Multilingualism has been defined in various ways in the literature. Kemp (2009) notes that monolinguals are speakers of one language, while bilinguals are speakers of two languages. Multilinguals generally use or know three or more languages, with varying degrees of proficiency in these languages (Barwell, 2018; Bose & Clarkson, 2016). However, scholarship in the field of multilingual research in the last decade has expanded the definition of multilingualism to include ‘various forms of social, institutional and individual usage as well as individual and group competences, plus various contexts of contact and involvement with more than one language’ (Franceschini, 2009, p. 29). This understanding of multilingualism acknowledges language diversity, includes sensitivity towards socio-cultural diversity, and appreciates society’s heterogeneity. It is now used as an umbrella term that encompasses research on bilingualism. For this article, a contextual understanding of multilingualism is adopted. Language plays a critical role in teaching, learning, and mathematical problem-solving (Barwell, 2014; Morgan et al., 2015; Moschkovich, 2017; Planas, 2018). Having knowledge of more than one language has been recognised as a ‘resource’ for learning mathematics (Adler & Ronda, 2015; Moschkovich, 2013, 2015), a recognition that was not the case in earlier years of multilingual studies in mathematics education research. Research on multilingualism has significantly influenced the field of mathematics education. Research on multilingual education has moved beyond deficit theories of multilingualism to address how the linguistic resources multilingual learners bring to class can be effectively harnessed to provide high-quality mathematics education. In other words, more research now focuses on multilingualism as a resource in multilingual mathematics classrooms (Barwell, 2018; Planas, 2018; Prediger, 2019; Ryan & Parra, 2019). However, many researchers highlight the complexity of learning and teaching in multilingual classrooms, where learners are simultaneously learning the language of learning and teaching (LoLT) and learning mathematics as both a discipline and a language. In multilingual settings, learners who share the LoLT with their home language are more familiar with the linguistic structures encountered in mathematics classrooms (Barwell, 2018). However, research indicates that this is not the case for learners whose home language differs from the LoLT (Gohel et al., 2007; Robertson & Graven, 2018). When learners’ home language differs from the LoLT, they must contend with the additional challenge of not being fluent in the language of instruction. Mathematics education involves problem-solving, which requires a firm grasp of mathematics vocabulary. This suggests that learners must develop proficiency in the language of instruction to comprehend mathematical terminology during problem-solving effectively. The LoLT is thus expected to play a crucial role in learners’ understanding of mathematical concepts. In multilingual contexts, research has demonstrated that while multiple language use in the mathematics classroom can support learning (Moschkovich, 2015), it is not a straightforward solution, as diverse language infrastructures within and around schools place varying demands on learners and their teachers. While a review of the literature on multilingual learners and mathematics highlights research conducted in various countries, there is a limited number of studies on this topic that specifically focus on the integration of Sfard’s (2008) commognition tenets and Polya’s (1945) four steps of problem-solving on how they complement each other in supporting multilingual learners’ understanding during problem-solving in geometry (e.g. Barwell, 2014, 2018, 2020; Moschkovich, 2002, 2007, 2012, 2018, 2019; Takeuchi, 2015, 2016). While many studies examine the socio-political dimension of language as a resource for gaining access to mathematical knowledge, they do not provide insights into how understanding of mathematical concepts is enhanced during geometry problem-solving (Planas & Setati-Phakeng, 2014; Setati & Barwell, 2006). The emphasis on language in these studies often overshadows learners’ understanding of mathematical concepts and the procedural framework that guides learners through geometry mathematical problem-solving, thus shifting the central aim of mathematics education research to the periphery. Therefore, the objective of this study is to build on existing research by providing an in-depth analysis of the commognition framework and Polya’s four steps of problem-solving during problem-solving for learners who study mathematics in a language that is not their native tongue. Their connection serves the different objectives; however, in this article, we argue that they can assist in analysing learners’ understanding of geometry concepts when solving geometry questions in multilingual settings.

According to Hanegem (2017), mathematics education is about encouraging mathematical thinking and understanding, and developing problem-solving abilities and research skills. Many mathematical problems can be solved using several different procedures (Gcasamba, 2014). Polya’s (1945) four steps of problem-solving are not designed to restrict learners to using a particular strategy to solve a task but, instead, provide a guide for understanding the problem and choosing an approach based on the learner’s understanding of the problem.

Polya’s four-step problem-solving process is shown in Figure 1.

This section provides a detailed overview of Sfard’s (2008) theory of commognition, with a specific focus on the theory that relates to this article. The section commences with an overview of the theory, its key elements, and its relevance to analysing multilingual learners’ thinking when solving geometry problems.

A brief explanation of the link between the two theories as perceived in the study suffices. The link between the two theories was explained based on the type of discourse participation provided by multilingual learners during geometry problem-solving questions. The focus was based on how the kind of discourse given by the participants affected the process of problem-solving as provided by Polya (1945). Firstly, the focus was on identifying the relationship between the discourse and the process. The focus was on the success or failure of multilingual learners in solving both word problem statements and problems with diagrams, because this would raise awareness of what influenced the discourse, enabling the participants to solve or fail to solve geometry problems in relation to Polya’s model. Based on the theoretical analysis, the failure of most participants to engage fully with Polya’s four steps to solve geometry questions was due to their inability to understand the geometry vocabulary, which, in turn, influenced their understanding of visuals. This analysis revealed a significant role these theories can play in improving multilingual learners’ comprehension of mathematical language (word use) in geometry, thereby enhancing their thinking and fostering active participation during problem-solving. According to Naidoo and Kampofu (2020), thinking becomes evident when learners can articulate their own understanding during the learning process, thereby promoting active participation. In analysing the two theories, the first step in Polya’s model is understanding the problem. In this case of geometry, this means understanding the geometry vocabulary, such as chord, arc, tangent, angles, and diameter. The failure to understand any geometry problem is related to a poor understanding of the above-mentioned words. In commognition, Sfard (2008) asserts that mathematics, on its own, is a language due to its word use. In this study, four elements were utilised: word use, visual mediators, routines, and narratives. Furthermore, this step aligns with two key aspects of Sfard’s commognition theory: word use and visual mediators. For word-based questions, word use plays a critical role in learners’ comprehension. In contrast, for questions accompanied by diagrams, learners integrate their understanding through visual mediators (such as diagrams) to develop a more precise conceptual grasp. Bautista et al. (2015) argue that using visual tools (visual mediators) in a structured and complementary manner can significantly enhance learners’ understanding of mathematical concepts (word use) and relationships during problem-solving. This relationship was clearly recognised even by participants who successfully solved the geometry problems. As a result, the failure to understand the word use resulted in multilingual learners finding it challenging to follow or construct coherent mathematical narratives due to their poor understanding of geometry vocabulary. The above finding reveals an essential gap in the literature that requires attention, which hinders learners’ visualisation skills related to the mathematical vocabulary. Hence, visualisation is regarded as a critical aspect for enhancing learners’ understanding (Lim et al., 2020), particularly for learners who may experience challenges in learning mathematics in a language that is not their native language. The second relationship was analysed based on step two of Polya’s steps (devising the plan), in relation to how word use and diagrams (visual mediators) influence the planning of the participants in selecting the correct procedure (routine) and another commognition tenet. It was found that participants’ devising of a proper plan was based on the influence of visuals and understanding of the problem statement, which relates to word use. This was the significant finding from the analysis of these two theories, which underscores the importance of visuals during problem-solving of geometry questions, which can be applied by teachers in trying to help multilingual learners comprehend geometry questions easily. This finding is also supported by the study by Govender and Amevor (2025), who support the view that learning geometry should incorporate visual representations, which enable learners to think geometrically and communicate geometric reasoning. Furthermore, learners plan by selecting from familiar routines, which depends on their prior exposure to mathematical vocabulary. For example, a multilingual learner who struggled to solve geometry word problems did so because they lacked understanding of word choice; as a result, no proper planning was done to solve the problem. In contrast, those who understood the geometry vocabulary managed to devise an appropriate plan and endorsed the narrative. Devising a plan or routine in geometry (explorative or ritualistic) comprises predicting a reasonable mathematical narrative: how applying specific rules, theorems, and properties of geometric shapes could lead to a solution. As a result, if a multilingual learner cannot formulate a coherent mathematical narrative due to a lack of understanding of vocabulary (word use) and diagrams (visual mediators), the planning is hindered. This finding concurs with Mudaly (2021), who argues that diagrams are visual tools that may help individuals make meaning of mathematical word problems (word use). Thus, it is necessary to find more ways to improve multilingual learners’ understanding of both visuals and mathematical vocabulary. In addition, Essien (2010) and Riccomini et al. (2015) found that, in multilingual settings, understanding mathematical word problems is challenging, especially for learners who are still learning the language of instruction while simultaneously mastering mathematical language. Hence, this study focuses on the potential significance of both Sfard’s (2008) commognition and Polya’s (1945) steps of problem-solving in enhancing multilingual learners’ understanding and spatial reasoning during geometry problem-solving. In support of this, Polya (1957) asserts that learners explore and assess their understanding before interacting with mathematical problems. The third relationship was analysed in step three of the Polya steps (carrying out the plan). Here, multilingual learners were expected to execute the selected routine (explorative or ritualistic), such as calculating angles and constructing diagrams. From the findings, it was noted that a break in commognitive routines, such as misapplying geometric properties or constructing wrong diagrams, derails execution. A significant finding was that this demonstrated the importance of commognitive tenets during geometry problem-solving in relation to Polya’s third step. These findings concur with Van Hiele’s (1999) theory of levels of understanding geometry, which explains that learners who have gone through the successive order can thoroughly compare different axioms and use these definitions and axioms in making deductive reasoning. In the context of this article, the analysis shows that understanding commognitive tenets improves geometric understanding, enabling them to engage in Polya’s (1945) four steps of problem-solving. For those participants who understood the geometry concepts well, it was easy for them to follow the plan of choosing the correct algorithm and applying the correct theorems and rules when solving geometry questions. Through the analysis of the two theories, it is easy to track what delays multilingual learners’ success and what facilitates their progress from one step to another when solving geometry questions. The last relationship was analysed based on the previous step of Polya’s steps (looking back; checking and reflecting). At this point, multilingual learners are expected to review their solutions for consistency within the mathematical narrative, like verifying that the sum of the angles of a triangle adds up to 180° and opposite angles of a cyclic quadrilateral are supplementary. Due to a lack of understanding of geometry vocabulary, multilingual learners are limited in their ability to articulate whether their solutions make sense or not. Precise terminology and clear visuals assist in communicating and evaluating the solution. Multilingual learners struggle to explain or justify their solutions, even if they are correct, due to linguistic or symbolic limitations. Notably, this step integrates the two elements of commognition: narratives and routines. Learners apply geometry theorems and axioms (narratives) to solve problems and develop systematic approaches (routines) during problem-solving. This analysis of both theories underscores the importance of using commognitive tenets, which are regarded as mathematical tools that enhance not only problem-solving but also the active, participatory involvement of learners in the teaching and learning of mathematics. Komatsu and Jones (2020) argued that learners’ competence in using mathematical tools not only supported spatial and geometric thinking but also enhanced meaningful learning and participation in the mathematics classroom.

Table 1 represents the summary of the relationship commognition tenets and Polya’s four steps of problem-solving discussed above.

The theoretical analysis of difficulties faced by multilingual learners in the study stems from a disconnect between cognitive elements and problem-solving steps, for instance:

By explicitly aligning instruction to develop these four elements, teachers can better support multilingual learners at each step of the problem-solving process. Figure 2 represents the above relationships based on the theoretical analysis of this literature.

The understanding of the geometry problem statement relies on the geometry vocabulary, which is referred to as word use in commognition. Multilingual learners must comprehend both the language of instruction and the geometry vocabulary. This could enable them to understand the entire geometry problem and what is required of them. Stephany (2021) concurs that at the beginning of the problem-solving process, the problem statement must be understood very well. Furthermore, understanding the rules of theorems and the terminology of geometry shapes, such as tangent, chord, diameter, circumference, and supplementary angles, is used in geometry learning and teaching, and in understanding geometry problems. When multilingual learners are only given problem statements, their ability to visualise the problem and construct a diagram depends on their understanding of the words used. According to the literature, a lack of knowledge of geometry vocabulary (word use) hinders multilingual learners’ visualisation abilities and hampers their understanding of geometry problems (Essien, 2010; Lariviere et al., 2025). Thus, multilingual learners who cannot understand the vocabulary of geometry fail to understand the geometry problem. In line with this study’s findings, Vula and Kurshumlia (2015) confirm that, for most learners, a lack of mathematical vocabulary is a serious challenge in solving mathematical word problems. In conclusion, based on the analysis of both theories, understanding word use is significant for solving geometry problems. Therefore, multilingual learners must be assisted in developing and advancing their mathematical vocabulary and register to master not only the mathematical content but also the mathematical genre.

In mathematical problem-solving, routines reflect the learner’s procedural knowledge, such as understanding the words used in the problem, applying theorems, and understanding visual mediators. Knowledge of word choice and visuals is critical for devising and executing the plan. At this stage, in this study, multilingual learners who failed to understand the problem due to a lack of mastery of geometry vocabulary had weak visualisation skills in geometry word problems and in problems with diagrams. The findings revealed that, for multilingual learners to devise and execute the plan, an understanding of geometric vocabulary is required, which enhances their visualisation abilities and enables them to develop the plan (exploratory or ritualistic). Those multilingual learners who demonstrated a good understanding of geometry vocabulary (word use) were able to construct accurate diagrams (visuals) and devise proper plans in both ritualistic and exploratory ways. In contrast, those multilingual learners who failed to understand the geometry problem statements could not formulate and execute adequate planning based on their misconceptions. These findings contribute to the existing literature by bringing new evidence from multilingual contexts and demonstrating the applicability of Sfard’s commognitive tenets and Polya’s problem-solving steps to geometry understanding. On the other hand, it provides concrete guidelines for navigating learners’ misconceptions in geometry learning using the two integrated frameworks. Lastly, narratives tie together logic and reasoning, which are critical in planning and reflecting on solutions. The ability to reflect on the solution requires multilingual learners to understand what informed their routines, enabling them to understand and verify their solutions. By knowing that a shape is a triangle, multilingual learners can verify their angles by proving that they add up to 180°. This stage of problem-solving requires multilingual learners to be clear about the narratives they provide before and after executing any routine so that they can verify their solutions. The narratives given during geometry problem-solving rely on understanding the problem statement and the influence of visuals, which inform the choice of any routine to be executed. Multilingual learners who were not fluent in geometry vocabulary demonstrated many misconceptions, providing incorrect narratives and angles that relied on the appearance of diagrams rather than understanding what was presented. As a result, they were unable to return and verify their solutions. From a cognitive perspective, these findings demonstrate that most multilingual learners rely on procedural memory rather than conceptual understanding. According to Hlongwana et al. (2025), routine-based learning develops short-term accuracy but not long-term conceptual understanding. Similarly, Campbell and Bean (2025)’s findings suggest a continuing decline in students’ self-confidence in mathematics, highlighting weak conceptual structures that contributed to uncertainty even in correct responses. These findings suggest that in multilingual contexts, challenges in mathematics learning are both cognitive, arising from fragmented conceptual structures, and pedagogical, stemming from limited understanding of mathematical vocabulary and metacognition reflections. In conclusion, the cognition framework and Polya’s four steps of problem-solving offer complementary strengths for supporting multilingual learners in geometry: while the cognition framework foregrounds the relationship between language, prior knowledge, and conceptual understanding, making visible the often-hidden cognitive and linguistic demands of geometric reasoning, Polya’s structured problem-solving steps provide a clear, procedural framework that guides learners through mathematical activities. Significantly, the cognition framework affords insights into meaning-making, discourse, and misconceptions that Polya’s model does not explicitly address. In contrast, Polya’s framework contributes a disciplined, transferable strategy for tackling problems that the cognition perspective alone does not operationalise. When integrated, these frameworks enable a more universal pedagogy that simultaneously supports how multilingual learners think, communicate, and act in geometry problem-solving contexts. In summary, this article confirms that the four elements of commognition by Sfard (2008) and Polya’s (1945) four steps of problem-solving complement each other during the geometry problem-solving process due to their strong connection, which serves as a key tool for identifying misconceptions, developing reasoning skills, building conceptual understanding, and designing personalised pedagogical strategies. It offers a more comprehensive approach to understanding learning and helps improve the quality of geometry education in Grades 10–12, especially in multilingual contexts.