Visualisation as a didactical and reflective tool in preservice teachers' work with arithmetic word problems

Teacher education has long been criticised for inadequately preparing future teachers for the challenges of contemporary classrooms (Lee, 2005). One response to this criticism has been the development of reflective teacher education, which emphasises teachers' ability to reason about their instructional choices and the consequences of those choices for student learning (Bergman, 2015; Harding et al., 2020; Wiens and Gromlich, 2018). Reflection is understood not only as reconsidering past actions (reflection-on-action) or adapting during teaching (reflection-in-action), but also as anticipating future instructional decisions based on prior experiences (reflection-for-action) (Farrell, 2014; Olteanu, 2017). These different types of reflection are particularly relevant in mathematics education because teachers' planning decisions often determine which aspects of mathematical content are made visible to students.

However, despite the central role attributed to reflection in teacher education, research has consistently shown that PSTs often struggle to engage in deep and meaningful reflection. For instance, their reflections frequently remain descriptive, experience-based, or focused on surface features of instruction rather than being analytically grounded in mathematical content or in the consequences of instructional choices for learning (Hatton and Smith, 1995; Jay and Johnson, 2002; Korthagen and Vasalos, 2005; Ulvik and Riese, 2016). These results indicate the need to understand how reflective practices can be supported in mathematics education to improve PSTs' opportunity to shape the visibility of mathematical structures and to develop sustainable problem-solving approaches.

Research has consistently shown that visualisations help students discern mathematical structures, make connections, and apply problem-solving strategies (Arcavi, 2003; Presmeg, 2006; Soni and Okamoto, 2020). In particular, structural visualisations such as bar models or number lines support students in moving beyond procedural strategies and towards sustainable problem-solving methods (Ng, 2004). In this study, sustainable solving methods are defined as problem-solving strategies that are transferable across tasks, conceptually grounded, and supported by external visualisations, enabling students to generalise, justify, and flexibly adapt their reasoning beyond a single problem situation. For teachers, visualisations therefore function not only as a didactical tool but also as a reflective resource (Olteanu, 2016a, b).

Despite extensive research on visualisation in mathematics education, relatively little is known about how PSTs employ and justify visualisations when preparing instruction. Existing studies have primarily focused on students' visualisation strategies or on categorising different representation types, without examining how PSTs integrate visualisation into their instructional design and reflective practices. This gap is relevant in the context of AWPs because the PSTs' ability to represent underlying mathematical structures is crucial for developing sustainable problem-solving approaches (Verschaffel et al., 2020). This study addresses that gap by investigating how PSTs use external visualisations when planning and reflecting on the teaching of AWPs. Specifically, it explores how external visualisations support both the development of sustainable solving methods for students and PSTs' reflection-on, -in, and -for-action. The research question guiding this study is: How do preservice teachers' uses of external visualisations in arithmetic word problem planning and teaching support the development of sustainable solving methods, and how does this process influence their reflection?

Reflective practice has long been viewed as a cornerstone of teacher education (Cruickshank, 1985; Dewey, 1933; Schön, 1987). Dewey (1933) defined reflection as the “active, persistent, and careful consideration of any belief or supposed form of knowledge in the light of the grounds that support it and the further conclusions to which it tends” (p. 9). Schön (1983) later distinguished between reflection-in-action, which occurs during teaching, and reflection-on-action, which takes place after teaching has occurred. Building on these perspectives, Killion and Todnem (1991) introduced the concept of reflection-for-action, later elaborated by Grushka et al. (2006) and Olteanu (2017), which refers to the anticipation of future instructional decisions based on accumulated experience.

Reflection-on and in-action, as well as the results of both, can be used to guide a teacher's future decisions (i.e. reflection-for-action). Reflection-for-action is particularly relevant in mathematics education, where teachers' choices and designs of examples strongly influence learning outcomes (Olteanu, 2016a, b). It is connected to the intended object of learning (the content planned for the classroom), the enacted object of learning (what is made available during instruction), and the lived object of learning (what students actually experience). For PSTs, reflection-for-action involves not only planning instructional content but also identifying which aspects of practice, such as the use of visualisation tools, require deliberate attention.

Despite its recognised importance, reflection remains underdeveloped in mathematics education within teacher education programmes. PSTs often report limited opportunities to engage in meaningful models of reflection (Stevenson and Cain, 2013), and reflection does not always translate into deeper understanding or improved instructional practice (Wopereis et al., 2010). In response, several strategies have been developed over the years, such as written reflections (Bergman, 2015), video recordings (Wiens and Gromlich, 2018), transcriptions (Harding et al., 2020), and group reflection (Farrell, 2014). The present study positions visualisation as an alternative reflective resource that enables PSTs, when working in groups, to anticipate how students might engage with tasks and to evaluate the critical aspects of their own instructional design.

Visualisation has been widely recognised as a central tool for mathematical thinking and learning (Arcavi, 2003; Presmeg, 1986, 2006) and to support students in grasping abstract concepts, making connections, and solving problems (Soni and Okamoto, 2020; Rellensmann et al., 2022). Research has examined visualisation from multiple perspectives, including its role in curriculum design, in technology-enhanced environments, and for different students' learning (Stylianou and Silver, 2004; Kozhevnikov et al., 2002). Meta-analyses show the positive effects of using visualisations. However, they specify that the positive effects depend on students' characteristics, task types, and the types of visualisations used (Ran et al., 2022; Sokolowski, 2018; Zhang et al., 2023).

Research in mathematics education indicates that visualisations can serve different functions depending on the distinction between cognitive and instructional opportunities (Arcavi, 2003; Kaur, 2019; Presmeg, 2006; Soni and Okamoto, 2020). In this context, a structured approach to visualisation that supports cognitive understanding and instructional planning, through its part–whole, comparison, and change models, is described as the Singapore Model (Kaur, 2019). Based on prior research, in this study is made a distinction between three functional uses of visualisation: (1) representational visualisations, which depict mathematical entities or problem situations (e.g. diagrams, bar models, number lines); (2) transformational visualisations, which allow learners to manipulate or transform representations to clarify relationships and structures; and (3) invariant structural visualisations, which emphasise underlying mathematical structures that remain constant across tasks, thereby supporting conceptual understanding and generalisation.

External visualisations are defined as cognitive and representational tools, including drawings, diagrams, graphs, and manipulatives (Schoenherr and Schukajlow, 2024). Arcavi (2003) distinguished three key dimensions of external visualisation: process (the selection, construction, and interpretation of visualisations), product (the final representation), and type (the form and resemblance of the visualisation). The external visualisations can also be grouped by physical resemblance and structural resemblance (Schnotz, 2005). Physical resemblance refers to mirror real-world appearances, and is visualised by pictorial representations or simplified models. Structural resemblance focuses on the depiction of abstract relationships between concepts or numerical calculations. It can be visualised by using length-based visualisations (e.g. number lines), area-based visualisations (e.g. tape diagrams), or relational visualisations (e.g. tree diagrams and concept maps). Within mathematics education, visualisations connected to structural resemblance are an important role in clarifying underlying connections in supporting arithmetic and algebraic reasoning (Presmeg, 1986; Ludewig et al., 2020). For teachers, the selection and design of visualisations are integral to reflection-on, -in, and -for-action, since they require anticipating how students will interpret and use these representations. Visualisation, therefore, functions not only as a didactical tool for students but also as a reflective resource for teachers (Olteanu, 2016a, b).

Research on PSTs' use of visualisation has examined the tools used in the classroom, such as digital environments (e.g. Zhang et al., 2025), as well as the spatial strategies employed by learners (Koyuncu et al., 2015). In solving AWPs, two main strategies are identified: the direct translation strategy, which focuses on extracting numbers and keywords, and the problem model strategy, in which the text is transformed into a coherent mental model of the situation (Hegarty and Kozhevnikov, 1999).

Research has shown that diagrams play an important role in the latter by helping learners identify relevant information, organise relationships, and determine the required operations (Ayabe et al., 2022). The bar model, which represents quantities and relationships through rectangular bars—making problem structures visible and guiding the choice of operations—is also especially effective in problem-solving (Kho, 1987; Ng, 2004). This method visualises three basic relations between natural numbers in a problem-solving context: the part-whole, the comparison, and the change (Baysal and Sevinç, 2020). Research has also shown that the use of bar models indicates a mathematical richness in solving arithmetic problems for PSTs (Sevinç and Lizano, 2024). Another approach is schema-based, in which repeated exposure to similar problems leads to schema formation, enabling solvers to recognise problem types and apply appropriate strategies (Christou and Philippou, 2002; Cook et al., 2020).

Nevertheless, PSTs often approach arithmetic word problems procedurally, focusing on steps and answers rather than fostering exploration or deeper understanding (Son and Lee, 2021; Jiang et al., 2022). They also demonstrate difficulties in solving non-routine problems and in using alternative strategies (Hiltrimartin et al., 2020; Guner and Ebay, 2021), which risk being perpetuated in their future teaching (Chapman, 2016). This study, therefore, addresses an important research gap: how PSTs employ external visualisations in arithmetic word problem contexts, and how this employment fosters both sustainable solving methods for students and reflective teaching practices for PSTs. Sustainability is often linked to problem-solving with real-world relevance (Makramalla et al., 2025). It emphasises critical thinking as well as teachers' design of tasks with environmental and social relevance. For example, students translate authentic problems into mathematical models, test solutions, and reflect on their implications by using mathematical modelling (Szabo et al., 2020).

The present study integrates concepts from variation theory (Marton, 2015; Marton et al., 2004) with concepts from reflection theory (Killion and Todnem, 1991; Olteanu, 2017; Schön, 1983, 1987) in order to analyse preservice teachers' (PSTs') professional learning processes as they plan, enact, and reflect on lessons designed for teaching arithmetic word problems (AWPs). Importantly, school students are not involved in this study; the focus is on PSTs' collaborative work in simulated teaching scenarios, with the aim of examining how instruction could foster sustainable problem-solving methods.

Figure 1 illustrates the integrative model guiding this study. The Professional Learning Cycle combines variation theory, reflection theory, and visualisation. At its centre, variation theory and reflection theory inform the analysis, while visualisation functions as a reflective resource, enabling PSTs to externalise and share their reasoning. The cycle is organised around three phases of reflection: planning (reflection-on-action), enactment (reflection-in-action), and post-lesson evaluation (reflection-for-action) and is framed by individual reflections, which underscore that professional learning is both collective and personal.

In this study, variation theory is applied both as an analytical tool and as a methodological framework. As an analytical tool, it provides concepts for examining how PSTs identify and make visible the critical aspects of arithmetic word problems through their planning and use of visualisations. Critical aspects are those features of a mathematical object or situation that must be discerned and understood to grasp the concept or solve the problem correctly. In the context of AWPs, this means anticipating which aspects students need to discern to develop sustainable solving methods, such as distinguishing between additive and multiplicative structures or recognising underlying part–whole relationships. Critical aspects are categorised as intended (identified during planning), enacted (made explicit through examples or visualisations), and lived (discerned by students) (Olteanu, 2016a, b). In this study, lived critical aspects do not refer to empirically observed student learning but to PSTs' reflective interpretations of what students appeared to, or would likely, discern in a hypothetical classroom context. To make critical aspects accessible, PSTs design patterns of variation: contrast, which highlights differences (e.g. why addition is inappropriate in a comparison problem); separation, which keeps one element invariant while another changes; generalisation, which connects structurally similar problems; fusion, which draws attention to multiple aspects simultaneously; and similarity, which represents the same problem through multiple forms such as bar models, number lines, or symbolic expressions (Marton, 2015; Marton and Pang, 2008; Olteanu and Olteanu, 2012). As a methodological tool, it explains the study's design by directing attention to how PSTs structured the teaching around patterns of variation that make critical aspects discernible. Central to variation theory is the idea that learning occurs when learners are able to discern the critical aspects of an object of learning (Marton and Booth, 1997).

Reflection theory complements variation theory by analysing PSTs' reflection on, in, and for action in relation to planning, enactment, and post-lesson evaluation. Reflection-on-action refers to anticipatory considerations during planning (Schön, 1983, 1987), including which critical aspects to address and how visualisations might facilitate their discernment. Lesson planning may follow different approaches, including forward planning (beginning with content), central planning (beginning with methods and activities), or backward planning (beginning with outcomes and working backward) (Farrell, 2014). During enactment, reflection-in-action occurs as PSTs adapt their use of visualisations in response to challenges that emerge while teaching. In post-lesson evaluation, reflection-for-action involves analysing experiences to inform future practice. This reflection occurred collectively through group discussions and individually through written accounts, and encompassed both theory and practice. According to Farrell and Macapinlac (2021), reflection on theory involves examining how teaching theories are used in practice, while reflection on practice focuses on observable classroom actions. In this study, reflection on practice refers to observable PSTs' actions during the enactment of the lesson. The dual focus of reflection ensures that PSTs' reflective thinking remains connected to collaborative processes while also supporting their personal and professional growth. In this way, reflection theory provides a coherent framework for understanding how preservice teachers evaluate their instructional decisions and connect them to broader pedagogical principles.

To analyse PSTs' use of visualisation as a reflective resource, this study employs a functional analytical distinction between three types of visualisation, drawing on research on visualisation in mathematics education (e.g. Arcavi, 2003; Kaur, 2019; Presmeg, 2006). Representational visualisations are identified when PSTs depict mathematical entities or problem situations, such as diagrams, bar models, or number lines. Transformational visualisations are identified when PSTs modify, reorganise, or adapt representations to make relationships and structures explicit, for example, by manipulating bar models or adjusting diagrams to clarify quantitative relationships. Invariant structural visualisations are identified when PSTs explain the mathematical structures that remain constant across problems, thereby supporting conceptual understanding and generalisation. By using this functional distinction, the study provides an analytical lens for examining PSTs' use of visualisations. It clarifies how different types of visualisations contribute to the development of sustainable problem-solving methods and facilitate PSTs' reflection-on-, in-, and for-action.

Within this framework, sustainable solving methods are conceptualised as strategies that are transferable, conceptually grounded, and flexible across tasks. They move beyond memorised procedures by relying on PSTs' ability to discern critical aspects connected to problem structures. Such methods support generalisation across tasks (e.g. recognising when multiplication rather than addition is required), make reasoning transparent through external visualisations (bar models, tables, or number lines), provide control and verification for justifying solutions, and foster flexibility by shifting between representations and strategies. They focus explicitly on mathematical structures, such as variant and invariant relationships in proportion, thereby promoting long-term understanding rather than task-specific answers.

This study employed a qualitative research approach to gain insight into PSTs' use of external visualisations in arithmetic word problem planning and teaching, how these visualisations support the development of sustainable solving methods, and how this process influences their reflection (Merriam and Grenier, 2009). In this study, a case study approach was used. A case study is grounded in the view that the validity depends on the theoretical framework guiding the analysis (Bogdan and Biklen, 1982) and has been successfully employed in reflective practice studies (e.g. Farrell and Macapinlac, 2021).

This paper is part of a larger study examining how PSTs develop problem-solving instruction through collaborative and theory-based practice participation. The study involved 22 first-year PSTs enrolled in a compulsory school mathematics education program at a university. Participation was voluntary. Written consent was obtained from the PSTs in accordance with the ethical guidelines of the Swedish Research Council (2024). Participants were informed of the study's purpose, their right to withdraw at any time, and the procedures for handling personal data. All data were anonymised, securely stored, and used solely for research purposes.

Upon completing the four-year teacher training program, the PSTs will teach students in grades 4–6 (ages 10–13). The program includes courses on general education, didactical knowledge, and mathematical content knowledge, with a primary focus on teaching school mathematics in grades 4–6. In their first year, they take a course on teaching mathematical concepts at the school level, covering different approaches to problem-solving and the use of various didactical theories, including variation theory. As part of this course, they are trained in three approaches to problem-solving instruction: (1) teaching for problem-solving, where students first learn skills or concepts and then apply them to solve problems; (2) teaching about problem-solving, which explicitly focuses on problem-solving strategies; and (3) teaching through problem-solving. During the course, PSTs were also introduced to different ways of using visualisations in problem-solving instruction as part of their regular teacher education. These instructional activities were not structured according to a predefined typology. In the analytical phase, PSTs' written and enacted uses of visualisation were coded according to the three functional categories defined in the theoretical framework.

The participating PSTs were divided into six groups. Five groups consisted of four PSTs each, while one group included two PSTs. Data collection for this case study took place over approximately one month. Each group was assigned an arithmetic word problem (AWP) from the national mathematics exam for grade 6 (Table 1). The table shows which problem types are most difficult nationally, the percentage of students who solved them correctly (column 2), and which group worked on each problem in the study (column 3). The reason PSTs worked on different types of problems was to capture variation in their use of visualisation and in the strategies they preferred when solving them.

The six AWPs differed in both structure and familiarity. Café Shopping and Cinnamon Roll Makers presented everyday contexts involving relatively simple additive or proportional reasoning. In contrast, Ball Throwing at the Market, Nuts, Bolts, and Screws, and The Village Animal Count required the coordination of multiple proportional relationships. Sweet Production and Ingredients combined direct and inverse proportional reasoning across two parts, necessitating shifts between different relational structures. While all problems could, in principle, be solved algebraically, PSTs were instructed to use only arithmetic operations. This constraint provided a consistent basis for comparing the application of visualisations and patterns of variation across tasks of varying complexity. Accordingly, problem characteristics were treated as a contextual factor in the analysis and were taken into account when interpreting PSTs' use of visualisations.

Data were collected from multiple sources (audio-recorded group discussions during the planning phase, video-recorded simulated enactments, written group notes, and individual reflective reports) that support triangulation and enhance the reliability of the study. During planning, each group analysed its assigned problem and designed three to four lessons aimed at promoting students' sustainable solving methods. This process exemplified reflection-on-action, as PSTs anticipated the critical aspects that students would need to discern and considered how different patterns of variation (contrast, separation, generalisation, fusion, and similarity) could make these aspects visible.

Following the planning stage, each group conducted a simulated lesson within its own group, which was video-recorded for review. The simulations resemble authentic classroom teaching, but do not involve school students. Instead, PSTs enacted the lessons as if teaching, allowing them to explore instructional strategies and anticipate student responses. Each group selected the final lesson they would enact. The enactments provided opportunities to observe PSTs' reflection-in-action, as they adjusted explanations, visualisations, and examples in real-time in response to emerging difficulties or unanticipated interpretations. The video recordings enabled the researchers to track how PSTs used visualisations to represent problem structures, anticipated students' discernment of critical aspects, applied patterns of variation, and emphasised particular solving methods. The recordings also served as material for PSTs' subsequent group reflections, linking enactment with post-lesson evaluation.

After the simulated lesson, PSTs engaged in a post-lesson reflection session, analysing the video together and providing feedback for potential revisions. This phase corresponded to reflection-for-action, during which PSTs evaluated how their teaching theories were translated into instruction, examined the actions they had taken, discussed alternative approaches, and considered ways to improve future planning. Finally, PSTs wrote individual reflective reports on the planning, enactment, and group discussions, providing an opportunity to capture how each PST connected theory, prior research, and insights gained from the iterative process.

Data analysis was carried out iteratively by the research team using a combined deductive–inductive approach. Deductive coding was informed by the theoretical frameworks of variation theory, reflection on, in, and for action, and the typology of visualisations proposed by Kaur (2019). This guided the identification of critical aspects, patterns of variation, and types of visualisations (representational, transformational, and invariant structural) used by PSTs in their planning, enactment, and reflection after the enacted lesson. Inductive coding complemented the deductive coding by capturing visualisation types and problem-solving methods used by PSTs. By explicitly linking deductive and inductive coding, the analysis clarifies how PSTs' choices of visual representations supported their reflection-on, -in, and -for-action, as well as how these visualisations are used to foster students' sustainable problem-solving methods.

The study has certain limitations, such as a small sample size, the specific arithmetic problem selected, and the simulated nature of the lesson enactments. Despite these limitations, the study provides valuable insights into PSTs' reflections and their use of external visualisations when planning and teaching AWPs. Through a thorough, multi-source, and iterative analysis, the study effectively addressed the research question in depth.

To address the research question, the results are organised into four sections: lesson planning, teaching enactment, group post-lesson reflections, and individual reflections. Each section integrates analytical findings with empirical evidence from PSTs' lesson plans, enactments, and reflections. References to figures indicate where visual evidence can be included to strengthen the results.

The analysis of each group's lesson planning shows that they chose different starting points for selecting visualisations to make intended critical aspects visible and for deliberately applying patterns of variation to support sustainable solving methods.

The analyses show that Groups 1 and 3 first identified lesson content and then selected methods and activities. Group 1 highlighted intended critical aspects such as recognising multiplicative relationships between three holes, partitioning a total score of 120 into eight equal parts, and validating solutions against the problem statement. To make these aspects discernible, they employed representational visualisation (line segments) to represent quantities proportionally (Figure 2).

This visualisation was employed to compare magnitudes while maintaining the method, referred to in this article as the comparative method, invariant across varying contexts. In this way, the PSTs intend to create opportunities to generalise the connection between visualisation and method use in solving the problem. The use of visualisation enacted separation by isolating one aspect at a time, while also coordinating multiple aspects, thereby generating a pattern of variation through fusion.

Alternative visualisations, such as the chocolate model, were considered but not adopted, showing PSTs' reasoning about unsuitable approaches (contrast) and highlighting similarity patterns in other visualisations:

Through collaborative negotiation, the preservice teachers (PSTs) reached consensus on using a length visualisation. Decisions about the visual representations were jointly negotiated and revised as the task progressed.

Reflection-on-action was visible in the anticipation of potential confusion, for example, with the division 120÷8, and in the selection of representations aligned with students' prior knowledge. Signs of reflection-in-action emerged as PSTs anticipated possible classroom contingencies, while reflection-for-action was directed towards developing students' habits to understand the information in the problem, and to check their solutions.

Group 3 focused on relational terms such as “more than,” “three times as much,” and “total,” and linked these expressions to the corresponding mathematical operations:

They used invariant structural visualisations (rectangles) and concrete manipulatives to support multi-step solutions and help maintain attention:

Patterns of variation were applied in several ways: contrast by modifying quantities, separation by focusing on relational terms, generalisation through the use of simpler analogous problems, fusion by combining multiple critical aspects, and similarity by employing multimodal representations. PSTs demonstrated reflection-on-action in anticipating how students might interpret the task: “We have to see which parts we need to go through … what they need to know beforehand, they probably already know well” (Group 3, Lesson plan). The problem-solving method they plan to use is also the comparative method.

Reflection-in-action appeared in considerations for supporting students during enactment, including peer support and monitoring understanding with manipulatives: “So they get to try it themselves … then you notice which students … have a grasp of it, somewhat, or not at all” (Group 3, Lesson plan). Reflection-for-action was evident in the post-lesson in refining examples and language: “You have to rephrase something to simplify it in some way. Or explain a certain word more clearly” (Group 3, Lesson plan).

Group 5 selected methods before content by focusing on the schema proportion method. The idea is first to identify the two related quantities, then arrange the known values and the unknown in a simple two-row table, with the known values in the top row and the unknown in the bottom row. Next, write the proportion as a fraction of the first quantity over the second quantity for both rows. Their intended critical aspects included the use of operations and unit conversions.

PSTs used transformational visualisations (ratio schemas) to manipulate quantities and illustrate proportional relationships. Patterns of variation included generalisation, contrast, separation, fusion, and similarity.

Reflection-on-action was evident in considerations of sequencing and scaffolding, reflection-in-action in planning for potential student adjustments, and reflection-for-action in creating opportunities for students to demonstrate their understanding.

Groups 2, 4, and 6 combined forward and backward approaches, choosing different entry points for lesson design depending on their focus on problem structure and intended critical aspects. Group 2 structured their lessons by starting with the desired solution and retracing the steps in reverse order (the backward method). They employed representational visualisations using magnets to highlight part–whole relationships and equivalence of fractions: “We can show 1/2, 2/4, 4/8 with magnets so students see they are the same.” (Group 2, Lesson plan). The backward approach allowed PSTs to clarify the connections between each step in the solving process, emphasising systematic reasoning rather than isolated calculations. Separation (isolating parts of the fraction) and generalisation (showing equivalence across different examples) were intentionally applied as patterns of variation.

Group 4 maintains a consistent solving method, the comparative method, which compares quantities across different problem contexts. They employed representational and invariant structural visualisations combining numerical representations, written explanations, and diagrams: “We can combine the number, we explain the task, and we write text and use the image at the same time” (Group 4, Lesson plan). They also applied fusion (combining multiple critical aspects), similarity (maintaining consistent structure across tasks), and contrast (highlighting differences between problem elements) as patterns of variation. Reflection-on-action was identified when PSTs anticipated potential misunderstandings and planned scaffolding strategies, while reflection-in-action was used to adjust visualisations dynamically: “What if we draw the objects vertically … or horizontally?” (Group 4, Lesson plan).

Group 6 structured lessons around the part–whole method, particularly for fraction problems. They planned tasks progressing from simpler to more complex scenarios:

Group 6 used representational visualisations (pie charts) and invariant structural visualisations (rectangles) to make the part–whole relationships explicit. Patterns of variation, including separation, fusion, similarity, contrast, and generalisation, were systematically applied to support consistent reasoning and transfer of methods across problems.

All three groups demonstrated reflective practice across the lesson planning and enactment stages. Reflection-on-action is used to anticipate challenges related to attention and prior knowledge, in sequencing tasks, or to plan the combination of writing and visualisation.

Reflection-in-action appeared in adjusting visualisations and scaffolding strategies in real time:

Reflection-for-action focused on promoting transfer through games, differentiated worksheets, or emphasised explanation:

During enactment, PSTs engaged in reflection-in-action by using different patterns of variation to explain the connection between the information provided in the problem through calculation and visualisation, enabling the problem-solving methods they had planned. Visualisations (representational, transformational, or invariant structural) became adaptive tools rather than static resources.

In Group 4, the PSTs discussed the use of different visualisations, such as bar models and tables. However, they realised that these visualisations tended to create confusion rather than clarity. Consequently, the group agreed to use representational visualisations (drawings) (Figure 3), which mimic real-world appearances. These visualisations were chosen to support additive and multiplicative comparisons and to “make the problem easier to imagine”.

By showing the relationship between the prices of different items through pictures, PSTs aimed to highlight how various purchases relate in terms of cost. This type of visualisation supports solving the problem using the comparative method.

In Groups 1 and 3, PSTs also employed the comparative method using representational (bar models represented as line segments, Group 1) and invariant structural (rectangles, Group 3) visualisations to make critical aspects visible while keeping the solving method invariant (Figure 4). Figure 4 shows a bar model that PSTs in group 3 used to represent a part–whole problem. During the task, they prepared colourful paper blocks and placed them on the board in no particular order, experimenting with different arrangements. One PST noted, “You can prepare them, you put them up there, not in any order … so I partially solve the task, but something goes wrong along the way,” highlighting how initial attempts revealed misunderstandings and prompted reflection. The group also discussed drawing blocks on the board to show multiple visual approaches: “So that you show several ways to apply … several visual ways … varies … varies the forms of representation.” By combining physical blocks and drawn models, PSTs tested alternatives, identified difficulties, and refined their representations. This process shows how visualisations served both as tools to make mathematical structures visible and as reflective resources that supported collaborative reasoning and strategy development.

Line segments (Group 1) as representational visualisation show proportional relations and how parts are related to the whole, whereas rectangles (Group 3) as invariant structural visualisation, highlight cumulative structures, supporting multi-step reasoning and a holistic view of part–whole relations. Both visualisations enabled generalisation across contexts and promoted sustainable problem-solving by linking method with relational reasoning rather than procedural memorisation.

In Group 5, PSTs used the schema-proportion method to map relations among sugar, dextrose, and sweets, highlighting a separation in reasoning. Ratio schemas as transformational visualisations were used to manipulate quantities and illustrate proportional relationships, supporting generalisation across tasks and PSTs' reflection-for-action (Figure 5). Visualisations functioned as mediators of structural reasoning, supporting sustainable problem-solving. In part (a), the schema visualised how dextrose corresponds to sweets by using the property of proportion and counting the quantity of sweets (Figure 5) directly.

In part (b), a similar transformational visualisation (ratio schema) was used to represent the relationship between sugar and dextrose, demonstrating how varying amounts of sugar influenced the required amount of dextrose. Using ratio schema visualisations, PSTs supported the proportion method, which helps solve problems involving fractional relationships.

In the part–whole method, PSTs used representational visualisation (pie charts) and invariant structural visualisation (rectangles) to support students' discernment of how parts combine to form a whole (Figure 6). PSTs used pie charts to illustrate how different fractions are related to the total number of animals. PSTs also employed area visualisations, such as rectangles, to structure problems and highlight relationships between fractions. For example, in the village animal count problem, asking how much 2/5 and 3/10 add up to, rectangles made the parts and their relations to the whole more tangible, enabling tracking quantities and their interactions step-by-step.

In Group 2, PSTs used line segments (representational visualisation) to depict the backward problem-solving process, starting with the final step and showing how the method and visualisation are connected across different contexts. This approach created a progression in the use of both the model and the visualisation, clarifying the calculations required at each stage (Figure 7).

The figure on the right illustrates a problem where Mia, Per, and Anna had a total of 12 apples. Mia ate 1/3 of the apples, Per ate 2/6, and the task was to determine how many remained for Anna. The figure on the left illustrates a different scenario where Peter, Ana, and Mia had 55 SEK. Peter had three times as much as Ana, and Mia had half as much as Peter. The task was to calculate the amount of money Ana had. After working with these examples, the PSTs applied the same model and reasoning to solve the “Cinnamon roll makers” problem. In this case, the backward method was used. The method began with the conclusion or final step (24) and then retraced each step in reverse order. By doing so, it identified how many rolls each person made before calculating the total number of rolls. This approach proved particularly useful when the direct solution was unclear, as it simplified complex problems by breaking them into smaller and manageable steps to make the problem-solving process easier to follow.

Across all methods, invariant structural visualisations (rectangles, schema diagrams) were particularly important for sustaining problem-solving. By emphasising relational structures that remain constant, PSTs enabled transfer of methods across tasks and promoted conceptual understanding rather than rote computation.

The post-lesson discussions across all PST groups show recurring patterns that illustrate the interplay between variation theory and reflective practice in supporting the discernment of critical aspects and the translation of teaching theories into classroom enactments. Three primary analytical foci emerged: (1) attention to critical aspects, (2) use and adaptation of visualisations, and (3) reflective considerations on theory and practice.

Across groups, PSTs demonstrated awareness of critical aspects as they were intended, enacted, and reflected upon when planning, simulating, and evaluating instruction for arithmetic word problems. PSTs' intended critical aspects were most clearly articulated during the planning phase (reflection-on-action). For instance, Group 2 explicitly reflected on whether their lesson give the opportunity to distinguish between additive and multiplicative structures: “We assumed students might add everything together, but we want them to see multiplicative structure” (Group 2, Lesson plan). Group 5 highlighted part–whole relationships when reflecting on fractions: “We planned tasks with blocks to show 2/5 + 3/10, but students often focused only on the numbers, not the fractions' structure” (Group 5, Post-lesson discussion).

PSTs' enacted critical aspects became visible during the simulated teaching enactments (reflection-in-action), when PSTs attempted to make their intended aspects explicit through visualisations and patterns of variation. Enactment sometimes revealed gaps between planning and student interpretation, highlighting the role of invariant structural visualisations in supporting consistent reasoning. In Group 4, for instance, PSTs initially assumed that students would discern the contrast between equal and unequal sharing. During enactment, they realised that this distinction had not become visible, leading them to redraw the visualisation: “We thought they would see the difference between equal sharing and unequal sharing, but they didn't notice until we drew it again.”

PSTs' lived critical aspects could be identified through post-lesson discussions (reflection-for-action) that focused on the gaps between intention and outcome. Several groups reflected that, despite targeted instructional efforts, students would likely rely on superficial cues rather than discerning the intended mathematical structures. Group 5, for example, noted that “Although part–whole relationships were emphasised, students seemed to focus on surface features rather than underlying proportional relationships.”. These reflections demonstrate the role of patterns of variation (contrast, separation, fusion, similarity, generalisation) in scaffolding students' understanding and supporting PST reflection.

The results show a shift in PSTs' reflective focus across the instructional cycle from what should be learned (intended critical aspects) to how these aspects were made visible (enacted critical aspects), and finally to the opportunities students have to discern them (lived critical aspects). This shift reflects PSTs' movement through reflection-on-action, reflection-in-action, and reflection-for-action, highlighting how visualisation and critical aspects served as catalysts throughout the learning process.

In PSTs' reflections, visualisations emerged as important didactical tools: they were used to address students' misinterpretations, to show different ways of representing the same concept, to provide opportunities for reasoning, and to support generalisation. Group 1 discussed their use of bar models and line segments (representational visualisation) to clarify part–whole relationships and maintain method consistency. Contrast and separation were applied to correct misinterpretations. Group 3 shows reflection-in-action when they discussed the use of a number line to scaffold student understanding: “We had to draw two separate lines because the students confused the units with the variable.” Group 6 also emphasised how the chosen visualisations shaped students' reasoning. They observed that the block model worked well for some students but hindered others who clung to surface-level similarities with previous tasks and discussed that the similarity-as-variation pattern had been overused, limiting students' ability to generalise. Such reflections show how PSTs not only evaluated the immediate didactic function of visualisations but also treated them as reflective resources that bridge theory and practice.

The group's discussion revealed a distinction between reflection on theory and reflection on practice. Reflection on theory was evident when PSTs discussed how the choice of visualisation influenced the enactment of the lesson. They also discuss whether procedural fluency should be prioritised over conceptual understanding when introducing multiplicative structures (Group 2) or explicitly linked their choice of visualisations to variation theory: “We used separation to make the denominator stand out, but we should have also tried fusion so they could see both numerator and denominator at once.” (Group 5). This illustrates how theoretical reasoning guided the choice of didactic strategy.

Reflection on practice was present when PSTs focused on students' misunderstandings, particularly confusion about the meaning of “sharing” in division tasks (Group 4). Reflection on practice was also evident when PSTs reflected on classroom interactions and acknowledged that their explanations were delivered too quickly, leaving students with limited time to express their reasoning (Group 1). Some groups connect theoretical insights to practical adjustments (Groups 3, 6). For instance, one PST in Group 6 observed: “Drawing helped me see what the students were missing, so next time I'll slow down and let them try their own representations first.”.

PSTs' written reflections illustrated three dimensions in which visualisations supported sustainable problem-solving and their own reflective practice. The first dimension is about clarifying proportional and part–whole relationships. PSTs highlighted in their reflections that students often struggle to discern how individual parts relate to the whole, for instance, when distributions of points, quantities, or fractions vary across a task. One PST noted:

PSTs identified visual methods, such as length or bar models, as effective for clarifying the concepts of parts, units, and fractions, and to give the students the opportunity to discern why totals are partitioned evenly. Even the use of concrete examples emerged as a key strategy for bridging these conceptual gaps. One reflection illustrated this point:

The second dimension concerns highlighting critical aspects through variation. PSTs frequently applied principles from variation theory, such as contrast and similarity, to enhance students' reasoning. Contrast was used by presenting incorrect solutions or counterexamples to provoke reflection.

Similarity was achieved by presenting consistent problem structures across tasks, enabling students to generalise principles and connect representations across contexts. This facilitated flexible problem-solving and reinforced sustainable strategies.

The third dimension is supporting dialogue and teacher reflection. PSTs noted that collaborative discussion fosters shared understanding and metacognitive awareness:

At the same time, visualisations shaped teachers' reflective practices, prompting reconsideration of instructional sequences, representation choices, and the integration of multiple strategies:

The reflections emphasised the importance of explicitly connecting “what to do” with “how to do”. They acknowledged that linking visual representations explicitly to problem structures helps students internalise reasoning processes:

PSTs' reflections-for-action indicate that the careful use of external visualisations supports sustainable problem-solving methods by clarifying proportional and part–whole relationships, highlighting critical aspects through variation, and promoting dialogue, while simultaneously fostering teachers' reflective practice.

To synthesise the findings and provide a coherent answer to the research question, this section integrates how preservice teachers (PSTs) used different types of external visualisations to support the development of sustainable problem-solving methods, and how these uses were connected to their reflective processes.

The Table 2 shows that PSTs' use of visualisations evolved across the learning cycle from making intended critical aspects visible during planning, to dynamically adapting representations during enactment, and finally to retrospectively evaluating visualisations as resources for generalisation and professional learning. Invariant structural visualisations played a particularly central role in supporting sustainable problem-solving by foregrounding relational structures that remained constant in the problem-solving process. Across all phases, patterns of variation functioned as a link between visualisation, method use, and reflection.

This study examined how preservice teachers' (PSTs') use of external visualisations in arithmetic word problem (AWP) planning and teaching supported the development of transferable problem-solving methods and influenced their reflection. The findings indicate that visualisations served a dual role. They served as instructional tools to make the problem-solving process visible and as resources for PSTs' reflection, helping them connect theory and practice, which was highlighted in early research (Arcavi, 2003; Olteanu, 2016a, b). These reflective processes were embedded in collaborative group work (see Farrell, 2014; Schön, 1983), where decisions about visualisations were negotiated and revised collectively. Collaborative negotiation enabled PSTs to identify the affordances and limitations of different visualisations, experiment with alternative models, and learn from initial misunderstandings (e.g. Groups 1, 3, and 4). Such iterative processes illustrate how reflection in-, on-, and for-action (Olteanu, 2017) is intertwined with group dynamics and decision-making.

PSTs used patterns of variation to highlight critical aspects of AWPs, in line with variation theory (Marton et al., 2004; Marton and Pang, 2008). While prior research shows that PSTs often prioritise procedural steps over structural understanding (Arcavi, 2003; Ludewig et al., 2020; Presmeg, 2006), the findings suggest that PSTs developed sensitivity to underlying mathematical relationships when working with visualisations. Representational and invariant visualisations (Kaur, 2019), such as line and area models, bar models, and schema-based diagrams, enabled PSTs to make proportional and part–whole relationships visible and connect problem contexts to broader solving methods, extending previous research on schema-based approaches to problem-solving (Christou and Philippou, 2002; Cook et al., 2020). The present study adds to this literature by analytically distinguishing between representational, transformational, and invariant structural uses of visualisation, showing how each type supports PSTs' reflective practice and the development of sustainable problem-solving strategies.

The PSTs' choices of visualisations varied across groups. Some groups focused on establishing a connection to the problem context (e.g. Group 4), whereas others prioritised the relationships between concepts (e.g. Group 5). Their work revealed several struggles, including misaligned ratios, incorrect partitioning in bar models, and inconsistent colour use. These struggles became productive sites for reflection, enabling the PSTs to reconsider the use of different visualisations and to refine their instructional decisions. They also underscore the pedagogical potential of collaborative reflection.

The study indicates that visualisations stimulated reflection at three levels: reflection-on-action during planning and post-lesson analysis; reflection-in-action during enactment, when PSTs adjusted visualisations in response to perceived student difficulties; and reflection-for-action, when experiences from collaborative work influence the organisation and decision-making in future instruction. PSTs used patterns of variation to highlight critical aspects of AWPs, in line with variation theory (Marton et al., 2004; Marton and Pang, 2008). The study indicates that visualisations stimulated reflection at three levels: reflection-on-action during planning and post-lesson analysis; reflection-in-action during enactment, when PSTs adjusted visualisations in response to perceived student difficulties; and reflection-for-action, when experiences from collaborative work influence the organisation and decision-making in future instruction. These findings extend prior work on reflective practice by showing that differentiating between functional uses of visualisation can help mediate reflection on instructional design and provide insight into PSTs' decision-making and reasoning processes (Olteanu, 2017; Schön, 1983).

The results also contribute to discussions of the sustainability of problem-solving in mathematics education. This study indicates that visualisations supported the selection of problem-solving strategies applicable across a specific context, highlighting the potential for sustainable teaching practices beyond real-world modelling (Makramalla et al., 2025; Szabo et al., 2020). This study also indicates that when visualisations are ambiguous or misapplied, such as when visual cues are unclear, sustainability in problem-solving does not emerge automatically and depends on constructive critique, comparison, and iterative refinement within PSTs' collaborative work.

The implications of this study are twofold. First, mathematics education should integrate visualisation during the university courses not only as a classroom tool for pupils, but also as a structured resource for PSTs' collaborative reflection. Such integration should create opportunities to test ideas, engage in discussion, and learn from imperfect or evolving representations. Second, sustainability in mathematics education should be conceptualised not only in terms of real-world applications, but also through the development of generalisable problem-solving strategies. These strategies can be supported by the reflective use of visualisations and informed by principles from variation theory.

The study makes three main contributions. First, it extends theories of reflective practice by showing how visualisations can serve as a resource for PSTs reflection, complementing established approaches such as text analysis, video, or peer dialogue. In addition, by distinguishing between representational, transformational, and invariant structural uses of visualisation, the study provides a functional-analytic framework for future research to examine the interplay among visualisation, reflection, and problem-solving. In particular, collaborative engagement with visualisations allows PSTs to negotiate, experiment, and learn from initial misunderstandings, which supports reflection in, on, and for action. Second, it provides empirical insights into PSTs' reasoning, with visualisations that demonstrate their potential to enhance conceptual understanding and didactical reasoning. Third, the study has practical implications for mathematics education by suggesting that the iterative use of visualisations, combined with collaborative discussion, can be incorporated into courses to help PSTs integrate theory and practice and promote sustainability in problem-solving strategies.

These conclusions should be considered in relation to the study's limitations. The small sample size, the simulated character of the lesson enactments, and the possible influence of researcher guidance limit the extent to which the results can be generalised. Despite these constraints, the detailed and iterative analysis provides important insights into how mathematics teacher education can strengthen both students' problem-solving and PSTs' reflection.

Due to Etikprövningsmyndigheten, Registration number 2024–05398–01, the research data supporting the findings of this study cannot be shared. However, the methodology and key findings are fully described in the manuscript, and any further inquiries can be directed to the corresponding author.

The authors wish to acknowledge the valuable support received during the development of this manuscript. This includes the contributions of colleagues who provided constructive feedback and assistance with language editing, whose input significantly enhanced the quality of this work. The authors also wish to extend sincere gratitude to the preservice teachers who participated in the study. Their engagement and thoughtful contributions provided essential insights that made this research possible.