I liked what the math guy said, but it's difficult to actually do, I think.
One thing I can say is there are a lot of kids I tutor who are decent at math, but have no idea what they are actually doing (in Calculus).
I do think the basic tenets of Calculus can be taught to interested children (but like others said, it may have to be someone who really understands this stuff in depth!). Calculus is about rates of change. One example off the top of my head, if one draws a curve on graph paper, if many rectangular boxes are drawn under the curve, we can show we have an estimate for the area under the curve, which is the integral.