cover article in Time magazine on gifted education

In article ,
(Herman Rubin) wrote:

Abstract ideas are NOT merely abstractions of more concrete
ones, but exist by themselves. Done that way, children
can understand them.

Only if they are developmentally ready. Claiming
that they exist independently does not suddenly make them
less abstract and more accessible.

It makes them MORE abstract, and hence more accessible.
The abstract idea, when understood, is simpler than
what it is an abstraction of, if presented that way.

I thought I'd mentioned this earlier, but apparently it is a distinctive of
gifted people that they work more easily from the abstract to the concrete,
from theory to practice. Average learners go the opposite way.

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

"Parenthood is like the modern stone washing process for denim jeans. You may
start out crisp, neat and tough, but you end up pale, limp and wrinkled."
Kerry Cue

In article .com,
Beliavsky wrote:

That is a sign of a deficient education, almost as bad as not knowing
the alphabet. I think kids who have not mastered their addition and
multiplication tables should not be permitted to join middle school.
Maybe there should be elementary school exit exams.

We don't have middle school, so I don't know what age you are talking about.
I was probably at about 80% accuracy by the end of 6th grade.
What would you rate as a pass? Funnily enough, I had no problems with
mathematical concepts, techniques or anything else, once calculators were
allowed in Year 8. In 6th grade, I would have told you that I was "bad at
maths" because I didn't know that arithmetic mathematics. Actually, I am
bad at arithmetic. I am rather good at maths.

In a school in
India attended by a niece, children learn the multiplication tables at
age 5.

Poor kids. I have already mentioned my son's classmate, who had addition
facts memorised at age 5. Unfortunately, she had no idea what to do with them
apart from recite them in order. IMO that is not mathematical thinking!

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

"Parenthood is like the modern stone washing process for denim jeans. You may
start out crisp, neat and tough, but you end up pale, limp and wrinkled."
Kerry Cue

In article . com,
Beliavsky wrote:

While you take a few seconds to reason out what 8 x 9 is, and perhaps
get it wrong (a process requiring more steps has a higher chance of
error), someone who has it memorized has progressed to something else.
In many math exams, time is a factor, and the person who has it
memorized will be at an advantage.

In my high school maths exams, marks were always given for correct working.
When I was not allowed access to a calculator, I still did well, as only small
proportion of marks was allotted to correct computation.

If your maths exams beyond primary school are really putting a higher premium
on accurate computation than on correct working of problems, which is what you
are implying, you guys are stuffed.

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

"Parenthood is like the modern stone washing process for denim jeans. You may
start out crisp, neat and tough, but you end up pale, limp and wrinkled."
Kerry Cue

Chookie wrote:
In article . com,
Beliavsky wrote:

While you take a few seconds to reason out what 8 x 9 is, and perhaps
get it wrong (a process requiring more steps has a higher chance of
error), someone who has it memorized has progressed to something else.
In many math exams, time is a factor, and the person who has it
memorized will be at an advantage.

In my high school maths exams, marks were always given for correct working.
When I was not allowed access to a calculator, I still did well, as only small
proportion of marks was allotted to correct computation.

If your maths exams beyond primary school are really putting a higher premium
on accurate computation than on correct working of problems, which is what you
are implying, you guys are stuffed.

In my experience, higher level math classes often allow
the use of calculators or give at least partial credit if a
correct approach was used.

Best wishes,
Ericka

Chookie wrote:
In article ,
(Herman Rubin) wrote:

Abstract ideas are NOT merely abstractions of more concrete
ones, but exist by themselves. Done that way, children
can understand them.
Only if they are developmentally ready. Claiming
that they exist independently does not suddenly make them
less abstract and more accessible.
It makes them MORE abstract, and hence more accessible.
The abstract idea, when understood, is simpler than
what it is an abstraction of, if presented that way.

I thought I'd mentioned this earlier, but apparently it is a distinctive of
gifted people that they work more easily from the abstract to the concrete,
from theory to practice. Average learners go the opposite way.

But even gifted kids have to scale the developmental
curve, and will not be ready for higher level abstractions
until they're ready for it. That might be a bit sooner than
for others, but it's not instantaneous.
Also, there's a difference between abstract concepts
and general/theory vs. specific/practice.

Best wishes,
Ericka

Chookie wrote:
In article ,
Ericka Kammerer wrote:

And what is your evidence that if they'd just
been exposed to these things earlier, they'd have grasped
them easily? What's to say that they wouldn't have been
equally confused earlier?

Well, my Dad insisted on teaching me Boolean logic at age 7 because he said
that if I were any older, I wouldn't be able to understand it!

;-) But that's not a particularly abstract concept,
and I suspect that to some degree it is true that it's easier
to get before you've had too many years of assumptions built
up.
Nevertheless, I don't think the notion that there
is no such thing as a developmental curve holds water.
Some things simply aren't grasped well until one is developmentally
ready, and when one *is* developmentally ready, they come quickly
and easily. Flogging them before that point is a waste of time.

Best wishes,
Ericka

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

Do not confuse the ability to solve with understanding.

In my book, it's a rather poor sort of understanding
that doesn't lead to an improved ability to *do* something.

It leads to the ability to formulate problems in the
appropriate language.

Only if you remember quite a bit more than the
basic concept.

Not much more, if that. If you understand the "real
world" problem in precise terms, and if those
concepts are sufficiently close to the concepts of
the mathematical system involved, you can formulate
the problem. You might not be able to solve that
formulated problem, and it may be that nobody can
do better than find an approximation. In many
situations, even that cannot be done. But what
good would it be to know how to solve if one does
not have the basic concepts to formulate?l

If you are faced with a medical
decision, you need to formulate your preferences in
quantitative terms, together with your probability
evaluations, both of which may take computing. Then
using the information from the medical people, you
can evaluate which procedure should be followed.

I would have difficulty doing this with my abilities
to calculate and my knowledge of probability and
decision making without going to a computer to take
my evaluation and tell me what I would consider to
be the best result. The best I can do with the help
of the computer would be an approximation.

If you understand the concepts, you can do this.
If you could compute perfectly in your head, you
could do no better.

Again, that very much depends, particularly
if you're still arguing that one never forgets concepts.
I had lots of probability and statistics. For a while, I
used it regularly. It's been quite a few years since then.
I do not retain enough to properly formulate anything but
relatively simple statistical problems. To do more, I would
have to go back and study.

Did you have the basic concepts? I very much regret
that probability is not taught without having the concepts
of measure and integral (one does not need calculus for
this) first, and in fact it is hidden. Also, attempts
to define probability are misleading, and the overuse
of "equally likely" makes things harder to learn. Again,
it is not about computing answers to simple problems, but
knowing how to formulate.

As for statistics, the basic principle of statistical
decision making is

It is necessary to consider all consequences
of the proposed action in all states of nature.

If you remember what was taught in the methods courses
you probably took, you can show that most of them violate
that one sentence on very simple grounds.

Here again, learning how to get answers makes if very
difficult to learn how to aks the questions.

Also, even without the memory issue, I don't buy
that there is no relationship between understanding a
concept thoroughly enough to formulate solutions well
and being able to solve the problem. Obviously, there
are problems that are essentially unsolvable by hand
due to their complexity, but my experience is that
working through at least some problems by hand (in a
somewhat simplified version, if necessary) generally
is very useful in helping people understand what they're
learning. No one has argued that every problem should
always be solved by hand.

The educationist using statistics puts his data
into a computer program. If he knows how to do
it by hand for a simple problem, it becomes no
easier to get the results. If he uses a poor
formulation of the problem, the computer may well
give him a poor answer; the computer is a
super-fast sub-imbecile, and does not think.

Well, I can hardly speak to any of that, as
I seem not to have met any of these "educationists"
you keep speaking of. Certainly, none of my children's
teachers have thought that it was ok for them not
to understand concepts or do problems mechanically,
incorrectly, or without any understanding.

I strongly question whether any of your children's
teachers understand the basic concepts of mathematics.
--
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

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.

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.

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.

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.

The concept of variable is an abstract concept. Do
not make it an abstraction of something else: even
though the idea may have evolved from less, it is
easier to understand it as the SIMPLE idea than to
try to build it up.

Again, you still have not backed this notion
up with anything other than your personal assertion.

I do not see that someone who has difficulty recognizing
that variable is a simple abstract concept can get
the point.

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.

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.

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.

When I say that the biggest problem is the use of
vocabulary which a third grader (or even a fifth grader)
would have difficulty with, I have some idea of what that
means. Also, there is a matter of presentation; the
notation in Suppes and Hill, which left out a particular
part, is harder to understand than that in the college book
by Suppes, which I used with that for my children, and the
notation in my late wife's book is simpler.

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.

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.

The rest of mathematics will not fall out of the sky, but
they will have the language to be able to see what is and
what is not a proof.

Ok. And?

Euclid came close for his geometry,
and the educationists have changed the geometry course to
learning facts and computations instead of 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.

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

I know (second hand; the person who told me had it first
hand) that when the group asked by SMSG to produce a high
school geometry book did an excellent job, with all the
axioms left out by Euclid added, they were told that this
book would be put on the list, but that they should produce
an easier one, using algebra and computation. Recall that
Euclid could not use algebra; he could use symbols for
geometric objects, but nothing else.

Even
a half century ago, it was known that the only real
mathematics course in high school was this geometry course.

You can call it real or not as you please. I don't
know that I buy the assertion that all students ought to learn
only "real math" by that definition, or even that following
your approach is the best way to teach the math that they
do need to know.

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.
--
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

In article ehrebeniuk-6C9865.22235707092007@news,
Chookie wrote:
In article ,
(Herman Rubin) wrote:

Abstract ideas are NOT merely abstractions of more concrete
ones, but exist by themselves. Done that way, children
can understand them.

Only if they are developmentally ready. Claiming
that they exist independently does not suddenly make them
less abstract and more accessible.

It makes them MORE abstract, and hence more accessible.
The abstract idea, when understood, is simpler than
what it is an abstraction of, if presented that way.

I thought I'd mentioned this earlier, but apparently it is a distinctive of
gifted people that they work more easily from the abstract to the concrete,
from theory to practice. Average learners go the opposite way.

I have to disagree. It is the gifted who have the ability
to generalize, and even most of them have problems in
getting rid of the garbage of the special cases. It is
still the case that most measure theory courses start with
measures on the real line, and cause problems thereby.

I do not think that anyone has difficulty with the concrete
after the basic abstract ideas are learned. But we cannot
teach abstract ideas the way we teach facts and methods.
One does not understand an abstract idea until the "light
bulb" goes on. Examples may be needed to do this, but the
general concept has to be there, and kept in mind. If the
examples are learned and used before the abstract concept
is presented, problems often arise.

I have seen far too many good students who seem to run
into a stone wall when they hit abstract concepts. In
set theory, and I do not mean set algebra, it is
necessary to take "set" as undefined!

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

--
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

In article ,
Ericka Kammerer wrote:
Chookie wrote:
In article ,
(Herman Rubin) wrote:

Abstract ideas are NOT merely abstractions of more concrete
ones, but exist by themselves. Done that way, children
can understand them.
Only if they are developmentally ready. Claiming
that they exist independently does not suddenly make them
less abstract and more accessible.
It makes them MORE abstract, and hence more accessible.
The abstract idea, when understood, is simpler than
what it is an abstraction of, if presented that way.

I thought I'd mentioned this earlier, but apparently it is a distinctive of
gifted people that they work more easily from the abstract to the concrete,
from theory to practice. Average learners go the opposite way.

But even gifted kids have to scale the developmental
curve, and will not be ready for higher level abstractions
until they're ready for it. That might be a bit sooner than
for others, but it's not instantaneous.
Also, there's a difference between abstract concepts
and general/theory vs. specific/practice.

What is a "higher level abstraction"? Generally, the
more abstract, the easier, IF one does not make a big
issue about what it means.

Even a weak learner can go from theory to practice. If
one understands something, and I do not mean knows the
words or even knows how to prove the theorems, it is
easy to apply. It may still be difficult to compute;
good mathematical computation is NOT taught.
--
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