We’re very used to representing quantities by numbers.   The cash register will tell us how much we have to pay, and the pump at the gas station will tell us how much gasoline we poured into our tank, and the scale will tell us how much we weigh – all by showing us a number.  We are used to interpreting these numbers, and we have acquired a whole arsenal of practices and distinctions around quantities and numbers, many of which we have come to take for granted.  When we watch children struggle with the very things we have come to take for granted, we have an opportunity to re-learn and re-appreciate the subtlety and the sophistication involved.  Watching children can make the familiar seem strange and fresh.  I intend to highlight some of these subtleties, starting in this post.

Not all kinds of quantities are the same, and not all kinds of numbers are the same.  We’ve all heard or read statements like “The average American adult male has 2.2 kids and .9 automobiles, and has been married 1.1 times.” (These specific numbers are completely made up.)  And we know, as adults, that there is no one person we could point to who is “the average American adult male”, and if we found that person, he wouldn’t have 2.2 kids.  We can be quite sure about that, because there is no such thing as a living and breathing .2 kid.
And yet we can make meaning of the statement “The average American adult male has 2.2 kids” by treating it as a short-hand for “American adult males, on average, have 2.2 kids” or “the number of kids that American adult males have, averages 2.2”.  We could visualize this in many ways, for instance: “if you took ten typical American adult males, they would have about 22 kids total.”  The phrase “2.2 kids” can be perfectly meaningful even if there is no such thing as a .2 kid.

We also know that certain kinds of numbers aren’t suitable to draw certain kinds of conclusions from.  For example, if you discovered that your telephone number is exactly twice what mine is, it is hard to see what this means for either of us. Telephone numbers aren’t set up for conclusions to be drawn about ratios.  For all intents and purposes, those ratios are meaningless.

This is the first post in an intended series about different kinds of quantity and the different kinds of conclusions you can draw from them.