I occasionally play at teaching Helen the cardinal numbers one to five, not in any organised fashion but every so often when she seems alert and curious and there are no more obviously interesting things for her to play with — during long car trips, baths, when I'm tired of building block towers for her to knock down, and so forth. I don't expect a 1.5 year old to do arithmetic, but it seems to me that understanding "two" and "three" is not necessarily any harder than being able to distinguish ducks from other kinds of birds (which she is now not too bad at).

Some people suggest that children learn counting, and thus ordinal numbers, first, but it seems to me that, at least for small numbers (up to five, or perhaps seven?) cardinals should be much easier to learn.

For those to whom the terms are unfamiliar, cardinal numbers represent the number of items in a collection of objects, while ordinal numbers are used to put objects into a sequence or to count them. If you present three blocks to a child and say "three", you're using the cardinal number three. If you count blocks "one", "two", "three" as you take them out of a box, you're using ordinal numbers.

The reason cardinals seem more straightforward to me is that they are simple, independent associations: it's not necessary to understand "two" before understanding "three", and learning the one won't interfere with the other. If you count one, two, three as you (say) put blocks into a box, in contrast, all the number words are associated with the identical same action — putting a block into the box — and the "three" will only have the right context if the "two" was comprehended. Obviously this only holds for small cardinals, within the human ability to comprehend directly, possibly no more than five and I suspect at most seven. The only way to get to ten is by counting (or possibly by comprehension of two groups of five or something like that).

For more on this topic, maths teacher Christopher Danielson explores some of the ways the ordinal/cardinal distinction affects learning. In contrast to my thinking above, he suggests "it could be said that most children learn ordinal numbers first", but that may be because of parental obsession with "one, two, three, ..." (which like the obsession with "a, b, c, d, e, ..." isn't necessarily a good thing). There's other good stuff on his web site.

**Addendum:** courtesy of Claire Bowern, I learn that the term I'm looking for is subitizing, "the rapid, accurate, and confident judgments of number performed for small numbers of items". See "Do young children acquire number words through subitizing or counting?" (*Cognitive Development*): "As a whole, subitizing appears to be the developmental pathway for acquiring the meaning of the first few number words, since it allows the child to grasp the whole and the elements at the same time."

Most higher mammals can tell the difference in a number of objects innately. For dogs it is 12, the number of sheep in an average flock.

You can train young children to rote learn lots of associations, but it is just operative conditioning. They will learn when they are good and ready.

Humans can certainly

countto 12, but they can't directly perceive a collection of 12 objects as different from one of 11 objects.The whole point of my post is that it is possible for humans to directly comprehend small cardinal numbers - and that's just learning by association, it makes no sense to label it anything fancy. And that presumably lays the basis for understanding later what it means to reach 12 rather than 11 when counting a collection of objects.

Interesting thoughts, I would rather raise the question if it is cost effective, meaning that spending time to establish knowledge when a biological mechanism (biological maturity) is about to be developed, is not the best thing to do. At that age language acquisition is the cetral domain, so I would say that at this age the best thing to do is stories, items, colors etc etc. Your reference to ''ducks and other birds' is sth different as it refers to assimilation and accommodation. On the other hand if it work its ok, but be careful not to take curiosity for interest, the time will come.

It is wonderful to know parents who want to spent time with their children. keep it up!

Panagiotis

"Most higher mammals can tell the difference in a number of objects innately. For dogs it is 12, the number of sheep in an average flock"

This appears t be a very localised insight. In Australia, 12 would not even count as a flock (or, as we would say, "mob"). If you called the sheared in. To shear 12 sheep, they'd reply "Yes … and what would their names be?"

Compare this (from a kelpie kennel's website):

"Some Noonbarra Kelpies have worked sheep mobs that number around 1500 on their own, which is a lot of work for just one dog. In general we'd use two or even three dogs on a mob that big. The average size mob on properties in our area is 250 - 500 sheep."

I am not aware of any comparative studies on canine numeracy, but I doubt Australian sheep dogs have a number sense up to the size of the average flock.

As to the cardinal vs ordinal thing, my instinctual feeling is that the cardinals would've harder precisely because they are "simple, independent" -- and so, rather abstract. I remember a Chris Langham sketch about the bloke who invented numbers trying to get any enthusiasm about the idea. "How can you have a … nine?" (air-miming nine strokes).

If I remember correctly, Richard Feynman's father laid numbers out on his highchair tray-table so as to reveal patterns in them.