The Academic Language of Mathematics is an eight-chapter open-access textbook primarily designed to help students studying mathematics through the medium of English develop their mathematical language and communication skills. Chapter 1 introduces general mathematical terminology that is relevant across all topics, whilst Chapters 2-8 each focus on a specific area of mathematics. The book progresses systematically from smaller to larger units of language: early sections focus on individual terms and notation, whilst later sections develop your ability to engage with extended spoken and written mathematical communication. No prior knowledge of linguistics is required; the book is designed to be accessible to any student encountering mathematics as part of their degree.

The eight chapters are as follows:

• Chapter 1: General Terminology
• Chapter 2: Differentiation
• Chapter 3: Integration
• Chapter 4: Sequences and Series
• Chapter 5: Complex Numbers and Functions
• Chapter 6: Vectors and Coordinate Systems
• Chapter 7: Lines and Planes in 3D Space
• Chapter 8: Systems of Equations and Matrices

Each chapter can be accessed directly from the main contents page on the left. For guidance on choosing a reading pathway through these chapters, as well as how each chapter works in practice and strategies on making the most out of this resource, see How to Use This Book.

Why these mathematical topics?

The chapter sequence reflects both the structure of a typical undergraduate mathematics module and the kinds of language that students most often need to use. Chapter 1 introduces core terminology that recurs across the discipline. Chapters 2, 3 and 4 focus on differentiation, integration, and sequences and series because these areas of calculus and analysis are taught early in many degrees, involve a rich mixture of notation, diagrams, and technical vocabulary, and provide natural contexts for practising how to describe change, accumulation, and limiting behaviour in English. Chapters 5, 6, 7 and 8 then extend this work to complex numbers, vectors, three-dimensional geometry, and matrices, which are central to later study in mathematics, physics, engineering, and data science, and exemplify further genres of mathematical communication, such as modelling explanations, geometric descriptions, and matrix-based reasoning. Together, these eight topics cover rudimentary mathematics that all students in STEM disciplines encounter as part of their degree, so the language studied in this book will support learning across a wide range of modules.

Structure of Each Chapter

Why is the book structured like this?

The book is organised thematically. Each of the eight chapters is dedicated to a particular mathematical topic and contains 7 sections. Within each chapter, the overall structure moves from ‘smaller’ to ‘larger’ units of language. The first two sections introduce you to technical terminology relevant to the mathematical topic. The third and fourth sections embed this language within mathematical notation and formalisms, principally equations and their constituent terms. The fifth and sixth sections focus on text-level language: spoken dialogues, email exchanges, and longer written texts. The final section is a glossary.

Throughout each chapter, you will encounter two types of focus box.

Description of each section

Section 1: Technical Terminology. The opening section introduces key mathematical terms relevant to the chapter topic, studied in isolation. Two activities develop your familiarity with each term’s meaning and form, including its spelling, pronunciation, and typical notation: the first asks you to match terms to definitions; the second asks you to select the more accurate of two descriptions for each term.

Section 2: Terminology in Context. Building on Section 1, this section develops your understanding of mathematical terminology and STEM expressions by studying them in context, i.e. within sentences and longer texts. A fill-in-the-gap activity asks you to complete mathematical texts using terms from Section 1 and the glossary. A further activity focuses specifically on STEM expressions, that is, multi-word phrases that are used across scientific disciplines.

Section 3: Introduction to Notation. This section develops your ability to read mathematical notation aloud; a key skill for lectures, tutorials, and mathematical discussion. Using the topic of the chapter as a context, the section introduces the relevant symbolic notation systematically, providing multiple ways to read each symbol or expression. Every example is supported by a voice recording.

Section 4: Equations and Diagrams. Building on Section 3, this section extends your ability to read mathematical notation aloud by applying it to complete equations. Three activities develop this skill progressively: the first presents equations for you to read aloud, with correct phrasings and voice recordings provided; the second reverses the process, asking you to convert spoken or written English descriptions into mathematical notation; the third asks you to label mathematical diagrams using the correct technical terms.

Section 5: Mathematics Communication. This section shifts the focus to interactional communication about mathematics. Two extended examples of mathematical communication are provided (either two spoken dialogues or one dialogue and one email exchange), each set in a realistic academic context, such as a peer study session or a student-lecturer exchange. A range of activities accompanies each text, developing your global understanding of the communication, your ability to identify and analyse useful phrases and sentence structures, and your confidence in producing similar conversations yourself.

Each spoken dialogue is accompanied by a video recording, designed to support a fuller understanding of the interaction and to make the academic context more immediate and realistic. You are encouraged to watch the video before working with the activities, as observing the dialogue being performed can provide important information beyond the words themselves. In addition to supporting listening comprehension and pronunciation practice, the recordings allow you to attend to features such as body language, gesture, facial expression, and the use of physical movement in mathematical explanation. Repeated viewing is recommended: you may first watch the whole dialogue to understand the overall exchange, then return to specific sections for closer listening and analysis, and finally watch the dialogue again after completing the activities to consolidate your understanding.

Section 6: Exploring Mathematics Through Text. This section focuses on extended written mathematical communication. Each chapter features a different type of text (such as an encyclopaedia entry, a welcome email from the Head of School, an assessment brief, an invitation to a mathematics talk, or a call for abstracts) giving you exposure to a variety of mathematical writing genres. A range of activities guides you through the text progressively, beginning with global tasks that examine the audience, purpose, and structure of the text, and gradually narrowing to focus on individual language items. Activities develop skills including summarising, paragraph analysis, identifying connections within and between paragraphs, and noticing and applying useful academic vocabulary and sentence structures.

Section 7: Glossary. The final section lists and defines all the key terms introduced in the chapter. For each term, the glossary provides the pronunciation, part of speech, related terminology, an audio recording, and example sentences. A Chinese translation is also provided for each term.