This section of The Academic Language of Mathematics is intended for the educators who use it in mathematics teaching contexts. These colleagues are likely to belong to one of two broad cohorts: STEM (Science, Technology, Engineering and Mathematics) subject specialists teaching mathematics, physics, engineering, computer science or related subjects; and English for Academic Purposes (EAP) lecturers whose role is to offer academic language and text-analytical support alongside subject specialists. Aimed at both groups, the section explains the concepts, approaches, models, and theories that underpin the design of the book, so as to support colleagues in making informed, effective use of it. For an overview of the aims, themes, and readership of the book, see Welcome to The Academic Language of Mathematics. For evidence of how students and lecturers have experienced the book in practice, see Testimonials.

The Nature of Communication in Mathematics

Mathematics communication and the professional mathematician

Among professional mathematicians, the term mathematics communication can describe a range of public-facing mathematics communication activities: public speaking, delivering workshops, working with school children, writing about mathematics for a general audience, and creating programmes and other media content. In the contemporary UK context, individuals such as Professor Hannah Fry, Matt Parker, Dr Katie Steckles, and Ben Sparks, among many others, have been described as mathematics communicators in this sense. It is clear that mathematics communication of this kind is an established part of doing mathematics among professional mathematicians.

However, in a second, broader sense of the term, any professional mathematician is involved in mathematics communication, not only with colleagues and fellow professionals (most saliently in collaborative research) but also with students in teaching and supervision (as well as external facing events such as open days). In short, mathematics communication is part of being a professional mathematician.

What, then, is the nature of mathematical communication? Murfett and Tran’s (2025) phenomenographic study of the conceptions of the term mathematics communication held by professional mathematicians and statisticians revealed a nuanced understanding of the phenomenon, articulated in two categories: qualities (the make-up of mathematical communication, what it consists of) and processes (the factors that shape how those qualities are deployed in a given communicative event). The three qualities which Murfett and Tran identify are:

1. Formal: the logical structure and mathematical rigour of mathematical communication
2. Pictures: ‘diagrams, pictures, graphs or images in mathematical communication’
3. Language: ‘the use of common language … to describe and connect the mathematics being communicated’

The three processes are:

1. Audience: who the audience of a particular mathematics communication event is
2. Medium: the various genres of mathematics communication
3. Identity: the academic’s own identity

Murfett and Tran note that of their 14 participants, ‘almost all … had conceptions of mathematical communication in each category’. They conclude that many of their participants had ‘varied and nuanced conceptions of mathematical communication’.

Alongside this academic investigation into the complex nature of mathematical communication, the richness of the phenomenon is evidenced through the copious training resources and symposia for professional mathematicians to support ongoing development in mathematics communication. In the UK context, among other examples, the Institute of Mathematics and its Applications and the London Mathematical Society both offer events dedicated to developing mathematics communication (IMA 2025; LMS 2026), and The Isaac Newton Institute for Mathematical Sciences dedicates resources to the theme of communicating mathematics. Taken together, the research literature and the proliferation of professional development resources point to the same conclusion: mathematical communication is a research-worthy concept and a non-trivial practice even for professional mathematicians. It is this that provides one of the key justifications for The Academic Language of Mathematics.

Academic Communication in Mathematics: Students as Communicators

The terms mathematics communication and mathematics communicator as used here refer to professionals’ use of language, symbols, and diagrams to communicate mathematics. However, students of mathematics in higher education also engage in communicating mathematics, and it is for this cohort that The Academic Language of Mathematics has been written. In the specifically pedagogical context of this book, the term academic communication in mathematics is therefore preferable to mathematics communication. This allows us to distinguish the typical communicative preoccupations of higher education students from those of professional mathematicians: students are not engaged in all of the activities associated with professional mathematics communication (writing peer-reviewed papers, creating and delivering lectures, conducting seminars, supervising doctoral students), nor are they prototypically communicating to public audiences, writing books for general readers, or undertaking media engagements as public-facing mathematics communicators do. Students’ focus is on written and spoken communication for the purposes of gaining a degree, in the context of this book, a degree taught through the medium of English.

Although in one sense narrower, academic communication in mathematics is no less complex than professional mathematics communication. Murfett and Tran’s (2025) two-part framework of qualities and processes applies as readily to student academic communication as to professional practice, and we return to it in discussing the approach of this book below. The unifying question underlying both can be formulated as: who is communicating what kind of mathematics, to whom, in what context, and for what purpose? There is, accordingly, a genuine overlap between the professional and pedagogical senses of mathematical communication.

Our approach to academic communication in mathematics

In addition to Murfett and Tran’s (2025) framing of mathematics communication noted above, in developing the theoretical underpinnings of The Academic Language of Mathematics, we also draw on Bruce and Ding’s (2026) more general model of academic communication. Based itself on Ferguson’s (1997) tripartite analysis of Language for Specific Purposes (LSP) teaching, Bruce and Ding propose a three-strand model of academic communication as follows:

• Social: the socially-driven communicative purpose of the academy, primarily to advance knowledge in the field
• Epistemological: knowledge of the ontology, epistemology and methodology of a discipline
• Discursive: knowledge of the genres, texts, language and dissemination practices of a discipline

In developing The Academic Language of Mathematics, we have taken inspiration from this model whilst adapting its three categories to the terminological and pedagogical distinction drawn above between mathematics communication (a professional and expert-facing concept) and academic communication in mathematics (a student-facing concept). Two adaptations are necessary. First, the social purpose of student cohorts differs from that of academic practitioners: students are not primarily engaged in advancing knowledge in the field, but in demonstrating a grasp of existing knowledge to expert audiences. Second, given the discipline-specific focus of The Academic Language of Mathematics, we narrow and reorder the three domains to foreground mathematics itself as a discipline. The Academic Language of Mathematics is therefore grounded in the following three complementary and mutually influencing domains:

1. the inherent structures and practices of mathematics as a discipline
2. the linguistic frameworks that shape how meaning is created and communicated
3. the epistemological approaches that guide how student learning is supported in EMI (English as a Medium of Instruction) mathematics contexts.

These three domains are treated as distinct but interdependent. Every discipline has characteristic epistemologies (how knowledge is constructed and validated) and often distinctive ontologies (the kinds of entities it treats as real or discussable). The language of the discipline both reflects and enacts these. At the same time, for a pedagogical resource such as The Academic Language of Mathematics, the third domain is indispensable: an epistemological framework for how students learn, not only what they are learning and how that knowledge is expressed.

The sections below address each domain in turn, drawing on established research in mathematics education, linguistics, and applied linguistics. Together, they explain why The Academic Language of Mathematics is structured the way it is, and why developing mathematical communicative competence requires attention to all three.

A. Disciplinary Foundations: How Mathematics Works as a Discipline

The first pillar of The Academic Language of Mathematics concerns the nature of mathematics itself: how mathematical knowledge is structured, what gives it its distinctive character, and what this means for how its language should be taught. Three principles are central.

The Cumulative, Deductive Structure of Mathematics

Mathematics is unusual among academic disciplines in the degree to which its knowledge is organised cumulatively and deductively (Sfard 1991; Tall 1991). Students must first understand foundational definitions of mathematical objects before meaningfully engaging with theorems, proofs, and applications (Alcock & Simpson 2002).

Contemporary theorising about disciplinary knowledge, most notably in the tradition of Bernstein (1999), developed further by Moore and Young (2001) and Young and Muller (2010, 2014), distinguishes disciplines according to the structure and grammar of their knowledge. Disciplines such as the physical sciences are characterised by hierarchical knowledge structures, in which new findings are progressively integrated into more general theoretical propositions. The humanities and social sciences, by contrast, tend towards horizontal knowledge structures, in which knowledge develops as a series of specialised languages operating alongside one another. Mathematics occupies a distinctive and perhaps surprising position in this framework: Bernstein classifies it not as a hierarchical discipline but as a horizontal knowledge structure (a set of discrete symbolic languages for particular problems) but one with the strongest grammar of any field, stronger even than the sciences, precisely because its languages do not require empirical referents and derive their authority entirely from formal rigour and deductive proof (Bernstein 1999). It is this combination of horizontal structure and exceptionally strong grammar that accounts for the cumulative, definitionally grounded character of mathematical knowledge and, consequently, for the pedagogical sequencing that The Academic Language of Mathematics reflects.

There is a small but growing body of work which attempts to understand how mathematicians view these structures in relation to communication and teaching, including emerging research on the nature of language, writing, and communication in mathematics as a discipline and within mathematics education (Cho 2015; Morgan et al. 2014). As noted above, we will continue to draw on Murfett and Tran’s (2025) work which identifies two organising categories in the conceptions of mathematical communication held by academic mathematicians: qualities (‘what mathematical communication consists of’) and processes (‘how academics put these ingredients together’). In constructing The Academic Language of Mathematics we have drawn on all six categories identified by Murfett and Tran, as in the following table. The left-hand column defines each Quality and Process from Murfett and Tran; the right-hand column indicates how The Academic Language of Mathematics addresses each one in turn.

For practical examples of how these qualities and processes appear in student-facing materials, see How This Book Is Organised and How to Use This Book.

The six categories in Murfett and Tran’s framework thus map directly onto the content and structure of The Academic Language of Mathematics. A further structural consequence of the cumulative nature of mathematical knowledge concerns pedagogical sequencing. Students cannot engage productively with the language of Taylor series before they have encountered the language of derivatives; the language of integration presupposes the language of limits. The Academic Language of Mathematics is therefore structured to follow the logical progression of mathematical knowledge, covering rudimentary mathematical topics foundational for all STEM subjects and beyond. Each chapter addresses one mathematical topic in the order in which it is encountered in that curriculum, with each chapter following the same internal structure throughout, as described in How This Book Is Organised, moving from smaller to larger units of language and communication.

Mathematical Abstraction and Decontextualisation

Disciplines vary widely from one another, and theorising these differences has a long history. Aristotle, in Metaphysics, divided knowledge into three categories: theoretical (e.g. mathematics), practical (e.g. politics), and productive (e.g. fine art). Medieval Europe organised knowledge into the trivium (logic, grammar, rhetoric) and the quadrivium (arithmetic, geometry, astronomy, and music); a division that notably did not treat mathematics as a single unified discipline. Contemporary theorising about disciplinary distinctions is more nuanced and evidence-informed, with multiple frameworks addressing the question from different angles.

One productive framework for understanding the distinctiveness of mathematics is Legitimation Code Theory (LCT), and in particular its concepts of semantic gravity and semantic density (Maton 2014). Semantic gravity (SG) refers to the degree to which knowledge is dependent on its context. Strong semantic gravity characterises disciplines in which knowledge is richly embedded in real-world examples and specific contexts (sociology, history, and practice-based fields such as nursing, for instance). Weak semantic gravity, by contrast, characterises disciplines in which knowledge is more abstracted from context; mathematics and the formal sciences being the clearest examples. Mathematical truths hold regardless of the context in which they are applied: the Pythagorean theorem is equally valid in architecture, astronomy, and carpentry, and its validity is not grounded in any of those applications. Semantic density (SD) refers to the degree of condensation of meaning within a symbolic or linguistic unit. Mathematics exhibits extremely strong semantic density: a single symbol or expression can compress a complex relationship or operation into a highly compact form, with each element carrying substantial inferential weight.

These two properties of mathematical knowledge (weak semantic gravity and strong semantic density) have direct implications for language instruction. Because mathematical knowledge is weakly context-dependent, situating language instruction in contrived real-world scenarios risks misrepresenting the discipline: it suggests that the validity of mathematical results depends on their application, when it does not. The Academic Language of Mathematics therefore grounds language instruction in the internal logic of mathematical discourse itself, in the linguistic functions that English serves within that discourse (defining, hypothesising, proving, concluding), rather than in artificial contextual framings. Because mathematical knowledge is strongly semantically dense, students need sustained engagement with the specific linguistic resources that make dense mathematical language interpretable: how to unpack a definition, how to read a symbolic expression in relation to its verbal framing, how to follow the logical steps of a proof. This is not an argument against application: many mathematical topics do have important real-world applications, and these are acknowledged. It is, rather, an argument for ensuring that language instruction respects the internal logic of mathematics rather than distorting it.

The Role of Language in Mathematical Communication

Following Murfett and Tran (2025), mathematical communication is best understood as multimodal: meaning is constructed through the coordinated use of symbolic notation, human language, and visual/diagrammatic representations, often alongside other semiotic resources (e.g., layout, gesture, digital tools). This is why research in mathematics education increasingly treats language not as a single channel but as part of a broader communicative system in which words and symbols are only part of what is going on and must be analysed in relation to other modalities.

Within this multimodal system, symbolic notation is central, but it does not operate in isolation: it is interpreted, framed, and made communicatively usable through discourse, genre conventions, and representational choices. In academic mathematics, writers and speakers routinely use natural language to define objects, signal logical relations, guide readers through inferential steps, delimit cases, and present argument structure; functions that are indispensable for participation in mathematical practices. In other words, notation can compress meaning efficiently, but language and other semiotic resources help establish what the notation is taken to mean, how it should be read, and why a step is justified within an argument.

For EMI contexts, these insights have a direct implication: students may have robust mathematical knowledge and procedural competence, yet still experience difficulty because the English‑medium discourse practices of mathematics (how definitions are worded, how assumptions are stated, how proofs are staged, how reasoning is narrated, and how diagrams and symbols are integrated into coherent argument) are not automatically available to them. Mathematics education research also notes that an under‑developed area has been the explicit development of the linguistic competencies required for participation in mathematical practices (Morgan et al. 2014), which is precisely the gap a resource like The Academic Language of Mathematics aims to address.

Accordingly, The Academic Language of Mathematics treats English not as a mere add‑on to mathematics, but as one of the resources through which mathematical practices become communicable and assessable in academic settings. The book therefore supports students in (i) linking established conceptual knowledge to English mathematical terminology, (ii) recognising and producing the recurrent language patterns that realise key mathematical rhetorical functions (defining, hypothesising, inferring, concluding), and (iii) integrating language with symbolic and visual representations in discipline‑recognisable genres. The bilingual glossaries at the end of each chapter are one practical mechanism to support this bridging work, but they operate within a wider goal: developing multimodal mathematical communicative competence in English‑medium higher education.

B. Linguistic Foundations: How Language Works

The second pillar of The Academic Language of Mathematics concerns the nature of language itself: how it is structured, how it carries meaning, and how it functions in academic mathematical contexts. Eight interconnected principles from linguistics and applied linguistics inform the design of every chapter.

B1. Language as a Hierarchy of Levels

A well-established finding of modern linguistics is that language operates simultaneously at multiple structural levels, which can be conceptualised hierarchically (Bloomfield 1933; Bolinger 1975; Fromkin, Rodman & Hyams 2014; Genetti 2014).

This hierarchical model has direct implications for The Academic Language of Mathematics. Rather than treating language as a collection of isolated words to be learnt separately, the book addresses language across all levels of the hierarchy; from the pronunciation and morphological structure of individual terms through to the organisation of complete mathematical texts. The progression from lower to higher levels reflects a broader pedagogical principle: that language competence develops from the recognition of individual forms towards the ability to communicate appropriately and effectively in authentic disciplinary contexts.

B2. Lexical bundles

Recent work on mathematical lexis includes Drury et al. (2022), who constructed a corpus of mathematical lexis drawn from spoken language in lectures, categorising items into four types. Whilst representing a valuable advance in understanding mathematical vocabulary, the corpus restricts itself to mono-lexical items. The Academic Language of Mathematics, by contrast, is grounded in the recognition that mathematical language is characteristically multi-lexical, and therefore draws extensively on the notion of the multi-word unit (MWU) or lexical bundle (Sinclair 1991; Salazar 2014; Walková 2024). A lexical bundle is a recurrent multi-word unit (such as we define… as, it follows that, or let X be) that occurs together more frequently than chance would predict. Lexical bundles are the functional building blocks of academic discourse; competent academic writers do not construct language word by word but draw on a repertoire of these ready-made units.

The nature of the lexical bundle implies co-occurrence relationships such as collocations, as well as morphological variation across any given lexeme. For example, the lexical bundle differentiate the function exhibits a co-occurrence relationship between the verb differentiate and the noun phrase the function. However, the same two-word roots recur in the phrase non-differentiable function with a change in both morphology and syntax, as well as the addition of the prefix non-. In The Academic Language of Mathematics, morphological and collocational dynamics are therefore integrated within the lexical bundles introduced to students, rather than treating vocabulary items as isolated units devoid of their combinatorial properties.

B3. Constructions as Schematics

Pedagogical Construction Grammar (Goldberg 2006; Ellis 2001; Boas 2022) offers a complementary perspective on how language is acquired and used. Rather than learning language as a set of rules applied to individual words, learners develop a repertoire of constructions; learned pairings of form and meaning that range from individual words through fixed phrases to complex syntactic patterns. A construction such as let X be… then Y is not merely a sequence of words; it is a schematic unit carrying a specific logical and communicative function; the set-up for a mathematical argument. Presenting language as constructions gives students a more powerful and transferable tool than memorising individual items, because a construction can be recognised and adapted across many different mathematical contexts.

The Academic Language of Mathematics uses constructions as a key unit of language instruction, presenting language patterns schematically so that students can both recognise them in texts they encounter and deploy them appropriately in their own mathematical writing and speech. The schematic representation of constructions (showing which elements are fixed and which can vary) supports students in moving from imitation of specific examples to flexible, independent use.

B4. Systemic Functional Linguistics (SFL)

Systemic Functional Linguistics (Halliday 1978; Halliday & Hasan 1989; Halliday & Matthiessen 2014) provides a framework for understanding how language choices simultaneously serve three functions: the ideational (representing the content of ideas), the interpersonal (enacting relationships between writer and reader), and the textual (organising discourse into coherent messages). This multi-functional perspective is particularly productive for mathematical language, where the same text must communicate precise mathematical content, position the writer within a disciplinary community, and organise ideas into a logically coherent sequence.

SFL’s attention to how grammatical and lexical choices realise meaning provides the theoretical grounding for how The Academic Language of Mathematics treats register and genre. The characteristic grammatical features of mathematical writing (high levels of nominalisation, passive constructions, impersonal syntax) are not arbitrary stylistic preferences; they are realisations of the discipline’s values of precision, objectivity, and logical rigour. Making these connections explicit helps students understand not only what mathematical language looks like but why it takes the forms it does.

B5. Genre Analysis

Genre analysis is well established in EAP contexts (Bruce 2008; Swales 1990; Swales & Feak 2004; Walková 2024). A genre is a recognisable type of text with a characteristic communicative purpose, structure, and set of language choices: mathematical proofs, definitions, problem statements, and worked examples are each distinct genres with their own conventions. Genre analysis examines the internal organisation of texts (the rhetorical moves through which communicative goals are achieved) and connects formal features to their functions for particular audiences and purposes.

In The Academic Language of Mathematics, each chapter includes a dedicated section (Section 6 of each chapter) in which students engage with an example of authentic mathematical text. Each text is representative of a genre that students are likely to encounter in their academic studies, and an intentional effort has been made to vary the genres across chapters both for interest, and to allow a variety of rhetorical and other features to be examined. Genres covered include an announcement from a head of a school of mathematics; an assessment brief; a call for papers for and participation in a student conference; an entry from a mathematics encyclopaedia; and an invitation to a mathematics talk. The activities adopt a Swalesian approach, focusing on rhetorical moves within a text and linking form and function to audience and purpose. Students are thereby equipped not only to understand the texts they read but to analyse and reproduce the generic structures they employ.

B6. Grammaring

Larsen-Freeman’s (2001, 2015) concept of grammaring reframes language competence as a dynamic skill rather than a static body of knowledge. True competence involves using linguistic forms accurately, meaningfully, and appropriately for communicative purposes; not merely knowing that a construction or bundle exists, but being able to deploy it flexibly in new contexts. Development moves non-linearly through a progression from noticing a pattern, through understanding its form-meaning logic, to practising it in contexts that serve a genuine communicative purpose, to applying it autonomously in novel situations.

The non-linearity of this process is pedagogically significant: students may simultaneously notice a new pattern, consolidate understanding of a previously encountered one, and deploy a well-established form with confidence. This means that revisiting language is not repetition for its own sake but a necessary feature of how competence develops. The Academic Language of Mathematics is designed with this developmental trajectory in mind (language is encountered repeatedly across different mathematical contexts and at increasing levels of cognitive demand) precisely because this is how genuine communicative competence is built.

B7. Fluency, accuracy, complexity and appropriacy (FACA)

Walková (2024) organises her account of linguistic approaches to EAP around three interconnected dimensions, which together describe what it means to be a fully competent language user:

1. fluency and range of expression
2. accuracy and complexity, covering both morphological and syntactic dimensions
3. appropriateness, encompassing genre conventions related to audience and purpose, as well as broader conventions of precision and effectiveness.

These three dimensions inform both the selection of language forms and the range, type, and ordering of activities within each chapter of The Academic Language of Mathematics. A focus on accuracy alone produces formally correct but communicatively limited language; fluency without accuracy produces confident but imprecise expression; neither without appropriacy produces language fit for mathematical academic contexts. The breadth of coverage afforded by FACA offers pedagogical benefits through its very variety, whilst linguistically it allows the material to capture the complexity and systematicity of academic mathematical language as a set of hierarchically ordered systems that mutually influence one another.

B8. Intercultural Rhetoric

Intercultural rhetoric, developed by Connor (1996, 2004, 2008) and colleagues and applied widely to student academic writing, for example, Ene et al. (2019), and Liu and Du (2018), examines how the rhetorical and writing patterns shaped by students’ first language and prior academic culture interact with the conventions of English-medium mathematical discourse. Connor’s framework reconceptualises earlier work in contrastive rhetoric, moving away from a deficit model towards a recognition that L1-shaped writing patterns reflect different but equally valid rhetorical traditions. A student whose academic formation has involved indirect argumentation, collective epistemic positioning, or different conventions for establishing proof is not writing incorrectly; they are drawing on a coherent rhetorical logic.

For The Academic Language of Mathematics, this has a direct implication: students’ prior academic writing experience and rhetorical knowledge are assets to be engaged, not errors to be corrected. Activities build bridges between L1 rhetorical conventions and English-medium mathematical discourse, making explicit the conventions of the latter whilst respecting and building on the former. This approach connects directly to the book’s broader commitment to treating students’ mathematical knowledge in their first language as a resource; a commitment expressed practically in the bilingual glossaries at the end of each chapter.

C. Epistemological Foundations: How We Support Learning

The third pillar of the framework underpinning The Academic Language of Mathematics concerns how students learn. Six interconnected principles shape the design of every chapter (listed here in alphabetical order).

C1. Academic socialisation

The academic socialisation perspective (Wenger 1998) recognises that learning mathematics is not simply a matter of acquiring facts and procedures; it is a process of becoming part of a disciplinary community with its own ways of thinking, arguing, and communicating. This involves learning not only what mathematicians say but how and why they say it: how arguments are structured, what counts as proof, how precision and rigour are expressed in language. The Academic Language of Mathematics makes these community practices visible and explicit, because for many students, particularly those working in a second language, they cannot be assumed. At the same time, following Lea and Street’s (1998) academic literacies perspective, the book does not treat these conventions as neutral or self-evidently correct; rather, it supports students in understanding them as disciplinary choices that can be examined and questioned; an approach developed further in the Criticality section below.

This is especially significant for students entering English-medium mathematics programmes from different educational traditions. The conventions of mathematical discourse (how a definition is introduced, how a proof is structured, what hedging is appropriate when conjecturing) are not universal; they are disciplinary norms that must be learnt. The Academic Language of Mathematics treats each chapter’s genre text and language activities as an apprenticeship into these norms, making the implicit expectations of the mathematical community accessible to all students, regardless of their prior educational context.

C2. Agency, autonomy, and activation

Deci and Ryan’s (2000) self‑determination theory rests on the principle that effective learning requires students to be active participants, not passive recipients. Throughout The Academic Language of Mathematics, activities ask students to make choices about language, justify those choices, engage with peers, and take responsibility for their own understanding. The goal is not to produce correct answers to closed exercises but to develop the capacity to make independent, informed language decisions.

This principle carries particular weight in mathematical language learning. Mathematics has a strong procedural tradition in which correctness is paramount, and students accustomed to that tradition may expect language learning to operate similarly: with single correct answers and clear rules. The Academic Language of Mathematics deliberately disrupts this expectation by including tasks that require students to compare formulations, evaluate alternatives, and articulate preferences. The aim is to develop the kind of reflective language awareness that allows students to make genuine communicative choices, rather than simply reproducing prescribed forms.

C3. Authenticity

This is the principle that the language, texts, and tasks in The Academic Language of Mathematics reflect  genuine mathematical communication; real texts, real genres with real communicative purposes (Gilmore 2007). Students encounter the language that mathematicians actually use, in the contexts in which they use it. This ensures that what is learnt is directly transferable to the lectures, textbooks, seminars, and written work students encounter throughout their degree.

In the context of mathematical discourse, authenticity has a specific meaning: it is not sufficient for a text to be real if it does not represent the genuine communicative purposes of the discipline. A worked example written to illustrate a grammatical point, but which no mathematician would ever produce, is not authentic in the relevant sense. The genre texts in Section 6 of each chapter are therefore selected or composed to reflect the actual genres of mathematical academic communication, preserving the rhetorical logic, register, and conventions of each.

C4. Bloom’s revised taxonomy

Anderson and Krathwohl’s (2001) revised taxonomy of educational objectives provides the framework for sequencing activities across cognitive levels, from routine tasks (remembering a definition, understanding notation) to non-routine ones (analysing the structure of a proof, evaluating competing formulations, producing original mathematical writing). Activities at the higher levels (analyse, evaluate, create) are not optional extensions; they are where genuine communicative competence develops. Every chapter of The Academic Language of Mathematics moves through this progression.

In the context of mathematical language learning, this progression has a specific shape. At the lower levels, students identify and reproduce the linguistic forms associated with mathematical objects and operations. At the intermediate levels, they examine how those forms function within mathematical arguments: how a conditional construction signals a logical relationship, or how nominalisation compresses a process into an object that can then be operated on. At the higher levels, students evaluate alternative formulations, analyse the rhetorical organisation of complete texts, and produce their own mathematical writing. This trajectory mirrors the cumulative structure of mathematics itself: each level of language competence presupposes and builds on the one below.

C5. Criticality

Criticality is grounded in critical EAP (Benesch 2001) and critical pragmatic EAP (Harwood & Hadley 2004). This reflects the principle that students are not simply taught to imitate the conventions of mathematical discourse but are supported in understanding those conventions as social choices that can be examined and questioned. The Academic Language of Mathematics supports students in participating effectively in mathematical communication and in reflecting on why it works the way it does.

In practice, this means asking questions that go beyond correctness: why does mathematical writing typically avoid the first person? What work does the phrase it can be shown that perform, and what does it conceal? Why is the passive voice so prevalent in definitions and proofs? These are not merely stylistic questions; they are questions about how knowledge is constructed, and authority is established in mathematical discourse. Engaging with them equips students not only to reproduce the conventions of mathematical writing but to understand them and, where appropriate, to make deliberate choices about when and how to deploy them.

C6. Reflexivity

Understood here as the epistemological reflex of cause-effect circularities and as an aspect of criticality, The Academic Language of Mathematics endeavours to integrate reflective-critical tasks in which students question themselves and the activities they are doing. Following Alley et al. (2015), we view reflexive practices as disruptive to the traditional unidirectional model of knowledge translation (KT), adopting instead a dynamic, self-referential, critical and contextualised view of KT. As unidirectional KT processes may be held to dominate mathematics pedagogy, the decision to integrate reflexive practice into The Academic Language of Mathematics may be seen as potentially innovative.

For students who have previously studied mathematics in another language and educational culture, reflexivity has an additional dimension: it invites them to notice where their prior knowledge and the conventions of English-medium mathematical discourse converge and where they diverge. Rather than treating such differences as deficits to be corrected, The Academic Language of Mathematics uses them as productive starting points for reflection; for example, asking students to compare how a concept is expressed in their first language with how it is expressed in English, and to consider what each formulation reveals about the underlying mathematical thinking. In this way, reflexivity connects directly to the book’s broader commitment to building on, rather than replacing, students’ existing mathematical knowledge.

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