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

.................

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.

They do know about pronouns, and the ambiguity in
their use. They also know of ambiguity in common
nouns, and there are quite of few of them such as
boy, girl, table, chair, raindrop, dog, cat, rabbit,
and enough more for them to realize that this is
not the case. They can handle a story in which
rabbits are named Flopsy, Mopsy, Cottontail, and
Peter. How hard is it to get across the idea that
they can have any other set of names.

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.

This is certainly true, and I have made the point quite
often here and on the mathematics and statistics groups.
However, I do not think that one can understand the
concepts of the integers without having some idea of
the simple proofs by induction, even if the details are
not remembered.

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.

Being able to prove something does not guarantee the
understanding of the underlying concept or concepts,
although it is more likely than being able to compute
answers. There are, in fact, simple theorems for
which the simple, but not so short, proofs give far
more of an understanding of the theorem than short
proofs using high-powered results, and I am quite
capable of both.

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.

That is what the goal of teaching should be. And
often these novel proofs are much better than the
ones previously known.

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.

There are, and in many, even if those courses exist,
not all good students get an exposure to it.

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.

Are they learning math? Or are they learning to calculate?

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.

At what age are children ready to understand the concept
that sequences of symbols can stand for ideas? Or that
one can attack a sequence of symbols by using rules of
pronunciation?

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?

The predicate arguments are the harder ones. Many logic
books do not even do sentential calculus from scratch, but
use truth tables. This cannot be done for the first-order
predicate calculus. There is a big difference between
being able to understand what a proof is, which should be
required of all, and being able to produce proofs.

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).

In that case, your county is quite unusual.

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.

Quite a few years ago, I taught a probability course with
the full calculus sequence as a prerequisite. This course
satisfied the probability requirement for a teaching major,
as did a lower course with fewer prerequisites, and it was
not intended for those. To make a long story short, on a
take-home part of the final (they never could have handled
it on an in-class exam), only 5 of the 21 such had any idea
how to set up problems involving calculus similar to the
example problems or homework problems, and these were
discussed in detail in class.

In addition, isn't the whole controversial "reform
calculus" (and reform math in general) supposed to focus
more on concepts and less on mechanics?

To do this, you have to go "all out". Doing it part way
achieves little. But I know of no such calculus courses
at the college level; the physicists and engineers want
their students to be able to solve applications using
calculus yesterday.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558