What is ten?

Consider the seemingly simple question What is ten?

Quantity. This refers to how many things there are. If ten is a quantity, then it refers to this many things: ***** *****

Numeration. This refers to how we write how many things there are. If ten is a set of symbols, then it refers to this: 10.

Number language. This refers to how we say how many things there are. If ten is a word, then it refers to this word: ten.

To illustrate the difference, ask a French person to read this number: 10. Then ask a fifth grader what this Roman numeral stands for: X. Finally ask a computer programmer what number this refers to in binary: 10.

In order, the French person’s dix illustrates that we can use different number language for the same numeration and quantity. The fifth grader’s ten illustrates that we can use different numeration for the same number language and quantity. And the programmer’s two illustrates that we can use the same symbols to represent different quantities.

If I weren’t so lazy, I’d link to a Karen Fuson reference for the research details. Maybe I’ll get to that sometime. But she’s the go-to person on this.

 

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Place value and language

Timon Piccini writes about a conversation he recently had with his niece.

My niece is in grade 1, and she is adept at adding single digits.  With little hesitation she can do her basic addition.  She even showed me that she could do things like add 100 + 100.  I thought this was really neat so I asked her some questions.

Me: What’s 1+1?
Niece: That’s easy it’s two.

Me: What’s 100+100?
Niece: It’s 200 duh!

Me: What’s 1000 + 1000?
Niece: 2,000 these are easy!

Me: What’s 10 + 10?
Niece: … I don’t know.

This leads Timon (TIM-in) to wonder about how language is related to early numeracy and later mathematical development. Interesting stuff.

I learned after commenting that his blogging platform doesn’t allow html code. So my comment is hard to read. I have reproduced it below. But go read his full post. And read the comments while you’re there.

My comment

This is where my mind has spent the last few years. So lovely to see that I’m not the only one intrigued by this sort of stuff.

I love that conversation with your niece. Just like we need to read aloud to our children, we also need to talk math with them. I don’t think we do any damage when we move to symbols (as you did in this conversation), but I don’t think we have any evidence that it’s really helpful, either. Like teaching a pig to sing, I suppose (wastes your time and annoys the pig).

What does seem to be helpful is that you’re interested in the child’s ideas. This can take many, many forms. One interesting activity for a curious teacher such as yourself is to take a moment to formulate a hypothesis and then a question to test it. Here you noticed that she could do 1+1 and 100+100 but not 10+10. Your hypothesis (which I also believe to be correct) is that this is language based. So ask her, What is 1 ten plus 1 ten? I’m curious whether she would say “two tens” or “two ten”. I can’t tell from your transcript whether she said “two hundreds” or “two hundred”.

Anyway, I think you and I would both be surprised if she had no answer for 1 ten plus 1 ten. When she offers it, follow up with How much is that? How much is two tens?

Many thanks to all the folks who have contributed references. Those will be helpful as I develop my own understanding of this territory. I’ll add my own (a bit self-serving, admittedly). I wrote a paper on relationships between quantity, numeration and number language (my bit starts a few pages into the file). That paper grew out of work I do with future elementary teachers and research from Karen Fuson. It contains several worthy references.