Characterising a Representation Construction Pedagogy for Integrating Science and Mathematics in the Primary School

Growing research evidence indicates student learning gains from guided representation construction/invention in school science and mathematics. In this inquiry approach, students address challenges around what features of a phenomenon/problem to attend to, what data to collect, how and why, and make collective judgments about multimodal accounts of phenomena. However, researchers to date have tended to focus on student learning rather than on the teacher’s role in guiding various phases of inquiry. In this paper we report on (a) analysis of Grade 1 students’ engagement in interdisciplinary mathematics and science inquiry practices in a classroom sequence in ecology; (b) the teacher’s role in guiding such inquiry; and (c) interpretation of these practices in terms of support of student transduction (connecting and remaking meanings across representations in different modes). Data from our study included video capture of two case study teachers’ guidance of tasks and classroom discussion and student artefacts. We examine the classroom processes through which the teachers used students’ invention and revision of data displays to teach the concepts of living things, diversity, distribution and adaptive features related to habitat in science. Mathematical processes included constructing and interpreting mapping, measurement and data modelling, sampling and using a scale. The analysis offers fresh insights into how teachers support student learning in these two subjects, through discrete stages of orienting, representation challenge, building consensus and applying and extending representational systems.


Introduction
Research on effective approaches to teaching and learning in science (Lehrer & Schauble, 2012;Manz et al., 2020;Prain & Waldrip, 2006;Tytler et al., 2013) and mathematics (English, 2012;Lehrer & Schauble, 2006;Mulligan, 2015;Oslington et al., 2020) supports pedagogies where the teacher's role is to guide students to invent, critique and revise representations to explain phenomena. In this approach, students, with support, engage in problem-solving, address challenges around what features of a phenomenon are worth attention, what data they should collect, how and why, and make judgments about the effectiveness and trustworthiness of their choices and findings. In this way, students emulate disciplinary inquiry processes and model-based reasoning, taking up opportunities to work confidently with, and discuss, disciplinary knowledge-building practices (Lehrer, 2021;Manz et al., 2020). In our parallel representation construction approach (Prain & Tytler, 2012, Tytler et al., 2013, 2021, teachers guide their students to respond to representational challenges as a basis for developing disciplinary practices and conceptual understandings. Through engaging with the purposes, structure and integration of multimodal representations in an authentic inquiry, they experience how and why knowledge in science and mathematics is constructed and communicated (Lehrer, 2021;Makar, 2016Makar, , 2018Prain & Tytler, 2012).
These approaches and theoretical analyses have encompassed research into both mathematics and science, with a growing literature on how this representational approach might be adapted to embed and link learning across both subjects (Lehrer, 2009(Lehrer, , 2021Lehrer & Schauble, 2012Tytler et al., 2021). In this paper, we report on findings from a three-year longitudinal project -Interdisciplinary Mathematics and Science (IMS) (Tytler et al., 2018) -which has developed an explicit pedagogical model based on the approach. We focus particularly on an ecology sequence in Grade One, implemented by two case study teachers. We consider the teacher's key role in guiding this interdisciplinary inquiry to establish (a) the concepts in science of living things, diversity, distribution and adaptive features related to habitat and (b) the concepts in mathematics of variation, measurement and data modelling of living things found in sample plots using tables and graphs and a scale, spatial orientation and mapping, area, coordinates, directionality and methods of sampling.

Conceptual Framework
Learning sequence design was informed by semiotic and disciplinary perspectives on how to support reasoning in mathematics and science . Following Peirce (1931/58), we understand meaning-making as the process of aligning representations (signs) with their referents, which can be real-world objects, processes and experiences, or other representations. In mathematics and science, teachers are expected to guide students to integrate meanings across verbal, visuo-spatial, embodied and symbolic modes of representations, what Kress (2000) described as "transduction" (p. 159). We claim transduction occurs when students recognise both correspondences between different modal representations of a phenomenon and coherence of meanings across representations and their referents . However, this account of transduction raises the question of how exactly teachers support this process, and how this support needs to vary, depending on the age of students and target concepts. Representations can be dynamic or static, old or new technology-mediated, gestural, material or embodied (such as a 3D model of a material system, or a roleplay). They can be spoken or written text or 'inscriptions' (Latour, 1990) that can be visual (a diagram, drawing or photograph), symbolic (force arrows or atomic symbols) or mathematical (graphs, equations). Our perspective draws on the widely accepted view of the key role of model-based reasoning in imagining, generating and testing claims and justifications in each domain (Gilbert, 2004;Lehrer & Schauble, 2006).
We consider this approach to learning is warranted on both epistemic and pedagogical grounds: aligning with the representational invention and refinements of scientists and mathematicians in making and communicating new claims (Latour, 1990) and enhancing student motivation and learning (Tytler et al., 2013) through, for instance, gaining first-hand experience of the aptness of representational conventions such as scale and base line or annotations on maps and tallies.
While science education research on inquiry approaches has focused on student reasoning and learning, there has been less focus on specific teacher pedagogical moves that enable a bridging between student ideas and accepted scientific framings (Unsworth et al., 2022), and the research in this tradition has tended to focus on the orchestration of classroom discourse, mainly talk (Mercer & Littleton, 2007;Mortimer & Scott, 2003;Tytler & Aranda, 2015). We argue that a broader semiotic perspective on multimodal representational transactions enables a clearer focus on student reasoning and learning in both subjects (Lehrer & Schauble, 2012;Lemke, 1990Lemke, , 2004Tytler et al., 2013) and clarifies teacher-student interactions that support this (Knain et al., 2021).

Research Questions
We address the following research questions: 1. How might a teacher support students' foundational learning in interdisciplinary learning sequences in which students construct, evaluate, and refine representations? 2. What insights into teacher guidance of student transduction emerge from analysis of this process?

Pedagogical Approach: Design of the Learning Sequences
The learning sequence design in IMS drew on our previous semiotically informed representation construction approach (RCA) (Tytler et al., 2013) focused on students' constructing and refining representations, grounded in material exploration, and explicit discussion of representational conventions and affordances (Prain & Tytler, 2012). We also drew on the work of  who described the principles and stages underpinning their approach as establishing the need to create/ invent representations, explore what they reveal, make decisions about appropriate representations and engage with an expanded set of representational tools. The approach is also consistent with other inquiry-based approaches where teachers stimulate and collaboratively work with students' ideas to achieve consensus (e.g. the 5E's model of Bybee, 2015, the model-based teaching approach of Windschitl et al., 2020) or approaches in mathematics based on problem-solving challenges and teacher orchestration of student work (Dorier & Maass, 2020;Pinto & Koichu, 2021;Sullivan et al. 2016). The IMS pedagogical model ( Fig. 1) differs from them in its extended explicit focus on teacher support of students' meaning-making with representational work and in its interdisciplinary nature. One aim in the IMS project was to identify productive activity and task design around the RCA representation construction and refinement principles, particularly interdisciplinary ones. To flesh out the pedagogy, we monitored the task-framing and teacher-student interactions created by competent and committed teachers. In this paper, we closely analyse a Grade 1 Ecology learning sequence implemented by two case study teachers, Anna and Colin (pseudonyms), from different cohort schools, who embraced the approach. From these two teachers, we saw how accomplished teachers can frame tasks in different ways, introduce extra activities and challenges and support students to develop connections between everyday ideas and mathematics and science representational systems. From this we developed and refined the pedagogical model which consists of four stages, each with a distinctive disciplinary purpose. The model was further refined in consultation with the full team drawing on analysis of other IMS learning sequences. We focused on its usefulness in conceptualising the pedagogical bases of further learning sequences, noting the ways in which the stages can circulate with different temporal patterns across a sequence, as well as overlap, as meanings shift between representations and The four stages of the model are: Orienting: Teachers pose questions, explore student ideas and guide students to focus their attention on what is worth noticing by asking for predictions, questioning what they have noticed, asking for ideas about what could be measured and introducing resources for the later stages of the inquiry.
Posing representational challenges: The teacher challenges students to explore and represent their ideas and practices verbally and in multimodal text, often involving material investigative work. For instance, they may be challenged to represent the distribution of living things in different schoolground habitats, deciding how to measure and record in and across sample plots, and explain patterns of difference.
Building consensus: This involves two stages. First, the teacher guides comparison and evaluation of students' ideas and representations to reach agreement about effective display of data patterns or to suggest explanations. Second, these ideas and representations are refined and consolidated in a teacher-guided process to establish a shared understanding of the key concepts and associated representations. Depending on student capabilities, this stage may entail indirect teacher instruction (such as task framing, questions, and prompts) and direct instruction (such as modelling and explanation).
Applying and extending conceptual understanding: Students are given new representational challenges to extend their new knowledge and practices in related situations, or further concepts are introduced through representational tasks, to repeat the cycle.
In IMS the disciplinary focus shifts back and forth between mathematics and science, with each iteration involving new questions and idea refinement , and productive knowledge-building in each subject. This pedagogy requires teacher flexibility and creativity. To support teachers' work with the approach, we need to develop explicit accounts of the nature and purpose of the teaching strategies associated with the model; what should count as student knowing in each subject, how to recognise this in student representational work and how to support its development (Tytler et al., 2013).

Research Methodology/Design
The project overall adopted a design-based research (DBR) methodology (Cobb et al., 2003;Kelly et al., 2014) based on a cycle of planning, trialling, data generation/evaluation and refinement of pedagogical sequences. Each cycle involved modifications to teacher support, including changes in framing of the pedagogy and the sequences, workshop design and development of ways effectively. The research team collaboratively worked with two metropolitan and two regional primary/elementary schools in Victoria, Australia, and one regional school in Wisconsin in the USA.
The school cohorts were from mainly middle-class areas, with students drawn from diverse cultural backgrounds. The schools were selected based on previous partnerships with the researchers, involving some teachers with an interest in science and mathematics and others who participated as part of grade level cohorts. For each cohort (between 1 and 6 classes each with 25-30 students), we worked with one case study teacher each year and between 16 and 21 case study students representing a range of abilities.
To support the DBR methodology, each year we held workshops for newly recruited teachers, reflective workshops at the end of the year and planning and review meetings during the topic implementation. The learning sequences were planned by the researchers and refined in consultation with the classroom teachers through these workshops and meetings. Detailed lesson guidelines and resources such as equipment and consumables were provided to teachers for each sequence, with sample student work from comparable cohorts where possible. Sequences were implemented in each of the four school terms, comprising between three and eight weekly lessons of 1-2 h duration, varying due to constraints in school timetables and demands of the topic. The research described in this paper draws on video and interview data from two Grade 1 case study teachers and their classes from separate cohorts (comprising 7 classes across two schools) engaged in a Grade 1 Ecology learning sequence that took place in the second term of the first year of the project.
Data were collected from consenting students and teachers following approved ethical procedures. For the pedagogical analysis, data sources included student work samples for each lesson from consenting students, classroom observations of cohort teachers and video capture of each case study teacher's pedagogical approach and their interactions with students during the lessons. An iPad camera on a tracking mount followed the teacher, and a hand-held camera was used for capturing group interactions. Analysis of the data has included micro-ethnographic discursive analysis of video capture to analyse teacher and student interactions and reasoning during lessons. The analysis process entailed a pair of researchers viewing video of the case study teachers, identifying key teaching and learning moves and coding these to establish the purposes of the key stages in the pedagogical sequence. Two further members of the team independently viewed these data and the interpretation, leading to discussion/debate on the key features of the pedagogy in terms of the purposes and reasoning support, drawing on our Peircean semiotic framing, and revisiting the video field to further explore and test these ideas. Out of this repeated cycle of viewing, discussion and theoretical analysis, data coalesced around particular episodes as insightful into the nature of the pedagogy in the different stages (see Ferguson et al. [2019] for more detail).

Designing the Ecology Learning Sequence
To frame analyses using the IMS pedagogical model, we describe the ecology learning sequence structure, largely developed by the research team and refined in consultation with the teachers during initial teacher workshop discussions and ongoing collaborative review. The intent of each phase in the sequence was interpreted and extended in situ by case study teachers Colin and Anna (pseudonyms). They decided how to engage students' interest and provide differential support through the shared discussion of student artefacts, introduction of extra tasks, targeted questions and approaches to referencing the different environments and organisms of their schoolgrounds. The teachers' engagement with, and flexible application of, the pedagogical model and their responsiveness to students' learning needs in each stage of the sequence comprise the analysis that follows. The sequence was built on and partly replaced a 'Primary Connections' (Australian Academy of Science, 2012) unit 'Schoolyard Safari' involving students investigating invertebrates such as ants or worms, in this case mainly through video presentations. While all the teachers had some experience of inquiry teaching through the 5E structured 'Primary Connections' resources approach and were familiar with using hands-on activities and material artefacts in their science and mathematics teaching, the structural stages and resultant pedagogical decision-making associated with the IMS learning sequence were new experiences for them. Colin and Anna in interview emphasised how their practice had been expanded over the year in the project. In the IMS Ecology learning sequence stages: 1. Orienting: Students discussed how to represent living things they might find in the schoolground, speculating on what and where they might be and how they might represent this information and determined which questions they might ask to assist with producing representations. As part of this, the class established what is meant, in science, by a 'living thing'. 2. Posing representational challenges: Students, in groups, investigated and individually represented the number and location of different living things in different habitats represented by sample plots. 3. Building consensus: Students evaluated and refined their data representations, tabulated class data, represented the distribution of particular animals across the different plots, and proposed reasons for variation in distribution. 4. Applying and extending: Students represented habitat characteristics, linked these with animal distribution and generated a wall display of 'living things in our school'.
The science concepts focused on living things, animal and plant diversity, habitat, structure and function and adaptation. Mathematics ideas and practices included spatial mapping skills, measuring and recording counts of living things, analysing and interpreting numerical data, data recording in tables and design of graphs to identify patterns of living things. In each stage, the mathematics and science practices were designed to be mutually re-enforcing; scientific questions about the distribution of living things in relation to habitat were raised and enriched by the mathematical representations, and mathematical practices such as mapping, data recording and measuring and graphical capabilities were enriched by engagement with science-related questions the class became invested in.

Results
We analyse the pedagogy at the different stages of the model from detailed analyses of video of Colin's and Anna's practice, to explicate in more detail the nature and purpose of each stage of the pedagogy, and the way that teachers respond to and strategically guide students' representational work.

Orienting
To illustrate the nature and purposes of this stage, we draw on Colin's practice over the first two lessons. Colin: a. Established student consensus on what is meant by 'living thing' through a challenge for students to classify objects as 'living' and 'non-living', followed by discussion and clarification b. Organised a preliminary exploration to alert students to what living things might be in the schoolground, where, and why c. Challenged students to represent areas of the schoolground they might explore d. Asked students to predict and represent what they might find and from these representations created a co-constructed list on the board The pedagogy opens up possibilities for creative engagement of students and demands flexible but purposeful responses from teachers. Here we draw on two short excerpts to illustrate how Colin's purpose drives his interactions with a student. Colin had challenged students to represent what living things they will find in the schoolground and affirms (1-2) and challenges (8-14) Samantha to extend her representations and thinking. could add to this picture to maybe give a bit more information about how they move and eat. 13. Samantha: Oh -maybe I could show it eating a leaf. 14. Colin: Ok -yes, maybe you could show it eating a leaf (Student continues refining representation) In examining this excerpt from a semiotic perspective, we can see that Colin is not simply imposing a representational convention concerning the situating of the caterpillar in its habitat but is bridging between Samantha's everyday sense of caterpillars and their diet (1-11), guiding her attention to the drawing as a sign that establishes new relations between the caterpillar, its needs and the habitat in which it is found (12-14). The drawing thus becomes a semiotic tool to establish new and expanded meanings of possible caterpillar/habitat relations. Here Samantha is guided to recognise a correspondence between the living referent and its abstracted re-representation.
In the second lesson, Colin took the students into the schoolground to orient them to the possibility of living things in different locations. After a student remembers noticing millipedes in the 'dirt' near where they were standing, Colin poses the question: would we find millipedes in the soil, more so than in sandy spots, and why? 15. Ben: Because there, there's more darkness and they need darkness to camouflage from the birds 16. Colin: OOOH Did you hear that? That was a really good answer. Ben said that they like the soil, they like the dark because they can blend in 17. Ben: So, they don't get eaten 18. Colin: That's one really good point -What happens when you start digging deeper in the soil? Does the soil feel dry or does it feel wet? 19. Student choral response: Wet… 20. Colin: Would there be more living things in wet soil? 21. Jules: YES! (loudly) 22. Colin: Why do you think that's -Jules? 23. Jules: They like water" 24. Colin (restates) they like water -they like the moist soil. … So also when you're looking now -I want you to look at the trees-I want you to look at the bark on the trees as well-because we need to not just look on the ground. We can look on the trees for other living things. You might find living things on living things … People saw some butterflies near flowers and insects near flowers…. and you might find living things in the trees and bark as well if you look very carefully… Again, rather than directly imposing formal science representational conventions, Colin provides verbal prompts to lead student reasoning about where animals might be found in relation to habitat conditions and animal behaviour (16,(18)(19)(20)(21)(22)(23)(24), expanding students' thinking about what constitutes a habitat. He links 'moist soil' in relation to millipede numbers and their 'liking' of moisture and 'bark' and 'trees' and 'flowers' in relation to butterfly and insect distribution (22-24), to signal bundles of correspondences that move students towards a coherent perspective based on the habitat/adaptation construct that drives the sequence. Students' observations are the starting point for prompting scientific ways to explore the schoolground.
In a subsequent lesson, Colin challenges students to represent where they think they will find living things. He uses various students' work to focus on the nature of living things (25); the form of the representation to show clearly where living things might be found (28-30); and the development of a shared picture of the variety of places they consider worth looking in to find living things (30-33). The excerpt below shows how he highlights the value of a map, its features, and how it can be interpreted for use (Fig. 2) Again, we see this pattern of the teacher introducing and refining the semiotic tools through which students will develop an understanding of relations between living things location and habitat. In this orientation stage, Colin achieves multiple purposes, including: a. Probing and engaging with students' prior ideas, for instance, clarifying views about 'living things' and developing a shared understanding of the features of living things from a science perspective or exploring their understandings of living things' needs in relation to habitat b. Establishing what is worth noticing about phenomena (the link between the distribution of living things and habitat features, the possibility of thinking about spatial distribution) from a scientific perspective, framing their observations and orienting their attention to engage them in and refine the semiotic reasoning tools that will prepare them for later productive representational work c. Establishing students' expectations about where they might find living things and developing knowledge and skills concerning representational conventions (mapping conventions, words, discursive patterns such as questions, drawings of adaptive relations) that students will need to engage productively with the appropriate scientific practices In these first lessons, Colin builds semantic bridges between students' everyday ideas about living things in the schoolground and the strategic introduction and use of the sign systems and modelling processes through which scientists interpret and justify their account of this topic and which form the basis of the sequentially and strategically organised, extended inquiry that is to come. This orienting phase, by bridging between students' natural interests and the scientific inquiry, is intended to establish in students a certain ownership of the question 'what lives in our schoolground'?

Posing Representational Challenges
Having established the pre-conditions for students' productive study of the schoolground, including curiosity and expectations of what they might find, the next stage of the sequence involved a challenge to explore and to represent their findings. Groups of students were assigned sample plots in different locations. In introducing the activity, discussion centred around: Where will you look within the plot for living things? Where might you find them? How will you make sure you are thorough in the investigation and ensure accurate data? How will you record as you find things, and where they are found?
The challenge to decide how to represent the variety and distribution of living things in the plots is not simply an after-exploration issue concerning communicating what is already discovered. The challenge of data representation is also the challenge of how to structure the investigation, and what is to be noted. Questions asked included: 'How will we be able to compare the plots?' and 'What will we need to make the same to ensure consistency and reliability for comparison?'.
During the group exploration of the plots, teachers were active in monitoring, questioning and suggesting refinements to their representational work. Anna constantly questioned the intent and clarity of students' work, illustrated by the excerpt below (see Fig. 3 In this excerpt Anna intervenes to shape the students' thinking towards a wider purpose with questions for clarification, challenges and suggestions. Again, we see how the representational task provides rich opportunities for the teacher to engage with students' ideas and competencies and to discursively intervene to respond to these and move thinking forward. She encourages Brad to refine his representation to clarify the number of particular living things (35-37) and establish conventions of clear labelling (37,40,42). She challenges Brad to represent location (42) and then to sharpen the sense of location in relation to the wider environment (46), followed by a further challenge to attend more closely to relative location (48). She responds positively and revoices the idea of 'label' to Brad's solution. This focus on details of the representation related to number, and location of living things establishes a bridge to the abstracted tools of graphical work and habitat identification that are to come, emphasising the purpose of mapping and measuring in a tangible way.
The representational challenge stage builds on the inquiry framing and skill preparation of the orientation stage. Its ultimate purpose of developing understandings of key topic concepts needs to be clarified and emphasised as students are engaging with the representational challenge. The need for strong teacher framing acknowledges that students do not yet understand what is at stake in this investigation, and the teacher's focus on representational form reflects the need to bridge to these semiotic tools as the basis for developing the clarity and precision of these abstracted representational systems as fundamental to scientific and mathematical perspectives.
After the data generation, consisting of tallies and location maps, students were challenged to generate ways to represent the data to show clearly what they found in their sample plots. Ideas were seeded and discussed in class discussion prior to this representational work. Students then worked on tables and graphs, with teacher encouragement and support to shape and refine them. Part of this support involved students observing others' work and subsequently refining their own. The teacher also gathered the class to discuss student work as it developed. Figure 4 shows a variety of student representational work including tallies, maps, and graphical representations of the number of living things in their sample plot. Many of the student representations had been refined during the processes described above, of sharing, and teacher evaluation and suggestions.

Building Consensus: Comparing, Evaluating, Synthesising
During and following this representation construction/invention stage, the teaching entailed orchestrating the evaluation of students' work, pointing out strengths and ambiguities, and different features of the displayed data, in order to develop understandings about how to effectively represent data and ideas. The teachers guided student discussion through comparison, questioning, challenging and offering suggestions to move their ideas towards scientifically and mathematically accepted ideas and practices. In the excerpt below, Anna first invites Sonya and Amy to share their graph of the number of living things in their plot. She proceeds to interpret the graph (in her left hand in Fig. 5), pointing to different features as she talks. 51. Anna: So, we've got two grubs -is that what that means? (students nod) So each box represents one -So 1 caterpillar, 10 ants 52. Amy: We made it up to ten (indicating that 10 was as far as they counted). 53. Anna: Because you've made the graph up to ten. 4 leaves, trees (reading on…) traps -are animal traps a living thing? (Students shake heads) What do traps tell us? 54. Student: (yelling out) Sign! (Sonya mumbles -'they're a sign') Anna: They're a sign -a sign of living things… Anna then asks another pair of students for their graph and holds it up, asking if it is as easy to read as the previous graph. 55. Students: No 56. Anna: Why is it not as easy to read? 57. Leah: Because the squares are different sizes -it's easier to tell if they're the same size you can just tell which ones smaller 58. Anna: OK I see you can see by telling which ones taller 59. Kim: It's a bit messed up and mucky 60. Tanner: That's how I could put them in order 61. Anna: That's ok -so the different sizes make it a little bit harder to compare the number of living things… Note the explicit interpretation of features of the graph as Anna models how to read it (51-53), the questioning of whether 'traps' are living things (53-55) and then the orchestration of student evaluation to establish the importance of a consistent scale for comparison (56-60). Anna here is drawing on clear examples of student work to establish the nature and purpose of graphical conventions, in this case the existence of a title, clear labelling and consistency of scale. This is linked closely to the students' experience of measuring and tallying (51-53), and the comparative evaluation also refers to the need to be clear about the patterns of living things which are the intent of the exercise (60). Thus, in this stage, the teacher orchestrates display and evaluation of student work to achieve consensus around representational conventions and further reinforcing links between the data and the interlocking representational systems.

Building Consensus: Refining and Consolidating
Following the groups' refinement of representational work to produce documented counts of living things in the different plots, teachers raised questions about variation in populations across the different sites to raise questions -Where are particular living things found? Why are the numbers of particular animals different across the plots? This led to further student interrogation of their data and the construction of a shared table on the whiteboard where groups entered their data on different types of living thing in their plots, enabling a further challenge assigned to each group -how can we represent the variation in numbers of a particular living thing (focusing now on animals) across the different plots? This provided an opportunity for students to further refine their graphical skills and to explore the features of the plots (shade, grass vs. soil, mulch, moisture) that might explain the distribution of organisms.
Anna worked with students to create a table on the board of the specific habitat features of each plot that students could refer to. She printed Google maps of the school and asked students to represent the location and features of the different plots and variation in the organisms that were found in each. From student input, she organised a table on the whiteboard identifying features of the different plots. Like other teachers at this point in the sequence, she focused discussion and representational work around variation Fig. 6 Students' representations of variation of organisms in plots, and suggestions for reasons (a). Woodlouse distribution (b). Ant distribution in numbers of organisms across the plots and the interpretation of variation in organism distribution as a function of habitat features. Figure 6 shows some students' graphical representations of distribution across plots and associated interpretation in terms of animal needs and habitat features. As with previous stages, in this refinement and consolidation stage, we see how the strategic focus of the teacher on student production, comparison and refinement of representational forms leads to supporting foundational learning of the adaptation concept and of students' data modelling competencies.

Applying and Extending Conceptual Knowledge
In the following excerpt (Lesson 4), Anna leads a discussion on the features of plots that would support living things and constantly refers students to the data concerning plot features and the distribution of living things. 62. Anna: I don't know what 'good tree' means though? So, we've got Amy saying protection. Are you saying it's a good tree for protection perhaps? 63. Tanner: Oh! I think I might know… because we have shade, there was more shade under the tree. 64. Anna: And what does that mean for the living things in your plot? … 65. Tanner: So, the tree gave the shelter to things in our square. … 66. Anna: Sean -do you have a good reason why maybe plot 2 might have been a good spot to be able to find lots of different living things? 67. Sean: Mmm 68. Anna: So, we've got trees, shelter for protection, we've got good soil for more living things to grow and be found in it and Amy said something about there were more mini beasts that spiders would eat, more food for mini beasts that you found. Is that what you're trying to say? 69. Students: YES! 70. Anna: Which plot had the most different living things in it? … Well, look at the data… Which plot had the most? So, plot 2 had the most living things, but which plot had the most DIFFERENT living things? You need to move to be able to actually see the data.
Here, Anna worked to draw attention to features that would explain the patterns of living things recorded in the table, focusing initially on Plot 2. She challenges students to be precise about language and introduces terms like 'protection' (61), draws attention to student suggestions such as 'shelter' and 'good soil' and interactions between animals and plants (67) and draws attention to the data to pose a question about variety of living things (69).
In this lesson, and across the sequence generally, we can see the productive use of mathematics (tallies, tables, graphs, maps) to trigger the interpretation of distributions through the scientific constructs of diversity, habitat, and adaptation. There were also opportunities to apply and extend the mathematics capabilities of tallying, tabular and graphical skills to explore new questions and the development of mapping capabilities to explore habitat features. Students were challenged to interpret links between a set of habitat features for each plot and the graphs representing distribution of different animals across the plots. The teacher framed and guided these linkages. Thus, students moved between the mathematics, their experiences of the plots and measurements the numbers represent and scientific concepts of diversity, habitat and adaptation that were embedded in the variety of representational resources students experienced.
The final part of the sequence involved the construction of a wall display of the class's work, as a visual record of living things in the school. This included maps, diagrams, photographs, tables and graphs showing living things found in various areas of the school and details of their distribution related to the features of their plots. Each group negotiated and prepared their own representation of their work. Figure 7 shows the display in one class. In this work, we can clearly see the range of semiotic resources introduced through the sequence, the meanings of which need to be coordinated/interpreted to realise the layered nature of understandings of living things in an environment.

Discussion
Our analyses indicate (a) the key learning processes involved in guided inquiry learning sequences built around students constructing/inventing, evaluating and refining representations across science and mathematics and (b) the key teacher role of encouraging and confirming student transductive meaning-making across Fig. 7 Anna's class display of living things in the schoolground modes and across both subjects in the different stages of the inquiry. In the sequence, the investigative needs of the science question 'what lives in our schoolground, and where?' gave a meaningful purpose to developing mathematical capabilities of measuring, data representation and interpretation and development of statistical concepts, variability and mapping. This rendering of the material world in numerical and spatial patterns raised further science questions related to diversity, habitat and adaptation, which again triggered further mathematical modelling to show the distribution of particular organisms across different habitats.
The broad features of the teaching sequence are compatible with other guided inquiry models in which the teacher guides students towards explanations (Bybee, 2015) and where modelling is a key element in students enacting mathematics and science processes (Leavy, 2008;Lehrer & Schauble, 2006, 2012Manz et al., 2020;Windschitl et al., 2020). However, this analysis of the IMS sequences, focused on representational work at a fine level of detail, indicates how the teacher can elicit and confirm student meaning-making across modes. For instance, in guiding students to interpret a pattern in their graph of living things in their schoolground, Colin encourages the students to recognise how features of this representational system relate both to their prior practical field inquiry and forward to understanding distribution from a science perspective.
The analysis opens up fresh insights into the nature of transduction processes and their crucial involvement in young children's learning. In encouraging and confirming students' representation construction and refinement, each teacher strategically builds a semantic bridge between students' everyday language, perspectives and interests in relation to living things in the schoolground and the network of abstract representational systems through which scientists make claims about adaptation. This perspective extends the analysis in relation to the role of everyday language in mediating scientific understandings (Hand et al., 2003), by focusing explicitly on multimodal meaning-making. The induction process involves not simply building competence in the abstracted scientific and mathematical representational forms through which explanations are constructed but strategic and individual attention to student reasoning with, through and from signs to support this learning at each stage. We thus argue that research in multimodality in mathematics and science needs to move beyond a focus on sign system grammars (Kress & Van Leeuwen, 2020) to ways in which meaningful links are made between representations, across modes. The analyses in this paper demonstrate how competent teachers intuitively support these links in a sustained way across the sequence. Anna and Colin, while not explicitly schooled in this semiotic perspective, intuitively supported students' meaning-making processes by linking what students notice to a developing set of disciplinary semiotic resources. They did this by providing contextual clues, encouraging possible inferences, providing judicious prompts and negotiating collective agreements to make the abstract scientific and mathematical representational systems purposeful and meaning-filled for their students.
Researchers have tended to focus on the way that learning sequences in science involve meaning shifts across multiple, multimodal representations and that students build knowledge through foregrounding or backgrounding disciplinary meaning possibilities in different modes (Svensson & Eriksson, 2020;Volkwyn et al., 2019). For us, transduction processes inevitably necessitate reasoning to establish coherence and correspondence between modes. We can see the way that Colin and Anna actively model and support this process, initially between the material world and material processes of measurement (the plots, the living things in them, and the process by which these were identified) and the material representations that students invent and refine. Strategies teachers employ to negotiate and support this reasoning process are demonstrated in each of the transcripts. These include introducing words pointing to links between living thing distribution and habitat (e.g. 12-14; 20-30); helping students achieve correspondence and coherence between their labelled maps and the distributions of living things in the data tables (79-86); aligning graphical conventions with tallies and the material reality of living things in plots (51-71); or realising correspondences between listed habitat features and living things counts to attempt a coherence through considering animal needs (87-95). At each point, the transcripts show how teachers create reasoning opportunities and support students in this process of meaning-making across and within modes.
Both Colin and Anna exemplify the strategic nature of teachers' support for student learning in this semiotically rich guided inquiry. Teachers need to frame activities and discussion purposely to engage students in questions of interest that will lead into significant knowledge and practice. In this approach, students are agentic and imaginative but strategically constrained to focus on questions of scientific and mathematical worth that lead to productive knowledge and practice. Teachers thus need to have a clear understanding of mathematics and science disciplinary concepts and reasoning processes, at a level that allows judgments of flexible responses to student contributions and interventions that move thinking forward. With appropriate support from the project resources, this was well within the range of the two case study teachers, neither of whom had special science or mathematics qualifications. The study has demonstrated a way of conceptualising discrete stages in a guided inquiry, semiotically informed learning sequence, progressively bridging from students' everyday perspectives and interests to open up access to a range of disciplinary semiotic tools. The engine of this knowledge-building process is the students' representational input, deliberately framed by the teacher to contextualise mathematical and scientific views.

Limitations
A limitation of this analysis is the restriction to one topic representing a particular set of relations between science and mathematics concepts. Other learning sequences developed by the research team in collaboration with teachers involved different cycling through the model stages.
The focus on a Grade One learning sequence meant that teachers needed to bridge between naïve, everyday perspectives to gently introduce the more abstracted perspectives and representational systems of science and mathematics. In later years, with an increasing focus on more formal abstracted representational systems that build on students' increasing access to scientific and mathematic conventions, it can be expected that the transduction processes will involve aligning meaning across scientific and mathematical representational systems of increasing complexity and abstraction.

Conclusion
The generative nature of the mathematics and science interdisciplinary model has significant implications for curriculum review and practice. Our social semiotic analysis offers fresh insights into how teachers can guide transductive meaning-making as these young students begin their induction into foundational concepts and processes in science and mathematics. This analysis highlights the precise ways that teachers can support student learning in and from this process. Further research is needed, more broadly, on how teachers can be oriented to guide student transduction in science topics across different grade levels. We need further research on how teachers at different grade levels can be supported to appreciate the multimodal nature of mathematics and science learning and practice, the nature and role of transduction processes in learning different concepts and topics and how to strategically support students to reason across modes.
Funding Open Access funding enabled and organized by CAUL and its Member Institutions.
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