Friends of mine have three precocious children. A ten year-old girl, an eight year-old boy, and a seven year-old youngest son. I often engage them in mathematical discussion.
This page is devoted to the impressions of numeracy from young children. I think teachers (and learners) can take something from such innocent insights.
The following are just snippets of mathematical conversation. Longer (and larger) conversations can be found on the main blog page under tag "talking with children".
The girl has just walked into a lengthy discussion the middle boy and I were having about Zeno's Half-way Paradox.
Me: What if there was a prize across the room, but your steps could only be as long as half the distance to the prize. Would you ever get the prize?
Girl: How many steps can I take?
Me: As many as you want.
Girl: So I could jump to here. And then here... and then here... and then... oh ... no... that sucks.
Having a dinner conversation.
Girl 1: Guess what problem was in my book this week?
Girl 1: Three kids bought 90 jars of mayonnaise and wanted to divide it evenly.
Me: And, how many would each get?
Girl 1: 30, but when are kids ever going to buy that much mayyonaise?
Me: Good point. What would be a better question?
Girl 1: I dunno. Maybe doing something that kids actually do.
We were making pizza for supper, and began setting the table. The middle child was given the task of setting the table. While trying to decide what he needed, I tired to get his impression of fractions.
Me: How many spots do we need?
Me: So how many slices should we cut the pizza into?
Me: Won't the pieces be too big?
Him: Well, you could cut them in half after and get 14.
Me: What if everyone wanted 3 pieces, and we had 2 whole pizzas to divide?
Him: Then we would cut them both into...(counts on fingers)... 21 pieces.
The oldest approaches me, launches herself onto my lap and begins to explain her weekly exploits in math class.
Her: "I can subtract 24 digit numbers."
Me: "When would you ever have to do that?"
Her: "Ummm... When I put it in my math activity book. Obviously it's not something you do."
The middle son got a hold of my magic 8-Ball and began asking multiple questions--upping the bravado each time.
Him: "What's that never ending number?"
Me: "Is there such thing as a never ending number?"
Him: "Yes. (pause) Infinity!"
Him: "When I grow up, will I be an Infinity-aire?"
The oldest daughter (who has been sick from school for a week) and I begin talking about school.
Me: "So do you forget everything from school yet?"
Her: "No. I remember my times 2's, times 3's, times 4's, times 5's, times 6's. Really all the way to times 9's"
Me: "Sounds like you are good to go"
Her: "Do you know the nine's trick?"
Me: "What do you mean?"
**Goes through the trick with her hands held in front of her**
Me: "How does that work?"
Her: "Because you have 10 fingers"
Me: "Would it work if you used 9 fingers to multiply by 8?"
**Tries trick to multiply 8x4 and gets a result of 35**
Me: "Is 8x4 35?"
**Tests the fact by counting four times on 8 different fingers**
Me: "What did you just do there?"
Her: "Multiplied 4 by 8. I touched 8 fingers 4 times each. You are supposed to do 3's like this, but I add one touch."
Me: "Would that trick work for your 9's too?"
Her: "That's a lot of touches."
The daughter is telling me of a book she just finished.
Her: "I read a 503 page book in two weeks."
Me: "Wow, it has taken me 2 weeks to read 100 pages of my book"
Her: "That's cause you are reading math books."
Me: "Why are math books harder to read?"
Her: "I don't know...because of all the terms and thinking and stuff."
The middle son and I began a conversation.
Me: "Did you have your first hockey game yet?"
Me: "How'd you do?"
Him: "I didn't have it yet, it's opposite day."
Me: "Oh, sorry. I didn't know?"
Me: "Of course I did, I thought it was opposite day?"
Him: "Oh right"
Me: "You mean wrong"
Me: "So can you ever know if it is really opposite day?"
Him: "I guess. If you say it is, then it is not if it is opposite day. And if you say it isn't then it is if it is opposite day."
Me: "So you can know for sure then?"
Me: "So... no??"
Me: "What's the opposite of black?"
Me: "What's the opposite of red?"
Him: "Umm...I don't think red has an opposite. Oh wait, the opposite of red is pink"
Me: "Does everything have an opposite?"
Me: "What's the opposite of zero?"
Me: "What's the opposite of 10?"
Him: "umm... 100"
Me: "oh... what is the opposite of -10?"