Press	5	6	4	15
Maintain	1	0	0	1
Confirm	2	1	0	3
Total	17	13	12	42

Teacher A was teaching Grade 9A at School A under Quintile 2.The teacher was female who had an honours degree in teaching mathematics, had 16 years of teaching experience and had been teaching Grade 9 for 11 years. Table 4 indicates her follow-up moves.

Table 4 shows that each type of move was recorded in Teacher A’s classroom discourse. The most dominant moves were elicit and press moves, which appeared a total of 19 and 15 times, respectively. There were four insert moves, one maintain moves and three confirm moves. Follow-up moves can be identified in the following extract from one of Teacher A’s observed lessons on the quadratic expression $x^2+5x+12$. The extract from Lesson 1 was chosen because all the teacher moves in the framework were recorded in this lesson and the total number of questions exceeded the number recorded in the other two lessons.

Excerpt 1

Table

4	T	Considering terms, what name do we give to this expression $x^2+5x+12$?	Elicit
5	L1	Trinomial
6	T	Why is it called a trinomial?	Press
7	L1	Eeeeh, because it has three terms and tri means three. Triangle for three angles (a round of applause from class).
8	T	Well explained. Let’s move on.	Confirm
9	T	Can someone give us the three terms? L2, are you able to do it for us?	Elicit
10	L2	Yes. Let me try. $x^2,5x\mathrm{and}12,$ so we have 1, 2, 3 terms.
11	T	Class, what else can we say about this trinomial?	Elicit
12	L3	It is called a quadratic expression.
13	T	Yes. Quadratic.	Maintain
14	T	What do you mean by quadratic? Explain to us.	Press
15	L3	Because of 2. I see 2 is placed on top of the first $x$.
16	T	On top of the first $x?$ Can you explain to us what you mean by on top?	Press
17	L3	Ummm, 2 is powering the first $x.$
18	T	Powering? What do you mean by this, L3?	Press
19	L4	x multiplied to $x$ double.
20	T	Double? Explain further.	Confirm
21	L4	Oh yes. I mean $x$ is multiplied by itself two times.
22	T	Is L4 correct, class?	Elicit
23	Ls	Yes. Twice (chorus).
24	T	Class, what other name is given to this powering number that L3 is talking about?	Elicit
25	Ls	Index, indices (chorus).

The teacher elicited moves from the learners that are evident in lines 4, 9, 11, 22 and 24. In line 4, the teacher was trying to elicit the fact that a three-termed expression is a trinomial. In response, L1 gave clear explanations which resulted in other learners giving a round of applause, acknowledging L1’s inspiring response. The teacher’s move in line 11 required learners to link the trinomial with other polynomials concerning its highest exponent. Furthermore, the teacher had to maintain learners’ responses by asking for additional input from them (line 13). The teacher confirms in line 8 that she had heard L2’s explanation and requested the class to proceed. The teacher continued with the lesson with press moves for learners to make their ideas explicit on the meaning of quadratic. The following explanations emerged: 2 is placed on top of the first $x$ (line 15), 2 is powering the first $x$ (line 17) and $x$ is multiplied to $x$ double (line 19). This resonates with Machaba’s (2017) view on scaffolding where the teacher is not the only source of information but supports learner-construction of knowledge in the form of, in this case, follow-up moves. This is in line with CGI, which emphasises learner involvement in their learning. Moreover, chorus answers were noticeable in Teacher A’s lesson delivery, showing that learners were part of the discussion. Teachers’ moves were in line with Chin’s (2007) suggestion that learners internalise principles as they describe, explain and justify their thinking, fulfilling the Department of Basic Education’s (2011) aim to produce learners who can use critical and creative thinking when solving problems. The finding concurs with Abdulhamid and Venket’s (2018) suggestion on the need for teachers to give suitable follow-up to learners’ responses that widen possible ways for mathematics learning.

In response to the interview question, ‘are there any advantages in utilising tracking moves in classroom discussions?’, Teacher A noted that, among other reasons, learners get involved in classroom discourse because follow-up moves encourage learners to provide clarity as they explain their ideas. She added that, as a teacher, she also learns from learner responses. Overall, her responsive teaching echoed Seung et al.’s (2011) notion that such teachers believe that learners are active participants in knowledge construction, concurring with CGI which encourages learners to become part of their learning. Notably, Teacher A’s moves were meant to involve all learners in the class. This is evidenced by different learners responding and the whole class also being involved. It is evident that Teacher A’s moves are skewed towards elicit and insert, which are closer to traditional practices. However, there is an indication of moving towards the press (‘Powering? What do you mean by this, L3?’), maintain (‘Yes. Quadratic’) and confirm (‘Double? Explain further’), which are reform-oriented moves.

Teacher B was female and teaching Grade 9B at School B under Quantile 2. The teacher had a tertiary diploma in teaching mathematics, seven years’ experience and had been teaching Grade 9 for four years. Table 5 shows her follow-up moves

Table 5. Teacher B’s follow-up moves

Teacher moves	Lesson 1	Lesson 2	Lesson 3	Total
Insert	3	2	3	8
Elicit	5	5	4	14
Press	1	2	2	5
Maintain	1	1	1	3
Confirm	4	0	1	5
Total	14	10	11	35

Teacher B’s most dominant follow-up move was to elicit (14) followed by the insert move (eight). The press (five), maintain (three) and confirm (five) moves were evident, but with limitations. The following is an extract of one of Teacher B’s observed lessons indicating teacher follow-up moves when teaching the classification of quadrilaterals based on their sides and angles. The extract from Lesson 2 was chosen because it was different from the other two lessons in the way the teacher responded when learners remained silent.

Excerpt 2

Table

4	L1	360 degrees
5	T	Okay. What is the relationship between the sum of angles in a quadrilateral and the sum of angles in any triangle?	Elicit
6	Ls:	(Silence from learners)
7	T:	Oh, okay, guys. Let’s go back to triangles. What is the sum of the interior angles of a triangle?	Elicit
8	Ls	180 degrees (chorus).
9	T	We can divide a quadrilateral into two triangles. (Teacher draws a quadrilateral on the board and inserts one diagonal of the quad [from field notes].)	Insert
10	T	How many triangles do we have from this quad?	Elicit
11	Ls	Two (chorus).
12	T	Each triangle has how many degrees?	Elicit
13	L2	180
14	T	With all this in mind, guys, what then is the relationship between the sum of the angles in a quad and the sum of angles in a triangle?	Press
15	L3	I think the degrees in the quad are two times the degrees in the triangle
16	T	Is there anyone else who can say it differently?	Press
17	L4	The sum of angles in a triangle is half of the angles in a quad.
18	T	Yes. Triangle has 180 degrees so the quad has 180 times 2 we get 360. Do we all agree?	Confirm
19	Ls	Yes (chorus).

This excerpt demonstrates Teacher B’s elicit moves in lines 3, 5, 7, 10 and 12 as she tried to engage learners. The elicit move in line 3 was correctly answered by L1. As the teacher continued with elicitation on a fact on the sum of interior angles of a quadrilateral, the learners remained silent. During the other Teacher B observed lessons, it was evident that the teacher sometimes answered her own question. In this scenario, she did something unique. When there was no response to her question (line 7), she revisited prior knowledge on the sum of interior angles of a triangle instead of providing learners with answers. Field notes show that the teacher drew a quadrilateral and divided it into two triangles. This strategic diagrammatic presentational pointed out certain features of polygons. This was meant to show awareness of how the result of 360 degrees as the sum of angles in a quadrilateral is built up, that is, the angle sum of a quadrilateral is twice the angle sum of a triangle. She was scaffolding in her follow-up move when she was supporting learners in an understanding of the relationship between the sum of angles in triangles and quadrilaterals. Hence, the practice was in line with the CGI framework which suggests that strong awareness of these underpinning mathematical connections and progressions is a critical part of responsive pedagogical follow-up.

Some of the follow-up moves were directed to the whole class and learners responded with chorus answers (lines 8, 11 and 19). This was one of the noticeable features in Teacher B’s classroom discourse. Although this confirms Nappi’s (2017) and Walsh and State’s (2017) idea that questioning and answering spreads the energy production in the classroom from the teacher to the whole class, at times learner attention was compromised when chorus answers emerged when there was reciprocal dialogue between the teacher and the whole class. However, from lines 12–17, subsequent questions were responded to by different learners, L2, L3 and L4. There was variation in Teacher B’s questioning practice and this is consistent with Hall’s (2016) idea of using a series of questions to explore an idea with the expectation of receiving answers and insights, in this case, from different learners. Learner involvement was in line with Darsih’s (2018) argument about the role of teacher practices that facilitate learners in becoming independent learners.

As stated before, Teacher B at times answered her own questions. In the follow-up interview questionm, ‘What do you do when learners do not respond to your questions?’, Teacher B explained her pedagogy of teaching by saying, ‘It depends on the topic or the time I have with the learners. If time allows, I will rephrase the question to make it simpler and more understandable and consequently involve learners’. Such practice in responsive teaching is evident in lines 7–19 where the teacher rephrased the question, unlike in other observed lessons, thereby scaffolding until learner involvement was manifested. Thus, it is evident that in Teacher B’s teaching, elicit and insert moves which might be more traditional are more prevalent. It is noted that Teacher B, in her teachings, brings in more of her knowledge of the discipline, asking questions that suggest answers (for example, ‘How many triangles do we have from this quad?’) or explaining concepts or making her points in the conversation (for example, ‘We can divide a quadrilateral into two triangles’).

Teacher C was a male teaching Grade 9C at School C under Quintile 3. He had a tertiary degree in teaching mathematics, 13 years’ experience and had been teaching Grade 9 for eight years. Table 6 outlines his follow-up moves as observed.

Table 6. Teacher C’s follow-up moves

Teacher moves	Lesson 1	Lesson 2	Lesson 3	Total
Insert	1	0	1	2
Elicit	5	5	4	14
Press	2	6	4	12
Maintain	1	1	2	4
Confirm	4	5	3	12
Total	13	17	14	44

Teacher C had a total of 44 moves recorded in his observed lessons comprising insert (two), elicit (14), press (12), maintain (four) and confirm (12) moves. The following extract is of Teacher C’s second observed lesson, indicating teacher follow-up moves when teaching the relationships of variables. Lesson 2 was chosen based on possessing many moves when compared with the other two observed lessons.

Excerpt 3

Table

7	T	Once again, what name do we give to the $x$ value? Yes, L1.	Elicit
8	L1	It is called an independent variable.
9	T	Yes, independent variable.	Confirm
10	T	Okay. What about the $y$ value?	Elicit
11	L2	$\mathrm{Yy}$ is the dependent variable or output.
12	T	Good. It is called output.	Confirm
13	T	Considering the first given table, what is the relationship between the value of $x$ compared with the value of $y$? (Pointing to the table on the board.)	Elicit
14	Ls	(Learners busy working out [from field notes].)
15	T	Does anyone have a solution? Tell us what you think.	Press
16	L3	Yes. I got $y=2x$.
17	T	How did you arrive at that solution? Can you explain to us your idea, L3?	Press
18	L3	Eeeeh, $x$ value has been multiplied by 2 to get $y$ value, for example
$0\times 2=0,1\times 2=2,2\times 2=4$ and so on.
19	T	That’s a good observation. It is correct when you say $0\times 2=0,1\times 2=2?$	Confirm
20	T	Is there a different method or way of getting the relationship? Show by raising your hand. L4, I see your hand was up first. Let’s hear what L4 has for us.	Press
21	L4	The common difference of the $y$ values is 2 so the value of $y$ is obtained by multiplying $x$ by the common difference to get $2x$. This gives us $y=2x$, the relationship between the independent variable $x$ and the dependent variable, $y.$
22	T	Well explained, L4.	Confirm
23	T	Where else have we used this idea of common difference which L4 has just explained to us?	Elicit
24	Ls	Number patterns (chorus).
25	T	Yes, we have applied this concept in number patterns. I hope we have all obtained the same relationship from the values in the given table.	Maintain

Teacher C had to elicit learner thinking as displayed in lines 7, 10, 13 and 23. In lines 7 and 10 the teacher was eliciting facts on $x$ and $y,$, being independent and dependent variables respectively. The teacher further elicited the relationship between the variables specifically represented on the table of values drawn on the board (line 13). For emphasis, the teacher had to physically point to the table (from field notes) to capture the learner’s attention. In addition, the elicit move in line 23 was connecting the topic being discussed with number patterns. The teacher tended to press for more information (lines 15, 17 and 20) when he requested learners to explain their ideas, for instance, ‘How did you arrive at that solution? Can you explain to us your idea, L3?’ (line 17). One interesting observation is when L4 (line 21) could explain the idea of linking it with number patterns. Confirm moves (lines 9, 12, 19 and 22) were built from learners’ ideas and simultaneously the teacher motivated learners by using words like yes, good and well explained. The presence of motivation was evidenced by positive gestures from learners after being praised by the teacher. Gestures noted in the field notes were nodding of the head or a smile from the acknowledged learner. A maintain move was recorded in line 25 when concluding the section under discussion. However, insert moves were not recorded in this excerpt.

Teacher C built questions following on from learner(s) ideas, as demonstrated in the discussion. It is evident in Teacher C’s teaching that he confirms (for example, ‘Good. It is called output’) and maintains and makes no intervention in the learner contribution, merely keeping it in the public space. Teacher C becomes a more neutral teacher where he holds back on his ideas in favour of enabling the conversation to continue or supporting a learner to work harder to articulate, clarify and deepen his engagement with the ideas. In some instances, he presses, trying to get the learner to intervene in his contribution. We see him probing learners more (for example: ‘How did you arrive at that solution? Can you explain to us your idea?’), to enable articulation and deepening of engagement.

Teacher pedagogy corresponded to Nappi (2017) and Welsh and Sattes (2017), who suggest that questioning and answering spreads energy production in the classroom from the teacher to learners and learners were benefitting from contributions from their peers. This was in line with CGI in incorporating learner contributions, as suggested by Carpenter et al. (2015). Scaffolding was evident in teacher-responsive teaching by involving one learner until the desired answer was reached. The pattern observed was directing two successive questions to one learner, followed by two more questions directed to a different learner, until finally concluding by addressing the whole class. In line with Jones’ (2019) view, learner participation was encouraged. Although learner inputs were short and communal, they served the purpose of allowing learners to make their thought explicit. Positive comments boosted learner confidence in agreement with Darsih’s (2019) remark which said that by being praised, learners’ self-worth and sense of capability were enhanced.

Cross-sectional Case Study Analysis of the Three Teachers

Figure 1 is the compound bar graph to visually compare the teacher moves across the five categories. A total of 121 moves were recorded. Overall, elicit moves were the most dominant followed by press moves.

Figure 1. Compound bar graph for teacher moves

Although all of Brodie’s (2010) follow-up moves were recorded, generally, all three teachers predominantly utilised elicit moves where teachers were mainly trying to elicit a fact or, at times, a connection. Nevertheless, most teacher moves were built on learner responses or ideas, conforming to National Council of Teachers of Mathematics’s (2014) notion of eliciting and use of evidence from learner thinking. Teacher C had a significant number of confirmed moves as compared with the other two participants. He coupled his confirm moves with complementary remarks that motivated the learners. This is a reflection of Abdulhamid and Venkat’s (2018) suggestion on the need for teachers to be aware of suitable follow-up responses that widen possible ways for mathematics learning. The trend for maintain and confirm moves was almost the same across the three teachers. Teacher B, with a lower teaching qualification and experience, tended to have fewer moves compared with Teachers A and C, who had a higher qualification and more years of experience. On this note, the level of qualification and teaching experience seemed to determine the frequency of moves.

Conclusion

The study showed a hybrid of teacher moves. Traditional practices promote learners to learn through memorisation and repetition and lack critical thinking whereas reform practices encourage learners to think critically and to justify their thinking.Although more teacher moves are skewed towards elicit and insert, which could be closer to traditional practices, there is a clear indication of a shift towards the press, maintain and confirm, which could be considered as reform-oriented moves. These findings confirm previous research (Brodie, 2010) that teachers are most likely to develop hybrid pedagogies in response to reforms and that they do so in different ways. These teachers’ hybrid practices could be regarded as a flexible and adaptive use of classroom talk in the service of a range of teaching purposes. These practices do reflect traditional and reformed teacher moves as polar opposite to each other, but teacher moves along a continuum.

A question was described as a follow-up question when it followed a learner’s response and was directly connected to the learner’s response or broad idea (Dong et al., 2018). This is consistent with Nappi’s (2016) idea of including a series of questions to explore an idea with the expectation of gaining answers and insights from learners. Indeed, teachers used different follow-up moves, namely insert, elicit, press, maintain and confirm moves, although the elicit move was the most dominant. In addition to these moves identified by Brodie, acknowledging or praising the learner for responding was another move that emerged in this study. We suggest that teachers who are interested in the productive mathematical involvement of learners use learner motivation as a move to increase learner alertness in the classroom discourse, as evidenced by the results of this study; hence, effectively strengthening teacher pedagogy. These suggestions complement Mhlolo et al.’s (2012) opinion on the need for teachers to be assisted in reinforcing the effectiveness of their pedagogical practices.

Since the study was limited to three participating teachers who were observed teaching different topics, we recommend one mathematical topic for future study because teacher actions are affected by various variables such as the nature of the lesson, the length of the topic, content knowledge about the concept and pedagogical content knowledge. Perhaps certain follow-up moves such as probing could be more prevalent in certain topics like Euclidean geometry, compared with other topics. Furthermore, the study included the analysis of three 30 min audios per teacher, which is a limited number of classroom audios to make inferences and decisions about Teachers A, B and C. It is much more advisable to view at least a six-hour classroom audio to be able to competently decide and criticise the instruction.

According to the findings of this study, teacher development interventions are crucial in assisting teachers in working with learners’ mathematical ideas. While the concept of follow-up moves is significant in research, it can also be employed by teachers and teachers’ educators in their teaching activities. While the framework we have offered is far from the sole option for responsive teaching, we believe it can and will provide teachers with a forum to discuss and reflect on their own practices of responding to and interacting with learners’ contributions. Being aware of the types of teacher moves and possible responses outlined in the framework is especially important within professional development programmes for supporting learner-centred teaching moves toward responsive teaching, as they are within the scope of current pedagogical practices in South Africa.

Disclosure Statement

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