Paradigm, No. 3 (July, 1990)
The Open University,
Milton Keynes MK7 6AA
David Eugene Smith lived from 1860-1944, and was professor of mathematics at Teachers College, Columbia University, from 1901-1926. Many of his works are still in print, notably his two-volume History of mathematics (1923) and his Source book in mathematics (1929). He was one of the most distinguished mathematical historians and mathematics educators of the century, a towering figure in a wide range of fields and activities. Although he is remembered as a historian, he was in the first instance a mathematics teacher, with the profound belief that in order to develop a good pedagogy for the subject one must know and understand its history.
Number stories of long ago is a very carefully crafted book aimed at children of elementary school age, and we can learn quite a lot from it about Smith's pedagogical conceptions, about the conceptualisation of childhood implicit in the text, about the role he sees for history at this level, and about the use of the book as an object. The pedagogical approach is remarkably clear: it is one of empathy and identification, indeed, complicity. Thus there are two prefaces, whose contrast is rather significant: 'Preface number one: just between us, and worth reading' and 'Preface number two: for the grown-ups, and not worth reading'.
The prefaces are written in quite different linguistic styles. The first preface is in the clear simple, rather beautiful, language in which the book as a whole is written:
You who read these stories should imagine ourselves sitting by the great log fire and listening to the Story-Teller. You should seem to see in the flames and the shadows the moving pictures of those who played their parts in Number Land when the world was learning as you do.
The second preface is written in Smith's usual adult style, noticeably less clear and flowing, explaining that the book is 'intended for supplementary reading in the elementary school'. 'Reading' is meant literally as the first preface makes clear. Smith assumed or expected that the child would read the book, using it to enter into a new imaginative world. (I don't know, by the way, an thing about the history of the convention of double prefaces for child and adult: can anyone help here? It would be interesting to know how far back it goes, what feature are characteristic and so on.)
To add to its attractions to a child, the book is very beautifully illustrated. There are eight coloured plates, three by E. M. Wireman, two by R. R. B. B., and the rest unsigned, on such subjects as: Ching and his three turtles; Hippias on the Acropolis of Athens; Robert Record learning the use of counters; Michael Stifel and his teacher; Ahmes studying fractions.
From these picture titles one may begin to see the book's strategy: Smith takes authors of past arithmetic texts and talks about their work through imagining the author as a child, how they would learn to calculate and so on, and describes the improvements made in their own books when they grew up. So several messages permeate the text: empathy with past mathematical writers as children; perhaps the message that the reader when grown up may create fresh work likewise, and certainly a message of tolerance and relativism:
"Did they really count like this?" asked the Tease. "Really," replied he of the curious book. "I think it is funny," said the Tease. "No funnier than your way would seem to Ching," said the Story-Teller.
The text is more than simply succession of straightforward stories. A complicated effect of further complicity is created by
the interaction between narrator, who is called 'The Story-Teller', and the children, called 'The Crowd', consisting of one main female child called 'The Tease', as well as Charles, Dor y, Edward, Fanny, George, Gertrude, Maude, and Will. This interaction can have a slightly arch quality, but may also contain instructional humour which has quite a ring of realism:
"Now we are getting near the end of our number stories", said the Story-Teller as the Crowd rushed into the room. "Which end?" asked the Tease. "A line has two ends." "Never mind which is the end of a line. One thing is sure, that I am near the end of my stories?" "Yes, but which end of the stories?" persisted the Tease. "The end that we are near," said the Story-Teller.
A further pedagogic feature built into the text is 'the Question Box', which is an elaborate way of incorporating end-of-chapter questions both into he text and into the use of the text by school-children. Smith says in the adult preface that the Question Box is a supply of questions that can be used either for conversation or for written work; they're questions like "What need have we for counting that Ching did not have?", "Which numerals seem to you the easiest, those of Chang, those of Lugal, or those of Ahmes?". Within the story-line of the book, they are a supply of questions that the children must tackle before they are told the next story.
They are part of the fun and play activities that the children enjoy, and which indeed moves into number puzzles and conundrums towards the end. So the genuine read-and-enjoy quality of the book is enhanced, or overlaid, depending on your perspective, by the presence of questions at the end of chapters, a reminder that a teacher is supposed to watch over the whole process and keep a check on how the child is developing knowledge and understanding.
Smith saw mathematical education not as an end in itself but as part of a much broader inculcation into culture and understanding. This is seen here in his remark at the close of his adult preface that he hopes the book "will create a new interest not merely in the study of arithmetic but in the story of the development of our civilization". One of the quotations at the head of the text rather strikingly appears to prefigure Foucault: 'There is no knowledge that is not power', but in fact this is a quotation from Emerson.