Tonight my wife asked me why 4 - (-3) = 7. Apparently my son had "explained" it to her earlier tonight, but she hadn't really understood his explanation. So I gave it a shot.
First I tried the same explanation that seemed to work for my son when he asked me what 4 - (-3) would be earlier today.
Me: "Imagine you had 4 blue circles, representing 4 positive 1s or 4 total. To this you add 1 positive circle and one negative circle. What would be the new total?"
My wife: "Still four. You added 0."
Me: "Okay, so let's add 0 like this 3 times so the total will still be 4, right?"
My wife: "I get it. I don't know why we are doing this, but okay."
Me: "So now, I take away the three negative circles, and therefore I am taking away a total of negative three from this picture and I'm left with 7. Therefore 4 - (-3) = 7."
My wife: "I don't get it."
I tried a number line representation.
My wife: "I like looking at things on the number line."
Me: "Now imagine I'm on the number line at 4. If I subtracted 3, I'd end up at 1, so if I subtract -3, I must do the opposite, and so I end up at 7."
My wife: "I don't get it."
Next, I tried operation consistency.
Me: "Okay, let's try again. You agree that 4 + -3 = 1, right?"
My wife: "Yes. I understand that."
Me: "And 4 + 3 = 7 is obvious to you, as is 4 - 3 = 1. 4 - (-3) can't be the same as 4 - 3, so it must be that 4 - (-3) = 7."
My wife: "That makes no sense."
I tried going back to the number line.
Me: "Okay, so what's the distance between 5 and 2 on the number line?"
My wife: "3."
Me: "Right, since 5 - 2 is 3. Basically, one way to think of subtraction is that it gives you the distance between two points on the number line."
My wife: "Aaaaaah, I get it now."
Me: "So what's the distance between 4 and -3 on the number line?"
My wife: "7. I get it, thank you so much. Who invented this rule anyway?"
(Aside: This is not quite true. -5 - -3 is a pretty good counter-example, but I'll talk to my wife about that tomorrow.)
I decided to go back to mathematical consistency in a different way.
Me: "Okay, let's look at the following pattern. 4 - 3 = 1, 4 - 2 = 2, 4 - 1 = 3, 4 - 0 = 4. What do you notice?"
My wife: "Well the thing you are subtracting is getting smaller, and the answer gets bigger."
Me: "What do you think would happen if I subtracted -1? What would make this pattern consistent?"
My wife: "Well 4 - (-1) would have to be 5 then."
Me: "Right. And so then 4 - (-2) = 6, and 4 - (-3) = 7."
My wife: "Okay. I'm going to ask our son to try and explain this to me tomorrow, pretending I don't get it. I'll let you know if his explanation makes more sense then."
Do you have any other models I can use, should this question come up again?
David is a Formative Assessment Specialist for Mathematics at New Visions for Public Schools in NYC. He has been teaching since 2002, and has worked in Brooklyn, London, Bangkok, and Vancouver before moving back to the United States. He has his Masters degree in Educational Technology from UBC, and is the co-author of a mathematics textbook. He has been published in ISTE's Leading and Learning, Educational Technology Solutions, The Software Developers Journal, The Bangkok Post and Edutopia. He blogs with the Cooperative Catalyst, and is the Assessment group facilitator for Edutopia. He has also helped organize the first Edcamp in Canada, and TEDxKIDS@BC.