Making the most of your textbook

It is possible that in most of your mathematical studies up to now you have only used your textbook as a source of questions (and probably to look at the answers in the back!).  If you are to become an effective student you must also learn to use your textbook to learn new mathematical concepts and techniques.  Effective use of textbooks is a vital skill for university study and for career development.  Mastering it now will be of great benefit to you in the future.

Believe it or not, mathematics textbooks are designed to be read by students.  The on-line resources are designed to be used alongside textbooks.  The textbooks together with the on-line resources have proved a very effective resource for supporting students studying Further Mathematics. 

How are your textbooks structured?

Mathematics textbooks are usually structured to a standard format.

• A chapter often begins with some background, which puts the mathematics it covers into context and perhaps gives a brief historical perspective

• The text then introduces and explains the new concepts covered in the chapter, often including concise definitions

• Next come examples, showing how you can use the new ideas that have been introduced to solve some problems

• After a few examples there is an exercise to enable you to practise the techniques that have been introduced

• Usually there are then some more notes and examples to develop the ideas further, then another exercise

• This pattern continues through the chapter until the ideas relating to the chapter topic have been developed to the extent of the module’s specification – (Be aware that often A level Mathematics and Further Mathematics barely do more than scratch at the surface of an area of mathematics

Some textbooks also contain discussion points and activities. These are designed to encourage you to think more deeply about the new ideas being introduced and to “learn by doing”.  This is a far more effective way of learning mathematics than just trying to absorb information – mathematics is much more about understanding and applying concepts than merely remembering information.

How should you use your textbook?

A page of mathematics in a textbook can look very intimidating.  If you are a reader of novels, it probably takes you between one and three minutes to read a page of typescript.  Reading a mathematics textbook is not at all like this:

• The ‘background’ section can be read through quite quickly – it is only concerned with giving you information.

• The text introducing new concepts and definitions needs to be read much more carefully.  Have a pencil and paper to hand (a pencil is better than a pen for this as you can rub out your mistakes).  Read it through slowly, two or three times.  Make a note of any sections you do not immediately understand.  Fill in any ‘gaps’ in workings, to make sure you understand all of the steps.  On the website you will find additional notes: read through these carefully and, if you feel it would be useful, print them out to include in your notes.

• Even if you still have unanswered questions about what you have read, don’t worry. Go on and work through the examples which follow (again, there are more examples on the website which you can work through as well).  To do this effectively you will need to work through them with a pencil and paper.  It can be a good idea to try to work out the examples first yourself, before looking at how they are done in the textbook.

• Often you will find that working through some examples will help you to answer some of the questions which arose when you read through the theory.  Re-read it again to see if you now understand it better.

• Once you feel you have a basic understanding of the ideas, make some brief notes, including key points and definitions.  There is no need to copy everything out.  Making some notes should help you to clarify the ideas.

Once you have worked through the notes and examples you are ready to tackle questions from the exercise which follows.  The initial questions in an exercise tend to be quite straightforward.  As you work through they get progressively more difficult.  The initial questions are usually planned to help you grasp the basics of the mathematics you have been learning.  The later questions may require you to use the new ideas you have learned with fluency.