I was interested in an issue and a couple of questions you had raised therein, namely:
"That's why we are particularly incensed, first by Common Core?s general neglect, and then by its shabby treatment of such an essential topic: the concept of Pi."
As mentioned in an earlier post at this Forum, I don't know much about 'Common Core' - but I believe that any "shabby treatment" the concept of Pi is most unfortunate (if your charge is accurate). Yes, it's an essential topic - and fundamental in many ways as well.
The important questions you've raised:
A: how would we at ccssimath.blogspot.com like Pi to first enter students' consciousness?
B: At what age is it appropriate for Pi to be introduced, in terms of both conceptual understanding and performing calculations?
Remarkably enough, the way the concept of Pi was brought home to me (well over 60 years ago!) was almost exactly as you've suggested:
QUOTE With centimeter tape measures in hand, groups of students carefully measure the circumferences and diameters of various round objects. The class makes a table of measured values and calculates the quotients. When they see time and time again the same answer result from division, whether it be from big circular objects or small ones?eureka!?they will have unwittingly discovered Pi for themselves. UNQUOTE [Of course, the tapes we used were marked off in feet, inches, etc].
It does appear that at least my kindergarten teacher even way back when was practicing something very like teaching math through 'discovery' (probably long before it became a bone of contention for the 'traditionalistas')!
I agree with you that the "Pi Day" (and etc) are essentially rather silly ways to go about it - just bumpf.
The question about the age at which Pi should be introduced:
Well, evidently it has to be after those excercises with the tapes and round objects. How long after? I don't know.
I'm fairly sure the tools described at the attachments to my post heading the thread "Discover: how to achieve it?"
could be of significant benefit for this purpose, if you were to set up a 'Mission' for the purpose and go the whole 'OPMS route'. It would not be a 'minor' Mission to work out, by any means. It would take quite some time and effort to work it all out to any degree of effectiveness. I believe the results would be of considerable benefit, in many ways.
(I should in all fairness note that Robert Hansen has been rather scepttical about this OPMS route - this has been, I know, because he has fundamentally understood nothing about it).