# Guest Notebook: Back to School special! (2) A little history of American math books

### by Robert Rosenfeld

###### Opening of Joseph Ray,* The Little Arithmetic* (1834)

*Read Part One of this post here!*

Nathan Daboll’s experience-based approach to teaching mathematics 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*. In the preface to *Emile* Rousseau writes, “the wisest writers devote themselves to what a man ought to know, without asking what a child is capable of learning.” Joseph Neef, a protégé of the Swiss educational reformer, Johann Pestalozzi, a follower of Rousseau who developed a curriculum in every discipline designed to imitate the way a child learns naturally in the real world, was invited to start a new school in Philadelphia after an American diplomat visited his orphanage school in Paris. Neef opened America’s first progressive school in 1809. Before it opened, he published the New World’s first strictly pedagogical book**, **its title showing both the influence of Rousseauean ideas and American national aspirations: *Sketch of a Plan and Method of Education founded on the Analysis of the Human Faculties and Natural Reason, suitable for the offspring of a Free People and for all Rational Beings.* Here are some passages from the description of mathematics instruction showing the influence of Rousseau:*From the moment a child learns to make the first use of its nerves, nature presents, unceasingly, to its eyes, a variety of objects; from which, at a very early period of its existence, it abstracts the notions of unity and plurality.*

*The power of combining numbers, is one of the noblest powers that have been given to man … how comes it then, that a power so noble has been debased in our schools to a merely mechanical, may I say, machine-ical operation?*

*As it is evident, that all our numerical notions proceed from objects, we shall, of course, begin our studies by them. Easily moveable things, as beans, pease, little stones, marbles, small boards, shall be our first instructors. To one bean we shall add one more, and after having carefully verified the sum resulting from this addition, we shall say, not one and one make two, nor one bean and one bean are two beans; but one time one bean, more one time one bean, is equal to two times one bean.*

His “easily moveable things” are the 1808 version of modern manipulatives. By the 1850s Rousseauean educational ideas were dominant in the US and have remained so.

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The first distinctly Pestalozzian arithmetic textbook, by Warren Colburn, appeared in the US in 1821. Warren Colburn had entered Harvard as an older student with experience in farm and then factory machinery work. Influenced by his professor John Farrar, an educational reformer, Colburn at his graduation ceremony presented an essay “On the Benefit accruing to an individual from a knowledge of the Physical Sciences,” in which he calls out the responsibility of those who know mathematics and science **“**to promote the diffusion of them among mankind.” Upon graduation Colburn opened a small private school in Boston and published his arithmetic book a year later.

Colburn’s book had a tremendous influence on all subsequent arithmetic textbooks. A vital point of the book was to consider mathematics a process of sense awareness, based on observations of real things, rather than an abstract ciphering procedure. The basic operations are first covered using only the names of the numbers and not the numeral symbols. A child could answer the questions by referring to objects or pictures of objects. Colburn stresses “mental” solutions—just think about it, reason it out. There is no writing down numerals. His book is tiny, almost just a compilation of exercises, but his heart is in how the material should be taught, laid out in a ten-page preface. One of the “first properties” a child discovers, he writes, as soon as the child “begins to use his senses,” is “the relation of number. He intuitively fixes upon *unity* as a measure, and from this he forms the idea of more and less; which is the idea of quantity.” We see that he is “able to make small calculations … and this we see them do every day about their playthings, and about the little affairs which they are called upon to attend to.” “The pupil should first perform the examples in his own way, and then be made to observe and tell how he did them, and why he did them so.” Other books and methods of instruction almost always fail because “the examples are so large, that the pupil can form no conception of the numbers themselves,” or they are “abstract” so the student can “discover but very little connexion between them, and practical examples.”

Colburn’s first word problem is a real break with tradition. It is about apples, not dollars: “James has two apples, and William has three; if James gives his apples to William, how many will William have?”

Meanwhile the “Common School Movement,” which established tax-based funding of public-school systems, was spreading from New York, the first state to implement it. Joseph Ray, who had begun teaching in rural schools in his native West Virginia when he was sixteen, joined the faculty of the brand-new Woodward High School in 1831, the first free public school to open in Cincinnati after Ohio’s Common School Law was passed 1825. Ohio had become the seventeenth state following the Louisiana purchase in 1803 and was rapidly becoming a major center for commerce as the West opened for settlement. Ray’s** **first math textbook appeared in Cincinnati 1834, responding to the demand for more coherent curriculum to implement across the system. In 1836 Woodward expanded to include a college, and a few years later William McGuffey, author of the McGuffey* Readers*, joined the faculty. Ray and McGuffey worked with the same local publisher, Truman, Smith and Co., which became a powerhouse in educational publishing, to produce two series’ of textbooks, the Ray *Eclectic Arithmetic*s* *and the McGuffey *Eclectic Reader*s. From 1836 to 1840 the Ohio school system implemented its first attempts at systematic grading and classification, creating further demand for textbooks adapted to the new approach.

Ray was a major education reformer in Ohio. He remained principal of Woodward until his death in 1855, and served terms as President of the Ohio State Teachers Association and as associate editor of the Ohio Journal of Education. His series of textbooks continued with updates for fifty years after he died. By 1913 more than 120 million copies had been sold. Today you can see newly revised editions of Ray’s books marketed particularly to Christian-based private and home schools.^{ }

Ray’s methods incorporated what he called “intellectual” arithmetic, an inductive approach where students themselves discover general rules by solving many specific problems first. The preface to the first book in his series, citing Pestalozzi, says that this approach teaches students “to reason, to analyze and to think for themselves; it gives them a confidence in their own reasoning powers, at the same time that it improves and strengthens their intellectual faculties,” leading the student “to an understanding of the principles from which the rules are derived, and teach[ing] him to regard rules as results rather than reasons: he will understand the ‘why and wherefore’ of every operation performed.” Ray’s book opens, like Colburn’s, with James and a playmate collecting apples. For the first lesson on fractions, Ray’s textbook explains in a “note to teachers” that “the teacher should be provided with a number of apples. When anything or any number is divided into two equal parts, one of the parts is called ‘one half’ of the thing or the number [illustration of half an apple].”

Over the whole series, Ray’s story problems portray “honest, hard-working men and women on the frontier, plowing fields, planting and harvesting crops, building walls, or buying and selling goods,” as educational historian David Kullman has described. The idea that public schools should foster moral civic behavior and to educate citizens to participate in democracy suffused the Common School movement.

It’s not surprising that many of us think that the arrival of the new math in the 1960s, when suddenly there were Cuisenaire rods in every classroom, marked the first time that there was a strong movement to change the way arithmetic should be taught. It’s tempting to think that the idea of hands-on experiential learning was new, prompted by the anti-authoritarian, creativity-embracing spirit of those times. But looking over the genealogy of American mathematics instruction, influential supporters of this approach appear as far back as the birth of the United States. The growing dominance of these methods corresponded with a changing vision, seeing mathematical reasoning not only as a tool for commerce but a discipline for clear thinking, of critical importance to a citizen in a democracy, and a growing belief that rote learning, Neef’s “machine-ical” approach, is debasing to a child’s education. Now teachers are trained to practice “differentiated instruction,” using both inductive and experiential approaches tailored to the individual student. But the impulse to define our dreams and our ambitions through how we teach these most basic acts of reasoning endures.

*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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edited Sep 13, 2023Very interesting (both parts 1 and 2). It was actually amazing to me to read that hundreds of years ago, people really did think of examples to try to show when or why you might want to do this calculation. The second is the use of real models (beans, etc). One generally thinks that math was taught solely by "show, then do" methods until fairly recently.

I read an interesting article once about how children can understand fractions starting at home, if they divide up real things (a. whole cake or some cookies for instance, as well as groups of items, which is quite different). I learned simple arithmetic learning to "play store" as a child (real groceries in our basement "store.") I don't remember what we used for money, but I do remember learning to give change by counting back, as well as learning to account for a person giving you coins to make the change simpler. I also played card games which involved being aware of total scores, such as Cribbage. I still enjoy number games.