cover article in Time magazine on gifted education

toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

Best wishes,
Ericka

On Sat, 01 Sep 2007 20:23:49 -0400, Ericka Kammerer wrote:

toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

Just thinking from my own experience. Once I got the concept, no more
practice did I do. Problems with a twist didn't seem to be very difficult.
If I didn't understand it, the practice helped.

"Chookie" wrote in message
news:ehrebeniuk-606636.20140602092007@news...
In article ,
(Herman Rubin) wrote:

toypup wrote:

Yes, I find writing is one thing that does improve with practice.
With
math, as long as the concept is understood, more practice in that area
does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others).[...]
I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

If one knows the concept of multiplication and its relation
to addition, leaning the multiplication tables only produces
speed. If one teaches multiplication by way of the tables,
only MISconception is produced.

I think it would be a very good idea after the concept of
multiplication is learned to have the students produce their
own copies of the multiplication table, and point out how it
can speed up the process. But there is no concept there.

ITA. I had to learn my times-tables by rote (as in, "Learn your
three-times
tables for homework". It was extraordinarily difficult and I never really
got
them down properly (I find it difficult to learn something that makes no
sense). Frequent times-tables races and so on did not improve matters;
continually rewriting wrong answers will not improve performance OR speed!

I am glad to see that DS1 is being taught multiplication in a much better
way.
They are learning multiplication as repeated addition (beginning with
concrete
materials and moving to the abstract). There has been no attempt to teach
them by rote, but they are obviously gradually learning some of the
equations,
or whatever they are called, by heart. And in different ways -- they have
found for themselves the equation for the area of a rectangle, for
example, as
well as working out the kids-with-three apples-apiece type problems.

I think you need both-learn the concept first, but then you really need to
learn the facts as well. Having to break every single problem down to
repeated addition, especially if the student still is thinking of addition
as counting, can lead to super slow, tedious calculation. And while some
students get the facts memorized simply by using them, many don't from what
I've seen. And even using a calculator isn't as fast as having the basics
memorized. We've gone so far in the direction of "concept, concept, concept"
that in many cases the students get bogged down on the arithmetic, which
they may understand but haven't truly mastered, when it comes to a harder
problem. In addition, not knowing the facts well often leads to errors in
arithmetic because there's simply more places to mess things up, especially
if you're using touch points or hash marks well into middle school, as quite
a large number of these "conceptually taught, little arithmetic" kids are.

In article ,
Ericka Kammerer wrote:
toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

If one knows the concept of multiplication and its relation
to addition, leaning the multiplication tables only produces
speed. If one teaches multiplication by way of the tables,
only MISconception is produced.

I think it would be a very good idea after the concept of
multiplication is learned to have the students produce their
own copies of the multiplication table, and point out how it
can speed up the process. But there is no concept there.
--
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 ,
(Herman Rubin) wrote:

toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others).[...]
I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

If one knows the concept of multiplication and its relation
to addition, leaning the multiplication tables only produces
speed. If one teaches multiplication by way of the tables,
only MISconception is produced.

I think it would be a very good idea after the concept of
multiplication is learned to have the students produce their
own copies of the multiplication table, and point out how it
can speed up the process. But there is no concept there.

ITA. I had to learn my times-tables by rote (as in, "Learn your three-times
tables for homework". It was extraordinarily difficult and I never really got
them down properly (I find it difficult to learn something that makes no
sense). Frequent times-tables races and so on did not improve matters;
continually rewriting wrong answers will not improve performance OR speed!

I am glad to see that DS1 is being taught multiplication in a much better way.
They are learning multiplication as repeated addition (beginning with concrete
materials and moving to the abstract). There has been no attempt to teach
them by rote, but they are obviously gradually learning some of the equations,
or whatever they are called, by heart. And in different ways -- they have
found for themselves the equation for the area of a rectangle, for example, as
well as working out the kids-with-three apples-apiece type problems.

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

"Parenthood is like the modern stone washing process for denim jeans. You may
start out crisp, neat and tough, but you end up pale, limp and wrinkled."
Kerry Cue

In article , Ericka Kammerer
says...

toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.


Well, the pure *concept* might be understood quickly; it's the application that
is practiced.

Banty

In article , Herman Rubin says...

In article ,
Ericka Kammerer wrote:
toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.

Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

If one knows the concept of multiplication and its relation
to addition, leaning the multiplication tables only produces
speed. If one teaches multiplication by way of the tables,
only MISconception is produced.

I think it would be a very good idea after the concept of
multiplication is learned to have the students produce their
own copies of the multiplication table, and point out how it
can speed up the process. But there is no concept there.

Right. (Good example, BTW.) Concept can be grasped quickly; it's the
application that's practiced. And application before concept is confusing.
Like in high school when I was "taught" trigonometry by opposite-over-hypotnuse,
etc. It wasn't until I picked up a college pre-calculus textbook that had a
simple graphical illustration of the unit circle that I understood all that.
And pretty much in an instant.

Actually creating the multiplication table is a great way to practice the
application and to internalize the concept.

But if those facts aren't at some kind of easy recall, further application is
hindered.

Banty

toypup wrote:
On Sat, 01 Sep 2007 20:23:49 -0400, Ericka Kammerer wrote:

toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.
Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.

Just thinking from my own experience. Once I got the concept, no more
practice did I do. Problems with a twist didn't seem to be very difficult.
If I didn't understand it, the practice helped.

Yet it is the case time and time again that people
who *think* they have the concept *don't*. And while I
have nothing to base this on other than my experience
in my kids' gifted classrooms, it seems that quite a number
of gifted kids are prone to thinking they've got stuff
before they've really thought it through all the way.
Many of them have minds that seem to leap to conclusions,
creating shortcuts that work with a subclass of problems
but not other subclasses of a problem. Or, sometimes they're
just so used to getting things amazingly quickly and so
they assume they have something solidly when they don't
quite.
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. After that, those who need more practice
should do more problems and those who've demonstrated
mastery should move on to more challenging variations.
After all, math is mostly used as a tool. The goal
isn't just to understand the mechanics of a particular
operation. The goal is to be able to solve problems.
There's a nearly endless supply of increasingly challenging
puzzles to solve, and continuing to work on those *does*
improve a person's thinking and problem solving abilities.

Best wishes,
Ericka

Chookie wrote:
In article ,
(Herman Rubin) wrote:

If one knows the concept of multiplication and its relation
to addition, leaning the multiplication tables only produces
speed. If one teaches multiplication by way of the tables,
only MISconception is produced.

I think it would be a very good idea after the concept of
multiplication is learned to have the students produce their
own copies of the multiplication table, and point out how it
can speed up the process. But there is no concept there.

ITA. I had to learn my times-tables by rote (as in, "Learn your three-times
tables for homework". It was extraordinarily difficult and I never really got
them down properly (I find it difficult to learn something that makes no
sense). Frequent times-tables races and so on did not improve matters;
continually rewriting wrong answers will not improve performance OR speed!

I am actually fascinated by this idea that there
are people who were simply taught multiplication tables
as memorization exercises with no discussion of what
multiplication *is*. While I was expected to memorize
my multiplication tables so that I could solve problems
quickly, I certainly was taught what multiplication was
and could have generated a multiplication table anytime
I needed to. My children were all taught the same way.
They knew what multiplication was, and could figure out
a multiplication problem in a number of different way
prior to memorizing their multiplication facts. They
were just slow until they had them memorized. And,
of course, it is *very* painful to do long division or
all sorts of mental arithmetic that one encounters
every day if one doesn't have those multiplication facts
memorized. I fail to see memorizing multiplication facts
as a Bad Thing, though obviously it's important for
children to understand multiplication as well as memorizing
their facts.
It's hard for me to fathom the notion of someone
memorizing a multiplication table without having been
taught multiplication. I certainly wouldn't gainsay
anyone's personal experience in that matter, but is it
really the case that there are loads of classrooms
in which this is the way teachers set out to teach
multiplication (i.e., by rote without actually teaching
what multiplication is)?

Best wishes,
Ericka

Banty wrote:
In article , Ericka Kammerer
says...
toypup wrote:

Yes, I find writing is one thing that does improve with practice. With
math, as long as the concept is understood, more practice in that area does
not improve anything.
Eh, I'm not sure I buy that at all. Even with something
as simple as multiplication tables, practice gets you speed
(which is useful in some cases, not so much in others). When
you get to anything beyond the basics, each problem is a little
puzzle to solve, and there are lots of wrinkles to be thrown
in. Each different perspective broadens one's problem
solving repertoire and one's ability to think about the
issues (up to a point, obviously). I'm not suggesting there's
much value to doing the *same* sort of problem over and over
again once you've got the concept, but it'd have to be a pretty
unimaginative teacher who couldn't come up with nearly endless
varieties of challenges once past the basics.


Well, the pure *concept* might be understood quickly; it's the application that
is practiced.

Yes, but in the practicing, one often finds that there
are nuances to the concept that weren't initially grasped,
especially once one gets beyond the very basics. I'm very
suspicious of this "all you need is the concept" argument,
because time and again I see students who "got" the concept
quickly, but it turns out, they didn't "get" it well enough
to apply it in a wide variety of situations.

Best wishes,
Ericka