When: at times you want to help teach counting (e.g., all the time)

Where: anywhere there's stuff to count

What we use: whatever is available, fingers if that's all we've got

Many of you have heard me say this before: "there are three kinds of mathematicians, those who can count and those who can't."

I never get tired of that (why?), so I should take the opportunity to publicly apologize to everyone who has (or will hear) me say it many times. Part of what lies behind the joke is how fundamental the basic counting skill seems to be for everything else in our standard math education curriculum. I've heard a high school-focused educator claim that many kids struggling at her level are really dealing with the lack of a firm grasp of one-to-one correspondence (a sub-skill for counting). If you really want to, you can see this skill highlighted as a foundation block in the US Common Core Standards: base skill level in counting and cardinality,

**almost all the time**until the kids are counting fluently. There are only some small points I can add to Amy's great post:

- start with 0. I make two balled fists and wiggle them when I say zero, unless I'm counting hands in which case I just say 0. The point is to make sure they realize 0 is also a number.
- be silly: this is a game for them kids, so feel free to make silly sounds and gestures.
- try to find things they can grab and move around as they count. My intuition is that the more of their body involved and the bigger the motion, the more they will remember.
- (optional) try counting in other languages. If you don't care which language, Chinese and Thai are good choices because of the logical naming system they employ (I think other Asian languages are similar).

**Combinatorics**

When I tell the joke and say "count," I'm internally thinking about combinatorics. I have long felt a bit weak in this area, with anxiety that my counts were either leaving out cases or double counting somewhere. That's the real reason I like this joke so much, because the self-deprecation has a meaningful kernel of truth.

I love #1 especially, since as adults we don't usually realize that zero is not a natural concept for our kids.

ReplyDeleteAnd I love #4. I have a draft post (early, early stages!) about language and number. It sounds like you have some insight into this, and if you ever care to share your thoughts, I'd love to hear them!

I'm heavily influenced by the book Mathsemantics by Edward MacNeal. It is a really good book and I'm disappointed that this field of study didn't take off as he had hoped.

DeleteMy other influence was a 5 minute discussion at the PROMYS program one summer about the definition of a convex region (in the real plane). Though I didn't fully get the point at the time, I had a sudden glimpse of the power of "good definitions" and have come to see this as a form of finding the right language (or languages) to describe what we are studying.

For the specific point about counting in multiple languages, here are some of my reasons:

(1) Using multiple languages draws a sharp distinction between the object and the name. This seems a critical mathematical habit as changing name can be such a powerful technique.

(2) Trying a language in which the parent isn't fluent will help make them more aware of the child's perspective.

(3) other languages, like the ones I listed, have a clear and logical pattern; since pattern exploration is such a core part of mathematical thinking, I like having basic building blocks follow a nice pattern, too.

(4) you can build additional games with this extra dimension. Here are some examples:

- take turns counting and each person alternates a language

- someone starts counting and the other person jumps in as soon as they figure out what language is being used and what's next in the count

- do all of these backward (can be especially hard in non-mother tongues).

- skip counting (forward and backward)