Another one from the archives.

About a year ago, Tabitha was doing a dot-to-dot drawing.

This is not the one she was doing. It is an example. Not a very good one, either.

Things were going well. She got to 11.

**Tabitha** (five years old at the time): I don’t know what twelve looks like.

**Me:** It’s a 1, then a 2.

With this tip, she cruised through the teens and got to 20.

**T**: What does twenty-one look like?

**Me:** A 2 and then a 1.

**T**: And twenty-two?

**Me:** A 2 and then a 2.

**T**: So twenty-three is probably a 2 and then a 3.

This got her going on to 29.

**T**: I don’t know what 30 looks like.

There are a lot of interesting things going on in this exchange. Among them:

Sometimes in mathematics, we need to live with new notation before picking its meaning apart too carefully. See also *fractions*, *functions* and* derivatives*.

Numeration and number language do not develop hand-in-hand. Tabitha knows the number language; she can count past thirty. She has not learned how to read or write numbers that high.

Patterns are powerful tools in mathematics. Tabitha’s experience in the teens gave her powerful intuitions for the twenties.

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*Related*

“Sometimes in mathematics, we need to live with new notation before picking its meaning apart too carefully.” cf: John von Neumann: “In mathematics you don’t understand things. You just get used to them.” One of my favorite thoughts about mathematics, and persistence, and playfulness.

This reminds me of when people argue vehemently that kids shouldn’t be introduced to decimal numbers until they’ve mastered fractions. I always think, “oh really? They’ve never seen a price tag?”