cover article in Time magazine on gifted education

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.

Banty

In article , Penny Gaines wrote:
Beliavsky wrote:
On Sep 5, 8:35 am, Banty wrote:

In article .com, Beliavsky
says...








On Sep 3, 10:27 pm, Rosalie B. wrote:

snip


So what? Nobody has been arguing that everybody should use their
fingers when adding. I think the oppurtunity to earn $1 million
would be enough to get anyone practising. But I would be surprised
if the only thing stopping most people from becoming an options floor
trader is their ability to add up "reasonably accurately".

Practice might not be enough. I probably could do it,
but others no, although experience would help.

The people who rely on precise multiplication don't use mental
arithmetic. They use computers or electronic calculators. Even
before electronic means were available, they would use non-electronic
aids.

The last extension of the computation of pi by manual
means was by Dase, doing it for Gauss. Gauss himself
did mental computations which I cannot, but Dase was
much faster, and he computed pi to 205 places, 200
correct.

After that, mechanical and later electromechanical
calculators were used. The last record that way
turned out to be wrong, as the terms of one of the
series beyond the previous point were not included.

--
Penny Gaines
UK mum to three


--
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 ,
Rosalie B. wrote:

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

I have read some research and also done some testing of my own which
shows that students can't grasp abstract ideas until they are ready.
Usually the students that aren't ready have trouble when it come to
algebra. So it wouldn't do any good for most students to give them
algebra earlier than they could actually understand it, and what
happens is that they get frustrated and learn to hate math.

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.

The problem is that they make algebra an abstraction
of arithmetic, and originally introduce only one variable,
with the idea of solving an equation. Making it merely
an extension of arithmetic, for which the concepts are
not known by the children, makes it difficult.

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.

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?

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.

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. 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.

--
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:

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.

--
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 in message
ps.com...
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.

And what do you do when the system is down? Or turning out nonsensical
answers?

I have seen far too many students accept what the calculator says is right
when it's pretty obvious it isn't. Now, these are usually user error, but
regardless, it should immediately set a thought of "something's off here".
And my husband deals regularly with so-called computer programmers who don't
recognize that the answers they're getting are gobbledygook and are going to
cause problems in the field (negative numbers are very possible
mathematically, but if your POS system is telling you that you just
sold -3.34 bananas, something is not right).

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.

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.
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!). A computer no more does the math than my saxophone plays for
itself. In both cases, if the machine is in good repair and operating
condition, in the hands of someone who knows what they're doing, the result
is greater than that the practictioner could reach alone-but if you don't
know what you're doing, it's just an expensive and bulky paperweight.

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.
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

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.

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?

Best wishes,
Ericka

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.

Best wishes,
Ericka

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.

Banty

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.