cover article in Time magazine on gifted education

On 2007-09-03, Banty wrote:
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.

Do you mean that he used the side-length ratios of the special right
triangles (45-45, 60-30, etc.) to calculate the values of the trig
functions, rather than the other way around? How did he handle angles
greater than ninety?

-nP

On Sep 3, 10:33 am, Ericka Kammerer wrote:
wrote:
Multiplicationis repetitive addition of the same number. It's
memorized skip counting. What additional understanding is necessary
before the child begins to memorize? It's not the quadratic formula or
the derivative of a polynomial or sonnet by Shakespeare.

I agree. That's why I'm questioning this notion that
people are "just learningmultiplicationby rote" with no
teaching of the concept. Given that the concept is so simple,
it's hard to imagine someone not bothering to teach it. However,
once you understand the concept, you are neither fast nor
particularly accurate without the memorization (generally
speaking--I'm sure there are some out there who can
skip count or add repetitively without error reasonably
quickly). One could claim that there's no need to knowmultiplicationfacts quickly as long as one can get to the
answer eventually (or use a calculator), but honestly, that
seems a bit petty to me. I doesn't seem too much to ask of
students to memorize theirmultiplicationtables, 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.

Best wishes,
Ericka

I think it's helpful to remember that our memory machine improves when
we train it by memorizing. All the better if what we memorize is
extremely useful, like the single digit multiplication facts. A good
memory, trained through the exercize of memorizing, will bring our
students much satisfaction in the future.

Would you take a look at www.xycharts.com please and give me your
opinion. Thanks.

Joe

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.

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.


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

On Sep 5, 8:35 am, 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.

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.

While you take a few seconds to reason out what 8 x 9 is, and perhaps
get it wrong (a process requiring more steps has a higher chance of
error), someone who has it memorized has progressed to something else.
In many math exams, time is a factor, and the person who has it
memorized will be at an advantage. Yes, if I need to compute 30*29 in
my head I will apply your logic, but IMO the multiplication tables at
least up to 10x10 must be memorized. I will ensure that my kids do so.

I have worked with options floor traders, for whom a key skill is
applying "put call parity" in their heads, which requires FAST and
reasonably accurate addition and subtraction of decimals in their
heads. Some of them have earned more than $1 million a year applying
such skills. They did not do basic addition and subtraction with
their fingers.

In article . com, Beliavsky
says...

On Sep 5, 8:35 am, 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.

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.

While you take a few seconds to reason out what 8 x 9 is, and perhaps
get it wrong (a process requiring more steps has a higher chance of
error), someone who has it memorized has progressed to something else.

Or not. Have you progressed to Fourier transforms and their application to
computer tomography, for example?

In many math exams, time is a factor, and the person who has it
memorized will be at an advantage. Yes, if I need to compute 30*29 in
my head I will apply your logic, but IMO the multiplication tables at
least up to 10x10 must be memorized. I will ensure that my kids do so.

It's an advantage where it's *set up* to be an advantage. Speed drills, for
example.

It's your right to foster your kids having the multiplication tables at their
recall. Indeed, I can't argue against it being a useful skill that needs to be
mastered at least to some extent. And to master it completely by recall is a
Good Thing, as long as the fundamentals are also understood.

But a couple of warnings - firstly you won't be able to distinguish between
recall and mental arithmetic. Secondly, you might be surprised by the
propensities and behavioral tendencies of your kids. Warning: you might get a
stubborn or academically challenged one


I have worked with options floor traders, for whom a key skill is
applying "put call parity" in their heads, which requires FAST and
reasonably accurate addition and subtraction of decimals in their
heads. Some of them have earned more than $1 million a year applying
such skills. They did not do basic addition and subtraction with
their fingers.


Which is fine for them. They would have gone into (or stayed with) that if that
wasn't a skill they had. (And they apply many *other* skills to earn those
seven figure incomes, I assure you!) They're not analysing timing for CMOS
circuits, though ...

Cheers,
Banty

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

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.

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.

While you take a few seconds to reason out what 8 x 9 is, and perhaps
get it wrong (a process requiring more steps has a higher chance of
error), someone who has it memorized has progressed to something else.
In many math exams, time is a factor, and the person who has it
memorized will be at an advantage. Yes, if I need to compute 30*29 in
my head I will apply your logic, but IMO the multiplication tables at
least up to 10x10 must be memorized. I will ensure that my kids do so.

I have worked with options floor traders, for whom a key skill is
applying "put call parity" in their heads, which requires FAST and
reasonably accurate addition and subtraction of decimals in their
heads. Some of them have earned more than $1 million a year applying
such skills. They did not do basic addition and subtraction with
their fingers.

If I have to add fast, and if I practice, I can do it. For instance I
had a job as a cashier when I was 18, and I had to process 200 people
in two hours. I did not have time to add the contents of people's
trays up on the cash register (which was an electrified one that still
had a handle that could be pulled manually so it was quite slow), so I
learned to do it quickly in my head even though they had items that
were 6 cents, 7 cents and 8 cents - not all of them were multiples of
5. I also learned to give change without the little machine that
fast food cashiers have now that calculates it for them.

But when I don't have a need to do that, I don't bother. Why should
I?

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

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.

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.


While you take a few seconds to reason out what 8 x 9 is, and perhaps
get it wrong (a process requiring more steps has a higher chance of
error), someone who has it memorized has progressed to something else.
In many math exams, time is a factor, and the person who has it
memorized will be at an advantage. Yes, if I need to compute 30*29 in
my head I will apply your logic, but IMO the multiplication tables at
least up to 10x10 must be memorized. I will ensure that my kids do so.

The "few seconds" is only relevant if you are doing that type of
calculation on a frequent basis. Keeping up to speed on this type
of puzzle takes a certain amount of time in practising: for most
adults the amount of time spent practising would far exceed the amount
of time saved when they needed to do the calculations.

Further, there is actually quite a high error rate. The researchers
of this paper had a 7.6% error rate in mental multiplication of single
digit numbers:
http://cocosci.berkeley.edu/tom/pape...iplication.pdf

I have worked with options floor traders, for whom a key skill is
applying "put call parity" in their heads, which requires FAST and
reasonably accurate addition and subtraction of decimals in their
heads. Some of them have earned more than $1 million a year applying
such skills. They did not do basic addition and subtraction with
their fingers.

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".

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.

--
Penny Gaines
UK mum to three

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

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

In article ,
Rosalie B. wrote:
Banty wrote:

In article , Bob LeChevalier says...

Beliavsky wrote:

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


I had to take a certain amount of SS in college (when we still had
distribution requirements). I had become somewhat dismissive of SS in
HS, but it was different in college. I didn't take any history or
geography in college because the history course that everyone wanted
to take was a two semester 6 hour class, and would only fulfill one of
the requirements even though it was two semesters. So I took
Economics (4 credits - one semester), and Sociology (4 credits - one
semester) which gave me the two courses worth 8 credits that were
required. (One of the things that we did in the Econ class BTW was
write reports)

A decent economics course will require at least
multivariate algebra.

How many reports should be required? Can one teach how to
write good reports? Frankly, the answer is "no", so all
that one can do is correct reports, and point out why they
are bad. If they are bad because of grammar or similar
errors, teach the errors. If they are bad because the person
does not know how to express clearly, you are likely to be
stuck, as we do not know how to teach it.

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?
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.

But we had them grouped for math and if they were ahead of their peers
in the grade, they went for math to the next grade (or two) up. One
of my children's friends was going over to the HS for Alg II when she
was in 8th grade. (In 7th grade, she went up to the 8th grade for Alg
I). That was unusual, and I don't know what good it did ultimately
because she got to the end of HS math early.

How many times do I have to say that this is still piddling
progress? In mathematics, cutting the time by more than half
for good students is VERY easy, maybe even for ordinary students,
by concentrating on understanding and not training for speed in
imitating machines.

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.

The abstract idea is grammatical, not numerical. Variables
can be used for anything, and should be. They are an
extension of language, excellent for precise communication.

As for solving problems in algebra, the early ones are made
difficult by not giving enough algebra, and by either not
giving reasonable rules or by making it too strict. The
key rule, not just for algebra, is that the same operation
done on equal entities gives equal results.

Abstract ideas are NOT merely abstractions of more concrete
ones, but exist by themselves. Done that way, children
can understand them.

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

With this, science can be greatly speeded as well.

Somewhat the same thing can be said of physics, except it is possible
to teach physics in a more concrete manner.

It was like my mom making me skip to Latin II as a sophomore when
languages were not started in my HS until sophomore year. I finished
Latin III as a junior and then there was no Latin IV for me to take.

What about finishing high school some years earlier?

I was given the option to take a Ford scholarship and skip my junior
and senior year in HS. My dad said I wasn't ready for college yet at
that age (although I could have done it intellectually), and he was
right.

We have had younger children going to college, and not
finding it particularly a problem.

--
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 ,
toypup wrote:
On Sat, 01 Sep 2007 16:45:52 -0400, Rosalie B. wrote:

How many reports should be required? Can one teach how to
write good reports? Frankly, the answer is "no", so all
that one can do is correct reports, and point out why they
are bad. If they are bad because of grammar or similar
errors, teach the errors. If they are bad because the person
does not know how to express clearly, you are likely to be
stuck, as we do not know how to teach it.

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

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. With writing, it gets better with practice.

This is if one can get over the first step. How do you
explain what is obvious to you? If it is not obvious,
you have a better chance.

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.

However, this is what the non-mathematician needs, to
formulate the situation. Teaching solution methods
without formulation is like giving a loaded gun to an
idiot.

I found that with time, I came up with some very good formulas to use in
certain situations and when I pulled them out, the teachers would marvel at
the reports. I think if I had to write more than one of such papers in
class, they would figure out my style, but I usually only had to write one.

Mostly, I did well in writing up labwork and persuasive writing, because I
was a bit unconventional and it always went over well. Maybe the teachers
were tired of reading the standard stuff.


--
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