Success with Mathematics

Heather Cooke

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781134426546
• Edition: 1
• Topic: Éducation
• Subtopic: Éducation générale

Many students find the leap between school and university level mathematics to be significantly greater than they expected. Success with Mathematics has been devised and written especially in order to help students bridge that gap. It offers clear, practical guidance from experienced teachers of mathematics in higher education on such key issues as:

* getting started
* ways of studying
* assessment
* mathematical communication
* learning by doing
* using ICT
* using calculators
* what next.After reading this book, students will find themselves better prepared for the change in pace, rigour and abstraction they encounter in degree level mathematics. They will also find themselves able to broaden their learning strategies and improve their self-directed study skills.

This book is essential reading for anyone following, or about to undertake, a degree in mathematics, or other degree courses with mathematical content.

1
Introduction—read this first!

This book is intended for people taking courses involving some mathematics. It is aimed at students in higher education, but it is also relevant for those working towards A level. How much of the book is relevant to you, and when you should use it, will depend on your own circumstances. You may just be getting used to organising your own study, perhaps have not studied for a long time, or may not feel very confident about your ability to learn mathematics—or all three.

LEARNING MATHEMATICS

Learning mathematics is not just about acquiring and mastering computational and problem-solving techniques, or solely about understanding definitions, arguments and proofs, or even simply about knowing how to work with examples and counter-examples. In addition to all of these things, it also involves you reconstructing the thinking and work of other mathematicians, so that it becomes part of your own thinking, as well as you undertaking your own mathematical activity and exploration. Learning mathematics requires you to develop ways of thinking mathematically while doing mathematics—for many, the most exciting and creative element of all.

Active learning

This is not the usual kind of textbook. For a start, it is designed for people working on their own rather than in a class setting. It is possible to learn quite a lot just by reading it, but for more effective, long-term learning, it is best to be actively engaged as well. This includes trying things out for yourself, making notes and talking to other people about what you have read (and possibly reread, for seldom can one reading be enough to gain a good understanding when mathematics is involved). To help you get into this possibly less familiar mode of learning, there are numerous tasks for you to do throughout this book: most have comments or solutions immediately following. You are strongly advised to try each task for yourself before looking at the Comment, so if your eyes tend to read on, you may need to develop a means of masking these comments initially—especially for the more mathematical tasks.

TASK 1.1

GETTING AN OVERVIEW

Read quickly through the contents pages, in order to get an overview of what is in the book. Then flick through the book as a whole to get a feel for what is in it and how it is organised, glancing at the general layout of the chapters. You can get an idea of the type of content by skimming over the headings and subheadings, without reading anything in detail.

Comment

These are some of the things you might have noticed.

There are more words than you might have expected in a book about mathematics. The book is written in nine chapters that have headings and subheadings. There are tasks, with comments immediately afterwards, and also notes in boxes. There are different types of diagrams, graphs and tables. The contents page lists the chapters and main sections within them and there is an index at the back.

When and how to use this book

The book is designed to help you prepare for studying mathematics at university level, but there are sections that will also be of help once you have embarked on your course. Things you may not think are important as you prepare may become more significant as you get into your course. So the book is not only for preparation but also for support while you study.

It may be a long time since you studied, so you may not be very confident or that sure about your current mathematical skill or your ability to learn mathematics. If this is the case, then start at Chapter 2 and work your way through the book from the beginning. But if you are feeling confident as a student responsible for your own learning, then it may be more appropriate for you to dip into different sections, for example Chapter 6, Learning by doing. The book can be drawn on in a variety of ways—it is yours, so use it in ways that help you.

Like any other, this book is unlikely to cover everything you need. It is primarily about how to study mathematics rather than providing you with an in-depth discussion of specific mathematical ideas and techniques. Nevertheless, it should support you in developing sufficient skill and confidence to be able to use other resources. (Some such resources are listed in Chapter 9.)

MATHEMATICS AT UNIVERSITY

What has your relationship been with mathematics up to now, and how would you assess your current relationship?

TASK 1.2

WHAT SORT OF STUDENT ARE YOU?

Which of the following most closely matches your current situation?

■ Mathematics pops up in the most unexpected places. I thought I wouldn't need it for my course, but there it was, in disguise.

■ I have always been reasonably good at mathematics and would like to study it further, but do not have that much confidence, especially recognising where to start with a solution.

■ I love mathematics and want to find out the most effective ways of working and strategies for coping when I get stuck.

■ I used to find mathematics easy, but have had a year or so away from studying it and can't really remember a lot.

■ I have always found mathematics hard to understand, but I need it for my job.

Comment

Your experiences of mathematics may differ from these mathematical learners, but basically the book offers some guidance on what to expect and how to proceed, no matter which of these individuals you feel closest to. Read the details of your course carefully and be prepared to ask for more information. Access and foundation courses, available at some universities, can also be used to bring your mathematical background and understanding up to the required level.

Topics in mathematics

It is perhaps helpful to see mathematics as a combination of what has been developed: (i) because of the need for solving particular, real-world problems and (ii) as the result of a more general and abstract search for and exploration of patterns and relationships; leading to the development of various ways of expressing and justifying them. These two aspects can be thought of as: (i) mathematics as a toolkit deployed for other human tasks and (ii) mathematics as a discipline and means of enquiry in its own right. What is taught at school is a mix of topics and approaches from this rich and complex tapestry, which has arisen from human engagement with mathematics for nearly 5000 years.

Lower-school mathematics mainly concentrates on the concrete operations and techniques for calculating. More abstract concepts and means are gradually introduced, such as ratio and proportion, algebraic manipulation and transformation in a variety of guises. A learner of mathematics has to become proficient at applying rules and techniques that mathematicians have developed over the years.

On the whole, universities are restricted in what is taken as prior knowledge by the nature of the upper-school mathematics curriculum. Although modules can additionally be taken in statistics, mechanics and discrete (decision) mathematics at school, the compulsory core units tend to be in the topics of pure mathematics and most university mathematics departments now make no assumption about non-core material.

What do universities expect?

This differs from university to university and certainly from course to course. For those courses with little overt mathematical elements (mathematics seen primarily as a toolkit), for example quantitative methods, the assumption is likely to be GCSE grade C, with some facility at re-arranging formulae being particularly useful. For programmes that specify A-level, most universities assume no other knowledge than the compulsory elements of the A-level syllabus. However, you are expected to be pretty fluent in your use and understanding of these elements, particularly algebraic processes and the nature of proof. Most mathematics departments would also expect that if you are studying mathematics as a discipline in its own right, you will have read some books and thought about the nature of mathematics. (Some examples of this growing literature about mathematics are listed at the end of Chapter 9.)

Here are some extracts from 2001 prospectuses for BSc mathematics degrees.

There is a basic ‘tool kit’ which every Mathematics undergraduate needs to master—such tools belong mainly to the realm of Pure Mathematics.

(University of Leicester)

At first the Mathematics course has considerable overlap with A-level.

(University of Durham)

However, there are a few university courses and modules that do not assume A-level mathematics. For example:

If you ha...

Table of contents

• Cover
• Half Title
• Full Title
• Copyright
• Contents
• 1 Introduction-read this first!
• 2 Getting ready
• 3 Ways of studying
• 4 Assessment
• 5 Mathematical communication
• 6 Learning by doing
• 7 Using ICT when studying mathematics
• 8 Using calculators
• 9 What next?
• Index