Logical level; perception of proportion

[CCHERPAS]

[From Chris Cherpas (970527.1650)

Logical level:
I've seen references to a logical or logic "level" that I assume
corresponds to some level in HPCT, but I'm not sure. Does somebody
have an HPCT translation of logical or logic level perceptions?

Perception of proportion:
This is a bit off the normal track, but...
Does anyone out there have experience analyzing "the perception
of proportion?" I recall Piaget having done some work that indicated
to him that children under a given age have difficulty "seeing"
proportions. I'm working on some computer-based instructional
exercises where grade-school children determine whether there is
a higher probability of getting red than green from an urn of red
and green balls (e.g., 1 chosen randomly on each trial...and, where
relevant, with replacement in some set-ups and without replacement
in others).

There's a jump when working with urns that are equivalent in
their mixes of red and green, but differ in their Ns, after working
with cases where the is only one urn and two colors (or 2 urns with
equal Ns) and one can just "additively see" (or count and then say) that
there are even more red than green in urn 1 than in urn 2.

Anyway, I'm trying to see how far you might have kids control for concepts
of probability without calculating, and bring in the calculations only after
it seems crucial or necessary.

Graphically one can partition the contents of these urns to show
pretty easily (I imagine) that, for example, "there are 3 red for every 1
green," and that comparing one group of these equivalent sets is like comparing
to a any other (or even the whole urn full) of such groupings. If sets of 3 red
to each green exhaust all the possibilities in both urns, but there is one,
say, green left over in one urn, then the "additive" sense of there being more
can then be exploited again to say that the urn with the extra green has a
higher P(green) and a lower P(red) than the other urn. Of course, this
is the "odds" ratio, rather than the probability ratio of favorable to total.

Spinners preserve proportions without worrying about Ns, but you can't fall
back on counting red balls if necessary, and measuring areas of pieces of pie
seems risky when differences of less than, say, 1/32 seem to be involved.

Best regards,
cc

[Bill_Powers1]

[From Bill Powers (970527.2108 MDT)]

Chris Cherpas (970527.1650)

Logical level:
I've seen references to a logical or logic "level" that I assume
corresponds to some level in HPCT, but I'm not sure. Does somebody
have an HPCT translation of logical or logic level perceptions?

It's (supposedly) level nine in the hierarchy, below principles and system
concepts and above sequences. The logical level is where you think in terms
of if-then or other logical rules. I think it's done mostly in terms of
symbols, like words or mathematical symbols. A logic-level perception would
amount to recognizing some logical relationship, like "and" or "or" or
"not", or even more complicated things like implication or syllogisms. But
it covers more than symbolic logic or mathematics -- I think of it as
simply the rule-driven manipulation of symbols. Paul Revere said "One if by
land, two if by sea," which is a rule stated as a logical proposition. If
you can recognize that as a rule, you're perceiving at the logic level.

Perception of proportion:
This is a bit off the normal track, but...
Does anyone out there have experience analyzing "the perception
of proportion?" I recall Piaget having done some work that >indicated to

him that children under a given age have difficulty >"seeing" proportions.

I'd conjecture that this is a relationship-level perception (level 6). If
you can see that A is "twice as large as" B, that's a relationship
independent of the scale of A and B. What is being perceived is the
relation between two magnitudes, not either magnitude by itself. It would
be interesting to know if this ability develops at a particular age. One
way to find out would be to provide a way to control a proportion, and see
if the children can do it. "Use the mouse to keep the red square half as
large as the green square" (the green square varies in size with time). The
question isn't whether they can estimate "half" accurately, but simply
whether they can maintain a constant proportion.

I'm working on some computer-based instructional
exercises where grade-school children determine whether there is
a higher probability of getting red than green from an urn of red
and green balls (e.g., 1 chosen randomly on each trial...and, where
relevant, with replacement in some set-ups and without replacement
in others).

"Probability" is a pretty advanced notion, isn't it? How can you find out
what the children understand from the terms you're using? This experiment
would seem to involve a heavy concentration at the logic level; it seems to
be mainly about the ability to understand what words mean in terms of
logical operations.

There's a jump when working with urns that are equivalent in
their mixes of red and green, but differ in their Ns, after working
with cases where the is only one urn and two colors (or 2 urns with
equal Ns) and one can just "additively see" (or count and then say) >

that there are even more red than green in urn 1 than in urn 2.

Anyway, I'm trying to see how far you might have kids control for
concepts of probability without calculating, and bring in the
calculations only after it seems crucial or necessary.

I guess I wonder why you want to have them control for concepts of
probability. You're dealing with pretty complicated ideas here, which
_require_ the use of logic but don't tell us much about how it works, when
it works. You have trouble expressing what you mean in words ("there are
even more red than green in urn 1 than in urn 2") -- how would you expect a
child to guess what you're talking about, when I, an adult, have a problem
with it?

I think that in order to _teach_ concepts like probability, you have to
boil them down to simpler ideas, like "are there more red in urn 1 than in
urn 2?" Or better, "If you took one handful of marbles from urn 1, and
another from urn 2, which handful would have more red marbles in it?"

It seems to me that teaching such things is a matter of teaching tricks of
mental computation, in this case teaching about comparing equal sample
sizes. There's no reason to think that people "naturally" know how to do
this. They may have an innate ability to compute, but it's not an innate
capability of computing anything in particular.

Best,

Bill P.