The teaching and history of mathematics in the United States

Front matter

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(2, I 3 [ ^fiole Number 107]

BUREAU OF EDUCATION CIRCULAR OF INFORMATION NO. 3, 1890

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coloinial times.

Elementary Schools.

On tlie study of mathematics iu elementary schools of the American colonies but little can be said. In early colonial days schools did not exist except in towns and in the more densely settled districts; and oven where schools were kept, the study of mathematics was often not pursued at all, or consisted simply in learning to count and to perform the fundamental operations with integral numbers. Thus, in Hampstead, N. H., in 1750, it was voted " to hire a school-master for six mouths in ye summer season to teach ye children to read and write." Arithmetic had not yet been introduced there. As late as the beginning of this century there were schools in country districts in which arithmetic w^as not taught at all. Bronson Alcott, the prooDinent educator, born in Massachusetts in 1799, in describing the schools of his boyhood, says: ■'Until within a few years no studies have been permitted in the day school but spelling, reading, and writing. Arithmetic was taught by a few instructors one or two evenings in a week. But in spite of the most determined opposition arithmetic is now permitted in the day school." This was in Massachusetts at the beginning of this C(^ntury.

In secondary schools, " ciphering" was taught during colonial times, which consisted generally iu drilling students in the manipulation of integral numbers. He was an exceptional teacher who possessed a fair knowledge of " fractions" and the " rule of three," and if some pupil of rare genius managed to master fractions, or even pass beyond the " rule of three," then he was judged a finished mathematician.

The best teachers of those days were college students or college graduates who engaged in teaching as a stepping-stone to something better. An example of this class of teachers was John Adams, afterwards President of the United States. Immediately after graduating at Harvard and before entering upon the study of law, he presided, for a few years, over the grammar scliool at Worcester. From a letter written by him at Worcester, September 2, 1755, we clip the following description of the teacher's daily work :

As a haughty monarch ascends his throne, the pedagogue mounts his awful great chair, and dispenses right justice through his whole empire. His obsequious subjects execute the imperial mandates with cheerfulness, and think it their high happiness to he employed in the service of the emperor. Sometimes paper, sometimes his penknife, now birch, now arithmetic, now a ferule, then ABC, then scolding, then flattering, then thwacking, calls for the pedagogue's attention.

School appliances in those days were wholly wanting (excepting the ferule and birch rods). Slates were entirely unknown for school use until some years after the Eevolution ; blackboards were introduced much later. Paper was costly in colonial days, and we are told that birch bark was sometimes used in schools in teaching children to write and figure. Thirty-six years ago a writer in one of our magazines* wrote as follows :

"There are probably men now living who learned to write on birch and beech bark, with ink made out of maple bark and copperas." But more generally " ciphering " was done on paper. Dr. L. P. Brockett says that on account of its dearness and scarcity, "the backs of old letters, the blauk leaves of ledgers and day-books, and even the primer books were eagerly made use of by the young arithmeticians."

Since few or none of the pupils had text-books it became necessary for the teacher to dictate the " sums." As in the colleges of that time, so in elementary schools, tnanuscript bools were used whenever printed ones were not accessible. To advanced boys the teacher would give exercises from his manuscript or " ciphering-book," in which the prob. lems and their solutions had been previously recorded. " With a book of his own the pupil solved the problems contained in it in their proper order, working hard or taking it easy as pleased him, showed the solutions to the master, and if found correct generally copied them in a blankbook provided for the purpose. * * * Some of these old manuscript ciphering-books, the best, one may suppose, having come down through several generations, are still preserved among old family records, bearing testimony to the fair writing and the careful copying, if not to the arithmetical knowledge, of those who prepared them. When a pupil was unable to solve a problem he had recourse to the master, who solved it for him. It sometimes happened that a dozen or twenty pupils stood at one time in a crowd around the master's desk waiting with * * * problems to be solved. Ther^were no classes in arithmetic, no explana tions of processes either by master or pupil, no demonstrations of principles either asked for or given. The problems were solved, the answers obtained, the solutions copied, and the work was considered complete. That some persons did obtain a good knowledge of arithmetic under sucli teaching must be admitted, but this result was clearly due rather to native talent or hard personal labor than to wise direction."t Those

* North Carolina University Magazine, Ealeigh, 1853, Vol. II, p. 452.

t History of Education in Pennsylvania, by James Pyle Wickersham, p. 205.

COLONIAL TLMES, 11

teachers who were the fortunate possessors of a, printed arithmetic used it as a guide in place of the old " ciphering-book."

In the early schools, arithmetic was hardly ever taught to girls. Eev, William Woodbridge says that in Connecticut, just before the Eevolution, he has "known boys that could do something in the first four rules of arithmetic. Girls were never taught it."* In the two " charity schools " in Philadelphia, which before the Eevolution were the most celebrated schools in Pennsylvania, boya were taught reading, writing, arithmetic; girls, reading, writing, sewing. Thus, sewing was made to take the place of arithmetic. Warren Burton, in his book entitled, " The District School as it was, by one who went to it," says that, among girls, arithmetic was neglected. The female portion " generally expected to obtain husbands to perform whatever arithmetical operations they might need beyond the counting of fingers." Occasionally women were employed in summer schools as teachers, but they did not teach arithmetic. A school-mistress "would as soon have expected to teach the Arabic language as the numerical science^"

The early school-books in IsTew England and in ail other Euglish settlements were much the same as those of Old England. John Locke, in his Thoughts concerning Education (1690), says that the method of teaching children to read in England has been to adhere to " the ordinary road of horn-book, primer, psalter, testament, and bible." This same road was followed in New England. We are told that books of this kind were sold to the people by John Pynchon, of Springfield, from 1656 to 1672 and after.! Eegular arithmetics were a great rarity in this country in the seventeenth century. The horn-book has been raised by some to the dignified name of a "primer" for teaching reading and impartin g religious instruction. If this be permissible, then why should we not also speak of it as an arithmetical primer ? For, what was the horn-book? It consisted of one sheet of paper about the size of an ordinary primer, containing a cross (called "criss-cross"), the alphabet in large and small letters, followed by a small regiment of monosyllables; then came a form of exorcism and the Lord's Prayer, and, finally, the Roman numerals. The leaf was mounted on wood, and protected with transparent horn,

" To save from fingers wet tbe letters fair."

It is on the strength of the Eoman numerals that we venture to propose the horn-book as a candidate for the honor of being the first mathematical primer used in this country. Horn-books were quite common in England and in the English colonies in America down to the time of George II. -They disappeared entirely in this country before the Eevolution. In early days the common remark expressive of ignorance was "he does not know his horn-book," This is equivalent to the more modern saying, " he does not know his letters."

* Eeminiscenses of Female Education, in Barnard's Journal of Education, 1864, p. 137. t Barnard's Journal of Education, Vol. XXVII.

George Fox, the founder of the Society of Friends, published in 1674, in England, a primer or spelling book, which was republished at Philadelphia in 1701, at Boston in 1743, and at Newport, R. I., in 1769.* Wickersham describes this little book as containing the alphabet, lessons in spelling and reading, explanations of scripture names, Roman nume als, lessons in the fundamental rules of arithmetic and weights and measures, a perpetual almanac, and catechism with the doctrine of the Friends. It may be imagined that a mere primer, covering such a wide range of subjects, could contain only a very few of the simplest rudiments of a subject like arithmetic. Fox's book was used little outside of the Society of Friends.^

Wickersham (p. 201) speaks of another book which is of interest as illustrating the book-making of those old times. It is entitled, " The American Instructor, or Young Man's Best Companion, containing Spelling, Reading, Writing, Arithmetic, in an Easier Way than any yet Published, and how to Qualify any Person for Business without the Help of a Master." It was written by George Fisher, and printed in Philadelphia, in 1748, by Franklin and Hall. This work never attained any popularity.

Dr. Brockett says that in New Jersey and, perhaps, also in Virginia, a book resembling the " New England Primer," but as intensely Royalist and High Church in religion as the New England Primer was Puritan and Independent, was in use in schools. It was called "A Guide for the Child and Youth, in two parts ; the First for Children, * * * the second for Youth : Teaching to write, cast accounts and read more perfectly; with several other varieties, both pleasant and profitable. By T. H., M. A., Teacher of a Private School, London, 1762." It does not appear that this book was reprinted here.

Wickersham gives another book of similar stamp but of much later date. " Ludwig Hocker's Rechenblichiein was published at Ephrata [Pennsylvania] in 1786. The Ephrata publication is an exceedingly curious compound of religious exercises and exercises in arithmetic. The creed, the Lord's Prayer, hymns, and texts of scripture, are strangely intermixed with problems and calculations in the simpler parts of arithmetic"!

One of the earliest purely arithmetical books used in this country was the arithmetic of James Hodder. It may possibly have fallen into the hands of as early a teacher as Ezekiel Cheever, "the father of Connecticut school-masters, the pioneer and patriarch of elementary classical culture in New England."! In a history of schools at Salem, Mass., we are told that " among our earliest arithmetics was James Hodder's."

* History of Education in Pennsylvania., l>y James Pyle Wickersliam, p. 194.

t After having been a faithful school-master for seventy years, he died in 170S, at the age of ninety-four, having "held his abilities in an unusual degree to the very last."

Hodder was a famous English teacher of the seventeenth century. Later writers have borrowed largely from his arithmetic of which the first edition, entitled " Hodder's Arithmetick, or that necessary art made most easy," appeared in London in 1661. An American edition from the twenty-fifth English edition was published in Boston in 1719. This is the first purely arithmetical book known to have been printed in this country.

In ]^ew York the Dutch teachers of the seventeenth century imported from Holland an arithmetic called the " Coffer Konst," written by Pieter Venema, a Dutch school-master, who died about 1612. So popular was the book that an English translation of it was published in New York in 1730. Yenema's appeared to be the second oldest arithmetic printed in America.

An English work almost as old as Hodder's, which met with a limited circulation in this country, is Cocker's Arithmetic. The first edition appeared in England after the death of Cocker, in 1677. According to its title page it was " perused and published by John Hawkins, * * * by the author's correct copy." De Morgan is perfectly satisfied that " Cocker's Arithmetic was a forgery of Hawkins's, with soipe assistance, it may be, from Cockers papers." Eegarding the book itself, De Morgan says :* " Cocker's Arithmetic was the first which entirely excluded all demonstration and reasoning, and confined itself to commercial questions only. This was the secret of its extensive circulation. There is no need of describing it; for so closely have nine out of ten of the subsequent school treatises been modelled upon it, that a large proportion of our readers would be able immediately to turn to any rule in Cocker, and to guess pretty nearly what they would find there. Every method since his time has been " according to Cocker." This book was found here and there in the colonies at an early date. Thus we read in Benjamin Franklin's Autobiography that (at about the age of sixteen; i. e., about 1722) " having one day been put to the blush for my ignorance in the art of calculation which I had twice failed to learn while at school, I took Cocker's trei^tise on arithmetic and went througii it by myself with the utmost ease." An American edition of the work appeared in Philadelphia in 1779. It contains the rude portrait of the author, " which might be taken for a caricature," and also the following poetical recommendation :

Ingenious Cocker, now to Rest thou 'rt gone, No Art can show thee fully, but thine own; Thy rare Arithmetick alone can show Th' vast Thanks we for thy labours owe.

Wickershamt mentions Baniel Yenmng^s Ber Oesohwinde Rechner as having been published by Sower in 1774.

* Article, " Cocker," Penny Cyclopaedia,

t History of Education in Pennsylvania, p. 200.

The first arithmetic written by an American author and printed here was that of Prof. Isaac Greenwood of Harvard College, in 1729. The book was probably used by the author in his classes at Harvard. We have nowhere seen it mentioned except in a biographical sketch of its author.* So far as we know, there are only three copies of Greenwood's Arithmetic in existence, two in the Harvard library and one in the Con. gressional Library. Prof. J. M. Greenwood, superintendent of schools in Kansas City, sends the writer the following descrij)tion of it :

The book is a small duodecimo volume of 158 pages^ exclusive of an advertisement (4 pages) prefixed, and the table of contents (4 pages) put at the end. The following is a transcript of the title-page : "Arithmetick, Vulgar and Decimal: with the Application thereof to a variety of Cases in Trade and Commerce. (Vignette.) Boston : N. E., Printed by S. Kneeland and T. Green, for T. Hancock at the Sign of the Bible and Three Crowns, in Ann Street, MDCCXXIX."

The headings of chapters are as follows : The introduction ; chapter 1, Numeration ; chapter 2, Addition j chapter 3, Subtraction ; chapter 4, Multiplication 5 chapter 5, Division ; chapter 6, Eeduction ; chapter 7, Vulgar FiBctions ; chapter 8, Decimal Fractions; chapter 9, Eoots and Powers; chapter 10, Continued Proportion; chapter 11, Disjunct Proportion ; chapter 12, Practice; chapter 13, Eales relating to Trade and Commerce.

From the preface : "The Author's Design in the following Treatise is to give a veryconcise Account of such Rules, as are of the easiest practice in all the Parts of Vulgar and Decimal Arithmfetick and to illustrate each with such esamples, as may be sufficient to lead the Learner to the full Use thereof in all other Instances."

" The Reader will observe that the Author has inserted under all those Rules, where it was proper, Examples with Blanks for his Practice. This was a Principal End to the Undertaking ; that such persons as were desirous thereof might have a comprehensive Collection of all the best Rules in the Art of Numbering, with Examples wrought by themselves. And that nothing might be,.wanting to favour this Design, the Impression is made upon several of the best sorts of Paper. This method is entirely new, * * *."

The paper used in the book is thick, the type large. Words and phrases to which the author desires to call special attention are printed in italic characters, and as more than half the book is, in the author's eyes, important, more than half the book is printed in italics.

In 1788, when Nicholas Pike published his arithmetic. Greenwood's book was entirely unknown, and Pike's was believed to be the first arithmetic written and i)rinted in America.

The first arithmetic which enjoyed general popularity and reached an extended circulation in the colonies was the School-master's Assistant, by Thomas Dilworth. The first edition of this was published in London in 1744 or '45. According to Wickersham, there appeared a reprint of this in Philadelphia in 1769. Other American editions were brought out at Hartford in 1786, New York in 1793 and 1806, Brooklyn in 1807,

* Apjsleton's Dictionary of American Biography.

Hew Londoii 1797, aud Albany 1824. At the beginning of tbe Eevolotion this was the most popular arithmetic, and it continued in use long after,

We have now enumerated all the arithmetics which were used to our knowledge in the American colonies. It may be instructive to give the last book which we have mentioned a closer examination ; for Dil worth's School-master's Assistant was the most noted arithmetic of its time. As an arithmetician Dilworth belonged to the school founded by Cocker, which scrupulously excluded all demonstration and reasoning. The School-master's Assistant gives all rules and definitions in the form of questions and answers. Let us turn to page M of the twenty- second London edition, 1784, and examine his mode of explaining proportion, or, as the subject was then called, the " Eule of Three,"

OF THE SINGLE RULE OF THKEB.

Q. How many Parts are tliera in the Eule of Three ?

A. Two : Single or Simple, and Double or Compound.

Q. By what is the single Eule of Three known ?

A. By three Terms, which are always given in the Question, to find'a fourth.

Q. Ave any of the terms given to be reduced from one Denomination to another?

A. If any of the given terms be of several denominations, they must' tse reduced into

the lowest Denomination mentioned. ^

Q. What do you observe concerning the first and third Terms I A. They must be of the same Name and Kind.

Q. What do you observe concerning the fourth Term ? ■ A. It must be of the same Name and Kind as the second. Q. What do you observe of the three given Terms taken togethel?f~ A. That the two first are a Supposition, the'last is a Demand. ' Q. How is the third Term known ? ♦

A. It is known by these, or the like Words, What cost ? Hoio many ? Eoxv much f Q. How many Sorts of Proportion are there ?

A. Two: Direct and Inverse.

And so on. We have quoted enough to give an idea of the book. It is not easy to see how a pupil beginning the subject of proportion could get clear notions from reading the above. Kor can we see how a boy who had never before heard of fractions could get any idea whpvtever of a fraction from Dilworth's definition, which is (p. Ill) : A fraction " is a broken number and signifies the part or parts of a whole number."

A closer examination of this arithmetic discloses many other strange things. It consists really of three parts, more or less complete in themselves, namely: Parti, on whole numbers ; Part II, on vulgar fractions; Part III, on decimal fractions. In Part I, the student is carried through the elementary rules, and through interest, fellowship, exchange, double rule 01 three, aliigation, single and double position, geometrical progression, and permutations. He is carried through all these without having as yet even heard effractions. The advanced and comparatively unimportant subjects, such as ailigation and progressionsj are mad© to precede so important and fundamental a subject as fractions. The teaching of decimal fractions after interest is illogical, to say the least. In Part II, after fractions have been explained, the rule of three is taken up a second time'; and in Part III, under decimal fractions, it is resumed a third time. Thus this rule is exi^lained three times ; the first time with whole numbers, the second time with common fractious, the third time with decimal fractions — thus leaving the impression that the rule is different in each one of the three cases.

The whole book is nothing but a Pandora's box of disconnected rules. It appeals to memory exclusively and completely ignores the existence of reasoning powers in the mind of the learner. Koticeable is the fact that in the treatment of common fractions, the process of" cancellation," which may be made to shorten operations so much, is not even mentioned. The book abounds in unnecessary and perplexing technical terms, such as " practice," '^ conjoined, proportion," "alligation medial," <' alligation alternate," " comparative arithmetic," "biquadrate roots," "sursolids," "square cubes," "secontlsursolids," "biquadrates squared," "third sursolids," and "square cubes squared." Under the head of duodecimals are given rules like these: " Feet multiplied by feet give feet; " "feet multiplied by inches give inches," etc. Tliese rules, taken literally, are absurd. We can no more multiply feet by feet than we can multiply umbrellas by umbrellas. These rules are in opposition to the fundamental ideas of multiplication in arithmetic. A concrete number can not be multiplied by a concrete number. It seems strange that so gross a^ error should not have been corrected in later editions of the book ; but still more strange is the fact that nearly all arithmetics down to the i^resent day should have persisted in making this mistake.

As an instance of the confusion of ideas to which it gives rise, I quote the following from an article "Early School Days " in Indiana, contributed by Barnabas C. Hobbs.* A law had just been passed requiring that teachers' examinations should be conducted by three county examiners instead of the township trustees, as had been the practice before. "I shall not forget," says Hobbs, 'imy first experience under the new system. The only question asked me at my first examination ■was, 'What is the product of 25 cents by 25 cents T * * * We were not as exact then as people are now. We had only Pike's Arithmetic, which gave the sums and the rules. These were considered enough at that day. How could I tell the product of 25 cents by 25 cents, when such a problem could not be found in the book ? The examiner thought it was 6i- cents, but was not sure. I thought just as he did, but this looked too small to both of us. We discussed its merits for an hour or more, when he decided that he was sure I was qualified to teach school, and a first-class certificate was given me."

* Tfie Indiana Schools, by James H. Smart, 1876.

We have spoken of Dilwortli's School-master's Assistant at some length, because from it we caa see what sort of arithmetics we inherited from the English. All arithmetics of that time were much alike. The criticisms upon one will therefore apply to all.

Before proceeding to another subject we shall examine briefly the '• Short Collection of Pleasant and Diverting Questions " in Dilworth. We shall meet there with a company of familiar friends. Who has not heard of the farmer, who, having a fos, a goose, and a peck of corn, and wishing to cross a river, but being able to carry but one at a ticne, was confounded as to how he should carry them across so that the fox should not devour the goose, nor the goose the corn? Who has not heard of the perplexing problem of how three jealous husbands with their wives may cross a river in a boat holding only two, so that none of the three wives shall be found in company of one or two men, unless her husband be present ? Many of us, no doubt, have also been asked to place the nine digits in a quadrangular form in such a way that any three figures in a line may make just 15? When these pleasing problems were first proposed to us, they came like the mofning breeze, with exhilarating freshness. We little suspected that these apparently new-born creatures of fancy were in reality of considerable antiquity; that they were found in an arithmetic used in this country one hundred years ago. Still greater is our surprise when we learn that at the time they were published in Dilworth's School-master's Assistant some of these questions for amusement had already seen as many as one thousand birthdays. The oldest record bearing upon this subject is found in a manuscript entitled Propositioncs ad acuendos jtivenes. The authorship of this paper has been generally attributed to Alcuin, whose years of greatest activity were spent in France, in the court of the great Charlemagne, and who was one of the most learned scholars and celebrated teachers of the eighth century. The MS. attributed to him contains the puzzle about the wolf, goat, and cabbage, which in the modern version is known as the "fox, goose, and peck of corn" puzzle.

In a MS. coming from the thirteenth century, two learned German youths, named Firri and Tyrri, are made to propose to each other problems and puzzles. Firri takes among others the hard nut of Alcuin about the wolf,-goat, and cabbage-head, and lays it before Tyrri in the modified and improved version of the three wives and the three jealous husbands. This same document contains also the following: "Firri says : There were three brothers in Cologne, having nine vessels of wine. The first vessel contained 1 quart (amam), the second 2, the third 3, the fourth 4, the fifth 5, the sixth 6, the seventh 7, the eighth 8, the ninth 9. Divide the wine equally among the three brothers, without mixing the contents of the vessels."

This problem admits of more than one solution, and is closely related to the last problem we quoted from Diiworth's collection. It is of spe- 881— Xo. 3 2

TEACHING AND HISTORY OF MATHEMATICS.

cial interest, since it gives rise to the follovring magic square, in which any three figures in a straight line have 15 for their sum.

2 9

7 5

I

The history of magic squares is a rich field for investig'ation. The Germans were by no means the originators of them. This honor must be given to the Brahmins in India. Later on the study of these curious problems was zealously pursued by the Arabs, who transmitted the fruits of their study to the Europeans.

Had we the time, we would attempt to trace the history of some other familiar puzzles. But enough has been said to show that many of them possess great antiquity. Nevertheless, when they were first proposed to us, they betrayed no signs of old age. May they continue perjjetually in their youth, and may they delight the minds of men for numberless centuries to come !

Colleges, haryakd college.

As early as 1636 the people of Massachusetts stamped their approval upon the cause of higher education by the founding of Harvard College. The nature of the early instruction given at this oldest of American colleges is of special interest to us. The earliest record bearing on the history of the rise of mathematical studies at Harvard is a tract entitled, '' New England's First Fruits." It was originally published in 1643, or five years after the college had opened, and contained the curriculum of studies then pursued. Whoever expects to find in it an extended course of mathematical studies resembling that in our colleges of to-day will be much disappointed.

In the first place, a student applying for admission to Harvard in 1643 was not confronted and embarrassed by any entrance examinations in mathematics. The main requirement for admission was Latin. Contrary to the practice of to-day, Latin was then taught as a spoken language. " So much Latin as was sufficient to understand Tally, or any like classical author, and to make and speak true Latin in prose or verse, and so much Greek as was included in declining perfectly the paradigms of the Greek nouns and verbs," \vere the necessary requisites for admission ; but in mathematics applicants were required to know not even the multiplication table.

When we come to examine the college course, which extended originally through only three years, we meet with other surprises. Boys did not receive that thorough " grinding" in the elements during the first years ot college that they do now ; on the contrary, no mathematics at all was taught except during the last year. The mathematical course began in the Senior year, and consisted of arithmetic and geometry during the iirst three-quarters of the year, and astronomy during the last quarter. Algebra was then an unknown science in the New World. It is interesting to notice that, in this original curriculum, the attention of each class was concentrated for a whole day upon only on6 or two subjects. Thus, Mondays and Tuesdays were devoted by the third year students exclusively to mathematics or astronomy, Wednesdays to Greek, Thursdays to "Eastern tongues." and so on. The importance attached to mathematical studies, as compared with other branches of discipline, may be inferred from the fact that ten hours per week were devoted to philosophy, seven, to Greek, sis to Ehetoric, four to Oriental languages, but only two to mathematics. According to these figures, Oriental languages were considered twice as important as mathematics. But we must remember that this course was laid out for students who were supposed to choose the clerical profession. For that reason, philosophical, linguistic, and theological studies were allowed to monopolize nearly the whole time, while mathematics was excluded almost entirely.