What you noticed
- Eyes have a limit. Up to about four or five, you see how many at once. This is called subitizing. Past that, you see it only when the things make a shape you know, or when one side is much bigger than the other.
- Moving things does not change how many. Shake them up and match again: the leftovers stay the same.
What you invented
- Matching. To compare two amounts, pair them up one for one and look at what is left over. No numbers needed.
- A mark for each thing. A notch stands in for a pebble, so you can carry the amount instead of the things. Move each thing out of the way as you mark it, and you never mark one twice.
- Groups you can see. A long row of marks is as hard to read as a heap of pebbles. Cut them in small groups, like tally fives or the five on a die, and many becomes a few groups and a few more.
You did all of this without counting. Every question was a comparison: is there enough, is there more, is it the same?
What comes next
Grouping in fives works for twenty or thirty. At a few hundred, the fives pile up and become hard to read again. The next module invents a mark for a whole group, then a mark for a group of groups, and follows that idea through numerals people really used, all the way to place value and zero.
The mathematics behind it
Matching one for one is what mathematicians call a one-to-one correspondence, or bijection. Two collections have the same size exactly when one exists between them. That is the definition of equal number that set theory starts from, and it needs no counting. "Shaking does not change how many" says that size does not depend on arrangement, a fact the set-theory view of the natural numbers will make use of later.
The rough sense of larger amounts that the eyes give you gets less accurate as the two amounts get closer, which is why big, close rounds felt like a coin toss.
Inspiration
The module follows Paul Lockhart's Arithmetic (Harvard University Press, 2017): counting is motivated by comparison, and numerals are invented, not handed over.
Further reading
- The Number Sense, by Stanislas Dehaene — A neuroscientist on the number sense we share with animals and infants: why small amounts are seen at once, and why close amounts blur
- Numbers and the Making of Us, by Caleb Everett — An anthropologist on cultures without exact numbers, and the case that numbers are a human invention, built on fingers and tallies
- Number Words and Number Symbols, by Karl Menninger — A cultural history of counting, with tally sticks, knotted cords, finger counting and counting boards, on the way to place value
- Frege's Theorem and Foundations for Arithmetic — Stanford Encyclopedia of Philosophy: how matching one for one (Hume's principle) can serve as the foundation of the natural numbers