Paradigm, Number 24 (December, 1997)
Division is as bad;
The Rule of Three doth puzzle me,
And Practice drives me mad.
A number of arithmetic copybooks or exercise books have survived from the nineteenth century and some from before then. Their contents are interesting in a number of ways. It is usually not too difficult to identify the source of the questions solved; Walkinghame's Tutor's Assistant is the most likely text and the author's preface to that book immediately throws some light on teaching practice in schools:
Having . . . drawn up a set of rules and proper questions, with their answers annexed. for the use of my own school... I found them by experience, of infinite use: for when a master takes upon him that laborious (though necessary) method of writing out the rules and questions in the children's books, he must either be toiling and slaving himself, after the toil of the day is over, to get ready the books for the next day, or else he must lose that time which would be much better spent in instructing and opening the minds of his pupils.
The copybooks reflect Walkinghame's words. Questions are written out, followed by their solutions, almost always in beautiful copper-plate hand-writing. When a new topic is begun, the elaborately decorated heading is followed by a brief statement of the topic's use, followed by a'Rule' (or Rules).
Seldom is any justification given for the Rule, either in copybook or textbook. It seems that the rules were learnt and practised in the contexts of problems to which they were applicable. Although Walkinghame's text, and others, contains sets of'Promiscuous Examples', (which should require the student to decide which rule to apply), I have seen no copybooks in which such sets of questions are used to that end.
Figure 1 from James Sharp, 1781.
Figure 2 (from James Sharp, 1782) shows an 'instrumental' approach; Pythagoras's theorem becomes two apparently distinct rules, rather than a single result to be applied to the different cases, as it would be in a modem textbook.
In some cases it appears that large sections of the textbook have been copied verbatim; example Figure 3 is an example (from William Bien, 1785) showing a 'dialogue' style reminiscent of Robert Recorde.
Questions in rhyme also appear. The text in this case is too faint to reproduce, but the calculation which follows it is given as Figure 4.
Girls seem to have studied the same topics as boys, perhaps with more relevant' questions. Figure 5 is an example from Elizabeth Bradford (n.d.) reads.
The question in Figure 6 appears in p. 93 in the 1821 edition of Walkinghame's Tutor's Assistant (incidentally it is wrongly placed in Walkinghame's text as it cannot be solved by 'Single Position').
A man overtaking a maid. Driving a flock of geese,
Said to her, How do you do, Sweetheart; where are you go-
ing with these 30 geese? No Sir, said see, I have not 30;
but if I had as many more, half as many more; and a geese
Befides, I should have 30&endash;how many had she?
The choice of topics in the copybooks reflects the contents of the textbooks. Some topics, such as book-keeping, have an obvious utilitarian purpose; others seem to have been included (as they might be today) chiefly because they are intriguing, or capable of solution by elementary methods. In general algebraic methods are avoided, but again this reflects textbook methods. As an example of style and method, the 'Single Rule of Three Inverse' illustrates many points. First we need the'Single Rule of Three Direct', which:
'Teacheth, by three numbers given, to find out a fourth, in such proportion to the third as the second is to the first.
RULE. - First state the question; that is place the numbers in such order that the first and third be of one kind, and the second the same as the number required;;... Multiply the second and third numbers together, and divide product by the first, the quotient will be the answer to the question...'
'Inverse Proportion is, where more requires less, and less requires more...
RULE. - Multiply the first and second terms together, and divide the product by the third; the quotient will bear such proportion to the second as the first does to the third.'
Readers will recognize the style of questions that the Rule of Three solves: 'If 10 men can dig a trench in 4 days, how long will 7 men take to dig a similar trench? The example of Figure 7 shows the working from a copybook dated 1781.
The implied criticism of a style of teaching which might be characterized as 'rules without reasons' should be tempered by the following (figure 8) example showing the extraction of a cube root . Thirty years ago (before the days of calculators) the extraction of square roots by a similar procedure was taught simply as a routine; many who taught it did not know the underlying theory themselves. Figure 9 shows another feature of solutions to problems; meticulous care for accuracy in calculation, but answers to 'practical' or 'real life' questions given with a ridiculous number of decimal places. The copybooks rarely show evidence of having been 'marked' by the teacher, which raises a number of questions. Did the student work the problem - or was the solution copied from the blackboard? Were the copybooks students' books, or did they belong to teachers?
Figure 9. From W.R. c1815. Text reads. "A Ladder 50 feet long, being placed in a street, reached a window 20 feet from the ground on one side, and by turning the foot, it reached another window 36 feet high on the other side: required the breadth of the street."
Figure 10 (Geo. Medlicot c 1858, Welchpool School) suggests that this book may be the work of a teacher.
Diagrams are usually carefully executed and frequently coloured; figures 12 and 13 show an incorrect drawing from a copybook and the same problem's solution from the 'Key' to the relevant textbook.
Figure 13. From Vyse Charles. Key to the Tutors Guide p. 224.1811. 10th ed.
The final example shows one worked example from a 1781 copybook.
Figure 14. James Sharp. 1781.
Before condemning the teacher for this nonsense we should pause and note that a similar calculation to that in figure 9 can be found in Leybourn's 'Arithmetick' of 1700. (William Leyboum, Arithmetick; Vulgar, Decimal, Instrumental, Algebraical. P. 136. 1700. 7th edition).
A Ladder 40 ft. Long may be so planted that it shall reach awindow 33 ft. From the ground on one side of the street andwithout moving it at the foot will do the same by a window 21 ft. High on the other side. The breadth of the street is required.