This section provides brief guidelines for the three primary groups of people for whom The Academic Language of Mathematics is intended:

1. Students studying mathematics as part of their degree, through the medium of English.
2. Academics from STEM disciplines (such as Mathematics, Physics, Engineering, and related subjects) whose focus is on the discipline of mathematics.
3. Academics from English for Academic Purposes (EAP) whose focus is on academic language and text.

For colleagues interested in the theoretical and pedagogical foundations of the book, see Theoretical Underpinnings. For evidence of student outcomes, see Testimonials.

The Academic Language of Mathematics is open access: it is freely available online, and you can access, download, and share it at no cost.

It is also designed to be interactive: the activities require active engagement rather than passive reading. Throughout the book you will find opportunities to:

• engage with simulated authentic mathematical texts
• practise expressing mathematical ideas in writing and speech
• collaborate with classmates on mathematical language
• reflect on your own language development

We strongly encourage collaboration. Learning mathematics can be a profoundly collaborative endeavour, and working with others helps you develop both communication skills and deeper mathematical understanding.

Structure of Each Chapter

Each chapter consists of seven sections. The early sections introduce mathematical vocabulary in isolation and then in context, before later sections develop the student’s ability to read, analyse, and produce extended academic texts in a range of mathematical genres. Audio recordings accompany key exercises throughout and video recordings accompany the dialogues. Full answers are provided for all activities, making the book equally suitable for classroom use and self-study. For a detailed description of the seven sections in each chapter, see How This Book Is Organised.

Reading Pathways

This book can be used in different ways depending on your role, needs and learning or teaching style. Whichever pathway you choose, it is worth noting that the eight chapters cover rudimentary mathematics that almost all students in STEM disciplines encounter as part of their degree. This means the chapters are relevant regardless of your specific course or specialism, and dipping into a chapter relevant to your current studies will still equip you with widely applicable language skills.

Note that you can search through the whole book using the search function in the top-right corner of each page.

Linear reading: This means working through the book chapter-by-chapter from beginning to end. Each chapter builds on concepts introduced in earlier chapters, giving you the most structured and progressive experience of the book.

Exploratory reading: This means making use of the chapters which address topics you are currently studying/teaching in your mathematics courses. Individual chapters are designed to stand alone in large part, though you may find it helpful to return to earlier chapters for reference, particularly in terms of the use of notation.

Iterative reading: This means taking a cyclical approach to the book. Work through some exercises in a chapter, then move on before returning to it later. Revisiting material you have already encountered strengthens your understanding of both language and mathematical concepts over time, and returning to a chapter after covering the same topic in your mathematics course will deepen your engagement with both.

Reference use: Use the glossaries and concept maps to locate specific terminology or language patterns. The search function helps you find relevant information quickly.

For students

The Academic Language of Mathematics supports your learning, understanding, and use of academic language in mathematics. You may be using this resource as part of a mathematics module or course, as part of academic English seminars, or as a self-study resource. You may be using it individually or in a small group. Each chapter offers graded exposure to mathematical concepts and language, and comprehensive answers and audio recordings are provided for every exercise, making independent study fully supported.

Consider whether you work best alone or with others (or a mix of both!), bearing in mind that mathematics is a collaborative discipline and that academic communication in mathematics is always an interaction. You may work through each chapter systematically, or dip into areas that are most relevant to your current study. As each chapter progresses from recognising and understanding language to using it actively in writing and discussion, working through each chapter in order will give you the most structured experience of the book.

Mathematics modules usually get harder as you move through your degree. But this book does not work in the same way. The Academic Language of Mathematics is organised around language skills, not around your year of study. This means your choice of what to read depends more on what you want to improve regarding the academic language of mathematics, than on how difficult your mathematics modules are. Even a strong mathematics student can benefit from the early parts of each chapter, where the focus is on key words and terminology. And a student who is still building their academic English may also benefit from later parts of each chapter, once they feel ready. In general, though, if you are early in your journey within the academic language of mathematics in English, the early sections in each chapter, which focus on terminology and notation, are likely to feel more useful at first. The later sections, which focus on dialogues, emails, and longer explanations, use richer and more complex language.

With this in mind, here are some simple starting points, though you should feel free to follow your own sense of what will help you most.

• If you are a first-year undergraduate, it often works well to start with Chapter 1 and move through the chapter in order, since this helps you build up key terminology step by step. If you already feel confident with basic vocabulary, however, you can move straight to the later sections of each chapter.
• If you are in your second or third year, you do not necessarily need to follow the chapters in order, if you are already comfortable with the mathematical language in them. Instead, choose the chapter that matches your current mathematics topic. Within that chapter, use the early sections if you want a quick reminder of key terms, or the later sections if you want to practise explaining your reasoning, for example in coursework, reports, or oral exams.
• If you are starting a postgraduate degree, you may already know a lot of mathematical terminology from your previous studies. In this case, the later sections of each chapter will probably be more useful, since they focus on how to write and speak about mathematical topics in more detail.

You do not need to complete every part of a chapter before moving on. Use the early sections when you want quick reference, and the later sections when you have more time to focus on how mathematical ideas are explained in English. Above all, choose what matches your own language goals, not simply your year of study.

Getting the most from each chapter

• Preview the chapter title and section headings and consider what mathematical topics and language you will encounter.
• Engage actively: write notes, discuss with classmates, and try explaining concepts aloud.
• Notice patterns in how mathematics is expressed, for example, the term illogical is composed of three parts: il + logic + al (this is known as morphology – less technically we might say the internal structure of words); start noticing similar patterns in mathematical terminology.
• Use the bilingual glossaries: mathematical terminology often has precise equivalents across languages, and knowing both helps deepen understanding.
• Keep a learning journal. See the section on Reflection Prompts and Journals below for guidance.
• Use the diagrams and pronunciation exercises in each chapter. See the section on Diagrams and Visual Aids and Pronunciation Exercises below for strategies.
• Watch the videos accompanying the dialogues in the fifth section of each chapter. See the section on Videos below for guidance.

Diagrams and Visual Aids

Throughout the book, you will find diagrams illustrating mathematical concepts with interactive labels. These serve multiple purposes:

• Help you understand the mathematical concept visually.
• Show you how mathematical ideas are expressed in writing.
• Provide opportunities to practise labelling and explaining diagrams.

As you work with each diagram, try to predict what a label might be before reading it; this active engagement will help you retain the terminology. Diagrams appear primarily in the fourth section of each chapter. For a structural description of that section, see How This Book Is Organised.

Reflection Prompts and Journals

Reflection activities invite you to think about your own language development. We recommend keeping a learning journal where you record:

• New vocabulary and phrases you encounter with relevant notes about their structure and use.
• Observations about how mathematics is expressed differently in different contexts.
• Your questions about language or mathematics.
• Your own progress in understanding and communicating mathematical ideas.

A learning journal is a personal record that you do not need to share with anyone. Even brief notes (a new term, a phrase you noticed, a question you could not yet answer) will help you consolidate what you are learning and track your own development over time.

Pronunciation Exercises

Reading mathematical notation aloud is a key skill for lectures, tutorials, and discussions with classmates. The third and fourth sections of each chapter provide dedicated pronunciation practice with audio recordings, and the video dialogues in the fifth section offer further listening and speaking opportunities. As you work through these exercises, try the following strategies:

1. Say it slowly first. Before attempting to read an equation at normal speed, read it very slowly, pausing between each symbol or word.
2. Record yourself. Use your phone or computer to record yourself reading equations. Compare your recording with the model answers provided.
3. Read with a partner. Take turns with a classmate, listening to how they read equations and learning from each other. Many activities in the book are designed for pair practice.
4. Pay attention to linking. Notice how words connect in natural speech. For example, the integral of flows as one phrase, not three separate words.
5. Do not worry about perfection. Pronunciation improves with practice. The goal is to be understood, not to sound like a native speaker. Mathematics and the English language are both global phenomena and are owned by everyone equally.

For a full description of what the different sections of the book cover, see How This Book Is Organised.

Videos

The fifth section of each chapter is centred on one or two spoken dialogues, each accompanied by a video showing the dialogue being performed. We strongly recommend watching the video before working with the activities. Seeing the dialogue enacted will give you a richer understanding of both the language and the context.

The videos offer more than audio alone. As you watch, notice the speakers’ body language and gesture: mathematical explanation often involves physical movement (pointing, writing, indicating shape or size) and observing this will deepen your understanding of how mathematical ideas are communicated. Pay attention to facial expressions, which signal emotions such as confusion, hesitation, and understanding that are central to academic dialogue.

We recommend watching each video more than once as you work through the activities:

1. Watch the whole dialogue first to get an overall sense of the exchange.
2. Then return to specific sections, particularly any you did not fully understand or where the pronunciation was unclear; replaying even a few seconds of a short section is a recommended practice.
3. Finally, watch the whole dialogue again after completing the activities to consolidate what you have learned.

We live in a global world, and the practice and teaching of mathematics in the 21st century is a transnational endeavour. As The Academic Language of Mathematics is a freely accessible online resource, you may wish to consider forming an international online study group. This would expose you to a wider range of Englishes and to teaching and learning practices from mathematics communities around the world. Approached in this way, the book offers a framework for truly international collaborative study.

We hope that The Academic Language of Mathematics leaves you with greater familiarity, control, and confidence in the academic language of mathematics, making your studies, and your future use of mathematics in employment or further study, more effective and enjoyable. For examples of how students and lecturers have used the book in practice, see Testimonials.

Good luck!

For Academics Teaching Mathematics

The Academic Language of Mathematics is designed primarily as a complementary text for academics teaching mathematics within STEM disciplines. Practical uses include:

• A short warm-up or close-down activity in a lecture to activate key terminology.
• A homework task alongside problem sheets: assign a section at the end of a lecture and revisit it at the start of the next, perhaps in quiz form.
• Pair or group activities in seminars; most exercises across all sections adapt readily to pair or group work.
• A basis for integrated activities that address both mathematical and communicative competence.
• As a resource to which to signpost individual students who may benefit from particular exercises.

The fifth section of each chapter focuses substantially on interactional communication and is particularly valuable for developing students’ confidence and competence in academic dialogue.

If you are co-teaching with an EAP practitioner, it is worth considering how to divide use of the book between content-focused and language-focused roles. You might identify mathematical topics your students find linguistically challenging and use relevant chapters as bridge materials, with EAP colleagues introducing language features and mathematics colleagues reinforcing them in context. We hope the book provides a productive basis for this kind of interdisciplinary collaboration.

When deciding which sections to use with a given cohort, it is worth noting that The Academic Language of Mathematics is organised primarily around language competency rather than year of study or mathematical difficulty. A mathematically strong final-year student may still benefit from the terminology-focused earlier sections of a chapter, particularly if English is not their first language, while a first-year student with strong academic mathematical language may be ready for the later, more discourse-focused sections. We would therefore encourage you to gauge suitability by your students’ familiarity with academic mathematical language, rather than assuming that later-year or more advanced mathematics students will not benefit from the early sections of each chapter.

Above all, we encourage you to experiment with what to use, when, and how, and to enjoy discovering the ways in which content knowledge and academic language can interact for students’ benefit.

For Academics Teaching Academic English

The Academic Language of Mathematics offers multiple affordances for EAP practitioners. Whilst the earlier sections assume some mathematical knowledge, all activities are highly structured and scaffolded, and answers are provided throughout. We therefore encourage EAP practitioners to engage confidently with the earlier sections, perhaps working through exercises collaboratively with students when concepts are less familiar.

If you have no prior familiarity with mathematical language or notation, Chapter 1: General Terminology is a good starting point, as it begins with concepts that will be familiar to most readers, providing a secure platform from which to build. The book is also useful for supporting students who are capable mathematicians but need more support with language. Assigning specific sections that address the language barriers your students encounter is a straightforward way to integrate the book into your teaching.

As a general guide, students with an IELTS Academic score in the region of 5.5 to 6.5 (often, though not always, those in the earlier stages of undergraduate study) are likely to benefit most from the terminology- and notation-focused earlier sections of each chapter. Students with stronger English proficiency, including those preparing for postgraduate study, may be ready to move more quickly to the later sections of each chapter, even if their mathematical background is more limited. These figures are offered only as a loose starting point; your own assessment of a student’s academic English, rather than their year of study or IELTS score alone, remains the better guide.

The final two sections of each chapter focus on academic communication and text. Though richly informed by mathematical terminology, they cover more general academic language features and are likely to feel most immediately familiar to EAP practitioners. Whatever your level of mathematical familiarity, we encourage adaptation: feel free to add to and modify the materials. One broadly applicable approach is to ask students to supply their own examples of the concepts and language covered (for instance, in the part on sets in Section 1.3, one can invite real-world examples of set relationships such as proper subset, superset, or complement).

Finally, on the reasonable assumption that most EAP practitioners are coming to the academic language of mathematics as relative non-specialists, we hope that The Academic Language of Mathematics will be a source of learning and inspiration around the discipline of mathematics. Of the three co-authors, one is an EAP practitioner with no specialist background in mathematics. The preparatory work for, and the co-writing of, this book has been a journey into the rich and complex world of mathematics; not always an easy one (a sentiment many mathematicians might share), but a rewarding one nonetheless. Our hope is that The Academic Language of Mathematics offers EAP practitioners a structured and enjoyable way into mathematics itself, alongside the academic language of the discipline.