I've spent a lot of time stressing the importance of counting by groups, too - since that's basic to place notation and to numbers generally. Went over concepts like "twenty means two groups of ten, a hundred is ten groups of ten" and so on. Counted by twos, fives, and tens; introduced the concept of even and odd numbers by writing the numbers 1 though 20 in two columns, with odd numbers on the left and even on the right.
Folded a piece of paper in half, counted the sections, and repeated the process to introduce the concept of powers of two. Reinforced this opportunistically with examples: "If Pinkie Pie jumps into the Magic Mirror Pond, how many Pinkie Pies will there be? And if both of them jump into the Magic Mirror Pond, then how many ...?"
Used the beam of a flashlight to illustrate conic sections. Holding the light at various angles to the floor, I taught her to identify the circle, ellipse, and parabola (and to say them!). Also had her identify these shapes as Mathematica plots, along with the sine wave, and pointed out the parabola in the shape of the cables of the Golden Gate Bridge*, the curve of water spouting from a drinking fountain, and the trajectory of a thrown ball. She won't need to know the mathematics of these things for some years yet, of course, but I want her to know that there is order and beauty in the universe, and that it can be grasped and understood. And I don't want her to be afraid of words like "parabola".
* A parabola, and not a catenary. With a load uniformly distributed along the cable itself, you get a catenary; but when the load is uniform along the horizontal (as in a suspension bridge, where the weight of the cable is negligible relative to the load) you get a parabola. Beer et al., Vector Mechanics ... Statics, 9th ed., pp. 385-395.
