Introduction

In New Zealand (NZ), the recent Trends in International Mathematics and Science Study (TIMSS) 2018/2019 results suggested that the average achievement of New Zealand year 5 student in geometry was at its lowest since 2006 (Rendall et al., 2020). Anthony and Walshaw (2007) proposed that the learners’ cultural knowledge provides plentiful opportunities to improve the teaching and learning of geometry. These reservoirs of learners’ cultural knowledge can be accessed by providing them with a supportive environment that appreciates their socio-cultural identities and language(s) (European Commission, 2015; Lo Bianco et al., 2016). In the NZ context, accessing this cultural knowledge of our diverse learners with multiple ethnicities and languages (Spoonley & Bedford, 2012) requires a broader understanding of multilingualism. Multilingualism is overt or covert presence of multiple languages in any classroom (Barwell et al., 2016). Currently, the most common languages spoken at home in NZ other than English are Te Reo Māori (indigenous language of NZ, which gained official status in 1987), Samoan, Mandarin, and Hindi (Statistics New Zealand, 2020), making the NZ English-medium classroom a multilingual context with culturally and linguistically diverse students.

The paper draws from a more extensive study that explores how children negotiate their meanings about 2D shapes, 3D shapes, and their properties in a year 5/6 NZ English-medium multilingual classroom and focuses on one aspect of the larger study—the mathematical construct of dimension. Very few studies in mathematics education research have explored children’s understanding of dimension in primary education. One aim of the paper is to reduce this gap by exploring how year 5/6 (9 to 11 years old) children discursively construct and negotiate their meanings about dimension in 2D and 3D shapes as they engage in classroom interactions during geometry lessons in an NZ English-medium multilingual primary classroom. Discursive construction refers to how people construct, understand, and display their understanding of the world around them as they interact with others (Potter, 2012). In this paper, discursive constructions refer to what children said and how they said to display their meanings about the mathematical construct of dimension.

Review of literature

The mathematical construct of dimension plays a crucial role in developing foundational skills in mathematics (Manin, 2006), more so for construing 2D and 3D shapes and their properties. In the NZ curriculum (Ministry of Education, 2007), 2D shapes are defined as plane shapes with only two dimensions—length and breadth. 3D shapes are solid shapes with length, breadth, and thickness/depth. These definitions may highlight different ideas about the dimension as a mathematical construct of shapes. Taking Euclid’s geometry measurement perspective, dimension is understood as the characteristics of length, breadth, and height held by an object. That is, if an object has only length, it is considered to have only 1 dimension, whereas if an object has length and breadth, it has 2 dimensions. Consequently, an object with length, breadth, and height will have 3 dimensions. Alternatively, Skordoulis et al. (2009) suggest that topologically speaking, linear shapes like line, rectangle, and curve are 1-dimensional; surfaces are two-dimensional and include sphere region, circular region, plane, and polygonal shapes. Three-dimensional shapes include solid objects like spherical region and cylindrical region (Manin, 2006; Ural, 2014). Following this perspective, the hollow sphere and the solid sphere will have different dimensions, two and three dimensions, respectively. Both of these perspectives can be interpreted from the definitions of 2D and 3D shapes provided earlier and may contribute to students’ understanding of dimension. Yet, this mathematical construct is seldom investigated in mathematics education with few exceptional studies (e.g. Morgan, 2005; Panorkou & Pratt, 2016; Tossavainen et al., 2017; Ural, 2014).

One of the earlier studies on exploring children’s conception of 2D and 3D shapes was done by Lehrer et al. (1998). They conducted a 3-year longitudinal investigation to explore how children’s conceptions of 2D and 3D shapes develop. The participants included 30 children in total, with 10 (who moved from Grade 1 to 3), 9 (who moved from Grade 2 to 4), and 11 (who moved from Grade 3 to 5). They printed shapes for 2D shapes and made use of wooden models for 3D shapes. Two triads of wooden models were used in the second and last years of study. The first triad of wooden models comprised a cube, a cone, and a pyramid. The second triad consisted of a cube, a triangular prism, and a rectangular prism. For the 2D shapes, they reported that children reasoned about shapes based on the “fat or skinny” dimension (p. 142). For the 3D shapes, children related 3D figures with known 2D figures and argued that the shapes could be morphed by “pulling” or “pushing” (p. 142). For example, students claimed that by “sitting” on a rectangular prism, it could be transformed into a cube. Lehrer et al. (1998) argued that this way of describing 3D shapes might indicate that students view dimension as a malleable quality of shapes or objects.

Panorkou (2011), in her phenomenographic study, studied twelve 10-year-old children’s experiences of dimension using three tools—Elica applications, Flatland the film, and Google SketchUp. She found a few more ways in which children constructed their understanding of dimension. These include dimension as (i) a material attribute of an object, as thickness; (ii) as a vector, expressing ideas of position, direction, and orientation; and (iii) as capacity, where objects with higher dimensions can contain objects with lower dimensions (e.g. a cube contains a square). Panorkou’s study shows how children may represent their understanding of dimensions in different ways (see Panorkou & Pratt, 2016).

On defining dimension, Morgan (2005) analysed how year 5 children (10 to 11 years) and their teacher described their understanding of dimension during a classroom discussion (see extract presented in Appendix in Barwell, 2005). She argued that the children and the teacher identified dimension as a multi-faceted notion that includes dimension as either “thickness” later noted by Panorkou (2011) too, or describing 2D as flat and 3D as fat, or something extra in 3D compared to 2D as found in Lehrer et al. (1998) study. Morgan (2005) suggested that these definitions about dimension during classroom interactions seem to be moment specific with layers of meanings attached to it as opposed to one specific unambiguous statement provided in policy texts. Using a discursive perspective, Barwell (2005) analysed the same classroom discussion data (analysed by Morgan, 2005) to explore ambiguities in defining dimension while describing 2D and 3D shapes. During this classroom discussion, the teacher has introduced the ambiguity about describing the plastic shapes as 2D shapes even though the plastic shapes had thickness as she mentioned “coz they look like three dimensional don’t they. They’re thick but they’re not meant to be, they’re meant to be two dimensional” (Barwell, 2005, p. 123). Barwell suggests that this introduction of ambiguity at this time provided learners with opportunities to engage in mathematical thinking which was evident in the following discussion, where a child construed that a line is like a rectangle filled in making it two-dimensional. Barwell argued that ambiguities in such discussions provide opportunities for children to use their linguistic resources, i.e. how children use their language to do mathematics.

Ambiguities, often referred as difficulties, in describing and defining dimension may also be amplified as reported in research by using the same word for the boundary of the shape and the space within the shape (Tossavainen et al., 2017; Ural, 2014). Ural (2014), in his study, with fifteen primary and secondary teachers from schools in Burdur Centre, Turkey, on how these teachers made decisions about dimensions of geometric figures, pointed out that this use of English terminology in teaching and learning shapes may hinder the understanding of geometry. The teachers were provided with a test based on eighteen geometric figures. For each geometric figure, teachers were asked to specify the number of dimensions of that geometric figure. For example, the teacher was asked if a point is 0-, or 1-, or 2-, or 3-dimensional. Other geometric figures included the line, angle, parabola, circle, triangle, and spherical region. Ural (2014) suggested that naming the rectangular region (showing enclosed space) and rectangular boundary as a rectangle may lead to an inadequate geometric understanding of shapes and their properties. He argued that it is essential to emphasise the difference between expressing shape as a region and as a boundary because it may influence one’s understanding of dimensions. For example, Bezgovšek Vodušek and Lipovec (2014) showed that in the Slovenian language, the boundary of a circle is not considered a 2D shape and is called Krožnica. In contrast, a disk is regarded a 2D shape of a circle and is called Krog. Having different terms for denoting circumference and circle may highlight the idea that 2D shapes have filled spaces within their boundaries, thus underscoring the crucial dimensional aspect of 2D shapes. Using the same term for the boundary of the shape and the shape itself may also lead to limited connections between interrelated concepts. For example, in their study with 82 Finnish pre-service teachers (typically 20-year-olds) on their definitions of the area and its dimensional aspect, Tossavainen et al. (2017) found that teachers had difficulty comprehending the two-dimensional aspect of the area.

In geometry education, research has focused on related concepts such as geometric thinking (e.g. Duval, 2017; Seah & Horne, 2019; Van Hiele, 1959/1985), visual-spatial thinking (e.g. Cohrssen et al., 2017; Hawes et al., 2017; Lowrie et al., 2017), geometric shapes (e.g. Hallowell et al., 2015; Roth & Gardener, 2012), and geometric proofs (e.g. Fujita et al., 2020; Ng et al., 2020), where the understanding of dimension is assumed to be already present without any explicit mention. The limited number of studies on this crucial construct highlights the research gap. This paper attempts to add to the knowledge base in this area of research. It seems that although few of these studies noted how language might influence teachers’ and children’s understanding of dimension, how children negotiate their meanings about the dimension as they interact with others is often neglected. This paper explores how children discursively construct and negotiate their meanings about the dimension as they participate during classroom interactions. In the next section, I draw on discursive psychology as an overarching perspective to inform this study’s theoretical framework and the framing of the research questions aimed at exploring children’s negotiation of meanings about the dimension.

Theoretical framework

Discursive psychology offers an anti-cognitive and poststructuralist account of meaning-making and underscores language as the primary mode of social activity. It approaches language-in-use (talk and text) as a domain of action in its own right rather than construing it as an outcome of mental states and cognitive processes (Edwards & Potter, 2005). Potter (2012) argued that studies from the discursive psychology perspective explore how people construct, understand, and display their understanding of the world around them as they interact in “everyday and institutional situations” (p. 113). Discursive psychology perspective investigates how participants discursively construct their thinking while interacting in particular situations. Within the discursive psychology perspective, this paper takes two theories—Ethnomethodology (Garfinkel, 1967) and Bakhtin’s (1981, 1986) dialogic theory to form the theoretical framework. Both theories construe language use and knowing as ongoing practical actions that occur within the interactions and simultaneously enable exploration of language use in both micro-conversational moment context and broader macro-context.

Harold Garfinkel (1967) developed ethnomethodology as an approach to investigate the indexical properties of language use that are constituted as practical actions by members when they engage in interactions. Indexicality is the aspect of language that helps us to interpret what is said by pointing to the local context of particular activities, group membership, and situations (Barwell, 2015). Ethnomethodology is the study of how reasoning and everyday activities are organised within a culture as identifiable events and occurrences (Heap, 1984). The ethnomethodological description provides a detailed account of how members make sense of any activity as it unfolds in its everyday manner. Barwell et al. (2019) discussed that it is the ordered nature of indexicality of the language-in-use that enables us to interpret what is said beyond the meanings of individual words. Thus, the ethnomethodological basis of this study allowed me to explore what is said and how it is said in one’s utterance within the interactional space.

Bakhtin’s dialogic theory allowed exploration of the interactional space in which broader socio-cultural meanings are attributed to a person’s utterances within a specific context. According to Bakhtin (1981), language provides us with shared spaces with the possibility of diverse meanings of utterances (Kazak et al., 2015), and the specification of meaning is dependent upon the preceding and succeeding dialogues (Wegerif, 2011). To show how different socio-cultural meanings from the interactional space seep through the utterances, Bakhtin suggested that the meaning of each utterance is negotiated as a result of the constant struggle between languages forces—unifying language and heteroglossia. The unitary language acts as a unifying language force that accounts for a system of norms that dictate the accurate use of language. This unitary language force aims to guarantee mutual understanding of the meanings of utterances by specifying meanings based on dominant disciplinary knowledge, limiting the possibility of divergent meanings of the utterances, and supporting the continuation of interaction (Barwell, 2018). One such example of unitary language can be an academic discourse that upholds Eurocentric or western concepts of mathematics in a school context. Driven by the decolonial theory, Parra and Trinick (2018) showed that the teaching and learning of mathematics in colonised countries such as Colombia and New Zealand still support the teaching and learning of Eurocentric enunciation of mathematics. Chronaki et al. (2022) further argued that the use of a particular account of mathematics in multilingual classrooms reinforces and sustains the hegemonic authority of monologic curriculum, as evident in their analysis of European states—Greece, Catalonia, and Sweden. To counter this hegemonic authority, Chronaki et al. (2022) suggest that heteroglossia provides us with democratising principle of expressing mathematical ideas freely without “blind reification of representing the knowing object through words, sounds and images” (p. 113). Heteroglossia acts as a diversifying force to decentralise the already established meanings of the utterances by embedding the use of language with individualised meanings, often from everyday discourse and also inclusive of different ways of expressing meanings. The ongoing play of the unitary language and heteroglossia contribute to the socio-cultural meanings of the utterance within a particular sphere of communication.

To seek answers to how children construct and negotiate their meaning of dimension in a New Zealand multilingual primary classroom, the following research questions are used to guide this study:

What discursive constructions do 9- to 11-year-old children use to represent their meanings about dimension in a year 5/6 NZ primary classroom?

How do 9- to 11-year-old children discursively construct and negotiate their meanings about dimension in a year 5/6 NZ primary classroom?

The first research question focuses on “what” meanings about dimensions are constructed and evident in children’s utterances during classroom interactions. In other words, discursive constructions are what words participants have used to communicate their meanings about dimension through their utterances. The second research question focuses on “how” these discursive constructions are said to convey meanings and negotiate their meanings. For negotiation, this research question explores how unitary language and heteroglossia during classroom interactions influence the process of negotiation of meanings about the dimension. The following section presents details on the research design with context and participants, data collection methods, and data analysis processes adopted for this study.

Research design

Qualitative research was designed for this study to explore children’s classroom interactions (both whole-class and group interactions) during geometry lessons on geometric shapes.

Context of school and participants

The study took place in a year 5/6 class at Rosemary School (pseudonym). Rosemary School is an English-medium state-run school in New Zealand. This school catered to the multilingual children population. A short five-question questionnaire was used to collect information about languages that the children used at home or school and how long they have been in New Zealand. The questionnaire was filled out by the parents of the children who consented to participate in this study. Participants included fifteen students with their mathematics teacher. Nine of the fifteen students were multilingual (1 Somali, 2 Tongan, 4 Māori, 1 Chinese, and 1 Filipino). The teacher had 7 years of teaching experience. Informed and voluntary consent to participate in the study was sought from the participants following the ethical approval from the University of Waikato Division of Education Ethics Committee.

Data gathering tools and processes

Six lessons on geometric shapes and their properties were observed at Rosemary School. Each lesson lasted for 45 to 50 min. Five data gathering tools were employed in addition to the questionnaire (mentioned earlier); these are (i) field notes, (ii) audiovisual recording of the lessons, (iii) semi-structured interviews with the teacher, (iv) semi-structured focus group interviews with children, and (v) relevant documents including New Zealand curriculum and resources, teacher’s unit plan, and students’ work samples.

Field notes

Field notes were taken for each of the lessons observed. Field notes are detailed descriptions of observations and interactions in the field that are kept as a chronological log. I noted as many details as possible to note what and how phenomena are unfolding (Punch & Oancea, 2014) while carefully listening and writing down keywords. These jotted field notes were later developed in fuller descriptive notes after each lesson observation. Field notes included a fuller description of settings and events and my analytic ideas, inferences, memos, personal feelings, and reflections (as suggested in Bailey, 2018).

Audiovisual recording of classroom and group interactions

All six geometry lessons were audiovisually recorded. A pilot study was conducted prior to this study to trial data gathering tools. Non-verbal cues like gestures, body movements, pointing, and facial expressions were found to be of particular relevance in revealing mathematical thinking and meaning construction in the pilot study, and audiovisual data provided opportunities for analysis of these aspects in data. Two directional cameras were used to audio and video record the whole-class and group interactions in the year 5/6 classes. I also used eye gear with an inbuilt camera and voice recorder to closely record moments of interaction that captured my attention. In addition, four to five audio recorders were kept on table tops to record talk-in-interaction in-group settings. These four to five recorders helped to record the interaction of all groups. Using different audio and video recorders enabled me to record interaction from different angles, which were later corroborated to produce thorough and detailed transcription of data for analysis.

Semi-structured interviews

Semi-structured interviews were conducted with the teacher after every second lesson was observed. In total, three teacher interviews were conducted, each for 18 to 20 min. The interviews were conducted using a semi-structured interview schedule. The purpose was to seek clarifications regarding the grouping of students, the structuring of the unit/lesson, tasks, and materials/manipulatives used. Semi-structured interviews also helped me to explore the teacher’s interpretations of settings, tasks, and purposes of the tasks for the observed lesson (Punch & Oancea, 2014). The interviews were audio-recorded, and the interview transcript was sent to the teacher for member checking (Denzin & Lincoln, 2018). Member checking is a recommended procedure to ensure rigour and trustworthiness in qualitative research.

Focus group interviews with the students

Focus group interviews with four groups of children were conducted for 15 to 20 min after all six lessons were taught. Each group had four to five children. The focus group interviews allowed further use of children’s interaction in a group setting as data (Ho, 2006) for exploring their understanding of dimension. The focus group interviews were audio-recorded and transcribed for analysis. The focus group interview setting allows children to voice their understanding in a comfortable environment, on the one hand, and provides researcher with a quick and cost-effective way to gather data on the other (Nuttavuthisit, 2019).

Relevant documents

I also collected teachers’ unit plan, resources used by the teacher to develop the unit plan, children’s worksheets, pictures, drawings, and other classroom artefacts. The New Zealand Curriculum (NZC) framework (Ministry of Education, 2007) was used to provide background information relevant to the teaching of mathematics, specifically geometry at levels 2 and 3 (year 5/6).

Data gathered through these different sources enabled me to provide a holistic and rich account of the data for the study, which is crucial for establishing the reliability of the findings (Grabowski & Oh, 2018). The detailed data description enabled me to provide evidence for the claims to support and corroborate the findings (Denham & Onwuegbuzie, 2013). Additionally, I, as a researcher, transcribed the audiovisual data following an interpretative and active process (Lester, 2019). The same segment of the audiovisual data was transcribed and re-transcribed several times using the same set of conversation analysis (CA) conventions, with a time gap of 3 weeks. Both the transcripts were matched, and when 91% similarity was found between the transcripts, the latest transcript was selected for analysis. This was done to ensure that I, as researcher, am transcribing audiovisual data as consistently as possible. Special attention was given to the consistent transcription of prosodic aspects (such as high pitch, low pitch, flat pitch, and emphasis). The difference (9%) between transcripts was often due to unclear words, latching, and overlapping speech. This transcription process allowed to maintain intra-rater reliability of the transcripts. Hence, the final transcripts are the product of an iterative process. Peer consultation was also sought to receive comments and feedback on the interpretation and analysis from the other CA practitioners and Māori colleagues to ensure the reliability of the findings. A detailed description of the transcripts and analysis procedure enabled me to safeguard the transparency and trustworthiness of the data and the analysis (Lester, 2019).

Data analysis process

Participants’ utterances were considered as the unit of analysis. Data analysis for the larger study included three steps: thematic, micro-level, and macro-level (see Fig. 1). The first step of thematic analysis involved repeated viewing of audiovisual data, field notes, semi-structured interviews with the teacher, focus group interviews with children, and documents to identify children’s ways through which they discursively constructed their meanings about shapes and their properties. This paper shows an analysis of one of the themes identified in the bigger study, which is the mathematical construct of dimension. In this theme, different ways in which children discursively constructed and expressed their meanings about dimension were coded. Children displayed a variety of ways of describing their understanding of dimensions. The codes also allowed the identification of relevant key moments for the next analysis steps—micro-level and macro-level. Key moments are those segments of audiovisually recorded classroom interactions during which children talked about dimension while engaged in geometry lessons. After multiple viewing of the audiovisual data of six lessons, two relevant key moments on dimension were selected for further analysis. These key moments included one instance of whole-class and one instance of group interaction. Each key moment was analysed at two levels—(i) micro-level and (ii) macro-level.

Fig. 1
figure 1

Data analysis process

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Micro-level analysis helped in answering the first research question. The first research question focused on what geometric understandings about dimensions are evident in children’s discursive constructions through their utterances. At this step, both key moments 1 and 2 were analysed using a selected feature of CA. In CA research, the term “turn” is used for participants’ utterances (Drew, 2013). The analysis of turn design explores how participants construct their utterances using words and prosodic features (which include pitch, the volume of voice, and silence) to convey the intended meaning and action. At this step, I transcribed audiovisual data using a few selected features of Jefferson’s (2004) transcript convention. Linguistic (such as words, umms, gaps) and paralinguistic features (such as high/low pitch, loud voice, creek voice) were transcribed. The use of intonation (or paralinguistic) features in children’s utterances was interpreted using insights from sociolinguistics research. For example, Ward (2019) has shown that English speakers use low pitch to signal authority over a knowledge claim. In the New Zealand context, New Zealand English speakers often use high rising terminal (HRT) intonation to check if the listener is following them instead of using it as a question (Warren, 2016) (please refer to Appendix for interpretations of other intonation features). Since the data included children from multilingual contexts, research insights from languages other than English were also used (which is referred to in the analysis section later in this paper). It is to be noted that the CA techniques have been borrowed to support the analysis.

The insights from the micro-level analysis were used to inform analysis at the macro-level. The second key moment was analysed at the macro-level to explore how unitary language and heteroglossia influenced the children’s negotiation of meanings about the dimension. The second key moment was selected for macro-level analysis as it provided more interactional time to highlight the influence of unitary language and heteroglossia. At this level, dominant discourses embedded in the utterances were identified based on the words used in the utterances and the intended meaning evident through the use of prosodic features in utterances (analysed at the micro-level). The analysis process shows how the utterances at the micro-contextual level influence the intended meanings at the macro-level, along with how dominant discourses at the macro-level inform the meanings in the specific utterances at the micro-contextual level.

Findings

In this section, I present the findings from each step of analysis: thematic, micro-level, and macro-level.

Theme: Discursive constructions about mathematical construct of dimension

The analysis of field notes and focus group interviews with children suggested that children express their understanding of dimension in three different ways—as “another world”, as “different ways to go”, and as “flat or fat”. These meanings are evident in the following utterances:

  1. (a)

    like another world. (Matiu, Fieldnote, Lesson 2)

  2. (b)

    it’s like a different world. (Alyssa, Focus Group Interview 3)

  3. (c)

    D is dimension. Like they are at different place. (Zara, Focus Group Interview 4)

  4. (d)

    [for 3D] Side to side, in and out. (Ethan, Fieldnote, Lesson 4)

  5. (e)

    3D is three ways to go and two 2D is two ways to go. (Matiu, Focus Group Interview 2)

  6. (f)

    2D is flat...and 3D is fat. (Zara, Fieldnote, Lesson2)

  7. (g)

    d is dimension. 2D is flat and 3D is fat. 3D has a lot of stuff. Like a 3D has some stuff in it. 2D is like flat and it has nothing. It's like his, his body was like he just, it's like squished over from the car. (Ozan, Focus Group Interview 1)

First three utterances (a), (b), and (c) construct dimension as “another world”; utterances (d) and (e) show dimension as “different ways to go”; and the last two utterances (f) and (g) talk about dimension in terms of “flat or fat”. In the audiovisual data, it seemed that “flat and fat” were interactionally constructed differently during classroom interactions to display the meaning of dimension. Thus, two key moments where children used “flat or flat” analogy to express their understanding of dimension were selected for further in-depth analysis at the micro-level (presented first) and macro-level analysis. Both the key moments are from the same lesson (lesson 2). The first key moment shows group interaction, whereas the second key moment is from the whole-class interaction.

Micro-level analysis

The lesson started with the teacher dividing the class into groups of three to four children and providing them with playdough or sticks with adhesive to make shapes they already knew. As children started working with playdough in their groups, the interaction occurring in one of the groups is presented in this key moment. Four children were sitting in a group and making shapes using playdough during this first key moment. As they started making shapes, one child (Matiu) stated that he was going to make a triangular prism. Figure 2 shows his shape of the triangular prism which he labelled as a pyramid (see the circled item in Fig. 2).

Fig. 2
figure 2

Matiu’s shapes with playdough

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The excerpt below shows the group interaction after Matiu announced his plan to make this shape. Matiu is an 11-year-old, male, bilingual student with Māori and English as his languages and has better proficiency in English.

Excerpt 1

  

12

Matiu

I am gonna trying to (1.0) its like they are ol just

13

 

three: d:: you cant like (.) make (.) not make a (0.8)

14

 

fat

15

Garry

yes you ca:n↑

In his first utterance, Matiu used stretching with “three d” to emphasise (Couper-Kuhlen, 2009) and used pauses (lines 12–14). Ward (2019) has argued that English speakers may use pauses to take time to think about what they want to say. In this utterance, Matiu mentioned that “you cant like (.) make (.) not make a (0.8) fat” (lines 13–14); it seems that Matiu was intending to say that it is not possible to make a flat shape. In the following utterance (line 15), Garry (a male 11-year-old Filipino-English bilingual child with English as his proficient language) used his late pitch peak to suggest that it is possible to make flat shapes. The use of the high pitch at the end of the utterance with the stretching of “can” (line 15) may be interpreted as indicating his confidence in his knowledge claim while correcting Matiu’s misperception (Ward, 2019). The discussion ended abruptly as the children started making shapes using the playdough. About the understanding of dimension, the analysis reveals that Matiu seems to suggest the understanding that it is not possible to construct 2D shapes using playdough.

The second key moment is part of the whole-class interaction during lesson 2. After the children made their shapes, the teacher asked them to come and sit on the mat to talk about their shapes and their properties using the “language of geometry” (teacher, lesson 2, field notes). During this key moment, the teacher invited Elie to describe the shape she had made using sticks and adhesive. Elie is a female, 10-year-old, bilingual student with more proficiency in English than Te Reo Māori. She had made a hexagonal skeleton or hexagon using sticks and adhesive (see Fig. 3). The teacher asked her if the shape made by her was 2D or 3D.

Fig. 3
figure 3

Elie making shapes using sticks and adhesive

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The following excerpt 2 shows the interaction presenting how the teacher and children displayed their understanding of dimension.

Excerpt 2

#

Speaker

Text

341

Teacher

>↑anyone else got some right< (.) um: Elie

342

 

with your sticks

343

Elie

um::: (0.4) I forgot what this shape's called

344

Teacher

very good? ↑so ↑how many (0.6) so (1.6) ↑so (.)

345

 

um: [describe it]

346

Elie

     °[its got] one two (2.0)° its: got one: two:

347

 

three four °five six° ↑its got six (0.2)

348

 

corne:rs

349

Teacher

got (.) six (1.5)

350

Elie

an::d its (.) go:t (1.5)

351

Teacher

Elie just hang on a minute (.) is it three d:

352

 

or two d: (1.0)

353

Elie

um:: I think its three d because °its not (.) a

354

 

two d° ((she was holding the shape and rolling it around her finger))

355

Teacher

put it down on a on the grou:nd (1.0) is it (.)

356

 

flat (.) or fat (0.5)

357

Elie

its fat (1.5)

358

Teacher

its fa:t (.) is it ↑coming ou:t towards you (1.0)

359

Elie

((looks at the shape holding it near the eye level))

360

Teacher

=okay lay it on the grou[nd (1.5)

361

Kimi

                                     [°no its flat°

362

Teacher

its its okay. so: its not actually coming out

363

 

of the ground or going through the grou:nd (.)

364

 

so we call so we call (.) ↑we call that a two

365

 

d? (0.5) okay [so:(.2)

366

Elie

                      [↓uhm::

367

Teacher

↑its ↑got six co:rners (.) yeah

368

Elie

and its got (2.8) ((counted the number of sides)) and it

369

 

got six si:des

370

Teacher

six si:des good girl. I like you brought that

371

 

language (0.5) Okhay (0.3) ↑can ↑anyone help um

372

 

Elie (0.4) on what has ↑six si:des and ↑six

373

 

corners and is a and it is a two d shape

374

 

(1.0)um::: (1.3) Yue

375

Yue

a hexa:gon?

376

Teacher

ka pai so um you have actually made a hexa:gon

377

 

um: (0.6)

378

Elie

I know that thats called a hexagon ((hold and shows

379

 

the shape to the Teacher))

380

Teacher

yeah a hexago:n has got six sides yeah (0.4) so°sort

381

 

of sort of a flat° (0.5) flat (2.0)um:

The teacher tagged Elie as the next speaker (line 341) and asked her to describe the shape she made using sticks and adhesive. In her response, Elie used pauses and “um” in her response (line 343). It seems she considered her response as dispreferred as she used “um” as a hedging deviceFootnote 1 (Pomerantz & Heritage, 2013). The teacher (line 344) provided a positive evaluation of the shape that Elie had made but did not attend to her dispreferred response. She used high pitch at several places in her utterance and pauses in her turn (line 345) to rephrase her question. Reed (2010) has shown that the use of high pitch occurs with the “interactional events that are designed as sequentially contrastive, or new” (p. 865). It seems that, through the use of high pitch, the teacher intended to design her turn to ask for new information about the shape. The teacher constructed her utterance with “how many?” probably signalling the number of sides, paused for 0.6 s, and rephrased the question as a command (line 345), “describe it” (Hayano, 2013). The teacher’s utterance seems to provide Elie with cues to direct her response in alignment with the teacher’s expectation of stating the number of sides. Elie may have noticed the teacher’s cue “how many” (line 344) as she started by counting the number of sticks that she had used to make the shape (line 346). At the end of her utterance, she stated that the shape had six corners. The teacher approved of Elie’s response (line 349) as she repeated “six” and started to write on the whiteboard. Hellermann (2003) has shown that teachers use repetition as a conversational marker to show their approval of students’ responses. At this moment, the teacher realised that Elie had not mentioned if the shape was 2D or 3D and asked if the shape was 2D or 3D (line 351). To this question, Elie responded that the shape she made was 3D (lines 353–354). Elie used a flat pitch for the first half of her utterance and a whispery voice for the second half. Research has shown that English speakers may use flat pitch to display their authority or confidence (Couper-Kuhlen, 2004; Ward, 2019). However, a whispery voice at the end of an utterance may indicate diffidence (Ward, 2019). The use of a flat and whispery voice may probably indicate that Elie was partially confident of her claim. The audiovisually recorded data (lesson 2) showed that Elie was holding the shape and spinning it around her fingers. During the first lesson, the teacher repeated children’s earlier explanation about the difference between 2 and 3D shapes during lesson 1, as “two d is flat. three d is fat. two d, straight onto the ground, three d, you can hold it, its fat, its solid” (teacher, lesson 1). It is possible that Elie understood the shape she made was 3D as she could hold it.

In the following utterance, the teacher asked Elie to put the shape on the ground (line 355). As the teacher did not repeat Elie’s previous utterance or used markers like “good girl”, it is probable that the teacher evaluated Elie’s response as incorrect (line 355). She stretched “ground” to emphasise it, probably to provide Elie with a cue. The fieldnotes (lesson 2) inform that during this activity, the teacher often stated that if the shape is coming out of the ground, it is 3D; otherwise, 2D. It seems that the teacher intended to use the same reasoning to help Elie to identify that the shape was 2D. The teacher rephrased her question and asked Elie if the shape was flat or fat (lines 355–356). The teacher did not emphasise “fat” or “flat” in her utterance. This lack of emphasis may imply that the teacher expected Elie to recall the “fat vs flat” distinction of shapes. Field notes and audiovisually recorded data (lessons 1 and 2) inform us that the “flat vs fat” analogy was often used in this class to describe 2D and 3D shapes. To this question, Elie (line 357) responded that the shape is fat. Elie’s flat pitch shows that Elie was sure of her answer (Ward, 2019).

The teacher (line 358) waited for 1.5 s before constructing her turn and then repeated Elie’s response (358); however, she stretched “fat” for emphasis. Hellermann (2003) has shown that silence in between turns can be interpreted as the current speaker’s (in this case, the teacher) orientation to the previous speaker’s (in this case, Elie) utterance as a dispreferred response. Moreover, the teacher used different intonation patterns (line 358) with the same words used by Elie (line 357). Using different intonational patterns with the same words often implies contrast rather than agreement (Hellermann, 2003). It seems that the teacher again evaluated Elie’s response as incorrect. She again provided Elie with feedback to reconsider her response (line 358).

The audiovisually recorded data inform that Elie held the shape at her eye level instead of verbally responding (line 359) to the teacher’s feedback in the previous turn. This may be interpreted as Elie’s way of restating that the shape is 3D as she could hold the shape in her hand. Noticing this, the teacher (line 360) asked Elie to put it on the ground. As the teacher was talking to Elie, Kimi (a female, 10-year-old, Tongan student) self-selected and offered a repair on Elie’s turn. Kimi structured her response in a whispery voice (line 361). In English intonation patterns, the use of a whispery voice may indicate doubt (Ward, 2019). However, in pacific languages, including Māori and Tongan, speakers often use low volume to provide their input without interfering with the speaker’s talk (Stubbe, 1998). It seems that Kimi used low volume in her utterance to provide her suggestion without interrupting the teacher’s utterance.

The teacher attempted to build an understanding of the shape as two-dimensional with Elie (lines 362–365). She used HRT (indicated by ?) as a way to overcome a barrier to comprehension and build solidarity (Warren, 2016). Therefore, through her utterances, the teacher attempted to develop a mutual understanding with Elie, as she explained that the shape was not “coming out of or going through the ground” (lines 362–365). Moreover, the teacher used the “so we call” phrase (line 364) twice in her utterance; this use of the phrase could be interpreted as her acknowledgement of the possibility of non-confirmation from Elie.

In the following turn (line 366), Elie uses “uhm” as a hedging device, probably to convey that she is not convinced (Drew, 2013). In addition, Ward (2019) has shown that a low/falling pitch may also be interpreted as a way to show declining interest in continuing a discussion. Thus, in this context, Elie’s use of low pitch may be interpreted as her way of indicating that she was not interested in carrying on with the conversation. With Elie’s response (line 366), the teacher reiterated the first information Elie had provided about the shape (lines 346–348). Ward (2019) has noted that high onset is often used in conversations to mark a change in the topic of conversation. Thus, the teacher’s use of high onset (line 367) may be interpreted as an intended action to change the topic of discussion. At this moment, it seems that the teacher picked on Elie’s withdrawal and thus attempted to change the topic of discussion. Elie picked up the cue and responded with the next property of the shape and stated, “it’s got six sides” (lines 368–9). The teacher accepted Elie’s response and showed her appreciation with the phrase “good girl” (line 370). The teacher explicitly commented on Elie’s use of language. Here, the teacher referred to the use of geometry-specific language. As mentioned earlier, the teacher explicitly asked students to talk about shapes and their properties in the language of geometry.

The teacher then constructed her turn (lines 371–374) as a question to know the shape’s name. She tagged Yue as the next speaker (line 374). It is interesting to note that Yue (a female, 10-year-old, Chinese bilingual speaker with English and Chinese as her languages) used HRT in her utterance (line 375). In other key moments in the more extensive data, Yue often used a flat pitch to give her answers when asked a question. Research has shown that bilingual Chinese speakers often use flat pitch while informing (Pickering, 2001; Wu, 2004), whereas English speakers often use pitch peaks like high onsets or HRT while responding to a question to seek confirmation or backchannel feedback from their listeners or addressee (Ward, 2019; Warren, 2016). Yue seemed to check if the teacher agreed with her answer. The teacher approved Yue’s response (line 376) and responded with “Ka Pai”, a Māori phrase which is used to imply positive assessment and means “well done”. The teacher (lines 380–381) acknowledged Elie’s response and stated the shape as “sort of a flat” instead of claiming it as two-dimensional.

The analysis of the two key moments reveals three discursive constructions about dimension. First, children may discursively construct dimension as (i) “flat vs fat”, (ii) as “another world”, and (iii) as “different ways to go”. The analysis suggests that prosodic features of children’s repertoire of languages contribute to their meanings during interactions. Macro-level analysis explores how different meanings are attributed to the use of terms—“fat” and “flat” as children engage in whole-class interaction. The next section presents macro-level analysis of the second key moment as it provides more interactional data for examining how the meanings of these terms were negotiated within the interactional space of whole-class discussion.

Macro-level analysis

Macro-level analysis helped in exploring the second research question, which focuses on how children negotiated their meanings about dimension using Bakhtin’s concepts of unitary language and heteroglossia. A pool of several dominant and not so dominant discourses can be identified in this key moment that draws attention to the heteroglossia as a possibility of diverse discourses influencing the negotiation of meanings within an interactional space. The analysis focuses here on the two dominant discourses that are identified and labelled based on the research discussed earlier. These are everyday discourse and Eurocentric-academic discourse. Everyday discourse included using informal language by children to represent their understanding of shapes and their properties in the geometry classroom. Eurocentric-academic discourse involved the use of geometry-specific vocabulary as suggested in the NZC for representing understanding of geometric ideas about shapes and their properties. The term “Eurocentric” highlights the presence of Western mathematical ideas in New Zealand’s academic mathematics as suggested by Parra and Trinick (2018).

Heteroglossia can also be appreciated in the different meanings that can be drawn from “flat” and “fat” in the utterance where the teacher asked Elie if the shape is “flat or fat”. For example, “flat” can imply either smoothness of the surface without any depth from the everyday discourse or a very thin object like paper cut-outs that are often used as resources in geometry classes for teaching 2D shapes from the Eurocentric-academic discourse. The use of the term “fat” could mean thick, thin, or something that one can hold as in everyday discourse, and in the case of geometry teaching within the use of Eurocentric-academic discourse may mean 3D shapes. This diversity of meanings here highlights the heteroglossia in the use of “flat” or “fat” and underscores how the meanings of a word is potential never final.

For providing a temporary finalisation of meaning to a certain word to support the continuation of interaction, the analysis suggests that unitary language force may support different discourses at micro-moments of interaction. The unitary language force, in this key moment, supported the two identified different discourses during different micro-moments within this interaction. The unitary language force supports the Eurocentric-academic discourse in the teacher’s utterance (lines 355–356, excerpt 2) as she asked Elie if the shape is flat or fat. Through this utterance, the teacher seems to use the analogy of flat vs fat for identifying shapes as 2D and 3D. The Eurocentric-academic discourse supports the use of “flat” for 2D shapes and “fat” for 3D shapes. In the following utterance, however, Elie stated that the shape is fat (line 357) using a flat pitch, displaying her confidence in her claim about the shape. Based on the micro-level analysis, it seems that Elie construed the shape she made as 3D as she could hold the shape and see its slight thickness (through her gestures by keeping the shape to eye level, line 359). From Elie’s perspective, it seems that she is using the appropriate geometry language to display her understanding of the shape; thus, it is the Eurocentric-academic discourse at work. Yet, it seems from the teacher’s perspective the meaning of “fat” is drawn from everyday discourse. Thus, for the teacher, the unitary language force seems to support the use of everyday discourse instead of Eurocentric-academic discourse to keep the flow of conversation. The analysis of Elie’s utterance again highlights its heteroglossic character. The interaction of unitary language and heteroglossia within the use of the “flat vs fat” analogy highlights that these words are laden with geometric as well as everyday meanings.

Three key findings can be drawn from this study based on the three levels of analyses. First, children used different discursive constructions to represent their understanding of dimension—“flat or fat”, “like another world”, and “different ways to go”. Second, children used prosodic features of their repertoire of languages that contribute to the meanings during interactions as evident in the utterances of Kimi and Yue. Third, the analysis revealed that the meaning of the words in any utterance depends upon the interaction of unitary language force and heteroglossia as well as the discourse supported by the unitary language force. Therefore, micro-moments within a broader interaction may suggest multiple meanings of the same words that contribute to meaning-making within mathematics classrooms.

Discussion

The paper aimed to explore 9- to 11-year-old children’s discursive constructions and their negotiation about the meanings of dimension in a New Zealand primary multilingual school. The analysis revealed three different discursive constructions—“flat or fat”, “like another world”, and “different ways to go”. It is possible that these discursive constructions were introduced to the children by their teacher in their previous years of schooling; however, in this study, children used these constructions during the six observed lessons. The first discursive construction, “flat vs fat”, for describing D in 2D and 3D is consistent with the research by Morgan (2005), Lehrer et al. (1998), and Panorkou (2011). These studies reported that children and teachers often describe “dimension” as one of the mathematical words that concern the thickness of the shape. However, the analysis exploring the interaction of unitary language and heteroglossia suggests that the meaning of such terms “flat” or “fat” may display diverse meanings, as seen in the macro-level analysis. From a discursive psychology perspective, it can be argued that it is the participants who interactionally treat any word, such as “fat”, in this instance, mathematically or colloquially during a particular interaction. Barwell (2005) noted this interactional construction of meanings around what makes a shape 2D or 3D and argued that the ambiguities in describing the flatness and fatness of shapes in moments like these during classroom discussions offer opportunities for further exploring the construct of dimension and considering it within the mathematical explanation provided. In addition, Bakhtinian dialogic theory provides evidence of how the meaning of the utterance is dependent upon the discourse supported by the unitary language force that aims to homogenise the meaning of the utterance to facilitate the flow of interaction (Barwell, 2018) within the milieu of discourses available in any interactional moment. This milieu of the plurality of discourses offers opportunities for diversity of meanings within interactions even when the language of interaction is only English. For identifying these opportunities in classroom interactions, Chronaki et al. (2022) call for dialogic translanguaging as a principle for teaching and learning in mathematics classrooms that acknowledges diverse ways of language use in classrooms along with diverse of knowing from anti-racist, indegenous, and anti-patriarchal ways.

The other two discursive constructions were not used often by children to express their understanding of dimension, yet are of significance. First, these discursive constructions were used to describe children’s understanding in an informal manner during lessons and focus group interviews. These discursive constructions appeared when they were not explicitly asked to use the “language of geometry” by the teacher. Second, these discursive constructions show that children use language in diverse ways to communicate their mathematical understanding. Discursively constructing dimension as “another world or place” may be interpreted as signalling the difficulty that children may have in expressing their understanding of dimension using language. This discursive construction of dimension has not previously been reported in research. Alternatively, it may construe dimension as a property of space instead of shape, which could be further explored. The other discursive construction evident in this analysis is the understanding of dimension as “different ways to move”, which is also reported by Panorkou (2011). She reported that children might express the difference between 2 and 3D shapes in terms of directions, positions, and orientations, as evident in this student’s statement: “because, that’s two D, you go only left and right and up and down, while if you are in three D you go everywhere. For example, let’s say a house if it was two D, you wouldn’t be able to go into the house” (Panorkou, 2011, p. 294). This way of expressing dimension seems to align with the measurement perspective often stated in curriculum documents, where 2D shapes are defined as planar shapes with length and width, and 3D shapes as solid shapes having a length, width, and height. Yet, this construction does not highlight the need for the “planes” to understand dimension.

This discussion of the discursive constructions highlights the need for developing a clear and comprehensive understanding of what dimension implies when presented as a mathematical construct in curriculum documents. It seems that a comprehensive understanding of dimension requires a topological and measurement perspective. Currently, curriculum documents define 2D and 3D shapes from a measurement perspective without drawing attention to the need for the “filled in space” to describe a shape as 2D or 3D. The topological perspective will enable learners to acknowledge the importance of the “filled in space” as “planes” for 2D and “solid” as 3D shapes. This inclusion of a comprehensive definition may call for teaching and learning resources and professional development opportunities for teachers. Moreover, such a description of dimension may also support teachers and learners to make connections between concepts such as area and plane for 2D shapes and volume and solid for 3D shapes, required at higher levels of geometry learning. This change in curriculum documents may also support teachers and children to acknowledge the contexts within which the construct of dimension is used and for what purposes.

The analysis at the micro-level highlighted the crucial role that prosodic features play in displaying children’s and teachers’ mathematical understanding. In addition, the study also suggests that multilingual children use a variety of prosodic features from across their repertoire of multiple languages. This paper provides concrete evidence to suggest that multilingualism of the classrooms like the one observed in this study can be realised in the linguistic resources (including prosody) used by multilingual children, challenging the monolingual assumptions of contemporary classrooms. Moreover, it is possible that prosodic features may be perceived differently by multilingual and English-speaking children; it cannot be denied that these features provide opportunities for potential meanings. From a discursive psychology perspective, it can be argued that the prosodic features of language use by participants during classroom interactions indicate how utterances become part of practical action and take a particular meaning amid potential meanings in an interactional space. The study, therefore, recommends professional development for teachers to develop an understanding of these prosodic features to understand how they are used by children while engaging in their negotiation of meaning during classroom activities.

Conclusion and future research

The study focused on exploring 9- to 11-year-old children’s discursively construction and their negotiation about dimension in a New Zealand year 5/6 classroom. The micro-level and macro-level analysis of two interactional key moments focused on exploring how meanings to discursive constructions are intended and expressed during classroom interactions. The study provides evidence of how the interplay of several aspects of context, including prosody in micro-context and discourses in macro-context, contribute to the meaning of any utterance. The study contributes to mathematics education research by adding to the limited knowledge base on children’s understanding of dimension with a few studies in the last 2 decades (e.g., Morgan, 2005; Panorkou & Pratt, 2016). Future studies could focus on teachers’ and children’s understanding of dimension in culturally and linguistically diverse contexts including New Zealand.