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Figuring It Out: Children's Arithmetical Manuscripts 1680–1880

MAA Review

My usual rule is: short books, short reviews. Let’s see whether I can adhere to that rule.

At first I thought the book might turn out to be boring. At first.

Well, there’s no really interesting math in it. And there’s a fair amount of history, my booby-prize subject throughout elementary and high school. True, there are plenty of pictures, but they’re not of flowers or people, and not in color; rather, they’re rectangles of writing, often unintelligible. In general, my first former-child’s impression was, dull dull dull, like a dictionary or Accounts Payable.

But then I got past the first couple of pages. Then I got past the next couple of pages. And then I realized I couldn’t put it down. It was 12:10 AM. I kept reading to the end.

I originally selected this book from this site’s list of “reviewables” because I’m interested in the math education, and the math itself, of young children. For eight years I was a home-schooling parent, or rather my two youngest were home-schooling children. Our approach was about as unstructured as a family can get. I did very little actual teaching of math, or arithmetic, reasoning that since arithmetic was a huge part of life, the kids would pick it up. We also played a card game involving arithmetic, only because we liked the game and not because I felt it would be educational. From that game the very youngest learned the basic arithmetical operations, including powers. (He was especially motivated to learn that one to any power is one, because each ace = one was worth one point out of a possible eleven.) Otherwise I didn’t do much math teaching, figuring also that they’d derive something out of their mother being a mathematician. At any rate, I’m interested in the history, not only of math, but of math education.

I learned a lot. The author is a retired math teacher and a collector of the manuscripts which are the subject of this book. Another name for them is “copybooks.” Teachers would teach, with or without textbooks, and the students, usually boys aged nine to fourteen, would do the exercises given them in these copybooks. The teachers often did not make comments or corrections. If they did, it was always orally; teachers never wrote in the copybooks. These little books belonged to the students. Each retained, at the end of his schooling, these copybooks, to use as souvenir or reference.

In this book the author frequently refers, with pride and joy, to his collection of copy-books. Page 3: “In my own collection there are books from all parts of England, from Cornwall and Kent to Northumberland and from Lincolnshire to Shropshire, with one or two from Wales and Scotland. I also have 7 from France, 6 from the USA, and 1 from Holland.” And on page 9:

I have a collection of some 70 arithmetical textbooks published before 1900, this is but a fraction of the total. Wallis & Wallis, in Biobibliography of British Mathematics, 1701–1760, list some 44 titles for the period 1701–1760 alone… the task of matching each manuscript with the corresponding textbook would be an immensely time-consuming undertaking. My success rate at the moment is a little over 50%.

The subject matter? Well, when I was in seventh grade, I got Cs in math. That was not only because seventh grade is, for many, a miserable year socially but also because, at least in my school, the math subject matter consisted of business math — and not the interesting calculus-derived formulas for compound interest or annuities. Much of the subject matter of the kids’ copybooks was of that ilk. No explanations and little, as I’ve said, “mathematician’s math.”

Proportions were taught using The Rule of Three, as well as the Double Rule of Three and the Double Rule of Three Inverse. The examples of problems from that era are interesting historically. Page 15:

‘If a Carrier receives £2. 2s. for the Carriage of 3 cwt 150 Miles how much ought he to receive for the Carriage of 7 cwt 3 qrs 14 lb for 50 Miles?’ He lays out the given quantities in two rows for the two parts of the question. He then reduces the weights to pounds and the money to shillings (Reduction was another process central to arithmetic); the rest of the calculation follows the stages: (i) Multiply 7 cwt 3 qrs 14 lb (i.e., 882 lb) by 50 (miles) to give 44,100; (ii) Multiply 44,100 by £2 2s (i.e., 42 shillings) to give 1,852,200; (iii) Divide 1,852,200 by the product of 336 (i.e., 3 cwt reduced to pounds) and 150 (miles) to give 36s and 9d, i.e.., £1 16s 9d (the answer)... Textbooks described a strictly rule-based approach to arithmetic and teaching followed down the same path…

Alligation and False Position were two other common topics. I would’ve gotten Cs, maybe Ds!

The book includes a description of school life and copybooks from the early 1800s by Rev. Warren Burton in The District School As It Was, by one who went to it, first published in 1833… The school was in New Hampshire. Pages. 6–7:

‘At the age of twelve, I commenced the study of Arithmetic, that chiefest of sciences in Yankee estimation… A new Adam’s Arithmetic of the latest edition was bought for my use. It was covered by the maternal hand with stout sheepskin… My first exercise was transcribing from my Arithmetic to my manuscript. At the top of the first page I penned ARITHMETIC, in capitals an inch high, and so broad that this one word reached entirely across the page. At a due distance below, I wrote… Addition in a large, coarse hand, beginning with a lofty A, which seemed like the drawing of a mountain peak, towering above the wilderness below. Then came Rule, in a little smaller hand, so that there was a regular graduation from the enormous capitals at the top, down to the fine running — no, hobbling hand in which I wrote off the rule. Now slate and pencil and brain came into use. I met with no difficulty at first; Simple Addition was as easy as counting my fingers. But there was one thing I could not understand — that carrying of tens. It was absolutely necessary, I perceived, in order to get the right answer; yet it was a mystery which that arithmetical oracle, our schoolmaster, did not see fit to explain. It is possible it was a mystery to him…’

Here are some other interesting passages from the book that kept me reading past midnight. Page 23:

Another example, by Benjamin Nichols (1799), is shown in Fig. 18. In the right-hand problem he realizes he is going to run out of room if he continues vertically downwards, so curls the working round to fit it in… In the next problem Benjamin could not fit the working into even two pages, so crept on to a third page adding the plaintive remark: ‘—the work was very long, and I did not fit it in to the Manuscript’.

Page 26:

Many of the manuscripts contained illustrations, drawing, doodles, and calligraphic elaborations, mostly unrelated to the text. It seems their purpose was more to enliven the book and to give opportunities for individual artistic invention… Three of the French manuscripts I have, in folio, are elaborately and colourfully illustrated. Jacques Castels (1752) was particularly fond of ships… A weird head at the top of one page from Melchior Veutre (1764) is shown in Fig. 25.

All this reminds me of some of my students’ drawings, for example, a cat on the bottom left of the homework sheet because “I had a feeling you’re a cat person.” I smile fondly as I re-read some of this book; kids will be kids, and kids were kids.

Pages 30–32:

…interesting enclosures and additions are occasionally to be found in manuscripts. William Murfitt (1790), for example, used the blank pages at the end of Vol. 2 as an account book to record bills for cobbler’s work… William Grave (1844) is a real treasure trove, containing… two visiting cards, a dinner menu on embossed card, and a pressed flower which can still be recognized as lily-of-the-valley… The menu is worth listing in full: October 29^th/ Clear Soup. Turbot / Oyster Patties /Saddle of Mutton / Pheasant, Wild Duck / Marachino Jelly, Vanille Ice / Anchovies / Cheese Straws.

Chapter 7, pp. 33–35, is titled “The Wider Curriculum.” Ah, now we’ve made it past seventh grade! “…Included somewhere in the manuscripts there was work on geometry, algebra, trigonometry, conic sections, fluxions, mechanics, and even the arithmetic of infinities…”

Pages 36–45, comprising a large fraction of this small book, are devoted to the lives of the manuscripts’ authors. He calls them by first name and gives details of their personal lives. He seems as fond of these kids as he is of his collection of their copybooks. Page 41:

We know how old Thomas was when he wrote his book since he gives his age in one of the examples in subtraction (Fig. 37). At 9 years old, he is the youngest of my authors to have stated his age. His writing is beautiful, without being over-formal, and remarkably mature… in fact, a visual delight…

Pages 43–45:

There were a number of comments and illustrations which had nothing to do with mathematics, but which shed light on what life was like in a boarding school at that time. Such as the rather gruesome drawing entitled ‘Picture of Billy French in bed with his toe tied’ (Fig. 42)… And on page 105: Hur-rah!!!!!! X Hurrah!!!!!! = 3 cheers for the midsummer holidays…’ These remarks are further evidence that manuscripts were not read by teachers after they had been written…

We, however, do get to read them, including the naughty juicy details of the kids’ lives and psyches. No, this book is not boring at all. Anyone teaching History of Mathematics can refer to parts of it and it will help enliven her course just as the kids’ illustrations enlivened their copybooks.

Marion Deutsche Cohen is the author of “Crossing the Equal Sign” (Plain View Press, TX), a collection of poetry about the experience of mathematics, as well as other literary works with images from math. She teaches at Arcadia University in Glenside, PA, where she developed the course “Truth and Beauty: Mathematics in Literature.”

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