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There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.

Bulletin of the American Mathematical Society, v. 69, p. 611, 1963.

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# The Right and Lawful Rood

Author's Note:  I believe that activities in the mathematics classroom should be as varied as possible. Not all students are passive learners - for example kinesthetic learners need to move around and touch things to learn. Not all my lessons cater for all types of learners, but I do like to vary the style of teaching to ensure that students experience methods that suit them some of the time. I consider the activity that follows well worth the effort of having students move around and organize themselves to take measurements. The fact that it also involves history of mathematics is a big bonus.

Figure 1.  Sixteen men leaving a church service measure their rood, as illustrated in the geometry book of Jacob Köbel (1460-1533), 1608 edition.

The word "rood" can be traced back to the Germanic "rute" and from there to the Old English rod. We know a rood today as a large crucifix often found beside entrances to old churches (for an example see Romsey Abbey (Hampshire, UK)). It is also a measure of land area of about a quarter of an acre or 40 square rods.

However the rood in which we are interested here is a linear unit which ranged from 16.5 to 24 feet in length at various times and in different countries. At 16.5 feet it was identical to the surveyor's rod. In a book on surveying by Jacob Köbel (1460-1533), he 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 the others, heel to toe. Then 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 and still makes division easy since 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 at the Mountbatten School, Romsey, UK as they left my lesson, repeated with our school staff (with their right feet) as they left a morning briefing, and then reiterated with some of the attendees at the Symposium X van de Historische Kring voor Reken Wiskunde Onderwijs (The historical group for arithmetic and mathematical education) in The Netherlands.

Figure 2. The teachers at The Mountbatten School, Romsey measure their rood.

The respective results were 4.14m, 4.40m and 4.68m, all well short of today's rod (5.03m) but longer than the old German rute (3.8m). Further data was obtained at the History and Pedagogy of Mathematics Conference held in Uppsala, Sweden during July, 2004. Here 16 mixed adults gave a result of 4.58 m and 16 males yielded 4.85m. This generated much discussion. Does this indicate that foot length has reduced over the last 5 centuries? (Perhaps manual labour in the fields leads to bigger hands and feet.) Has shoe length reduced? (The illustration 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 this experiment with different year groups and measure the sexes separately.  We will then be able to use the data to compare the groups, an excellent opportunity for statistical coursework.

Opportunities for using history of mathematics and real data in the classroom do not come often. By involving pupils and using their data they feel ownership of the data and are eager to see how they measure up to other groups. Readers who send me their results (pransom@btinternet.com) will receive a complete set of the data sent to me in due course. Please let me know the age, sex (female, male, mixed) and geographical location of the data sent.  If enough data is received, there may be scope for a wide range of statistical investigation. With a large enough data base, means and standard deviations can be found and ages and/or sex can be compared.