Since the fall of 2014 I’ve been slowly (ever so slowly!) working my way through the Introduction to Arithmetic written by Nicomachus of Gerasa, who was born in the first century after Christ, and along the way I’ve also been reading through the supplemental materials that are available in this PDF version of Nicomachus, where I found the following passage:
Arithmetic is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after Pythagoras, however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, λογιστικη [logistic, or calculation]; of this division of the science no Greek treatise has been transmitted to us. In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children.
“Studies in Greek Mathematics” by Frank Egleston Robbins and Louis Charles Karpinski
This was my first clue that what we call arithmetic today isn’t necessarily the same thing as what Plato called arithmetic, and that brings me to chapter two of the Klein book, which summarizes the very few references in classical literature to teaching arithmetic—excuse me, calculation—to children. One of the books mentioned was Plato’s Laws, so I looked up the passage in my copy of the book:
All freemen I conceive, should learn as much of these branches of knowledge as every child in Egypt is taught when he learns the alphabet. In that country arithmetical games have been invented for the use of mere children, which they learn as a pleasure and amusement. They have to distribute apples and garlands, using the same number sometimes for a larger and sometimes for a lesser number of persons; and they arrange pugilists and wrestlers as they pair together by lot or remain over, and show how their turns come in natural order. Another mode of amusing them is to distribute vessels, sometimes of gold, brass, silver, and the like, intermixed with one another, sometimes of one metal only; as I was saying they adapt to their amusement the numbers in common use, and in this way make more intelligible to their pupils the arrangements and movements of armies and expeditions, and in the management of a household they make people more useful to themselves, and more wide awake; and again in measurements of things which have length, and breadth, and depth, they free us from that natural ignorance of all these things which is so ludicrous and disgraceful.
Plato, Laws, Book VII  (emphasis added)
The second chapter of Klein also quotes a passage from a commentary on Plato’s Gorgias by Olympiodorus the Younger (c. 495-570) where he says in passing that “even little children know how to multiply.” And in chapter three Klein reiterates that children must be taught calculation through play, “thus making it possible for children to acquire correctness in counting and in combining numbers painlessly.”
This is such an exciting line of inquiry! I wish all my kids were babies again so I could start over.