Is Long Division a Useless Skill?
David Kaufman
Club Weekly for the week of April 25 to May 2, 1995.
Sorry, but I haven't had the time to polish or add flavor to
the previous pages so far posted to make them true first
drafts. But this is an opportunity for others who think what
I have to say is worth such an effort by them.
*********(Page 1 of 1)*********(For Page 8 of 10) ********
Why Waste Time Teaching or Practicing Long Division?
Purpose: How can we get elementary school teachers to stop
wasting their student's precious time learning and
practicing useless skills like long division and long
multiplication.
The practice of these skills is extremely tedious and
serves no practical need in solving any real world problems
because the inexpensive $3 calculator does the job to 8
significant figures in a split second.
Resolution: Elementary school teachers should teach their
students how to mentally divide a one digit number into a 2
digit number and the usual times table.
These 2 skills (of simple multiplication and division)
allow for the estimation of the correct decimal place (and
number size range) of numerous useful problems solved on the
calculator that require multiplication and division.
Even this estimation (to check if we punched in the
correct decimal places on the calculator) could be done by
the calculator also. But I don't recommend this because it
is easy to look and estimate the answer if we know the times
table.
Interest: Numerous useful problems in everyday life and
science require the calculator for multiplication and
division. For example, most constant rate problems.
In the past 11 years, I've probed numerous systems in
math, science and everyday life (simple and complex) solving
endless useful problems on the calculator and I never saw a
need for long multiplication or long division.
So teachers: why waste your time and your student's
time teaching long division and long multiplication?
Nurture: Note: Learning to multiply by one digit or divide
by one digit is easy to do and does give the practice needed
to multiply and divide by one digit.
Students should encourage their teachers to have them
practice dividing 1 digit numbers (or numerals if you
prefer) into 2 digit numerals mentally.
When it comes to doing long division or long
multiplication, students should encourage their teachers to
show them how to multiply and divide on the calculator by
solving useful constant rate problems.
___________________________________________________________
(End of possible page 8)
These constant rate problems which usual require both
long multiplication and long division for their solution are
more commonly known as the solution of a proportion (or 2
equal fractions).
____________________________________________________________
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| Remember: Where there is a will there is a way. |
| |
| Create your 1-10-100 Network by getting 10 people |
| who each get 10 new people for some Public Good. |
| |
| Or just be a connector only so others can focus. |
|__________________________________________________________|
C by David Kaufman, Apr. 30, 1995. (dav...@panix.com.)
Be good. Do good. Be one. And go jolly.
Herman Rubin
Melissa N. Matusevich <mmat...@pen.k12.va.us> wrote:
>don yost (don yost ) writes:
>> I read some research receintly about long division. Teaching long
>> division is not just a waste of time, it is harmfull. If a student
>> learns the algorithm before they learn the concept, they will confuse a
>> skill with understanding and the concept will be difficult or impossible
>> to learn.
>Why throw the baby out with the bath water? Why not allow
>students to discover the allogrithm after they learn the
>concept? One mathematics program which does a great job doing
>this is CSMP [Comprehensive School Mathematics Program]. In
>our school division, studenst who were in the CSMP program in
>grades one through six blew the socks off the algebra prognosis
>test given at the end of sixth grade. These kids have a much
>greater understanding of mathematical concepts than kids taught
>in the traditional manner. Their computation skills are also
>up to snuff. I highly recommend CSMP; it is a *mathematics*
>program, not just arithmetic with a geometry add on.
Computational skills are no more basic to understanding mathematics
than the ability to type is basic to understanding language. This
does not mean that computational skills may not be useful.
Learning computational procedures, and more importantly WHY they
work, based on the coneptual knowledge, is a relatively efficient
way to develop the skills, and also to see their utility.
Mathematical concepts can be learned at an early age, provided the
confusion introduced by calling the acquistion of skills as knowledge
is kept out. But it is difficult for anyone to learn that what has
been taught as mathematics is really only arithmetic, and has nothing
to do with understanding the concepts.
It is very difficult to teach statistical concepts to those who have
learned statistical methods. This includes those with PhDs.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
That's just the way it goes
>>> I read some research receintly about long division. Teaching long
>>> division is not just a waste of time, it is harmfull. If a student
>>> learns the algorithm before they learn the concept, they will confuse a
>>> skill with understanding and the concept will be difficult or impossible
>>> to learn.
I guess I got lucky. I needed to do some division for some reason
while visiting my father in 2nd grade (yes, I was a geek at that
age, even), and so I tried to figure out how many times x went into y.
Well, after I spent a good half hour adding and multiplying x so that
it got big enough for y (I could multiply), dad saw the results, and
decided to teach me how to long divide. When we were taught it in
school the next year, it was a breeze for me, but the point is, I knew
*why* x went into y however many times, because I had taught myself
the concepts, and dad taught me the algorithm.
Wishing he still thought math was fun,
James
Kevin Bruce Pease
Because, despite your opinion, calculators aren't always
readily available.
> The practice of these skills is extremely tedious and
>serves no practical need in solving any real world problems
>because the inexpensive $3 calculator does the job to 8
>significant figures in a split second.
I'll agree with that, provided a calculator is handy. I can't
count the number of times where I've had to do out long division or
multiplication at work, when I'm filling out a return/refund sheet, or
taking inventory. Sometimes, it's more work than it's worth to carry
around a calculator with you. Sometimes, there just isn't a
calculator handy. I'd LOVE to try explaining to a customer "Sorry,
sir, but I can't give you your money back yet, because I can't figure
out what 63.99 divided by 6 is." Or, better yet, "Sorry, sir, but I
can't give you your money back yet, because I can't multiply 12.99 by
8."
Granted, I can do a lot of the math I need at work in my head,
but sometimes, it's easier to do it out on paper. (12.99 x 8 = 13 x 8
- .08 = 80 + 24 - .08 = 103.92) I'm glad I have the skills to do
it. Even though it *is* tedious, it's still a useful skill to have. I
say teach the kids how to do it, and *then*, once they've demonstrated a
proficiency at it, let them use calculators.
----------
Kevin
Kes...@wpi.wpi.edu
"Growing darkness taking dawn, I was me but now he's gone..."
don yost
division is not just a waste of time, it is harmfull. If a student
learns the algorithm before they learn the concept, they will confuse a
skill with understanding and the concept will be difficult or impossible
who can divide, they assume the student also knows the concept, so they
dont teach it. The end result, is that in a high school physics class,
most of my students have never been taught the concept of division and
the have no idea of the concept. "Tell me what things to divide", " I
hate word problems", "what goes first?"
This may seem far fetched, but trust me. Test students on the concept
some time and check it out.
Martin & Anne Kane
don yost <doy...@cello.gina.calstate.edu> wrote:
>I read some research receintly about long division. Teaching long
>division is not just a waste of time, it is harmfull. If a student
>learns the algorithm before they learn the concept, they will confuse a
>skill with understanding and the concept will be difficult or impossible
>to learn.
Well, it never hurt me! :-)
Cheers!
Martin
---------------------------------------------------
Martin & Anne Kane
Jeng Jia Hung
>Jeng:
>
>Do you really think long division is a "basic" skill? It's a neat
trick,
>but I wouldn't call it a "basic" skill. I'd rather have my students
>understand what the meanings of divisions may be, have the ability to
>estimate, etc. It seems perfectly reasonable that division can be
taught
>in combination with multiplication. Then, put more emphasis on
>estimation and mental computation, both strategies and skills, e.g.
258/3
>can be thought of 240/3 and 18/3. Most kids who try to mentally do this
>computation imagining the long division process gets lost. Estimation
>and mental computation are much more powerful "basic" skills than long
>division, and if teaching long division harms students masering more
>powerful skills, isn't it reasonable to put less emphasis on teaching
>long division?
>
>Tad Watanabe
>Towson State University
>Towson, Maryland
I think long division is a basic skill because I've learned it early in
elementary and I've haven't forgotten how. What you're proposing,
however,
is that the children not even know how, and instead, calculate through
estimation or that $3 calculator. I don't think it is right to change the
traditional ways of teaching unless it's actually physically damaging the
kids. This is because Asian countries, and much of the European countries
are using traditional sit-behind-your-desk-and-learn, and are doing much
better at math than the U.S. Don't change because an idea just sounds
plausible.
All I can say is, other countries have taught their long division to
their
elementary schoolers, and on the recent estimation of each countries'
educational worth, those other countries don't seem to be doing too
badly.
--Jeng Jia.
Tad Watanabe
: I think long division is a basic skill because I've learned it early in
: elementary and I've haven't forgotten how. What you're proposing,
: however,
: is that the children not even know how, and instead, calculate through
: estimation or that $3 calculator. I don't think it is right to change the
: traditional ways of teaching unless it's actually physically damaging the
: kids. This is because Asian countries, and much of the European countries
: are using traditional sit-behind-your-desk-and-learn, and are doing much
: better at math than the U.S. Don't change because an idea just sounds
: plausible.
I really don't know too many things schools do that "physically" damage
children. I do know some things schools do that harm children mentally
and intellectually, though. I don't say we should abandon long division
algorithm completely, but I think we can save alot of trouble (for both
teachers and students) if we delay the formal instruction of long
division until after children develop the concepts of operations and a
variety of strategies for estimation and mental computation.
I do agree that we shouldn't change for the sake of changes, but I also
believe it is just as bad not to change "because that's what I've learned
while I was in school."
: All I can say is, other countries have taught their long division to
: their
: elementary schoolers, and on the recent estimation of each countries'
: educational worth, those other countries don't seem to be doing too
: badly.
We have taught long division to our elementary schoolers, and our kids
are not doing too well. So, maybe we should change.
By the way, contrary to common belief, paper and pencil are not always
available for children (or adults) to do long division in a grocery
store. So, the argument that calculators are not always available is not
quite sufficient to argue for teaching paper and pencil method like long
division. The one thing we can be sure that children have with them is
their ability to think. Therefore, mental computation and estimation are
much more "basic" skills that long division.
Tad Watanabe
: we cannot be a society dependent upon calculators. Why would it be
: good to lose yet another basic skill? .....
: --Jeng Jia.
Jeng:
Do you really think long division is a "basic" skill? It's a neat trick,
but I wouldn't call it a "basic" skill. I'd rather have my students
understand what the meanings of divisions may be, have the ability to
estimate, etc. It seems perfectly reasonable that division can be taught
in combination with multiplication. Then, put more emphasis on
estimation and mental computation, both strategies and skills, e.g. 258/3
can be thought of 240/3 and 18/3. Most kids who try to mentally do this
computation imagining the long division process gets lost. Estimation
and mental computation are much more powerful "basic" skills than long
division, and if teaching long division harms students masering more
powerful skills, isn't it reasonable to put less emphasis on teaching
long division?
Tad Watanabe
Joe Keane
``This week we'll divide four-digit numbers by two-digit numbers, but don't
try dividing by three-digit numbers because we haven't gotten to that yet.''
Otherwise, they'd have to teach *understanding* of division, and they can't.
The idea is that kids are little machines, and we want to burn code into them.
Of course they'll still know less than a calculator; how many can even compute
square roots, never mind trig functions? Plus they're a million times slower.
--
Joe Keane, amateur mathematician
Melissa N. Matusevich
> division is not just a waste of time, it is harmfull. If a student
> learns the algorithm before they learn the concept, they will confuse a
> skill with understanding and the concept will be difficult or impossible
> to learn.
Why throw the baby out with the bath water? Why not allow
students to discover the allogrithm after they learn the
concept? One mathematics program which does a great job doing
this is CSMP [Comprehensive School Mathematics Program]. In
our school division, studenst who were in the CSMP program in
grades one through six blew the socks off the algebra prognosis
test given at the end of sixth grade. These kids have a much
greater understanding of mathematical concepts than kids taught
in the traditional manner. Their computation skills are also
up to snuff. I highly recommend CSMP; it is a *mathematics*
program, not just arithmetic with a geometry add on.
Melissa Matusevich
Kevin Bruce Pease
>We have taught long division to our elementary schoolers, and our kids
>are not doing too well. So, maybe we should change.
Maybe it's not the teaching technique, but the attitude of the people
being taught. [shrug]
>By the way, [...] paper and pencil are not always readily
>available for children (or adults) to do long division in a grocery
>store.
No, but it's a lot easier to carry around a pen and a few
pieces of paper than it is to cart around something that is fragile,
like a calculator (let's be honest - kids generally aren't known for
their great care and concern with fragile items, and when you get
right down to it, your average adult can be pretty rough on things
like that, too... I've broken probably about a half dozen calculators
(whether it be the LCD, keys, casing, etc.) since I got "my own" back
in 8th grade (I'm now a sophomore in college). I carry a pen or a
pencil on me at all times... I carry my calculator with me when I need
it.
Call me reactionary, but I think it's a useful skill to have,
irregardless of how available a calculator is.
> So, the argument that calculators are not always available is not
>quite sufficient to argue for teaching paper and pencil method like long
>division.
I know lots of people who have a pen or pencil on them 90% of
the time, and either have some paper, or could get some paper, 90%
of the time. Conversely, I know relatively few (probably 1 or 2)
people who carry a calculator on them 24 (or even 20) hours a day.
I still maintain that teaching children long division is an important
skill that should be taught before they are moved on to letting the
calculator "do the job."
----------
Kevin
Kes...@wpi.wpi.edu
"Growing darkness, taking dawn, I was me but now he's gone..."
Gary Dyrkacz
>From: dav...@panix.com (David Kaufman)
>Subject: Is Long Division a Useless Skill?
>Date: 30 Apr 1995 18:32:04 -0400
> Below is a possible page 8 for a proposed 10 page Cube
>Club Weekly for the week of April 25 to May 2, 1995.
>Sorry, but I haven't had the time to polish or add flavor to
>the previous pages so far posted to make them true first
>drafts. But this is an opportunity for others who think what
>I have to say is worth such an effort by them.
> *********(Page 1 of 1)*********(For Page 8 of 10) ********
> Why Waste Time Teaching or Practicing Long Division?
>Purpose: How can we get elementary school teachers to stop
>wasting their student's precious time learning and
>practicing useless skills like long division and long
>multiplication.
> The practice of these skills is extremely tedious and
>serves no practical need in solving any real world problems
>because the inexpensive $3 calculator does the job to 8
>significant figures in a split second.
>Resolution: Elementary school teachers should teach their
>students how to mentally divide a one digit number into a 2
>digit number and the usual times table.
> C by David Kaufman, Apr. 30, 1995. (dav...@panix.com.)
> Be good. Do good. Be one. And go jolly.
I all to often see people divide numbers like 1000/25 on a calculator, when
it could be done several times faster by just looking at the problem.
Ignoring long division would accentuate the ludicrous of this. Maybe I am one
of the few, but I don't carry a mechanical calculator with me. Frequently, I
need to calculate a number and at least I can resort to long division. I
usually do carry my brain. :-)
Gary
[][][][][][][][][][][][][][][][][][][]
Gary Dyrkacz Hinsdale, Illinois
dyr...@mcs.com
[][][][][][][][][][][][][][][][][][][]
Sara Sinclair
K-12Chatgroup!! It is for KIDS!! We don't [alas.. sniff sniff :'-(] have a
moderator. I'm sure the stuff your about division and whatever else you keep
posting is all grand and swell, but it is NOT very interesting to kids. DON'T
invade our group!! We don't go filling your's with postings about Power
Ranger and Barny assasination plans, or keypal searches, do we?!?!?!?
Hummmmmmm?!?!?!!? Have a little RESPECT!! You always go wining for it from
kids, now it's our turn. So there. :-P
Thanx for cooperating and not making that very annoying error of being
boring around kids again, SABS %-)
Alberto C Moreira
>>>don yost (don yost ) writes:
>>>> I read some research receintly about long division. Teaching long
>>>> division is not just a waste of time, it is harmfull...
>I guess I got lucky. I needed to do some division for some reason
>while visiting my father in 2nd grade (yes, I was a geek at that
>age, even), and so I tried to figure out how many times x went into y.
>Well, after I spent a good half hour adding and multiplying x so that
>it got big enough for y (I could multiply), dad saw the results, and
>decided to teach me how to long divide. When we were taught it in
>school the next year, it was a breeze for me, but the point is, I knew
>*why* x went into y however many times, because I had taught myself
>the concepts, and dad taught me the algorithm.
Gosh, no, I disagree. I don't remember when I learned division, but it
was before I got into first grade; my father taught me. Just like a lot
of other things in mathematics, the concept walks side by side with
the skill, and after a while they're undistinguishable; the concept
becomes irrelevant, and the skill automatic. That, I believe, is the
level of competence that should be aimed at.
It is fundamental a kid knows how to divide well, specially if
he or she is going into one of the mathematical sciences; I can't
imagine how a student could learn physics if he or she can't, for
example, divide two polynomials; and the algorithm is the same as
the algorithm that divides two numbers. Worse, if you get something
like
.314592 / .0007
during a chemistry lab, any mathematically competent high school
student should be able to solve it in his/her head without the
need for a calculator. Otherwise, life in more advanced disciplines
will be miserable; at every corner the teacher will be needing
skills that haven't been cultivated at an earlier age.
I get examples of this lack of cultivated skills every day with my
daughters, pity I didn't jot down some examples to pass back to you.
The consequence of not having the skill is that usually they take twice
as long - or more sometimes - to solve physics or chemistry problems.
How many times I saw my daughter feeding a whole equation into her
calculator, just like her teachers told her, and fumble, fumble, fumble,
in twice the time it would take a skilled student to do it in his/her
head, the answer comes, and more often than not it's wrong. Wrong
because they typed something wrong or because they forgot a decimal
point or put it in the wrong place, or because they don't know the laws
of associativity very well and don't put the parenthesis in the right
places; or because they set the calculator's range wrong and the
answer lacks precision. But more important and down to the point, THEY
DON'T HAVE THE COMPUTATIONAL SKILLS TO DO A SANITY CHECK;
whatever the calculator vomits, out it goes into the sheet; I
call this lottery, not knowledge.
_alberto_
Herman Rubin
Alberto C Moreira <alb...@moreira.mv.com> wrote:
>In article <1MAY1995...@vms1.tamu.edu> jem...@vms1.tamu.edu (That's just the way it goes) writes:
........................
> It is fundamental a kid knows how to divide well, specially if
> he or she is going into one of the mathematical sciences; I can't
> imagine how a student could learn physics if he or she can't, for
> example, divide two polynomials; and the algorithm is the same as
> the algorithm that divides two numbers. Worse, if you get something
> like
> .314592 / .0007
> during a chemistry lab, any mathematically competent high school
> student should be able to solve it in his/her head without the
> need for a calculator. Otherwise, life in more advanced disciplines
> will be miserable; at every corner the teacher will be needing
> skills that haven't been cultivated at an earlier age.
There is a point in learning how the mechanical operations follow from
the theory. But it is only useful, and not essential, to develop
speed in this. Knowing what to do is far more important than knowing
the mechanics.
The problem you have above is "short" division. It is, unfortunately,
usually taught before long division. But short division is merely
long division with the intermediate numbers handled purely mentally.
Now I can do short division with arbitrary two-digit divisors, and
if I really want to, with three-digit ones. This is useful, but not
of major importance. The same holds for the ability to do long
division.
Teach the concepts, and children will want to learn to get the
answers themselves. Teach the methods, and little is accomplished.
Dave Hayden
>>>don yost (don yost ) writes:
>>>> I read some research receintly about long division. Teaching long
>>>> division is not just a waste of time, it is harmfull. If a student
>>>> learns the algorithm before they learn the concept, they will confuse a
>>>> skill with understanding and the concept will be difficult or impossible
>>>> to learn.
>I guess I got lucky. I needed to do some division for some reason
>while visiting my father in 2nd grade (yes, I was a geek at that
>age, even), and so I tried to figure out how many times x went into y.
>Well, after I spent a good half hour adding and multiplying x so that
>it got big enough for y (I could multiply), dad saw the results, and
>decided to teach me how to long divide. When we were taught it in
>school the next year, it was a breeze for me, but the point is, I knew
>*why* x went into y however many times, because I had taught myself
>the concepts, and dad taught me the algorithm.
>Wishing he still thought math was fun,
>James
For another reason why long division should be taught, look no further
than a Pentium processor near you. The Pentium, in case anyone hadn't
heard, had a flaw in its division algorithm.
As a computer programmer, I need an intimate understanding of the
fundamental algorithms if arithmetic. I need to work with numbers in
base 2, 10, 8, 16, and sometimes others. I frequently hear numbers
thrown around (mostly on TV and radio) that, after a little quick
mental math and some common sense, are clearly ridiculous. For these
reasons, I think a firm understanding of the concepts behind
arithmetic are very important.
Dave
--
David Hayden da...@babel.ho.att.com
AT&T Bell Laboratories (908)946-1107
943 Holmdel Road
Holmdel, NJ 07733
Alberto C Moreira
>> .314592 / .0007
>There is a point in learning how the mechanical operations follow from
>the theory. But it is only useful, and not essential, to develop
>speed in this. Knowing what to do is far more important than knowing
>the mechanics.
>The problem you have above is "short" division.
You can replace the numbers by anything you want; the problem
is still the same. The way I learned, the algorithm is called
"division", it doesn't matter how many digits you have where.
>It is, unfortunately,
>usually taught before long division. But short division is merely
>long division with the intermediate numbers handled purely mentally.
>Now I can do short division with arbitrary two-digit divisors, and
>if I really want to, with three-digit ones. This is useful, but not
>of major importance. The same holds for the ability to do long
>division.
It is important if your time is precious and you're trying to use
it in a more profitable occupation than computing a division.
And yet, if it comes out wrong it has the potential to ruin one's
work! The skill is necessary exactly for that reason, so that
an advanced student doesn't waste time with less important
things, and yet he/she doesn't let them get in the way and ruin
the results of the problem or experiment.
>Teach the concepts, and children will want to learn to get the
>answers themselves. Teach the methods, and little is accomplished.
I believe students must learn both. And convert concepts into
skill. Otherwise, every problem is an uphill struggle, and there's
never enough time to tackle the important things because one
keeps tripping in lesser stuff one can't handle.
It's a bit like trying to play Liszt's 6th Hungarian Rhapsody without
be a master of octaves; just the concept isn't enough, it takes that
enormous amount of skill to get through.
_alberto_
Jeng Jia Hung
>I do agree that we shouldn't change for the sake of changes, but I also
>believe it is just as bad not to change "because that's what I've
learned
>while I was in school."
I don't believe it is a bad to change, but not knowing something is
harmful no matter which way you put it.
>By the way, contrary to common belief, paper and pencil are not always
>available for children (or adults) to do long division in a grocery
>store. So, the argument that calculators are not always available is
not
>quite sufficient to argue for teaching paper and pencil method like long
>division. The one thing we can be sure that children have with them is
>their ability to think. Therefore, mental computation and estimation
are
>much more "basic" skills that long division.
Well, you certainly aren't a person who thinks practice makes perfect.
I remember how I hated my math classes because the teachers would always
assign homework. They were tedious to do, yet I remember those same
skills
today. The very act of putting something on paper makes people remember
how to do things. Can't work it out in your head?--use that piece of
paper and everything seems to come easier. Use a calculator, and some
magic number pops up, and you write that down. What concepts do you
learn?
You learn never to leave home without your calculator again.
If you don't think that kids can handle long division at elementary
school,
then children should be taught at an earlier age to better prepare them.
--Jeng Jia.
Graham Pulford
[...]
> Why Waste Time Teaching or Practicing Long Division?
>
>Purpose: How can we get elementary school teachers to stop
>wasting their student's precious time learning and
>practicing useless skills like long division and long
>multiplication.
The idea that learning something like long division is obsolete
just because we can now do it with a calculator or PC is a fallacy
in my opinion. How many other parts of a basic school curriculum would
fall to the axe if we chose to enforce this kind of teaching philosophy?
I know for a fact that many final year students who seek outside
tuition in maths/physics do not have the basic skills in algebra
(eg indices, surds, logs, division...)
to be able to learn the more advanced topics that they seek tuition
for. Algorithmic work like long division and multiplication is a good
discipline and a sound basis for learning. I took a long time to learn
it but it was worthwhile.
Australians and North Americans would probably be surprised to know
that in France, for instance, school students learn about vector
spaces in year 10 (what they call 3rd year). How many people need this
kind of knowledge for their daily lives, you may ask? Not many, but
at least it gives the student an appreciation of the depth of
knowledge required for subjects like maths.
It seems to be a modern trend to replace the hard with the easy simply
because it is apparently more "relevant".
* Graham Pulford g...@mullian.ee.mu.OZ.AU *
* Department of Electrical & Electronic Engineering *
* University of Melbourne, Parkville, VIC 3052 *
* Tel: (+61 3) 344 4752 Fax: (+61 3) 344 6678 *
David K. Davis
: Jeng Jia Hung (jjh...@sfsu.edu) wrote:
: : I think long division is a basic skill because I've learned it early in
: : elementary and I've haven't forgotten how. What you're proposing,
: : however,
: : is that the children not even know how, and instead, calculate through
: : estimation or that $3 calculator. I don't think it is right to change the
: : traditional ways of teaching unless it's actually physically damaging the
: : kids. This is because Asian countries, and much of the European countries
: : are using traditional sit-behind-your-desk-and-learn, and are doing much
: : better at math than the U.S. Don't change because an idea just sounds
: : plausible.
: I really don't know too many things schools do that "physically" damage
: children. I do know some things schools do that harm children mentally
: and intellectually, though. I don't say we should abandon long division
: algorithm completely, but I think we can save alot of trouble (for both
: teachers and students) if we delay the formal instruction of long
: division until after children develop the concepts of operations and a
: variety of strategies for estimation and mental computation.
: I do agree that we shouldn't change for the sake of changes, but I also
: believe it is just as bad not to change "because that's what I've learned
: while I was in school."
: : All I can say is, other countries have taught their long division to
: : their
: : elementary schoolers, and on the recent estimation of each countries'
: : educational worth, those other countries don't seem to be doing too
: : badly.
: We have taught long division to our elementary schoolers, and our kids
: are not doing too well. So, maybe we should change.
: By the way, contrary to common belief, paper and pencil are not always
: available for children (or adults) to do long division in a grocery
: store. So, the argument that calculators are not always available is not
: quite sufficient to argue for teaching paper and pencil method like long
: division. The one thing we can be sure that children have with them is
: their ability to think. Therefore, mental computation and estimation are
: much more "basic" skills that long division.
I wouldn't focus on long division necessarily, but in general I think we've
done a great deal of harm by neglecting paper and pencil calculation.
Arithmetic is the first and most familiar 'formal system' that we learn.
I've seen so many students who are totally devoid of any feeling for number.
Yes, there should be understanding, but in mathematics there is an issue of
technique and dexterity, and I don't see that we can neglect or jump or
arithmetical dexterity.
When I was young there was a method for calculating square roots very similar
to long division that we learned without understanding. I don't remember it
and I don't miss it. But the old ways gave me a feeling for numbers that I've
never totally lost.
And in the case I don't have a calculator, and I have a column of numbers
to add up, I can do it. I add it down, then I add it up, and finally take
the average of the two sums.
Dave D.
Michael K. Murray
Pease) wrote:
>
> No, but it's a lot easier to carry around a pen and a few
>pieces of paper than it is to cart around something that is fragile,
>like a calculator (let's be honest - kids generally aren't known for
>their great care and concern with fragile items, and when you get
>right down to it, your average adult can be pretty rough on things
>like that, too... I've broken probably about a half dozen calculators
>(whether it be the LCD, keys, casing, etc.) since I got "my own" back
>in 8th grade (I'm now a sophomore in college). I carry a pen or a
>pencil on me at all times... I carry my calculator with me when I need
>it.
My calculator is on my watch!
Michael
Michael Murray Fax: 61+ 8 232 5670
Department of Pure Mathematics Phone: 61+ 8 303 4174
University of Adelaide Email: mmu...@maths.adelaide.edu.au
Australia 5005 http://macpure.maths.adelaide.edu.au/mmurray/mmurray.html
Martin Bright
|> I read some research receintly about long division. Teaching long
|> division is not just a waste of time, it is harmfull. If a student
|> learns the algorithm before they learn the concept, they will confuse a
|> skill with understanding and the concept will be difficult or impossible
|> who can divide, they assume the student also knows the concept, so they
|> dont teach it. The end result, is that in a high school physics class,
|> most of my students have never been taught the concept of division and
|> the have no idea of the concept. "Tell me what things to divide", " I
|> hate word problems", "what goes first?"
|>
|> This may seem far fetched, but trust me. Test students on the concept
|> some time and check it out.
Of course, this philosophy could be applied to many other areas of maths as
well. Why bother going through all that boring stuff about triangles and ratios
when you're teaching trigonometry? Students are never going to need that - just
press the 'sin' button and out comes the answer. Who cares whether the square
root of a number, multiplied by itself, gives the original number again? It's
just that funny sort of tick sign on the calculator.
Why should we bother teaching differentiation from first principles? It's easy -
just take the exponent, put it in front, and decrease the exponent by one. No
problem.
Calculators now can even manage integration (by Simpson's rule), sketching
graphs, eigenvalues, matrix inverses, solving systems of linear equations, and
so on. Take one with you everywhere - to the shops, in the bath... and whatever
you do, don't let the batteries run out - you'd be completely stuck.
:-)
--
Martin Bright, Pre-University Employee | martin...@uk.ibm.com
Warwick Development Group, IBM UK | 9BRIGHM at CROVM3
(My views are not necessarily those of my employers, or for that matter of
anybody else apart from myself.)
Rune Aasgaard
> Ignoring long division would accentuate the ludicrous of this. Maybe I am one
> of the few, but I don't carry a mechanical calculator with me. Frequently, I
> need to calculate a number and at least I can resort to long division. I
> usually do carry my brain. :-)
I carry a sliderule to! (But no calculator) :-)
> Gary
Rune
--
Rune Aasgaard, dr.ing. Tel : +47 51805853
Statoil Fax : +47 51805670
N-4035 Stavanger, Norway E-mail : ra...@statoil.no
Shane Story
before I learned the concept - needless to say learning the
concept was not a tremendous stretch of my cognitive
abilities. The concept is as important as the algorithm, and
the algorithm is as important as the concept. In my
opinion the beauty is in the algorithm and not in the concept,
however.
As an aside, no one cares if you can do long division using a
calculator. If you have the
patience and intelligence to crank it out by hand and understand
what you are doing, you have evolved to a higher cognitive state.
If you can do it by hand, you can do it using a calculator with
little extra effort. The converse is not true. The same
is true of graphing and graphing calculators. Pushing buttons
is a very very small part of learning mathematics; algorithms
and concepts are far more important.
As a professional computer scientist/mathematician, the need
to understand both the concept and algorithm is obvious. I might
understand the physics behind a particular differential equation .
I might even utilize the 4th order Runge-Kutta method to find
the solution at a particular value (algorithmic).
In checking my solution, I find that the error is not h**4 but
instead is totally out of whack. With some further study I
discover that the differential equation is stiff, and the
stepsize is too large (concepts). I reduce the stepsize
and all is well. I double check my solution with a multi-step
method (NEW algorithm). Time is short and the boss wants
the answer, so I don't investigate the concepts behind the
various multi-step methods.
Thanks goes to the early mathematics/physicists (Newton, Leibnitz,
Fourier, Euler ...) who approached challenging problems both
algorithmically and conceptually. And thanks to those (Cauchy, Hilbert,
Galois,...) who refined and expanded the old ideas and created
new ones.
-----
sh...@ssd.intel.com
--
Shane Story
Martin Bright
Rubin) writes:
|> In article <3o4t0t$i...@zen.hursley.ibm.com>,
|> Martin Bright <martin...@uk.ibm.com> wrote:
<snip>
|> >Of course, this philosophy could be applied to many other areas of maths as
|> >well. Why bother going through all that boring stuff about triangles and ratios
|> >when you're teaching trigonometry? Students are never going to need that - just
|> >press the 'sin' button and out comes the answer. Who cares whether the square
|> >root of a number, multiplied by itself, gives the original number again? It's
|> >just that funny sort of tick sign on the calculator.
|>
|> What do you think is being taught in high school mathematics these days?
|> That the trigonometric functions are ratios is mentioned in the first
|> week, and thereafter ignored. This even happened in the past, when
|> teaching was better, for this particular subject. After the words, the
|> students memorized algorithms for solving plane and spherical triangles,
|> and the large number of identities, and did not understand them.
<snip> lots more in a similar vein
|> --
|> Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
|> hru...@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
My post wasn't being entirely serious. Maybe I should have made this more
obvious. :-)
Martin
Miriam Levinson
On 30 Apr 1995, David Kaufman wrote:
> Why Waste Time Teaching or Practicing Long Division?
>
>
> Purpose: How can we get elementary school teachers to stop
> wasting their student's precious time learning and
> practicing useless skills like long division and long
> multiplication.
>
> The practice of these skills is extremely tedious and
> serves no practical need in solving any real world problems
> because the inexpensive $3 calculator does the job to 8
> significant figures in a split second.
>
What if your battery dies and there's no sunshine to run your solar cell
and you REALLY need the answer? I have done both long division and
multiplication many times in work environments when I needed an answer
and no consumer electronics were handy at the time. These are basic
skills. Don't worship them, but teach them, say I.
--Miriam Mir...@charm.net
IMRE BOKOR
when cheep calculators ah available. Hoo kneads it?
Just like learning two spell. With spelling chequers
yore better of knot doing it.
Thoroughly Modern Millie
Neal Plotkin
in favor of the following position: it's important for kids to learn how
to do long division by *some* method, but the method doesn't have to be
the standard algorithm (as taught in American schools; I don't know how
it's taught elsewhere). They just have to be able to divide one number
by another *somehow*, with only pencil and paper.
IMHO, the standard long division algorithm is highly unintuitive, makes
very little sense, doesn't seem to relate to anything else, etc., etc.
Compare it to the "long multiplication" algorithm, which is basically a
series of repeated 1-digit multiplications and then putting together the
results. If you think of it this way, you can reconstruct how to do long
multiplication; you don't actually have to remember it (in fact it's
easier this way; you don't have to do any carrying while multiplying,
only while adding).
You could teach long division by trial and error, closing in on the right
answer as you go. You could combine this with estimation, to make the
process shorter.
Maybe you could teach it by repeated subtraction. In fact, if you did
that, maybe elementary school teachers might even find out that the
standard algorithm is actually repeated subtraction with shortcuts; it's
not just a memorized sequence of steps! Imagine that! Maybe they could
even figure out how to teach the standard algorithm in a logical,
sensible manner, and this whole argument would go away!
-
Neal Plotkin nplo...@umich.edu (313)747-4088
University of Michigan Law School
313 Hutchins Hall
Ann Arbor MI 48109-1215
R1S...@vm1.cc.uakron.edu
>wasting their student's precious time learning and
>practicing useless skills like long division and long
>multiplication.
>
worked with their hands. There are many, many occupations that require
a quick and accurate long division (and multiplication). Carpentry comes to
mind. When a carpenter is designing and building there is a lot of
mental calculations involved -- including long divsion. Such long division
may even involve divsion by *TWO* digits. I can't see a carpenter carrying
a calculator in his/her kit, constantly pulling it out to divide 27' 5" by
12; or simple converting inches to feet. In carpentry/layout work such
calculation are made second nature.
Tony Tosca
or else pretty soon there won't be anyone left who knows how to program
those calculators (then again, we could stress recycling calculators as we
"teach" recycling aluminum cans).
To me, this suggestion (that we do away with teaching yet another skill)
is part of a scheme that many in the public education arena have developed
as a clever art form: If students are failing, avoid admitting to that
failure by re-defining success. Shame, shame. Point is that many
students _can't_ do long division, and that's sad.
My opinion only.
Ciao,
Tony
----------------------------------------------------------------------
Anthony V. Toscano Cambria Union School District
1350 Main Street Cambria, CA 93428 805-927-4400 (Voice)
Internet: ato...@telis.org Internet: atos...@netcom.com
Internet: atos...@slonet.org CIS: 71640,2520 AOL: Tony Tosca
eWorld: AToscano GEnie: A.Toscano3 Delphi: AVTOSCANO
----------------------------------------------------------------------
Jay Cummings
: In article <3o1354$1...@panix.com>
: dav...@panix.com (David Kaufman) writes:
:
: >
: >
: >Purpose: How can we get elementary school teachers to stop
: >wasting their student's precious time learning and
: >practicing useless skills like long division and long
: >multiplication.
: >
: Many of the people who have replied to this proposal obviously have not
: worked with their hands. There are many, many occupations that require
: a quick and accurate long division (and multiplication). Carpentry comes to
: mind. When a carpenter is designing and building there is a lot of
: mental calculations involved -- including long divsion. Such long division
: may even involve divsion by *TWO* digits. I can't see a carpenter carrying
That's not long division, unless one really does all that multiplying of
terms and subtraction and bringing down the next digit in one's head,
which would be unusual. It's a facility with quick estimates, possibly
checked by a quick multiplication.
: a calculator in his/her kit, constantly pulling it out to divide 27' 5" by
: 12; or simple converting inches to feet. In carpentry/layout work such
: calculation are made second nature.
And why shouldn't a carpenter use a calculator? Costs a lot less than a
tape measure and weighs less too. Answer: Sometimes it is faster to do
a quick estimate or an easy problem in one's head. But a calculator will
be quicker than long division.
The only question is if learning long division helps a child develop the skills
of quick estimation. I don't know. It's not a useless algorithm; certainly
someone needs to know it. But do kids, when they can learn the method much
more easily as an adult? Is it a better developmental tool than lots of
practice with a calculator? Kids will get benefits from studying almost
anything, even if it's only improved study habits.
Herman Rubin
><doy...@cello.gina.calstate.edu> writes:
>|> I read some research receintly about long division. Teaching long
>|> division is not just a waste of time, it is harmfull. If a student
>|> learns the algorithm before they learn the concept, they will confuse a
>|> skill with understanding and the concept will be difficult or impossible
>|> to learn. Also, since middle and high school teachers expect students
>|> who can divide, they assume the student also knows the concept, so they
>|> dont teach it. The end result, is that in a high school physics class,
>|> most of my students have never been taught the concept of division and
>|> the have no idea of the concept. "Tell me what things to divide", " I
>|> hate word problems", "what goes first?"
>|> This may seem far fetched, but trust me. Test students on the concept
>|> some time and check it out.
This is the case with most of the teaching of mathematics. It was even
observed long ago. That even those adept at the manipulations often did
not have the slightest idea of what anything meant was the impetus behind
the "new math." That it failed was because the teachers were almost
entirely in that group, and could not even learn what anything meant.
>Of course, this philosophy could be applied to many other areas of maths as
>well. Why bother going through all that boring stuff about triangles and ratios
>when you're teaching trigonometry? Students are never going to need that - just
>press the 'sin' button and out comes the answer. Who cares whether the square
>root of a number, multiplied by itself, gives the original number again? It's
>just that funny sort of tick sign on the calculator.
What do you think is being taught in high school mathematics these days?
That the trigonometric functions are ratios is mentioned in the first
week, and thereafter ignored. This even happened in the past, when
teaching was better, for this particular subject. After the words, the
students memorized algorithms for solving plane and spherical triangles,
and the large number of identities, and did not understand them.
>Why should we bother teaching differentiation from first principles? It's easy -
>just take the exponent, put it in front, and decrease the exponent by one. No
>problem.
How do you think calculus is being taught? If it was taught from first
principles, the engineers and physicists and chemists would object that
the students are not getting the formulas fast enough.
The use of the binomial theorem to carry out the expansion MAY be used,
but these other departments are going to complain if it is on the test.
As for obtaining the derivatives by induction, the knowledge of this
cannot be assumed, even for those with undergraduate mathematics degrees.
>Calculators now can even manage integration (by Simpson's rule), sketching
>graphs, eigenvalues, matrix inverses, solving systems of linear equations, and
>so on. Take one with you everywhere - to the shops, in the bath... and whatever
>you do, don't let the batteries run out - you'd be completely stuck.
Calculators can often manage closed form integration. But if the closed
form procedures do not work, not many mathematicians, other than numerical
analysts, can carry out numerical procedures and have some idea of the
accuracy of the results.
Jeff Suzuki
: What if your battery dies and there's no sunshine to run your solar cell
: and you REALLY need the answer? I have done both long division and
: multiplication many times in work environments when I needed an answer
: and no consumer electronics were handy at the time. These are basic
: skills. Don't worship them, but teach them, say I.
That's not a bad idea, but the problem is that the line between
teaching and worshipping gets really blurred.
What's the purpose of teaching mathematics? If we're trying to teach
people "real life" skills, then why not go with the calculator? To
say the calculator might die and you can't get a replacement and you
need the answer NOW is like trying to say that anyone who goes on a
cruise ship has to learn how to survive on a desert island, since the
ship might sink and the radio beacon might go out and you might get
stranded for twenty years.
On the other hand, with calculator in hand, you can do all sorts of
math experiments, involving neat concepts like divisibility,
primality, recursive functions, etc., etc. You might even get one or
two interested in higher mathematics this way, which you wouldn't be
able to do if you say "Today we'll learn how to divide three digit
numbers".
Jeffs
Caroline Pachaud
I once had a math/science teacher who taught me my most valuble lesson in
learning: in order to be able to complete a process, one must fully
understand it. Basically, it's pointless to learn how to do long
multiplication if you don't understand it, because then you'll forget.
That year, we didn't get all the way through our mathbook, but by golly,
we knew exactly what each problem meant, and therefore, knew how to solve
it. Long division and long multiplication are valuble skil because they
help people understand mathematics. In a world like today, we may have
calculators to do the work for us, the same as we have computers that
correct grammar, spelling, and geography, but we still learn it because
society must advance. That doesn't mean that we forget. I must say, I
don't think I'll ever really need to know how to reduce a radical number,
but I know how so that I can add to the knowledge that I already have,
and pass it on to others, so that they (or I) can explore further.
In America, we are trying to upgrade the calibur (sp?) of education so
that we are not left in the dust of a rapidly advancing world. How can
we expect to do this, if we don't even understand *why* three times four
equals twelve? I'll admit, maybe we don't need to have long
division/multiplication in order for us to function, but we *do* need the
comprehension, in order to move ahead. In the post above, Joe said that a
teacher CAN'T teach
*understanding* of division. That is not true. Division is the opposite
of multiplication, which is a slightly more complicated form of
addition. *Understanding* is the key to learning. Maybe the US public
school system thinks that kids only need to memorize to learn, but it's
like learning the alphabet and the phonetic sounds. How do we learn how
to read if we don't understand that certain sounds placed with other
sounds create words? Children are not robots. They need the
comprehension to develop and add to today's knowledge.
There's my bit for the day.
Eleanor
Robert Brault
dy> division is not just a waste of time, it is harmfull. If a student
Bring the doctor in....i've been subjected in my early childhood to long division...and even forbidden to use the calculator (O cruel world !)
8-)))
>>>>>>>>>>>>Robert
Rick Link
>I read some research receintly about long division. Teaching long
>skill with understanding and the concept will be difficult or impossible
>to learn. Also, since middle and high school teachers expect students
>who can divide, they assume the student also knows the concept, so they
>dont teach it. The end result, is that in a high school physics class,
>most of my students have never been taught the concept of division and
>the have no idea of the concept. "Tell me what things to divide", " I
>hate word problems", "what goes first?"
>
>This may seem far fetched, but trust me. Test students on the concept
>some time and check it out.
So what if you don't have a calculator, and have to divide something? I
know it rarely happens, but consider Murphy's Law. It's happened to me, and
am I ever glad that my old fossil education kicked in.
Rick.
--
SNAIL: Combustion Dynamics Ltd. Phone: (403)529-2162
#203, 132 4th Avenue S.E. FAX: (403)529-2516
Medicine Hat, AB T1A 8B5 EMAIL: ri...@combdyn.com
DISCLAIMER: All opinions expressed are mine and *NOT* my employers!
Jeff Suzuki
: For another reason why long division should be taught, look no further
: than a Pentium processor near you. The Pentium, in case anyone hadn't
: heard, had a flaw in its division algorithm.
I think this says that it's more important to spend time doing
estimation and short division than anything else. You're not going to
catch a computer on any small error unless you perform all the
calculations yourself --- and if you do this, why bother with the
computer? But if your pentium say 389,291,382 divided by 3,289 is
42,389, you don't need long division to know this is wrong.
Jeffs
don yost
arguement differed slightly. I think long division is a usefull skill, I
even use it myself. My point is that it should follow the concept, not
be taught before the concept. I teach high school physics, and ALL my
students know long division, but many of them don't know what it means.
They don't understand the concept. They should have been taught what it
means in grade 1-5, then taught the algorithm.
David Kastrup
>Of course, this philosophy could be applied to many other areas of maths as
>well. Why bother going through all that boring stuff about triangles and ratios
>when you're teaching trigonometry? Students are never going to need that - just
>press the 'sin' button and out comes the answer. Who cares whether the square
>root of a number, multiplied by itself, gives the original number again? It's
>just that funny sort of tick sign on the calculator.
Using a calculator, of course, multiplying the square root with itself does
not always give the old number. If your calculator does not handle imaginary
numbers, it will produce errors. In addition, the calculator usually would want
you make think that there is only one square root (even if calculator designers
are by now thinking about branch cuts, they usually decide for one. Multi-valued
expressions are usually not implemented).
The truth of numbers does not lie in calculators, just as the truth of elections
does not lie in polls. You still need a grasp of maths in order to make sense
of the things you can do easier with a calculator, mostly good and fast
numeric approximations.
--
David Kastrup, Goethestr. 20, D-52064 Aachen Tel: +49-241-72419
Email: d...@pool.informatik.rwth-aachen.de Fax: +49-241-79502
Jeff Suzuki
: The idea that learning something like long division is obsolete
: just because we can now do it with a calculator or PC is a fallacy
: in my opinion. How many other parts of a basic school curriculum would
: fall to the axe if we chose to enforce this kind of teaching philosophy?
Speaking of axes, do you (generic) know how to use one to cut down a
tree? Start a fire with flint? Butcher a hog? Not very long ago,
these were considered "essential" skills, and any teacher (which
usually meant parent) worthy of the name passed those skills down to
their students.
On the one hand, long division is a nice skill to know. OTOH, is it
nice enough, and does the prevalence of calculators, justify spending
as much time as is spent on long division? You can't keep students in
ignorance of computing devices forever, and sooner or later, one of
them is going to say, "But why do we need to do it this way, when we
have a calculator?"
To say "Because some day your calculator might break" is specious.
It's like saying that everyone should know how to parachute because an
airplane might have all its engines fail and the wings fall off ---
and students are aware of this. From this, they extrapolate to "Math
teaches me useless skills that I can duplicate on a calculator" ---
which leads to the single most common complaint I've heard about math:
what good is it?
Jeffs
Jimbo
>don yost (don yost ) writes:
>> I read some research receintly about long division. Teaching long
>> division is not just a waste of time, it is harmfull. If a student
>> learns the algorithm before they learn the concept, they will confuse a
>> skill with understanding and the concept will be difficult or impossible
>> to learn.
>
>
>students to discover the allogrithm after they learn the
>concept? One mathematics program which does a great job doing
>this is CSMP [Comprehensive School Mathematics Program]. In
>our school division, studenst who were in the CSMP program in
>grades one through six blew the socks off the algebra prognosis
>test given at the end of sixth grade. These kids have a much
>greater understanding of mathematical concepts than kids taught
>in the traditional manner. Their computation skills are also
>up to snuff. I highly recommend CSMP; it is a *mathematics*
>program, not just arithmetic with a geometry add on.
>
>Melissa Matusevich
Where can I get some more information on this CSMP, is this public domain
information? I'm a homeschool parent always looking for new and
innovating ideas for my children. Any help would be appreciated.
Cheryl Damschen
Tad Watanabe
: Many of the people who have replied to this proposal obviously have not
: worked with their hands. There are many, many occupations that require
: a quick and accurate long division (and multiplication). Carpentry comes to
: mind. When a carpenter is designing and building there is a lot of
: mental calculations involved -- including long divsion. Such long division
: may even involve divsion by *TWO* digits. I can't see a carpenter carrying
: 12; or simple converting inches to feet. In carpentry/layout work such
: calculation are made second nature.
:
I do question whether or not they are indeed doing long division (as long
division algorithm). Rather, I suspect, they are using an invented
algorithm that is different from what we learn in school. For example,
if you are doing 4178/23, I would do something like 23x100=2300, leaving
1878. 23x50=1150 (a half of 2300), so that leaves 728. So far I did
150. 23x20=460 with 268 left. 23x10=230, leaving 38. So one more 23
with 15 remainder. That means I used, 100, 50, 20, 10, and 1, or 181
with 15 remainder. You may think this is rather cumbersome method, but
it appears that this is more "natural" way kids deal with division
situation (by building up through multiplicatio). If kids (and adults,
too) practice this type of thinking, this method can be very efficient, too.
Anyway, just because the situation can be thought of invovling division,
does not necessarily mean the people are indeed using the long division
algorithm.
Tad Watanabe
Towson State University
Towson, Maryland
Herman Rubin
Well said. If they know what it means, then they can at least use a
calculator to get an answer which they can understand. But neither
here nor anywhere else is there much evidence that learning any kind
of technique will give any insight into what is going on.
This was the motivation behind the "new math." The demise of that
was due to the inability of the elementary school teachers, well
versed in teaching arithmetic manipulations, to be able to grasp
the concepts. The concepts are neither the manipulations nor the
proofs.
David Longley
hru...@b.stat.purdue.edu "Herman Rubin" writes:
> In article <3o72lo$6...@cello.gina.calstate.edu>,
> don yost <doy...@cello.gina.calstate.edu> wrote:
<snip>
> >They don't understand the concept. They should have been taught what it
> >means in grade 1-5, then taught the algorithm.
>
> Well said. If they know what it means, then they can at least use a
> calculator to get an answer which they can understand. But neither
> here nor anywhere else is there much evidence that learning any kind
> of technique will give any insight into what is going on.
>
> This was the motivation behind the "new math." The demise of that
> was due to the inability of the elementary school teachers, well
> versed in teaching arithmetic manipulations, to be able to grasp
> the concepts. The concepts are neither the manipulations nor the
> proofs.
> --
> Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
> hru...@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
>
I urge everyone to consider whether it is in fact *possible* to learn
anything which is not in fact an 'effective procedure'. I'm not sure
that our concern to inculcate *understanding* is anything other than
a desire to ensure that the skills can be effectively applied. We now
know for example, that we will get a better grasp of what a MD or any
other professional *knows* by observing his behaviour rather than ask
ing him to give an account of what he is doing (as reviewed in the
series 'Fragments of Behaviour..'). So why are we so keen to get kids
to 'understand concepts' rather than show ythat they can apply them
appropriately?.
My main point is that I no loneger believe that learning is anything
but the acquisition of mechanical skills which can then be applied to
specific contexts, and I think there's a lot of evidence to support
this thesis now. We would naturally wish for *insight* and *understa-
nding*, but I suggest that these are additional behaviours which are
more apposite to the requirements of *teaching* than learning. Such
skills may have little to do with application in fact.....but I have
said this all elsewhere...
--
David Longley
Robert Israel
|>
|> My main point is that I no loneger believe that learning is anything
|> but the acquisition of mechanical skills which can then be applied to
|> specific contexts, and I think there's a lot of evidence to support
|> this thesis now. We would naturally wish for *insight* and *understa-
|> nding*, but I suggest that these are additional behaviours which are
|> more apposite to the requirements of *teaching* than learning. Such
|> skills may have little to do with application in fact.....but I have
|> said this all elsewhere...
Is your posting intended to communicate some insight or understanding of
the topic, or are you just applying some mechanical skills you have
acquired? Just wondering...
Anyway, if you want to put it in behaviouristic terms, I'd be quite happy
with students who are able to use their arithmetic skills in contexts that
are significantly different from those that they have seen before. This is
a skill, but I don't think I'd call it a "mechanical" skill.
--
Robert Israel isr...@math.ubc.ca
Department of Mathematics
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Herman Rubin
David Longley <Da...@longley.demon.co.uk> wrote:
>In article <3o7u0g$1d...@b.stat.purdue.edu>
> hru...@b.stat.purdue.edu "Herman Rubin" writes:
>> In article <3o72lo$6...@cello.gina.calstate.edu>,
>> don yost <doy...@cello.gina.calstate.edu> wrote:
...................
>> >They don't understand the concept. They should have been taught what it
>> >means in grade 1-5, then taught the algorithm.
>> Well said. If they know what it means, then they can at least use a
>> calculator to get an answer which they can understand. But neither
>> here nor anywhere else is there much evidence that learning any kind
>> of technique will give any insight into what is going on.
>> This was the motivation behind the "new math." The demise of that
>> was due to the inability of the elementary school teachers, well
>> versed in teaching arithmetic manipulations, to be able to grasp
>> the concepts. The concepts are neither the manipulations nor the
>> proofs.
>I urge everyone to consider whether it is in fact *possible* to learn
>anything which is not in fact an 'effective procedure'. I'm not sure
>that our concern to inculcate *understanding* is anything other than
>a desire to ensure that the skills can be effectively applied.
It is quite possible to get this understanding in situations in which
there is no hope of applying the skills. At best, one can hope that
in the future they can be applied. But often the understanding is
not only currently useless, but requires computers larger than the
univers to carry out. Nevertheless, the concepts can be quite clear,
and one can then try to find engineering-type approximations.
In teaching elementary service courses in probability, I point out
to the students that there is no way they will compute the solution
to a practical problem. But they have to understand the language,
so that they can communicate the problem to the computer programmer.
And there are places in abstract mathematics where even any kind of
visualization is highly subjective.
Learning the skills is like learning to build an automobile. Few
of us know how to do this; does that mean we cannot understand
the purpose of an automobile and how to use it?
We now
>know for example, that we will get a better grasp of what a MD or any
>other professional *knows* by observing his behaviour rather than ask
>ing him to give an account of what he is doing (as reviewed in the
>series 'Fragments of Behaviour..'). So why are we so keen to get kids
>to 'understand concepts' rather than show ythat they can apply them
>appropriately?.
>
You will not get a grasp of what a theoretical scientist does that way,
or even someone applying knowledge. There is the story about the
trouble-shooter who comes into a non-working factory, looks around,
takes out a wrench, and swats a pipe, and everything is working.
He submits a bill for $2500.00. This is questioned, and then he
submits a detailed bill:
Hitting with the wrench .25
Knowing where to hit 2499.75
You will not observe how he decided where to hit.
>My main point is that I no loneger believe that learning is anything
>but the acquisition of mechanical skills which can then be applied to
>specific contexts, and I think there's a lot of evidence to support
>this thesis now. We would naturally wish for *insight* and *understa-
>nding*, but I suggest that these are additional behaviours which are
>more apposite to the requirements of *teaching* than learning. Such
>skills may have little to do with application in fact.....but I have
>said this all elsewhere...
One can learn concepts, and how to formulate problems, without learning
majipulative skills. One can then build on this. The extent to which
one should acquire strong manipulative skills is not clear, but there
seems to be little basis for assuming that understanding will come from
such acquisition.
Observing how an MD makes a diagnosis will tell you nothing about medicine.
It is the underlying knowledge which is important. And this knowledge has
lots of structure, which both improves the understanding and even the
acquisition of facts.
Alan DeVries
: In article <3o72lo$6...@cello.gina.calstate.edu>,
: don yost <doy...@cello.gina.calstate.edu> wrote:
: >They don't understand the concept. They should have been taught what it
: >means in grade 1-5, then taught the algorithm.
: This was the motivation behind the "new math." The demise of that
: was due to the inability of the elementary school teachers, well
: versed in teaching arithmetic manipulations, to be able to grasp
: the concepts. The concepts are neither the manipulations nor the
: proofs.
I'm not sure we're correct placing all the blame on the teachers. You
might also ask whether the average child in grades 1-5 is capable of
dealing with the concepts abstractly. From the little I know of psychology
I believe children that age are far better at dealing with concrete
problems at that age, memorizing & maniuplating things.
Alan DeVries
David Kastrup
>Miriam Levinson (mir...@charm.net) wrote:
>What's the purpose of teaching mathematics? If we're trying to teach
>people "real life" skills, then why not go with the calculator? To
>say the calculator might die and you can't get a replacement and you
>need the answer NOW is like trying to say that anyone who goes on a
>cruise ship has to learn how to survive on a desert island, since the
>ship might sink and the radio beacon might go out and you might get
>stranded for twenty years.
If cruising was to be an important class subject, teached for 12 years
or more, I would *definitely* want the pupils to be able to handle
emergency cases as well.
If you think that basic calculation skills are not what a school should
teach, because a calculator might be used, then what is school supposed
to teach? We are not teaching just for university, are we? At least
people not studying will be able to help themselves when there
calculator breaks down.
In that case they might be better off knowing long division than they
are solving symbolic integrals.
And don't think that I do not need basic skills (including long division)
not all the time when I m juggling with integrals. It just wouldn't do
if I had to grasp for a calculator for every 1440/12 creeping up. Of
course, you can use shortcuts to solve things like that, but you will
not *notice* the shortcuts available if you never concern yourself
with problems of the kind at all.
It is tedious tasks which get you to learn how to abbreviate them, not
ingeniousness. When you are pondering ingenious things, you have no
time to think for yourself, and few motivation.
And if something as basic as division is involved, I don't think it
can harm to master it.
Way Sun
> Why Waste Time Teaching or Practicing Long Division?
>Purpose: How can we get elementary school teachers to stop
>wasting their student's precious time learning and
>practicing useless skills like long division and long
>multiplication.
> The practice of these skills is extremely tedious and
>serves no practical need in solving any real world problems
>because the inexpensive $3 calculator does the job to 8
>significant figures in a split second.
I always thought students attend a math class to learn math skills
and not calculator skills......hmmmm?
Maybe we should have CAL101: Introduction to Calculators
Soon, someone will write : "Why Waste Time Thinking? Use a Computer!"
Way
Gwen Hyatt
On Tue, 2 May 1995 R1S...@VM1.CC.UAKRON.EDU wrote:
> Many of the people who have replied to this proposal obviously have not
> worked with their hands. There are many, many occupations that require
> a quick and accurate long division (and multiplication). Carpentry comes to
> mind. When a carpenter is designing and building there is a lot of
> mental calculations involved -- including long divsion. Such long division
> may even involve divsion by *TWO* digits. I can't see a carpenter carrying
> a calculator in his/her kit, constantly pulling it out to divide 27' 5" by
> 12; or simple converting inches to feet. In carpentry/layout work such
> calculation are made second nature.
Let's see...12*2 is 24 and 12*3 is 36 so it must be between 2' and 3'.
After the 24 feet are accounted for, 3' 5" remain which is 3*12+5=41".
This is just a little more than 3" when divided by 12. So 2' 3" will be
pretty close. (I could continue for more precision.)
Of course, what I'm doing _is_ long division, but I am able to do this
from understanding the concept of what it means to divide and without use
of the coveted algorithm.
Incidentally, calculator wrist-watches are relatively inexpensive, and a
credit-card calculator fits in a back pocket as easily as a pencil (and
never needs sharpening.)
Gerald Diamond
>> Why Waste Time Teaching or Practicing Long Division?
> Because, despite your opinion, calculators aren't always
>readily available.
>> The practice of these skills is extremely tedious and
>>serves no practical need in solving any real world problems
>>because the inexpensive $3 calculator does the job to 8
>>significant figures in a split second.
I disagree. If you don't know how you'll never have a feel for how bog
or little things should be. Thus every number that comes out of your
calculator or computer will have to be assumed to be correct and heaven
help you when you make an entry error. You will be "divisionally"
illiterate.
It is also very useful to be able to do the calculation to one oor two
places in your head ( or on paper) just to get a feel for whether you are
in the right ball park.
A>I'll agree with that, provided a calculator is handy. I can't
>count the number of times where I've had to do out long division or
>multiplication at work, when I'm filling out a return/refund sheet, or
>taking inventory. Sometimes, it's more work than it's worth to carry
>around a calculator with you. Sometimes, there just isn't a
>calculator handy. I'd LOVE to try explaining to a customer "Sorry,
>sir, but I can't give you your money back yet, because I can't figure
>out what 63.99 divided by 6 is." Or, better yet, "Sorry, sir, but I
>can't give you your money back yet, because I can't multiply 12.99 by
>8."
.....
Amen
>----------
> Kevin
> Kes...@wpi.wpi.edu
--
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+|\
Gerald Diamond | \
Ontario Ministry of Environment and Energy | \
| \ ^. .>
Herman Rubin
>> Why Waste Time Teaching or Practicing Long Division?
>>Purpose: How can we get elementary school teachers to stop
>>wasting their student's precious time learning and
>>practicing useless skills like long division and long
>>multiplication.
>> The practice of these skills is extremely tedious and
>>serves no practical need in solving any real world problems
>>because the inexpensive $3 calculator does the job to 8
>>significant figures in a split second.
>I always thought students attend a math class to learn math skills
>and not calculator skills......hmmmm?
The purpose of mathematics classes should be to learn the concepts,
so that one can intelligently formulate the problem. Only then can
one decide what manipulations to use. If they are simple enough, it
may pay to carry them out by hand. I can multiply 20 digit numbers
by hand if I have to. I did not practice this. Nor do I consider
this as particularly important.
>Maybe we should have CAL101: Introduction to Calculators
>Soon, someone will write : "Why Waste Time Thinking? Use a Computer!"
I suggest you read Wiener's _Cybernetics_. Computers do not think
at any respectible level; they are superfast subimbeciles. People
should not try to be poor imitations of computers, but to think,
to be able to apply what has been learned in unanticipated ways.
Andrew
Long Division is not a useless skill. It helps to be taught how to do
it early, not only to reduce reliance on calculators, but you need long
division to find the roots of cubic polynomials and roots of higher order
equations.
The pages below are utter claptrap!
Andrew
^
> *********(Page 1 of 1)*********(For Page 8 of 10) ********
>
> Why Waste Time Teaching or Practicing Long Division?
>
>
> Purpose: How can we get elementary school teachers to stop
> wasting their student's precious time learning and
> practicing useless skills like long division and long
> multiplication.
>
> The practice of these skills is extremely tedious and
> serves no practical need in solving any real world problems
> because the inexpensive $3 calculator does the job to 8
> significant figures in a split second.
>
> Resolution: Elementary school teachers should teach their
> students how to mentally divide a one digit number into a 2
> digit number and the usual times table.
>
> These 2 skills (of simple multiplication and division)
> allow for the estimation of the correct decimal place (and
> number size range) of numerous useful problems solved on the
> calculator that require multiplication and division.
>
> Even this estimation (to check if we punched in the
> correct decimal places on the calculator) could be done by
> the calculator also. But I don't recommend this because it
> is easy to look and estimate the answer if we know the times
> table.
>
> Interest: Numerous useful problems in everyday life and
> science require the calculator for multiplication and
> division. For example, most constant rate problems.
>
[a lot cut out]
Seeyoulater
Andrew. . .
/\ . .
/\ / \ /\ . .
/ \/\ / \ . . |>>
/ \ \/ \ . . . |
/ \/ \ . . . |
/ \ \ . ...|
------------------------------------------------------------------------
Andrew Dunlop Dept. of Mathematics
Collingwood College Durham University
E-mail: A.J.D...@Durham.ac.uk
[5mWWW page: http://www.dur.ac.uk/~d421e8 [0m
------------------------------------------------------------------------
Chris Lusby Taylor
>
> R1S...@VM1.CC.UAKRON.EDU wrote:
> : Many of the people who have replied to this proposal obviously have not
> : worked with their hands. There are many, many occupations that require
> : a quick and accurate long division (and multiplication). Carpentry comes to
> : mind. When a carpenter is designing and building there is a lot of
> : mental calculations involved -- including long divsion. Such long division
> : may even involve divsion by *TWO* digits. I can't see a carpenter carrying
> : a calculator in his/her kit, constantly pulling it out to divide 27' 5" by
> : 12; or simple converting inches to feet. In carpentry/layout work such
> : calculation are made second nature.
> :
> I do question whether or not they are indeed doing long division (as long
> division algorithm). Rather, I suspect, they are using an invented
> algorithm that is different from what we learn in school.
Why does it matter whether they perform division by multi-digit divisors
by the exact algorithm you learned at school or by some other algorithm?
Surely what matters is that they have a mental method of getting the
correct answer, and if they understand the concept behind the method,
then so much the better because this makes them unlikely ever to forget
it.
What annoys me is that children are taught mumbo-jumbo methods by teachers
who don't know why they work, but merely that they do. These children,
I argue, can't be expected to understand the concepts.
For instance, even before you get to long division, you need to be able
to do long subtraction.
In England this is taught by a variety of mumbo-jumbo methods.
I was taught that one may need to "borrow" 1 from the top number
(diminuend?) but must then "pay back" 1 to the subtrahend
(the bottom number - I remember that one).
E.g.
256
-37
---
Symonds
>I read some research receintly about long division. Teaching long
>division is not just a waste of time, it is harmfull. If a student
>learns the algorithm before they learn the concept, they will confuse a
>skill with understanding and the concept will be difficult or impossible
>who can divide, they assume the student also knows the concept, so they
>dont teach it. The end result, is that in a high school physics class,
>most of my students have never been taught the concept of division and
>the have no idea of the concept. "Tell me what things to divide", " I
>hate word problems", "what goes first?"
>
>This may seem far fetched, but trust me. Test students on the concept
>some time and check it out.
Well I didn't understand all of that but I'm 8 years old and learning long
division and I think it's booooooooooooooooooooooooooring. When I asked my
mother why do I have to do it she said because it's in the book. Any
suggestions about how to make it more interesting?
bye
David Symonds
Sy...@resnet.fmhi.usf.edu
>I read some research receintly about long division. Teaching long
>division is not just a waste of time, it is harmfull. If a student
>learns the algorithm before they learn the concept, they will confuse a
>skill with understanding and the concept will be difficult or impossible
>to learn.
I think you've hit upon a primary issue...should learning "concept" be a
prerequisite to learning a skill. I have taught many adults and many
children how to use various computer programs. If you were to observe a
typical adult, you would see them rubbing their chin, thinking, and staring
at the screen for extended periods before actually trying to type, click a
mouse, or interact with a computer. On the other hand, watch kids, they
grab the mouse and just start clicking here and there. SOme behaviors are
reinforced by something happening with the particular application they are
using, other not reinforced by nothing or a beep occuring after an error.
Guess which group learns the fastest, is more fluent and more likely to
experiement with new applications or try different things within an
application?
While kids make many more errors, they learn the basic princples of an
application much faster than thier adult conterparts who spend time trying
to "understand" how a program works. Kids become much more fluent and are
more likely to generalize skills learned to other programs...
In this case it is more apparent that behavior (Skills) preceeding
conceptualization is a much more effecient learning strategy and seems to
leand to a more integrated conceptualization of learning software
applications.
Forgive the anecdotal nature of this post, but what the heck...My father
taught me Multiplication by rote memorization of tables. For me, when I
mentally multiply numbers I immediately just know the product. Some of the
folks I work with were taught via some conceptual based (translated
"counting") approach. It is always interesting to me to see many of them
have to move their fingers, or silently move their lips while counting to
themselves to figure out simple multiplication problems.
ibo...@metz.une.edu.au
Useless Skill?
From: Alan DeVries, dev...@ripco.com
Date: Wed, 3 May 1995 22:02:59 GMT
In article <D80vx...@rci.ripco.com> Alan DeVries,
dev...@ripco.com writes:
>I'm not sure we're correct placing all the blame on the teachers.
You
>might also ask whether the average child in grades 1-5 is capable of
>dealing with the concepts abstractly. From the little I know of
psychology
>I believe children that age are far better at dealing with concrete
>problems at that age, memorizing & maniuplating things.
>
>Alan DeVries
>
It's lucky children don't "know" as much about psychology as you
do! If they did, they might never learn the difference between the
first, second and third person before completing their BS degrees.
Or they might never learn to count, or the names of colours, or know
what "love" "fear", etc. are. If these concepts and the use of
language
are *not* abstarct, what, pray tell, is?
d.A.
Herman Rubin
>Herman Rubin (hru...@b.stat.purdue.edu) wrote:
>: In article <3o72lo$6...@cello.gina.calstate.edu>,
>: don yost <doy...@cello.gina.calstate.edu> wrote:
>: >They don't understand the concept. They should have been taught what it
>: >means in grade 1-5, then taught the algorithm.
>: This was the motivation behind the "new math." The demise of that
>: was due to the inability of the elementary school teachers, well
>: versed in teaching arithmetic manipulations, to be able to grasp
>: the concepts. The concepts are neither the manipulations nor the
>: proofs.
>I'm not sure we're correct placing all the blame on the teachers. You
>might also ask whether the average child in grades 1-5 is capable of
>dealing with the concepts abstractly. From the little I know of psychology
>I believe children that age are far better at dealing with concrete
>problems at that age, memorizing & maniuplating things.
The approach was tested for several years with mathematicians, and some
teachers who knew mathematics, doing the teaching. Your view of what
children can do very definitely was the prevailing view among educationists
at that time, and it could not have been generally introduced without clear
evidence that the children could learn it. My own opinion is that it was
nowhere near abstract enough; one can base properties of the integers with
full mathematical precision on counting alone, and at the level of these
children.
Everything I have personally seen in mathematics and statistics indicates
that knowing manipulation makes it very much harder to learn concepts.
As the adage goes,
It ain't what you don't know that hurts you;
It's what you know that ain't so.
Try teaching statistical concepts to a college student who has had a
statistical methods course, or try teaching the general concept of an
integral to someone who has had a cookbook calculus course, and the
problem becomes clear. Try teaching the fundamental idea of variables
to the majority of high school mathematics teachers, and the problem
will clearly show itself.
But it was a complete surprise that few of the teachers could handle it.
After both lowering the level and attempting to educate the teachers, it
finally was dropped.
Herman Rubin
David Longley <Da...@longley.demon.co.uk> wrote:
>In article <3o8tog$11...@b.stat.purdue.edu>
> hru...@b.stat.purdue.edu "Herman Rubin" writes:
>> >My main point is that I no loneger believe that learning is anything
>> >but the acquisition of mechanical skills which can then be applied to
>> >specific contexts, and I think there's a lot of evidence to support
>> >this thesis now. We would naturally wish for *insight* and *understa-
>> >nding*, but I suggest that these are additional behaviours which are
>> >more apposite to the requirements of *teaching* than learning. Such
>> >skills may have little to do with application in fact.....but I have
>> >said this all elsewhere...
>> Observing how an MD makes a diagnosis will tell you nothing about medicine.
>> It is the underlying knowledge which is important. And this knowledge has
>> lots of structure, which both improves the understanding and even the
>> acquisition of facts.
.....................
>At a guess, the reason so many educated people want to go for abstract
>skills is because they get impatient trying to teach all of the mechanical
>steps - nowhere is this clearer than in mathematics and logic....yet there
>is no excuse for intutive leaps here - all of the procedures are in fact
>'effective'. hence my albeit cryptic remarks about abstraction being a
>teacher's 'crutch' (I'm as guilty in practice as anyone else of course!..
>good teaching is *hard* work.
What are you even hinting at? Many of the supposedly educated people
have never had an abstract course. There are skills in dealing with
abstract ideas, but what one gets are concepts. But the entire present
curriculum avoids concepts, and as for abstract courses, they do not
exist. There may be formalisms, but they are taught in the form of
skill acquisition as anything else; they are not abstract at all.
One can also develop effective procedures, and understand their
limitiations, if one has the concepts first. The time to learn
manipulations is when the concepts are already there; otherwise
the manipulations are pure training, not education.
Keith O'Rourke
<At a guess, the reason so many educated people want to go for abstract
<skills is because they get impatient trying to teach all of the mechanical
<steps - nowhere is this clearer than in mathematics and logic....yet there
<is no excuse for intutive leaps here - all of the procedures are in fact
<'effective'.
It might be interesting to change the discussion to University level.
As a motivating example think of differentiation in R^n
Spivak M Calculus on Manifolds - almost half the space devoted to the
chain rule versus partial derivatives
Note his important comment "Finding Df(x) [by chain rule], however may be
a fairly formidable task [pre-computer algebra days]. .. Fortunately, we
will soon discover a much simpler method of computing Df(x). [partial
derivative approach]
Even with this text, most courses would tend to focus energy and TESTING on
the much simpler method of computing. The role of the chain rule gets
lost?
Then look at Marsden and Tromba Vector Calculus -- the Chain rule is in
the starred (for omission) chapter "some technical diff theorems".
I believe in mathematics at all levels there is this tendency to concentrate
on calculations (including formal proofs as calculations) rather than the
"roles that the concepts play".
Perhaps it is hard to know when this can be taught and to whom?
Keith O'Rourke
The Toronto Hosp.
Angela C. Schmidt
As a "university level" student, I feel that I shoule say something here.
Our school has been working out a program to find better ways to teach
(and learn) calculus. It is a controversial subject with the MAth dept.
I went through a program (most of it failed) where we were taught the
concepts and applications on the computer using a text based around M
Mathematica software. It was very good at teaching the concepts and
uses of all those things that we were learning to do. Problems were
taken from textbooks in varoius engineering fields (most of us are studing
one sort of engineering or another) and we were shown how to use what we
had learned.
That part worked and gave me a much better understanding of the concepts.
But, I still was not very good at doing the things on paper. That is
where our program failed. There was no practice on paper. When I got to
other classes that expected me to know how to do these things, I was in
trouble. I have since taken a traditional class and now have a better
handle than many of my peers on the concepts of what we are doing in
our classes.
Angela
Bobincio Batiller Pasion
: As a "university level" student, I feel that I shoule say something here.
: Angela
Personally, I think that only by repeatedly doing problem after
problem will you understand the concepts of any mathematical
theory... I too am a university student, and have had more than
my share of math that I didn't like. You _must_ do the problems
to understand the concepts. _Especially_ when dealing with
multi-variate calculus or eigenvalue subspaces of differential
equations (Ouch!). You can easily find a solution for mechanical
problems that actually deal with _actual_numbers_, but when asked
to prove an abstract theory, you need the practice.
MY 2cents
Bobby
-------------------------------------------------------------------------------
Login name: bbpasion In real life: Bobincio Pasion
Plan: E-Mail: pas...@cpsc.ucalgary.ca
not FLUNK OUT bbpa...@acs.ucalgary.ca
..::''''::..
.;'' ``;. DEEP THOUGHT:
:: :: :: :: "It takes a big man to cry...
:: :: :: :: but it takes a BIGGER man,
:: .:' :: :: `:. :: to LAUGH at that man."
:: : : :: Jack Handey
:: `:. .:' ::
`;..``::::''..;'
``::,,,,::''
David Longley
isr...@math.ubc.ca "Robert Israel" writes:
> In article <799514...@longley.demon.co.uk>, David Longley
> <Da...@longley.demon.co.uk> writes:
>
> Is your posting intended to communicate some insight or understanding of
> the topic, or are you just applying some mechanical skills you have
> acquired? Just wondering...
>
> Anyway, if you want to put it in behaviouristic terms, I'd be quite happy
> with students who are able to use their arithmetic skills in contexts that
> are significantly different from those that they have seen before. This is
> a skill, but I don't think I'd call it a "mechanical" skill.
I've reviewed the evidence bearing on this in a series entitled 'Fragments
of Behaviour ...25/4/95'. All effective procedures are mechanical skills..
and yes, if you take the *OR* in your first paragraph as inclusive, you
will have taken my point.
--
David Longley
David Longley
> >but the acquisition of mechanical skills which can then be applied to
> >specific contexts, and I think there's a lot of evidence to support
> >this thesis now. We would naturally wish for *insight* and *understa-
> >nding*, but I suggest that these are additional behaviours which are
> >more apposite to the requirements of *teaching* than learning. Such
> >skills may have little to do with application in fact.....but I have
> >said this all elsewhere...
>
<snip>
>
> Observing how an MD makes a diagnosis will tell you nothing about medicine.
> It is the underlying knowledge which is important. And this knowledge has
> lots of structure, which both improves the understanding and even the
> acquisition of facts.
> --
> Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
> hru...@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
>
A good methods section in a scientific paper should let a colleague do all
that is required to replicate the experiment......
Well, what I was referring to was the literature on actuarial vs clinicial
decision making. If one builds a model from the signs & symptoms an MD
*uses* along with the presence or absence of the illness (ie a logistic
regression model) one invariably finds that the actual regression weights
do not bear a very good relation to reported weights. That is, it has been
widely reported that it's better to observe what they do, rather than take
too seriously what they claim they do when making a diagnosis.
Nisbett & Wilson (1977) published a classic in this field entitled 'Telling
more than we can Know: Public Reports on Private Processes' (Psych Rev.).
I think this is why 'Expert Systems' are so hard to build - the 'knowledge
enginee s think that they just have to ask experts - but ironically, the
experts just don't seem to know.
Fortunately for nearly all of us, when we give an account of ourselves, we
can generally rest assured that our verbal accounts are often way out of
sync with what we actually do......
At a guess, the reason so many educated people want to go for abstract
skills is because they get impatient trying to teach all of the mechanical
steps - nowhere is this clearer than in mathematics and logic....yet there
is no excuse for intutive leaps here - all of the procedures are in fact
'effective'. hence my albeit cryptic remarks about abstraction being a
teacher's 'crutch' (I'm as guilty in practice as anyone else of course!..
good teaching is *hard* work.
--
David Longley
David Longley
> Herman Rubin (hru...@b.stat.purdue.edu) wrote:
> : In article <3o72lo$6...@cello.gina.calstate.edu>,
> : don yost <doy...@cello.gina.calstate.edu> wrote:
> : >They don't understand the concept. They should have been taught what it
> : >means in grade 1-5, then taught the algorithm.
> .
> : This was the motivation behind the "new math." The demise of that
> : was due to the inability of the elementary school teachers, well
> : versed in teaching arithmetic manipulations, to be able to grasp
> : the concepts. The concepts are neither the manipulations nor the
> : proofs.
>
> I'm not sure we're correct placing all the blame on the teachers. You
> might also ask whether the average child in grades 1-5 is capable of
> dealing with the concepts abstractly. From the little I know of psychology
> I believe children that age are far better at dealing with concrete
> problems at that age, memorizing & maniuplating things.
>
> Alan DeVries
>
>
I think you'll find that true of people of any age. In 'Fragments...' I made
a start at pointing to a reseach backlash against the teaching of abstract
principles...elsewhere I have suggested that there is a danger of confusing
the skills we teach, teach, learn and practice (at any age) & how we *talk*
about those skills.
--
David Longley
ibo...@metz.une.edu.au
Da...@longley.demon.co.uk writes:
>
>I urge everyone to consider whether it is in fact *possible* to
learn
>anything which is not in fact an 'effective procedure'.
There goes the Axiom of Choice!
There goes the Intermediate Theorem of Calculus!
There goes the Fixed Point Theorem for contracting
maps in a complete metric space!
There goes the Fundamental Theorem of Algebra!
Whew!! That sure makes maths easier!!!
d.A.
Jeff Suzuki
: I disagree. If you don't know how you'll never have a feel for how bog
: or little things should be. Thus every number that comes out of your
: calculator or computer will have to be assumed to be correct and heaven
: help you when you make an entry error. You will be "divisionally"
: illiterate.
: It is also very useful to be able to do the calculation to one oor two
: places in your head ( or on paper) just to get a feel for whether you are
: in the right ball park.
But this isn't "long division". Long division would be how you get
the answer for 38925 divided by 1963. Short division would say
"that's about 40,000 divided by 2000, so it should be _about_ 20."
The problem is, students who know long division DON'T have a feel for
whether they're in the right ball park. "My answer is 32.895, but the
book says 32.894. What's wrong?" is a fairly common question I hear.
Or, even worse, "I divided 7832 by 32, and got 12. Now we take the 12
and..." A lot of my students can't estimate, and I get blank looks
when I do.
Jeffs
Jeff Suzuki
: Sew ah yew saying that wee don't knead too lean two
: spell because their ah spelling chequers? Ore do ewe
: think it ay waist of thyme too learn to right because
: wee have typwriters?
Spell checkers have not gotten to the point where they can replace
good spelling; calculators _have_ gotten to the point where they can
replace most human calculation. Ask me in twenty years if I think we
need to spend time teaching "i before e except after c, except when
pronounced 'ay', as in neighbor and weigh, and about two dozen
exceptions..."
: You are confusing arithmetic with mathematics. While arithmetic
: skills seem to be indispensible to mathematics, they do not comprise
: it.
I'm not the one who's confusing them. What do students call things
like long division? "Math". Most people when they say "I hate math"
mean "I hate arithmetic". I don't blame them.
: >which leads to the single most common complaint I've heard about
: math:
: >what good is it?
: The same is true of almost any school subject. What is the use of :
: learning geography when there are atlases? Why learn history when :
: there are libraries full of history books? Why learn foreign languages
: when : there are translators? Why learn to walk when we can drive or
: be : wheeled : around in a chair?
You can't answer a question with a question. Unless the _student_
thinks that the subject is important, they're going to resent having
to learn it --- and a resentful student is the most difficult to
teach. Answering "What good is it?" with a snippy answer like that
will be a sure way to lead to resentment. Worse yet, what if the
student says "Yeah, exactly my point?"
The student who learns best is one who is motivated to learn --- and
if they see that long division is something that takes them five times
as long to arrive at the wrong answer, they're not going to be very
motivated to learn it.
Jeffs
Jeff Suzuki
: je...@math.bu.edu (Jeff Suzuki) writes:
: >(g...@ee.mu.OZ.AU) wrote:
: >: The idea that learning something like long division is obsolete
: >: just because we can now do it with a calculator or PC is a fallacy
: >: in my opinion. How many other parts of a basic school curriculum would
: >: fall to the axe if we chose to enforce this kind of teaching philosophy?
: >Speaking of axes, do you (generic) know how to use one to cut down a
: >tree?
(Stuff about how he can do these things deleted)
Okay, so you know. I don't. I could probably figure them out, but I
certainly wasn't taught how to do them.
: I think the big problem here is, I don't really understand how
: you can teach the concept of something, and, at the same time, divorce
: it from any and all practical, concrete applications.
Have you ever been in a second semester calculus course? When I took
the course, we spent all of our time learning how to do trig
integrals, etc., etc., and a snippet on series. The course was all
manipulation, and no application.
Whether you understand it or not, the problem is that math IS
routinely taught without reference to applications. Or the
applications are specious: "If John takes 3 hours to mow the lawn,
and Bill takes 2 hours, how long does it take for them both to mow the
lawn?"
The problem isn't convincing ME that math is useful (look at my return
address). The problem is convincing STUDENTS that math is useful. If
you can convince a high schooler that algebra is at least as useful as
learning how to pick up members of the opposite sex, they'd start with
x + y = 3 and work their way to calculus in no time.
Jeffs
John Brevik
>
>Unless the _student_
>thinks that the subject is important, they're going to resent having
>to learn it
I can say from experience that this is not true. I teach fifth grade
math, and when I taught long division, the question whether it would
serve the students in later life _never_ came up. (Believe me, my
students are mouthy enough to say something like this when it occurs
to them!) Kids enjoy learning new skills. We probably spent all of a
week and a half on long division itself, and now they ungrublingly do
it whenever a problem calls for it. In fact, their skill with long
division has enabled them to understand how repeating decimals repeat,
and they all love repeating decimals because they seem kind of
mysterious (notwithstanding the fact that their future earning power
will be in no way enhanced by understanding repeating decimals).
Using a calculator would never give them this insight.
ibo...@metz.une.edu.au
From: Jeff Suzuki, je...@math.bu.edu
Date: 3 May 1995 11:58:01 GMT
In article <3o7r49$i...@news.bu.edu> Jeff Suzuki, je...@math.bu.edu
writes:
>Graham Pulford (g...@ee.mu.OZ.AU) wrote:
>
>: The idea that learning something like long division is obsolete
>: just because we can now do it with a calculator or PC is a fallacy
>: in my opinion. How many other parts of a basic school curriculum
would
>: fall to the axe if we chose to enforce this kind of teaching
philosophy?
>
>Speaking of axes, do you (generic) know how to use one to cut down a
>tree?
Yes. I even managed to finish the job when the axe handle broke
by using the axe-head as a wedge, a metal rod and a rock as a hammer.
It took longer, but it worked.
>Start a fire with flint?
Yes, or even with broken glass from a bottle.
> Butcher a hog?
No. But i guess two out of three is not too bad!
>Not very long ago,
>these were considered "essential" skills, and any teacher (which
>usually meant parent) worthy of the name passed those skills down to
>their students.
I don't ever recall being taught these as school subjects.
Just as I know how to tie my shoelaces without that being
part of a school subject.
On the oter hand, had I gone to a tech. college to become a butcher,
then it would have been on the syllabus to know how to butcher a hog.
>
>On the one hand, long division is a nice skill to know. OTOH, is it
>nice enough, and does the prevalence of calculators, justify
spending
>as much time as is spent on long division?
Absolutely. In the same way as knowing how to knead dough by hand
is justified even though there are machines to do it.
>You can't keep students in
>ignorance of computing devices forever, and sooner or later, one of
>them is going to say, "But why do we need to do it this way, when we
>have a calculator?"
Sew ah yew saying that wee don't knead too lean two
spell because their ah spelling chequers? Ore do ewe
think it ay waist of thyme too learn to right because
wee have typwriters?
>
>To say "Because some day your calculator might break" is specious.
>It's like saying that everyone should know how to parachute because
an
>airplane might have all its engines fail and the wings fall off ---
>and students are aware of this. From this, they extrapolate to
"Math
>teaches me useless skills that I can duplicate on a calculator" ---
You are confusing arithmetic with mathematics. While arithmetic
skills seem to be indispensible to mathematics, they do not comprise
it.
>which leads to the single most common complaint I've heard about
math:
>what good is it?
The same is true of almost any school subject. What is the use of
learning geography when there are atlases? Why learn history when
there
are libraries full of history books? Why learn foreign languages when
there are translators? Why learn to walk when we can drive or be
wheeled
around in a chair?
d.A.
Kevin Bruce Pease
>(g...@ee.mu.OZ.AU) wrote:
>: The idea that learning something like long division is obsolete
>: just because we can now do it with a calculator or PC is a fallacy
>: in my opinion. How many other parts of a basic school curriculum would
>: fall to the axe if we chose to enforce this kind of teaching philosophy?
>Speaking of axes, do you (generic) know how to use one to cut down a
>tree?
'Deed I do. So should anyone with about three functioning
brain cells. I've chopped down trees with several different
implements. A saw is by far the easiest... it's lightweight, easy to
use, and works fast. This would be analogous to
using a computer to solve a division problem. In the absence of a
saw, an axe is the next easiest way... It's heavier, requires more
work, and is more likely to break, but it works pretty well. Sort of
like the equivalent of using a calculator. In the absence of a saw or
an axe, it's possible to use a knife on small trees (I've done it
before... slow, time-consuming, but there are payoffs.)... if I don't
have an axe or a saw with me (which is *definitely* possible), I can
always use a knife with a decent sized blade, which is something I
*always* carry with me when I go out camping, or hiking, or on a
training exercise with the Army. I *don't* always carry an axe or a
saw (nor a computer or a calculator). This would be equivalent to
doing it on paper... it's easy in that I don't have to carry around
something I might lose, break, or just get sick of carrying, and it
works just as well, but slower.
>Start a fire with flint?
Yes, I could. I also have a little firestarter that's made of
magnesium, which lights and burns easily when struck by sparks from a
flint. I just scrape some shavings of magnesium off the block, and
then strike the flint, and the fire starts a lot easier. But I still
have the necessary knowledge to build a decent fire hole & set up my
fuel properly, and start it with only a flint. It's easier to use the
little fire starter, but that doesn't mean I shouldn't have the basic
skills.
>Butcher a hog?
Never done that, but I've butchered a chicken & a rabbit
before, and I'm assuming that a hog would be similar to the rabbit,
just bigger. Skin it, remove the organs, wash it out, cook it up, eat
it. The principle is the same, it's just a matter of applying it.
>Not very long ago,
>these were considered "essential" skills, and any teacher (which
>usually meant parent) worthy of the name passed those skills down to
>their students.
And I think it's sad to see that people don't have some basic
"survival" skills... they're useful to have, and they can make you a
lot more comfortable if you DON'T have access to the "creature
comforts" we take for granted.
>On the one hand, long division is a nice skill to know. OTOH, is it
>nice enough, and does the prevalence of calculators, justify spending
>as much time as is spent on long division?
Probably not, but they should be able to demonstrate a
proficiency in it before they are allowed to use calculators, IMHO.
>To say "Because some day your calculator might break" is specious.
What if it does? Why is it so specious to suggest that a
fragile, mechanical object might break, and there might not be a
replacement handy?
Why bother learning the highly mechanical skill of writing
letters on paper, when we can type them in? Why learn to write or
type at all, when voice recognition is looking more and more likely to
be something in wide use? Why bother doing ANYTHING, if it's just
going to be made easier in the future?
>It's like saying that everyone should know how to parachute because an
>airplane might have all its engines fail and the wings fall off
You're forgetting something: The likelihood of my calculator
breaking, or the batteries running down, or me not having a
calculator, when I need it, is FAR greater than the chance that I'll
be in a plane and have to parachute out of it. Would you say it's
useless to know how to change a tire, when it's just as easy to call a
tow truck or a garage and have them fix it?
>and students are aware of this. From this, they extrapolate to "Math
>teaches me useless skills that I can duplicate on a calculator" ---
>which leads to the single most common complaint I've heard about math:
>what good is it?
You're saying something here that doesn't quite gel with what
your original point was. A lot of math skills that I've heard
criticized are those that "have no use in real life." I suspect I'm
not the only person who learns most easily by example. You can talk
concepts and plans and methods all day long to me, and I still
probably won't understand what you're talking about. However, if you
explain the concept AND show me an example as you explain, I'll have
the concept down in about five minutes, and be 3/4 of the way towards
being able to apply the concept to "real-life" problems.
I think the big problem here is, I don't really understand how
you can teach the concept of something, and, at the same time, divorce
it from any and all practical, concrete applications. The concept and
the application SHOULD go hand in hand, and once mastery of both is
accomplished, THEN the shortcuts and tricks should be taught.
When it came to derivatives, my teacher made us learn the
long, boring, "take the limit" way, which really really sucked,
especially since my brother had already taught me the power rule to
check my homework with. I'll tell you what, though, Knowing the "long
way" to do it helps, because you're not always going to be able to
derive things by using the power rule. I found that out when I
started taking my college-level calculus courses last year... I
learned the "long boring way" first, and THEN learned the shortcut to
find the answer, and then learned how to apply those answers to
different types of problems.
Like I said... I don't see how you can divorce the concept
from the application. Do you think the student is going to understand
it better if you say, "The whole concept of division is [this]
. . . ", and then say, "And, if you just punch this sequence of keys
on your calculator, the answer pops right out!" The student will,
most likely say, "Why bother worrying about the concept when this
nifty little machine can do it all for me?" And the whole effort of
teaching the concept is wasted.
----------
Kevin
Kes...@wpi.wpi.edu
"Cover me, cover me, give me shelter from the storm..."
Truman Prevatt
wrote:
>
> Jeng Jia Hung (jjh...@sfsu.edu) wrote:
> : By your reasoning, learning long division is useful simply because
> : we cannot be a society dependent upon calculators. Why would it be
> : good to lose yet another basic skill? .....
>
> : --Jeng Jia.
>
> Jeng:
>
> Do you really think long division is a "basic" skill? It's a neat trick,
> but I wouldn't call it a "basic" skill. I'd rather have my students
> understand what the meanings of divisions may be, have the ability to
> estimate, etc. It seems perfectly reasonable that division can be taught
> in combination with multiplication. Then, put more emphasis on
> estimation and mental computation, both strategies and skills, e.g. 258/3
> can be thought of 240/3 and 18/3. Most kids who try to mentally do this
> computation imagining the long division process gets lost. Estimation
> and mental computation are much more powerful "basic" skills than long
> division, and if teaching long division harms students masering more
> powerful skills, isn't it reasonable to put less emphasis on teaching
> long division?
>
Back in graduate school when I was taking algebraic topology the instructor
made a comment on homological algebra. While cohomology groups are very
improtant their "brute fource" calculation by chasing diagrams is very
mechanical and tedious, but it is something every student should do once in
his life to understand how it is done. I believe the same can be said for
long division. Make every student do it once then get on with teaching
concepts and don't dwell on mechanics.
______________________________________________________________________________
The race is not always to the swift, but to those that keep running.
pre...@lds.loral.com
______________________________________________________________________________
William Lorimer
Link) wrote:
> In article <3o1nd0$k...@cello.gina.calstate.edu> don yost
<doy...@cello.gina.calstate.edu> writes:
> >I read some research receintly about long division. Teaching long
> >division is not just a waste of time, it is harmfull. If a student
> >learns the algorithm before they learn the concept, they will confuse a
> >skill with understanding and the concept will be difficult or impossible
> >to learn.
I've been following this thread (more or less) for some time, and I think
it's time I put in my 2 cents worth.
In my spare time I do volunteer work as a math tutor for the John Howard
Society, a charitable organization which tries to reduce crime by helping
convicts, ex-convicts, and others who have come into conflict with the
law, to break their existing self-destructive patterns of behaviour.
One of the first students I had was a grizzled old recovering alcoholic
who essentially wanted nothing more than to have a reasonably steady
paycheque, a one-room apartment, and a job as a carpenter on a
construction site. He had enrolled in the JHS carpentry job-training
program, and part of this training involved a course in mathematics (how
to compute how many 2x4s would be required to construct a floor of a given
area, for example).
He was very frustrated because of the "difficult" questions he was
expected to answer - his attitude to a question like "divide 182,596 by
513" was that no one could do something like that without a calculator.
When I taught him, in less than 10 minutes, to solve problems like that
with a pencil and paper, his confidence level improved dramatically. He
was actually astonished to find it was so easy. By the time he finished
the program a few months later, he was quite comfortable working with
fractions, decimals, and pretty much any other basic maths required for
carpentry.
Another student, also in the carpentry program, waited about three weeks
until he felt he could trust me, and then asked, ("not right now, but
maybe - someday - when you have time") what the little marks on the ruler
meant - you know, the ones in between the numbers. This same student, I
discovered after 3 weeks, didn't know the multiplation tables past 3x3.
That is, he could not multiply 3x4 in his head. Oh, he could do it with a
pencil and paper - he knew that 3x4 was the same as 3x2 plus 3x2, which he
could do - but he simply had never learned the multiplication tables. And
this was a grown man in his mid-to-late 20s.
It seems to me that most of the people who hang out in this newsgroup have
no real understanding of what it must be like to have absolutely *no* math
skills of any sort. I've seen what that can do to people, and I've seen
how much of an improvement in their lives even something as "basic" as
long-division, or memorizing the multiplication tables, can sometimes
make.
Sorry about the diatribe, but I think those who think we should scrap
basic math education should be very certain that we're not going to
produce a generation of people who think that long division is something
only a handful of geniuses can comprehend.
--
WR Lorimer
(I apologize if the above questions have already been addressed.)
"For lust of knowing what should not be known\We take the Golden Road to Samarkand." J. Elroy Flecker
Kevin Broderick
Jeff Suzuki (je...@math.bu.edu) wrote:
: Graham Pulford (g...@ee.mu.OZ.AU) wrote:
: : The idea that learning something like long division is obsolete
: : just because we can now do it with a calculator or PC is a fallacy
: : in my opinion. How many other parts of a basic school curriculum would
: : fall to the axe if we chose to enforce this kind of teaching philosophy?
: Speaking of axes, do you (generic) know how to use one to cut down a
: tree? Start a fire with flint? Butcher a hog? Not very long ago,
: these were considered "essential" skills, and any teacher (which
: usually meant parent) worthy of the name passed those skills down to
: their students.
: On the one hand, long division is a nice skill to know. OTOH, is it
: nice enough, and does the prevalence of calculators, justify spending
: as much time as is spent on long division? You can't keep students in
: ignorance of computing devices forever, and sooner or later, one of
: them is going to say, "But why do we need to do it this way, when we
: have a calculator?"
: To say "Because some day your calculator might break" is specious.
: It's like saying that everyone should know how to parachute because an
: airplane might have all its engines fail and the wings fall off ---
: and students are aware of this. From this, they extrapolate to "Math
: teaches me useless skills that I can duplicate on a calculator" ---
: which leads to the single most common complaint I've heard about math:
: what good is it?
Yes, but knowing how to long divide is necessary to learn more advanced
concepts (divding polynomials is the one that pops into my head, as I'm
learning that right now). My teachers generally make us learn how to do
something on our own, but will let us use calculators if we know how to
do something, but it's just tedious. JSYK, my math class *is* moving
into the current decade--we usually spend our Friday class learning to
use spreadsheets and graphing on the computers. We'd probably do more,
but we don't have the available computers. Understanding *how* is
important; imagine trying to teach someone who didn't understand how to
switch cables on the back of a computer how to hook a computer into a
network? They could understand a virtually infinite amount of knowledge
on software, architecture, etc, but they still wouldn't be able to plug
the coax cable into the back of the machine.
--
--KTB aka "Sparty"
web: http://m-net148.arbornet.org/~kbroderi/
mail: kbro...@m-net148.arbornet.org
spa...@genesis.nred.ma.us
* Ski Fast, Live Forever.
Kevin Broderick
: Dave Hayden (da...@edo.ho.att.com) wrote:
: : For another reason why long division should be taught, look no further
: : than a Pentium processor near you. The Pentium, in case anyone hadn't
: : heard, had a flaw in its division algorithm.
: I think this says that it's more important to spend time doing
: estimation and short division than anything else. You're not going to
: catch a computer on any small error unless you perform all the
: calculations yourself --- and if you do this, why bother with the
: computer? But if your pentium say 389,291,382 divided by 3,289 is
: 42,389, you don't need long division to know this is wrong.
Alrighty then. Let's say you use Windows calculator. Try any division
or subtraction (and possible addition, I'm not sure) with two or more
decimal places, then try it out on paper or in your head. Example:
2.01-2=0. Try it. Though there is a fixed version out (I think), I
believe most people with Windows 3.1/3.11 have the erroneous calculator.
Friesen
Many things said, some of them clever, some not quite so clever... Given
Kaufman's (the original poster's) troubles with net administrators, it's
entirely possible that the whole thing was cooked up as an attempt to
incite one last flamewar and claim some kind of petty victory.
Unfortunately, it seems that many folks thought he was serious.
Have you noticed? The original diametrical positions have all softened
since the whole brouhaha began. I suspect we all agree more than we
thought we did.
I'd like to offer two tidbits for your consideration. They both touch
on the issue of skills/mechanics/understanding.
1. A few days ago, a gentleman posted to an electronic music list, asking
for the binary-hex translation of 11111010. Apparently, this had never
been covered in his math classes. Or he'd forgotten. Or he was too
stinkin' lazy to look up (or work out) the problem for himself.
What made matters worse was that, along with a few kindly helpful
messages, one well-intentioned individual punched the numbers into a
programmer's calculator and informed the entire list that the answer was
something like R98R62. He admonished that this answer could be had from
any $20 calculator.
If the original question was cause for dismay, this dismally incorrect
response provides a spectacular vindication for those who advocate
tossing the electronics entirely.
The person had simply punched a number into a calculator and transcribed
the answer. It is likely that NO THOUGHT went into the process -- beyond
the mind-numbing task of punching buttons and looking at a pair of
displays.
The original question asked for a binary-hex conversion: the culprit
seems to have misread the question -- instead trying to convert the
DECIMAL number 11111010 into hex. Then another answer was made when the
answer was transcribed: there's no "R" digit in hex; it should have been
an "A".
THAT'S what happens when we allow technology to supplant the mind.
2. Robert Heinlein has an interesting suggestion as regards what people
should and shouldn't need to be able to do.
"A human being should be able to change a diaper, plan an invasion,
butcher a hog, conn a ship, design a building, write a sonnet, balance
accounts, build a wall, set a bone, comfort the dying, take orders, give
orders, cooperate, act alone, solve equations, analyze a new problem,
pitch manure, program a computer, cook a tasty meal, fight efficiently,
die gallantly. Specialization is for insects."
R.A. Heinlein (as "Lazarus Long")
I might quibble on a few points, but I think he's onto something there...
3. (I know, I said I'd offer only two. My calculator must have misled me.
Or maybe I'm a member of the Spanish Inquisition...)
The brain is a muscle. It gets better from use. Even exceptionals (human
calculators) admit that they practice. They think about numbers. They
think about thinking about numbers. Rigor is not the enemy. Even tedium
is not the enemy. Giving in to laziness is the enemy.
4. The proper purpose of technology is not to supplant man. It is to make
man more powerful. Any application of technology that lessens man is
wrong. Using a calculator to supplant an average mental process is bad.
Long live homo intelligens!
Share and enjoy,
Michael Friesen
David Nelson
"ploughing" through long division. I am a school principal - no I was not
snooping outside the classroom! - with a still vivid memory of being taught
long division and totally not having the slightest understanding of what I
was doing!
With the widespread use of hand held calculators in primary (elementary)
schools I cannot see any earthly use in wasting valuable teaching time in
teaching long-division processes to students who don't have the basic
division concept. I agree that the four number process concepts MUST be
taught to students who are able to grasp the concepts at an appropriate age
and grade level. I also agree that if there is a mandated syllabus
requirement then that syllabus must be taught.
HOWEVER, let's get teaching in focus! What are we really on about in Years
1-6 in teaching mathematics. As my own 16year old son often says to me "Get
real, Dad!" I suggest taking heed of that message! Get real!
What use is there REALLY in teaching long division.
I have another "hobby horse" too!! Why waste valuable time in teaching
students processes in rational number! Who really wants to know how to
divide seventeen twenty thirds by six sevenths? And what use is it!
Immortal
s...@neon.chem> <3obgti$a...@ds2.acs.ucalgary.ca>
Bobincio Batiller Pasion (bbpa...@acs2.acs.ucalgary.ca) wrote:
: Personally, I think that only by repeatedly doing problem after
: problem will you understand the concepts of any mathematical
: theory... I too am a university student, and have had more than
: my share of math that I didn't like. You _must_ do the problems
: to understand the concepts. _Especially_ when dealing with
: multi-variate calculus or eigenvalue subspaces of differential
: equations (Ouch!). You can easily find a solution for mechanical
: problems that actually deal with _actual_numbers_, but when asked
: to prove an abstract theory, you need the practice.
OUCH!
I hope you're not implying that repeatedly doing problem after problem
will teach people the theory behind the madness. I've had calculus
students who just liked that formula for the volume of the area under a
curve rotated around the x-axis, and could plug in function after
function. Often they'd make some trivial mistake and out would pop a
negative answer, like -6000pi, where the curve is y = x^2 + 1 on [0,1] or
some such, which they'd then merrily circle as their answer, and be on
their merry way. The didn't even realize what it was they were doing.
I think computers are a valuable tool. In the above example, you could
have a computer draw the curve and shade the area under, and then whip it
around the axis and produce a nice shaded object. But I can do this with
a bit of coloured chalk. Not to the impressive degree, though, but it's
certainly possible. I think theory should be introduced and constantly
reinforced by example.
Justin
Ron Stoloff
This reminds me of the days when I was using a slide rule (for the
young, this is what people used before the calculator.) To do the
problem, you *had* to know the scale of the expected answer. This is
what you mean by "short division." I agree that this would be a better
skill to teach our kids since they can use a $3 calculator but still get
the wrong answer unless they know what the answer is supposed to be
Ron Stoloff, Philly
Jan Gunnar Moe
Problem in this post of mine is the equivalent of an exercise, in this post it
does NOT mean a real PROBLEM requiring special creativity, fantasy, problem
solving ability.
>Personally, I think that only by repeatedly doing problem after
>problem will you understand the concepts of any mathematical
>theory...
No, No, No!!!!!!
Repeatedly doing problems _is_ important, to automate techniques, learn
operate fast without doing sloppy errors and things like that.
What you say is that large numbers of problems done is a _necessary_ condition
for understanding. This is simply wrong. The _necessary_ condition for
understanding is that reflective afterthought is invested when doing each
problem. To develop understanding, you should all the way while doing the
exercise control that you are able to answer questions like
What am I doing?
Why am I doing it?
How do I get nearer to the solution of my problem by doing it?
Why am I allowed to do it?
When the problem is solved, it is time to ask questions like
What is the similarity between this problem and others that I have done
earlier?
What is the difference between this problem and others that I have solved
earlier?
What are the similarities between this problem and others that I have earlier
solved by using the same strategy, method or technique?
What are therefore the characteristic traits of problems which may be solved
using this strategy, method or technique? In other words: What is
characteristic for the FAMILY of problems for which this special problem is a
special case?
How can I myself generate other problems belonging to this problem family?
What does the teacher/instructor/textbook author do when he/she generates new
problems belonging to this family?
By reflecting around questions like these (no attempt on my side to present
all relevant questions) one prepares oneself to meet new problems.
What is so dangerous when you say that solving of lots of problems is
necessary, is that some people may start believing the inequivalent
proposition that it is sufficient.
I have seen all to many examples that students may correctly solve lots of
'problems' without learning anything. Why is this so?
It is so because after a certain technique is presented in textbooks,
exercises are given massaging this special technique, and students do all the
exercises, not reflecting over the adequacy of the technique. After having
solved problems following chapters delivering several techniques this way,
they are simply unable to solve mixed problems. Part of the responcibility is
of course on the hand of the teacher/textbook author. But much of the cause is
the all to strong belief on the importance and usefullness of solving lots of
problems.
My point: Quantity is only important after quality is guaranteed.
Jan Gunnar Moe
__________________________________________
I too am a university student, and have had more than
>my share of math that I didn't like. You _must_ do the problems
>to understand the concepts. _Especially_ when dealing with
>multi-variate calculus or eigenvalue subspaces of differential
>equations (Ouch!). You can easily find a solution for mechanical
>problems that actually deal with _actual_numbers_, but when asked
>to prove an abstract theory, you need the practice.
> MY 2cents
David Longley
> In article <3obgti$a...@ds2.acs.ucalgary.ca> bbpa...@acs2.acs.ucalgary.ca
> (Bobincio Batiller Pasion) writes:
<snip>
>
> >Personally, I think that only by repeatedly doing problem after
> >problem will you understand the concepts of any mathematical
> >theory...
>
> No, No, No!!!!!!
>
> Repeatedly doing problems _is_ important, to automate techniques, learn
> operate fast without doing sloppy errors and things like that.
>
> What you say is that large numbers of problems done is a _necessary_
> condition for understanding. This is simply wrong. The _necessary_
> condition for understanding is that reflective afterthought is invested
> when doing each problem.
<snip>
RATTUS POPPERICUS?
I'd like to suggest a way of reconceiving this which shows why *intensional*
(folk-psychological/cognitive) talk tends to mislead us, and why behaviour
analysis can be so powerful. I hope to show that he second author may well
be *agreeing* with the first, but that the latter doesn't realise this. This
theme is developed elsewhere in 'Fragments of Behaviour..', and is the basis
of a behaviour modification and assessment system called Sentence Management
in the field of 'Corrections'.
When one begins Operant Conditioning work with rats, one has to get the
animals to notice where the food pellets they are going to bar press for
are going to be delivered. This is often referred to as 'magazine training'
because the little food pellets are dispensed one at a time from a magazine.
After a few deliveries, the rat quite happily munches away after each pellet
pops down the food shute. The next task is to get it to go near the lever,
so one watches for the rat to go towards the lever, and as soon as it moves
in the right direction, one can press a button to deliver a food pellet. As
the rat moves closer and closer one ceases to deliver pellets when it is at
a relatively remote site, and only reinforces behaviour which brings the rat
almost on to the lever. Finally, the rats brushes against, or falls upon the
lever, and the mechanism of lever press - pellet delivery takes over. Then,
the rat 'learns to press the lever'.
Now, what is often not fully appreciated, is the fact that an enormous amount
of behaviours are being learned here. Each approximation that is learned is a
contingency:
IF Such_and_such_behaviour THEN food_pops_out_over_there
IF So_and_So_behaviour THEN NOT food_pops_out_over_there.
'Pressing the lever' per se is an abstraction which the trainer makes.
The rat learns a lot of behaviours, and progressively some are selectively
reinforced and others not (they are extinguished). In fact once one gets the
animals to repeat tthe required behaviour often enough it does become stereo
typed (mechanical)..and the longer the animal is trained, the better it is
able to stop when food is no longer contingent (this is called extinction).
The amount of lever pressing in extinction can be shown to be a function of
how much training the animal gets during acquistion. One could say that the
rat progressively 'homes in' on the required invariant class of behaviours.
Elsewhere, in standard classical conditioning paradigms, this is called
'configuring', and in a slightly different guise, 'blocking' (in either case
some elements of the behavioural array drop out).
The point is that stripped of the 'Rattus-Norvegicus-falsificationist' talk,
what the animal does is perform a set of behaviours which can be 'construed'
cognitively *from the teachers point of view*, but which are probably best
*seen* as a set behaviours which can be shaped up to the required behaviour
through differential feedback.
What is important is practice, so that the 'effective' strategies can be
configured. To talk about 'understanding' being necessary apart from this
may well just be a failure to appreciate the subtelty of sound behaviour
analysis and management.
--
David Longley
Herman Rubin
IMRE BOKOR <ibo...@metz.une.edu.au> wrote:
>Jeff Suzuki (je...@math.bu.edu) wrote:
>: ibo...@metz.une.edu.au wrote:
....................
> But a society like ours which depends (whether
>necessarily or otherwise) on our mathematically based technology
>and economics, is one where neumeracy is a political imperative.
>To understand the data presented, to check and verify it
>requires numeracy and a minimum of mathematical reasoning,
^^^^^^^^^^^^
to say
>the least.
With this I agree, but NOT with learning to manipulate numbers.
The essentials of mathematical reasoning are the use of variables,
so that "real world" problems can be translated into formal systems
where the power of the logical system can be unambiguously applied,
and the ability to understand that assumptions have consequences,
which can be checked in a formal system. THIS, which is not even
taught to matematics majors, is the essentials part, not calculation.
And calculation should be based on understanding. The time to teach
long division is AFTER the structure of the number system, and the
meaning of multiplication and division, are known, and definitely
before teaching short division. None of this logical order occurs
at present.
......................
>: Unless the _student_
>: thinks that the subject is important, they're going to resent having
>: to learn it --- and a resentful student is the most difficult to
>: teach. Answering "What good is it?" with a snippy answer like that
>: will be a sure way to lead to resentment. Worse yet, what if the
>: student says "Yeah, exactly my point?"
This is totally the fault of the current miseducational system. It used
to be the case that teaching was for the future, not the test.
>The student who asks "What good is it?" in those words is likely to
>be resentful already.
This now happens with students at all levels, not just children. Even
prospective researchers ask this question. But it may be important
tomorrow, even if there is no use for it today. And if it is conceptual,
rather than just a statement of fact, it can help in understanding.
Facts and routine methods can be easily added. Structure takes time to
build up.
IMRE BOKOR
: : Sew ah yew saying that wee don't knead too lean two
: : spell because their ah spelling chequers? Ore do ewe
: : think it ay waist of thyme too learn to right because
: : wee have typwriters?
: Spell checkers have not gotten to the point where they can replace
: good spelling; calculators _have_ gotten to the point where they can
: replace most human calculation.
The fact that calculators can perform many arithmetic operations
faster than most humans has never been at issue. What has been at
issue is whether this fact renders superfluous the ability/skill
in doing so without a calculator. Of course calculators are limited
in their accuracy and blind dependence on them --- which is an
unavoidable consequence of not being able to perform the calculations
without their assistance --- is soemthing we should prefer to eschew.
We perform many "useless" tasks in life which are better done by
machine, or even better, not even done at all. Mankind survived
many generations without long division, and would certainly
survive without it. But a society like ours which depends (whether
necessarily or otherwise) on our mathematically based technology
and economics, is one where neumeracy is a political imperative.
To understand the data presented, to check and verify it
requires numeracy and a minimum of mathematical reasoning, to say
the least. To say that we can rely on the pocket calculator is
equivalent to saying we should forget geography, because we can rely
on personal organisers, or forget spelling because we can rely
on spelling-checkers (if we know what we wish to write).
I see this as a recipe for disaster. Note, however, that I am *not*
saying that that is all there is to mathematics, quite the contrary.
But the long division algorithm provides an excellent opportunity
to introduce pratical methods for solving problems. The thrust of
it is "Make an intelligent guess, and see how far off the mark you are.
Try to improve your guess." In this case the procedure is a finite one,
but it opens the philosphical/mathematical questions of effective
procedures, of iterated approximations (do they converge? If so
do they cnverge to the required answer?) and these are genuine
*mathematical* problems. Long division is within the scope and
reach of primary pupils, the fixed point theorem for contracting
self-maps of a complete metric space is probably not, yet they
share some common features.
: Ask me in twenty years if I think we
: need to spend time teaching "i before e except after c, except when
: pronounced 'ay', as in neighbor and weigh, and about two dozen
: exceptions..."
The Hungarians solved that problem about a cenury ago when a spelling
reform was introduced, making the spelling almost completely phonetic.
If that happens within the next twenty years in American or in English
then I *know* what the answer will be.
: : You are confusing arithmetic with mathematics. While arithmetic
: : skills seem to be indispensible to mathematics, they do not comprise
: : it.
: I'm not the one who's confusing them. What do students call things
: like long division? "Math". Most people when they say "I hate math"
: mean "I hate arithmetic". I don't blame them.
I don't blame them either, but that does not mean they are correct in the
belief that all there is to mathematics is finding the right software to
solve the problem for you. I have so often been asked "What is there
left to do in mathematics now that we have computers?", or "I thought
that evrything there is to be known about mathematics has been discovered
long ago", when I confess to an interest in mathematical research.
Most people confuse other related things as well. So what?
: : >which leads to the single most common complaint I've heard about
: : math:
: : >what good is it?
: : The same is true of almost any school subject. What is the use of :
: : learning geography when there are atlases? Why learn history when :
: : there are libraries full of history books? Why learn foreign languages
: : when : there are translators? Why learn to walk when we can drive or
: : be : wheeled : around in a chair?
: You can't answer a question with a question.
Why not?
: Unless the _student_
: thinks that the subject is important, they're going to resent having
: to learn it --- and a resentful student is the most difficult to
: teach. Answering "What good is it?" with a snippy answer like that
: will be a sure way to lead to resentment. Worse yet, what if the
: student says "Yeah, exactly my point?"
The student who asks "What good is it?" in those words is likely to
be resentful already.
: The student who learns best is one who is motivated to learn --- and
: if they see that long division is something that takes them five times
: as long to arrive at the wrong answer, they're not going to be very
: motivated to learn it.
Some students are more motivated when they *can't* do something and
are bored when they find the task too easy. Many like challenges to
show how good they are. Others shun challenges. There is no hard and
fast rule. Pupils do not come in DIN-Norm formats.
d.A.
Truman Prevatt
>
edit..
>
> The problem isn't convincing ME that math is useful (look at my return
> address). The problem is convincing STUDENTS that math is useful. If
> you can convince a high schooler that algebra is at least as useful as
> learning how to pick up members of the opposite sex, they'd start with
> x + y = 3 and work their way to calculus in no time.
>
> Jeffs
Now I think you got something there. When I was teaching a probably and
statistics course in to the engineering students at USF, the first comment
was "what is this stuff good for". "The rewards coming from exploring the
beauty of mathematics" didn't really wash with these people. But if you
could figure out how to tie mathematical skills to sex - then the
Government could classify it, put it into practice covertly and the US
technology would be second to none.
______________________________________________________________________________
The race is not always to the swift, but to those that keep running.
Truman and Mystic "The Horse from HELL" Storm
pre...@lds.loral.com
______________________________________________________________________________
Jeff Suzuki
: In article <3ob014$3...@news.bu.edu>, Jeff Suzuki <je...@math.bu.edu> wrote:
: >
: >Unless the _student_
: >thinks that the subject is important, they're going to resent having
: >to learn it
: I can say from experience that this is not true. I teach fifth grade
: math, and when I taught long division, the question whether it would
: serve the students in later life _never_ came up...
My experience with college (and older) students is that they do resent
learning something they perceive is not "useful". Your point is
taken, however.
Still, doesn't this suggest that 1) kids can, 2) kids want to, and 3)
kids do learn a great deal when they are young? They're not going to
be young forever --- and if we fill up this time teaching them things
that can be done by other means, are we wasting their potential?
(This is not meant as sarcasm; it's a serious question)
Now, before I get flamed (and, BTW, if you must flame this post, have
the decency to do so in a public forum, so that you can show the world
what you're made of -- private email flames will be posted) I should
point out that I have not suggested we abolish teaching long division;
in fact, I don't think anyone has. The question is how much time do
we spend teaching it.
Or, as I like to put it: there are many things your calculator can
do. There are many things only a human brain can do. Do you spend
time teaching things a calculator can do? Or should you spend time
teaching the things that only a human brain can do?
Jeffs
John Lewis
Referring to a message dated 02 May 95, sent by Neal Plotkin to (crosspost 2)
All
on the subject of Is Long Division a Useless Skill?
NP> Maybe you could teach it by repeated subtraction. In fact, if you did
NP> that, maybe elementary school teachers might even find out that the
NP> standard algorithm is actually repeated subtraction with shortcuts;
NP> it's not just a memorized sequence of steps! Imagine that! Maybe they
NP> could even figure out how to teach the standard algorithm in a logical,
NP> sensible manner, and this whole argument would go away!
In the late 50's in UK, school kids used log tables, technicians used slide
rules and commerce had comptometers. The most common calculator in our firm's
costing office was a little gem by FACIT. You set up the dividend on levers
which ran from 0 to 9. There was a 1's lever, a 10's lever, a 100's etc etc.
You wound it into the machine by turning the handle once. Next you set up the
divisor and wound the handle backwards, and again, and again until the bell
rang. You'd gone too far - go forward once. Slide the whole cylinder
carrying the divisor one place to the right and wind backwards to keep
subtracting till the bell rang etc etc. Eventually the answer was seen on a
counter that noted how many times you'd been able to subtract the divisor x
100, then x 10, then x 1. Pretty close to long division.
When I went to training college in 1964 we still had to use log tables or a
slide rule. I was pretty good at long multiplication and long division, and
could add and subtract flawlessly until an eccentric Maths lecturer (Tom
Brissenden) said we'd lost a thumb on a buzz saw and would have to work in
base 9 for that lesson. I've never felt so helpless! All the little skills
I'd used so effortlessly and thoughtlessly were suddenly no use at all.
Good bye, _/()/-//\/ /_
David Kastrup
:With this I agree, but NOT with learning to manipulate numbers.
:The essentials of mathematical reasoning are the use of variables,
:so that "real world" problems can be translated into formal systems
:where the power of the logical system can be unambiguously applied,
:and the ability to understand that assumptions have consequences,
:which can be checked in a formal system. THIS, which is not even
:taught to matematics majors, is the essentials part, not calculation.
:And calculation should be based on understanding. The time to teach
:long division is AFTER the structure of the number system, and the
:meaning of multiplication and division, are known, and definitely
:before teaching short division. None of this logical order occurs
:at present.
:This is totally the fault of the current miseducational system. It used
:to be the case that teaching was for the future, not the test.
Don't forget that the future for many people does *not* include
college education, and a mathematical occupation. So you should
teach them that math which they are likely to need in future life.
Manipulating variables will not solve their day-to-day problems.
Manipulating numbers is an important skill, even if you do not
think it is basic to mathematics. However, in a while lot of
calculations and integrals things like 1440/4! and similar terms
crop up. It would seriously hamper progress of working on such
thigns if you would have to stop for the calculator each time
an expression like that crops up. So I deny even the advanced
math users the irrelevance of basic calculation skills.
It is a dream of one too advanced in math that you can get
along without basic skills, or even understand what you are
doing.
It is similar to demanding that one *first* teach children
proper grammatical constructs, *then* the words they are
going to use in them.
Without words to work with, the grammar makes no sense, and
cannot be properly taught (at least if there is no other
language you can start from, like in *second* language
acquisition).
--
David Kastrup, Goethestr. 20, D-52064 Aachen Tel: +49-241-72419
Email: d...@pool.informatik.rwth-aachen.de Fax: +49-241-79502
Herman Rubin
David Kastrup <d...@hathi.informatik.rwth-aachen.de> wrote:
>hru...@b.stat.purdue.edu (Herman Rubin) writes:
>:With this I agree, but NOT with learning to manipulate numbers.
>:The essentials of mathematical reasoning are the use of variables,
>:so that "real world" problems can be translated into formal systems
>:where the power of the logical system can be unambiguously applied,
>:and the ability to understand that assumptions have consequences,
>:which can be checked in a formal system. THIS, which is not even
>:taught to matematics majors, is the essentials part, not calculation.
>:And calculation should be based on understanding. The time to teach
>:long division is AFTER the structure of the number system, and the
>:meaning of multiplication and division, are known, and definitely
>:before teaching short division. None of this logical order occurs
>:at present.
>:This is totally the fault of the current miseducational system. It used
>:to be the case that teaching was for the future, not the test.
>Don't forget that the future for many people does *not* include
>college education, and a mathematical occupation. So you should
>teach them that math which they are likely to need in future life.
The person who does not understand the concepts will not be able
to decide which procedure to use. I did not say that long division
should not be taught, but it should be taught only as a means to
get explicit numerical answers when the problem is understood.
The original posting in this thread was in teaching the formal
manipulation without any indication of why, and using massive
amounts of drill. When something is understood, little additional
drill is helpful.
>Manipulating variables will not solve their day-to-day problems.
>Manipulating numbers is an important skill, even if you do not
>think it is basic to mathematics. However, in a while lot of
>calculations and integrals things like 1440/4! and similar terms
>crop up.
Now how is a mathematical ignoramus going to recognize such a
problem? Is someone who should not be learning the language
of mathematics going to encounter such a problem? Are we going
to hire squads of arithmetic clerks?
It would seriously hamper progress of working on such
>thigns if you would have to stop for the calculator each time
>an expression like that crops up. So I deny even the advanced
>math users the irrelevance of basic calculation skills.
To some extent, this is the case. But how many people are
going to be able to divide an 8-digit number by a 3-digit
one in the length of time it takes to pick up the calculator
and put in the numbers? I am quite capable of doing these
manipulations more quickly than most, and I suspect that I
would use the calculator.
>It is a dream of one too advanced in math that you can get
>along without basic skills, or even understand what you are
>doing.
What are basic skills? Maybe we should go to base 2, so that
the manipulations are easier to carry out, although more time
consuming. In base 2, it becomes clear that the bit-by-bit
square root takes about the same time as division.
>It is similar to demanding that one *first* teach children
>proper grammatical constructs, *then* the words they are
>going to use in them.
Surprisingly enough, it seems that this is almost what happens!
>Without words to work with, the grammar makes no sense, and
>cannot be properly taught (at least if there is no other
>language you can start from, like in *second* language
>acquisition).
It is not taught, but somehow learned.
Beyond a sufficient number of words to communicate, this is
what actually happens in the native language, until the whole
word or whole language people get in and try to stop it.
What is needed for the acquisition of the simple and powerful
mathematical grammar is a crude understanding of the grammar
of the vernacular, and possibly some knowledge of the counting
process. The Peano Postulates are about counting. Even the
weakest form of them enables the characterization of the
arithmetic operations; characterization should usually be
used instead of definition for other purposes; that was one
of the weaknesses of the "new math".
Alberto C Moreira
[stuff deleted...]
>I believe in mathematics at all levels there is this tendency to concentrate
>on calculations (including formal proofs as calculations) rather than the
>"roles that the concepts play".
Yet when people really teach concepts, people who decry computation
seem to be the first to cry wolf. If you teach people that an integral
is nothing but a sum of products, that a sum and a product are just
two artificially defined functions that characterize a few algebraic
structures like commutative group and field, and that a function is
nothing but a set of pairs - you'll be hearing a lot of people scream
because it doesn't teach them how to balance their checkbooks.
No, I read it different; the tendency is to concentrate on the easy
side of the fence. Anything out of the ordinary, be it conceptually or
computationally complex, is reacted against by a lot of people.
_alberto_
Benjamin J. Tilly
bbpa...@acs2.acs.ucalgary.ca (Bobincio Batiller Pasion) writes:
> Personally, I think that only by repeatedly doing problem after
> problem will you understand the concepts of any mathematical
> theory... I too am a university student, and have had more than
> my share of math that I didn't like. You _must_ do the problems
> to understand the concepts. _Especially_ when dealing with
> multi-variate calculus or eigenvalue subspaces of differential
> equations (Ouch!). You can easily find a solution for mechanical
> problems that actually deal with _actual_numbers_, but when asked
> to prove an abstract theory, you need the practice.
As a grad student in math, I agree that doing some problems is
essential. But just chugging through so many problems is not, IMHO,
even close to sufficient. For instance with multi-variable calculus I
noticed (while tutoring) that one simple point that students were
hopelessly confused on was the distinction between giving a surface
implicitly, as a graph, or in parametrized form. For each way of giving
a surface there are formulas for various things. However the three
formulas for, say, the tangent plane are different.
The problem is that the distinction is so trivial for the prof that the
prof usually does not stress the distinction, or try to sort the matter
out for the students, and so the students will merrily apply whatever
formula they happen to remember to the problem. This works well on
homework where the last formula that they heard is probably the right
one, but it causes problems on tests, and I shudder to think about what
their long-term learning of the material is like.
So I think that there might be value in (and I have never seen this
done so this is just an idea) giving homework and every so often having
the students record in their homework "where everything lives" (as
geometers say). ie In what fashion is the surface that they are talking
about specified by the formula that is written down, what is the nature
of the answer that they are looking for, verbally repeat what the
connection is between the two. I would think that if the students had
to do a few problems and provide such a running commentary on paper,
then the students would begin to provide such a running commentary on
their own in their heads while they are doing problems, and I find it
hard to think of a more useful habit to get anybody into while you are
trying to get them to understand what they are doing.
What does everybody else think about this?
Ben Tilly
Herman Rubin
>As a "university level" student, I feel that I shoule say something here.
>Our school has been working out a program to find better ways to teach
>(and learn) calculus. It is a controversial subject with the MAth dept.
>I went through a program (most of it failed) where we were taught the
>concepts and applications on the computer using a text based around M
>Mathematica software. It was very good at teaching the concepts and
>uses of all those things that we were learning to do. Problems were
>taken from textbooks in varoius engineering fields (most of us are studing
>one sort of engineering or another) and we were shown how to use what we
>had learned.
A text should not be based on a particular computer software. I am still
unconvinced that you learned the concepts, but rather how to use this
computer program to grind out answers.
There is far too much emphasis on getting answers. The concepts are what
are needed to ask the questions.
>That part worked and gave me a much better understanding of the concepts.
>But, I still was not very good at doing the things on paper. That is
>where our program failed. There was no practice on paper. When I got to
>other classes that expected me to know how to do these things, I was in
>trouble. I have since taken a traditional class and now have a better
>handle than many of my peers on the concepts of what we are doing in
>our classes.
I still get the feeling that you are in the computational stage. A concept-
oriented course might have you doing some things by computer, but not that
much. There is no point in having the computer differentiate x, or even
sin(x) or 1/x as a rule. But it still is all right if you know when to do
this.
The traditional 2-year calculus course can be done in one semester called
"hand computation in calculus" if the concepts are known. But the concepts
are even harder to teach to someone who has had the traditional course
than to teach to someone who has not.
Alberto C Moreira
>From: immo...@wam.umd.edu (Immortal)
>Subject: Re: University level math Was Re: Skills vs. concepts
>Date: 5 May 1995 03:09:12 GMT
>s...@neon.chem> <3obgti$a...@ds2.acs.ucalgary.ca>
>Bobincio Batiller Pasion (bbpa...@acs2.acs.ucalgary.ca) wrote:
>: Personally, I think that only by repeatedly doing problem after
>: problem will you understand the concepts of any mathematical
>: theory... I too am a university student, and have had more than
>: my share of math that I didn't like. You _must_ do the problems
>: to understand the concepts. _Especially_ when dealing with
>: multi-variate calculus or eigenvalue subspaces of differential
>: equations (Ouch!). You can easily find a solution for mechanical
>: problems that actually deal with _actual_numbers_, but when asked
>: to prove an abstract theory, you need the practice.
>OUCH!
>I hope you're not implying that repeatedly doing problem after problem
>will teach people the theory behind the madness.
He is absolutely right, it takes a lot of problem solving. But there
is a big, big difference between solving one problem 50 times and
solving 50 problems once each. If you solve one problem only once,
and well, the more problems you solve the more you learn.
> I've had calculus
>students who just liked that formula for the volume of the area under a
>curve rotated around the x-axis, and could plug in function after
>function. Often they'd make some trivial mistake and out would pop a
>negative answer, like -6000pi, where the curve is y = x^2 + 1 on [0,1] or
>some such, which they'd then merrily circle as their answer, and be on
>their merry way. The didn't even realize what it was they were doing.
There may be something missing in the conceptual part; and like I
said, application of some formula is something you need only solve
once. Not even once per formula; I said ONCE. The moment your
student applied ONE formula to ONE problem, that's enough; let's
get to more serious stuff. How about DERIVING the formula ?
>I think computers are a valuable tool. In the above example, you could
>have a computer draw the curve and shade the area under, and then whip it
>around the axis and produce a nice shaded object.
I would contend that the mathematics of it doesn't need the curve.
It's the abstract concept that matters. If you can't handle it as
ax^2+bx+c it's kind of irrelevant - as far as mathematics learning
goes - whether you can handle it as 2x^2+4x+6. I find computers
totally irrelevant in the teaching of mathematics, even harmful;
a good teacher must try to develop abstract reasoning in their
students, and the less it's anchored in graphics and pictures
the better for the student in the long run.
> But I can do this with a bit of coloured chalk. Not to the impressive
>degree, though, but it's certainly possible. I think theory should be
>introduced and constantly reinforced by example.
Yes, but not by easy-to-digest examples. Theory must be
backed by non-numerical examples as much as possible.
If some of your students are going to be computer scientists,
or computer engineers, for example, pictures help nothing;
even examples help nothing: it's far more important to know
how to represent an arbitrary function than how to figure out
what shape the curve has and how cutely one can shade the
area under the curve.
Anytime you teach mathematics to computer people, think of
this: imagine you had to write a computer program that takes
an arbitrary n-argument function and n arbitrary, possibly
non-numeric arguments, and applies the arbitrary function
to the arbitrary arguments. THAT is what I, a computer man,
call applied mathematics; I don't need to know whether my
function is a parabola or whatever, but I need to know how to
represent it and how to apply it to an arbitrary set of arguments.
_alberto_
Alberto C Moreira
>With this I agree, but NOT with learning to manipulate numbers...
[the rest deleted]
I cannot see how one can have real insight without a skill base. I'll give you
an example, I got it from the book "The USSR Olympiad Problem Book", by
Shlarsky, Chentzov and Yaglom (Dover, $10.95) (by the way, this is a real
treasure chest of creative mathematics problems!) The problem is very
simple:
if n is an integer, prove that n^3 - n must be divisible by 3.
This is in my opinion, a problem that most junior high students in an upper
mathematical bracket should be able to solve in half a minute. It needs no
calculator, no graphics; it entails a few insights, one piece of skill and two
more advanced "A+" sort of reasoning. But the problem gives such a lot to its
solver, in terms of mathematical technique and vision, and in teaching how to
properly "milk a cow", in mathematical terms.
I'll present the solution in steps to illustrate my thoughts.
1. Insight: we're talking about divisibility, so let's see if I can
factor that expression somehow into a product.
2. Skill: it's obviously equal to n(n^2-1)
3. Further skill: it's equal to n(n+1)(n-1)
4. Insight: Aha! it's a product of three consecutive numbers.
5. Skillful Insight: in three consecutive numbers, one must be divisible
by 3: problem solved.
Now, the advanced steps; it takes a superior student to get beyond this point:
6. What if n is negative ? no problem, out of three consecutive negative
numbers one must be divisible by three as well.
7. What is one of these numbers is zero ? No problem either, the product
will be 0 and 0 is divisible by three.
Now, this is an easy problem, it can be solved in less than a minute by a
capable high school student. But it cannot be solved without the skill or
without the insight; both are needed, side by side. And the skill must come
easy and fast, because if a student takes five minutes to expand n^3-n into a
product (it should take the blink of an eye in the student's mind!) he or she
will have been sidetracked from the mainstream and take a lot more time and
energy than necessary to solve the problem. Incidentally, the SAT is full of
such problems, it almost seems that they're trying they hardest to sidetrack
the unwary in an attempt to identify the real good math talents; a lot of SAT
problems have two ways to be solved, one easy and long one and the other
fast way that needs insight and skill. And although the whole test can be done
by the easy method, it cannot be done in the time allocated!
What this problem gives is a little workout for one's creative powers. I would
even argue that items 6 and 7 are what Herman Rubin calls "problem
formulation", it actually takes a student that can formulate such alternatives
to achieve the complete solution of the problem. It also teaches the good
student a little fact, to be kept in the back of one's mind because who knows,
one day it may be useful: if a number has the form n^3-n, chances are it is
the product of three consecutive numbers, all having the same sign.
It's this kind of knowledge that allows people to use mathematics to describe
complex physical systems and then predicts the systems behavior by looking
at the equations and massaging them. Cow milking is an essential part of
any science professional's set of skills; it is far more important that tying
up equations to curves or integrals to shaded areas.
I'll leave you with the next problem, it's also food for thought: prove that
n^5-n is divisible by 5. (This one needs a tiny bit more skill, and a different
insight). To that one, I'd add a little one of my own: can you prove or
disprove that n^m-n is divisible by m for any positive m ?
Fantastic book, I recommend it warmly.
_alberto_
David Longley
alb...@moreira.mv.com "Alberto C Moreira" writes:
> In article <3oc4so$5...@cville-srv.wam.umd.edu> immo...@wam.umd.edu (Immortal)
> writes:
> >From: immo...@wam.umd.edu (Immortal)
> >Subject: Re: University level math Was Re: Skills vs. concepts
> >Date: 5 May 1995 03:09:12 GMT
>
>
> >s...@neon.chem> <3obgti$a...@ds2.acs.ucalgary.ca>
>
> >I think computers are a valuable tool. In the above example, you could
> >have a computer draw the curve and shade the area under, and then whip it
> >around the axis and produce a nice shaded object.
>
Just an oportunistic request really. Does anyone know of any graphical
tutorial on distributions, SDs, CIs, p levels etc? For the PC. Anything
which can be given to students to work through at their own pace,
preferably which is interactive to a degree, always seems seesm a useful
teaching aid I find. I have a good public domain PC tutor for DOS and
hardware etc, but nothing for probability, statistics and .....
sigh......."significance testing"
--
David Longley
Albert Y.C. Lai
d...@tabaqui.informatik.rwth-aachen.de (David Kastrup) wrote:
> Using a calculator, of course, multiplying the square root with itself does
> not always give the old number. If your calculator does not handle imaginary
> numbers, it will produce errors. In addition, the calculator usually would w
> you make think that there is only one square root [...]
A long time ago on a local BBS a highschool student once posted about
what 0/0 "should" be. He said it was infinity, because you can punch
0/0 into the calculator and check that it gives you "E" just like
other calculations involving infinity. He also said he was arguing
with his classmates on this issue, and he commented that his
classmates were stupid because they did not think of using the
calculator.
After I read that I posted back and said something like he did not
have a brain.
Oh, before you flame me, let me tell you that the university-level
users of the BBS applauded my act. :)
--
Albert Y.C. Lai tre...@io.org
00La...@wave.scar.utoronto.ca L...@titania.scar.utoronto.ca
http://www.io.org/~trebla/
Albert Y.C. Lai
tony...@aol.com (Tony Tosca) wrote:
> Well, Folks, I sure hope *some* of us teach *some* students long division,
> or else pretty soon there won't be anyone left who knows how to program
> those calculators (then again, we could stress recycling calculators as we
> "teach" recycling aluminum cans).
As a last resort, we can always teach long division at the PhD level.
:)
If that also fails, I am sure one day some PhD student will
re-discover it as a PhD thesis. :) :) Oh, maybe that student will
also get a Fields Medal for the re-discovery too... Sorry, I can't
resist, but maybe people figure that this re-discovery (they think it
is a new breakthrough) is so important to science, economics, and
world peace that in that year all five Nobel Prizes in Physics,
Chemistry, Medicine, Economics, and Peace are awarded to the student.
I am not sure whether I should put smileys here.
Is it just me, or is the quality of students in general degrading?
When I was in secondary school, it always occurred to me and actually
the whole class that past exam papers are always more difficult than
current exam papers; yet more senior students get higher marks, e.g.,
more students in Form Upper 6 of 1987 get high scores than those in
Form Upper 6 of 1988, but the 1987 exam is harder than the 1988 exam.
Similar situations seem to occur when I am in university too. The
past papers look so unfathomable. Worse, the past papers always cover
a much wider and deeper scope.
I start to believe that this is fate. Resistance seems futile. While
no one wants to, we have to accept the trend that students can take
less and less. I hope I am wrong here. I mean I wish.
David Kaufman
the idea that long division (by paper and pencil) is a useless skill
which I presented in my post titled, "Is Long Division a Useless Skill?"
Am I wrong in posting this to you students? I would like you to
respectfully educate your math teachers who are (unfortunately for you and
them) poorly trained in useful math ideas and skills.
If you want to understand what ideas and skills are important
about division, please read my current post titled, What is Important
About Division?" Read it carefully over and over until you see its
extreme significance for your proper education.
Most math teachers don't know how to use numbered words properly,
and this is I believe the most powerful skill for thinking clearly and
concisely about numerous valuable problems in math, science and everyday
life.
Remember, we are here to make a contribution that is useful to
others. So be kind in helping your poorly trained math teacher about
using numbered words in solving constant rate problems.
Remember, where there is a will to do good, there is a way.
You can make a significant contribution with my ideas that you
develop with hard work and then share with others.
Good look in this great adventure to become useful and therefore an
extremely attractive person to others of high character and good
quality. Imagine the possibilities!
Barry Kissane (kis...@murdoch.edu.au) wrote:
: Jeng Jia recently wrote:
: >Why would it be good to lose yet another basic skill?
: Long division is NOT basic, although it is a skill. It is not basic,
: because competent performance of the skill doesn't gain you anything else
: important. You can get by perfectly well without it, and a cheap calculator
: is a quite acceptable substitute. although it is a skill, it is a skill
: that isn't worth the time it takes to develop it.
: If you stop teaching long
: > division in favor of using that calculator, why stop there? Why not
: > quit teaching most math because most math can be done with a generic
: > graphing calculator? I mean, you certainly don't use algebra, geometry
: > or calculus much in the real world, so can all math be done with a
: > calculator because we could move on to more important things?
: Unless you regard the skilfull execution of standard algorithms as math (an
: argument that is not possible to seriously defend IMHO), it is not true
: that a generic graphing calculator does math. Whether or not you use math
: in the real world (is there a complex world? an unreal world??) is
: irrelevant to this issue. You still need to know mathematics to use a
: graphics calculator well, and in fact it is already clear that students can
: learn a great deal about mathematics with the help of one.
:
: > Even learning how to do square roots on paper isn't a waste of time.
: Why isn't it?? I guess that faithfully following a set of arcane
: instructions could be regarded as a good use of time (does it train us to
: follow rules?), but there are much better uses of the same amount of time.
: There is too little time for most kids to get a decent mathematical
: education without wasting it on silly things that have only historical
: significance (i.e., did you REALLY have to do that when you were a kid??)
: > These are concepts that must be learnt and are not just numbers that
: > can be punched into a calculator.
: Not so. There are no CONCEPTS of importance in the execution of the
: standard algorithm for finding square roots, unless one looks beyond the
: actual practice and sees some significance in training people to do
: thoughtlessly what their teachers tell them to do. I can imagine this
: argument being advanced by people who see schooling as a training for life
: (where people often tell their underlings to do as they are told; this
: occurs not only in the military, regrettably). But I have great difficulty
: seeing an argument related to getting a good mathematical education in any
: of this.
: In fact, teaching such things may well be dangerous, in the sense that
they : demand the thoughtless execution of a set of bizarre rules; if
students : come to regard such things as important aspects of mathematics,
it should : not surprise us that they turn away from mathematics as soon
as they get : the chance. And they do.
Sorry, I hit the wrong key to mess up the above paragraph.
Good luck. From: dav...@panix.com (?) May 6, 1995
Remember: Be good. Do good. Be one. And go jolly.