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I’ve long been fascinated by how young children are taught the concept of negative numbers.  Here’s a conversation I had with one of my granddaughters when she was around six or seven.

“If you had five oranges and I asked you to give me two, how many would you have left?”
“Three.”
“Yes.  If I asked you to give me four, how many?”
“One.”
“Yes.  How about if I asked for all five?”
“None.”
“Yes.  What if I asked you to give me six?”
“I’d give you five and have none left.”
“No.  You’d give me five and owe me one.  We call that minus one.”
“But that’s silly.  How can I give you more than I have?”
“Just like a bank can loan you money you don’t have in your savings!”
“Hmm…”

I understand that, these days, the concept of negative numbers is taught when a child reaches around seven or eight, usually using the idea of a debt, as above, or temperature going below 0 ⁰C, the freezing point of water.  In my recent book, The Dream Guardian, I decided to explore the teaching of this concept a little further.  One of my child protagonists, Nikki, has a conversation with her grandfather, Grandpa, in which she expresses difficulty in understanding the concept and annoyance at the taunting of her older brother, Tommy, who chides her for not understanding.  Grandpa explains about oranges and then invents a new card game to help her.  Here is the extract in which he introduces Nikki to the game.

“Grandpa, I’m still having trouble with those minus numbers,” Nikki said, as she marched in through the front door and into the kitchen and flopped down in the easy chair, dropping her school bag by the side.
“Hello, Nikki.  I’m fine thanks.  How are you?  I’m fine too, Grandpa,” said Grandpa.
“Oh, sorry Grandpa.  Sorry.  Yes, hello and how are you, Grandpa?”
“I’m fine,” Grandpa said with a laugh.  “Now what seems to be the problem?”
“Well, I spoke with Mr Clark (Nikki’s teacher) about the five oranges and trying to give six to Tommy and he said hold your horses; we’ve not done negative numbers yet.  I said I know, but I need to understand now ‘cos Tommy keeps asking me for the minus one orange I owe him.”
“Ah, naughty Tommy.  He’s doing that because he’s older and thinks he’s cleverer than you, but it’s all relative.  Here’s a fun way you can learn about negative numbers.  You play cards, right?”
“Yes,” replied Nikki.  “Sometimes we play Chase the Ace and another game called Last Card.”
“How about Pontoon, also called Blackjack or Twenty-One?” asked Grandpa.
“Is that the game where you start with two cards and you can add some more to try and get a score of twenty-one, or very close?” asked Nikki.
“Yes.  You play against the dealer and you can add up to three more cards but your score must always be twenty-one or less.  If you go over twenty-one, you go bust.”
“Stick or twist,” Nikki shouted.  “I remember now.  If you want another card from the dealer, you say twist.  If you don’t want another card, you say stick.”
“Yes, that’s right.  Now here’s a variation that will help you with your negative numbers.  First, you take the pack of cards and remove all the picture cards and any jokers.  That will leave you with just the number cards, aces through to tens.  That makes forty cards altogether; twenty black cards and twenty red cards.  The numbers on the black cards are positive and the numbers on the red cards are negative, just like the money in a bank account: black for when you are in credit and red for when you are overdrawn.   So, for example, the seven of clubs counts as plus seven, and the nine of diamonds counts as minus nine.”
“Next, the dealer deals two cards, just like in Pontoon, but what you have to do is add the black numbers up and then subtract the red numbers.  So, if you had the seven of clubs and the nine of diamonds, you would be holding seven take away nine, equals minus two.  Do you get it, Nikki?”
“Yes, I think so.  It’s like the five minus six oranges.  Can I stick or twist?”
“Yes, just like in normal Pontoon, but your objective is to try and make the total in your hand equal zero.  If you had the minus two I just mentioned, you might twist another card.  If the third card was, say, the two of spades, your total would now be exactly zero—plus seven from the seven of clubs, minus nine from the nine of diamonds, and plus two from the two of spades.  But, if your third card was, say, five of hearts, your total would be… well, you tell me, Nikki,” Grandpa said.
“Um.  Let me think…  Plus seven from the seven of clubs, minus nine from the nine of diamonds…  What was the next card, Grandpa?”
“Here, let me show you,” said Grandpa getting up, opening a drawer, and picking up a pack of cards.  “I’ll just remove the picture cards and the jokers.  I’ll be the dealer.”

Grandpa sorted and shuffled the cards and then dealt two to Nikki and two to himself.

“Take a look at your cards, Nikki, but don’t let me see them.”

Nikki looked at her cards.  She had a seven of spades (plus seven) and four of diamonds (minus four) which gave a total of seven minus four, equals plus three.  She decided to twist another card and Grandpa handed her the ten of hearts (minus ten).  Her total was now plus seven minus four minus ten equals minus seven.  She twisted another card, the six of spades (plus six).  Here’s how the game worked out.

“I won, Grandpa!” Nikki shouted.
“Yes, you did, but more importantly you have learned how to add and subtract positive and negative numbers, so well done, twice.”
“What is this card game called?” she asked.
“I don’t think it has a name.  I’ve made it up.  Shall we call it Make Zero?”
“Yes.  That’s a good name.  I’m going to teach Tommy and then maybe he’ll stop going on about the minus one orange.”
“Good idea, Nikki.  Good idea.  Now, how about a story?  …”

If you have children, or grandchildren, who are struggling with the concept of negative numbers, and if you try playing Make Zero with them, I would be interested in how it works out.  E-mail me at ben@ben-bennetts.com

(^_^)

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