Maria, thanks for the detailed reply and the examples.
The embodied exercises, and an infinity of elephants (etc) makes it all so concrete, embodying the abstract. I can really see it working.
In a sense all abstraction is, by definition, emergent, or derivate from the embodied and the concrete, but without these kinds of exercises, and without a range of degrees of freedom, abstraction is often too big a jump, and appears to have no 'grounding' and therefore no utility.
What strikes me too is that the footprints thinking needs to emphasise the value of offering a range of degrees of freedom, as a range of entry points - exercises, inquiry, (edge of) chaos, which ideally should be available simultaneously, all of the time.
This could be difficult to deliver, but its a key element of really open learning design, because if there is no suitable entry point, or point of engagement, no amount of emergent learning design will mean anything to that student. And as you correctly point out, "different people have different personal preferences for each of these mixtures, and these preferences change from day to day, or from minute to minute" - that's a challenge, and its an important one.
My experience of mathematics is much more limited - but this is in principle so similar to the work I did in LOGO (with kids) many years back, which provided the opportunities for kids to use complex variable programming to draw beautiful shapes - they were concentrating on the shapes, and iterative variations in shapes, and had no idea they were learning about variables and functions.
