cover article in Time magazine on gifted education

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. 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.

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.

--
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:

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?

Could you prove it? Starting with the self-contained
Peano Postulates, it can be proved, but not right
away. Yet the Peano Postulates can be understood by
a kindergarten child.

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.

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. This is a major problem with students in
all fields which require precise formulation, and more
and more are requiring it, including literature.

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.

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. One can
teach sentential calculus quite quickly; Suppes and Hill
go through a development of that from a set of rules,
which is slow and tedious. Other books do it quickly by
formal procedures, which can be shown to be equivalent.

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. Euclid came close for his geometry,
and the educationists have changed the geometry course to
learning facts and computations instead of proofs. Even
a half century ago, it was known that the only real
mathematics course in high school was this geometry course.

--
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:

I think there's a fine line. Abstract concepts are
ahaky in early childhood largely for developmental reasons.

There is a HUGE difference between learning an abstract
idea directly, or attempting to lead up to it by more
concrete examples. The first is what I am proposing;
the second can be quite difficult, and even painful.

Again, what's your basis for claiming this?
Why would you suggest that boatloads of research
indicating that abstract reasoning is a developmental
skill is all wrong and the only problem is that
folks having been teaching the abstractions directly?

I repeat, it is easy to learn a concept directly, but
much harder to carry out the process of abstraction.
How hard is it to teach the concept that letter sequences
can be used to represent words?

Actually, before a child is developmentally ready
to grasp that concept, it's *very* difficult. And it's
not particularly abstract, because there is nearly a 1-1
correspondence between sounds and symbols. And I'm sorry,
but your simple assertion that "it is easy to learn a
concept directly" doesn't provide much evidence to me
that abstract concepts are easily grasped by children
before they've reached a stage of development associated
with the ability to deal with abstractions if only folks
bypass those pesky concrete analogies.

Even with the whole word method, it was used that letter
sequences represent words, and the sound correspondence
was deliberately avoided, so much so that a seventh grade
mathematics book had the word "rug" italicized as a word
the students would not know how to read.

--
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 om,
Beliavsky wrote:

At a
higher level, I wonder if the time spent in calculus on teaching what
variable transformations should be used for what integrals should be
reduced in favor of teaching students how to use Mathematica or Maple.
Students ought to do a few exercises to learn the concept of change-of-
variables, but practising to the point of gaining proficiency is less
important than it was only 30 years ago.

Um, we learnt the concept of change-of-variables (if you do actually mean
"substitution", as Herman suggests) a lot earlier than we learned about
calculus. I don't recall any song and dance being made about it when we got
to calculus. Is this some kind of serious issue in your mathematics classes?

--
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

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.

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.
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.

Best wishes,
Ericka

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

I think there's a fine line. Abstract concepts are
ahaky in early childhood largely for developmental reasons.

There is a HUGE difference between learning an abstract
idea directly, or attempting to lead up to it by more
concrete examples. The first is what I am proposing;
the second can be quite difficult, and even painful.

Again, what's your basis for claiming this?
Why would you suggest that boatloads of research
indicating that abstract reasoning is a developmental
skill is all wrong and the only problem is that
folks having been teaching the abstractions directly?

I repeat, it is easy to learn a concept directly, but
much harder to carry out the process of abstraction.
How hard is it to teach the concept that letter sequences
can be used to represent words?

Actually, before a child is developmentally ready
to grasp that concept, it's *very* difficult. And it's
not particularly abstract, because there is nearly a 1-1
correspondence between sounds and symbols. And I'm sorry,
but your simple assertion that "it is easy to learn a
concept directly" doesn't provide much evidence to me
that abstract concepts are easily grasped by children
before they've reached a stage of development associated
with the ability to deal with abstractions if only folks
bypass those pesky concrete analogies.

Even with the whole word method, it was used that letter
sequences represent words, and the sound correspondence
was deliberately avoided, so much so that a seventh grade
mathematics book had the word "rug" italicized as a word
the students would not know how to read.

I don't see what you're trying to get at here?
What does that have to do with the preceeding discussion?

Best wishes,
Ericka

In article ,
(Beth Kevles) wrote:

Times tables (and addition, subtraction and division): It turns out to
be hard to keep up with examples given by the teacher in class, and hard
to follow math examples in textbooks, if you can't keep up with basic
computations. Not impossible, but difficult.

This sounds to me like a fault in the teaching method.

There are fun ways and
dull ways to get kids to learn basic math facts. Mine both did "mad
minutes" where they had to go for a combination of speed and accuracy on
a sheet with 100 problems on it. This was sufficient incentive, so the
kids learned their basic math tables.

These sorts of races ranked high in my list of reasons to hate maths in
primary school. After a year of times-table races in sixth grade, I felt I
was no more accurate nor fast than I had been at the start of the year, and my
confidence was way down. Giving a child "incentive" (I'd call it
"humiliation") is no good without giving them the tools to achieve the target.
I still don't know why it's easy for me to memorise some things and not
others. I still don't know why I made (and probably continue to make) errors
of the 2+2=2 variety when writing arithmetical problems out.

Computation: Doing it on paper teaches the kids to write math out
neatly (a real-life skill). If taught in conjunction with
"reasonableness", as in "is this answer reasonable based on the numbers
you were given?" then it deepens understanding. I've also seen that
kids who show their work often show holes in their basic conceptual
understanding that can then be addressed by teachers. (Oddly, a concept
taught early in the year can be mis-remembered or mis-applied later in
the year. Hence doing the computation and showing one's work are, in
fact, excellent learning toos.)

That I certainly agree with. My marks were certainly improved when my working
indicated that I'd understood the problem and its solution; I'd just not given
the correct answer!

Gifted education is NOT just about learning content and concepts as fast
as you can.

That is usually not what is meant by the term. What we do know is that too
much repetition actually reduces accuracy, probably because the kids become
stale. Another issue is what should be done if a gifted child knows something
and the rest of the class do not. I had an interesting moment early this year
in a discussion with DS1's teacher.

Chookie: I see in the syllabus that children learn to tell time to the minute
in 3rd grade -- [I was about to ask whether this was when most of them would
be able to do it anyway, as I wanted to get a feel for how pushy the syllabus
was]
Teacher: Oh, don't worry about that; none of them can do it at this age!
Chookie: Er, DS1 can!

Now, DS1 is in 1st grade. You can see what his advanced knowledge means in
terms of classroom management. If he is expected to sit still and be retaught
something he has known for years, then do multiple examples of it, he will
become contemptuous of his teacher, his classmates, and even of learning
(because he isn't doing it) and most likely disruptive. The best option
pedagogically is to give him something that he has to work at -- that's what
the other children are getting, after all. This teacher's solution is that
the children are put into ability groups and each group is getting work at the
appropriate level. I hope all Ds1's future teachers take the same approach!

--
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 ,
"Donna Metler" wrote:

If you have never gained proficiency yourself, you are very unlikely to
recognize errors. It's like a friend's child, who recently went through all
steps of an algebra problem, and couldn't figure out what was wrong. The
problem she had was simple-at some point, she'd effectively divided by zero.
I saw it once I looked at her problem steps, in a very short time, my
husband glanced at it and immediately knew what had happened somewhere. And
the reason both of us could recognize it is that we've both made that same
mistake in long hours of practicing algebra problems, so know to look for it
almost automatically.

But that sounds like a logical error, not a computational one.

--
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

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?

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.

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.

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.

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.

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?

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...).

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.

Best wishes,
Ericka

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!

--
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