Arithmetic and Numeracy

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Numerical proficiency in the American population varied in the late colonial decades, depending on occupational needs, but overall, training in arithmetic was arcane, difficult, and limited. Simple enumeration and addition were probably widely imparted to colonial children by parents, but the formal study of written arithmetic was confined to boys ten years old and older who studied it at a district school or with a master. British textbooks and their few American derivatives presented hundreds of abstract rules of calculation, each illustrated by an example. Arithmetic was a heavy labor involving memorization of the rules and close study of the examples. With explanatory text at a bare minimum, arithmetic was rightly judged to be an arduous subject, too difficult for young children.

Arithmetic found its main application in commerce and trade for the figuring of prices and the measurement of goods, the compounding of interest, and the sharing of risks. Denominate numbers—pounds and shillings, pints and gallons—added complexity, as did the differently valued monetary systems in place in the various colonies. Geometry and trigonometry had yet narrower application, namely in surveying, navigation, and gunnery. As a result, arithmetic was seen as a practical tool and thus not part of the classical curriculum of college-bound boys. Harvard College did not require basic arithmetic as an entrance requirement until 1802.

Two dramatic changes marked arithmetic instruction in the early Republic. The first arose out of the adoption of decimal-based money, and the second out of pedagogical innovations. Dollars, dimes, and mills of decimal coinage began to circulate in the mid-1790s, sparking the publication of scores of new textbooks aimed at the "Columbian Arithmetician" or the "Federal Calculator." Authors explicitly linked decimals with republicanism, positioning simplified arithmetic as a challenge to the indecipherable bookkeeping and taxation policies of tyrants. As textbooks proliferated, authors sought to distinguish their works through changes in pedagogy. Simplified arithmetic, it was argued, could now be taught to younger children, and more elaborate explanations of abstract rules began to appear. The most radical change in pedagogy arrived in 1821 with Warren Colburn's book, First Lessons, or Intellectual Arithmetic on the Plan of Pestalozzi. In this and in several more books (published in 1822, 1825, and 1826), Colburn banished rote-learned rules and targeted children of ages four to six to learn "mental arithmetic" in their heads, before they could read or write. He championed an "inductive" method, in which students would puzzle out carefully chosen problems, inventing their own techniques and eventually discovering the rules of arithmetic for themselves. Colburn's ambitious ideas created an instant sensation and gained a popular following.

Predictably, by 1830 a backlash developed, with some educators calling for a return to traditional rule and example learning. Well into the mid-century, the debate continued, greatly energizing arithmetic instruction. The combination of decimal money, new pedagogy, a rise in textbook publication, and the steady spread of common schools all combined to transform numeracy into a basic skill that spread rapidly, along with literacy, in American culture.

See alsoEducation: Elementary, Grammar, and Secondary Schools .


Cohen, Patricia Cline. A Calculating People: The Spread of Numeracy in Early America. Chicago: University of Chicago Press, 1982. Reprint, New York: Routledge, 1999.

——. "Numeracy in Nineteenth-Century America." In A History of School Mathematics. Edited by George M. A. Stanic and Jeremy Kilpatrick. Reston, Va.: National Council of Teachers of Mathematics, 2003.

Patricia Cline Cohen