I like to point people to class forums on the Elitist Jerks web site for an example of that. There are several studies contrasting high success on situated tasks (e.g. administering drugs to patients) to low success on similar formal tasks on tests (in this case, problems about proportions).

I try to mention free play multiple times everywhere I talk about math. It's very important. It's the foundation.

"Inspired by calculus" is what I like to call our activities. Both because the goal is to get kids inspired by calculus, and to distinguish the activities themselves from formal calculus courses.

I am aggregating some (healthy, not mnemonic) ideas for memorizing times tables efficiently, based on patterns. This will be one of the next projects for Natural Math.

Cut a piece of paper in half... and again... and again... Kids can understand that you can keep halving the paper. They also realize that at some point you'll have to stop, because your scissors are too large for tiny papers. But then they can imagine how you COULD keep going and going and going - in your mind.

I love these examples as bridges to algebra. "Two hours from now" can land on different times, because "now" is variable. You can tackle these problems as arithmetic, or you can tackle them as algebraic.