cover article in Time magazine on gifted education

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.

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.)

In the earlier grades (K-6, roughly ages 5-11) it actually does make
sense for most kids, even most gifted kids, to shapes, computation,
fractions, decimals, basic graphs and plots, etc. That is, lot of
pre-algebra skills. It deepens their sense of numeracy, how numbers fit
together, and so forth. And then when they hit algebra at about age
11-12, it's easy. (That's the age at which they need to learn to sit
down and work through a problem until it's done. Focus and extended
concentration! Or as my grandparent used to say, "sitsfleish".)

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.

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.

--Beth Kevles

http://web.mit.edu/kevles/www/nomilk.html -- a page for the milk-allergic
Disclaimer: Nothing in this message should be construed as medical
advice. Please consult with your own medical practicioner.

NOTE: No email is read at my MIT address. Use the AOL one if you would
like me to reply.

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 ;-)

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....?

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.

Banty

On Sep 6, 8:58 am, Banty wrote:

snip

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 ;-)

I think Mathematica and Maple are the two symbolic math programs with
the biggest market share. But other things being equal it is better
for students to use free tools, so that they always have access to
them in the future. SAGE http://www.sagemath.org/ fits the bill.

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.

Spreadsheets are useful, but they are not intended for symbolic
calculations, and I think they are over-used because they are only
math software many people have bothered to learn.


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....?

To solve an integral (compute the antiderivative) one often makes a
substitution such as y = exp(x).

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.

Banty

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.

In article ,
Ericka Kammerer wrote:
Banty wrote:
In article , Ericka Kammerer
says...
Herman Rubin wrote:
In article ,

There are a few basic ideas in algebra. The most important
one is the LINGUISTIC use of variables. This can, and
should, be taught with beginning reading.
This flies in the face of quite a bit of research
in the area. You can call it linguistic or mathematical
or whatever you wish, but the essential concept of algebra
is a layer of abstraction that kids aren't ready for until
they have reached certain developmental milestones. Flogging
the concept before then is just beating one's head against a
brick wall.

I read him to mean that the idea of a letter referring to a variable should be
introduced with the rest of language. I don't know if it can or not (need to
think on it, maybe there is research), but that's not the same as actually
teaching algebriac concept or algebra manipulation. It goes to what we were
talking about before - that x + y = 5 if x=2 and y=3 as an answer to a homework
not being linguistic enough for elementary teachers, when actually it's
perfectly linguistic.

'=' meaning "same as", perhaps, being taught along with 'cat' meaning them
mice-catching critters. Variables are more 'algebraic' so I think that would
take some wherewithall to understand the abstraction, but kindergarteners know
something can be sometimes big and sometimes small, for instance.

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.

Restricting variables to numbers initially can make
it difficult later to use variables otherwise. I
have been told by someone that he had no problems
with variables for numbers, but when they were used
for functions, he had major problems.

Variables should not be used only for nouns and
pronouns, as they usually are, but for verbs and
other things which are used for communication.
The idea is that a variable can stand for anything,
but only one thing in a given context.

You can introduce some things that push the limits a bit,
but it's only going to go so far. If we're just talking about
the notion that letters or other symbols can stand for numbers,
that seems commonly taught in very early grades (at least from
first grade on, for my kids--and that was before the GT center,
so we're talking in the mainstream classes). Much beyond that
just isn't going to fly developmentally for most until they're
ready for more abstract concepts.

Best wishes,
Ericka


--
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

Banty wrote:
In article , Ericka Kammerer
says...
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
I see no point in teaching for the test. Concepts are
not forgotten once learned, but rote often is.
I don't think that's true (or if it is, it's in
such a limited sense as to be useless). I had lots of
advanced math in college. I don't use much of it in any
regular fashion anymore. I understood the concepts quite
well at the time. While I retain a very basic notion of
what the concepts are, it's certainly not enough to actually
solve any reasonably complex problem. I could spin up again
fairly quickly with a little refresher, but I sure as heck
have forgotten the meat of many of the concepts due to the
simple fact that I haven't used them in nearly 20 years. And,
of course, that's true of any field. If you don't use it,
you lose it--including concepts, if it goes on long enough.

As I said, what you really keep from the concepts is how
to formulate the problems, not how to solve them. Figuring
out how to solve a complex problem is unlikely to be learned,
but must be deduced. Not all are that capable of deduction.

Do you know what limit, derivative, and integral (not
antiderivative) are? Now you can "speak" calculus, even
if you have forgotten all the formulas.
I remember what they are at a very basic level, along
with rings, groups, fields, and assorted theorems associated
with computational theory and so on and so forth. That said,
I would be next to useless in applying that knowledge to problem
solving, beyond perhaps identifying that a solution might have
something to do with one concept or another. In any sort of
practical terms, that knowledge and those skills are inaccessible
to me, without time and resources to spin up on them again.
The things that I use with any regularity are
much more accessible to me. The things that I laid a firm
foundation for with regular practice have remained more
accessible after being neglected, though nothing is a perfect
safeguard given enough disuse.


As an engineer in the microelectronics industry, I regularly use algebra for
spreadsheet computations, statistical analysis (but a lot of that is packaged
into vendor programs, but basic computations I might do for some things), and I
refer to the concepts of Fourier analysis regularly as a lot of what I do refers
to imaging in photolithography. But the associated analysis is very complex and
computationally resource-consuming, and we go to model simulations which have
been developed either in-house or available from a vendor. I refer to the
concepts of calculus regularly. But I'd have to crack open my texts to actually
do much more mathematical manipulation than the algebra and basic stats.

But the concepts I constantly call upon.

I don't even call on most of the concepts with any
regularity, so I really wouldn't claim to even know them
anymore--which is why I contest the notion that one never
forgets a concept. I could perhaps "pick the concept out
of a lineup," but no way no how is it available to me for
anything useful. And it wasn't that I just sort of learned
the stuff and immediately forgot it. I learned it well, and
understood it, and used it for a while. It just isn't really
there anymore after nearly 20 years of disuse. It's been
equally long since I was in the groove of serious musical
practice every single day, but because I periodically use
those skills, I retain most of them.

Best wishes,
Ericka

Best wishes,
Ericka

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

Variables are usually presented as mathematical, with
all the baggage that carries. Present them as linguistic
entities which can stand for anything, give a few examples
which are kindergarten level, and the idea is their.

For example, instead of the rabbit children being named
Flopsy, Mopsy, Cottontail, and Peter, name them a, b, c,
and d or f, m, c, and p, or whatever. THAT has the
essence of variables. Are they not ready for that?

To what end? Where are you going with this?
Yes, some kindergarteners will get that a thing can
have different names. That's different from an abstraction
or the concept of a variable.


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.

A variable is merely a recognizable symbol or collection
of symbols, not used for something else, which has a
fixed meaning in a section of discourse. It is no more
and no less.

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.

In fact, it is those who have been taught through facts
and manipulations who seem unable to understand abstract
ideas at any age.

Where do you have any shred of evidence for this,
particularly with early elementary aged students?

I have seen it in graduate students;
they can calculate, but cannot get the basic ideas.
Unfortunately, basic ideas are NOT taught, because of
the mistaken belief that one has to work up to them.

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? What's to say they weren't
taught these things and just didn't get them the first
several go arounds?

The fact that these have been taught to children.

"Children" covers a lot of territory.

The book by Suppes and Hill has been used to teach
formal logic, which includes variables but not in
a mathematical context, to fifth graders.

There's a lot of developmental change over
the years leading up to and beyond fifth grade. What
a fifth grader can do is very different from what
a third grader can do, or a kindergartener. There's
variation among individuals, of course, but there is
a developmental curve.

The
biggest problem is likely to be Suppes' tendency
to be sesquipedalian. I believe my late wife's
book (for college students) would be easier if
merely some of the exercises were omitted, and
could be done for most no later than third grade.

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.

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.
--
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:

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?

Best wishes,
Ericka

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

I see no point in teaching for the test. Concepts are
not forgotten once learned, but rote often is.

I don't think that's true (or if it is, it's in
such a limited sense as to be useless). I had lots of
advanced math in college. I don't use much of it in any
regular fashion anymore. I understood the concepts quite
well at the time. While I retain a very basic notion of
what the concepts are, it's certainly not enough to actually
solve any reasonably complex problem. I could spin up again
fairly quickly with a little refresher, but I sure as heck
have forgotten the meat of many of the concepts due to the
simple fact that I haven't used them in nearly 20 years. And,
of course, that's true of any field. If you don't use it,
you lose it--including concepts, if it goes on long enough.


As I said, what you really keep from the concepts is how
to formulate the problems, not how to solve them. Figuring
out how to solve a complex problem is unlikely to be learned,
but must be deduced. Not all are that capable of deduction.

Do you know what limit, derivative, and integral (not
antiderivative) are? Now you can "speak" calculus, even
if you have forgotten all the formulas.

I remember what they are at a very basic level, along
with rings, groups, fields, and assorted theorems associated
with computational theory and so on and so forth. That said,
I would be next to useless in applying that knowledge to problem
solving, beyond perhaps identifying that a solution might have
something to do with one concept or another.

That is all that an understanding of the concepts CAN
give. As for solving problems, the concepts can provide
the tools, but not tell how to do it.

For the non-mathematician, the task is to formulate.
There may or may not be algorithms for solving a
particular type of problem, and algorithms can be
memorized and practiced. There are algorithms which
had to be on the back burner until the speed of
computers made them usable, and the algorithmic
process of Tarski for finding all real solutions of
a system of algebraic equations and inequalities for
real numbers has not yet been done, although many
want it.

Teaching algorithms rarely teaches any understanding
of anything except algorithms. To go beyond this
takes native ability, which must not be dragged down
in the educational process to operate.

In any sort of
practical terms, that knowledge and those skills are inaccessible
to me, without time and resources to spin up on them again.

Whether the ability to produce solution procedures is
accessible for you is something I cannot assess.

The things that I use with any regularity are
much more accessible to me. The things that I laid a firm
foundation for with regular practice have remained more
accessible after being neglected, though nothing is a perfect
safeguard given enough disuse.

Do not confuse the ability to solve with understanding.
--
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

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.

In my comment above, I was speaking primarily of
*problem solving*, not just mechanics. If you can solve
a handful of 3-digit addition problems, you can likely
solve a hundred of them. What is questionable is whether
you can look at a variety of different problems and realize
that addition (or whatever else) is the appropriate technique
to use to solve it and that the problem can be formulated
correctly in a solvable form. Often people *think* they
get a concept because they can solve a couple of clearly
defined equations. When they actually have to apply the
concept and determine what out of their repertoire of
techniques is appropriate to use, they find they didn't
understand the concept as well as they thought they did.
There needs to be enough practice in problem solving to
know that the students are correctly identifying how to
solve a variety of problems.

Best wishes,
Ericka