Ideally I think parents should make understanding organic to daily occurrences. If they treat math like spinach (we have to get through your multiplication flash cards before you can have fun, because it's good for you) their child will grow to resent it. Gaining real understanding of logic, structure, measurement, relationships, and numbers, done well, should be a creative, joyful process.

These are some things I've noticed while puttering around the house. I hope they are helpful. For all of these ideas, I would:

- Know at least a little about the concept/solution before posing it (I put very broad categories after each one to possibly make research a little easier.)
- Respond to the child's curiosity - don't force her to do anything.
- Don't try to turn everything into a textbook math problem. It's ok to start with undefined terms and not enough information. Learning how to ask clarifying questions is part of learning how to solve problems.
- It's ok to start with too-complicated a problem. Learning how to pose a simpler problem is a part of learning how to solve problems.
- Don't be afraid to leave questions unanswered for another day - give the ideas time to marinate. Cultivate patience with irresolution.

Notice two cylinders side by side and wonder aloud which uses more material, and which holds more. (Canisters for oatmeal, tennis balls, and Pringles come to mind.) Follow up: roll up a 8.5 x 11 piece of copier paper both the tall way and the short way. Wonder which way holds more. (Volume/surface area)

Do you frequent a particular ice cream place? I bet they offer well-defined options for ice cream flavors and toppings. Wonder how many possible different sundaes you could order. (Or the related questions, such as how long would it take you to eat every possible sundae, if you had one a week?) If the initial problem is too complicated, solve a simpler one. (Combinatorics)

Does your family ever clink glasses at the dinner table? Next time you do that, wonder how many clinks. (They don't have to be clinks. Do they smooch all their relatives at family gatherings? High five them?) If you only have 4 people, this is easy and boring, but if you can get them to wondering at the pattern for 5 people, 6 people... (Combinatorics)

When you are cutting a cake, or pancakes, wonder how you can get the most pieces out of the fewest cuts. (Quadratic relationships)

While traveling, set them to figuring out "How much longer?", using the data from highway mileage signs and the speedometer. (Distance/Rate/Time)

If you are playing a game with two dice, wonder what outcome is most likely. What implications does that have in the game? (Probability/sample space)

"I need 1 and 3/4 cups of butter. Can you get them for me?" When she does this successfully, talk about how she divided by a fraction to arrive at 3 and 1/2 sticks. Draw a picture, write it out with symbols. Notice how weird it is that a division can yield a result larger than the thing you were dividing. Be on the lookout, together, of other examples around the house where one might divide by a fraction. (Division/fractions) (Thanks for this one, Jackie.)

Find number-y fun in your child's favorite sport. These examples are from baseball because I know more than nothing about it: How many outs in a perfect game? Minimum/maximum at-bats in a scoreless game? (Multiplication/division)

Take advantage of anything a child asks about. Sometimes they will hear an older friend or sibling refer to something exotic: prime numbers, or square roots, and wonder about it. I have seen parents give a definition and not exploit the opportunity! Don't stop at "A prime number is only divisible by 1 and itself." Say something tantalizing like, "I wonder if we could find all the prime numbers between 1 and 100? That seems impossible!"

And finally, I can't recommend highly enough a collaborative math circle (as opposed to the type geared toward training for competitions.) If I had kids, I'd be googling "my city math circle" to see what's out there, and starting one up if it didn't already exist.

I'd love to hear about other things you and your kids have come up with! Happy math-ing!