The idea that all children should learn arithmetic seems to have blossomed in the western countries around 1800. Prior to this period arithmetic training had been reserved for a small group of boys bound for mechanical or business fields. The training didn't usually begin until at least the age of twelve, and featured rote memorization of the written numerals and the rules for special cases to solve arithmetic operations.

Michalowicz and Howard (2003) have argued that “students of the 18th century rarely had a textbook,” that those who studied arithmetic “wrote in a ‘cipher’ book,” and that “the textbook was mainly for the teacher or for individuals who were self-taught.”pg 79).

Another writer quoted an educator in Boston around 1810 as stating that “printed arithmetics were not used in the Boston schools” until after he left there (p. 45). Rather, teachers set “sums” for

their pupils out of ciphering books that they had prepared at school, or had copied from textbooks or from the ciphering books of other teachers.(Monroe, 1912, pp. 5-16).

When mass arithmetic education started to become popular in the British Ilse and US, these rote memorization approaches continued and were used on even younger students. One of the first educators to influence Britain, and the US away from this structured approach toward a "mental/experiential" approach to understanding arithmetic was the famous Swiss educator, J. H. Pestalozzi. One of the tools he used as a primary instructional item was the horizontal abacus, or counting frame. But the path that brought it from its vertical Roman roots to the horizontal classroom model had a long and winding route that balanced on a narrow turn of events in the life of a French mathematician/soldier in the Napoleonic campaign in Russia. And his is the story I wish to tell here.

The abacus has been around, in one form or another, since at least the ancient Greeks and Romans. The image at the top appears on a second-century CE funerary relief that depicts the deceased young man reclining beneath his dead father's portrait, with his grieving mother seated on the right. The slave standing on the left is operating an abacus, which symbolizes the family's success in business. (Barbara F. McManus, The Roman Abacus

*on the web*)

It spread throughout the European and Asian continents and by 1300 was common throughout both continents. Then the sweeping adaptation of Arabic numerals and improved computational methods led to its complete disappearance in western Europe, so that by the late 18th century it was unremembered. But in the later part of the 18th century (my best guess) a horizontal form of the Roman abacus became common for Russian classrooms (who apparently discovered the idea of "understanding arithmetic" slightly earlier). These abaci, called the schoty (счёты) were not only horizontal, the wire frames bowed out forward of the frame allowing the teacher to hold them up in front of a class without disturbing the order. In addition they were made with the fifth and sixth beads colored differently to make it easier to recognize numbers.

The single row of four beads was for working with a fractional quarter-ruble coin that existed at that time. Why it was place four rows from the bottom (commonly) has never been explained to me.

So the stage is set for an 1809 graduate of the E'cole Polytechnique in Paris, and student of Gaspard Monge to return to his home in Metz in the Alsace-Lorraine region. Then in 1812, he was called to duty with Napoleon's forces to invade Russia. If you don't remember, that didn't go well for Napoleon, and not too well for our young mathematician either. He was captured in November of that year, and sent off on a forced march of "hundreds of miles". Keep in mind that Russia was cold and snowy that November, and weary soldiers on forced marches were prone to, and in fact did die.... but not our hearty hero. For over a year, he was kept in a Russian prison, and while there decided to reconstruct and improve and some old ideas of his teacher Monge, and the great Lazare Carnot. These writings would become a classic work in projective geometry, Traite' des propiertes projectives des figures (1822). When he was freed and repatriated to France, he returned to his home in Metz, and brought along a Russian abacus. He gave it to a teacher in Metz, and suggested that it might be useful in teaching small children. The item had been so forgotten that it was treated as a novelty as it slowly began to be reintroduced in France, then more quickly into Britain and the US under supporters of Pestalozzi.

And the weary warrior/mathematician whose survival made it all possible? If you didn't get the clue with the title to his classic work in projective geometry, it was Jean Victor Poncelet. As a mathematician, his most notable work was in projective geometry, in particular, his work on Feuerbach's theorem. He also made discoveries about projective harmonic conjugates; among these were the poles and polar lines associated with conic sections. These discoveries led to the principle of duality, and also aided in the development of complex numbers and projective geometry. And if you ever happen to be visiting the Eiffel Tower, look up, there are 72 names of scientists around the 1st stage of the tower, and yes, our hero/warrior/mathematician is one of them.

If you teach, tell this story to your students. His greatest influence may not be the mathematical writing he was recognized for, but a small item that he passed along to a teacher in his home town thinking, "It might be useful for teaching children."

John Golden points out in a comment that a small type of counting frame is becomming very popular in elementary education again. It is called the rekenrek (which seems to have been semi-americanized from the Dutch rekentuig for abacus. The root is the same Germanic root that gives us the English term "reckon" for counting or doing arithmetic. Interestingly, its proto-Germanic root was for "motion in a straight line" which seems perfect for a counting frame.

The version I have seen promoted has only two horizontal wires, but there seem to be some available with multiple rows as well.

## 12 comments:

I've had great experiences with a modern form of the abacus, the rekenrek (also AL abacus), which is put to great use in the Context for Learning series. Cf. http://www.contextsforlearning.com/gradesK_3/sequenceInstruction.asp or this sample from their materials that talks a bit about their use: http://schoolwires.henry.k12.ga.us/cms/lib08/GA01000549/Centricity/Domain/4011/RGEarlyAddSubMinContOver.pdf

I was thinking to admit my child for an Abacus course and wasn’t really able to make a decision since long. But after reading your Abacus blogs, I plan for no further delays. Your posts are amazing and make me realize Abacus is worth being taught to kids.

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Hi,

really a nice post about Abacus history. keep posting...

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OMG!!! Did you really blog about the abacus without mentioning the Chinese? White-washing history much? Sad that Wikipedia is more historically accurate than an “expert.”

The Romans and Greeks didn’t invent the abacus, they used boards and they were not the first, even. There’s a difference between beads rolling along a board versus wired beads (which is what an abacus truly is and it’s powerful because it’s much faster than marks/beads on boards). The Russian’s version came later than the Chinese and think where they are geographically and take a gander at how they were influenced.

It’s okay if you disagree, but at least your readers should know your biased Supremacy.

I’m sure you’ll delete my comment, but that only proves you’re a racist.

Vamp, yes,I disagree. I just reread the Wikipedia article and it says"The abacus, also called a counting frame, is a calculating tool that was in use in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the written Hindu–Persian numeral system. The exact origin of the abacus is still unknown. ".

I might also point out that my note is not a history of the abacus, but of its use in American Education. Nor do I, or have I, ever claimed to be an "expert". I'm a retired math teacher with an interest in the history of math and math education.

Thank you for sharing your comments, and opinion. Sorry you felt you had to be abusive to try to teach me something. Most of what I share here came from comments and information shared by far more learned folks than I, and I've tried to acknowledge each of them. You may know lots, but nothing you wrote seemed to enhance my edu ation about either math,or math education

Thank you for taking the time to read and respond to my comment, Pat. Knowing where you’re coming from, I do regret my tone and should not have assumed that you were a historian of mathematics.

Under my assumption, I didn’t understand why your website did not cover mathematical contributions of Eastern and other societies, given how monumental and influential they were to Western societies. A simple Google on “Chinese mathematical contributions” opens a labyrinth of interesting facts and domino effects to Western theories.

Of course, America also opened her door and lifted the Chinese during their stale growth, elevating Eastern mathematical advancements to this day.

I understand now that you were not intentionally snubbing other culture’s mathematical contributions. Thank you for your clarification.

Good talk.

Thank you for sharing this informative post. As a parent, I found it very helpful and up-to-date. It's important to stay informed for the sake of our children's well-being.

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