cover article in Time magazine on gifted education

Donna Metler wrote:

I have never understood why practice is considered desirable and necessary
in music, in dance, in sports-but is somehow a bad thing in other fields.

I absolutely agree.

And I strongly suspect that one reason so many students with music
backgrounds are successful in math-intensive disciplines is that anyone who
has played an instrument to any degree of proficency does not question the
idea that practicing to automaticity is necessary, ever again (it only takes
ONE time of getting up to perform a piece and messing up big time to teach
that lesson!).

Well, maybe some of us are a bit slower on the uptake
than others in that regard ;-) However, I do agree that classical
music instruction does instill a certain kind of discipline that
has always stood me in very good stead. Without music, I don't
think I would have learned those skills so that they were available
down the line when academics didn't come as easily.

Best wishes,
Ericka

Beth Kevles wrote:

Gifted education is NOT just about learning content and concepts as fast
as you can. It's also about learning how to learn when the learning is
difficult or has boring bits. So many gifted kids don't know how to
WORK at learning, or don't have the self-confidence to ask 'dumb'
questions in front of a class of peers.

That's part of the reason that a musical education for
my children is so important to me.

I hope I haven't rambled too much. I spent quite some time last night
convincing one son that he needed to work more on understanding
quartiles and range, and that he'd mixed up median and mode. He really
thought that since he'd understood the teacher's explanation in class,
that meant that he understood the concepts fully and didn't have to
concentrate on the problems in his homework. And he's very typical of
gifted kids.

I agree--seems very common in my experience as well.

Best wishes,
Ericka

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.

Best wishes,
Ericka

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?

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.

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?

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.

Best wishes,
Ericka

In article om,
Beliavsky wrote:
On Sep 2, 3:13 pm, Ericka Kammerer wrote:

So, in my opinion, it is helpful for kids to
go through a reasonable set of exercises that hit upon
different variations of the problem to verify that they've
really got it.

Because of modern technology, I think certain kinds of practice in
math should be reduced in favor of instruction in software tools.
Although I think kids should memorize the multiplication tables up to
10x10, so that they can estimate quantities in their heads, I don't
think their accuracy rate of multiplying 3-digit numbers (I remember
doing such worksheets) is important -- they can use a calculator. 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.

As you have seen from some of the other posters, it
is quite possible to use mathematics in practice and
not know the multiplication tables. That part of the
computing is done reasonably well.

Alas, Mathematica and Maple are not as good as you
might think, but they are getting better. Unfortunately,
their notations (and those of Maxima, Axiom, etc.) are
all different and highly arbitrary. There is a running
discussion of their weaknesses in several newsgroups by
giving problems which they cannot handle, or which they
handle incorrectly, or in which they do some complicated
things, but cannot handle what the calculus student is
likely to be able to do.

But you are right, acquiring proficiency in solving should
not be the goal. Understanding what is meant, and being
able to formulate, are the important parts. Most calculus
students can do manipulations, but do not have the least
idea of the concepts. The graduate students we get in
mathematics and statistics might have had as little as
5 minutes instruction total in the concepts.

--
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 ,
Banty wrote:
In article om, Beliavsky
says...

On Sep 2, 3:13 pm, Ericka Kammerer wrote:

So, in my opinion, it is helpful for kids to
go through a reasonable set of exercises that hit upon
different variations of the problem to verify that they've
really got it.

Because of modern technology, I think certain kinds of practice in
math should be reduced in favor of instruction in software tools.
Although I think kids should memorize the multiplication tables up to
10x10, so that they can estimate quantities in their heads, I don't
think their accuracy rate of multiplying 3-digit numbers (I remember
doing such worksheets) is important -- they can use a calculator. 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.

Mathematica? Maple? I use neither. Never heard of the latter. Why not
MathStats ;-)

Those are more powerful. In fact, they are somewhat competing
to do the whole job correctly, and sometimes that does not work.
I sometimes use them because if I went through a long process
I might well make errors.

Some exposure to spreadsheets would be good, as there are one or two truly
widespread applications, and they are used in many fields. But it wouldn't be
particularly useful to get too far into any specific math program.

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.


"Change of variables" is....?

Possibly called "substitutions" in your book, such as
using x = cos(theta).

In helping out my son, one thing that I notice is that the names of the concepts
have changed. So, I have to see what he's doing to understand which concept
he's trying to learn. And, since he has only been exposed to that current
terminology, it took a while to get him to understand that I know the math, even
though I don't know the current gibberish for it.

Probably, there is some terminology I've forgotten, long having been
dissassociated from the actual concept in my mind.

Is it the mathematical terminology, or the terminology
introduced by the educationists to "make it easier"?

Banty



--
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 . com,
Beliavsky wrote:
On Sep 6, 8:57 am, (Beth Kevles) wrote:
Hi --

Having watched two highly-gifted sons take successive math classes, I
actually think that learning the times tables and learning to compute by
hand (pencil and paper) are, in fact, useful skills on the pathway to
learning mathematics, even if they aren't goals in and of themselves.

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. 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.

I agree with you and would add that when a student needs to factor
(x^2 + 10*x + 21) on an algebra test or differentiate f(x)=7*x^3 on a
calculus test, he needs to have 7x3 = 21 MEMORIZED.

I disagree. How hard is it to compute 7x3, which is all
that is needed in the second case; besides, why are so
many of these easy to grade rote problems given instead
of seeing if the students have any understanding?

As for the algebra problem, and again why are so many
of those given, x^2 + 10*x + 21 = (x + 5)^2 - 4 =
(x + 5 + 2) * (x + 5 - 2). If this is not thought
of, it is not too hard to see that any solution must
be of the form (x+a)*(x+b), where a+b = 20 and a*b=21,
and if an integer solution exists, there are not too
many choices for a+b=10, and all except 3 and 7 can
be ruled out quickly.
--
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:

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?
--
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

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.

Best wishes,
Ericka

Best wishes,
Ericka

"Herman Rubin" wrote in message
...
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.

Well, here's an example. My 2 1/2 yr old has a good concept of counting, of
number, and understands that when you put two groups together you get a
bigger number, that you can take items away and have a smaller number, that
you can divide into groups so that multiple care bears each get the same
number of cookies, and similar things. And if you watch her play and listen
to her talk, she demonstrates this easily and coherently. She seems more
than ready for math, right?

However, if you do the test Erika suggests, she'll fail UNLESS she counts.
Similarly, she still fails the test where, if she sees me pour 1/2 cup of
water in two different sized containers, she'll point to the one with the
higher water level as containing more, and doesn't see a problem with saying
that the water is the same in the measuring cups but different in the coffee
mug vs the glass.


Piaget is old-but his stages of development largely still apply. And while a
child (especially a gifted child) will often pick up bits of knowledge and
understanding beyond their years, until they've gotten the developmental
stage down, they're not able to fully apply that knowledge. Some kids do it
faster, some slower, but until they're there, they're not ready.

My gut feeling is that when my DD finally really gets the concept of
differing size and quantity down (which usually happens in the preschool
years), she'll probably race through basic arithmetic-but until then, she's
going to be limited to what she can physically count and manipulate. There
is no way she could handle the concept of a variable until she's got the
abstraction down-and right now, she's simply not there yet. Maybe she will
be by age 5, maybe not.