Student–teacher dialectic in the co-creation of a zone of proximal development: an example from kindergarten mathematics

ABSTRACT

This paper reports on a case study which explores the co-creation of a zone of proximal development (ZPD) in a mathematics teaching-learning activity in a Norwegian kindergarten. To capture the complexity of teaching and learning mathematics in kindergarten the study uses qualitative methods within an interpretative paradigm. The findings illustrate how a five-year-old girl and a kindergarten teacher co-create a ZPD by expanding each other’s action possibilities, and how the co-creation is fundamentally based on mutual trust and responsibility. The results give insights into how mathematical learning possibilities may be promoted in kindergarten. The study illustrates the importance of being receptive to children’s contributions and, above all, to trust children’s abilities to take responsibility for moving mathematical teaching-learning activities forward.

KEYWORDS:

• Joint activity
• kindergarten
• mathematics
• teaching-learning
• togethering
• zone of proximal development

Introduction

The zone of proximal development (ZPD) is one of the most noted concepts that Vygotsky (1987) introduced. The concept has been frequently cited, interpreted and elaborated on in a variety of ways. To understand the ZPD, many scholars emphasise that the concept needs to be interpreted and understood in relation to Vygotsky’s overall view on mental development (Levykh 2008; Meira and Lerman 2009; Valsiner and van der Veer 1993; Veresov 2004; 2017; Wells 1999; Wertsch 1984). However, following Vygotsky’s methodology of relating knowledge to its historical and cultural setting, the ZPD needs to be related to one’s current perspectives and empirical data (Meira and Lerman 2009). This is also what I attempt to do by applying the concept in an empirical example based on my overall understanding of Vygotsky’s cultural-historical and dialectical perspective.

Co-creation of ZPD and ‘togethering’

Conceptualisations of the ZPD have changed from looking at it as an attribute of an individual, to a view that the ZPD as a collective process (John-Steiner 2000; Levykh 2008), or a collective ‘space’ (Hussain, Monaghan, and Threlfall 2013; Abtahi, Graven, and Lerman 2017; Mercer 2000; Roth and Radford 2011). Although several scholars regard the ZPD as being collective, they have different approaches to what it means to be collective. Mercer (2000) considers the ZPD as being part of real-life activity where both participants (teacher and student) contribute to its creation. However, the focus remains on an asymmetrical relationship between the participants; the teacher (the more knowledgeable) and the learner (the less knowledgeable). Goos, Galbraith, and Renshaw (2002) on the other hand, argues that the ZPD always has a two-way character because the teacher and the students always appropriate each other’s ideas. Instead of looking at a teaching-learning activity as an expert-novice interaction, they move towards equal status interaction. Similarly, Zack and Graves (2002) emphasise that both the teacher and the children are learning in problem-solving situations and argues for a conception of the ZPD as an intellectual space where the children’s and the teacher’s knowledge and identities are formed and transformed in moment-to-moment interaction. Some scholars have extended the notion of the ‘more knowledgeable other’ to include artefacts. From their study on the emergence of a ZPD in the interplay between a 5-year-old girl, her mother and remote control, Abtahi, Graven, and Lerman (2017) suggest that the ZPD should be considered as multi-directional, instead of a bi-directional. The study illustrates how the role of the more knowledgeable other alternates between the child, (the cultural properties of) the artefact and the adult. Meira and Lerman (2009) investigates the emergence and maintenance of a ZPD involving a 2.5 years old boy, Pedro, and a kindergarten teacher (KT) in a nursery. The paper views the ZPD as an ever-emergent semiotic field where the participants are learners and teachers of each other. The field is a sign-mediated, intersubjective field that emerges from the participants’ interaction, where learning-leads-development.

This study draws on Roth and Radford’s (2011) symmetrical view on the ZPD. As with many of the above-mentioned studies, this paper argues that both participants learn in a teaching-learning¹ activity and therefore the ZPD must be regarded as a symmetrical space. Roth and Radford (2011) illustrate how a ZPD emerges from the joint activity arising between a teacher and a fourth-grade student working on an algebra task. The paper illustrates how the ZPD arises from and is part of the activity itself, where the participants, through their concrete actions, mutually work to expand each other’s action possibilities. In Roth and Radford’s (2011) view, the ZPD is a space where the participants’ room to manoeuvre is expanded by their interaction. The participants, through their joint activity and by the way, they mutually expand each other’s action possibilities, create a symmetrical space where they both are teachers and learners of each other. In line with both Roth and Radford (2011), and Meira and Lerman (2009), I argue that the ZPD cannot be decided in advance of the activity, rather it emerges at the moment, as part of the joint object-oriented activity.

To realise an event, the participants make a mutual ethical commitment which Radford and Roth (2011) call ‘togethering’. Togethering is ‘a theoretical category … that aims to account for the teacher-students embodied-, sign-, and artefact-mediated interaction that includes both co-knowing and co-being’ (244), and involves the way the participants ethically engage and attune to one another in the joint activity, involving, for example, trust and responsibility. Aligning with Radford’s (2014) approach to ethics, I conceive trust as an attitude participants have towards each other (and themselves). It involves a willingness to rely on other’s (and one’s own) actions to attain the object of the activity. Trust also involves some level of risk-taking (the participants cannot know, in advance, what actions will move the activity forward), and an idea of success (Radford, Marin-Tamayo, and Simbagoye 2018), without which the participants would take the risk to initiate an event. Responsibility is a fundamental basis of intersubjectivity and for relationships with others. ‘Being responsible means living and acting with and for the other’ (Radford, Marin-Tamayo, and Simbagoye 2018, 81, my translation). Responsibility is characterised by the way the participants respond to each other in interaction and is linked to trust: if a person is trusted to act, he/she may or may not accept to act (to take responsibility). Responsibility is in this sense concerns actions that are not yet performed.

Although several studies cited above illustrate how we may conceive the ZPD as a symmetrical phenomenon arising from interaction involving young children, none of them illustrates the significant role that a young child (5-years old) plays in co-creating and maintaining the ZPD in the context of an organised mathematical learning activity. This is, I hold, the unique contribution of this study. This study advances our overall understanding of the ZPD as a phenomenon and in particular, how it may be co-created and maintained through togethering in an organised mathematical learning activity. The following research question has guided the study: How is a ZPD co-created and maintained through togethering between a KT and a child in an organised mathematical learning activity in a Norwegian kindergarten?

Setting

The case study reported in this paper is situated within a research and development project called the Agder Project, the aim of which is to investigate how mathematical activities² designed in the project stimulate preschool children’s mathematical learning. The segment examined in this study is selected from a whole-group session in one of the project’s kindergartens where the participants (a group of nine children, age 4–5 years, and a KT) work with an addition problem (5 + 7 + 8). Norwegian children attend kindergarten in ages of 0-6, and the children in this study are in their last year of kindergarten before they enter school. The segment examined in this study focuses on the interaction between a five-year-old girl (Ada³) and a kindergarten teacher (KT) while solving the addition problem.

Methodology

To investigate the co-creation of a ZPD in kindergarten I use qualitative methods within an interpretative paradigm. The case in this case study was ‘a KT and a 5-year-old child (in a group of kindergarten children) engaging in a mathematical activity involving the counting-on strategy for addition’. The selected segment is part of a larger dataset containing 16 sessions implemented by four KTs. All observed sessions were video-recorded and field notes were written. The audio and video data from all observed sessions were transformed into tables with columns: time; description of the interaction, which includes utterances (with speaker) and observable actions (supplemented by video stills). The selected segment occurs within a whole-group session where the oral interaction is between the KT and Ada; the other children are watching/listening to the ongoing interaction. The other children participate with ideas before and after this particular segment. The segment was selected because of its suitability for the focus of this paper; student-teacher dialectic and the co-creation of a ZPD through togethering. The selected segment was transcribed⁴ focusing on the sequence of utterances (incrementing the utterance number by one whenever a new speaker entered the discourse) but commenting on facial expressions, gestures (hand movements), bodily actions and tone of voice.

In cultural-historical activity theory, the unit of analysis is activity (a process or system of relations), thus it is the process that unfolds that I focus on in my analysis. Following Radford (2013) I regard the multimodal nature of the activity as important and I bring various semiotic means into my analysis (spoken words, gestures, facial expressions, other bodily actions and the use of artefacts). In addition, Vygotsky’s dialectic approach orients my analysis to social interaction and therefore I always consider two subsequent turns in relation to one another, or I consider a turn in relation to the following activity (several turns in a row). Analysis includes augmenting verbal transcripts with other co-temporal semiotic means (gestures, facial expressions, etc.). This analysis was discussed (and refined) whilst watching the video with four research colleagues in the Agder Project. Extracts from the transcript were also discussed with an external expert in the field. This analysis was not an end in itself but was a step towards interpreting the segment as a whole.

Results

Before the selected segment, the KT introduces the session by using a doll called Super Sigurd as a pedagogical tool. Super Sigurd has built three towers in three different colours (yellow, blue and red) and needs help to figure out how many building blocks the three towers consist of all together. The children and the KT, in cooperation, find out that there are five building blocks in the yellow tower, seven in the blue tower and eight in the red tower. Then the KT asks ‘How can we figure out how many there are all together?’, and encourages the children to contribute with different strategies to solve the problem. Several children contribute with their ideas before the KT invites Ada to explain her idea:

127

KT Yes ((looks at Ada))

128

Ada I know … ehm … if we say … ehm … ((moves her body up and down))

do not count eight and then we just count further

129

KT ((Excited facial expression)) Oh, did you hear what she suggested?

((whispers)) Would you like to show us?

130

Ada ((Starts to count the yellow tower in the middle)) One, two, three, four,

five, count without eight

131

KT OK. Now you counted one, two, three, four, five and then, what do you

want to do next? ((She uses her index finger to count the five building

blocks in the yellow tower and then she moves her finger to the blue

tower when she says ‘and then’)).

132

Ada Ehm … To count similar as we counted the yellow

133

KT ((Questioning look)) Start to count from one at the bottom here? ((points

on the building block at the bottom of the blue tower))

134

Ada Mm ((agreement))

135

KT One, two, three, four, five, six, seven, and then … ?

136

Ada Ehm, we can just count like this all the time, without eight

137

KT ((Questioning look)) I think you have to show me, because I don’t really

understand what you mean. Maybe you can show … 

138

Ada We count this one first ((points at the yellow tower)) and then this one

((points on the blue tower))

139

KT Yes, maybe you can count it? Do as how you think, Ada

140

Ada One, two, three, four five ((counts the yellow tower)), six, seven, eight,

nine, ten, eleven, twelve ((continue to count the blue tower)). And then

we just find it without counting … 

141

KT Oh, we have to continue?

142

Ada Mm ((agreement))

The co-creation of a ZPD by creating action possibilities

In line 127 the KT invites Ada to explain her idea. In line 128 Ada moves her body up and down while she says, ‘I know … ehm … if we say … ehm … ’. Ada’s bodily actions suggest that this is not easy, and that she tries hard to choose suitable actions to explain properly. Figure 1⁵ illustrates Ada’s and the KT’s stances when Ada says, ‘do not count eight’ (line 128). The building blocks are placed in front of the KT, which means that Ada does not have direct access to them and must therefore use language and gestures to explain her idea. The figure illustrates how Ada kind of ‘holds on to’ the tower of eight with her right hand when she emphasises ‘not’. I interpret the gesture as an iconic gesture, where Ada indicates the tower’s height and makes it ‘a whole’. Further in line 128, when Ada says, ‘and then we just count further’, she swipes her index finger from the first tower to the last tower. From the whole utterance in line 128 it seems that Ada says that one should not count each building block in the first tower but, rather, regard the tower as ‘a whole’, and then count further from the whole (from eight).

Figure 1. Ada’s and the KT’s stances when Ada says, ‘do not count eight’ in line 128. (Ada sits with her back to the camera. The KT (and Lea) focus on Ada).

The KT closely watches and listens to Ada’s explanation in line 128 (illustrated in Figure 1). The KT does not know what strategy Ada will bring into the activity, and she needs Ada to explain it to her.

In line 129, the KT reacts with excitement, and the utterance ‘Oh, did you hear what she suggested?’. I interpret this as indicating several things: the KT recognises a quite sophisticated counting strategy (counting on) which she may have opportunity to share with the whole group; the KT appreciates Ada’s suggestions; and she promotes the other children to pay attention. Further in line 129 the KT promotes Ada to share her idea with the other children by asking ‘Would you like to show us?’. This time the KT wants Ada to use the building blocks to show her idea, which creates action possibilities for Ada.

In line 130, Ada leans forward to the building blocks and starts to count from one on the yellow tower. She does not count the eight blocks in the red tower; however, she does not continue from eight (starting on nine) on the yellow tower either. Ada’s explanation does not correspond (directly) with the counting strategy that the KT recognised in line 128, and it seems that the KT understands that she needs to guide Ada in her explanation, that is create new action possibilities for Ada. In line 131 the KT repeats Ada’s actions (brings Ada’s previous actions into her awareness again) and when she asks Ada, ‘and then, what do you want to do next?’, she moves her index finger from the yellow tower to the blue tower, which suggests a possible next move. I interpret the KT’s gesture as a careful hint to count further on the blue tower.

Figure 2 illustrates the KT’s stance when Ada explains the next move in line 132, ‘Ehm … To count similar as we counted the yellow tower’. It also illustrates how the KT still holds on to the blue tower (which was her ‘hint’ for a possible next move), but Ada does not choose an action which corresponds to the KT’s intention.

Figure 2. The KT’s stance when Ada explains her next move in line 132. (The KT (and Lea) focus on Ada who sits with her back to the camera.).

Figure 2 illustrates how the KT’s eyes are focused on Ada and her facial expression, ‘thoughtful’, suggests that she is concentrating on following Ada’s explanation. At this moment it seems that the KT is confused and does not know how to contribute in the joint activity. In line 133 the KT reformulates Ada’s suggestion, ‘Start to count from one at the bottom here?’, but her reformulation is turned in to a question and I interpret that the question is asked to make sense of Ada’s explanation, not to create anything new in the activity. Ada replies ‘Mm’ (line 134), and confirms that this is what she wants the KT to do. The KT then does exactly as Ada suggests, counts from one on the bottom of the blue tower. The KT says ‘One, two, three, four, five, six, seven, and then … ?’ (line 135), and she does exactly as Ada suggests without giving any indications for a possible next move. The KT does not contribute with any new actions, which can move the activity forward. Through her question ‘and then … ?’ at the end of line 135, and her still thoughtful facial expression, it seems that the KT does not have any clear, conscious action possibilities and asks Ada to guide her actions.

In line 136 Ada does not respond in a manner which is helpful for the KT. Ada just repeats an earlier suggestion, ‘Ehm, we can just count like this all the time, without eight’ (line 136). The suggestion is similar to the suggestion in line 130 and does not create new action possibilities for the KT. This appears to confuse the KT even more, and she expresses her confusion by her facial expression (she knits her eyebrows and tightens her mouth) and her utterances in line 137; ‘I think you have to show me, because I don’t really understand what you mean. Maybe you can show … ’. Before the KT is finished asking for an explanation, Ada raises her body which may be a sign that she has already interpreted the KT’s request for an explanation before the KT has asked the question verbally. Ada responds, ‘We count this one first ((points at the yellow tower)) and then this one ((points on the blue tower))’ (line 138). This explanation is different from the previous one. Now she considers the two towers and explains how to move from one tower to the next. However, even if Ada has contributed with a completely new explanation, the KT once again asks Ada for a further explanation. The KT says, ‘Yes, maybe you can count it? Do as how you think, Ada’ (line 139). The ‘yes’ indicates that the KT now has an idea of Ada’s suggestion again, but it seems that she is not quite sure. This time the KT asks Ada to use the building blocks (to count them), perhaps to make sure that it is easier for Ada to explain. The KT’s continuing request for an explanation and her recent request for Ada to use the building blocks in her explanation indicates that the KT really wants to understand Ada’s idea and continue to participate in the activity.

Even though the KT does not fully understand Ada’s idea, she continues in the joint activity. Both in line 137 and 139, the KT asks Ada to use the building blocks in her explanation, not only to explain verbally. She probably realises that Ada needs to use the building blocks in her explanation, at least to be able to explain in a different way. The KT’s request for an explanation with the building blocks, I hold, creates action possibilities for Ada.

In line 140, Ada comes forth and explains her idea by using the building blocks. I interpret that Ada chooses actions which (hopefully) can create new action possibilities for the KT and bring the KT into the joint activity again.

Figure 3 illustrates Ada’s stance in line 140. Ada carefully counts the yellow tower and then continues on the blue tower. She talks quite slowly and is precise when she moves her finger from one building block to the next, apparently to ensure that her actions are clear. Even if Ada does not count further from eight, that is, starting on nine on the yellow tower, she is able to count further from the yellow tower to the blue tower; it is precisely those actions that are necessary to create new action possibilities for the KT. In line 141, the KT says, ‘Oh, we have to continue?’ which I interpret as the KT thinks she once again understands, but she also asks Ada for confirmation. Ada confirms the KT’s understanding by responding ‘Mm’ in line 142.

Figure 3. Ada’s and the KT’s stances in line 140. (Ada sits to the right and the KT to the left).

Ada cannot know the influence of her actions in line 140 in advance, but it seems that she knew she needed to change her explanation (choose actions other than the previous ones), to bring the KT into the activity. Although the KT wants to understand and continue to participate in the activity, she is dependent on Ada’s explanation to do so. Ada’s actions are crucial for bringing the KT into the joint activity again. From line 141 and onward, the KT is once again able to productively participate in the joint activity and to create new action possibilities for Ada. Together they complete the counting strategy, ‘counting on from eight’ and show it to the whole group.

The co-creation of a ZPD through togethering

The analysis above illustrates how Ada and the KT, by their concrete actions, co-create and maintain a joint object-oriented activity, where they both expand each other’s action possibilities. In what follows, I discuss how the ZPD is co-created based on togethering. In line 128, when Ada first explains her strategy, the KT closely watches and listens to Ada’s explanation (illustrated in Figure 1). The KT does not know what strategy Ada will bring into the activity, and she needs Ada to explain it to her. The pauses in Ada’s speech and her body movements suggest that she uses a lot of effort to explain and she is dependent on the KT’s support in her further explanation. From this moment, Ada and the KT stand in a constant relation to each other and are both dependent on each other to continue the activity, and this is when togethering is first established. Ada takes the risk to share her idea, which seems not fully appropriated, and she trusts the KT to support her in her further explanation. However, the KT does not know exactly what strategy Ada will bring into the activity, and she needs Ada to explain it to her. Thus, both need to trust one another (that is rely on each other’s actions) and take responsibility (that is continuously respond to each other and produce actions) for moving the activity forward.

A significant passage in the segment starts in line 132/133, when the KT gets confused. From that moment and in what follows, the ZPD is in danger of collapsing, which illustrates how fragile the ZPD is. However, this is also where the strength and impact of the togethering work between Ada and the KT can be appreciated. From line 132/133 Ada shows great responsibility, she chooses different actions and works hard to create new action possibilities for the KT to keep the KT in the joint activity. However, as indicated by the KT’s responses, many of Ada’s attempts do not serve that purpose. But Ada does not give up and she continues to work hard to bring the KT into the activity again. Finally, in line 140 Ada succeeds. From the effort that Ada makes to explain her idea, it seems that she knows that her actions are important – and her actions are important. If she fails to bring the KT into the activity again, the whole activity (and the ZPD) could collapse. It is important to note that the activity could collapse, but we cannot know if it would. Maybe the KT would have made another effort to restore the activity.

Both the KT and Ada show perseverance in the activity. Even if the KT, at some point, is not able to productively contribute with new actions in the activity, she is part of the joint activity all the time, in the sense that she does not ‘drop out’ and that she constantly struggles to understand. This illustrates how the KT also takes responsibility in the activity. Although the KT is not able to productively participate with new actions, she responds to Ada and communicates that she needs Ada to guide her. However, it is Ada who, through her actions, creates possibilities for the KT to once again productively participate in the activity. Ada’s cautious and responsible actions (however not necessarily fully conscious actions) make sure that the activity keeps moving and the ZPD continues to be maintained. It is also important to notice how the KT when she starts to get confused, trusts Ada’s ‘capabilities’ for bringing her into the joint activity again. I hold that Ada and the KT make an implicit ethical commitment to one another and show responsibility to keep the activity moving. The KT and Ada could have given up the conversation, and the activity could have collapsed, but they did not. They kept on adjusting to one another.

Discussion

The analysis above illustrates: how Ada and the KT co-create and maintain a joint object-oriented activity, where they both expand each other’s action possibilities: how the co-creation is based on an ethical commitment that the participants make. To be able to expand each other’s action possibilities they must trust one another and take responsibility in the activity.

The ability to listen and be receptive to others’ initiatives have been identified as crucial features for co-creating ZPDs (John-Steiner 2000; Meira and Lerman 2009; Wells 1999). The KT’s capability to listen to (and observe) and be receptive to Ada’s ideas, lays at the heart of the findings in this study too. John-Steiner (2000) argues that it is important to ‘see’ the other in the activity and that ‘mutual care-taking’ lays the ground for learning. Similarly, Wells (1999) found that the most important thing was to listen to the students. In the segment analysed above, Ada risks sharing her ideas with the KT and the other children, and perhaps this is because the KT is able to create a ‘trustful environment’. The KT clearly listens to Ada and expresses (both verbally and non-verbally) that she wants to understand Ada. This is similar, however not identical, to the results in Meira and Lerman’s (2009) study where the ZPD between the 2-year-old boy and the KT emerged and was maintained because the KT was receptive to the child’s way of expressing himself. In this study, the KT was not only receptive to Ada’s way of expressing herself, but she was also dependent on Ada’s explanation to productively participate, and both needed to trust one another to move the activity forward. This is clearly not one-way communication. The activity (and thus the ZPD) and the co-construction of knowledge are dependent on all participants’ contributions and engagement (c.f. Abtahi, Graven, and Lerman 2017; Hussain, Monaghan, and Threlfall 2013; John-Steiner 2000; Levykh 2008; Mercer 2000; Roth and Radford 2011; Zack and Graves 2002). Moreover, the ZPD is not determined in advance of the activity, rather it emerges at the moment, as part of the joint object-oriented activity. The ZPD is an ever-emergent intersubjective interaction space (cf. Meira and Lerman 2009; Roth and Radford 2011) which, in the above episode, emerges as the KT and Ada interact with each other.

This study explores, in particularly, the significant role that Ada (the presumed less knowledgeable participant) plays in co-creating and maintaining a ZPD in an organised mathematical learning environment. This study illustrates the responsibility that Ada takes for co-creating and maintaining the ZPD, and how the KT trusts Ada in guiding her in the activity. According to Radford and Roth (2011), togethering is the mechanism that makes the participants commit and attune to one another in the joint activity. Through an ethical commitment based on trust and responsibility, Ada and the KT engage and attune to one another to move the activity forward. Without such an ethical commitment, the movement of the activity cannot occur, and the object cannot be realised through the activity.

The findings show how a ZPD may be co-created and maintained through togethering and suggest: that ethical considerations are crucial for this; that it is important to invite children to actively participate in mathematical learning activities; and, above all, to be receptive to children’s contributions and trust their abilities to take responsibility in teaching-learning activities. KTs may benefit from positioning themselves as ‘learners’ and let the children guide them in the activities.

Disclosure statement

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

Notes

1 In this research study, the term teaching-learning is used in line with Vygotsky’s construct ‘obuchenie’ which refers to the mutually constitutive relationship between teaching and learning (Roth and Radford 2011).

2 ‘Activity’ in ‘mathematical activities designed in the project’ does not have the same meaning as ‘activity’ in activity theory. Researcher designed activities are planned scenarios.

3 All names of the people in this manuscript are pseudonyms

4 Transcription codes: (()) denotes non-verbal actions or contains explanations and interpretations necessary to understand the dialogue; _ denotes that the underlined word is emphasised; … denotes a pause in the verbal utterance.

5 Sketches are made to anonymise persons, and simultaneously illustrate body positioning and facial expressions.