# Best foot forward

Students can be turned on to maths by rediscovering ancient units of measurement. Peter Ransom explains how

Most of the imperial units of measurement used in the UK date from the Romans. With the Claudian invasion of ad43, these units became firmly established. Units of length were based on body measurements (conveniently so, as one always had these to hand, so to speak), and on the Latin word uncia - one twelfth.

From this root, we derive the ounce (in the apothecaries' system there are 12 ounces in a pound) and the inch (one twelfth of a foot). The foot was based on the length of a human foot. This of course varied from person to person and so some standard had to be agreed. And because standards varied from country to country, making trade difficult, so further standardisation became necessary.

The ell, the unit used in measuring cloth, varied so much that Charles Vyse in his 1799 book The Tutor's Guide, states that "Scotch and Irish linens are bought and sold by the Yard: but Dutch linens are bought by the Ell Flemish, and sold by the Ell English" (one Ell Flemish was three quarters of a yard, one Ell English was five quarters of a yard). Most traders had their own "ellsticks" with which to measure the cloth.

Units of time have remained constant: an hour divided once by 60 is a minute, and one minute divides into 60 seconds. A second gets its name because it is the second time the hour is divided. There are even "third" and "fourth" units of time, with 60 thirds in a second and 60 thirds in a fourth.

Recently, I gave the closing talk at the Symposium X van de Historische Kring voor Reken Wiskunde Onderwijs (the Historical Group for Arithmetic and Mathematical Education) in the Netherlands. My presentation revolved around units used in English textbooks from the 18th century up till the Second World War, and the imperial system of weights, measures and money.

This was supplemented with classroom activities, one of which I hope readers will try in their classrooms and send me the result. I have not found any reference to this experiment having been performed since the 16th century.

The 'right and lawful' rood

One definition of rood is a large crucifix, often found beside entrances to old churches (eg at Romsey Abbey in Hampshire). The word can also be traced back to the Germanic rute and from there to the Old English rod. Another definition is a measure of land area of about a quarter of an acre, or 40 square rods. However, the rood under consideration here is a linear unit, which ranged from 16ft 6ins (5.03m) to 24 feet (7.32m) at various times and in different countries. The rood, or pole, declared in the reign of Queen Elizabeth I, was 16ft 6ins feet - Jidentical to the surveyor's rod which dates back to Anglo-Saxon times inBritain.

In his 16th-century book on surveying, the German Jakob Koebel (1460-1533) mentions that the surveyor should request that, on leaving the church service, 16 men should stop as they come out and stand in a line with their left feet touching, heel to toe. The length of the 16 feet gives the "right and lawful" rood. Dividing by 16 then gives an average foot. (Why 16? Perhaps this gives sufficient people to produce a valid sample while division is easy because 16 is a power of two and so four successive halvings gives the mean foot.) This method of random selection was used with my Year 7 class as they left a lesson, and repeated with school staff leaving a morning briefing and again with attendees at the conference. The respective results were 4.14m (13ft 7ins), 4.40m (14ft 5ins) and 4.68m (15ft 4ins) - all well short of today's rood of 16ft 6ins (5.03m) but longer than the old German rute of 12ft 512ins (3.8m). Further data were obtained at the History and Pedagogy of Mathematics Conference in Uppsala, Sweden, during July, when 16 mixed adults gave a result of 4.58m (15ft) and 16 males yielded 4.85m (15ft 11ins).

This generated much discussion. Does it indicate that foot length has reduced over the past five centuries? (Perhaps manual labour in the fields leads to bigger hands and feet.) Has shoe length reduced? The illustration below, taken from a compendium of surveying by Koebel, shows the men wearing shoes, but these seem similar in shape to today's footwear. How much does age matter?

In future years we will do the experiment with different year groups and measure the sexes separately, so we will be able to use the data to compare groups - an excellent opportunity for GCSE maths data-handling coursework.

Opportunities for using history of maths and real data in the classroom do not come often. By using their own data, pupils feel ownership of the project and are eager to see how they measure up to other groups.

Peter Ransom is a leading mathematics teacher at the Mountbatten School and Language College in Romsey

* The History and Pedagogy of Mathematics Newsletter is an international publication available free in electronic or paper form.

British Society for the History of Mathematics: www.dcs.warwick.ac.ukbshm

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