Language is central to mathematics education because it shapes how mathematical ideas are communicated, conceptualised, and learned. What can be meaningfully expressed is constrained by language; mathematics classrooms foreground how concepts are constituted through linguistic, symbolic, and representational systems. Teaching and learning are, therefore, language-mediated processes in which meaning is negotiated through talk, symbols, gestures, and inscriptions (Schleppegrell, 2007). Mathematical discourse is also highly specialised, characterised by technical vocabulary, dense symbolism, and precise syntactic relations. It supports rigour but can disadvantage learners still developing proficiency in the language of instruction or academic registers (Planas et al., 2018). Importantly, mathematics-specific language predicts achievement beyond general language skills (Peng & Lin, 2019). Classroom discourse is largely shaped by teachers, whose talk introduces and regulates mathematical meanings (Himmelsbach et al., 2023).

Within this landscape, teachers act as linguistic mediators, bridging students’ everyday language and informal representations with the mathematics register (Pöhler & Prediger, 2015). Language-responsive teaching emphasises intermediate linguistic forms that foreground mathematical structure without oversimplification, and such practices are closely tied to teachers’ professional knowledge, such as Mathematical Knowledge for Teaching (MKT) (Ball et al., 2008), including subject matter knowledge and pedagogical content knowledge (Shulman, 1986). Empirical studies suggest that stronger MKT is associated with more precise and conceptually aligned mathematical language, higher instructional quality, and improved student learning.

In parallel, teacher noticing has become a key lens on professional competence in mathematics teaching, typically defined as teachers’ capacity to attend to mathematically relevant features of instructional situations, interpret students’ thinking, and respond in ways that advance learning (Sherin et al., 2011). Noticing is often theorised as a mechanism linking teacher knowledge to instructional action and student learning (Krauss et al., 2020). Yet, evidence indicates it is related to, but not reducible to, teachers’ mathematical knowledge, marking it as a distinct component of teaching expertise (Copur-Gencturk & Rodrigues, 2022).

Despite this conceptual proximity, language use and noticing have largely been studied separately. Noticing research, especially in cognitive–psychological traditions, often analyses teachers’ interpretations of students’ strategies while bracketing the linguistic and semiotic resources through which thinking is expressed and teachers’ interpretations are articulated (König et al., 2022). Conversely, language research has frequently examined registers and discourse patterns without explicitly theorising how noticing processes shape teachers’ language in instructional moments. Only limited work has begun to connect language use and noticing, leaving open questions about how teachers notice language-related aspects of students’ mathematical activity and how this is reflected in explanations, evaluations of solutions, and references to mathematically critical ideas/quantities (Planas & Pimm, 2023). Recent syntheses therefore call for integrated approaches that treat language, teacher knowledge, and noticing as mutually constitutive aspects of professional competence, especially in tasks with multiple valid solutions that require teachers to recognise key ideas and quantities and communicate distinctions with mathematical precision and pedagogical productivity.

Division tasks with correct answers expressible as a mixed number (e.g., 4 1/3) or as quotient–remainder form (e.g., 4R1) illustrate this complexity. Both may be mathematically correct, but they foreground different interpretations of quantities and ideas. Supporting students’ understanding requires teachers to move beyond merely naming relevant quantities at a surface level and to use language that makes underlying structures visible and meaningful. Against this background, the present study investigates the interrelations among teachers’ mathematical knowledge, language use, references to ideas and quantities, and content-specific noticing when responding to a division problem analysis task with two correct solutions. The research question is: How are teachers’ mathematical knowledge, evoked mathematical ideas and quantities, language use, and content-specific noticing skills associated when responding to a division problem analysis task?

Data come from 139 teachers working primarily in public schools in the United Arab Emirates. The context is analytically relevant because, while classroom instruction in public schools is typically not multilingual, the broader environment is widely bilingual, and scientific subjects commonly foreground English terminology, which may shape teachers’ reliance on technical vocabulary as a marker of mathematical legitimacy (Planas & Pimm, 2023). Participants completed an open-ended questionnaire targeting specialised content knowledge for teaching (Ball et al., 2008). The instrument was a task, as seen below. TASK: A division problem represented in the diagram below has the following two answers: 4 1/3 and 4R1 (R means remainder). (picture of division-algorithm) a) In such a division problem with the given two answers, the dividend would be ……. and the divisor would be …………. because.... b) What underlying mathematical ideas does a student need to understand to be able to attack and make sense of such a division problem situation? Please explain. Teachers were asked to justify both solutions and to state the mathematical ideas/quantities students would need to understand to interpret them. The analysis used a two-level approach. First, teachers’ written responses were coded qualitatively along four dimensions. Each response was coded along four dimensions. (1) Language register was coded as technical, everyday, or meaning-related, following work on language-responsive mathematics teaching and intermediate registers that foreground structure while remaining accessible (Pöhler & Prediger, 2015). (2) Mathematical knowledge was coded as CCK or SCK, where SCK was indicated by explanations that unpack relations among quantities and meanings beyond computation (Ball et al., 2008). (3) Content-specific noticing captured the extent to which teachers attended to and articulated the targeted relationship between the remainder and the fractional part of the quotient, consistent with frameworks viewing noticing as selective attention to mathematically salient content (Copur-Gencturk & Rodrigues, 2021; Sherin et al., 2011). (4) Evoked ideas/quantities captured surface-level references to concepts, quantities, constraints, and representations that teachers named while working on the task, including those prompted by the prompt’s “what students need to know” component. The qualitative analysis then informed the second-level analysis. At this level, codes were converted into variables and analysed via path analysis to examine how evoked ideas/quantities, knowledge, noticing, and language use were associated. Model interpretation followed established SEM reporting expectations and attention to fit and plausibility (Kline, 2016). A theoretically motivated model was evaluated alongside an alternative specification to support interpretive warrants.

The study offers an integrated account of teachers’ explanations for a division situation, with two valid answers, by modelling the relations among evoked ideas/quantities, mathematical knowledge, content-specific noticing, and language use. Descriptively, many teachers could compute correctly and employ technical terms, yet meaning-related language and explicit articulation of the relationship between the remainder and the fractional quotient were uncommon. This pattern suggests a potential gap between procedural/terminological performance and relational sense-making, aligning with broader concerns that teachers may attend to surface features without making mathematically generative relations explicit (Weyers et al., 2023). The path model is consistent with the view that competence dimensions are interdependent rather than isolated. In particular, the findings support an interpretation in which evoked ideas/quantities relate to teachers’ knowledge and noticing, and these, in turn, relate to the language used in explanations. This aligns with competence perspectives that position noticing as a proximal mechanism through which knowledge becomes consequential in instruction and discourse (Krauss et al., 2020). The results also reinforce the instructional importance of meaning-related language as an intermediate register that can make structure visible without relying exclusively on formal terminology. For professional learning, the implications are that strengthening procedural fluency or technical vocabulary alone is unlikely to be sufficient. Teacher education and professional development may need to couple (a) explicit work on relational noticing in mathematically rich tasks with (b) supported practice in articulating those relations through meaning-related language that links technical terms to conceptual meanings. Future research could extend this approach across multiple tasks and test interventions that treat noticing and meaning-related language as coordinated targets of teacher learning.

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