Introduction

Research on young children’s mathematical thinking, understanding, and learning has come a long way from the seminal work of Jean Piaget to its present state. Indeed, studies on these topics can be effortlessly found in key international journals and significant books (e.g., English & Mulligan, 2013; Perry et al., 2012). The contribution of the Australasian research to this global endeavour has been systematically overviewed every 4 years by the members of the MERGA community (for the latest review see Downton et al., 2020). But research on young children’s mathematics has been not only prolific but also diverse. “Young children’s early modelling with data” (English, 2010), “Supporting early mathematical learning: building mathematical capital through participating in early years swimming” (Jorgensen, 2017), “Exploring the use of iPads to engage young students with mathematics” (Attard & Curry, 2012), and “Young children’s mathematical learning opportunities in family shopping experiences” (MacDonald et al., 2018)—these titles illustrate the wide range of epistemic objects, research aims, questions, methods, and situations that shape studies in this area (cf. Bikner-Ahsbahs, 2019). This is not to speak about different conceptual frameworks and theories that researchers employ.

On the one hand, this diversity speaks both to the complexity of young children’s mathematics and to the research maturity of the area that embarks on this phenomenon in its complexity. On the other hand, diversity imposes a substantial challenge for the research community to build on, synthesize, and put existing research in use (e.g., Cross et al., 2009). This challenge draws attention to methods and tools researchers implement to communicate their findings. In the words of Presmeg and Kilpatrick (2019), “communicating one’s research clearly and adequately is at least as important as conducting it proficiently” (p. 347).

Our point of departure is that visual tools (e.g., schemes, diagrams) can assist researchers to communicate their findings “clearly and adequately”. For instance, Bakker et al. (2021) use an artistic image to present common themes that emerged from their investigation of future directions in mathematics education research. The researchers explain that they created the image after alternative visuals, such as “star” and Venn diagram, were considered. Notably, Bakker et al. (2021) refer to the issue of finding an appropriate presentation as “a recurring question during the analysis” (p. 4). This reflection indicates that considering means to present findings can inform the processes of data analysis and findings generation. Accordingly, to paraphrase Presmeg and Kilpatrick (2019), we propose that visuals can also be useful for researchers to “conduct research proficiently”. Furthermore, we relate the communicational and analytical merits of visual tools with the emerging interest in innovative methodologies that “‘make visible’ the ways in which young children experience mathematics” (Downton et al., 2020, p. 235).

In this paper, we introduce the Grove of Realizations as a tool for constructing diagrammatic visuals that can be used to generate and share research findings on young children’s mathematics learning in small groups. Our aim is to introduce this tool and present its affordances for data analysis and research communication. We structure the paper as follows. In the “Background” section, we depict the sort of studies to which the focal tool may be relevant. The theoretical background is presented in the “Commognitive background” section. We introduce the Grove of Realizations in the “The Grove of Realizations” section and illustrate its analytical affordances in the “Illustrating the affordances of the Grove of Realizations” section. The “Summary and discussion” section concludes with a discussion of the merits, limitations, and directions for further development of the tool.

Background

The first part of this section demarcates the line of research in which we situate the work in-hand. The second part overviews a dialogical approach to learning.

Delineating relevant lines of research

To delineate the sort of studies for which this paper may be of interest, we turn to the construct of research pentagon (see Fig. 1). Drawing on her previous work on networking theories (e.g., Bikner-Ahsbahs & Prediger, 2014), Bikner-Ahsbahs (2019) developed this construct as a heuristic tool for researchers to structure their studies, reflect on their components, and explore the coherence between them. The elements of a pentagon are epistemic objects under scrutiny, the research aims and questions pursued, the implemented methods, and situations designed to undertake the study. In tune with this Special Issue, we appeal to studies of which the epistemic objects encompass such constructs as mathematical behaviours, thinking, and learning of young children. More often than not, these studies employ qualitative methodologies to explore how the epistemic objects in their focus come about. Of our special interest are situations where children work in small groups through collaboratively engaging in a mathematical situation (we elaborate on this aspect in the “Dialogical learning and its caveats” section).

Fig. 1
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Research pentagon showing main aspects of doing research (Bikner-Ahsbahs & Prediger, 2014)

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Bikner-Ahsbahs (2019) notes that “theoretical background assumptions also enter the scene, which are not shown in the pentagon but which are equally important as a pre-requisite for a coherent framework” (p. 158). Many schools of thought have acknowledged the crucial role of theory in conducting a research study in mathematics education (e.g., Bikner-Ahsbahs & Prediger, 2014; Sfard, 2008). For instance, Presmeg and Kilpatrick (2019) highlight that theory underpins all aspects of the research process, “from the initial choices of questions to be addressed, through suitable methods of data collection and analysis, to the conclusions reached as a result of the research, and future directions that the research might take” (p. 348). Accordingly, we see theories as the glue that hold research pentagons together.

We do not wish to limit the line of studies depicted above based on the theories that they employ. To the contrary, as it was argued in the “Introduction” section, the overarching motivation of this work is to support meaningful research communication and engagement, including cases where research studies grow from different theoretical grounds. Nevertheless, we consider young children’s learning from the commognitive viewpoint (Sfard, 2008). The international mathematics education community has recognized the commognitive framework for its conceptual and analytical apparatus that accounts for the specificity of mathematics and the insights it offers into its learning and teaching (e.g., Morgan, 2020). Commognition has been used in numerous studies into mathematics learning of young children (e.g., Lavie & Sfard, 2019; Rodney, 2019; Sfard, 2007; Shinno & Fujita, 2021).

Dialogical learning and its caveats

The belief that mathematical dialogues that unfold in collaborative situations facilitate learning seems to be gaining in popularity in terms of research, developments in classroom practice, and educational policy (e.g., Hill et al., 2019; Hunter et al., 2020; Mercer, 2002; Mercer et al., 2019; NCTM, 2000; O’Connor & Michaels, 2019; Topping & Trickey, 2015; Sofroniou & Poutos, 2016; Webb et al., 2019). Lefstein and Snell (2013) describe dialogical education or discourse-based pedagogies as learning that is collective—teachers and children address learning tasks together; reciprocal—they share ideas and consider alternative viewpoints; supportive—they help each other reach common understandings; cumulative—they build on their own and each other’s ideas; and purposeful—the talk has specific educational goals. That is to say, multiple voices, authored by individual students, are expressed and listened to with the goal of advancing collective thinking and reaching consensus. The benefits of dialogical approaches to learning have been described as, “non-hierarchical, democratic, pluralistic, ethically and culturally sensitive, and inherently egalitarian” (Kennedy, 2009, p. 71) and as supporting the emergence of “deep connected mathematical understandings” (Hunter et al., 2020, p. 178).

Nevertheless, Sfard (2019) observed that the connection between communication and learning is often assumed and oversimplified. She criticizes a general “unshaken confidence” (p. 89) in the benefits of students’ talk and draws on her own empirical studies to show that student–student collaborations can yield disengaged interactions and unproductive communication with limited advantages for learning (e.g., Chan & Sfard, 2020; Sfard & Kieran, 2001). Hence, a need was signalled for theoretically informed methodologies to provide insights into how the processes of learning through a dialogue can be documented, analysed, and shared within the research community. We embark on this call from the commognitive standpoint and with special attention to dialogical learning in small groups.

Commognitive background

The first part of this section lays out commognitive terminology and underpinnings that are later used to discuss the tool in the focus of this paper. The second part focuses on analytical instruments on which we build to introduce our tool.

Commognition in a snapshot

The commognitive framework (Sfard, 2008) construes mathematics as a form of discourse. Discourses are defined as, “different types of communication, set apart by their objects, the kinds of mediators used, and the rules followed by participants and thus defining different communities of communicating actors” (Sfard, 2008, p. 93). Operationally speaking, discourses are distinguishable through four characteristics: keywords (e.g., “odd number”) and their use, visual mediators (e.g., Numicon tiles in Fig. 2, diagrams) and their use, narratives endorsed as correct (e.g., “the sum of odd numbers is even”), and routines (e.g., substantiating that a number is even).

Fig. 2
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Numicon tiles

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Learning is defined as a lasting change in the communicational patterns that a person exhibits when participating in a mathematical discourse, i.e., a change in at least one of the four characteristics (keywords, visual mediators, narratives, routines). Commognition distinguishes between object-level and meta-level learning. The former is associated with an expansion of one’s discursive repertoire; for instance, when the learner starts using a new keyword or implementing a previously foreign routine. In comparison, the more complex meta-level learning requires revising the rules that underpin one’s discourse. For instance, whereas the words “odd” and “even” may initially be used as adjectives describing specific numbers, learners may eventually use these words as signifying general terms or mathematical objects in their own right (e.g., a generic symmetrical/paired array for even and asymmetrical/paired array with a singleton for odd, or algebraic expressions 2n and 2n + 1). This process is called objectification, which involves both reification—replacing talk about processes with talk about objects—and alienation—presenting phenomena impersonally, as if they were occurring independently of human actions (Sfard, 2008, p. 44).

Commognitive conflicts—instances where the interlocutors use the same words to talk about different things—are considered to be a key mechanism for triggering meta-level learning (Sfard, 2008). Resolving a commognitive conflict involves one of the interlocutors making sense of the other person’s thinking and gradually accepting and adopting the new, initially incommensurable, discourse and abandoning his or her own. At this point, we acknowledge that the notion of a commognitive conflict does not discriminate between instances whereby interlocutors are aware of clashes in their communication, and ones which may only be revealed via discursive research. Accordingly, we expand the notion of a commognitive conflict to include (i) communicational clashes—instances whereby interlocuters are cognizant of interpersonal disagreement—and (ii) discursive gaps—differences in interlocuters’ words (and the ways in which words are used), endorsed narratives and/or routines which are researchable via tracing their signifiers and realizations. Note also, that even when interlocutors are cognizant of interpersonal disagreement, they may still remain unaware of how or why they disagree.

What makes a mathematical discourse distinct is that, while in many academic disciplines, there is a natural separation between the objects under scrutiny and the talk about them (e.g., in exact sciences), from the commognitive viewpoint, “in mathematics the objects of talk are, in themselves, discursive constructs, and thus constitute a part of the discourse” (Sfard, 2008, p. 129). On the level of an individual learner, mathematical objects are viewed as personal constructs that become researchable through the learner’s participation in public communication. Then, their mathematical objects become accessible through perceptually graspable signifiers (e.g., words, symbols, gestures) that the learner uses and realizes into other signifiers. It may be tempting to equate the notions of signifier and realization with “representations” that feature in many theoretical frameworks. We advise to resist this temptation for two reasons. First, the use of “representation” tacitly implies that there exists a “true” mathematical object, with “representations” being its avatars. In commognition, the relation between a particular signifier and its realizations emerges from the patterns that researchers identify in the collected data. Accordingly, almost any mathematical realization can act as a signifier, which bears the potential for it to be realized further (an example of the signifier-realization relationship follows in the “Tools for capturing mathematical objects” section and is visualized in Fig. 3). Second, other frameworks often construe mathematical objects as intangible and abstract when “representations” are accessible and concrete. Within a commognitive definition, a mathematical object is defined an aggregate of signifiers and realizations that a person uses as part of their participation in a mathematical discourse.

Fig. 3
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A realization tree for the signifier “even number”

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Tools for capturing mathematical objects

Sfard (2008) has introduced a realization tree (p. 165) as an analytical tool for visualizing one’s mathematical objects. A realization tree presents a hierarchical structure of one’s signifier-realization relations by placing a key signifier at the “root” and its branching realizations that are realized further. For instance, in natural numbers, the mathematical object “even number” may be realized as “0, 2, 4, 6, 8, 10…”. In turn, this node fulfils the double role of a realization and a signifier which is realized by nodes underneath it such as numbers “divisible by 2”, as a number made of multiples of 2 (2 + 2 + 2 + 2…), “a double”, “any number that has 0, 2, 4, 6, or 8 as a ones digit”, or as “2n” (see Fig. 3). Each of these realizations may then be realized further. Overall, the notion of realization tree is not very different from the chain of realizations in semiotics (e.g., Peirce, 1955).

Sfard (2008) contends that researchers could use realization trees in at least three ways. The first one pertains to assessing the quality of one’s discourse on a particular mathematical object. For instance, researchers can attend to the richness of the branches, their depth, and stability across communicationally different situations. Second, attending to the developments in one’s tree (e.g., introduction of new signifiers, changes in the hierarchical structure) appears as an organic tool for tracing their discourse development, i.e., learning. The third usage emerges from the fact that interlocutors may realize the same signifier in different ways, which is likely to hinder the effectiveness of their communication. Accordingly, generating interlocutors’ realization trees through discourse analysis bears the potential to offering explanations for commognitive conflicts.

Realization trees have since been adapted and further developed in several studies. While focussing on functions, Nachlieli and Tabach (2012) used a realization tree to capture the complexity of this mathematical object, describe its historical development in the mathematics community, and show how students initially used its realizations (specifically formulas, tables, and graphs) separately before referring to them with the same signifier—“function”. Weingarden et al. (2019) introduced a Realization Tree Assessment (RTA) as a tool to capture whole-classroom discussions on mathematical objects. The RTA provides a visual means to tell a story about an object’s realizations, the addressed links between them, and the information on whether it was a teacher or students who contributed to the realization process. Weingarden et al. (2019) acknowledge that the RTA tool does not capture how many students participated in a lesson, meaning that a visualization of a lesson where many students generated the signifier-realization pairs would look the same as the lesson where one student had done all the discursive work. This was a concession deemed necessary by the researchers to create an efficient assessment tool that produced a simple and clear image.

The Grove of Realizations

In the first part of this section, we describe general principles for constructing Groves of Realizations. We then exemplify the construction of a grove using data coming from a group of students as they classified odd and even numbers and reasoned about their sums.

General principles

Whereas Sfard’s (2008) tree of realizations has been intended to analyse discourses of individuals, we expand this tool to a collaborative small-group setting. We aim to adapt the use of realization trees to account for how the same signifier might be realized in different ways by different interlocutors. We are interested in a means to show dominant realizations, consistencies and inconsistencies within a group, and the course of development for different group members within one collective visual. Since our goal is to visualize the effectiveness of interpersonal communication and map the discursive development of the group members, we perceive the inability of Weingarden et al.’s (2019) RTA tool to capture how many students participated in a discussion and who articulated realizations to be limitations that need to be addressed (see “Tools for capturing mathematical objects” section for details).

We introduce the Grove of Realizations as a tool to map discursive data to visuals representing mathematical objects of the group members and these objects’ development through a mathematical activity. We use “grove” metaphorically to refer to analytically constructed visuals that combine trees of realizations of individuals. In this way, groves provide information about dominant realizations featuring within a group, overlaps and demarcations between individual students’ objects, and the courses of their development through collaborative work.

Building group members’ individual realization trees from the collected data is the first step in constructing the grove. Decisions about the signifier at the root node of the tree should be informed by the research aims and questions. For example, in our study of the student’s grasp of odd and even numbers the root of the tree of realizations is simply labelled, “even” and “odd” (see Figs. 5–8). Realization “leaves” are then established within the tree. To “grow” leaves, the commognitive researcher closely analyses the data. The inclusion of a realization leaf in an individual’s tree means that the person has been generating this realization (i.e., via their use of keywords, gesture, or a visual mediator) as they engaged in mathematical activity.

Fig. 4
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The “odd + odd” concept cartoon (left) and “even + odd” concept cartoon (right)

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Once the group members’ trees of realizations are established, growing a grove for the group becomes a matter of combining their trees. Here, the researcher examines the data to establish analytical categories of realizations (leaves) observed within the group. In doing so, it is important to not only look for how expansive the categories of realizations are (where the width of the grove reflects the richness of the discourse) but to also identify the degree of objectification (where the depth of the grove reflects hierarchies of realizations).

For the grove to show who generated which realization, student-specific branches are formed using a differently coloured or coded lines per student. This means that each student’s personal tree of realizations (the total number of observable realizations connected by student-specific branches) can remain visible within the group’s grove. Numbering the leaves (according to the order in which they were observed in the data) on the grove also enables the researcher to directly refer the reader to particular points on the tool where the data (dialogue) and accompanying analysis is visualized.

An in-depth exemplification

To exemplify the construction of a grove, we use data collected as part of the first author’s doctoral research (Knox, 2021). That research explored primary-students’ first steps in a deductive discourse on numbers, with a focus on parity that aimed to answer the following research question: “What are the sources of communicational conflict within the group with regards to their talk about odd and even numbers and their substantiating universal narratives involving the sums of odds and evens?”.

Seven groups of year 4 students (aged 8- and 9-year-old) from New Zealand classes worked collaboratively to classify numbers as odd or even. They were then presented with concept cartoons (Keogh & Naylor, 1999) featuring three characters that made contradicting statements (see Fig. 4). In each cartoon, two statements were universal, while the third statement foregrounded the “sometimeness” of the previous statements (see Knox & Kontorovich, 2022, for details about the designed learning setting). The students were asked to decide which character they agreed with and justify why. The group work was video-recorded with two cameras and students’ written artefacts were added to the data corpus. Each group session was transcribed in its entirety and analysed to identify instances of students’ discursive development (i.e., changes in the use of keywords, visual mediators, narratives, or routines) regarding odd and even numbers. Accordingly, the root of the grove is labelled “Even” and “Odd”.

Table 1 shows data from a single group which we use to exemplify how the principle of grove construction allows us to illustrate three central affordances of the tool for (i) visualizing gaps in communication between group members, (ii) mapping students’ discursive growth, and (iii) showing whether and how gaps are bridged. To track discursive developments within group work, it is necessary to establish an initial grove of the students’ pre-existing realizations (i.e., what they already knew before the group session) as a baseline from which to compare more developed groves as learning takes place. We provide quotes in Table 1, which are illustrative of larger patterns of discursive data for Danny, Jane, Zara, and Robert (all pseudonyms) that occurred initially. We show how this discursive data—each student’s use of the words “odd” and “even” according to the narratives they used and endorsed, and the visual mediators (Numicon tiles, diagrams) they used within their reasoning—is used to inform the construction of each student’s tree of realizations within the grove of pre-existing realizations (in Fig. 5).

Table 1 Representative quotes from students’ classifications of odd and even numbers
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Fig. 5
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Group’s Grove of Pre-existing Realizations for “Odd” and “Even”

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To construct the grove of pre-existing realizations for the group, we categorize Robert’s—“two plus two”—and Zara’s—“they’d both get two each”—justifications of four as even, along with Jane’s “eleven and eleven” justification of why twenty-two is even, as all articulating a realization related to the symmetrical additive structure (or double) of even. We represent these realizations on leaf 1a in the grove. To show that all three students articulated a realization of this kind, we add student-specific branches connecting to this leaf. Zara also articulated a different realization of even via her “in twos” and “adding two on” utterances. Although we consider these realizations to be similar to those of leaf 1a in that they are related to the symmetrical structure of even numbers, we present them in a different leaf because they were in the form of a multiple of twos (e.g., where 2 + 2 + 2 or 3 \(\times\) 2 substantiates six as even) rather than a double, and so they are represented in a new leaf (1c). Accordingly, Zara’s branches connect with both leaves 1a and 1c for even. With regard to realizations of odd numbers, we show leaves 1b and 1d as odd counterparts for the even realizations in leaves 1a and 1c: 1b is of an asymmetrical additive form and 1d is of the form multiple of twos plus one. Jane’s “two, two and one” justification for classifying five as odd gives us reason to connect her red branch with leaf 1b. We also attach Zara’s branch to leaves 1b for her “[extra] one there” narrative and to 1d for her “Instead of adding two on, you add on one” narrative.

Realizations of even and odd represented in leaves 1a, 1b, 1c, and 1d are all considered to be from the same family realization because they were all about the structure (the symmetry or asymmetry) of numbers. Therefore, these realizations all stem from the main structure-based arterial branch “1” for realizations of odd and even. Danny’s realizations for odd and even do not stem from this family. For him, the keywords “odd” and “even” were about a number’s place in a sequence of “even–odd, even–odd” consecutive numbers. We add sequence-based realization leaves 2a (for even) and 2b (for odd) to the grove and show Danny’s realizations as stemming from a different arterial branch: 2. Furthermore, Danny’s justifications for “odd + odd = even” and “even + odd = odd” show that he managed to reduce sequence-based realizations of infinite odds and evens numbers to two finite sets—numbers that end in 1,3,5,7,9 (for odd) and 0,2,4,6,8 (for even). His demonstrations for the sums of the last digits in odd and even numbers suggest more generalized realizations of all naturals being reduced to 10k, 10k + 2, 10k + 4, 10k + 6, 10k + 8 for even and 10k + 1, 10k + 3, 10k + 5, 10k + 7, 10k + 9 for odd. Accordingly, we add further leaves for generalized realizations of even (2a(gen)) and odd (2b(gen)) on the grove and show the advanced hierarchy of these realizations via the vertical depth of their positioning.

Illustrating the affordances of the Grove of Realizations

Having exemplified the method of coding and constructing Groves of Realizations, we now move to show the main affordances of the tool.

Visualizing gaps in communication

When asked to classify numbers, all four students agreed which numbers were even and which ones were odd. Indeed, if the students had been asked to classify without being pushed for substantiations, one might have prematurely assumed an overlap in their mathematical objects. However, Episode 1 shows the emergence of a communicational clash as the students substantiated their classifications.

Episode 1 The communicational clash
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In this episode there are several instances of Danny communicating his disagreement with the other students: he questions [34; 58; 60; 89], refutes [32; 34; 79], and shows exasperation towards [60] their substantiations and provides alternative substantiations for classifying numbers as odd and even [79]. Commognitive analysis of the collected data revealed discursive gaps between Danny and the other students (Zara, Jane, and Robert) both at the object-level and the meta-level. These two types of gaps are described next and visualized in the grove in Fig. 5, which shows substantial differences in their realizations.

Object-level gaps

The origins of Danny’s branches, stemming from the arterial branch 2 within the grove to leaves 2a and 2b, illustrate the object-level difference between Danny’s sequence-based realizations and those of the other group members. Jane, Zara, and Robert’s realizations all stem from the structure-based arterial branch 1, and the dominance of structure-based realizations of odd and even numbers within the group is visually illustrated by the thickness of this branch made from several student-specific branches. Jane, Zara, and Robert shared structure-based realizations of odd and even, meaning that they could “talk the same talk” as one another. In contrast, Danny’s realizations are isolated from the others: only a single branch (his) is attached. Representing this discursive divide using the Grove of Realizations tool helps to visually expose the groups’ object-level discursive gap.

Meta-level gaps

In terms of the students’ pre-existing discourses, the data showed that Danny was operating on a higher discursive level than the rest of the group. While the rest of the group was mostly concerned with specific numbers and their discourse rendered them short of discursive tools to substantiate universal narratives, Danny operated with more general terms (e.g., “even numbers”, “odd numbers”, “sequence”, “pattern”). Correspondingly, Danny already had appropriate discursive machinery to account “odd + odd” and “even + odd”. Accordingly, the grove visualizes the meta-level difference between Danny’s discourse and that of the other three students’ discourse via the depth to which the branches in their realization trees extend. Danny’s generalized realizations of even (2a(gen)) and odd (2b(gen)) are represented as extensions of realizations 2a and 2b in Danny’s tree within the grove (Fig. 5), whereas Jane, Zara, and Robert’s branches do not extend to generalized realizations of odd and even. Hence, whereas object-level gaps in communication are visible through branches to different realizations of the same level (shown horizontally on the grove), meta-level gaps are visible through hierarchy (shown vertically on the grove).

Mapping discursive development

As indicated in the “Commognition in a snapshot” section, observing an expansion of realizations for an individual signals learning at the object-level. This expansion would be evidenced by an individual’s endorsement of new narratives about these objects and modifications in their routine ways of substantiating these narratives (e.g., using different keywords or visual mediators). On the Grove of Realizations tool, this would appear as the (horizontal) growth of new branches to new realizations. Meta-level learning (for example, leaners shifting from empirical to deductive discourse on natural numbers) would be evidenced by a change in routine whereby learners had objectified “odd” and “even” as signifying general terms or mathematical objects in their own right (e.g., as algebraic expressions 2n+1 and 2n). This would appear as the (vertical) extension of existing branches on the Grove of Realizations. We now present two episodes to show the students’ discursive developments within group work: (i) Jane and Zara’s metal-level learning and (ii) Danny’s object-level learning.

Jane and Zara’s meta-level learning

Episode 2 captures a development in Zara and Jane’s substantiation procedures. Jane chooses a numeric example (Numicon 3-tile) to “add to” the odd generic strip and, by running her finger around the exterior of the latter, she visually mediates the generic “even” rectangular shape created by the two odds. In turn, Zara introduces the generic word “square” [341] and instantly replaces it with Jane’s “rectangle” [342]. These words and drawings point to a degree of “objectification” of the discourse—replacing talk about processes with talk about objects: the objects (even and odd) have changed from specific numbers to generic shapes. This is a key step: to be able to objectify odd and even as abstract entities requires a means to endorse odd and even numbers as being conflated into one all-encompassing generic unit (see “Commognition in a snapshot” section). Zara’s narratives underscore that the structure of the sketched rectangle determines its parity (i.e., “if you have like two circles on each side” and “if you added an extra one”), when the precise number of paired circles is irrelevant and can “just keep on going down”. This may be interpreted as a marker of generic talk on numbers’ parity.

Episode 2 Jane and Zara substantiate “odd + odd = even” using new discursive tools
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The growth in Jane and Zara’s realizations of odd and even is shown visually as growth of leaves on their trees of realizations (Fig. 6). Both of their trees now have newly-grown branches that extend vertically to generalized realizations of odd and even (realization leaves 1a(gen) and 1b(gen)). Jane and Zara’s trees of realizations show that meta-level development took place: the girls now have discursive tools to account for the signifiers “odd” and “even”.

Fig. 6
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Jane and Zara’s meta-level learning

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Object-level learning

To illustrate how object-level learning is visualized by the tool, we turn to Danny and the growth of his realization tree.

Episode 3 shows expansions in Danny’s realizations for odd and even. Danny now endorses sharing or “divided by two” [493] and “multiple of two”, substantiations of even “like two, two, two, two, two, two, two” [649], and a rectangle-like structure to visually mediate evenness [608; 620]. Similarly, Danny’s “one sticking out” and “always a gap” [598] substantiations of odd, and the figures “like a gun [620] he uses to visually mediate odd, show his realizations of odd are of the form 2n + 1 [620] and 2n + 1 [649]. In essence, Danny has encapsulated his “last-digit” realizations of even (10k; 10k + 2; 10k + 4; 10k + 6; 10k + 8) and odd (10k + 1; 10k + 3; 10k + 5; 10k + 7; 10k + 9) into realizations that are also structurally different: even are symmetrical and odd are asymmetrical. Accordingly, there is now evidence to put a branch between Danny’s generalized sequence-based realizations and generalized structure-based realizations in the grove of realizations (shown by a solid green branch from 2a(gen) and 2b(gen) to 1a(gen) and 1b(gen) in Fig. 7). The horizontal growth in Danny’s tree of realizations via the additional green branches to the new leaves within his tree of realizations mark learning for Danny at the object-level.

Episode 3 Danny’s “Structure-and-Sequence-Based” Routine for “Even + Odd = Odd”
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Fig. 7
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Danny’s object-level learning

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Showing whether and how discursive gaps within a group of learners are bridged

Monitoring growth in groves makes it possible to observe whether and how learners within a group bridge discursive gaps (see Fig. 8). While the “before” Grove of Pre-existing Realizations (Fig. 5) visualized the discursive gaps between Danny and the other students; now, the “after learning” Grove of Realizations visualizes the nature of the learning made by the group’s members. Jane and Zoe have extended branches to generalized realizations of even and odd in the form of 2n and 2n + 1 (leaves 1a(gen) and 1b(gen)) and Danny has newly-grown structure-based and generalized realizations of even and odd (i.e., to the same leaves). Namely, it shows that Jane and Zara’s meta-level learning and Danny’s object-level learning enabled the group to bridge their gaps. At this point in the story, these three students were able to talk with similar discursive tools and according to the compatible rules. In other words, there are evidence to suggest that their discourses had reached a point of commensurability.

Fig. 8
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The “after learning” Grove of Realizations

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Summary and discussion

The adage “One picture is worth a thousand words” conveys that complex and sometimes multiple ideas described verbally can be communicated more efficiently by a single still image. While this old wisdom has been corroborated in numerous studies, psychological research also suggests that it is a combination of words and images that leave a greater mark in the longer run (see studies on the Picture Superiority Effect, e.g., Paivio & Csapo, 1973; Nelson et al., 1976). This is the analogy that was on our minds when introducing the Grove of Realizations as a tool to complement typical commognitive analyses. Indeed, prior commognitive studies demonstrate myriad details that emerge when dissecting mathematical discourses of individual students and dyads (e.g., Lavie & Sfard, 2019; Kontorovich et al., 2019; Kontorovich & Greenwood, 2022; Steiner, 2018). In our larger research with numerous groups of students, the number of details expanded even more (Knox, 2021; Knox & Kontorovich, 2022). Thus, the Grove of Realizations was conceived to generate eloquent visuals that supplement expansive findings.

In this paper, we used data from a single group of young learners to both exemplify the construction of their grove of realizations and illustrate the affordances of the tool for (i) visualizing gaps in communication between group members, (ii) mapping their discursive development, and (iii) showing whether and how discursive gaps within a group of learners are bridged. Regarding our first aim, we expanded Sfard’s (2008) notion of a commognitive conflict to distinguish between communicational clashes and discursive gaps. Where the former term describes disagreement that interlocutors are cognizant of (but may remain incognizant of the reasons behind it), the latter describes differences in words (and their uses), visual mediators (and their uses) endorsed narratives and/or routines which are researchable via tracing their signifiers and realizations. In terms of showing where effective communication and clashes in communication occurred, we showed how, by mapping the extent of students’ pre-existing realizations, the tool helped reveal object-level and meta-level discursive gaps within the group. Thicker branches to realizations showed which students shared realizations, whereas isolated thin branches to realizations showed which students held divergent realizations. In doing so, the tool helped clarify the effectiveness of the group’s interpersonal communication and visualize potential sources of communicational conflicts. Once the grove was constructed to show the student’s pre-existing realizations, it provided a baseline for tracking the development of each student’s discourse or the lack there off. New growth of branches to additional realizations (horizontal growth of branches on the tool) showed learning at the object-level while extensions to existing branches in the students’ trees (vertical depth of branches on the tool) showed learning at the meta-level. It was then possible to visualize the bridging of the group’s discursive gaps in the grove via new connections between students’ once-isolated branches at shared leaves of realizations.

We hope that the visual affordances of the introduced tool will assist commognitive researchers to share their research findings with a broader community. Indeed, a glance at the visuals may be sufficient to notice whose take on mathematical objects in the group was richer, where the group members’ understandings overlapped and where they diverged. Juxtaposing groves from “before” and “after” a collaborative activity foregrounds the change in members’ discourses, when horizontal and vertical developments correspond to object-level and meta-level learning. The notions of learning, objects, realizations, and such, have very specific commognitive definitions, which makes them challenging to communicate quickly and efficiently to researchers who are not fluent with this framework. Indeed, Sfard (2008, 2019) argues that commognition is a research discourse that is set aside from other theoretical frameworks. Then, metaphorically speaking, visuals may act as a bridge on which newcomers and visitors to this discourse could step to get a closer and more meaningful look at commognitive analyses and findings.

The Grove of Realizations enriches the commognitive apparatus by expanding Sfard’s (2008) tree of realizations. This expansion is in tune with contemporary calls for student-centred and communication-based instruction (e.g., Hill et al., 2019; Hunter et al., 2020; Lefstein & Snell, 2013; Mercer, 2002; Mercer et al., 2019; NCTM, 2000; O’Connor & Michaels, 2019; Topping & Trickey, 2015; Sofroniou & Poutos, 2016; Webb et al., 2019) and corresponding instructional practices in primary-school classrooms where students are often invited to work on mathematical tasks in small groups. In this way, the Grove of Realizations appears as an adaptation of the original Sfard’s tool to contemporary teaching practices with young children.

The Grove of Realizations addresses some of the limitations of Weingarden et al.’s (2019) RTA. Indeed, the introduced tool provides information about the students who generated a particular realization as part of their discursive repertoire. Accordingly, a grove of a group where all the realizations come from the same student will look differently from the one where several students contributed towards the emergence of the same set of realizations. This feature appears especially useful to assess the mathematical contribution of each group member to the discussion.

Let us conclude with five limitations and directions for further development of the Grove of Realizations. First, so far, we have used this tool in the single context of number parity with seven groups of 8-year-olds. Thus, we consider the tool as a work-in-progress and expect it to develop through usage in future studies. This line of studies is also crucial to evaluate whether the discussed affordances of the tool come into existence. Second, in our larger research, the tool visualized the realizations of four-member groups rather neatly (for examples, see Figs. 5, 6, 7, and 8). Yet, we do not discard the possibility that in larger groups or in groups with exceptionally rich and diverse realizations, growing groves may become too unwieldy and the resulting visual may become too confusing. Third, generating groves requires researchers to make nontrivial analytical decisions, such as whether similar but not identical realizations should be collapsed into the same leaf. Making these decisions inevitably yields a loss of information. For instance, being mostly concerned with similarities and differences between the realizations, the groves are silent about how frequently a particular realization re-emerged in the group discussion. In this sense, the generation of a grove appears not very different from any analytical endeavour, where researchers decide on what deserves to be taken into account and what can be omitted. These decisions should be informed by the aim and questions of the study (cf. Bikner-Ahsbahs & Prediger, 2014; Mason & Waywood, 1996), shaped by the intended usages of the generated visuals and communicated to the reader. This conception of the tool leads us to the fourth point. When introducing the Grove of Realizations, we envisaged verbal utterances as the primary source of students’ realizations. In relation to the focus of the Special Issue, this vision may better speak to research with older children. That said, we acknowledge Rodney (2019), who grounded her commognitive approach in the sphere of gestures to explore mathematics learning of a 5-year-old. Accordingly, further research with younger children may consider adapting the tool to account for both verbal and non-verbal realizations.

The fifth point relates to Robert—the fourth member of the group in the focus of our paper. Readers may well be wondering what course his learning took and why his student-specific branches feature so rarely in our groves. By commognitive underpinnings, researchers study human thinking and learning through attending to publicly accessible communicational patterns of a discussant. And since Robert remained largely passive throughout the session, little can be said about his take on odd and even numbers. However, the absence of a student’s contributions does not necessarily imply that he did not engage in mathematical discourse, for instance through self-communication (thinking). This is in tune with another adage—“Absence of evidence is not evidence of absence”.