# Textbooks and the Maths of Days Gone By

In recent weeks the theme of Maths textbooks has been doing its rounds in my head, aided by the propicious timing of several meetings. At a recent NCETM meeting I met someone from Hodder’s publishing team (they publish the MEI A level books) and we talked at length about the new A level structure, the need for revised textbooks and the value/potential for well thought out electronic supporting resources. Coming up this week, there is the ICMT2014 at Southampton University. (Did you know there was an international conference dedicated to Mathematics textbooks?!)

After reading a research paper by Geoffrey Howson (free-access link likely to expire in August 2014) and a number of other related articles and papers, I have now become much more conscious about the content, structure, use, effect and value of textbooks. I’ve accumulated a fair number of maths books since taking my A levels and so scanned my bookshelves looking for an interesting place to start making my own comparisons.

I have a few books from the turn of the 20th Century which, if I remember rightly, an aunt passed on to me after going to flea markets. Two focus on arithmetic, one on algebra and two volumes on geometry:

• Arithmetic in Theory and Practice, J. Brook-Smith, 1884
• Arithmetic for the Preliminary Certificate Examination, R. H. Chope, 1906
• Elementary Algebra, Hall and Knight, 1897
• Elementary Geometry Books I and II, B. Arnett, 1904

Interesting to see that even a century ago, some books were targeted for specific assessments. (I honestly don’t know if this is the corresponding assessment, but here is a 1911 Arithmetic paper.) For the remainder of this post, I’ll focus on Arithmetic. The others – and more modern books – are another story to be told another time.

### Arithmetic in Theory and Practice

The author’s preface includes some interesting comments regarding metric and imperial units:

The tendency of the present day is to make use of the Metric System in international transactions and scientific pursuits; but to the retail dealer and his customers our present system presents so many advantages, that I almost doubt the possibility of uprooting it. They first divide into halves, into quarters, and into eighths, then into thirds and into sixths, but very rarely into fifths or tenths. The probability therefore is that, in this country, the two systems will exist side by side, the scientific and the practical; just as we have two systems of logarithms, and two ways of measuring angles.

And then, regarding proportional reasoning and unit cost etc:

In the earlier part of the work I have used the Method of Reduction to the Unit, but I am far from advising an exclusive adherence to that Method; when the student has gained a clear and firm grasp of ratio, it would be unwise of him to neglect the powerful instrument that has come into his possession.

Here is an image of the contents of this nearly-400 page book.

Of particular curiosity value to me was Chapter VII Evolution. This section discusses written methods for extracting square roots of integers and decimals and then even extends this to cube roots and then (by repeated application) fourth, sixth and ninth roots! The details will surely make the content of a future blogpost.

The book finishes with a number of ‘past papers’ from the 1860s and 70s. The barrier for modern readers is the difficulty of the units used: £ s d; tons cwt lb oz dwt; roods poles yds ft. I had no idea what a rood is, but apparently it’s quarter of an acre (which obviously makes it 1210 square yards).  Even the titles of the papers themselves are interesting: some are labelled with universities (presumably the equivalent of today’s exam boards?) such as “University of London. Matriculation Examination. Jan 1870”, “University of Oxford. Responsions. Hilary Term 1871” and others are for public service: “Army Examination. For Direct Commissions. July 1879” and “India Forest Department. Jan 1870”.

### Arithmetic for the Preliminary Certificate Examination

This book identifies itself as a smaller edition of The Tutorial Arithmetic, adapted specifically for the requirements of the Preliminary Certificate Examination. In the preface, the author writes:

The special features of the book are the use of algebraic symbols and the omission of recurring decimals, square and cube root (except by factors, see §78), true discount, and foreign exchanges (except as given in §133).

The portions of the book which are not required for the obligatory Arithmetic in Part I of this examination are Chapter XX (Practice), Chapter XXI §§138-140 (Ratio), and Chapter XXV (Shares and Stocks).

Once again, here is a photo of the contents:

I think Chapter 28 is interesting, categorising different problems as:

• Problems on Work
• Problems on Supply Pipes
• Problems on Trains
• Problems on Races
• Problems on Clocks
• Problems on Calendar

An example from the races section:

A can beat B by 20 yards in a race of 200 yards, and in a race of 250 yards B can beat C by 10 yards. By how many yards can A beat C in 100 yards, assuming that their respective rates are the same in the various races?

Finally, here’s the sample examination paper provided at the end of the book. Don’t worry ladies, this one is appropriate for “Men and Women”!

It’s a bit fiddly to photo pages from these books as they are quite fragile, but if there is anything you’re especially interested in or curious about then let me know and I’ll see what I can do.

## Think, Pair, Share

• How different are these books from today’s? What has remained, what has gone, what has changed?
• Why have these changes occurred? Clearly there has been a change (reduction) in our need for arithmetic proficiency. How well does the content of our current maths books reflect the needs of our future workforce?
• Have the presentation of material and methods of assessment changed? These books have no diagrams (let alone inane clip art..) but they do contain past papers.
• What would your ‘ideal’ maths textbook look like?

## Good Old-fashioned Homework – no calculators!

• Yesterday I bought some aquarium rock from a local store. The rock is priced at £3.25 per kg and the pieces I chose weighed approximately 4.4kg. Not having a calculator to hand, the slightly-flustered guy serving me said “Is £12 ok?”. To the nearest integer, what was the effective percentage discount that he gave me?