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cover article in Time magazine on gifted education



 
 
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  #441  
Old September 9th 07, 01:13 AM posted to misc.kids,misc.education
Rosalie B.
external usenet poster
 
Posts: 984
Default cover article in Time magazine on gifted education

(Herman Rubin) wrote:

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.

I'm not sure what it means to not make a big issue about what it
means.

However .. I had a boyfriend in hs who skipped his senior year in hs
and went directly to Johns Hopkins as a math major. Subsequently by
the time I graduated from college (5 years later), he had his PhD in
math. In his first year he was taking English Comp (required),
French, Physics (which he insisted was just applied math) and four
math courses (two kinds of calculus, some advanced algebra and a name
like Rieman Spaces).

Now I'm absolutely not oriented to higher math at all, but I do
consider that I am somewhat gifted. This boyfriend would come over to
my house to study. I basically had to hand-hold him through English
(type and rewrite his papers), and listen while he complained about
French and Physics, but I had no expectation that I would be able to
help at all with the math. This proved not to be true. He have
something he didn't understand and would try to explain the problem to
me, and I would ask a question (without understanding ANYTHING that he
was talking about), and my question would almost instantly give him
the answer to his problem.

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.

  #442  
Old September 9th 07, 01:38 AM posted to misc.kids,misc.education
Ericka Kammerer
external usenet poster
 
Posts: 2,293
Default cover article in Time magazine on gifted education

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


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.


Yes, I had plenty of probability and stats (and
calculus, and formal logic, and "modern algebra," and
so on and so forth). I got it. I simply have forgotten
much of it, thanks to disuse. I have also forgotten
enough Latin that I can't sit down and read Ceasar easily,
enough music theory that I can no longer easily take
dictation for more complicated chord progressions, and
enough accounting & finance that I'd need to brush up before
attempting any serious valuation of a business. It's
use it or lose it for most of us.

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.


Of course they do. Living in the real world, however,
one does not expect that most, if any, real life studies will
achieve that lofty goal. One does the best one can, and then
one attempts to recognize the limitations on any results imposed
by the failure to achieve perfect methods.

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.


Feel free to question. I have felt quite
comfortable with most of their approaches, and feel
quite comfortable with my children's resulting achievements
in math. Do they all have a background identical to yours?
Probably not, but I don't find that to have been an
impediment. Most of my math teachers in elementary
school didn't have that background either, and it didn't
seem to lead to any serious issues in my learning (or
applying) math either. So, I must say, I'm not terribly
concerned about my children's future at the moment. Now,
I realize that we are blessed with good schools here,
so I'm not suggesting the scenario is as rosy everywhere,
but my experience at this point doesn't really lead me
to put a lot of faith in the notion that the strategies
you advocate are required to teach mathematics.

Best wishes,
Ericka
  #443  
Old September 9th 07, 03:13 AM posted to misc.kids,misc.education
Ericka Kammerer
external usenet poster
 
Posts: 2,293
Default cover article in Time magazine on gifted education

Herman Rubin wrote:
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.


No, it's not. The notion of a variable that
can represent a wide variety of things is a pretty serious
abstraction. At that age, language is much more concrete,
usually representing a 1-1 correspondence between the
word and that which it represents.

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.


I do not for a moment believe that one has to
be able to prove something in order to grasp a concept.
The world is far too full of exceptions to that rule.
I will agree that if you can prove something, you likely
understand something at a higher level, but not that
it is essential to understand everything at that higher
level from the get-go.

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.


Well, I sure as heck didn't produce memorized
proofs, since the proofs I was assigned for homework
hadn't been given to me previously. Seeing as the
neighbor kids seem to have rather similar homework,
at least around here, they still seem capable of
producing novel (to them) proofs.
Now, are there areas where proofs aren't taught
anymore? There may well be. As far as I can tell,
here isn't one of them.

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.


Well, my kids haven't been to calculus yet (nor
have the neighbor kids), so I can't for sure say what
they are teaching in calculus here.

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.


And yet somehow they manage to begin reading and
learning math despite not yet being able to manage more
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.


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.


I have not said anything about how the teaching
of abstract concepts should be approached. I have said
that young children are not ready to deal with abstract
concepts until they have reached a certain point developmentally.
I don't particularly care *how* you attempt to convey
the concept.

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.


I'm familiar, thanks. And note that "symbolic"
and "abstract" are not the same thing.

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.


Again, what is the relevance here? You made a
claim about formal logic:

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.


I said that formal logic was not sufficient for teaching
math (nor do I think it is necessary at the elementary
level) and expressed skepticism that kindergarteners would
hit the ground running with it. Then, you come back with
formal logic not being just the sentential calculus. What's
your point here?

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.


Well, I have no idea what it is like everywhere.
I can tell you what it was like when I was taking geometry
(plenty of proofs, thank you very much). In my county,
proofs are a required part of geometry, according to county
standards (including for non-honors courses).

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


Well, so did I, but Mrs. Montagna was a very old-
fashioned teacher and she did believe in such things.
While it was annoying, I don't think it was particularly
harmful. Every teacher has his or her peccadillos. I'm
willing to spot 'em a few as long as they don't interfere
with the learning.

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.


So far, I have yet to see that that lack of understanding
is pervasive here. Perhaps it is elsewhere. I recall a
study a few years ago comparing advanced high school calculus
students from Japan and the US. IIRC, the both groups of
students performed equally well on more conceptual questions,
but (given that the test did not allow calculators), the
Japanese kids beat the pants off the US kids when it came
to problems requiring more challenging computation (with
many of the US students not being able to solve the problems
at all without a calculator). Doesn't sound like there's
a huge emphasis on plug'n'chug to me.
In addition, isn't the whole controversial "reform
calculus" (and reform math in general) supposed to focus
more on concepts and less on mechanics?

Best wishes,
Ericka
  #444  
Old September 9th 07, 03:18 AM posted to misc.kids,misc.education
Ericka Kammerer
external usenet poster
 
Posts: 2,293
Default cover article in Time magazine on gifted education

Herman Rubin wrote:
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.


A higher level of abstraction means more removed
from the concrete. And young children do not think the
same way adults or older children do, particularly in
terms of how they are able to reason about abstractions.
So, while it may be the case that your assertions hold
true for those who are developmentally able to deal
with abstractions, I seriously doubt it holds true for
those who are not yet at that point.

Even a weak learner can go from theory to practice.


I'm not sure that assertion holds up. Some
studies at least suggest that weaker learners do better
the other way 'round. Personally, I don't have an
opinion there--haven't looked at the issue enough.
*I* prefer to go from theory to practice when learning
known material, but don't know that's representative.

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.


Well, sure. I don't believe anyone has disputed
that. (Though I have suggested that believing one
understands something is not always the same as understanding
something, and thus it is necessary to test one's
understanding through application.)

Best wishes,
Ericka
  #445  
Old September 13th 07, 07:58 AM posted to misc.kids,misc.education
Donna Metler
external usenet poster
 
Posts: 309
Default cover article in Time magazine on gifted education


"Herman Rubin" wrote in message
...
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:


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

No, it's not. The notion of a variable that
can represent a wide variety of things is a pretty serious
abstraction. At that age, language is much more concrete,
usually representing a 1-1 correspondence between the
word and that which it represents.


They do know about pronouns, and the ambiguity in
their use. They also know of ambiguity in common
nouns, and there are quite of few of them such as
boy, girl, table, chair, raindrop, dog, cat, rabbit,
and enough more for them to realize that this is
not the case. They can handle a story in which
rabbits are named Flopsy, Mopsy, Cottontail, and
Peter. How hard is it to get across the idea that
they can have any other set of names.

It is exactly this which can make it difficult. Very young children
overgeneralize. If something's furry, they may call it "doggie" or if they
see a woman, they may call her "mommy".

Later, they add specifics. That doggie is actually a cat, and his name is
Tom. That mommy is Stephen's mommy, and her name is Mrs. Jones. And
everything is in relation to the child.

To a very young child, EVERYTHING is a variable. As they grow up, they start
getting the idea that some things are fixed, can be trusted, can be depended
on and which can't. Because that dog is NOT a cat, and he's not a horse
either, even though all three are items in the set of "furry animals".
Stephen's mommy is not the same as Jamie's mommy or Kevin's Daddy. They're
not interchangable parts. And, no, the world doesn't stop when you're not
there, and no, it doesn't revolve around you.

Until a child is through this stage, which continues into early elementary
school, I don't think the concept of "they can have any name you give them"
is going to have the right effect-because young children already believe
this, and are slowly but surely learning that this ISN'T the case for most
of the things they encounter in day to day life.





  #446  
Old September 13th 07, 07:46 PM posted to misc.kids,misc.education
Herman Rubin
external usenet poster
 
Posts: 383
Default cover article in Time magazine on gifted education

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


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

No, it's not. The notion of a variable that
can represent a wide variety of things is a pretty serious
abstraction. At that age, language is much more concrete,
usually representing a 1-1 correspondence between the
word and that which it represents.


They do know about pronouns, and the ambiguity in
their use. They also know of ambiguity in common
nouns, and there are quite of few of them such as
boy, girl, table, chair, raindrop, dog, cat, rabbit,
and enough more for them to realize that this is
not the case. They can handle a story in which
rabbits are named Flopsy, Mopsy, Cottontail, and
Peter. How hard is it to get across the idea that
they can have any other set of names.

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.


I do not for a moment believe that one has to
be able to prove something in order to grasp a concept.


This is certainly true, and I have made the point quite
often here and on the mathematics and statistics groups.
However, I do not think that one can understand the
concepts of the integers without having some idea of
the simple proofs by induction, even if the details are
not remembered.

The world is far too full of exceptions to that rule.
I will agree that if you can prove something, you likely
understand something at a higher level, but not that
it is essential to understand everything at that higher
level from the get-go.


Being able to prove something does not guarantee the
understanding of the underlying concept or concepts,
although it is more likely than being able to compute
answers. There are, in fact, simple theorems for
which the simple, but not so short, proofs give far
more of an understanding of the theorem than short
proofs using high-powered results, and I am quite
capable of both.

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.


Well, I sure as heck didn't produce memorized
proofs, since the proofs I was assigned for homework
hadn't been given to me previously. Seeing as the
neighbor kids seem to have rather similar homework,
at least around here, they still seem capable of
producing novel (to them) proofs.


That is what the goal of teaching should be. And
often these novel proofs are much better than the
ones previously known.

Now, are there areas where proofs aren't taught
anymore? There may well be. As far as I can tell,
here isn't one of them.


There are, and in many, even if those courses exist,
not all good students get an exposure to it.

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.


Well, my kids haven't been to calculus yet (nor
have the neighbor kids), so I can't for sure say what
they are teaching in calculus here.


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.


And yet somehow they manage to begin reading and
learning math despite not yet being able to manage more
abstract concepts.


Are they learning math? Or are they learning to calculate?

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.


I have not said anything about how the teaching
of abstract concepts should be approached. I have said
that young children are not ready to deal with abstract
concepts until they have reached a certain point developmentally.
I don't particularly care *how* you attempt to convey
the concept.


At what age are children ready to understand the concept
that sequences of symbols can stand for ideas? Or that
one can attack a sequence of symbols by using rules of
pronunciation?

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.


I'm familiar, thanks. And note that "symbolic"
and "abstract" are not the same thing.


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.


Again, what is the relevance here? You made a
claim about formal logic:


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.


I said that formal logic was not sufficient for teaching
math (nor do I think it is necessary at the elementary
level) and expressed skepticism that kindergarteners would
hit the ground running with it. Then, you come back with
formal logic not being just the sentential calculus. What's
your point here?


The predicate arguments are the harder ones. Many logic
books do not even do sentential calculus from scratch, but
use truth tables. This cannot be done for the first-order
predicate calculus. There is a big difference between
being able to understand what a proof is, which should be
required of all, and being able to produce 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.


Well, I have no idea what it is like everywhere.
I can tell you what it was like when I was taking geometry
(plenty of proofs, thank you very much). In my county,
proofs are a required part of geometry, according to county
standards (including for non-honors courses).


In that case, your county is quite unusual.

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


Well, so did I, but Mrs. Montagna was a very old-
fashioned teacher and she did believe in such things.
While it was annoying, I don't think it was particularly
harmful. Every teacher has his or her peccadillos. I'm
willing to spot 'em a few as long as they don't interfere
with the learning.


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.


So far, I have yet to see that that lack of understanding
is pervasive here. Perhaps it is elsewhere. I recall a
study a few years ago comparing advanced high school calculus
students from Japan and the US. IIRC, the both groups of
students performed equally well on more conceptual questions,
but (given that the test did not allow calculators), the
Japanese kids beat the pants off the US kids when it came
to problems requiring more challenging computation (with
many of the US students not being able to solve the problems
at all without a calculator). Doesn't sound like there's
a huge emphasis on plug'n'chug to me.


Quite a few years ago, I taught a probability course with
the full calculus sequence as a prerequisite. This course
satisfied the probability requirement for a teaching major,
as did a lower course with fewer prerequisites, and it was
not intended for those. To make a long story short, on a
take-home part of the final (they never could have handled
it on an in-class exam), only 5 of the 21 such had any idea
how to set up problems involving calculus similar to the
example problems or homework problems, and these were
discussed in detail in class.

In addition, isn't the whole controversial "reform
calculus" (and reform math in general) supposed to focus
more on concepts and less on mechanics?


To do this, you have to go "all out". Doing it part way
achieves little. But I know of no such calculus courses
at the college level; the physicists and engineers want
their students to be able to solve applications using
calculus yesterday.
--
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
  #447  
Old September 13th 07, 08:19 PM posted to misc.kids,misc.education
Ericka Kammerer
external usenet poster
 
Posts: 2,293
Default cover article in Time magazine on gifted education

Donna Metler wrote:
"Herman Rubin" wrote in message
...
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:

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

No, it's not. The notion of a variable that
can represent a wide variety of things is a pretty serious
abstraction. At that age, language is much more concrete,
usually representing a 1-1 correspondence between the
word and that which it represents.

They do know about pronouns, and the ambiguity in
their use. They also know of ambiguity in common
nouns, and there are quite of few of them such as
boy, girl, table, chair, raindrop, dog, cat, rabbit,
and enough more for them to realize that this is
not the case. They can handle a story in which
rabbits are named Flopsy, Mopsy, Cottontail, and
Peter. How hard is it to get across the idea that
they can have any other set of names.

It is exactly this which can make it difficult. Very young children
overgeneralize. If something's furry, they may call it "doggie" or if they
see a woman, they may call her "mommy".

Later, they add specifics. That doggie is actually a cat, and his name is
Tom. That mommy is Stephen's mommy, and her name is Mrs. Jones. And
everything is in relation to the child.

To a very young child, EVERYTHING is a variable. As they grow up, they start
getting the idea that some things are fixed, can be trusted, can be depended
on and which can't. Because that dog is NOT a cat, and he's not a horse
either, even though all three are items in the set of "furry animals".
Stephen's mommy is not the same as Jamie's mommy or Kevin's Daddy. They're
not interchangable parts. And, no, the world doesn't stop when you're not
there, and no, it doesn't revolve around you.

Until a child is through this stage, which continues into early elementary
school, I don't think the concept of "they can have any name you give them"
is going to have the right effect-because young children already believe
this, and are slowly but surely learning that this ISN'T the case for most
of the things they encounter in day to day life.


In addition to that, while there are similarities
between variables and things like pronouns, that doesn't
necessarily make pronouns all that abstract. Young children
are still using pronouns to refer to fairly concrete things--
people, objects, etc. The ability to deal with abstraction
isn't just about understanding that there are "catch all"
names for concrete things that would otherwise be referred to
by a specific name. An abstract concept, on the other hand,
is abstract regardless of how it's referred to.

Best wishes,
Ericka
  #448  
Old September 18th 07, 08:25 PM posted to misc.kids,misc.education
Herman Rubin
external usenet poster
 
Posts: 383
Default cover article in Time magazine on gifted education

In article ,
Donna Metler wrote:

"Herman Rubin" wrote in message
...
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:
Herman Rubin wrote:
In article ,
Ericka Kammerer wrote:


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

No, it's not. The notion of a variable that
can represent a wide variety of things is a pretty serious
abstraction. At that age, language is much more concrete,
usually representing a 1-1 correspondence between the
word and that which it represents.


They do know about pronouns, and the ambiguity in
their use. They also know of ambiguity in common
nouns, and there are quite of few of them such as
boy, girl, table, chair, raindrop, dog, cat, rabbit,
and enough more for them to realize that this is
not the case. They can handle a story in which
rabbits are named Flopsy, Mopsy, Cottontail, and
Peter. How hard is it to get across the idea that
they can have any other set of names.


It is exactly this which can make it difficult. Very young children
overgeneralize. If something's furry, they may call it "doggie" or if they
see a woman, they may call her "mommy".


Later, they add specifics. That doggie is actually a cat, and his name is
Tom. That mommy is Stephen's mommy, and her name is Mrs. Jones. And
everything is in relation to the child.


To a very young child, EVERYTHING is a variable. As they grow up, they start
getting the idea that some things are fixed, can be trusted, can be depended
on and which can't. Because that dog is NOT a cat, and he's not a horse
either, even though all three are items in the set of "furry animals".
Stephen's mommy is not the same as Jamie's mommy or Kevin's Daddy. They're
not interchangable parts. And, no, the world doesn't stop when you're not
there, and no, it doesn't revolve around you.


Until a child is through this stage, which continues into early elementary
school, I don't think the concept of "they can have any name you give them"
is going to have the right effect-because young children already believe
this, and are slowly but surely learning that this ISN'T the case for most
of the things they encounter in day to day life.


Are you sure? This has NOT been tried, and I did not
even try it at all with my first child, and only did
it with my second child later. The difference in their
abilities was great, and they both would have benefited
from programs which went at their mental levels.

The first one got it without my teaching, from the books I
have mentioned before which use difficult vocabulary, when
in kindergarten. The second got in in connection with
learning the order of the alphabet; variables for the
various problems were used in alphabetic order, with no
regard to what was being discussed.

As I have said, the material being used for teaching was
not intended for the age at which they were taught, and
there were other drawbacks with them.

Children can learn abstract ideas if they are taught.
How hard it is to teach them is not at all clear, but
the idea that the process of abstraction, which is
quite difficult, is needed to understand abstract ideas,
is just plain false.

Another point; it is not necessary to be able to prove
theorems, or to carry out calculations, to understand
the underlying principles or the results.





--
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
 




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