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.