Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:

If you show kindergarteners
a bunch of blocks, let them count them and determine that
there are 10 of them, and then push some of them to one
side and the rest to the other *while they're watching and
can see that you didn't remove or add any blocks*, and then
ask them how many blocks there are in total, *most* of them
will not know that there are still 10 blocks. They're
not going to get the notion that a symbol can be a representation
for the abstraction that is a variable.

What does the above have to do with the concept of
variable? A much more derived result of mathematics
than the simple concepts is involved here. The
fact, that if a set is divided, the number of objects
in the two sets together equals the original number
is a theorem, which is harder to prove from the
axioms than you seem to think if the easier ordinal
approach is used.

My point is that it is something that is very
basic and easily understood and demonstrated by children
just a few months older when they are developmentally
able to deal with the abstraction required. Up until
that developmental turn has been taken, it is difficult
even for very smart kids. If they can't get something
that simple (they're not being asked to prove it, after all),
how are they going to deal with even more abstract concepts?

What you think is basic is something I see as having many
simple but not yet understood steps. The idea that symbols
can represent objects, actions, descriptions, etc., is not
of that form. It is pure simple language.

No, it's not. The notion of a variable that
can represent a wide variety of things is a pretty serious
abstraction. At that age, language is much more concrete,
usually representing a 1-1 correspondence between the
word and that which it represents.

Could you prove it?

First of all, what would it matter in this
context? You asserted that young children (before the
age where they're typically understood to have a firm
grasp of abstract concepts) can learn abstract concepts
easily if only one refrains from attempting to lead up
to the abstract concept by way of more concrete examples.
I suggested one abstract concept that most kindergarteners
demonstrably do not grasp, but typically do grasp a short
time later as they begin to move up that developmental
curve. Whether or not you or I can prove that particular
mathematical theorem is largely irrelevant to the issue
of whether or not this is an abstract concept that a
young child can grasp.

If you cannot prove that theorem, or even have an idea of
how to go about it, are you sure that you can properly
present the idea? As I have repeatedly stated, the
attempts to teach mathematical concepts to teachers have
been extremely unsuccessful, and that includes those who
have become high school teachers of mathematics.

I do not for a moment believe that one has to
be able to prove something in order to grasp a concept.
The world is far too full of exceptions to that rule.
I will agree that if you can prove something, you likely
understand something at a higher level, but not that
it is essential to understand everything at that higher
level from the get-go.

When they come out of high school now, they do not have
the development to prove it, or even indicate a proof.
I believe that a good program would enable a child who
has learned the concepts and what addition is could
sketch a proof.

I rather suspect that most high school students
could swing such a proof if that were something that
was taught. I doubt most kindergarteners (or even
first or second graders) could.

They might produce a memorized proof.

Well, I sure as heck didn't produce memorized
proofs, since the proofs I was assigned for homework
hadn't been given to me previously. Seeing as the
neighbor kids seem to have rather similar homework,
at least around here, they still seem capable of
producing novel (to them) proofs.
Now, are there areas where proofs aren't taught
anymore? There may well be. As far as I can tell,
here isn't one of them.

When I started teaching, the binomial theorem, and the
derivatives of powers, were proved by induction. Now, the
difficulties of teaching induction are so great that this
has been dropped. Hand waving, and argument by fiat, are
used. So the student gets the idea that calculus methods
are to be memorized, and plugged in. Those students, even
if they remember all the formulas, cannot do anything but
compete poorly against computer packages.

Well, my kids haven't been to calculus yet (nor
have the neighbor kids), so I can't for sure say what
they are teaching in calculus here.

I have not denied that a variable is a simple
abstract concept. I have said that until children are
developmentally ready, they are not going to master even
simple abstract concepts.

If they cannot understand the notion of a variable, they
are in no position to attempt mathematics. I am not
even sure that they are ready to read.

And yet somehow they manage to begin reading and
learning math despite not yet being able to manage more
abstract concepts.

Again, based on what evidence? You're just
basically asserting that something that has worked
with 5th graders will automatically work with 3rd
graders. How do you know that?

Partly because I understand what is in it and what the
problems are. I used it to teach my children, one before
age 6, and the other somewhat later.

And what is your evidence that these two
cases are representative?

Because it is SIMPLE. Putting it as late as that is
because a certain amount of vocabulary is needed.

Again, there's a whole body of research regarding
the development of abstract thinking. Where is your critique
of this literature to say that you are right and it is wrong?
Seeing as precision is of interest.

If one starts with the view that abstract thinking can only
come through the process of abstraction, you will not make
any attempts to teach abstract concepts directly. The
only traditional mathematics course which made any such
attempt is the "Euclid" geometry.

I have not said anything about how the teaching
of abstract concepts should be approached. I have said
that young children are not ready to deal with abstract
concepts until they have reached a certain point developmentally.
I don't particularly care *how* you attempt to convey
the concept.

However, there is the game "WFF 'N PROOF", which starts
out with versions for small children, which teaches
well formed expressions (formulas) and proofs, and
everything is symbolic; the notation is Polish, which
has no connection to Poland except it was developed
by a Pole. It has no parentheses.

I'm familiar, thanks. And note that "symbolic"
and "abstract" are not the same thing.

And there are plenty more sources that teach
formal logic, some even in child-friendly ways. Nevertheless,
I rather doubt you will find many kindergarteners who are
ready for it, nor do I think that if you teach them formal
logic that the rest of mathematics will just fall out of the
sky and bonk them on the head.

Formal logic is not just the sentential calculus.

I'm sorry. Did I say that it was somewhere?

Arguments with quantifiers are the hard part. In fact,
some books teach the sentential calculus through truth
tables; whatever method is used, the connectives and
quantifiers are the basic concepts. What Aristotle
did is NOT adequate.

Again, what is the relevance here? You made a
claim about formal logic:

These books teach formal logic, not any other subject,
through the first-order predicate calculus. This is
what is needed for mathematics, but does not require
mathematics to understand.

I said that formal logic was not sufficient for teaching
math (nor do I think it is necessary at the elementary
level) and expressed skepticism that kindergarteners would
hit the ground running with it. Then, you come back with
formal logic not being just the sentential calculus. What's
your point here?

Again, I'm beginning to wonder if these "educationists"
are mythical beasts. Proofs are still a core of geometry around
here, and were when I took geometry as well. I recall fondly
[cough] Mrs. Montagna and her rules about precisely how proofs
were to be written up (on white, unlined paper, folded just so,
in ink...).

How many students now take the proof oriented geometry
course? Check in any high school which is not of the
honors variety; you will find it small.

Well, I have no idea what it is like everywhere.
I can tell you what it was like when I was taking geometry
(plenty of proofs, thank you very much). In my county,
proofs are a required part of geometry, according to county
standards (including for non-honors courses).

BTW, I object to the rules about what paper to use, etc.

Well, so did I, but Mrs. Montagna was a very old-
fashioned teacher and she did believe in such things.
While it was annoying, I don't think it was particularly
harmful. Every teacher has his or her peccadillos. I'm
willing to spot 'em a few as long as they don't interfere
with the learning.

If we wait until the teachers understand the basic concepts
of mathematics, they will never learn the basics, and only
the geniuses will have a chance to understand them. The
math that you think they need to know can be done for them,
and more and more is. Understanding concepts and formulating
are what can be human; the rest is merely mechanical.

So far, I have yet to see that that lack of understanding
is pervasive here. Perhaps it is elsewhere. I recall a
study a few years ago comparing advanced high school calculus
students from Japan and the US. IIRC, the both groups of
students performed equally well on more conceptual questions,
but (given that the test did not allow calculators), the
Japanese kids beat the pants off the US kids when it came
to problems requiring more challenging computation (with
many of the US students not being able to solve the problems
at all without a calculator). Doesn't sound like there's
a huge emphasis on plug'n'chug to me.
In addition, isn't the whole controversial "reform
calculus" (and reform math in general) supposed to focus
more on concepts and less on mechanics?

Best wishes,
Ericka