Re: Pure-cardinal approach *is* possible! (was: Mathematical concepts)

> From: hru...@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
> it is necessary to separate the understanding of the concepts from
> the ability to produce proofs.

I agree. But how would you propose testing whether a child really does
understand a concept, before declaring this lesson finished and moving
on to the next lesson that teaches a different concept?

> >Which is one reason why I don't advocate even mentionning anything
> >infinite, not even the set of natural numbers, on the first day of
> >introductory pre-school cardinality and ordinality.

> Then they will have difficulty later.

That's fine with me. Your alternative seems to be to push an entire
graduate-level foundations course at a 3-yr-old the very first day they
ever hear of the concept of cardinal numbers, so they get so frustrated
they give up already and never want to ever try any mathematics of any
kind ever again. By my method, the first lesson is easy, the second
adds to it, etc. until after a few lessons the graduate-level stuff is
merely difficult/interesting, not impossible/frustrating.

> > HUNDREDS TENS ONES
> This is introducing base 10 too early.

Third grade is too early??? You gotta be kidding!
There's plenty of time for the 3-yr-old to learn cardinal and ordinal
number concepts, and Peano's axioms, etc., and for the 4-yr-old to
learn how to construct proofs, and for the 5-yr-old to solve previously
unsolved problems, before finally getting base ten.

> The use of the different columns can be developed from the
> abstract standpoint than using different symbols in the
> different places, or different marks for each power of 10.

Why do you propose teaching something the child will need to un-learn
later?? With popsicle sticks, each of which looks like this: | you can
have a number that looks like this ||||||||||||||||||||||||||||||||||
and then break it up into meaningful pieces any way you want. For work
on computers, it might be best to group the sticks by |||| and then to
group those bunches by |||| |||| |||| |||| and then group those bunches
of bunches by |||| |||| |||| |||| |||| |||| |||| ||||
|||| |||| |||| |||| |||| |||| |||| |||| etc. So that random-size
clump I cited earlier would be grouped and laid out this way:
|||| |||| |||| |||| |||| |||| |||| |||| ||
(It wouldn't really look like that, because each group of |||| would be
clumped together with a rubber band around it, then each next-level
likewise, so it'd be easy to recognize the distinctly different sizes
of standard groups.) Then the teacher could mention that the
traditional way (centuries before computers were invented) is more like
this: ||||| then ||||| ||||| then
||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||||
then
||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||||
||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||||
etc. So that random-size clump I cited earlier would now be grouped and
laid out this way: ||||| ||||| ||||| ||||| ||||| ||||| ||||
Now we have the foundation for oldstyle tallies, newstyle tallies,
oldstyle Roman numerals, and Arabic numerals in a positional system.
.