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This was the question:

Hi,
 
I spoke with you last summer about my just-turned-4-yr-old daughter and her love of math. She is now 5 and I can't slow her down. She wants to push ahead and I am a bit stuck for words to explain.
I wanted to ask you how do you think I should explain the following:
 
Lets say I am teaching:
 
3 x 4 = 12        4 x 3 = 12
 
12 / 3 = 4    12 / 4=3
 
I was wondering how you think I should approach this in words. I know she will get the multiplication easily but I am not sure exactly how to word the division part of these in relation to the multiplication. This is from someone who hated math and is learning a whole new way to do things.

She already understands the concepts of multiplication and division, but before we go further I want to teach her the inverse operation of division at the same time. I believe then she will see that one multiplication fact  is automatically four math facts learned. I believe then she will see and understand the concepts rather then just memorize facts.
 
I hope I am making myself clear.

Thank you for any help here,
 
A Mom from Cal.

Professor Homunculus' Answer:

Thanks for the great question. You made yourself very clear, and I am thankful for the opportunity to address this important issue. I've had some notes on this for months, and haven't gotten around to putting them up on the site yet.

You bring up a interesting and important point when you mention that one "fact" is really more, if you understand the concepts, rather than just accept the facts on their faces.

First, it's important that we make clear that this lesson is about multiplication and division of whole numbers, only. In another lesson, I'll explain multiplication and division of fractions in simple words.

We use the words "multiplied by," or "divided by" too often, without explaining to children what they mean in words that they understand. Young children should understand that "multiplied by" means "many-ed" by (as in "this many of that") and that "divided by" means "chopped up by," (as in "this chopped up by that.")

Once they understand that, you could replace that with "groups of," or "parts of," in most cases, and it would solve a lot of problems. (Remember, these things apply to whole numbers only.)

If you teach :

• "4 groups of 3 gives you 12," or
• "Three groups of four things in each group, gives you twelve things"

(for whole numbers), it then makes sense to say

• "4 parts of 12 gives you 3," or
• "Twelve things, in three parts, gives you 4 things in each part."

Please send me some feedback on this. I've been giving a lot of thought to simple language for the basic operations, and this is as streamlined as I have been able to get it. Your thoughts are appreciated.

I'd also like to give you a big Hotcha! for getting your child to love math at such an early age, and encouraging her.

Yours truly,

Professor Homunculus