My last post, A Little Bit of History, was essentially written for my own benefit. I needed to summarise what I had been reading about the introduction of A levels historically, and what examinations preceded them. The next part of my journey has been to get a feel for trends in A level teaching through the prefaces of numerous textbooks.

I already have a fair collection of such books myself, but I am especially indebted to my Twitter colleagues who helped out with this part of the research. Not only was I reassured that I’m not the only mad collector of old books out there, but it’s also a testament to teachers’ goodwill that people were willing to take a little time out to help me without really knowing why! I hope I don’t overlook anyone, but would like to thank @MathematicQuinn (he has *so* many books), @KathHodgson, @MrBenWard and @GLRunshaw for their contributions.

## Why the Preface?

The preface of a book usually comprises a single page which is rarely read. I doubt I am alone in going straight to the contents list, if not directly to chapter one. However, when you pay attention to what the authors write in the preface, you begin to appreciate its value. Moreover, reading through a succession of them over a period of decades, you begin to notice clear trends in education and mathematics. Indeed, as the most successful books are revised, the authors include a supplementary preface detailing the changes and their rationale.

I’m going to keep this post relatively short, but I’ve taken the time to summarise some of the common themes that are discussed by the different authors. Moreover, I’ve picked out a few of the trends that interested/amused me.

I’ve used nigh on forty books from pre-1990 as the basis for this investigation. If you’re interested in seeing the details I’ve picked from them, then they’re all summarised in this Google spreadsheet.

## Common Themes

- How the book relates to specific examinations; if it covers the requirements of several boards; or if it exists independently from syllabuses but is still appropriate for use
- The need for the book: perhaps others are deemed to lack mathematical rigour; or have syllabus changes meant the need for new works
- The relevance to higher education: a number of books indicate that they will also be of value to first year engineers and students of other disciplines. for example
- The use of the book for revision and exam preparation. Indeed, some were specifically written just for that purpose.
- Inclusion of material beyond the syllabus, or of a more challenging nature, eg with an eye on S level papers.
- The philosophy of the authors’ approach to the development of material and rigour of explanation; the decisions about whether to include or exclude proofs
- The ordering of topics: almost all admitting freedom to the teacher to choose their own path; one notably not recommending this (Loveday’s Statistics) and one begging for a ‘branched’ approach (Backhouse’s Pure Maths 1)
- The use of references in writing: only a few include such remarks, but they credit academic reports and writings that have influenced the presentation of topics
- Graded exercises with enough ‘accessible’ questions early on to engage students. Often more challenging problems marked with an asterisk or dagger
- Miscellaneous questions and past exam questions: almost all books pride themselves on the quantity of exercises included; miscellaneous exercises are typically at the end of chapters and contain past exam questions
- Second reading: a number of books imply that they will be referred back to a second time, and that certain material should be left until then

## Trends over Time

- A number of books, especially those covering mechanics, needed revising for metrication (approx 1970) and the introduction of SI units of measurement
- Complex numbers were included as an additional chapter in revised editions, implying they were introduced as a syllabus change (approx 1966)
- The ‘modern’ and ‘traditional’ mathematics issue. I know little about this currently, but a number of books emphasised their synthesis of the two and the resulting ‘integrated’ course (approx 1980)
- References to vectors are still puzzling me: I’m wondering if they were part of the ‘modern’ mathematics debate and book readers needed to be reassured about the value of vector methods. Quadling & Ramsay, 1971, wrote:

“Vector methods, still regarded as an innovation fifteen years ago, are now widely used by quite young pupils at the school”

- Probability must have had an introduction onto the GCE A level syllabus as a number of more recent books emphasise its inclusion.
- Space travel: between editions of Quadling & Ramsay’s mechanics books, man went to the moon. What a great theme to include in teaching mechanics.
- Calculators: in 1982 Perkins and Perkins make explicit reference to students having access to a scientific calculator
- Reference to the student’s own use of a book: it is interesting to see which books make explicit reference to a student using a book by themselves. I think the first to refer to such a student as ‘he or she’ was Backhouse in 1963.
- Introductory work: a number of books include preliminary work as the calibre of students beginning A level study is unreliable. This is a trend from the 1980s onwards with this notable quote from Bostock and Chandler, 1990:

Now that GCSE courses have been introduced it can no longer be assumed that all students enter an A-level course with the algebraic skills and geometric knowledge that used to be expected. Many more students now move in to sixth-form colleges to do A-levels and hence come from a variety of backgrounds, including those who wish to embark on an A-level course from intermediate level GCSE.

- It is also Bostock and Chandler who first make reference to the use of computer software to aid understanding in mathematics. They recommend this book from the Mathematical Association, written to use BBC Basic:

## Think, Pair, Share

- There is often much debate about whether textbook use in class is good or bad. Do they really only exist as a source of practice problems? Are the included explanations useful? And useful for whom, teachers or students working by themselves?
- Do you ‘train’ your students how to use a textbook? I noticed myself doing that this year: explicitly showing them how to find information; where to find appropriate work to complete independently; and even how to revise with the aid of the textbook.
- What would your ideal A level textbook contain? @DrBennison began this debate with his “What do we want in an A level textbook?” post. He’s also blogged about Bostock & Chandler.

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