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Mental Arithmetic

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This post was written by 2021 Junior Fellows Sean DiLeonardi and Amal Charara.

In 1930, concerned about the effect of industrialization on the arts, literary critic Donald Davidson looked to the previous century for an explanation. In his essay, “A Mirror for Artists,” appearing on page 37 of I’ll Take My Stand: The South and the Agrarian Tradition, he wrote, “The humanities have fought a losing battle since the issue between vocational and liberal education was raised in the nineteenth century.”

teacher teaching children in a classroom practicing math exercises with blocks with mathematics equations on the blackboard.
Classroom scene in Washington, D.C. elementary school – children working with blocks and at blackboard in mathematics class, Frances Benjamin Johnston, 1899.

A major component of the “battle” that Davidson pointed an accusatory finger at was the rise of “practical” arithmetic—an idea, increasingly popular in the latter half of the nineteenth century, that arithmetic prepared students for a variety of professional career paths. The learning of arithmetic had long been vital to earlier forms of specialized vocational training, as in manuals for those entering mercantile apprenticeships. However, practical arithmetic standardized professionalization by treating all students as potential workers in fields that required numeracy skills–from banking and farming to masonry and carpentry. The business application of arithmetic education became so ingrained–apparently to Davidson’s dismay–that by 1895, the West Virginia State Series curriculum defined arithmetic as: “the solving of problems from its various applications to business life.”

Cover image from
Walsh’s Graded Mental Arithmetic, (1909).

But is it true that vocational training diminished the role of the humanities in education? To grapple with this question, it is necessary to think about a concurrent development in nineteenth-century education circles that focused on the improvement of skills and concepts rooted in humanistic thinking. “Mental arithmetic” (as its proponents often described it) was meant to sharpen internal processing, memory, literacy, and the power of imagination in students–what would turn out to be desirable qualities in a child’s future occupation. Textbooks encouraged teachers to hide away the chalk and slates, requiring pupils to instead practice arithmetic in their heads. In this way, mental arithmetic and practical math became linked: performing calculations in one’s head, proponents reasoned, enables faster and more efficient calculations. As one nineteenth-century textbook explained, students must practice mental arithmetic because, “Business men demand of their employees speed and accuracy.”

To augment the goals of mental arithmetic, numerous techniques emerged that facilitated mental development, thus making the nineteenth-century classroom a space for imaginative thinking. These include one of the oldest numeracy techniques–the word problem–which had been used since antiquity to apply number skills to familiar contexts, often using narrative elements that resembled stories. But by the end of the nineteenth century, word problems more frequently required students to imagine themselves in different industries, from banking to agriculture, as in the illustration below of a man estimating the amount of wood in a pile of lumber; or, to take an example from The North American Arithmetic (1832): “A farmer went to the city with 8 barrels of cider, which he sold at 4 dollars a barrel. He then purchased 3 hogsheads of salt, at 3 dollars per hogshead, and paid an old debt of 12 dollars, how many dollars had he to carry home?”

Illustration shows a man sitting on a log facing a large stack of wood with a train and factory and smokestack in the background.
Illustration from Orton’s Lightning calculator, and accountant’s assistant, 1869.
image consists of two pages forty two and forty three
Business word problems from The North American Arithmetic , 1832.

This connection between arithmetic, imagination, and business preparation extended to many other techniques introduced into nineteenth-century classrooms, including flash cards, memorization charts, and perhaps most surprisingly, even poetry. As L.J. Woodward explained in the introduction to Number Stories (1888), an arithmetic primer for young learners, poems allow for “the use of familiar words, frequent repetitions, and short sentences, in order that the child’s attention may be wholly given to the combinations.” Woodward suggests, somewhat remarkably, that by reading poems to children, they might be better conditioned to memorize their multiplication tables.

There's no dew left on the daisies and clover, there's no rain left in heaven; I've said my seven times over and over - seven times one are seven. I am old, so old I can write a letter, my birthday lessons are done; the lambs play always - they know no better; they are only one time one by Jean Ingelow no rain left in heaven;
Poem from Number Stories, 1888.

Throughout these examples, the development of mental activity aided internal calculation and, by extension, business training. But in doing so, mental arithmetic also affirmed the value of such fundamental humanistic concepts as imagination, literacy, story, and poetry–those very phenomena that vocational training had ostensibly replaced. Indeed, to emphasize this very point, one might examine a final example, a photograph of a Washington, D.C. classroom in 1943, in which students are performing arithmetic calculations to practice food rationing. Donald Davidson might have complained about the utilitarian effects of preparing students for the business of shopkeeping or conditioning them for a wartime economy. But at the same time, how could such objectives be achieved without the arts? Are the students not playacting? Pretending? Imagining? Creating a fiction in which arithmetic is vital to the story?

Two boys behind a counter with canned and box goods on the shelves behind them with three customers buying goods
Buying foods with war ration book two becomes real to pupils in this model store. Referring to the posted list, pupils learn practical arithmetic when they figure both prices and points, United States, Office of War Information. Alfred T. Palmer, photographer, 1943.

It continues to be the case today that there are those who undervalue the humanities, not only within the practice of education but throughout contemporary society. But if one insists on arguing that this status is because the humanities have little to offer the driving interests of vocational training and business preparation, they might want to take a closer look at the role of imagination in these studies before trying to make the numbers add up.


  1. Neat post…Thank You

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