The Academic Language of Mathematics is a free, open-access web-book, funded by The Libraries Open Books Programme at the University of Leeds in the United Kingdom. It has been developed as a rich and practical resource for students and lecturers engaged in the learning and teaching of mathematics through the medium of English.

What Is This Book About?

As the title suggests, this is not a traditional mathematics textbook. The primary intended readership is students who study mathematics as part of a STEM degree, and the book contains a considerable amount of mathematics. The authors hope that you will learn mathematics from it! However, the core aim of the book is different: to place key mathematical topics within a linguistic and textual context, and as a result to address the kinds of academic language required to participate effectively in communicating mathematics in academic contexts, primarily in higher education.

In other words, although the primary reader of this book is likely to be a student studying mathematics as part of their degree, the focus is on the related but separate matter of academic communication in mathematics. Any practising or developing mathematician is also a mathematics communicator, and The Academic Language of Mathematics has been created as a handbook for developing the competencies which this role requires.

Who Is This Book For?

The Academic Language of Mathematics is designed for students studying mathematics as part of their degree through the medium of English, for academics from STEM disciplines, and for English for Academic Purposes (EAP) practitioners. The mathematical topics covered are fundamental across STEM disciplines, and the book is particularly relevant in English-medium instruction contexts, where the ability to communicate mathematically in English directly shapes academic outcomes and a sense of belonging within the mathematical community. To read more about how The Academic Language of Mathematics can be used by each of these three groups see How to Use This Book.

Why Study the Language of Mathematics?

Many students are mathematically capable yet find it challenging to express their mathematical thinking clearly in English. This gives rise to a genuine paradox: strong mathematical understanding can be undermined by an underdeveloped ability to communicate it.

More fundamentally, language and mathematical thinking are not easily separated. Research into how professional mathematicians communicate shows that mathematical communication is a rich, multi-layered activity, one that involves formal rigour, visual and diagrammatic representation, and the use of language to describe and connect mathematical ideas. For students in higher education, developing communicative ability is not supplementary to learning mathematics; it is part of learning mathematics. For a full account of these ideas, see Theoretical Underpinnings.

Themes of the Book

Several central themes run throughout The Academic Language of Mathematics:

1. Mathematics has distinctive language practices: the way we express mathematics reflects its definition-first, deductive, and symbolic character.
2. Language operates at multiple levels: from individual terms and their notation through to complete texts.
3. Language development is progressive: proficiency in mathematical English grows through sustained engagement with mathematics, not as a prerequisite to it.
4. Prior knowledge is a resource: whatever your linguistic background, your existing knowledge and experience are an asset.
5. Communication is always contextual: understanding mathematical language means attending to both the words themselves and the disciplinary contexts in which they are used.

For a full account of the theoretical and pedagogical principles behind the design of this book, see Theoretical Underpinnings.

How This Book Is Organised and How to Use It

The Academic Language of Mathematics consists of eight chapters, each dedicated to a core mathematical topic and structured to move progressively from individual mathematical terms and notation through to extended mathematical communication. The book is open access, interactive, and suitable for both classroom use and independent study. It can be read linearly, used for reference, or dipped into according to your current studying needs.

For a full description of the chapter structure, see How This Book Is Organised. For guidance tailored to your role as a student, mathematics lecturer, or EAP practitioner, see How to Use This Book.

The Authors

The Academic Language of Mathematics has been co-authored by three colleagues whose expertise spans mathematics and English for Academic Purposes. Moreover, all of the co-authors have studied mathematics in at least two languages.

Dr Costas Loizou, FHEA is a Lecturer in Mathematics and Director of Year 1 at the SWJTU‑Leeds Joint School in the Faculty of Engineering and Physical Sciences at the University of Leeds. His mathematical background is in hyperbolic PDEs with multiplicities and generalised functions. His teaching and pedagogical research focus is on empowering transnational engineering students to thrive in English‑medium STEM curricula by integrating problem‑solving with disciplinary language development and a strong sense of belonging to Leeds.

Deak Kirkham is a Lecturer in English for Academic Purposes (EAP) at the University of Leeds, UK. After several years working on in-sessional and pre-sessional English for Business, Deak has been building in-sessional provision in the School of Mathematics at the University of Leeds across UG and PGT cohorts since 2020. His expertise lies primarily in the pedagogy and delivery of academic English for mathematics. In addition to these interests and background in academic English in a disciplinary context, Deak draws on his background in formal linguistics to deliver on a number of UG linguistics modules. Deak is a passionate polyglot and has competence in a number of languages.

Dr Shuwei Chen is an Associate Professor in Applied Mathematics at the School of Mathematics, Southwest Jiaotong University, China, and has taught at the SWJTU-Leeds Joint School since 2020. His research specialises in decision-making and logic-based reasoning, reflected in over 90 peer-reviewed publications, two national patents in China, and roles including Area Editor of the International Journal of Computational Intelligence Systems. Alongside his research, Shuwei brings an insider’s understanding of the linguistic and cultural transition facing Chinese students moving into English-medium instruction, informing the book’s design, examples, and relevance to this cohort.

The Academic Language of Mathematics is the product of a collaboration that extends well beyond its three co-authors. The authors are grateful to all those who have contributed to its development. See Acknowledgements for details.

Feedback

The three co-authors have found the development of this resource both stimulating and rewarding. Testimonials reported improved knowledge, confidence, and independence in the academic language of mathematics, and this response has confirmed both the value of the resource and the importance of continuing to develop it. Suggestions and comments from students and colleagues are actively welcomed and may be directed to:

• Costas Loizou, c.loizou@leeds.ac.uk
• an anonymous feedback form.