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).
____________________________________________________________
| |
| 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
john holford
doy...@cello.gina.calstate.edu says...
>
>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.
The dumbing down of America is based on rationalizing cop-outs, laziness,
and not doing our job. This is pathetic.
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
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.
Henry Choy
: 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.
Obviously this poster has forgotten or has not even taken high school
algebra
: The practice of these skills is extremely tedious
Yeah I can agree with that. I did long division on a big factoring
problem just a couple days ago. Wow! Time to hit the shower after.
: 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.
It would be neat if students have a bit of time on their own to
discover the long division algorithm from a definition of division.
--
Henry Choy "Math class is hard" - Barbie
e-mail: ch...@cs.usask.ca "Stupid is as stupid does." - Mrs. Gump
Henry Choy
: 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.
Harmful - anything like cigarettes can be harmful? Probably not.
If a student has trouble with math it probably doesn't stem from long
division coming early. However if there are a lot of algorithms taught
ahead of concepts, watch out.
: 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?"
K12 math is so simple you can teach it all in any grade. I kid you
not though a student might find the work brisk (not breakneck).
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 $3 calculator does the job to 8
>significant figures in a split second.
>Resolution: Elementary school teachers should teach their
>students how to mentally divide a one digit number into a 2
>digit number and the usual times table.
> C by David Kaufman, Apr. 30, 1995. (dav...@panix.com.)
> Be good. Do good. Be one. And go jolly.
I all to often see people divide numbers like 1000/25 on a calculator, when
it could be done several times faster by just looking at the problem.
Ignoring long division would accentuate the ludicrous of this. Maybe I am one
of the few, but I don't carry a mechanical calculator with me. Frequently, I
need to calculate a number and at least I can resort to long division. I
usually do carry my brain. :-)
Gary
[][][][][][][][][][][][][][][][][][][]
Gary Dyrkacz Hinsdale, Illinois
dyr...@mcs.com
[][][][][][][][][][][][][][][][][][][]
Henry Choy
: In article <3o1nd0$k...@cello.gina.calstate.edu>,
: 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
Henry checks Martin for a temperature.
Actually, it didn't hurt me, but when I look back at my K12 math it
was pretty stupid in a lot of ways--that could hurt.
Barry Kissane
>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.
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?"
|>
|> 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
David Kaufman
: Doesn't anybody check the Newsgroups line?
: Please stop cross-posting to the k12.chat.* groups. This discussion is
: not appropriate for these groups.
: --
: Dave Radcliffe radc...@alpha2.csd.uwm.edu
: A great many people think they are thinking when they are merely
: rearranging their prejudices. -- William James
What do you think students with Dave's idea?
Do you just want to do long division and not know whether
it is a waste of time or not?
Just beware. Most of the people supporting long division are just
talking nonsense.
So how can you tell?
I repeat. I live and breathe math and science so I can bring
college ideas to the elementary level. I have never had to
divide a polynomial by a polynomial or divide by long division.
My TI-30 Solar calculator that I bought for $35 in 1982 still
works fine even in dim light.
I think short division is fun because it is easy to do.
There are far more important mathematical forms that are
neglected that would serve you better. But most teachers
themselves are just copy cats and don't know how to think about
math or much else. Maybe they had too much long division
kind of practice.
I apologize to those who do make an effort to think about
important things.
But what is important? And where is it in K12.ed.math?
Do you think I'm boring children? I must go. Should I come back?
Caroline Pachaud
On Mon, 1 May 1995, Joe Keane wrote:
> I think that teachers like long division because it takes up a lot of time:
> ``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
>
>
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
Rick Link
>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.
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
Gordon Royle
>I note that a lot of college freshmen (whom I teach in a general education
>math course using the COMAP stuff, at a decent four-year college, rather
>than at IIT) cannot tell me how to solve a problem like
> Sally wants to put a row of 3/4 inch square tiles across the
> bottom border of an artwork that is 2 feet wide.
>They don't see that division is relevant.
I'm not really surprised, seeing as you haven't actually given a
problem.
This is a single factual sentence. Normally a problem requires some
sort of question to be answered.
Gordon
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
Gregory A. Sanders
math course using the COMAP stuff, at a decent four-year college, rather
than at IIT) cannot tell me how to solve a problem like
Sally wants to put a row of 3/4 inch square tiles across the
bottom border of an artwork that is 2 feet wide.
They don't see that division is relevant.
A number of them cannot tell me the remainder for 11 divided by 3
(the whole concept is unfamiliar--I don't just mean the method of
obtaining it). Notice that calculators don't tell you this.
If I ask, "how much is 124 divided by 4" perhaps one in fifty don't know how
to find the answer without a calculator. If given the answer, they don't
know what it means (it's news to them that, for example, 4 times the answer
equals 124, and similar inferences).
Some are a bit hazy on adding and subtracting negative numbers and on
dealing with negative signs on terms in general.
2 - (x - 3) = ?
These students are actually reasonably bright and most are traditional-age
undergrads. The explanation doesn't seem to be innate ability. Somehow
they graduated high school and got admitted to a somewhat selective
college. They are typical business/arts/humanities/social-sciences majors,
but not science/math/cs majors.
For what it's worth, the students who know how to do long division normally
can solve these problems and the students who can't are typically relying on
calculators.
Now, to get to the point, many of the respondents in this thread have said
long division is just a technique. So is punching in numbers in a calculator.
For better or for worse, students who learn long division generally seem to
understand division to some degree (even if they can't explain *how* long
division works). I really think they came to understand division because
one has to do a number of short divisions in the process of doing long
division, and you have to learn what division means in order to develop
any confidence at all in applying the long division algorithm--or for
that matter, to have any confidence about remembering the algorithm
correctly. Empirically, they seem to have been helped rather than hurt.
If you propose to get rid of long division, I'd like to hear counterproposals.
Substituting mechanical use of a calculator for mechanical use of long
division hardly inspires my confidence about that substitution leading to
increased understanding. How would you teach students division? What
would you teach them?
-- Greg
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.
Barry Kissane
R1S...@VM1.CC.UAKRON.EDU wrote:
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 agree. Only an incompetent carpenter would use long division or a
calculator to divide 27'5" by 12. Clearly the answer is (almost) 27.5". But
this is not an argument FOR long division; it's an argument AGAINST it!
The smart carpeneters, of course, would not use feet and inches :-) ... but
that's another story.
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.)
Tad Watanabe
: If you propose to get rid of long division, I'd like to hear counterproposals.
: Substituting mechanical use of a calculator for mechanical use of long
: division hardly inspires my confidence about that substitution leading to
: increased understanding. How would you teach students division? What
: would you teach them?
: -- Greg
I am proposing that we dealy the instruction of long division algorithm
until, say 6th or 7th grade, and use the time made available to do more
work on estimation, mental computation, proportional reasoning (not
learning how to set up the proportion and solve it by cross multiplying),
informal geoemtry investigations, etc. Division as an operation can be
taught without teaching algorithms. An algorithm serves the operation,
not the other way around. Many children naturally think of "division"
situation as missing factor problem: e.g. if you have 56 cookies and
sharing them among 13 people, how many would each get? They would do
something like: if each get 3, that would be 39, 4 would be 52. Can't
get 5 because then we would need 65. So, the answer is 4.
Elementary mathematics should be the time where children can explore
their world without having to memorize and master meaningless (to them)
procedures that were imposed upon them. Formalization of their
experiences can (and should) come later.
Nicolas Graner
> Elementary mathematics should be the time where children can explore
> their world without having to memorize and master meaningless (to them)
> procedures that were imposed upon them. Formalization of their
> experiences can (and should) come later.
Many (most?) young children *love* to memorize and master meaningless
(to them) things such as poems, songs and procedures, and are
extremely good at it. It is probably much more difficult to motivate
older children to learn such a boring thing as long division, and more
difficult for them to memorize it. Let's teach each age what they are
good at: younger children should memorize language and drills, older
children should build up abstract models and formalise what they've
learned before (I am simplifying to the extreme, of course). There is
a lot of common sense in traditional curricula (and a lot of crap that
should be cleaned, too :-) )
Nicolas
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
Thomas W. Cowdery Jr.
Distribution:
: : >They don't understand the concept. They should have been taught what it
: : >means in grade 1-5, then taught the algorithm.
: 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
Actually, I agree with both of you. The kids *should* be taught what
division means and then taught the algorithm. Their level of abstraction will
be limited at that age (shoot, I run into high school seniors whose ability to
think abstractly is very limited), but there are ways to get the basic idea
across to the typical 4th or 5th grader.
Truthfully though, as a high school teacher who teaches predominantly
freshmen, I don't see many students who don't understand the basic concept of
division, at least for integers. I do see many who are not very skilled with
the algorithm. They seem to have difficulty doing anything that is repetitive
and involves multiple (if similar) steps. Even if they can get the answer on
the calculator, I believe the mere fact that multiple steps and iteration are
inherent in long division makes it a valuable lesson. (NO, I don't believe
that is the ONLY reason we should continue to teach long-division, but it is
one reason.)
A bigger problem (IMHO) is difficulty with fractions & decimals. Those
concepts do often seem weak, and the skills working with the related
algorithms are equally weak (especially with fractions). I have argued many
times against what I percieve to be the *overuse* of calculators. If all the
calculators were used for was integer operations, I would be less concerned.
But especially with the advent of calculators capable of doing fractions, I
fear those skills will be the next thing people start dropping from the
curriculum. (I can hear it now - they never get it any way, a calculator is
faster and more accurate). Which is fine, until they get into Algebra and are
supposed to combine 3x/4a^2c and 5y/6ab^2 (or something analogous). Then the
kids who cannot even find common denominators for halves and thirds run into a
brick wall. That is something they need to *do* for themselves often enough
that it is indelibly etched into their memory. There simply is no substitute
for experience.
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
Thomas Cowdery || TOP TEN REASONS TO BECOME A TEACHER
twc...@rs6000.cmp.ilstu.edu || #7 Having your own child in class
thomas....@deskmic.com || counts as *quality* time.
\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/
Tad Watanabe
school math curriculum. I don't know about the original poster of this
thread (David Kaufman), but I am not advocating the abandonement of long
division, or any algorithms. I am saying that we should put much less
emphasis on it in elementary school and possibly postpone the formal
instruction of algorithms until middle school.
Someone has said learning algorithms and concepts are both important, and
I agree. However, I would like to say that, in most cases (if not
always), concepts lead the way. Unfortunately, when it comes to teaching
of arithmetic in elementary school, it is more less algorithm-driven.
For example, most textbooks (and teachers) write simple single digit
addition vertically, but I don't see why we do that other than to lead
the students toward the standard algorithm. Research shows that children
do not completely understand place values until second or even third
grade level, but understanding of place value concept is essential for
understanding algorithms. So, for those students who don't understand
place values, the addition algorithm is a magic. They just try to
memorize the process. The greatest advantage of the algorithm is that it
is relatively easy to memorize. Therefore, we get students who have no
conceptual understanding of what they are doing but are experts in
carrying out meaningless (to them) process. I am sure most people here
would agree that this is not desirable.
Also, someone said (and I heard many teachers tell me that, too) that
multiplication is the same as addition. Multiplication is not the same
as addition (repeated). It CAN be interpreted as such, but as an
operation they are different in nature. For example, when you multiply
length and width of a rectangle to obtain the area, both factors have the
unit of length (say inches), but not the product (square inches). The
sum of two numbers have the same unit as addends. There are several
important differences between multiplication and addition.
Multiplicative reasoning is essential for more complex mathematics, but
when we treat multiplication as simply repeated addition, we are limiting
our children to additive world. The result is often those kids who can't
make sense of fractions, cannot deal with proportional reasoning, etc.
As someone else has said, there are MANY important mathematical ideas
primary grade children can learn without having to worry about memorizing
division (or any arithmetic operations) algorithms. Early mathematics
education is essential, and it should be filled with opportunities to do
mathematics through exploration and experimentation. Formalizing of
their experiences can wait till later, when teachers (collectively) can
spend much less time to accomplish more.
Gary Tupper
RE: skills/concepts & long division:
When in doubt, I generally blame the teacher.... (the kids just dunno
and the parents are out at bingo).
Nevertheless, if we percieve math as simply a 'game' with pretty clear
rules, procedures, definitions, etc.... then no harm should come of the
long division game (some students may exel, some students may be bored etc).
The 'insight' games, the 'proof' games and the 'derive' games are all fun
too - at various developmental stages and for certain individuals.
The 'problem' seems to arise when we forget its a game and think its about
reality (ie 'real-world' physical apple-pie reality). We have succumbed to
this siren call when we attempt to 'explain' (rather than exemplify) a
concept such as addition of integers by pulling out the marble bag....
My grade 11 students are usually surprised at the question "What is the
volume of a litre of pure H2O mixed with a litre of pure alcohol?" Or maybe
its the answer that surprises....
Anyway, I blame the school system.....
Gary Tupper, Terrace
MELVIN BILLIK
I would suggest you invest in a cheap
calculator watch for those emergencies when a calculator
is not handy.
Sorry -- there must be a better reason for teaching long
division than saying "because calculators may not be handy."
MEL
Terry Moore
> 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.
> 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.
In a sense this is true. But there are different kinds of effective
procedure. One is an algorithm for performing a division by
hand, another is an algorithm for deciding what operations are
required to solve a specific problem. Unfortunately, there is no
known algorithm for doing this, but any effective problem solver
is using some personal algorithm even if s/he cannot describe it.
However, there can be no guarantee, in advance, that a problem
solver's personal algorithm will solve the problem. Good
problem solvers are good at guessing whether a problem is
tractable and choosing promising approaches. This is a difficult
skill that we should be helping our students to aquire. Teaching
any sort of algorithm as a bag of tricks to be learned may, perhaps,
sometimes be useful, but it certainly has nothing to do with the far
more important task of teaching students to apply their knowledge
to solving problems.
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan
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:
>: >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..."
Kevin Bruce Pease
>David G Radcliffe (radc...@alpha2.csd.uwm.edu) wrote:
>: Doesn't anybody check the Newsgroups line?
>: Please stop cross-posting to the k12.chat.* groups. This discussion is
>: not appropriate for these groups.
>What do you think students with Dave's idea?
He's probably right. Personally, I don't care.
>Do you just want to do long division and not know whether
>it is a waste of time or not?
Yes. I want to do long division for the rest of my life. I
want Santa, the Easter Bunny, and Charlie Brown to lock me up in a room with
stack after stack of books full of long division problems. I want
them to threaten to beat me with garden hoses if I don't finish all of
the divison problems in a specified time. I just want to do long
division all the time, because it's so FUN!
David, you make plenty of broad sweeping statements about how
"long division is Evil! It's harmful! Tedious! Useless!" Yet you
provide very little evidence, and very few examples, to support your
claims. Long division has always served me well, and it has never
hindered my ability to comprehend the concept of division, which, when
you get right down to it, is NOT really that big a deal.
>I repeat. I live and breathe math and science so I can bring
>college ideas to the elementary level.
Strange... I've met plenty of people who "live and breathe
science" who can't explain concepts to elementary students to save
their souls. They're so into the concepts that they forget that the
kids don't know the basics, and can't explain things worth a damn.
>I have never had to
>divide a polynomial by a polynomial or divide by long division.
I find that *extremely* hard to believe, and I would even go
so far as to say it's bordering on an outright lie.
>My TI-30 Solar calculator that I bought for $35 in 1982 still
>works fine even in dim light.
Doesn't mean it always will, and doesn't mean you'll always
have it handy when you need it.
>I think short division is fun because it is easy to do.
Personally, I think all math is pretty crummy, and don't like
it one bit. It's useful, and I see the need for it, but I hope to
never have to take another math course again in my life. :)
>There are far more important mathematical forms that are
>neglected that would serve you better.
Instead of just saying, "Division, like . . . uh, sucks, or
something." Why don't you offer up some of these "far more important"
forms as alternatives? It's easy to criticize and find problems.
It's infinitely harder to come up with a solution.
>But most teachers
>themselves are just copy cats and don't know how to think about
>math or much else.
I take great exception to that statement. I know plenty of
teachers who are excellent, and very capable thinkers. Teachers can
only do so much when it comes to learning - if you don't want to
learn, you're not going to, even if you have thirty Nobel prize
winners hammering the concept into your head 24 hours a day. There's
a lot of dependency on the learner's attitude when it comes to
learning.
>Maybe they had too much long division kind of practice.
Gee, how mature. Not only can he bore people around the
world, he can insult them, too.
>Do you think I'm boring children? I must go. Should I come back?
Answers in order: Yes. No.
I find your ideas to be fairly trite, and, in all honesty, a
lot of stuff you talk about isn't all that novel and wonderful.
[shrug] I don't know how other teachers feel about you, but as a
student, I don't think I've turned out all that badly, and I've been
subjected to all the "evils" you seem to see... I'm a graduate of
public schools (had PLENTY of *excellent* teachers there), I did lots
of long division (I'm still perfectly capable of understanding
concepts), and, quite honestly, I think I'm a fairly capable thinker.
Now I'm no Einstein, but despite all of the things you seem to think
are counter-productive, I think I've turned out okay.
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
David Kaufman (dav...@panix.com) wrote:
: David G Radcliffe (radc...@alpha2.csd.uwm.edu) wrote:
: : Doesn't anybody check the Newsgroups line?
: : Please stop cross-posting to the k12.chat.* groups. This discussion is
: : not appropriate for these groups.
: What do you think students with Dave's idea?
Bullsh**.
: Do you just want to do long division and not know whether
: it is a waste of time or not?
I think that long division is a very useful skill, and it can be used in
mental math. This, BTW, is coming from an 8th-grader who understands
long division and is learning dividing polynomials in Algebra I. I
learned it 4th grade and have used it repeatedly. I do definately use it
in mental math sometimes, often with quicker answers than someone using a
calculator. There are some numbers that look like--AAGH! I can't do *that*
in my head!--when you first look at them, then you notice that it's really
easy if you try it (ie 255/3-- 25/3=8R1, 15/3=5, so 255/3=85). I will say,
though, that knowing the mathematical concepts behind the algorithm
you're using is very, very handy, especially when you get into more advanced
stuff. For instance, transferring fration skills into rational expression
skills.
: Just beware. Most of the people supporting long division are just
: talking nonsense.
Give me several examples of 'nonsense', please.
: So how can you tell?
: I repeat. I live and breathe math and science so I can bring
: college ideas to the elementary level. I have never had to
: divide a polynomial by a polynomial or divide by long division.
: My TI-30 Solar calculator that I bought for $35 in 1982 still
: works fine even in dim light.
Ah, I see. You carry a calculator with you at all times, you can draw it
quicker than a gunfighter can have his gun out, and you can punch in
whatever you need to do every time, without error, even before anyone
could do it mentally. For example: when I'm at the computer and it's tied
up in a download. Sometimes I know that a file is 412028/7002022 done and
has taken 27/60 of an hour. I can divide though that and get an accurate
answer, or I can say well, 4120 goes into 7002 once, leaving 2882, and
4120 goes into 28820 five times, so 412028/7002022 is roughly 15/100 or
3/20. Then I'd probably round 27/60 to 1/2 and say that the file would take
an hour to do 3/10, so it would take roughly 3 1/3 hrs. to download the file.
: I think short division is fun because it is easy to do.
: There are far more important mathematical forms that are
: neglected that would serve you better. But most teachers
: themselves are just copy cats and don't know how to think about
: math or much else. Maybe they had too much long division
: kind of practice.
: I apologize to those who do make an effort to think about
: important things.
: But what is important? And where is it in K12.ed.math?
: Do you think I'm boring children? I must go. Should I come back?
I think that k12.chat.junior is definately an okay place for discussions
on matters like this, as most JHS kids are getting pretty damn close to
entering 'the real world' if they haven't already, and its a good skill
to be able to pick apart somebody else's pro/con arguments. In other
words, you shouldn't just sponge in everything somebody says to you. For
example, my SS teacher was debating mostly with me, but also with an
invitation to the class to participate, Affirmative Action. She was
spewing out stuff that I'd think I'd hear on ultra-right wing radio talk
shows, and most of the kids in the class just sat on their rear ends and
said *nothing* about this. JSYK, my teacher later said that the class
should wake up, since everything she was saying was *totally* against
what she believed.
--
--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
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.
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.
Levy4
concept, then the student will be able to regurgitate numbers without the
slightest idea of how to apply their new found skill. Also, equations
that are memorized tend to become unretrieveable when they are not
cemented into the brain cells with the logic behind the algorithm. The
logic and underlying concept helps the student to visualize and make sense
out of the abstract algorithms and formulas that are thrown at them at a
dizzying pace. The best students end up being those who have the best
memory recall, rather than those who are best able to apply the concepts.
We do not need human computers who are capable of spewing out mathematical
facts, but thinking beings who can apply concepts, use logical thinking,
and resolve complex problems. This can only happen when the emphasis is
put on teaching the concepts.
For example, when my son was in fourth grade and learning long division,
he asked his father for assistance. My husband could not do the problem
without the aid of a calculator. My husband was taught long division in
elementary school, but had lost the ability to recall the algorithm.
Because he was not taught the logic behind the algorithm, he was not able
to reconstruct it in order to help out our son. They slaved for hours on a
homework assignment that should have taken 15 minutes and were still not
able to do the work. Since I was out for the evening, my son put his
homework away and got up extra early the next morning to ask me for
assistance.
I explained that division is just a short cut for subtraction, just like
multiplication is a short cut for addition. I showed him how to break the
long division down into smaller groups so that he could do the problem
more easily. He was in a hurry to finish the work and did not want a long
explanation behind the algorithm. He just wanted me to refresh his memory
of the algorithm so that he could get his work done quickly. I refuse to
teach an algorithm without an explanation of the underlying concept. I
want my son to be able to figure out the algorithm on his own if he has
forgotten it.
If students are not taught the logic behind an algorithm that is as simple
as long division, then where will our mathematicians come from who will
give us the algorithms of the future? Where will our scientists and
engineers come from who must find new methods and procedures for solving
complex problems? I am not interested in teaching children to be
automatons, but thinking, reasoning humanbeings.
Andrea L.
Raymond Cottrell
Hello!
I am moving to Las Vegas and would like to get some info on job
prospects, etc. Please e-mail me with any info you may have. Thanks!
Sam Powell
To me, learning the steps was a huge step in learning mathematics
overall. I still look back at those days at learning long division,
and believe it helped launch my strong interest in mathematics today.
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
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