Guest Notebook: Back to School special! (1) A little history of American math books
by Robert Rosenfeld
Frontispieces of Summario Compendioso and Hodderās 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 HernĆ”nĀ CortĆ©s 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 Hodderās Arithmetick, a Boston copy of a book first published in London. Hodderās 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 āschoolā 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 Wardās Young Mathematicianās Guide, a monster at some 500 pages, covering material from how to read and write numbers to Newtonās 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 āciphering book.ā 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 Lincolnās 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 āNote to the Readerā he says his goal is to help make youth successful āas to Clerkship and Trades.ā The Table of Contents begins with āThe definition of Numbers & Numerationāāhow 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 ādeductiveā approach, was dominant for at least another hundred years.
Hodderās first āappliedā 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 āpencilsā 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 ā¦ 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 operationsāaddition, subtraction, multiplication, and divisionāprimarily 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 āis not studious to get rid of the Money of one Place before he arrives at another, he is sure to be a considerable Loser.ā Not until 1794 was there an official US dollar coin, the new standard unit. The closest to a standardized coin before then was the SpanishĀ silver dollar cast in Mexico, sometimes called a āpiece of 8ā (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 Hodderās Book one starts to see āAssistantsā for schoolmasters. The School Masterās Assistant by Thomas Dilworth, first published in London in 1743, eighty-two years after Hodderās 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 āvulgarā 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 theĀ Coinage ActĀ or theĀ Mint Act of 1792 established the silverĀ dollarĀ as the unit of money in the United States and created aĀ decimalĀ system for US currency. Dilworthās 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 masterās use of any book at all as an āassistant.ā 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 Masterās Ignorance can never be brought in question, who can begin and end it readily; and most certainly that Scholarās 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. āThese little Forgeries are soon detected by the Diligence of the Tutor.ā
In the early editions the preface is followed by an essay, āOn the Education of Youth,ā directed not to the schoolmasters but to the parents. The essay emphasizes the role of education to preserve āReligion and Virtueā 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, āto be sensible of their Childrens Defects,ā and to compel children āto submit to the little (Imaginary ) Hardships of the School,ā noting that, āwhile the Master endeavours to keep Peace, good Harmony, and Friendship among his Scholars, they are generally taught the Reverse at Home.ā
The essay closes with a āwhile Iām at itā plea for the full equal education of girls. Why? Both because a general āliberalā education makes for a better life, but also because a women may need to manage on their own if āleft 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.ā
Immediately after the Revolution the period of elated nationalism was reflected in new schoolbooks, and we see āUnited Statesā 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. Pikeās 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 āHow can we more essentially benefit our country than by instructing and giving a proper direction to the minds of our youth?ā In addition to the four basic operations and their applications, there are chapters on algebra and conic sections and even applications of Newtonās rules on the force of gravity. In the first two editions of Pikeās book, the type of money problems called āreduction,ā 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 āeaglesā and āmills.ā) 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.) Pikeās book became extremely popular. It was reissued in revised editions for another fifty-five years. Competitors entered the marketplace, one of which apparently used ācopious extractsā of Pikeās material. Of this incident, the preface to Pikeās second edition (1797) notes, āas this circumstance is notorious, the time of the reader need not be taken up with the recital.ā
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 ācharactersā of āhigh rank in the literary world.ā Washington did however praise the finished book as an outstanding example of āAmerican Genius.ā
The much smaller Schoolmasterās Assistant, by Nathan Daboll, designed to bring exercises about federal money to lower-grade schools, also achieved great popularity and longevity. Daboll emphasizes an āintire newā 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 ā¦ I find, in the greatest part of [other authors], a scarcity of examples near the beginning, where they are most wanted ā¦ 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.
Dabollās style is often playful, using cute names for his characters like Timothy Tailor and James Paywell in a problem about a tailorās bill.
His textbook editions circulated so widely and for so long that Dabollās 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 āvoters had been cheated by false returnsā that āI learn[ed] to figure according to Daboll, and if these gentlemen will come and sit down with me I will show them figures they canāt get around. [Applause.]ā
Dabollās approach is a harbinger of a new philosophy of education growing from European Enlightenment ideas expressed in Jean-Jacques Rousseauās controversial 1762 book, Emile, or On Education ā¦
Read Part Two of this post here
Robert Rosenfeld is a retired professor of math and statistics, formerly of the University of Vermontās Vermont Mathematics Initiative. He is the author and co-author of several textbooks in algebra and statistics.
Book PostĀ is aĀ by-subscriptionĀ book review service, bringing snack-sized book reviews by distinguished and engaging writers direct to our paying subscribersā in-boxes, as well as free posts like this one from time to time to those who follow us.Ā We aspire to grow a shared reading life in a divided world.Ā Become a paying subscriber to support our work and receive our straight-to-you book posts. Adrian Nicole LeBlanc, Jamaica Kincaid, Marina Warner, John Banville, Nicholson Baker, Ingrid Rowland, Ćlvaro Enrigue, more.
Detroitās Source Booksellers is Book Postās Autumn 2023 partner bookstore! We partner with independent bookstores to link to their books, support their work, and bring you news of local book life across the land.Ā Weāll send a free three-month subscription to any reader who spends more than $100 with our partner bookstore during our partnership. Send your receipt to info@bookpostusa.com. Read more about Sourceās story in here in Book Post.
Follow us:Ā Instagram,Ā Facebook,Ā TikTok,Ā Notes,Ā Bluesky, Threads @bookpostusa
If you liked this post, please share and tell the author with a ālike.ā