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I've become somewhat obsessed with teaching Katherine math.  At almost 5 years old, I don't believe that multiplication and division are concepts too difficult to understand.  I've blogged in the past about the "postfix trick", where you put the times at the end to make it more understandable — "How much do you have if you do 3 2 times?"  The mental switch there is easier to grasp and you can see "3 and 3".  But this trick gets unmanageable quickly since it's hard for little brains to grasp numbers much larger (especially if you run out of fingers).

Lately we've started working with some flashcards that Nanta and Grampy got us.  At first we were going to set aside the multiplication cards as too hard, but after going through the various shape/color/letter cards Katherine said, "I want to do those."  She seems to have quickly grasped the tricks for 0 ("if you have 0 of anything, you've got 0") and 1 ("If you only have one of something, then that's what you have, so if you have one 7, then the answer is 7.")  And I tried the postfix trick for various combinations of multiplying by 2 or 3.  Part of the problem is that you quickly run out of fingers, you see.  There's not much that you can multiply by anything greater than 4×2 or 5×2 and still get the answer on both hands.

Here's a game I plan on trying when I get a moment.  I thought of it last night right at bed time so we didn't have a chance to play it, and I'm bored on the train so I'm writing it up.  You'll need:

• 10 little plastic Dixie cups, like the sort you hang by the bathroom sink to rinse when you brush your teeth
• at least 20 coins, beans, or other small countable items
• two big bowls (optional)
• marker.  Number the cups 1-10.

I chose 20 countable items because I think that children my daughter's age can reasonably work with numbers in that range.  The game could easily be expanded up to 100 by simply providing more countable things.  So you have to be careful to only choose combinations that result in an answer less than 20

Put all of your countable things into one of the big  bowls.  You could get by with just piles on the table or floor if you want, I figured the bowl just keeps it a little neater.  My 3yr old has a tendency to walk through the middle of such games and mess up the piles.

Pick a flashcard, or just make one up.  Say your flashcard is "5 x 3".  The child counts out 5 cups (using the numbers to help, if necessary), and then proceeds to take the items from the big bowl and distribute them, putting 3  in each cup.  Naturally it's important at this stage that the child can count to these numbers, but I'm assuming that she can, otherwise teaching her multiplication is a bit overkill.

Once that's done, dump all the cups into the other big bowl. Now count what's in that bowl.  There's your answer!

Sure it's a lot of steps, but the name of the game is for the child to eventually memorize the answers are predict the outcome, and then see if she's right.  You can take turns and let your toddler fill up the cups with beans, and then the adult has to guess at the answer.  Then you can figure out together whether you were right by dumping everything out and counting them up.

Division can be taught this way as well, but it's substantially harder because of fractions.  With the help of an adult you can count out a large number of coins, then pick a number of cups that you know divides that number evenly (for instance, pick 12 beans and 4 cups).  Then the child can distribute one coin each into the cups until they're all gone, and look at how many are in each cup.  If there aren't the same number in every cup, that means that the numbers don't divide.

Like I said, haven't tried it yet, but I hope to try it soon.  I'll post the results.