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**Academic English for Mathematics** aims to improve your ability to study Mathematics effectively in English. It is a joint effort of a English for Specific Academic Purposes Instructor and a Mathematician. It is written for international students who are planning to embark on an undergraduate programme of Mathematics and speak English as a foreign language. With this course, you will develop your knowledge of academic and scientific conventions and you will improve your skills in the following areas:

- reading and understanding of Mathematical articles, theorems, proofs, axioms, definitions and word problems in English
- listening to lectures, understanding sign-posting language, main points and improve your note-taking skills
- noticing writing conventions for different audiences and purposes within the same discipline and improving academic writing skills such as paraphrasing, formality and citations
- contributing effectively in seminar discussions
- preparing and giving effective scientific presentations
- improving academic vocabulary and prominent language features
- improving study skills such as planning, note-taking and summarising
- improving your critical reading and writing skills with peer-review evaluations

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**Book Review**

1 Real Numbers**Themes**

Properties of real numbers; whole numbers, natural numbers, integers, fractions,

rational numbers, square roots, irrational numbers**Vocabulary**

Definitions of types of numbers; properties of real numbers, derivatives

Common procedures followed by mathematicians: factoring, using reciprocals/

division, collecting like items, using average, converting to decimal or fractional

notation**Writing**

Paragraph structure, cohesion and coherence**Note-taking**

Identifying main points in a lecture: “Imaginary and complex numbers” contents

2 Algebraic Expressions, Equations and Functions**Themes**

Algebraic expressions, equations and functions**Discussion**

Skimming and scanning, identifying main points**Vocabulary**

Using a variety of words to describe arithmetic processes; translating “text” to algebra;

reading functions and equations**Note-taking**

Linear style abbreviations and symbols**Style**

Introducing formality by comparing two excerpts

3 Exponents and Exponential Functions**Themes**

Exponents and exponential functions; exponential decay and growth; geometric

progressions and scientific notation**Academic vocabulary**

Definitions;

Writing the product as a monomial in a standard form

Numbers of Science: Conversions from Metric system to English

Metric prefixes for powers of 10

Comparing objects of widely different sizes: orders of magnitude**Academic writing and style**

Introduction to paraphrasing;

Expressing cause and effect as a paraphrasing tool

4 Introduction to Reasoning and Proof **Themes**

The language of proofs regarding segments; inductive and deductive reasoning,

theorems, logic tables, axioms, methods of proof**Discussion**

Useful phrases**Vocabulary**

Definitions of Logic Theory terms**Writing**

The language and symbols of proofs**Note-taking**

Writing

Mathematical proofs by induction, identifying the base, hypothesis and conclusion

5 Introduction to Probability **Themes**

Introduction to probability, random events, random variables and their categorisations,

density and distribution functions, notation and axiomatic definition of probability**Vocabulary**

Definitions of probability related terms; collocations; notation in probability, adverbial

phrases**Academic style**

Introduction to academic caution; modal verbs and the lack of tentative language in

Mathematics**Note-taking**

Peer-reviewing notes on probability webinar

6 Introduction to Statistics**Themes**

Data, sample population, numerical descriptors, rational equations and functions;

descriptive and inferential statistics, statistical significance; standard deviation,

coefficient of variation, mean, median; graphs**Vocabulary**

Statistics: Definitions

Word formation and use-in-context**Language**

The use of gerund and infinitive in Mathematics**Presentation/Writing**

Reporting graphs and charts**Writing**

Writing a report following a chart or graph

Plagiarism, citations and references; why and how we use them

7 Geometry connections**Themes**

Introduction to Geometry; lines, points, planes and angles, parallel and perpendicular

lines**Discussion**

Expressing agreement, disagreement or acknowledgement**Academic vocabulary**

Use in context, definitions of popular mathematical terms

Geometric shapes: 2D and 3D definitions

Giving examples**Academic writing**

Summary guidelines and practice**Note-taking**

Identifying the moves in a lecture; using reference verbs

8 Properties of Triangles**Themes**

Triangles; main and secondary elements of a triangle, types of triangles by lengths

of sides, classification according to internal angles, the Pythagorean Theorem, the

concepts of congruence and similarity**Vocabulary**

Definitions; use-in-context: types of triangles

Writing a two-column proof

Making comparisons**Writing**

Passive voice**Presentation**

Opening/closing phrases and transitions

9 Introduction to Trigonometry**Themes**

Right triangle Trigonometry, ratios, sine, cosine, tangent,

trigonometric identities, function graphs**Academic vocabulary**

The language of theorems, axioms and proofs**Note-taking practice**

The unit circle**Academic presentation skills**

Signposting language that engages your audience enhances the impact of your speech

10 The Geometry of the Circle**Themes**

Properties of circles; chords, tangents, secants, equations and graphs of circles**Vocabulary**

Use-in-context; definitions

Geometry tools**Note-taking**

Equations and graphs of circles**Writing**

How to write an argumentative essay on “Pure vs. Applied Mathematics”

Using evaluative language to unfavour someone’s view

11 Polygons and Quadrilaterals**Themes**

Polygons and quadrilaterals; parallelograms, rhombi, rectangles, squares,

kites and trapezoids

Ratios, proportions and similarity applied to polygons and quadrilaterals,

proportionality with parallel lines, dilations and fractions**Vocabulary**

Definitions; use-in-context; adjective suffixes**Academic language focus**

Subject verb agreement; quantifiers**Academic style**

Avoiding wordiness and repetition**Academic presentation skills**

From text to slides

Using visuals to enhance the impact of your presentation**Note-taking**

Tessellations

12 The Geometry of Three Dimensions **Themes**

Introduction to the geometry of three dimensions, points and lines in space, coordinate

systems, polyhedra and solids of revolution**Academic vocabulary**

Definitions: use in context**Academic style**

Formality and complexity;

Comparing texts written for different audiences**Introduction to critical reading**

Common pitfalls when writing a scientific paper

Appendix

Glossary

Evaluation criteria

Transcripts

Bibliography

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