Guest Notebook: Back to School special! (1) A little history of American math books
by Robert Rosenfeld
Frontispieces of Summario Compendioso and Hodder’s Arithmetick
The oldest American arithmetic textbook is Sumario Compendioso, printed in Mexico City, the capitol of New Spain, two hundred years before the American War of Independence. It was written in 1556 by Brother Juan Diez Freyle and consisted primarily of rules and tables to help traders of silver and gold, but it also has some straightforward arithmetic and algebra instruction. Brother Juan had first come to the New World in 1518 with the conquistador Hernán Cortés and over the years held various important administrative positions. The printing house was an arm of the Spanish government and church. The first English language arithmetic book did not appear in the New World until 1719, 163 years later. It was James Hodder’s Arithmetick, a Boston copy of a book first published in London. Hodder’s book was influential in America for a long time.
Schooling in the 1600s and early 1700s in the English colonies was quite variable, best organized in the larger towns in the Puritan colonies of New England and least so in the more rural southern colonies. Many children simply had some version of home schooling or instruction from an educated person (often a minister who may have been a college graduate in America or England). A man or woman could set up a “school” at home and tutor several children. Some communities set aside a building as a school. The push for public schools for all children did not take root until well into the nineteenth century. Elementary education often did not include anything about mathematics, only reading and writing. Arithmetic was primarily intended for later use in commercial life and in some specific trades. It could be learned in apprenticeships. If you were among the elite and attended a college, perhaps Harvard or Yale or William and Mary, and wanted to study mathematics, you might use a book from England such as John Ward’s Young Mathematician’s Guide, a monster at some 500 pages, covering material from how to read and write numbers to Newton’s new calculus.
When arithmetic was taught at the grammar school level, most students did not have a textbook. Quite often their schoolmaster did not either. The master would dictate a problem and the student would write it down and then work it out on a piece of paper (or slate, or birch bark, or anything he or she could write on). When the teacher said it was correct the student would copy the problem and the solution into his own “ciphering book.” The problems often were taken from an arithmetic book the teacher owned, or from his own childhood ciphering book. The same problems could be assigned for decades. Historians looking at Abraham Lincoln’s ciphering book from the 1820s have identified the original sources for some of his exercises, including one book from 1743.
Hodder was a private schoolmaster in London who taught arithmetic, writing, shorthand, and accounting. His Arithmetick came out about a year after his text on writing, The Penman's Recreation. In a “Note to the Reader” he says his goal is to help make youth successful “as to Clerkship and Trades.” The Table of Contents begins with “The definition of Numbers & Numeration”—how to read and write numbers. In 1661 it was still new for many people to work with the written Arabic numeral system. The abacus was still in use in the marketplace if not among mathematical scholars and professional accountants. Arithmetic books had begun with numeration since Leonardo of Pisa (a.k.a., Fibonacci) wrote Liber Abaci in 1202 to introduce the Hindu-Arabic numeral system to the west. Hodder begins teaching the operations by giving a rule and an illustration of that rule. He presents a set of five numbers to be added. He says to line up the numbers and start adding mentally with the rightmost column. There is no reference to real objects or situations. Not the way we start today with a young person. This sequence, rule first and applications last, sometimes called a “deductive” approach, was dominant for at least another hundred years.
Hodder’s first “applied” calculations in addition are about money. Written arithmetic with English currency presents challenges of denomination: Hodder even warns about the messiness of writing out the calculations in ink. (Erasable “pencils” were not mass produced at the time. Graphite, a form of carbon, had been discovered in England around 1565 and one could dig it up, saw off slices, and wrap them in string to create a homemade pencil.)
I need not here to acquaint you that four Farthings make a Penny, twelve Pence a Shilling, and twenty Shillings a Pound … You may make a Prick with your Pen at every 4 in the Farthings, and at every 12 in the Pence, and at every 20 in the Shillings: But this Way is neither so neat nor commendable; for if you once prick false, you must prick it all over again, which will look like so many Blots, and make you more subject to mistake. Therefore I recommend these two Tables following to you, to begotten perfectly by heart, before you adventure upon Addition as 1 Shilling is 12 Pence, 2 Shillings is 24 Pence, and so on.
Starting with numeration and proceeding through the four basic operations—addition, subtraction, multiplication, and division—primarily with applications in commerce, was the standard approach in all the arithmetic books of the period.
Dealing with money in the colonies before 1776 was indeed a mess. Each colony printed its own paper money. One traveller noted that if one “is not studious to get rid of the Money of one Place before he arrives at another, he is sure to be a considerable Loser.” Not until 1794 was there an official US dollar coin, the new standard unit. The closest to a standardized coin before then was the Spanish silver dollar cast in Mexico, sometimes called a “piece of 8” (because the coin was worth 8 of a smaller coin called a reale). That dollar had been circulating since about 1550 and was recognized all over America. Only in 1857 did the Coinage Act end the legal use of foreign coins in the US.
Fifty or more years after Hodder’s Book one starts to see “Assistants” for schoolmasters. The School Master’s Assistant by Thomas Dilworth, first published in London in 1743, eighty-two years after Hodder’s Book, was extraordinarily long-lived and used in many schools, with fifty-seven American editions and revisions that would be published for almost a hundred years. Dilworth goes into more detail regarding the mathematical concepts and provides a question-and-answer approach for each one. The teaching of “vulgar” fractions, one whole number over another, such as 3/4, preceded the chapter on decimal notation, such as 3.75. This sequence would prevail until the Coinage Act or the Mint Act of 1792 established the silver dollar as the unit of money in the United States and created a decimal system for US currency. Dilworth’s fourth section is a collection of miscellaneous problems for review and entertainment, a trove for the schoolmaster who needs a variety of exercises.
Dilworth includes a preface going to some lengths to justify a master’s use of any book at all as an “assistant.” He apparently thought that some potential customers would worry about appearing incompetent. He promises that any such reservations will be mitigated by the greater success of his students.
He indeed (if any such there be) who is afraid his Scholars will improve too fast, will undoubtedly decry this Method. But that Master’s Ignorance can never be brought in question, who can begin and end it readily; and most certainly that Scholar’s Non-Improvement can be as little questioned, who makes a much greater Progress by This, than by the common Method.
He argues, in the name of efficiency, for giving the same problem to all the students and says not to worry about cheating. Why should the master make extra work for himself. “These little Forgeries are soon detected by the Diligence of the Tutor.”
In the early editions the preface is followed by an essay, “On the Education of Youth,” directed not to the schoolmasters but to the parents. The essay emphasizes the role of education to preserve “Religion and Virtue” in the world. One mark of the Protestant Reformation in England in the sixteenth and seventeenth centuries was the elevation of the role of schooling to promote Christian morality. The primary goal of Reformation education was that every person should be able to read the Bible for himself, hence Puritan New England colonies were quick to set up community-supported (but not free) schools. For example, in 1647 Massachusetts passed a law that any town with fifty or more families had to hire a teacher to teach their children to read and write. Towns with more than a hundred families needed to build a school. The smaller, more rural, and less religiously homogeneous mid-Atlantic and Southern colonies depended more on individual churches and individual tutors for schooling.
Dilworth appears to have written the essay because he is thinks that lax parenting is frustrating the efforts of schoolmasters. He reminds parents of the importance of regular attendance, of not contradicting the teacher at home, of insisting on truthfulness and even-temperedness, “to be sensible of their Childrens Defects,” and to compel children “to submit to the little (Imaginary ) Hardships of the School,” noting that, “while the Master endeavours to keep Peace, good Harmony, and Friendship among his Scholars, they are generally taught the Reverse at Home.”
The essay closes with a “while I’m at it” plea for the full equal education of girls. Why? Both because a general “liberal” education makes for a better life, but also because a women may need to manage on their own if “left to shift for themselves in the melancholy State of Widowhood (and what Woman knows that she shall not be left in the like State?) obliged to leave their Business to the Management of others; sometimes to their great Loss, and sometimes to their utter Ruin.”
Immediately after the Revolution the period of elated nationalism was reflected in new schoolbooks, and we see “United States” starting to appear on their covers. The first of these new United States arithmetic books, A New and Complete System of Arithmetic: Composed for the use of the Citizens of the United States, was published in 1788 by Harvard graduate Nicolas Pike, master of the Newburyport, Massachusetts, grammar school. This is a gigantic book, over 400 pages, was first designed as a college text but later abridged for elementary school use. Pike’s title suggests that he wanted his book to be the signature math book of the new country, covering a myriad topics, not only ones practical to the trade. A Latin quotation from Cicero on the title page translates as “How can we more essentially benefit our country than by instructing and giving a proper direction to the minds of our youth?” In addition to the four basic operations and their applications, there are chapters on algebra and conic sections and even applications of Newton’s rules on the force of gravity. In the first two editions of Pike’s book, the type of money problems called “reduction,” where you convert mixed monetary units to just one, used only British values, but by the third edition in 1808 they also featured US currency. (Two of the original US units were “eagles” and “mills.”) There were also exercises involving foreign units like length: 6 points make 1 line and 12 lines an inch, and, best of all, 3 Barley Corns make 1 inch. (We still use barleycorn in shoe size. For example, a size 11 shoe is one barleycorn shorter than a size 12.) Pike’s book became extremely popular. It was reissued in revised editions for another fifty-five years. Competitors entered the marketplace, one of which apparently used “copious extracts” of Pike’s material. Of this incident, the preface to Pike’s second edition (1797) notes, “as this circumstance is notorious, the time of the reader need not be taken up with the recital.”
When he began to put together the book, Pike wrote to George Washington asking for permission to dedicate it to him. Washington turned him down and suggested he submit it to some “characters” of “high rank in the literary world.” Washington did however praise the finished book as an outstanding example of “American Genius.”
The much smaller Schoolmaster’s Assistant, by Nathan Daboll, designed to bring exercises about federal money to lower-grade schools, also achieved great popularity and longevity. Daboll emphasizes an “intire new” presentation of decimal fractions early. His preface shows him to be an experienced teacher of young people.
It has been my province, through the greatest part of my life, to be employed in teaching arithmetic … I find, in the greatest part of [other authors], a scarcity of examples near the beginning, where they are most wanted … In teaching the first rules of arithmetic, I have ever found it best to encourage the attention of scholars with a variety of easy and familiar questions, which may serve to strengthen their capacities, before they moved to that which is more difficult.
Daboll’s style is often playful, using cute names for his characters like Timothy Tailor and James Paywell in a problem about a tailor’s bill.
His textbook editions circulated so widely and for so long that Daboll’s name entered American jargon as a synonym for sound mathematical reasoning. For example, in an Indiana newspaper from November 1871, a Mayor Kalbfleisch in Brooklyn, NY, assured listeners fearing that “voters had been cheated by false returns” that “I learn[ed] to figure according to Daboll, and if these gentlemen will come and sit down with me I will show them figures they can’t get around. [Applause.]”
Daboll’s approach is a harbinger of a new philosophy of education growing from European Enlightenment ideas expressed in Jean-Jacques Rousseau’s controversial 1762 book, Emile, or On Education …
Read Part Two of this post here
Robert Rosenfeld is a retired professor of math and statistics, formerly of the University of Vermont’s Vermont Mathematics Initiative. He is the author and co-author of several textbooks in algebra and statistics.
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