cover article in Time magazine on gifted education

In article ,
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.

But if only the application is taught, will the concept
ever be learned? Those who have tried to teach mathematical
concepts to teachers have not succeeded.

For one thing, they say that they know how to do arithmetic,
so there is no point in someone teaching them what it means.
They cannot understand that in mathematics things have to be
proved, not just learned, and that there can even be different
concepts for the same objects.

One of my former colleagues, teaching in a summer institute
for high school teachers, commented that at most 10% could
learn "abstract" algebra and foundations of analysis under
any circumstances. This should be a prerequisite, and I
do not mean just passing the courses.



--
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:

Also, does writing too many reports make it difficult to
express things precisely, which requires "mathematical"
notation? This is equivalent to writing declarative
sentences and short paragraphs in English, but in a
language with little fixed vocabulary and strict grammar.

Why on earth would it? Most of my life I have had
to do a great deal of writing, and never had any trouble
with mathematical notation (or proofs, for that matter).
That's not to say that all good writers are also going to
be good at mathematical notation or proofs, but I can't
imagine what about writing would inhibit developing fluency
with mathematical notation. (And, of course, at least in
most expository writing, precision is a virtue.)

Best wishes,
Ericka

Herman Rubin wrote:
In article ,
Rosalie B. wrote:

Of course one can be taught to write reports. It's one of the things
that is normally done in HS English.

It may be on the list of topics, but is it TAUGHT?

Sure. My kids have been in the process of being
taught how to write reports since they were in early elementary.
Each year they learn a bit more.

In fact, CAN it be taught? What I said still goes;
one might be able to teach what is wrong with a
report, but as to how to express oneself, the main
point of a report, we have no idea how to teach it.

Well, my kids' teachers don't seem to have had
much of an issue. Of course there's a level at which
the kids have to come up with their own way of expressing
their ideas, some of which will be better than others, but
there are loads of techniques that can be taught (and are,
at least some places).

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.

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.

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?

Best wishes,
Ericka

In article , Penny Gaines wrote:
Donna Metler wrote:
"Chookie" wrote in message
[snip]
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.

There are really two parts to what is being taught. There is the
conceptual understanding of multiplication, and there is being
able to do practical multiplication sums. These are actually
quite different.

The ability to do practical mental or paper multiplication
is from practical standpoint limited. You would not expect
someone to multiply two hundred-digit numbers without error
in a reasonable length of time. How many would you expect
to multiply two ten-digit numbers manually? One can use
calculators and computers, if the concepts are known.

Over the centuries people have used numerous different techniques
to get the answer to a multiplication sum, some of which promote
more understanding of the underlying mathematics then others.
In addition some of the methods involve just knowing one or two tricks -
how to double - and others require a knowledge of the full
multiplication tables.

For computations involving non-integers, which are the
situation in surveying, physics, engineering, astronomy,
etc., multiplication was, for practical purposes, tedious.
So we have tricks, such as logarithms (for those who do
not know, the logarithm of a product is the sum of the
logarithms) and the derivative slide rule. There is also
Napier's bones, which essentially put the appropriate part
of the multiplication table on a the screen, so that one
can multiply a number by a digit by merely adding, and keep
this up until the usual method of multiplying numbers is
done. So it isn't vital to be able to know the multiplication
tables to do arithmetic.

When most children are expected to learn the multiplication tables by
rote, the underlying assumption is that they will be doing mental/pen
and paper calulations often enough to make it worth while to devote a
lot of educational time to learning the tables.

Even then it may not be necessary. I do a fair amount of
hexadecimal arithmetic manually, as the computer
facilities, while they should be easy to use (the computers
do it directly), are not. Have I learned the hexadecimal
tables? No, I recompute them each time, as I do when using
base 60, the oldest multiplications on record. I do not
know what devices the Sumerians and Babylonians used to do
this, but we know they did.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

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

toypup wrote:

I'm sure this child will be learning the multiplication tables by rote and
he will not know what it means. No one will try to explain it more than
the teacher explains it in class, because they will be so busy trying to
catch him up.

But will that be because the teacher isn't teaching
what multiplication means, or because he just isn't ready
to understand it yet? Obviously, either way is a problem.
I'm just asking whether in all these classrooms where
multiplication is only taught by rote, is it *really* only
taught by rote, or is it just that it's being taught but
for one reason or another the kids aren't getting it? If
there's this argument that kids are somehow being held back
from progressing at an appropriate rate through the math
curriculum, it sort of makes a difference! Honestly, for a
kid who's ready for the concept, multiplication just isn't
all that complicated. It's hard to imagine that teachers
simply aren't covering it at all, and instead are jumping
straight to memorizing multiplication facts with nary an
explanation of the concepts. It may be that there's not
*enough* conceptual work, or that some teachers aren't
conveying the concepts *well*, but are they really not
covering it at all? And if they are covering it but the
kids aren't getting it, then one wonders if it actually
is the case that these kids are ready to progress so much
more quickly than they are....

If the struggles my son had with the way they taught ratios is any indication,
they teach the concepts, but proceed rather quickly to various means and tricks
to get the homework and test answers right.

No, they don't. This definitely should be done with
algebraic notation. One can demonstrate the rational
numbers by introducing a/b with the usual terminology, a is
the numerator and b the denominator. That is, b, which
has to be non-zero, names the "size" of the parts, and a
counts them, so a/b + c/b = (a+c)/b. Now one needs that,
if c is not zero, that a/b = (axc)/(bxc).

Multiplication is a little harder, but by using divided
rectangles, one can show that (a/b)x(c/d) = (axc)/(bxd).

Also, the use of the lowest common denominator should
be deemphasized, and they should learn how to compute
it, both by factoring and by the Euclidean algorithm,
not just by guessing.

I can't swear that my trig teacher never presented the unit circle. But he
pressed and pressed with opposite-over-hypotenuse, etc. Perhaps because he saw
it as the way for most students to handle the material in its application.

This is the classical definition. Just put it in the
unit circle and you get the result. But similarity of
triangles is needed for that, and it might not be taught.

Banty

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

In article ,
Bob LeChevalier wrote:
Ericka Kammerer wrote:
I doesn't seem too much to ask of
students to memorize their multiplication tables, and frankly,
I think it's one of the more commonly used skills. Every day
I run across instances where I'm multiplying in my head to
figure something or another, and I sure would hate to have
to drag out a calculator every time! I'd have to have one
tethered to my hip.

The problem is that memorized things don't always stay memorized.
Oft-repeated story: I used a flash card program with my kids that
recorded what they did. The standard for an "A" in 3rd grade was to
be able to do a worksheet of a 100 problems in 5 minutes and make no
more than 5 errors. My son trained, so that he could do 100 random
flash card problems in under 3 minutes and miss no more than 3, and
usually 0 or 1. When he did this for a week straight, I figured he
had them down. A month later he did poorly on a math test. I had him
do the flashcards and he only got 50% right taking 10 minutes. I had
him train up to 97% again, and a month later he had again dropped back
to 70% (and I never got him to sustain higher than that after a month
of non-practice). About then I also started noticing that he was
counting on his fingers under the table for many problems even when
doing them at the top speed.

Counting on fingers does correspond to the ordinal
characterization of addition.

Memorizing the tables at best gains speed. It shows
nothing about understanding. You are right, that
many memorized things do not stay memorized. This
is a common feature with students at all levels.

One of my former colleagues (not in a mathematical area)
told me of a graduate student who came in to question
his B in a graduate course. After going over the student's
paper, my colleague asked him about one problem where he
got the first part right, how he would proceed. The answer:
"The course was over too weeks ago; I have forgotten."

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

What is important about a course is how the material will
be used many years down the line, not the grade at the end.
We should teach to that, and not assume they are the same.

He later proved equally able to memorize and quickly forget memorized
poems and songs - what he did to songs that he loved but hadn't sung
for a while was sometimes a bit humorous. The only thing he seemed to
be able to memorize and have it stick was music, once he started
playing an instrument.

lojbab


--
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:

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.

Best wishes,
Ericka

In article .com,
Beliavsky wrote:
On Sep 3, 10:27 pm, Rosalie B. wrote:

snip

I'm not sure what I was taught, but I probably never would have
memorized the nines table except that I had to recite it for 'parent's
night' in the third grade.

I do not trust myself to do sums in my head or even on paper. I've
never been able to reliably balance my checkbook. When I had a job
that required multiple calculations, I set up a computer spreadsheet
so I could put in the number from the test equipment and the computer
would apply the formula and give me the right answer. Once I got the
formula right, the computer did it correctly every time. I didn't.

Actually for simple addition I still count on my fingers.

That is a sign of a deficient education, almost as bad as not knowing
the alphabet. I think kids who have not mastered their addition and
multiplication tables should not be permitted to join middle school.
Maybe there should be elementary school exit exams. In a school in
India attended by a niece, children learn the multiplication tables at
age 5.

On the contrary, memorization is not important.

One of the things we have found is that Indians are
excellent at memorization. One of my co-authors,
who is from South India, told me when I asked about
it, that when he was four, he had to memorize the
Vedas in Sanskrit, a language which he did not know
except for a few borrowed words.

For many of them, this produced a block to using
their knowledge except as taught. Learning the
tables implies nothing.

BTW, since this property of Indian learning was
found at a previous university, I have never
given memorization examinations. In addition,
the evidence seems to pile up that memorization
does little, if any, good.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

In article ,
Banty wrote:
In article .com, Beliavsky
says...

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

snip

I'm not sure what I was taught, but I probably never would have
memorized the nines table except that I had to recite it for 'parent's
night' in the third grade.

I do not trust myself to do sums in my head or even on paper. I've
never been able to reliably balance my checkbook. When I had a job
that required multiple calculations, I set up a computer spreadsheet
so I could put in the number from the test equipment and the computer
would apply the formula and give me the right answer. Once I got the
formula right, the computer did it correctly every time. I didn't.

Actually for simple addition I still count on my fingers.

That is a sign of a deficient education, almost as bad as not knowing
the alphabet. I think kids who have not mastered their addition and
multiplication tables should not be permitted to join middle school.
Maybe there should be elementary school exit exams. In a school in
India attended by a niece, children learn the multiplication tables at
age 5.

There should be elementary school exit exams; can they read,
and do they know WHEN to do the arithmetic operations.

I think you miss the point.

If I were to be confronted with a test of the multiplication tables, I would
pass. But I would pass by, for some facts on the table, doing some mental
arithmetic using commutative and associative laws (8 x 9 = 80 - 8 = 72 , for
example), rather than going by recall. My son used his fingers for awhile for
some numbers, I guess Rosalie still does.

Confronted with the need to calculate a tip for a restaurant check or change
from a 20 dollar bill, I sometimes get mixed up, and dont' mind correction.

And I have a PhD in an engineering discipline.

It's not about not knowing one's way around mathematics and arithmetic and their
applications (I know that Rosalie in fact has held at least one job demanding a
lot of that); it's about how one goes about getting to the multiplication facts.
And about how our handy-dandy portable carbon-based computation units have a
certain miscalculation rate.

Banty


My late wife, who was a professor of mathematics and
certainly knew what addition and multiplication meant,
would not even trust herself with balancing the checkbook.
Sometimes she gave me the figures and asked me to add
them, which I did largely mentally.

Mathematicians do not have to be good at arithmetic;
some are, and some are not. I happen to be rather
good, good enough to know that it does not matter.
--
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
says...

Herman Rubin wrote:

Also, does writing too many reports make it difficult to
express things precisely, which requires "mathematical"
notation? This is equivalent to writing declarative
sentences and short paragraphs in English, but in a
language with little fixed vocabulary and strict grammar.

Why on earth would it? Most of my life I have had
to do a great deal of writing, and never had any trouble
with mathematical notation (or proofs, for that matter).
That's not to say that all good writers are also going to
be good at mathematical notation or proofs, but I can't
imagine what about writing would inhibit developing fluency
with mathematical notation. (And, of course, at least in
most expository writing, precision is a virtue.)

Three times in my undergraduate career I encountered people degreed in both
physics and English Literature. Both my first college physics prof and my first
college higher-level math prof had their first degrees in English literature. I
did an expository report on the Second Law of Thermodynamics, thinking it would
fly over my English prof's head. She gave it to her husband, also an English
prof.

Lo and behold he had two degrees in physics, had gone to Cambridge University
for research, and discovered Chaucer (or James Joyce or something like that...).
You never saw a paper marked up the way my 2nd Law Of Thermodynamics paper was -
questions and equations about the case of refrigerations, and stuff like that.
Curled my hair!

I had changed my major from History to Physics at the end of my second year of
college - actually I found it amazing and comforting that those two worlds do
meet and people make amazing changes in direction of their study, just like I
did.

Banty