Keywords

Introduction

The studyFootnote 1 reported here addressed kindergarten teachers’ discursive practices as we were focusing on the mathematical competencies that a kindergarten teacher (KT) drew on in her teaching of a mathematical activity designed by researchers. Analyses were conducted based on the Knowledge Quartet (KQ) developed by Rowland et al. (2005). According to Maher et al. (2022), the KQ encompasses both the static and the dynamic aspects of (mathematics) teachers’ knowledge. We are particularly interested in the dynamic features of kindergarten teachers’ mathematical competencies. Norwegian kindergartens are situated within a social pedagogical tradition (Norwegian Directorate for Education and Training, 2017). Thus, a kindergarten teacher is supposed to nurture and empower mathematical explorations amongst the children, let them get familiar with and achieve experience with respect to mathematical concepts within number, geometry, and measurement. The study reported here had a forerunner in a study conducted by Hundeland et al. (2017).Footnote 2 In that study, we found that the four dimensions of the KQ were intertwined in the dialogues, as one move might simultaneously exemplify more than one dimension. In this study we are broadening our scope in scrutinizing the teaching of a KT even further, as we aimed to elaborate the ways that a KT revealed her subject knowledge when teaching a mathematical activity for 5-year-olds. As will be seen, the re-analysis also resulted in new insights into the subtleties of KT’s revealed mathematics knowledge.

Even though the research literature is sparse when it comes to preschool teachers’ knowledge for teaching mathematics, there are a number of studies addressing preschool teachers’ pedagogical content knowledge and competencies more broadly. For the sake of space, we will here only mention two relevant studies (Other studies may be found in Levenson et al. (2011) and in the two studies’ literature reviews).

Oppermann et al. (2016) studied how preschool teachers’ mathematical content knowledge, among others, are related to their sensitivity to mathematical issues arising in play-based situations. These researchers argue that “offering high quality mathematical education is a challenging task for preschool teachers and requires a number of competencies” (p. 174). In particular, the preschool teachers’ mathematical content knowledge and pedagogical content knowledge are important.

Bruns et al. (2017) examined the effects of a professional development course involving early childhood teachers by conducting a pre-test/post-test study. These authors found “that the course affected teachers’ mathematical pedagogical content knowledge” (p. 76). Furthermore, they argue that early childhood teachers’ mathematics-related competence is highly relevant to children’s mathematical learning. Bruns et al. argue, based on the research literature, that mathematical pedagogical content knowledge needed for teachers of early childhood comprises “knowledge about ways to analyse mathematical development, to create mathematical learning environments for young children and give adaptive support in natural learning settings.” (p. 78).

We argue, based on the study of Oppermann et al. (2016) and Bruns et al. (2017) that insights into KTs’ knowledge for teaching mathematics in kindergarten are highly relevant and important. Moreover, we argue that the KQ adds to these insights being a suitable framework for analysing KTs’ knowledge for teaching mathematics.

This study aimed to scrutinise some of the subtleties involved when a KT carried out and involved a group of children in a mathematics activity. We deliberately use the term teaching of mathematics to address the KT’s practice, even though the tasks of teaching in kindergarten are different from tasks of teaching in school (cf. Erfjord et al., 2012). For example, in the Norwegian kindergarten mathematics lessons of individual work is rare. The children do neither have explicit learning goals to achieve as pupils have in school. The Framework plan for the enterprise of kindergartens in Norway (Norwegian Directorate for Education and Training, 2017) rather emphasises that “Kindergartens shall highlight relationships and enable the children to explore and discover mathematics in everyday life, technology, nature, art and culture and by being creative and imaginative. The learning area shall stimulate the children’s sense of wonder, curiosity and motivation for problem-solving.” (p. 53). This quote is in line with what Wells (1999) calls inquiry, as a “willingness to wonder, to ask questions” (p. 121), and so forth. Nevertheless, the role of a KT in teaching a mathematical activity carries several similarities with the role of a mathematics teacher in the classroom: The KT plans the teaching of the activity, leads the mathematics work, acts in the moment in order to adapt questions and tasks for the children. Furthermore, she organises the activity and the interplay of the different children’s contributions as a plenary session comparable to a classroom situation in school. The Framework plan for the enterprise of kindergartens in Norway (Norwegian Directorate for Education and Training, 2017) does not use the term ‘teaching’, but as argued above, a KT’s practice may still be labelled teaching (cf. Sæbbe and Mosvold (2016) for a further discussion of the concept of teaching in Norwegian kindergartens).

The Knowledge Quartet as an Analytical Framework

The Knowledge Quartet was launched by Rowland et al. (2005) in order to address and characterise mathematics teachers’ knowledge in mathematics teaching. Originally, Rowland et al. used videotapes from pre-service teachers’ mathematics classroom lessons as well as post-teaching stimulated-recall interviews in their analyses of mathematics teachers’ utilization of pedagogical and mathematical knowledge. We adopted the KQ as our analytical lens in the characterisation of kindergarten teachers’ revealed mathematical and pedagogical knowledge when teaching 5-year-olds. Our particular use of the KQ addressed one KT’s teaching activity involving two-dimensional geometrical shapes.

As the term indicates, the KQ encompasses four dimensions which address mathematics teachers’ knowledge revealed in the classroom (Rowland et al., 2005). Rowland et al. took a grounded approach to their data and identified these four dimensions along which the mathematics pre-service teachers’ “mathematics-related knowledge” (p. 255) was analysed. The KQ focuses on teachers’ observable mathematics-related knowledge emerging in situations in the mathematics classroom. These four dimensions are Foundation, Transformation, Connection, and Contingency.

Foundation is a dimension of the Quartet used to address the knowledge background of the (kindergarten) teacher, knowledge of both mathematics and mathematics education. Furthermore, Foundation is informing the three other dimensions. Foundation addresses the propositional knowledge, i.e. the KT’s knowledge of relevant mathematical concepts and their inherent relationships, the KT’s knowledge of pedagogical and mathematics education research informing the practice of mathematics teaching in kindergarten, and the KT’s view upon the purpose and relevance of mathematics education for kindergarten children and these children’s mathematics learning. We analytically use this dimension to evaluate the mathematics and the didactical insights revealed by the KT’s teaching.

Transformation is one of the two dimensions used to address knowledge-in-action, i.e. the (kindergarten) teacher’s knowledge of mathematics and mathematics education as revealed in orchestrations of mathematical activities. Transformation comprises the KT’s choices regarding demonstrations given, representations used, and examples provided in her teaching. Furthermore, transformation addresses the KT’s ability to transform the mathematics “in ways designed to enable students to learn it” (Rowland et al., 2005, p. 265). We analytically use this dimension to evaluate the KT’s choices of representations and examples when characterizing various two-dimensional geometrical shapes.

Connection is the other dimension of knowledge-in-action as it addresses how the KT draws connections between involved mathematical concepts, overtly making connections between involved mathematical procedures, discussing various meanings for the involved concepts and diverse ways of carrying out the involved procedures. We analytically use this dimension to evaluate how the KT characterises the different shapes, also by names, as well as how she makes mathematical connections between the geometrical shapes and their features.

Contingency is the dimension used to address knowledge-in-interaction, i.e. the (kindergarten) teacher’s unfolding of knowledge of mathematics and mathematics education in interaction with the children. Contingency thus addresses the teacher’s ability to respond mathematically appropriate to situations that have not been planned for or anticipated (Rowland & Zazkis, 2013), i.e. to act in the moment. Furthermore, Contingency encompasses whether the KT takes advantage of emerging mathematical learning opportunities and whether the KT deviates from her goals of the activity. We analytically use this dimension to evaluate the KT’s ‘on her feet’ responses regarding the children’s suggested ideas and to what extent she makes the children aware of particular mathematical ideas.

In the mathematics education literature, there are a number of studies who have utilised the KQ as an analytical lens (see for example Petrou and Goulding (2011), Liston (2015), Rowland et al. (2015), and Maher et al. (2022)). Maher et al. studied mathematics teaching comprising differential calculus and discrete probability. They found that “there is a complex interplay among aspects of the Knowledge Quartet, including the impact of foundational knowledge on contingent moments” (p. 233). However, none of these have investigated the mathematics teaching of a kindergarten teacher.

Methods and Context

The research design of our study bared characteristics of a case study (Bassey, 1999), as we delved into particularities and details of the KT’s mathematics teaching providing a thick account. We observed and videotaped the KT’s teaching and transcribed the videos for analytical purposes. The analytical process was driven by our use of the analytical codes associated with the four dimensions of the KQ. Phase 1 consisted of collective reflection on our data material, which encompassed videotapes of two sessions of four KTs, having the KQ’s dimensions and associated codes in mind while making a first attempt to use the codes in analyzing our data. In phase 2 we collectively watched video excerpts from the four KTs, a phase resulting in choosing one of the KTs, called Wilma, as our analytical case. This choice was due to Wilma’s teaching as being suitable for employing KQ as an analytical tool, and thus in answering our research question. In particular, we chose to analyse Wilma’s teaching because it more than the others revealed all the four dimensions of the KQ. Furthermore, Wilma’s teaching, more than the others, revealed more mathematical concepts and ideas, and her teaching, more than the others, explicitly revealed mathematical dialogues between the KT and the involved children. In watching the video of Wilma, we were able to observe her revealing of foundational knowledge, knowledge-in-action and knowledge-in-interaction. In phase 3 we transcribed in detail the video of Wilma’s teaching. Phase 4 consisted of our collective conducting of in-depth analyses of the teaching.

Context

Wilma is a well experienced kindergarten teacher at the age of 45. She is educated as kindergarten teacher from university training (180 ECTS). The mathematical activity explored in this study concerned children’s inquiries into features of two-dimensional geometrical shapes, the various shapes’ names, and their conceptual relationships with each other. The session lasted for 26 minutes, and it involved six five-years-old children. In our design of the activity, materialised as a written activity description, we emphasised the aims of the activity, gave suggestions for how to teach the activity, provided explicit examples of mathematics questions to ask the children, and were explicit about the manipulatives to use, triangles, squares, rectangles, circles, trapezium, and rhombus. As regards the aims, we wrote: The children are supposed to get experience in recognizing properties to different two-dimensional shapes. Furthermore, the children are supposed to practice mathematical argumentation with respect to features of the various shapes. Concerning the teaching, we wrote: Let the children investigate the shapes and their characteristics. Let the children discover the shapes’ differences. Moreover, we also encouraged the KT to make the activity her own, benefitting utilisation of her own experience and competence. Drawing on the KQ, earlier research and our methodical approach, we want to find answers to the following research question:

In what ways do a kindergarten teacher’s subject knowledge come into play when teaching a mathematical activity for 5-year-olds?

Analysis and Results

We argue in accordance with Rowland and colleagues that “the quartet is comprehensive as a tool for thinking about the ways that subject knowledge comes into play in the classroom” (Rowland et al., 2003, p. 97). In the following, we present an analysis of Wilma’s teaching of the geometry activity informed by the four dimensions of the KQ.

Foundation

The analytical contributory codes of Foundation are: awareness of purpose; identifying errors; overt subject knowledge; theoretical underpinning of pedagogy, use of terminology; use of textbook; reliance on procedures (Rowland et al., 2005, p. 265). We found examples of Wilma’s foundation revealed in the initial phase of her teaching:

Dialogue 1

Wilma::

Today I have brought this box (Shakes a cubic box so that it makes sounds)

Sam::

Oh, yes. The one you showed us before. But I don’t remember what’s inside it

Jack::

It’s shapes

Wilma::

Yes, that’s correct. And with mathematical terminology we call them geometrical shapes. Are you able to pronounce that?

Sam::

I think it is cookies (Smiles as he says it)

Wilma::

John, are you able to pronounce that? Geometrical shapes?

John::

Geometrical shapes (Several children repeat and say simultaneously “Geometrical shapes”)

Wilma::

Yes, that is what they are called with mathematical terminology. Inside this box there are several of such shapes (She opens the box and shows it to all the children so that they may look inside the box)

John::

It looks like a puzzle

Wilma::

Yes, it looks like a puzzle. That’s true

Sam::

Yes. Are we going to puzzle with them?

Wilma::

At least we are going to work with them, yes we are

Ken::

Can you pour them out?

Wilma::

I was thinking pouring them out. Then I want you to take a look at them. Currently, there are quite a few shapes and some of them are almost identical. Now you may take a look at them. (She pours the shapes out on the table; the children take some shapes each and say “that is small” and “a triangle”)

In this dialogue we argue that particularly the codes awareness of purpose, theoretical underpinning of pedagogy, use of terminology, and overt subject knowledge may be used to analyse Wilma’s teaching. Wilma’s shaking of the box with shapes inside, showing the shapes and eventually pouring them out on the table, establish curiosity and engagement among the children. Moreover, the similarities made between shapes and puzzles nurture the children’s interest and curiosity. Wilma’s playful way of teaching this initial phase, certainly demonstrates her awareness of the overall purpose of the activity. Wilma furthermore attempts to make the children inquire into the different shapes and the shapes’ features. Wilma wants the children to “take a look at them”, and by doing that she signals that she wants the children to study the various shapes, distinguish between different shapes, recognise the shapes’ characteristics, similarities between the shapes and so on. Wilma is thus empowering the children in using inquiry as a tool to make sense of the mathematics, simultaneously as she nurtures the children’s curiosity and interest. These actions testify as exemplifying her theoretical underpinning of pedagogy.

Moreover, Wilma introduces congruent and similar shapes, and a large variety of shapes (various triangles, various quadrilaterals, circles of numerous sizes, ellipses, hexagons and octagons). This variety exceeds the variety suggested by us. Thus, Wilma makes her subject knowledge regarding geometrical shapes overt when elaborating and deviating from the written description.

Wilma emphasises mathematical terminology by her twice expressing of the term “geometrical shapes”. She wants the children to appropriate the term. This emphasis on terminology also occurs a few minutes later, where Wilma emphasises that what Susie calls an oval shape mathematically is called an ellipsis. Wilma is making a link between shape (oval) and name (ellipsis) overt:

Dialogue 2

Susie::

(Picks up a small oval shape and shows it to the other children) This is oval

Wilma::

Yes, that’s true. That one is oval. Do you know what it is called with mathematical terminology?

Susie::

(Susie shakes her head)

Wilma::

Sam and Jack, look at the one Susie now has in her hands. Susie said that it was oval. With mathematical terminology that shape is called an ellipsis.

Susie::

Ellipsis?

Wilma::

Yes, ellipsis.

The codes associated with Foundation are applicable and useful when analyzing this dialogue. By drawing on the codes of KQ, we get glimpses into parts of Wilma’s foundational knowledge with respect to two-dimensional geometrical shapes and how she utilises this knowledge in her teaching of five-year-olds in kindergarten. Both mathematical and didactical insights are revealed.

Transformation

The analytical contributory codes of Transformation are: choice of representation; teacher demonstration; choice of examples (Rowland et al., 2005, p. 265). The dialogue below exemplifies how Wilma’s knowledge-in-action was revealed. The children picked up various shapes that they found interesting, pentagons, ovals, and quadrilaterals; shapes they were not familiar with from before:

Dialogue 3

Wilma::

Do you know what? These two quadrilaterals actually have other names with mathematical terminology. They have four edges (She counts “one, two, three, four” aloud while simultaneously pointing at the edges).

Susie::

But what are they called then?

Wilma::

That one is called a rhombus (she points at the rhombus while speaking).

Susie::

Rhombus.

Wilma::

Rhombus. And that one, do you notice that two and two edges are equal (she points at the parallelogram she shows). That edge and that edge (slides her finger along the two opposite parallel edges), are equal, and that edge and that edge are equal (slides her finger along the two other opposite parallel edges). Its name is actually a parallelogram.

Sam::

A paragram?

Wilma::

Yes, a parallel o gram.

This dialogue shows how Wilma transforms the mathematical content involved in the activity. She focuses at discussing two particular shapes which the children are unfamiliar with, a rhombus and a parallelogram. We interpret her utterance “These two quadrilaterals actually have other names” as an attempt to elaborate the children’s conceptual reasoning concerning quadrilaterals. She makes a conceptual juxtaposition by counting the edges of these shapes aloud, making it overt that the shapes are indeed quadrilaterals but at the same time particular kinds. Wilma implicitly distinguishes these quadrilaterals from the more familiar quadrilateral shapes rectangle and square.

Wilma’s teaching is interpreted as to illustrate how the dimension of Transformation is unveiled in a kindergarten setting. The associated codes were useful in characterizing how she transforms the mathematics to create appropriation opportunities for the children. Wilma chooses to use concrete, manipulative materials (choice of representation) and clearly shows the two shapes to all the children while she focuses on their mathematical names as well as their features (teacher demonstration). Wilma’s action of showing one example of each of the new quadrilaterals while simultaneously describing the shapes as quadrilaterals by counting their edges (choice of examples), further illustrates her transformation of the mathematics involved.

Connection

The analytical contributory codes of Connection are: making connections between procedures; making connections between concepts; anticipation of complexity; decisions about sequencing; recognition of conceptual appropriateness (Rowland et al., 2005, p. 265). We interpret Dialogue 3 to also exemplify how the dimension of Connection characterises Wilma’s teaching. Wilma is making connections between concepts when she elaborates on the children’s conceptualization of quadrilaterals. She establishes a shared focus of attention by pinpointing particular features of the parallelogram. Wilma uses her sliding index finger to emphasise that two and two edges are pairwise parallel. She says, “that edge and that edge are equal”, and we interpret this as an attempt to make a connection between the characteristics of the edges and their fundamental role for classifying the shape as a particular quadrilateral. Wilma uses the term “equal” and not “equal length” and “parallel”. The parallelism and equal length of the pairwise edges are in that sense only implicitly communicated. However, using the term “equal” together with the gesture of sliding her finger along the edges, we interpret as communicating that the edges are of equal length. From a mathematical point of view, if a quadrilateral has two and two opposite edges of equal length, these edges necessarily are parallel, and the quadrilateral is indeed a parallelogram.

We interpret Wilma’s choice regarding variety of shapes as well as congruent and similar shapes as exemplifying how she anticipates the internal mathematical complexity and how she recognises the conceptual appropriateness of these two-dimensional geometrical shapes. Wilma’s decisions regarding variety and number of shapes also testifies to how the dimension of Connection characterises Wilma’s teaching.

At the end of Dialogue 3 we once again find an example of Wilma’s focus on mathematical terminology, a focus demonstrating how she uses her foundational knowledge. Wilma offers opportunities for the children to appropriate the name “parallelogram” twice, showing her eagerness in naming mathematical objects correctly. The dialogue below further demonstrates how Connection characterises Wilma’s teaching:

Dialogue 4

Wilma::

Yes, Ken. Do you want to show that shape? Then we first have to discuss the shape of it

Ken::

One, two, three, four edges (Ken shows, while rotating, a trapezium, close to a square)

Wilma::

Four edges, Yes, it does have that. But what is different with this? If we compare it with that (She picks up a square). What is different? We may put them down at the table for all to see (She puts down the square while Ken puts down the trapezium next to it in the middle of the table). What is different?

Susie::

That one is more askewed (points at the trapezium)

Wilma::

Yes, that one is more askewed. It looks like there are two lines that are tilted… And then there are two lines that are equally straight (Slides her index finger along the parallel edges). This shape is called a trapezium. That is a difficult word.

Sam::

You are quite precise

Wilma::

Do you think I’m precise? Well, that’s good. It is important to be quite precise.

In Dialogue 4 we once again see how Wilma tries to make the children appropriate connections between concepts, this time by comparing and distinguishing between a square and a trapezium. Ken has chosen a shape that he wants to show the other children, a shape that he finds out has four edges by counting. Counting has been used by Wilma (Dialogue 3) as a strategy to characterise shapes. Ken here adopts that strategy and implicitly argues the shape to be a quadrilateral, an implicit claim confirmed by Wilma. Then Wilma continues and starts to compare the chosen trapezium with a familiar and close to congruent square. She obviously wants the children to look at the two shapes that are almost identical, in order for the children to come up with the features that distinguish the trapezium from the square. In doing that, Wilma obviously also has anticipated the complexity and involved concepts to be appropriate for these children.

Three times Wilma asks the question “What is different?”. Susie recognises that the trapezium “is more askewed”. We interpret this utterance as Susie’s way of telling the others that two of the edges are not parallel and that this fact makes the trapezium differ from the square. This interpretation is supported by the response Wilma gives, that “there are two lines that are tilted” and that “two lines that are equally straight. Comparing the two shapes edge by edge, the trapezium has two edges that are not parallel, a feature that makes that shape having a separate name: “This shape is called a trapezium”. Interestingly, Sam comments on Wilma’s argument and claims her to be quite precise. We interpret Sam’s utterance as his way of showing that he has recognised that Wilma is mathematically accurate in her way of reasoning and orchestrating the activity. Sam seems to be involved in an initial process of appropriating the stringency and accuracy of mathematics.

Contingency

The analytical contributory codes of Contingency are: responding to children’s ideas; use of opportunities; deviation from agenda (Rowland et al., 2005, p. 266). From the four dialogues analysed, we saw several examples of how Wilma responded to the children’s contributions. Now we want to illustrate how Contingency characterised her teaching. In the following we do this by inclusion of single moves Wilma made, where she responded to the children through repetition, questioning and affirmation to make the children aware of the mathematical ideas involved.

On several occasions, Wilma was responding to children’s ideas by acting in the moment. The children came up with two mathematical ideas in their dialogic contributions, the concept of sorting and the concept of geometrical shapes. Concerning sorting, Sam asked the question: “Can we sort them?”. Some moves and seconds later, Wilma responded to Sam’s question by asking a question in return: “Sam, what does it mean to sort?”. She nurtured Sam’s reasoning and wanted him to be explicit about his thinking regarding the mathematical concept of sorting. Later in their dialogue, Wilma followed up the emphasis on the concept of sorting by affirming one child’s actions: “But he has sorted. That’s excellent”. As regards the concept of geometrical shapes, Wilma emphasised the mathematical features of the various shapes. This was exemplified in Sam’s utterance: “Yes, but these are small (points at the short edges of the rectangle). These two are equally long”. Wilma’s response to this contribution was to address a question to all six children: “Does anybody know what the shape is called when two edges are quite long and two edges are shorter?”. Wilma’s response, in the form of a question, we argue, nurtured curiosity and interest among the children. Furthermore, her question also testified to how she contingently thought “‘on her feet’ and respond appropriately” (Rowland et al., 2005, p. 266) to the children’s contributions. Wilma took the opportunity to involve all the children in collective reasoning regarding this shape.

Similarities and differences between the shapes were also focused on in Wilma’s teaching, as we saw in Dialogue 4. Additionally, on another occasion, Susie argued that two congruent triangles may be joined in order to make a rectangle: “Jack’s shapes have such…, but Ken’s do not have such when he puts them together”. Wilma responded to Susie’s contribution by giving a question in return: “What happens when you put them together?”

In Wilma’s teaching, we also found instances where she did make use of opportunities. Wilma made the children pay attention to the mathematical concept of counting as a strategy to classify shapes: “John, perhaps you can count how many edges they (two regular hexagons and one regular pentagon) have?”. Another example where Wilma made use of opportunities occurring in the midst of teaching, was when Wilma nurtured the children’s mathematical reasoning through questioning: “How did you figure out that one (points at one of the hexagons)?”; and “Do you want to tell the other children?”. Additionally, Wilma’s emphasis on naming unfamiliar shapes always occurred after the children had sorted the various shapes and a discussion had been going on regarding the shapes’ mathematical features.

Discussion

We set out in this study to explore the following question: In what ways do a kindergarten teacher’s subject knowledge come into play when teaching a mathematical activity for 5-year-olds? We used the Knowledge Quartet (Rowland et al., 2003, 2005) as an analytical lens through which we analysed one kindergarten teacher’s teaching of a mathematical activity on two-dimensional geometrical shapes. From the analyses it was evident that the KQ was a powerful and useful framework when characterizing a KT’s orchestration. From the analyses we saw that the KT employed her foundational knowledge (Foundation), knowledge-in-action (Transformation and Connection) and knowledge-in-interaction (Contingency) in several ways. In this sense, our study empirically confirms the claimed importance of early years teachers’ mathematical content knowledge and pedagogical content knowledge (Bruns et al., 2017; Oppermann et al., 2016).

Almost all analytical contributory codes of KQ were found useful and illuminating in our analyses. The KT’s foundational knowledge was, as far as what was revealed in the analyses, characterised by her emphasis on the use of mathematical terminology and her substantial subject knowledge regarding two-dimensional geometrical shapes. She also revealed her foundation knowledge through the overt acknowledgment of the activity’s purpose and her theoretical underpinning of pedagogy through establishment of curiosity, interest, and engagement on behalf of the children. The children were given opportunities to inquire into the mathematics (Wells, 1999). Thus, Wilma’s foundational knowledge fundamentally informed her teaching.

The codes (cf. Rowland et al., 2005) have furthermore proven to be analytically valuable tools, showing that the KT revealed her knowledge-in-action through use of manipulatives and purposeful choice of examples in demonstrating the features of various geometrical shapes. Moreover, the knowledge-in-action revealed was characterised by Wilma making connections between the concepts of quadrilaterals such as square and trapezium as well as drawing attention to features such as number of edges, length of edges and parallelism between edges associated with the various shapes. Furthermore, these utterances and actions showed how she “capitalises on these contingent situations” (Rowland & Zazkis, 2013, p. 137), in order to create opportunities for the children to appropriate the involved mathematical concepts.

Reflecting on the analyses made by Rowland et al. (2005) of mathematics teaching in a British school classroom, we observed in the analyses above that the KQ cannot be directly used as a template for analyzing a KT’s teaching of a mathematical activity in a Norwegian kindergarten setting. Some of the codes were not found applicable. KTs working in accordance with the Norwegian curriculum for kindergarten (Norwegian Directorate for Education and Training, 2017), rarely teach mathematical activities characterised by long theoretical introductions and demonstrations. That would be regarded as inappropriate. In our analyses we rather observed that the KT’s moves were quite short concerning time, often about 5–10 seconds. Additionally, it is rare in the Norwegian kindergarten setting to give children extensive time for inquiring into mathematical ideas without the KT interfering occasionally. The KQ analytical codes use of textbook, reliance on procedures, teacher demonstration, and making connections between procedures, are thus partially inapplicable since these are argued to be rarely present in mathematics teaching in kindergarten. In our analyses, we have thus not found these codes to be useful.

Our analyses have also shown that it was challenging to apply only one code for some of the moves and parts of the dialogues. In the dialogues analysed above, we observed that the dimensions were to some extent intertwined. This result is in accordance with the finding of Maher et al. (2022). This is also an argument in accordance with the developers of KQ (Rowland et al., 2005). One move may be argued to exemplify several codes, and hence, dimensions, simultaneously. The four dialogues analysed were thus not mutually exclusive adopting the KQ as an analytical lens. Nevertheless, they are argued to be characteristics of the kindergarten teacher’s revealed foundational knowledge, knowledge-in-action and knowledge-in-interaction.

Studies of KTs’ mathematical competencies are important since it is those competencies they have to draw on in order to nurture the children’s processes of appropriating the mathematical concepts involved in activities (cf. Moschkovich, 2004; Rogoff, 1990). Wilma’s use of knowledge-in-action and knowledge-in-interaction established opportunities for the children to make the mathematical concepts of various two-dimensional shapes, their names and features, their own. Future research opportunities may be found in scrutinising participating children’s appropriation processes with respect to the mathematics offered. Further studies of KTs’ mathematical competencies and how these are revealed in practice is also a promising road ahead.