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Language and Conceptual Understanding

5/23/2012

13 Comments

 
I love my nephews and nieces.  They are great fun, and there is nothing more fun then getting them to do math.  I don’t know why I do this, but I love having them do little things here and there.  Count this, count that, what is this plus that?  At one point on my spring break my nieces were doing addition practice all on their own and showing me their work!  Seriously, what’s wrong with me?  I was really intrigued by my niece’s ability to add large numbers together and it has caused me to reflect on the nature of language and our ability to reason.

Ones, Tens, Hundreds, Problems?

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.

She has not learned place value, and therefore her ability to reason this was not developed.  I tried to walk her through it.  I got her to draw dots, but miscounting the dots led to incorrect answers.  I tried having her write out the answers to see a pattern.

1 + 1 = 2
10 + 10 = ?
100 + 100 = 200
1000 + 1000 = 2000

She found the process confusing.  I did not know what to do.  I’ve never taught math to someone this young.  This was just some fun we were having, and then I felt that her confusion might result in a math phobia that would predominate the rest of her life (I may be a bit overdramatic, but I was worried). We decided to take a break from math for the moment, and worry about that problem later, but I find myself still thinking (months past) about that moment.

Language and Thought

I remember in my intro to psychology class where we talked about what came first the ability to think or the ability to talk.  Can we share thought or elaborate our thought without language?  How does language affect our ability to think? Can we even think without language? Are we limited in thought by the language that we are given?

My niece gave a curious example for me to ponder upon.  She was able to reason, with ease, how to add hundreds and thousands together, even though she has little understanding of place value.  My hypothesis is that she can do this because the language of hundreds and thousands still uses our single digit counting; she is simply counting a unit that happens to be named hundred.  One plus one equals two, one apple plus one apple equals two apples, one hundred plus one hundred equals two hundred.  For my niece hundred was just another quantity that we can add together.  Ten, however, did not have the single digit number in front of it. We do not call ten one-ten, or twenty two-ten, and she was thus unable to reason it in any way.  Why is that? Why do the tens have this magical new way of naming numbers? Every other number that does not require a ten type number, uses single digits (i.e. 1,000,000 - one million; 100,000 - one-hundred thousand; 1,000,000,000 - one billion). Is this a fault in our language of mathematics, or just another hurdle that we must overcome?  Is this language deficiency making our sense of place value more difficult to grasp?

I don’t know, but it’s interesting to ponder... 
13 Comments
Kristina link
5/23/2012 10:40:11 am

If you look at Math-U-See, they have a unique way of thinking about place value that makes it very clear. He suggests naming the numbers one-ty instead of the the teens. This, to me, is a bit confusing, since we do not talk this way. However, using tens blocks is a great way to teach place value. There's only so many unit blocks that will fit in the units house before they have to move next door into the tens house. This is a Math-U-See concept that I really like.

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Timon Piccini link
5/23/2012 03:22:37 pm

Ya I understand the ten blocks perfectly fine, but my question here is not "Are we able to teach place value?" so much as "Does our language inhibit intuitive knowledge of place value?" Just because one-ty sound funny does not negate the fact that it might be more helpful for a natural understanding of place value.

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Nathan Kraft link
5/23/2012 11:09:13 am

I think I read about this in Malcolm Gladwell's Outliers - comparing counting systems and how other languages use a format more like "two-tens" rather than "twenty". This helps these children learn to count higher at a younger age. I think it's about time we change the names of numbers to something that makes more sense. (I think the teens are really confusing.)
You should have asked your niece what one hundred and one plus one hundred and one is? Two hundred and one?

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Timon Piccini link
5/23/2012 03:26:33 pm

I'd love to research into other languages. I know French is completely wacky when it comes to numbers; 60-90 always blow my mind. I wonder how much resistance to change there would be if teachers decided we needed new names for numbers. Imagine saying 233, two hundred three ten three.

Ya I wasn't fully prepared to teach a lesson in place value to my dear niece, as I said she was doing addition all on her own, and I decided to give her a challenge.

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Timon Piccini link
5/24/2012 04:24:54 am

This is what French students have to deal with (I am translating into english to show the bizarreness).

1 > one
10 > ten
11 > eleven
12 > twelve
20 > twenty
21 > twenty and 1
31 > thirty and 1
61 > sixty and 1
71 > sixty eleven
81 > four twenty and 1
91 > four twenty eleven

Is this natural?

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LSquared
5/25/2012 02:06:51 am

There's an interesting article on this (place value language, etc) at http://rightstartmath.com/resources/research-summary (especially the first page). (And I wouldn't worry about your niece--I think she'll do just fine!)

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Kevin link
5/25/2012 08:00:11 am

I think I read that Asian languages in particular have this advantage which is one of the reasons cited for the compartive success in Maths by these students.

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Timon Piccini link
5/25/2012 08:20:18 am

Well it makes sense to me. Understanding units helps place value; understanding place value helps combining like terms; this helps algebra, etc. etc. etc.

Math builds so much on itself, and naturally a house with a terrible foundation is destined to collapse at some point.

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Fawn Nguyen link
5/27/2012 02:49:23 am

@Nathan Yes, Gladwell mentioned it in Outliers.
@Kevin In Vietnamese, we do say two-tens for 20, three-tens for 30, etc. I can personally vouch for the "comparative success" by Asians since I was whacked with a stick on my palm if I didn't give the correct answer to "5 times 7" by the teacher. Learning math was literally painful! :) and :(

God Bless the French, they use base 20, oui?

Also interesting is how we say fractions differently. There is nothing in saying "three fifths" to help a child understand that it's the same as 3 divided by 5. Language helps and hinders.

Thanks, Timon, you got us thinking here.

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Timon Piccini link
5/27/2012 04:03:35 am

Thanks Fawn, I hope your hand has healed. I think our minds are fascinating, and I am glad that the folks here are engaging in this. I look forward to reading all the resources.

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TImon Piccini link
5/27/2012 04:36:02 pm

I think the French use base WTF.

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Christopher Danielson link
5/27/2012 02:52:16 am

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. <a href="http://christopherdanielson.wordpress.com/2012/03/31/just-sayin-literacy-and-numeracy/">Just like we need to read aloud to our children</a>, we also need to <a href="http://christopherdanielson.wordpress.com/category/talking-math-with-your-kids/">talk math with them</a>. 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, <i>What is 1 ten plus 1 ten?</i> 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 <i>How much is that? How much <strong>is</strong> two tens?</i>

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 href="http://dl.dropbox.com/u/47041687/04-Communications.pdf">a paper on relationships between quantity, numeration and number language</a> (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.

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Timon Piccini link
5/27/2012 04:06:31 am

Thanks Christopher. I like the idea of reading math aloud. I'll be getting a bit more into this when I blog about my fractions unit, because having students say out loud what they understand division to be, really helped them figure out how to use it in regards to fractions.

I wish those links turned out well I want to read your posts, I'll see if there is anyway to fix that but for now copy-paste should do the trick.

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