Introduction

‘Exploring data, especially tangible and contextualised data from areas of interest to the specific audience, and building upon the intuitive knowledge of young learners, are examples of key aspects that will facilitate a trajectory for embracing big data.’ (Howley et al., 2021, p. 42).

As data science processes become essential to automation processes in all areas of life, education systems must also be responsible for introducing their students to data science perspectives that can promote meaningful and innovative problem solving in the classroom, even in the early years. Engaging students in statistical investigations involving STEM contexts is one way to highlight meaningful application of data science and statistical skills to students, in contexts that are part of real life. Being data literate is important in this digital world more generally, to not only understand data we encounter but also to understand how one’s information is translated through datafication (Pangrazio & Sefton-Green, 2020). Progress has been made in supporting teachers to equip young people with developing the skills they need to be data literate (Gould, 2017; IDSSP, 2019; Wolff et al., 2016), including experiencing messiness when working with complex data in meaningful contexts through statistical investigation.

Managing the messiness of real-world problems involves skills that primary children typically do not encounter when solving problems using mathematics and statistics in school. We were interested in providing children with experiences that reflected the authentic work of STEM professionals as they might initially generate models from contexts involving messy data. The aim of this study was to better understand the changing nature of complex data in one classroom statistical investigation, which involved connections between complex mathematical relationships, by examining the ways communication processes supported students to understand and meaningfully work with that data (as a concept). Cognitive tuning (Wit, 2018) provided insights into the classroom communication processes between the teacher and her students as she scaffolded the statistical investigation.

The classroom illustrations depicted in this paper are part of a statistical investigation to generate a model to predict the amount of fruit a passionfruit vine might produce. The class partnered with a STEM professional who was an agricultural scientist. His visit sparked an inquiry into the ways to determine how much energy a plant might absorb through its leaves to create sugar needed to make plant parts. Computer modelling used to predict crop production involves a series of algorithms, the first of which focuses on the relationship between how much sunlight a plant receives (through solar radiation, based on leaf area), measured as megajoules per square metre. To predict crop production, agricultural scientists use their contextual knowledge to select key elements for their model, one of which is the relationship between the length and the width of a leaf to estimate its area. The Year 4 students in this class focused on this initial aspect of the problem, to form the basis of their statistical investigation into How big is a leaf?; this helped them identify and generalise ways to accurately and efficiently predict the sizes of leaves on a passionfruit vine. The processes the children went through simplified that of the agricultural scientist. This investigation of leaves constitutes the first step towards creating an algorithm used by computers involving human thinking, when modelling the production of crops. Such an algorithm builds on informal inference, or the conclusions drawn through statistical inference, which extend beyond the data (Leavy & Hourigan, 2021). It is hoped that an outcome of this paper is to provide teachers a meaningful illustration of practice that highlights the kinds of communication processes teachers employ in complex statistical investigations.

Literature

Data science

More than statistical literacy, data science is distinguished by Wilkerson and Polman (2020) as ‘intensely interdisciplinary’ and embedded within social and environmental contexts, involving data of a relational nature. Data science explorations integrate statistical thinking with the content of whichever subject is being investigated. LaMar and Boaler (2021) explain how a data perspective can be taken with all subjects, and this interdisciplinary perspective will be important to support even young students to participate in their data-filled world. Attempts to incorporate data-science literacy concepts into statistics education do so to keep pace with the skills needed to engage with emerging complex data sets and new forms of data (Bargagliotti et al., 2020). Of interest to mathematics education researchers are purposeful opportunities to explore age-appropriate data science concepts with children that are authentic to solving problems involving messy data (Howley et al., 2021). In data science situations, data can be traditional, for example, distributions of measurements of leaves as a continuous quantitative variable; or non-traditional. Non-traditional data is multidimensional and may include pictures, text and gestures, which can be transformed through data moves to produce different representations of the same data (Bargagliotti et al., 2020; Erickson et al., 2019). The multidimensional aspect of the data depicted in this study is seen in the leaves the students worked with. Students were to work with different aspects of the leaves to generate data and useful patterns to learn about plant growth. An example of messy or unclear data to be found in natural settings, presented in the Pre-K-12 GAISE II Framework for Statistics and Data Science Education, was to do with ladybugs (Bargagliotti et al., 2020, pp. 31–35). Just as the pictures of ladybugs presented multiple numerical and categorical variables such as size, shape and colour, data decisions about leaves might at first seem unclear to children. In the ladybug example, students worked with secondary data in the form of pictures of ladybugs, and this may have drawn attention to features displayed by the insects. These kinds of statistical investigations involving the natural world, however, can support student development of skills ‘in manipulating and restructuring data, transforming provided variables into new variables, and querying the origins and suitability of data for the purpose at hand’ (Bargagliotti et al., 2020, pp. 9–10).

In the effort to include data science education in schools, previously established statistical investigation cycles remain relevant and are re-evaluated to incorporate data modelling and specialist data-handling skills (Noll et al., 2021; Wolff et al., 2016). At the primary level, exposure to data science concepts can support students to become familiar with non-conventional data and prepare them for exploration of more advanced data science processes, involving working with large amounts of collected data or data which are non-standard (Howley et al., 2021). Insights into classroom explorations of data science concepts can support teachers with identifying ways to integrate meaningful classroom investigations involving these data science concepts.

Data modelling and conceptions of data

Research depicting data modelling in middle and primary school classrooms focus on the construction and use of data. For instance, Doerr and English (2003) report how students generated a list of factors to be considered when deciding which pair of sneakers to buy, applying a frequency-based strategy to rank the factors identified. These authors also describe a range of data modelling experiences set in authentic contexts for children aged 10 to 13 years of age, to extend ideas about quantitative and non-quantitative data. Although English (2009) explains that modelling that focuses on mathematising realistic situations often goes beyond what is taught in traditional mathematics, the operations to do with data that she describes are applicable to statistical thinking processes. Specifically, these operations involved data collection processes, representation of data and interpreting tables of data. Classroom activity sequences involving model-eliciting activities and subsequent exploration and application activities, set in meaningful contexts, can bring focus to central statistical concepts (Doerr et al., 2017; Gal, 2019). The early introduction of data modelling ‘provides the important foundations for dealing effectively with statistics that govern our world’ (English, 2021, p.8).

In this article, we focus on the characteristics of data as relational and embedded in environmental contexts, to guide the selection of data science experiences. Prior to the classroom investigation, little teaching of statistics had taken place in the classroom. Just prior, however, students had explored ways to represent plant growth (of a capsicum https://youtu.be/gRS80BqZ0dc) using a height-time graph to tell the story. This was to introduce the idea that plants make plant parts. Due to the age of the students, and the importance of guiding students through all stages of a statistical investigation, this paper focuses on the developing conceptions of data, from the leaf on the passionfruit vine, to the related measurements of leaf size, transformed to include the abstract generalisations students formed which included proportional relationships between leaf sizes (see Fig. 1 in “Methodology” section).

Fig. 1
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Developing conceptions of data

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Problem solving

Although the previous sections describe processes of coming to know, or understand, a situation, it is important to further articulate the nature of statistical problem-solving and data modelling involved in this study. In the approach used by Doerr et al. (2017), modeling activities value ‘the problems faced by the modeler, the contexts being modeled, and the modelers themselves’ (p.90). The activities described by Doerr and English (2003) were intended to be non-routine, supporting students to build explanatory relationships between the data they created, operated on and transformed. Similarly, the investigation into How big is a leaf? was problematic to the students in many ways. The students did not know how to determine the amount of sunlight a plant might get to produce plant parts such as flowers and fruit. Once leaves were identified as the part of a plant that gathered light energy in lesson one, students were interested in knowing the direction of the sun, whether the plant was getting enough sunlight, where the plant was situated in the garden and how this might affect the amount of sunlight it would get. Schoenfeld (2016) articulated the importance of problems being problematic (when the problem solver does not know how to proceed), even to mathematicians when they approach problems as a ‘guessing game.’ The lesson sequence described in the “Methodology” section of the paper reflects the careful structuring by the teacher to not reveal a solution process. Although the teacher in this study posed the investigation question, the challenge was in structuring the investigation to guide students towards the data-driven statistical processes used in agricultural science contexts to establish proportional relationships between leaf measurements, to do with plant growth. Such an investigation exposes young students to data science processes relevant to modelling used by agricultural scientists in authentic contexts.

Theoretical framework: cognitive tuning

Important to successful problem solving in the mathematics classroom can be collaboration between students, with an emphasis on communication and mathematical reasoning (Fielding-Wells, 2014; Goos, 2004; Makar & Allmond, 2018). Although students may independently record and represent their ideas and solution processes, in investigations involving inquiry, much problem solving is completed by students in small groups, reflecting the sociocultural approach (Allmond et al., 2010). Wit (2018) recognised the importance of communication in group work and applied the process of cognitive tuning when looking at interactions between committee members in task groups, when making progress towards a socially anchored representation or understanding of a task. Cognitive tuning explained the communicative processes involved in arriving at a common understanding and acknowledged the existence of strong normative conformity pressure. Communication processes involving an element of ‘pressure’ seemed to support the nature of the statistical investigation outlined in this paper which was underpinned by an inquiry approach (Makar & Allmond, 2018). Such a focus may emphasise the important role of the classroom teacher in scaffolding or guiding student problem solving in these settings. Few applications of cognitive tuning by Wit (2018) were found in mathematical research. In analysing emerging norms and social interactions within groups working on challenging mathematics tasks, Kontorovich et al. (2012) used Wit’s earlier work to describe group dynamics and interactions. Whereas Kontorovich et al. (2012) used the framework to understand between-group differences, the intention of this work is to characterise the communication processes in a statistical investigation, focused on data.

Three basic modalities are offered towards building a commonly shared frame of reference: normalisation, conformity and innovation—through ‘group members’ attempts to form, maintain and change their common frame of reference, respectively’ (Wit, 2018, p. 378). At the moment of normalisation, when the task is introduced, participants do not yet have a shared interpretation (of the task); a normative frame of reference is thus required for the group to work together. Previous experiences of participants in a group can influence the communicative process of cognitive tuning. For instance, an inexperienced participant may feel unsure about their own ideas and experience cognitive conflict when they face conformity pressures from the majority of the fellow members who share a common belief. However, an individual can resist the pressure, and so, conformity in a group is not always easily achieved. When members of a minority do challenge a group norm held by the majority, innovation can result. Wit’s (2018) framework highlights that innovation relies on achieving some level of conformity in order to have a benchmark to compare the novel idea proposed. Conformity is essential to authentically achieving consensus (as opposed to reaching consensus by exertion of authority), and ‘assumes prior normalization’ (Wit, 2018, p. 378). Cognitive tuning seems to align well with a knowledge building culture such as inquiry, where peers and teachers can influence those in the group who are seemingly unsure, to conform to their own solution processes/ideas. When a dissenter’s voice expresses an opinion that is different to the majority pressure, a new innovative frame of reference can be introduced. ‘Productive learning dialogues require students to mix individual thinking (to form voices) and interaction with others (to change voices)’ (Abdu et al., 2021, p. 2).

Research into mathematical inquiry classroom contexts is framed by socio-constructivist perspectives (Goos, 2004; Hunter & Hunter, 2018; Makar & Fielding-Wells, 2018) with analysis expected to take into consideration the very practical nature of the research as it is conducted in school communities. In this exploratory case study, a qualitative approach is used to consider the two questions guiding our research; (1) How can a teacher’s communication processes support students’ emerging understandings about the abstract concept(s) of data? (2) In what ways can cognitive tuning offer insights into the statistical thinking processes students use in the primary mathematics classroom? This paper will look more closely at how a teacher’s communication processes supported her year 4 students’ developing conceptions of data to solve a STEM problem authentic to agricultural science. Cognitive tuning (Wit, 2018) presented a useful framework for characterising the kinds of dialogue used between students and their teacher, as they engaged with non-traditional forms of data.

Methodology

Participants, background, and context

The students were in year 4 (8–9 years) at a metropolitan public primary school in Australia. There were 26 students in the class (12 girls and 14 boys) — all with different learning needs, special needs and achievement levels. Four students who did not consent to participate in the research still engaged in the learning experience. However, they were not filmed, nor their learning efforts captured to respect their wishes. The first author co-taught the class and worked with the classroom teacher to plan the inquiry based on conversations with their STEM professional, an agricultural scientist from a government scientific organisation in the area.

The agricultural scientist and the teachers had met three times (once face-to-face and twice over the phone) to discuss the key mathematical concepts involved in the modelling of agricultural scientists to predict crop size. Briefly, this included a need to measure the area of leaves as a relationship between the length and/or width of a leaf and its area (in crop plants). A passionfruit vine on the school ground provided the opportunity for the class to explore this relationship. However, with passionfruit vines, this relationship is different for differently shaped leaves (leaves with one, two or three lobes) but constant for a given leaf shape. Previous experiences with comparing and ordering shapes and objects based on area in year 2 (age 7) involved using uniform informal units; in year 3, centimetres and metres were formally introduced to measure length. At this grade level (year 4), students were expected to be introduced to familiar metric units of area and to compare areas of shapes by informal means (ACARA, 2010). The teachers saw an opportunity for their year 4 students to informally calculate the area of leaves of a passionfruit vine found on the school grounds to meet this curriculum requirement. The students had not been formally introduced to processes for calculating area, and the teacher requested that the visiting scientist not introduce this term to the class when discussing how plants get their energy from the sun to create ‘plant parts.’

With a focus on leaf size, the unit consisted of 8 lessons (L1-8, 60–90 min each) and a further visit by the STEM professional. Although the lesson sequence was drafted before it was taught, the specifics of each lesson were formulated after the previous lesson and adapted in response to student thinking. A key focus of the lesson sequence was the design of experiences that would support students’ developing conceptions of data (see Fig. 1).

Lesson 1 Building context and responding to the provocation left by the visiting scientist, ‘The more light the plant gets, the more sugar it makes. The more sugar it has, the more plant parts it makes.’ Formulation of the inquiry question How big is a leaf? Considering ways to measure ‘big.’

Lesson 2 A focus on exploring ways to measure surface area

Lesson 3 Introducing the unit cm2 and estimating area by counting square centimetres using grid paper

Lesson 4 Developing a relationship between length and width using a rectangular frame around the leaf. What fraction of the box (frame) is covered by leaf?

Lesson 5 Creating fractions to represent relationships. Using a common denominator to make fractions easier to compare

Lesson 6 How are the fractions, representing the relationship between the leaf area and the area of the box around it, similar or different? Finding errors in measuring area.

Lesson 7 How can we organise the fraction data in a useful way to aid interpretation, and to find patterns to generalise from?

Lesson 8 Making predictions from the data. Forming statements about predicting the size of a leaf

Data collection and analysis

The exploratory case was the first of two classroom units in a post-doctoral, design research (Cobb et al., 2003) study. The study data that were collected consisted of digital videos of the sequence of eight lessons; field notes regarding the visit by the STEM professional who partnered with the class and discussions during teacher planning; research journal reflections and classroom artefacts including student notebooks. The first author videoed each lesson using one camera accompanied by a small blue-tooth microphone to focus on the teacher’s voice and to capture instances when the teacher joined student conversations in table groups. The microphone was also repositioned to capture discussions between students in small table groups (up to 6 students) as they progressed in problem solving. When whole class teaching and conversations took place, the camera was focused on the classroom, placing the microphone close to the teacher to capture her voice. When students worked in small groups, the microphone was centrally positioned to capture these discussions with the camera refocused on the student speakers. Small group work was common so different student conversations could be captured. Video data with accompanying audio depicting teacher communication processes were analysed using the theoretical framework of cognitive tuning to illustrate how one class built common understandings about the concept of data, emphasising the classroom’s communicative processes of normalisation, conformity and innovation (Wit, 2018). Teacher dialogue reflecting normalisation and conformity about the concepts of data was used to explore how the processes of cognitive tuning made the connections more visible between the teacher’s support and the students’ emerging ideas.

Specifically, the process of analysis followed an adaptation of Powell et al. (2003), who provide a model for analysing video data, designed for work in mathematical learning communities to inquire into students’ mathematical activity. The approach supported the process of analysis of classroom videos by providing opportunities for the researchers to revisit recorded events, offering a glimpse into the non-linear learning progression of students anticipated in inquiry settings. First, the authors viewed the video sessions to familiarise themselves with the lessons and to briefly describe the content by creating video logs. In videotape methodology, critical events are those moment of significant change, or possibly those visible and audible moments that reflect conceptual leaps from earlier understandings (Powell et al., 2003). In this study, critical events were identified where the authors could see potential examples related to cognitive tuning during discussions about area, proportional reasoning, and data to solve the problem. This included both discussion between students and between the teacher and her students (whole class or small group). The critical events presented in the findings highlighted instances of were crucial to consider in terms of Wit’s (2018) cognitive tuning processes. Transcriptions of these critical events were coded and annotated for themes based initially on the three modalities of cognitive tuning, and patterns, enabling the researchers to make sense of the data and to construct a basic storyline by selecting excerpts that illustrated the key ideas efficiently. This included consideration also of how the excerpts revealed students’ statistical thinking processes. The narrative below is composed of the storyline revealed through analysis of the classroom data.

Findings

We share excerpts from lessons in this investigation in which students were wrestling with an idea related to the data being explored, and the teacher supported them in progressing their ideas. In the first set of excerpts (from Lesson One), students are seeking a method for finding the size of a leaf. Each student received a different leaf to study, with either mono-, bi- or tri-lobes. Although students investigated the properties of their own leaf, they collaborated in table groups on solution processes. In the second set of excerpts (from Lessons Six and Eight), we focus on how students were making sense of their leaf data. By this stage, students used information about all the leaves in the class. In each case, we also focus on how the theoretical concepts of cognitive tuning helped to understand how the students were supported to move their ideas forward in the problem-solving context.

How can we find the size of a leaf?

In order to address the inquiry question, How big is a leaf?, the students worked in groups to explore how they might devise a way to find the size of their leaves. In open-ended problems set in a context, there are a number of elements involved, and it is not always clear to students which elements are relevant and which are irrelevant. In this case, the teacher has to make a call whether the elements students are focusing on are worth wrestling with, or if they need to be redirected. For example, in the excerpt below, Ella was concerned that the photocopy of their leaf (see Fig. 2) may have distorted the size of the leaf (a reasonable concern).

Fig. 2
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Coloured photocopy of a tri-lobed leaf from a passionfruit vine, including some annotations of length

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1 Teacher Can you tell everybody what you just said Ella?

2 Ella Well, the leaf might not be as it is actually …

3 Teacher When you do a photocopy of a leaf, like this one here that we just did, the leaf is exactly, the photocopier will make it exactly the same size if you don’t, if you don’t enlarge it or shrink it. When you said take a photo, that’s different. But a photocopy is exactly the same size as the leaf.

4 Student They’re copied

5 Teacher Damara?

6 Damara What if it’s crinkled, like Daniella’s? … It’s like crumpled, so it is not exactly the same.

7 Teacher I know. Even if it’s slightly crumpled you’ve just gonna go with the surface area you’ve got. (01:11:30, Lesson 1)

It would be ambitious for students at a year 4 level to explore the range of concepts required to predict crop yield. However, through normalisation, the teacher guided her students to focus on exploration of one element, the surface area of a leaf, to build a shared frame of reference around the question How big is a leaf? From this excerpt, we see that one element of normalisation in a task is sorting through the elements that are the focus of the investigation and those that divert from the purpose of the task. In this case, the teacher made a call not to encourage students to explore the impact of the photocopier or crumpling of the leaves. The teacher anticipated that these elements would be distracting and not progress students’ ideas in line with the goals of the investigation, so she simply stepped in and redirected their attention. This is an important skill in managing students’ ideas in exploring an open-ended task, some of which are unanticipated. Although Wit (2018) describes strong group members exerting pressures to ensure a group arrives at a common understanding, cognitive tuning in this mathematics inquiry classroom is reflected in the decisions made by the teacher to guide her students to converge on a common frame of reference. In this instance, a focus on calculating leaf area is supported by the teacher’s normalisation processes, to bring focus to the part of the plant absorbing sunlight. The crumpled leaf presents a cognitive conflict to Damara, and the teacher reassures her that the photocopy is the same size.

As students began working in their groups, they were trying to work out how they might find the size of the leaf. Liam, Kyrie, Isla, and Helena were looking at their photocopy and trying a few ideas.

8 Liam Wait! (He puts his hand in Kyrie’s desk and they pull out a plastic ruler.) …

9 R (Researcher asking Liam to think aloud his idea) What are you thinking of?

10 Kyrie Well, me and Liam are thinking of, well how can we measure it because it’s curved. (Kyrie is trying to place his ruler on the edge of the leaf. Liam takes Kyrie’s ruler to try).

11 R Oh, it is curved. …

12 Liam They’re curved.

13 Kyrie Well, that makes it really hard.

14 Liam Wait!

15 Kyrie Have you got something curved in your desk? A piece of paper or something? (Liam shakes his head but opens his desk to take a look. They rip a small corner from a page in their book, measure the scrap of paper and then line it up with the edge of their leaf.) So that’s one cm, one cm, another cm, one cm. 3cms, 3cms. (Kyrie continues counting in centimetres around the perimeter of the leaf).

(01:14:37, Lesson 1)

In this excerpt, Liam and Kyrie are working together to explore an initial idea for finding the area of the leaf. They have no clear consensus so far on the approach they will take. They use a ruler to measure the leaf and soon recognise a problem. How do they measure a curve? This suggests that their preliminary idea may have been to consider the perimeter of the leaf rather than the area. Although this pair is not yet clear on a direction, they appear engaged in exploring possible approaches. This suggests that they were comfortable with engaging in productive struggle (Warshauer, 2015), where they are given time and support to explore their own ideas. Wit (2018) argues that having a common social frame of reference for working together is a critical foundational aspect of effective collaborative work. We therefore consider this initial time and routine of engaging in productive struggle provided by the teacher a second element of her normalisation of the task. Although there is no audible communication by the teacher, the routine of working in this way implies a knowledge-building culture has been established and that she is communicating through her expectations and time for exploration.

As the students were working, the teacher had been circulating and listening to students’ progress. She pauses the class at regular intervals to conduct a ‘checkpoint’, where students provide an update on their progress.

16 Teacher (Teacher claps to pause the class.) Ok, if you’re measuring the surface area of that leaf, how – has anybody got any idea how they’re going to do it? Anybody? (Points to Daniella, Sandy and Will’s group) Um, Daniella? No? Sandy? Can you tell me how your group was talking about that?

17 Sandy Hmmm. I’m not sure.

18 Teacher Ok, what about you, Arlo? Are you thinking how you were going to do that? (no response) Has a group come up with an idea on how they’re going to actually measure the area of that leaf? Do you need another 2 minutes? (Class agree). Ok, two minutes then I will pick anyone in your group that I like to report back to me.

(01:16:47, Lesson 1)

It may have been more efficient for the teacher to step in at this juncture and tell the students how to progress, but she chose to give them more time to wrestle with the problem first. By doing so, she continued to show them that she valued their ideas. The pedagogical tool of a checkpoint allowed the teacher to maintain momentum and accountability, value students’ interim ideas, and normalise productive struggle. The process of the teacher listening to groups and pausing to check in and share progress could also be considered her tacit hand in working towards conformity of the task. In the process of conformity, there exists cognitive conflict as individuals’ personal frames of reference on a potential solution differ and, in this instance, students had very different ideas about how to progress. Once conformity is achieved, this will enable the teacher to progress to the next milestone of the lesson, finding a way to calculate an irregular area. Isla and Helena, sitting nearby, suggest to Liam and Kyrie their ideas for finding area.

19 Isla [I would] measure the stem, from here to here (she points to the bottom of the stem to the top of the leaf’s (furthest point).

20 Helena (inaudible, demonstrates she is suggesting measuring across the leaf also, the width)

21 Liam Don’t we have to measure the whole thing? (Moves his hand across the surface of his tri-lobed leaf.) I think we should measure that, and that, and that (Points from the centre of his leaf along each of the three lobes).

22 Isla (points to the lobes of her leaf in agreement) That and that and that?

23 Liam But how can we find around this bit? (Runs his finger along the curved edge)

24 Isla I don’t know.

25 Helena I’ve got this idea, if we just measure across the middle. We should measure it through the middle.

26 Isla Yeah but if you want to know (points) there to there then you have to measure it that way (points from the bottom of the stem along to the end of a leaf lobe). Because then it’s not the actual leaf. (01:18:39, Lesson 1)

To make progress in the problem-solving aspect of the task, conformity around a conception of surface area was essential. Although Liam’s response (moving his hand across the surface of the leaf) suggests he was also thinking of area, he then pointed to the lengths of the lobes of the leaves, partially swayed by Isla and Helena’s suggestions. Liam recognised that more of the leaf than the length was needed to address the area, so looked for additional linear measures to add to the single length. However, the curvature led him back to perimeter, pointing around the curved edge of the leaf.

Helena’s contribution demonstrated her own awareness of two-dimensional shapes (where both length and width were relevant), suggesting that as a group the four of them were working towards conformity. Their capacity to do so relied on their willingness to actively listen to peer ideas and compare them to their own to collectively move forward (implying evidence of normalisation), another collaborative skill that was likely developed over time, but they were still in a stalemate. As the group was working towards converging on an idea to measure the lengths of the three lobes of the leaf, Helena pressed the group again to consider the width. In doing so, she is challenging their emerging conformity to suggest an innovation in an attempt to influence the direction of the group. This is another indicator that a classroom culture that the teacher engendered valued such intellectual risks. At this stage, a teacher would often generate conformity through authority by directly instructing students how to progress. Yet, in this case, the teacher resisted.

Although the teacher is not present in this episode, there are inferences to be made from the students’ discussion. The students demonstrated skills in working collaboratively by responding to one another’s ideas without the teacher present. Their own process of negotiation and wrestling suggested they were seeking a common understanding of how to approach the task. The scaffolding that would have gone into enabling students to collaboratively negotiate a shared interpretation of the task as well as the time the teacher provided for them to struggle and engage in this negotiation could also be considered her tacit hand in normalisation of the task as well as supporting students to make progress towards conformity of a direction to pursue. Without these two elements (scaffolding to collaboratively negotiate with peers, and time provided that values this negotiation and struggle), normalisation of the task by the students would not be possible. Without this normalisation, there would also be no possibility for students to contribute to conformity (Wit, 2018). And without conformity, there is no option of innovation.

By the end of the lesson, collectively, students had generated multiple approaches to finding the surface area of their leaf including measuring its length and/or width, measuring around the outside of the leaf with flexible rulers or bits of torn paper, adding up or multiplying various linear measurements (e.g., the lobes of the leaf). Through generating these ideas, their productive struggle sensitised them to the challenge of not knowing how to measure area, the ‘inside space’ of the leaf. At the end of the lesson, one of the students placed a circular object on top of her leaf to partially cover its area. It was through the diversity of students’ ideas that allowed the teacher to capitalise on turning this innovation into conformity by clarifying the difference between perimeter and area and learning to use square centimetres to measure area in the next lesson.

In this first section, we observed explicit and tacit communication by the teacher reflecting Wit’s (2018) three processes of collaboration—normalisation, conformity and innovation—to scaffold students’ ideas in a direction that supported the teacher’s curricular goals. Although seemingly a classroom problem involving a mathematical focus on area, the teacher wanted to support the students with exploration of the relationship between the length and/or width of a leaf and its area and so guided the classroom investigation towards statistical processes. Early conceptions of data involved photocopied leaves and measurements of area, yet no relationships had been identified between such measures. Ideas about data needed to progress in order to identify any relationships, which could present students with a meaningful mathematical model of a biological system to explore.

Area of the leaf as a fraction

We jump forward in the unit to Lesson 6, where students are starting to make sense of their data. Prior to this, students had gone through a process of measuring the areas of their leaves by counting the number of squares (and partial squares) the leaf covered using square centimetre grid paper (Lesson 3). From one group’s innovation, students noticed how much easier it was when a frame was drawn around the leaf as shown in Fig. 3. This made the process of finding area more efficient and set a context to discuss the relationship between the leaf area and the area of the frame (Lesson 4). This deconstruction of the space into parts was non-trivial for students, who had not yet learned a rectangular area formula or how to find areas of composite shapes (both of which would appear in the curriculum later).

Fig. 3
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Using grid paper to find the area of the leaf in a rectangular frame

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The teacher took the opportunity for students to compare the leaf relative to (as a decimal fraction of) its frame, which was the way that biologists developed a partial model for a plant’s sun exposure from the length and width of its leaves. This provided an opportunity to introduce and deepen students’ understanding of decimal fractions in an authentic context. A symbolic representation of the fraction now depicted both the leaf area and the area of the frame around it, and student understanding of this data (fraction measures depicting leaves) would be critical if patterns were to be identified in the data. Normalisation, towards a commonly shared frame of reference about the newly abstracted data (concept), would be important if students were to progress in the problem-solving task. This also illustrates a mathematical shift in the investigation, from surface area to data representing a relationship, in a statistical sense. Previously, the mathematical focus was on surface area. The problem focus had also shifted to the specific task of finding the relationship between the area of the leaf and the area of the frame around it. This refinement of the problem was significant in guiding students to focus on ratio.

One issue that came up was in trying to compare the ratios of a leaf to its frame given that the leaves and frames were not the same across the class. For example, if one leaf was 146 cm2 and its rectangular frame was 248 cm2, how would you compare its ratio (146/248) to a leaf and frame ratio of 72/118? By converting the fractions to decimals (truncated to hundredths, since hundredths was aligned with the curriculum at this age), students could see that it was easier to compare the fractions (Lesson 5).

The abstraction at this point was challenging, and it was important to continually have students report what the fractions represented, allowing them to discuss and make sense of the mathematical ideas in terms of the context. For example, the teacher had students revisit what the numerator and denominator of a fraction meant within the context, and then what the fraction represented. Through these discussions, students’ observations allowed them to extend the investigation into richer mathematical territory. For example, some decimal fractions were similar even though the original leaves were different sizes. However, this was not usually the case. The variability of the fractions initiated a discussion about how to look for patterns in the data they had now collected (see Fig. 4).

Fig. 4
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Collation of data from ratio of leaf to frame

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27 Teacher When we look at data, we try and make sense of what the data means. We look for patterns. We look for things they have in common or things that are really different. Things really stand out, things that are similar. … What does this data mean? I know we’ve got all of these fractions all over the board, but what are they? And what do they actually represent? (Pulls out Timothy’s name from a box of popsicle sticks with each of the names in the class) …

28 Timothy It represents 72 out of 93.

29 Teacher What does it represent though? That’s just a number. If someone came in and said ‘Whoa! What is all that?’ What would you tell them? …

30 Rafael They’re representing how big our leaves are.

31 Teacher (Nods) Precisely what is it measuring?

32 Akayla How much of the box is taken up by the leaf.

33 Teacher Great! How much of the box is taken up by the leaf. Does this actually relate to your leaves? Is this here (points to a fraction) about your leaves?

34 Students Yes.

35 Teacher So if that’s how much of the box the leaf takes up, how can that be of any use to us? That's what we want to know. How is that useful? … Can you talk with the people in your group?(00:07:26, Lesson 6)

In this excerpt, the teacher is first ensuring that students are staying connected to the task. By asking students to interpret the data, she is seeking normalisation to ensure that the students have a shared understanding of (and recall the purpose of) the task. Before they are ready to consider the direction of how they will work with their data, conformity around the meaning of the data being explored would support students to be able to consider, discuss and generalise about the proportional relationships they could see. What is important in her questioning in this excerpt is that she recognises that solving a complex problem requires that students keep the problem purpose in mind when they develop their approach. This thread is often lost, and students can easily end up with an answer that does not address the problem (Hancock et al., 1992; Makar & Rubin, 2009).

Zack and Reid (2003) explained that when children address complex problems, they are able to work with ‘good enough information’ to maintain momentum rather than attempt to see the full solution process from beginning to end. They emphasised that although students’ interim ideas were often ‘incomplete, tentative, and sometimes inconsistent’ (p. 28), the children in their study continued to explore, clarify, question, explain, entertain multiple possibilities and discard unproductive ideas as they worked towards a stable position. In our study, the class reached a point where the complexity of the problem was masking the big picture—what problem in context was being addressed. The teacher could have told them what to do next, but instead chose to pause the class to (1) help students refocus on the problem, (2) demonstrate that she valued sense-making, (3) normalise confusion as a regular part of addressing complex problems and (4) draw on their small groups as a resource and to ensure students had an opportunity to articulate their thinking.

36 Teacher Are there any you can put together. Could you put any of these fractions together? Which are the easiest to compare? Daniella?

37 Daniella (Hard to hear, she indicates the pink ones)

38 Teacher Why is it easier to compare the pink fractions? …

39 Kyrie Because the pink ones have the same denominator. (00:14:13, Lesson 6)

As students sorted through the fractions expressed in hundredths, the teacher continued to pause and ask for their insights. Over the next lesson, they located similar fractions and compared these leaves in front of the class, worked through some calculation errors that arose and sorted their collated class data into bins from 20/100 (‘20 s’) to 80/100 (‘80 s’), grouped by tenths (ten hundredths).

In Lesson 8, students began to notice some of the data were clustered in bins. For example, Naomi had written in her book, ‘This data shows us that a lot of people have there desimal [sic] fraction between 40 and 49’ (see Fig. 5). This led to a discussion about whether they could be more specific. The teacher asked for anything they noticed.

40 Sonny There’s not as many in the 20s.

41 Teacher Can you be precise? …

42 Oskar Not many.

43 Teacher Not many, not many. [Be] precise and exact. Use numbers.

44 Sonny There’s only one in the 20s.

45 Teacher There’s only one in the 20s. That’s more accurate, that’s precise. Who [else] have we got? Oriana?

46 Oriana Um, for Daniella’s data, … she actually wrote a statement of that there’s actually more, most people are between 40 and 49.

47 Teacher Great! So that’s being more specific. There are, the most people are between 40 and 49. (00:01:38, Lesson 8)

Fig. 5
figure 5

Naomi’s data organised into clusters

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Just as the problem-solving process shifted to making precise statements involving proportional relationships to predict leaf size, so to the teacher brought continued focus to the shared understanding of the task through normalisation: that ratios could be used to determine leaf size generally and that precise statements could be made about this. The cluster of ten leaves in the 40 to 49 bin was explored, and they noticed that although the leaves were all different sizes, all of the leaves in that bin were all tri-lobed. In contrast, the leaves in the 70 s (70/100–79/100) bin were all mono-lobed. The children recorded their findings in their notebooks and prepared to report their findings to the scientist (sometime after Lesson 8) and yet found the task difficult. One statement written by Naomi was shared (below) and was pivotal in supporting her peers to generate their own responses.

Most boxes are 40% to 60% taken up by the leaf. Then you can fill the box up 40% to 60% and that could be close to the area of the leaf. But with a mono leaf, it will take up between 70% and 80%. (Naomi)

A smile from her teacher was enough ‘pressure’ to convince others with little confidence (or still experiencing cognitive conflict), to conform their thinking about relationships between the data and how to form a generalisation. Such a statement relied on the focused decomposition aspect of statistical data and patterns that could be seen, as well as responding to the specific problem of using the relationship between the area of the leaf and the area of the frame around it, to predict the size of a leaf.

Discussion

Wit’s (2018) modalities of cognitive tuning assisted in characterising the ways one teacher guided her students’ developing conceptions of data, incorporating classroom communication processes, and shedding light on how the teacher supported her class to progress problem solving in the statistical investigation. An emerging pattern of modalities of cognitive tuning (Wit, 2018) seemed to align with and potentially support students’ perceptions of the changing conceptions of data (summaries of critical events reflected in Table 1), and to abstract understandings from the mathematical topics they explored. The initial problem of predicting how much fruit a passionfruit vine might produce sparked the inquiry, yet the enormity of such a task (broad mathematical concepts) required normalisation (N1) by the teacher to bring focus to specific statistical elements of the problem related to data, the surface area of a leaf, and generating data about this. Next, conformity processes (C1) were required to shift students’ personal frames of reference of the concept of surface area, to ensure the class was discussing the same concept. Innovative thinking (In1) progressed this conformity. This supported students to reach a level of abstraction (A1) which involved representing the ratio between two different areas (leaf and frame around the leaf). Normalisation (N2) processes now involved supporting students to shift their mathematical thinking and to use the relationship between the area of the leaf and the area of the frame around it, to predict the size of a leaf. Conformity (C2) by the teacher and peers (Naomi) supported students who struggled with describing this relationship. Further normalisation (N3) processes would support students to write statements (A2, In2) including making generalisations about the data.

Table 1 Characteristics of cognitive tuning phases throughout the classroom investigation aligned with changing conceptions about data
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Similar to Wit’s application of cognitive tuning in a group setting, the classroom depicted in this study was considered the ‘group’. Wit described strong normative conformity pressure as essential for success in solving group tasks, and in a classroom, the teacher scaffolds or guides their students towards common understandings as outlined in the curriculum content they are required to teach. Indeed, assessment processes focus on a student’s ability to recall, apply or transfer those common understandings. In this study, conversations involving students and their teacher also offered insight into students’ statistical thinking processes as they developed understandings about the abstract concept of data. The level of abstraction reached by the students required being able to express relational understandings about data, connecting a range of mathematical concepts including statistical thinking, which built throughout the investigation. This exemplifies the complexity of messy data generated in authentic contexts. Of interest is the pattern of normalisation, conformity and innovation (final row, Table 1) that lead to abstraction. This pattern is worth further investigation in similar contexts, to see how classroom communication patterns may support abstraction in statistical investigations. Findings from further studies will strengthen the insights into the statistical thinking processes students demonstrated in this study, related to conceptions of data, and the subsequent construction and use of data. Further to this, this research illustrates the kinds of generalisations (abstractions) about data that children at this age can make when participating in statistical investigations that are problem-based and involve rich mathematical connections. Cognitive tuning processes supported innovation and sought to build a common frame of reference about data to further the statistical problem-solving process. This emphasised the transformative nature of the data throughout the investigation and illustrates the dynamic nature of complex statistical data being explored in classrooms which is beyond definitions of standard data. One limitation of the study is the teaching experience held by the classroom teacher, and the supports she received around statistical investigation processes by the first and third authors. Not all classroom teachers have worked with researchers with the aim of improving mathematical pedagogy in their own classroom. The teacher herself was a researcher who had much experience with teaching statistics through inquiry. Established classroom norms may have supported the communication processes that led to student-centred problem solving, presented here. Her passion is teaching and learning in statistical investigations through inquiry. Further to this, not all classes are attached to a STEM professional, who is able to visit and spark such investigations; the authors encourage classroom teachers to seek out such relationships. Future research could explore the outcomes of teacher-STEM professional relationships informing similar investigations in other classrooms, with different teachers.

The notion of a teacher pursuing commonality in their students’ thinking when teaching mathematical concepts is often a goal of teaching and a goal of research generally. It is well known that teachers may use various scaffolds to help their students reach mathematical learning goals. The authors of this paper found that this notion of pursuing commonality was not dissimilar to Bruner’s (Wood et al., 1976) notion of scaffolding to support student thinking towards learning goals. Further research involving cognitive tuning in statistical investigations is warranted to gain further insights into the communication processes used.

  • This manuscript represents further exploration and analysis of a MERGA presentation (2022) by the authors:

How Big is a Leaf? Using Cognitive Tuning to Explore a Teacher’s Communication Processes to Elicit Children’s Emerging Ideas about Data