Memorization comes before understanding
Patrick Reany
I want to start a conversation about the role
of memorization in the learning of math and
physics. I did this sometime ago and found
the discussion interesting.
My basic premise is that:
Memorization comes before understanding.
But to fully appreciate what this means,
one has to have a good definition of what it
mean to "understand" something, right. Also,
is wouldn't hurt to have a definition of what
memorization means, either.
Now, I want to counter the obvious charge
that all I'm advocating is for students to
memorize their textbooks word-for-word.
This I do NOT do. And I'm sure that even
my strongest critiques will agree that some
memorization of new material must be
accomplished somehow, if conceptual
understanding is to be achieved. So, the
question is, "How to find the happy
balance?"
See,
http://www.ajnpx.com/html/Memorize.html
Patrick
Daniel McLaury
entire higher-math career, since I started Calculus my Sophomore year
of highschool, I have made it a point *never* to memorize anything.
The sole meaning of mathematics, and life itself (Math is life;
everything else is just details...) is understanding and proof. To
memorize is to go against this. Even those who are not so in love w/
math as I should take heed of this; as an example, my understanding
made it possible for me to be the *only* student to score an A on my
last university math test.
Flip Flippy
news:3D29CB81...@asu.edu
> I want to start a conversation about the role
> of memorization in the learning of math and
> physics.
Hello Patrick.
This is an interesting question indeed and I can understand
the lop-sided results.
I would like to say that you need to take into account what the
universities are after, "the almighty profit". Universities have
become businesses and as such they must graduate (i.e., keep) as
many students as possible.
What this means to me is that "memorization" of mathematics is
key to succeed in that goal (profit). It would be fantastic to
believe that all students wants to understand proof, concepts and
algorithms, but alas that is not the case.
Many students want a good GPA to get a good job, period. They really
have no interest in understanding the material over memorizing
something. Note: in some cases there may be nothing wrong with this.
The danger that I have seen is to those poor unsuspecting students who
want to further themselves. For example, a friend of mine tried taking
the first three exams to become an actuary and failed them all, twice!
The problems expected that the student could "think", "deduce" and
"work" the problem. Many students will fail this because all they
did was "memorize" rather than "understanding".
Also, if professors took the time to review all of the proofs in
sections,
they would not be able to complete all of the material, but that
should be okay. I feel this because a student who can "think" and
"solve" problems by having an "approach" and understanding of the
"concepts" is worth ten of the others that have received good grades.
This is a real dilemma these days for universities because they are
diploma mills in search of the almighty dollar at the expense of
graduating what can be termed "mathematicians" as opposed to whatever
the other can be called.
I hope we can see some of your results in this area.
Flip
--
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Patrick Reany
Daniel McLaury wrote:
> Nothing could possibly be farther from the truth.
I take it you read nothing on my website.
> Throughout my
> entire higher-math career, since I started Calculus my Sophomore year
> of highschool, I have made it a point *never* to memorize anything.
So, you're saying that you have nothing of
math content in your head? Then you must
know absolutely NO math.
> The sole meaning of mathematics, and life itself (Math is life;
> everything else is just details...) is understanding and proof.
How can one understand a concept C if one has NOTHING
at all in one's memory related to that concept?
> To
> memorize is to go against this. Even those who are not so in love w/
> math as I should take heed of this; as an example, my understanding
> made it possible for me to be the *only* student to score an A on my
> last university math test.
Memorization of X means to undergo a ANY process
to change your brain states such that, at the end of that
process, X can be retrieved from memory on demand
and without error, whereas before the end of that
process, this couldn't be done. Memorization
has no necessity to any particular methodology
of making such a change in brain states, which you
seem to believe.
Understanding concepts is a PROCESS of finding
relationships between parts. If you don't have the
parts memorized, you have no hope of finding the
relationships among them.
Patrick
Patrick Reany
Flip Flippy wrote:
> "Patrick Reany" <re...@asu.edu> wrote in message
> news:3D29CB81...@asu.edu
> > I want to start a conversation about the role
> > of memorization in the learning of math and
> > physics.
>
> Hello Patrick.
>
> This is an interesting question indeed and I can understand
> the lop-sided results.
>
> I would like to say that you need to take into account what the
> universities are after, "the almighty profit". Universities have
> become businesses and as such they must graduate (i.e., keep) as
> many students as possible.
I am not really interested in teaching people how
to play the game of getting through formal
education in this thread. Yes, that is an important
subject, but not really what I want to deal
with here.
> What this means to me is that "memorization" of mathematics is
> key to succeed in that goal (profit). It would be fantastic to
> believe that all students wants to understand proof, concepts and
> algorithms, but alas that is not the case.
Sometimes one has to memorize just to work
perfunctory problems on tests. But that's not of
interest to me here. I am interested in teaching
people how to get a deep understanding of
concepts in math and physics. If one can't get
the basics memorized one can't quickly build
on top of that knowledge the next layer of
understanding.
>
>
> Many students want a good GPA to get a good job, period. They really
> have no interest in understanding the material over memorizing
> something. Note: in some cases there may be nothing wrong with this.
I have no interest in reaching those kinds of students. I'm
interested in getting to those few students that really
want to get a deep understanding of deep concepts
of math and physics.
> The danger that I have seen is to those poor unsuspecting students who
> want to further themselves. For example, a friend of mine tried taking
> the first three exams to become an actuary and failed them all, twice!
> The problems expected that the student could "think", "deduce" and
> "work" the problem. Many students will fail this because all they
> did was "memorize" rather than "understanding".
That's right. I NEVER preach that memorization
by itself is good for understanding. One memorizes
the parts of a concept and then actively constructs
logical relationships on those parts: That is the
process of understanding!
> Also, if professors took the time to review all of the proofs in
> sections,
> they would not be able to complete all of the material, but that
> should be okay. I feel this because a student who can "think" and
> "solve" problems by having an "approach" and understanding of the
> "concepts" is worth ten of the others that have received good grades.
>
> This is a real dilemma these days for universities because they are
> diploma mills in search of the almighty dollar at the expense of
> graduating what can be termed "mathematicians" as opposed to whatever
> the other can be called.
>
> I hope we can see some of your results in this area.
>
>
I'm not sure what you mean. My webpage on this
has a lot more details on all this. I don't do
research on this topic, so I have no results to
share.
Patrick
Neil W Rickert
>Daniel McLaury wrote:
>> Nothing could possibly be farther from the truth.
>I take it you read nothing on my website.
I take it that he probably read a lot of your web site and
sincerely disagreed.
>> Throughout my
>> entire higher-math career, since I started Calculus my Sophomore year
>> of highschool, I have made it a point *never* to memorize anything.
>So, you're saying that you have nothing of
>math content in your head? Then you must
>know absolutely NO math.
Know, that is not what he said at all. It is dishonest of you to
misrepresent what was said.
>> The sole meaning of mathematics, and life itself (Math is life;
>> everything else is just details...) is understanding and proof.
>How can one understand a concept C if one has NOTHING
>at all in one's memory related to that concept?
You have it completely backwards. One cannot have a memory of
what one does not understand.
In the case of mathematics, the understanding is usually sufficient and
there is no need to memorize.
>> To
>> memorize is to go against this. Even those who are not so in love w/
>> math as I should take heed of this; as an example, my understanding
>> made it possible for me to be the *only* student to score an A on my
>> last university math test.
>Memorization of X means to undergo a ANY process
>to change your brain states such that, at the end of that
>process, X can be retrieved from memory on demand
>and without error, whereas before the end of that
>process, this couldn't be done.
I expect that the theory of memory implicit in that assertion is
bogus.
>Understanding concepts is a PROCESS of finding
>relationships between parts. If you don't have the
>parts memorized, you have no hope of finding the
>relationships among them.
Naive.
Mathematicians are concerned with relations between infinitudes of
things. If they had memorize all of those uncountably many things as
prerequisite to finding relations, they could never get started.
Ben Golub
example something very simple (though very beautiful), the definition of a limit.
* * *
Let f(x) be defined for all x in some open interval containing the number a, with
the possible exception that f(x) need not be defined at a. We will write
lim_(x->a) f(x) = L
if for any number epsilon > 0 there exists a number b > 0 such that
|f(x) - L| < epsilon if 0 < |x-a| < b
* * *
Okay. Now, that's a bunch of words and symbols at the surface level. The
definition could be memorized by somebody with no mathematical knowledge
whatsoever as a series of English words, or by somebody with no knowledge of
English or math as a series of sounds (let eff of ex bee dee find, etc.) Of
course, such a memorization is useless and not even worth discussing.
Nevertheless, anybody who wants to do calculus at any real level needs to know
this definition intimately. No question, he should be able to write it on a
blackboard without pausing, and there is some element of memory in that. But it
shouldn't be the kind of trivial memory described above, which is just mindless.
The kind of memory the student must have is a memory of the CONCEPT, which is
significantly distinct from a memory of WORDS or SYMBOLS or anything of that sort.
Though we can translate a concept into words, the concept does not equal the words
that express it, in the same way that the keyboard on which I am typing this is
not equivalent to the word "keyboard."
In order to attain a proper conceptual understanding, the student must explore the
idea of what a limit means with a good teacher or a good book. He must twist it
around in his head, investigate familiar functions, interpret |x-a| and |f(x) - L|
as distances, and so forth. He must play mental games with the definition, test it
for holes. I think this is an experience familiar to a lot of people involved in
math, so it's not anything earth-shattering, but when I think of a limit I think
of the familiar drawing where we see epsilon and beta intervals on the y and x
axes, respectively. It sort of moves and wiggles so that I can see that changing
one of the intervals affects the other. I am keenly aware that we can make the
interval epsilon infinitely small, which is the whole trick, and so forth.
Now, if somebody asked me right now to write the defintion, I would translate the
CONCEPTUAL understanding I have in my head back into English words and
mathematical symbols. The memory I have is indeed memorized, but not in the
mindless sense that a lot of people hate (what comes after x in that formula!?
darn!).
So in my opinion in order to memorize math for later use in anything other than a
dronish, trivial, and useless fashion, we have to translate English and math
symbols into concepts, internalize them, and then do the reverse when asked to
remember. Thus, in math, understanding and useful memorization are inseparably
intertwined.
Just a few cents from a math student.
Ben
Patrick Reany <re...@asu.edu> wrote in message news:3D29CB81...@asu.edu...
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Bill Taylor
|> Memorization comes before understanding.
Blimey, watchit mate!
You're going to cop a load of old cobblers from Herman Rubin for this heresy!!
I would not go quite as far as the other fellow in denouncing memorizing
anything AT ALL, but there is a grain of truth in what he said.
If you want a quick and dirty epigram to cover the case, I would suggest this:
Manipulation comes before understanding.
=======================================
Oh dammit - now I'm going to cop it from old Hermy as well! Oh well.
Eggshelly, my epigram isn't quite fully correct, there CAN be understanding
without manipulation, but rarely, and it is likely to be a more complete grok
afterwards. Until you've done a few manipulations, usually called "examples",
your understanding is likely to be patchy at best.
As for the "no-memorizing" part of the rule - I would soften it to this:-
"Don't memorize anything until you have redone the deduction of it for yourself."
Only then will you fully grok what it is you're memorizing, and attempting
to understand. Then, after you have achieved this step, you can go ahead
and memorize as much of it as you think fit. That'll then be a big help.
There is no chance for this advice.
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Free advice is worth what you pay for it.
------------------------------------------------------------------------------
Robert Israel
>My basic premise is that:
> Memorization comes before understanding.
"... comes instead of ..." is more like it, in many cases.
Once it has the conceptual framework, the mind finds it easy to
fit in the individual pieces, because they all fit together neatly
and logically. Without that framework, you have a jumble of unrelated
facts and techniques that take a lot of effort to memorize. If you
understand the basic ideas, the actual amount of memorization needed for
the average math or physics course is very small. I always tried to
memorize as little as possible - as long as I could readily reconstruct
some idea when needed, there was no point in memorizing it. In addition,
this type of attitude makes it easier to tackle new problems that don't
fit a memorized pattern.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
The World Wide Wade
wrote:
> > Throughout my
> > entire higher-math career, since I started Calculus my Sophomore year
> > of highschool, I have made it a point *never* to memorize anything.
>
> So, you're saying that you have nothing of
> math content in your head? Then you must
> know absolutely NO math.
One need not "memorize" to remember.
--WWW.
me...@cars3.uchicago.edu
>Patrick Reany <re...@asu.edu> writes:
>
>|> Memorization comes before understanding.
>
>Blimey, watchit mate!
>
>You're going to cop a load of old cobblers from Herman Rubin for this heresy!!
>
>I would not go quite as far as the other fellow in denouncing memorizing
>anything AT ALL, but there is a grain of truth in what he said.
>
>
>If you want a quick and dirty epigram to cover the case, I would suggest this:
>
> Manipulation comes before understanding.
> =======================================
Now you're talking. I'll second that.
>
>As for the "no-memorizing" part of the rule - I would soften it to this:-
>
>"Don't memorize anything until you have redone the deduction of it for yourself."
>
And this as well.
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
Dr Arm®
>
> Mathematical memorization in its proper form is a type of understanding. Take for
> example something very simple (though very beautiful), the definition of a limit.
Of course. It's how your algebra teacher instilled the product of two
negative numbers into your head.
But mathematical memorization can also be the equivalent of beating your
kids: Works good only for solving previously addressed problems.
da
Ben Golub
> But mathematical memorization can also be the equivalent of beating your
> kids: Works good only for solving previously addressed problems.
Well, I disagree. For instance, a true understanding of the definition can be used
to prove a wide variety of limits, some of them quite nontrivial. This is very
distinct from the true inability to solve new problems gleaned from a rote
memorization of formulas and words, not concepts.
When we commit mathematical concepts to memory by understanding them, we enable
ourselves to link them later or to use the "tricks" we learn to solve completely
new problems.
I agree with OP to an extent in that you can't ever do any interesting math unless
you have a certain critical mass of information conceptually committed to
memory -- but it must be conceptual. (And the more advanced the math you want to
do, the greater the critical mass, of course). Otherwise, your garden variety
amateur mathematician (read: crank or JSH) really could prove FLT (AAAH! AAH!)
In addition, to do interesting math, you need a creative mind that can make good
use of those concepts while seeking out new ones with which to grow its abilities.
So its an intertwined process.
Ben
Ben Golub
> One need not "memorize" to remember.
Memorization: let eff of ex bee dee find for all ex in sum oh pen in ter vahl....
Remembering: knowing what a limit is.
As others have said pretty well, you need to manipulate before memorizing, to make
the concept our own. Only then will the remembering be worth a darn.
Neil W Rickert
>If you want a quick and dirty epigram to cover the case, I would suggest this:
> Manipulation comes before understanding.
> =======================================
That sounds right.
David C. Ullrich
>In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>
>>My basic premise is that:
>
>> Memorization comes before understanding.
>
>"... comes instead of ..." is more like it, in many cases.
>
>Once it has the conceptual framework, the mind finds it easy to
>fit in the individual pieces, because they all fit together neatly
>and logically. Without that framework, you have a jumble of unrelated
>facts and techniques that take a lot of effort to memorize. If you
>understand the basic ideas, the actual amount of memorization needed for
>the average math or physics course is very small.
Maybe people are talking past each other. The people who say there
should be less emphasis on memorization, etc, are right. But I
suspect maybe we're talking about memorizing different things:
For example when I teach linear algebra one of my big problems
is getting the kids to believe that they actually do have to
"memorize" the definitions - I try to use a different word
because they find the word "memorize" so scandalous. But there's
no way to figure out that the definition of "basis" is
"independent spanning set", you simply have to memorize that
definition.
Of course you don't have to memorize the definition verbatim,
but you _do_ have to memorize in some sense exactly what the
definition _says_. Dealing with students who have essentially
no experience manipulating abstract concepts, it seems that
suggesting they memorize the definitions verbatim at first
is the easiest way to get them to memorize exactly what the
definition says. Gets frustrating - they have no idea that
if they want to show something is a basis they need to
show that it's independent and show that it spans, when
if they'd just concede and memorize the definition then
they _would_ know that this is what is required.
What they learn instead is garbled and/or wrong versions
of the definitions. If they _knew_ the definitions then
they could eventually see how the concepts fit together,
which is of course the goal. But it's impossible for them
to see how the concepts fit together, because with their
garbled versions of the definitions they _don't_ fit
together.
>I always tried to
>memorize as little as possible - as long as I could readily reconstruct
>some idea when needed, there was no point in memorizing it. In addition,
>this type of attitude makes it easier to tackle new problems that don't
>fit a memorized pattern.
Of course. But you _did_ memorize _something_ - in my experience
with kids these days they don't believe that they should have
to memorize _anything_, and that attitude makes it much _harder_
to tackle new problems. The reason you found you could tackle
new problems is you _did_ believe that it was necessary to
know exactly what the words meant. Making it easier to learn
new things is exactly why I wish I could convince them that
it's necessary to memorize _some_ things - when I say that
memorization is much more important than a lot of people
admit these days I'm certainly not saying that they should
memorize the sort of thing I suspect you're alluding to as
things you didn't memorize, just the opposite; the point
to memorizing exactly what things mean is to allow them
to figure things out without memorizing the answers to
millions of problems.
>Robert Israel isr...@math.ubc.ca
>Department of Mathematics http://www.math.ubc.ca/~israel
>University of British Columbia
>Vancouver, BC, Canada V6T 1Z2
David C. Ullrich
James Hunter
Bill Taylor wrote:
> Patrick Reany <re...@asu.edu> writes:
>
>
> If you want a quick and dirty epigram to cover the case, I would suggest this:
>
> Manipulation comes before understanding.
> =======================================
>
> Oh dammit - now I'm going to cop it from old Hermy as well! Oh well.
>
> Eggshelly, my epigram isn't quite fully correct, there CAN be understanding
> without manipulation, but rarely, and it is likely to be a more complete grok
> afterwards. Until you've done a few manipulations, usually called "examples",
> your understanding is likely to be patchy at best.
>
> As for the "no-memorizing" part of the rule - I would soften it to this:-
>
> "Don't memorize anything until you have redone the deduction of it for yourself."
Better yet, you never listen to anything a school teacher
says about *memory*, since they're all obviously
lying assholes.
>
>
> Only then will you fully grok what it is you're memorizing, and attempting
> to understand. Then, after you have achieved this step, you can go ahead
> and memorize as much of it as you think fit. That'll then be a big help.
>
> There is no chance for this advice.
>
> ------------------------------------------------------------------------------
> Bill Taylor W.Ta...@math.canterbury.ac.nz
> ------------------------------------------------------------------------------
> Free advice is worth what you pay for it.
> ------------------------------------------------------------------------------
You paid nothing, so you got nothing, which
is the only thing assholes actually deserve,
so it's not only free advice, it's fair advice.
Nico Benschop
Me too! There's nothing like manipulation! -- NB
Patrick Reany
Neil W Rickert wrote:
> Patrick Reany <re...@asu.edu> writes:
> >Daniel McLaury wrote:
>
> >> Nothing could possibly be farther from the truth.
>
> >I take it you read nothing on my website.
>
> I take it that he probably read a lot of your web site and
> sincerely disagreed.
>
> >> Throughout my
> >> entire higher-math career, since I started Calculus my Sophomore year
> >> of highschool, I have made it a point *never* to memorize anything.
>
> >So, you're saying that you have nothing of
> >math content in your head? Then you must
> >know absolutely NO math.
>
> Know, that is not what he said at all. It is dishonest of you to
> misrepresent what was said.
It is not dishonest to get the poster to define his
or her terms.
>
>
> >> The sole meaning of mathematics, and life itself (Math is life;
> >> everything else is just details...) is understanding and proof.
>
> >How can one understand a concept C if one has NOTHING
> >at all in one's memory related to that concept?
>
> You have it completely backwards. One cannot have a memory of
> what one does not understand.
You seem to have no idea how the human mind
accomplishes "understanding." It starts off as
bootstrapping: One memorizes and then
builds on top of that. [snip]
>
> Mathematicians are concerned with relations between infinitudes of
> things. If they had memorize all of those uncountably many things as
> prerequisite to finding relations, they could never get started.
Obviously no one is supposed to memorize
an infinite number of things. Where did I ever
say one should?
Patrick
Patrick Reany
The World Wide Wade wrote:
To have something in memory is to have
it "memorized."
Patrick
Patrick Reany
Ben Golub wrote:
> The World Wide Wade wrote:
>
> > One need not "memorize" to remember.
>
> Memorization: let eff of ex bee dee find for all ex in sum oh pen in ter vahl....
>
> Remembering: knowing what a limit is.
>
> As others have said pretty well, you need to manipulate before memorizing,
What are you manipulating? Manipulating with what?
> to make
> the concept our own. Only then will the remembering be worth a darn.
>
> Ben
Patrick
Patrick Reany
Ben Golub wrote:
> Mathematical memorization in its proper form is a type of understanding. Take for
> example something very simple (though very beautiful), the definition of a limit.
>
> * * *
> Let f(x) be defined for all x in some open interval containing the number a, with
> the possible exception that f(x) need not be defined at a. We will write
>
> lim_(x->a) f(x) = L
>
> if for any number epsilon > 0 there exists a number b > 0 such that
>
> |f(x) - L| < epsilon if 0 < |x-a| < b
> * * *
>
> Okay. Now, that's a bunch of words and symbols at the surface level. The
> definition could be memorized by somebody with no mathematical knowledge
> whatsoever as a series of English words, or by somebody with no knowledge of
> English or math as a series of sounds (let eff of ex bee dee find, etc.) Of
> course, such a memorization is useless and not even worth discussing.
I have NEVER advocated memorization of material
well beyond one's competency, so this is irrelevant.
But if a student is in any calculus class like those I
took, he or she had better set the above definition,
and dozens of others as well, into memory. The sooner
the better, and very accurately.
BTW, if one does want to memorize the definition
of a limit to gain understanding of it (NOT that I
ever claimed that memorization ALONE can
accomplish this), one had better already have
memorized what the hell absolute values
mean. Memorization comes before understanding.
It's the OBVIOUS starting point.
>
> Nevertheless, anybody who wants to do calculus at any real level needs to know
> this definition intimately. No question, he should be able to write it on a
> blackboard without pausing, and there is some element of memory in that.
It's either memorized or its not. There is no "some element"
about it. Why the equivocation about this?
> But it
> shouldn't be the kind of trivial memory described above, which is just mindless.
I NEVER advocate mindless anything.
> The kind of memory the student must have is a memory of the CONCEPT,
Usually it's the CONCEPT that eludes the student. Memorization
comes first and then the student figures out what the concept
means by inventing relationships among the parts of the
concept. That is what the PROCESS of understanding really is
in the first place. If you contest this then present your
own explanation of the understanding of difficult concepts.
> which is
> significantly distinct from a memory of WORDS or SYMBOLS or anything of that sort.
That HAS TO COME FIRST.
>
> Though we can translate a concept into words, the concept does not equal the words
> that express it, in the same way that the keyboard on which I am typing this is
> not equivalent to the word "keyboard."
>
> In order to attain a proper conceptual understanding, the student must explore the
> idea of what a limit means with a good teacher or a good book.
Exploring is just one of MANY methods of getting
something in MEMORY. If "exploring" involves
repetition it is also a form of rote, by definition. BTW,
what is to be accomplished by "exploring"? Just what
I've been claiming all along! The discovery/invention
of relationships among conceptual "parts."
> He must twist it
> around in his head, investigate familiar functions, interpret |x-a| and |f(x) - L|
> as distances, and so forth. He must play mental games with the definition, test it
> for holes. I think this is an experience familiar to a lot of people involved in
> math, so it's not anything earth-shattering, but when I think of a limit I think
> of the familiar drawing where we see epsilon and beta intervals on the y and x
> axes, respectively. It sort of moves and wiggles so that I can see that changing
> one of the intervals affects the other. I am keenly aware that we can make the
> interval epsilon infinitely small, which is the whole trick, and so forth.
>
> Now, if somebody asked me right now to write the defintion, I would translate the
> CONCEPTUAL understanding I have in my head back into English words and
> mathematical symbols. The memory I have is indeed memorized, but not in the
> mindless sense that a lot of people hate (what comes after x in that formula!?
> darn!).
I NEVER advocate mindless or purposeless
memorization, although I do believe that
all memorization is good for you -- it's
just like physical exorcise. It's good for you
if not taken to an extreme.
>
>
> So in my opinion in order to memorize math for later use in anything other than a
> dronish, trivial, and useless fashion, we have to translate English and math
> symbols into concepts, internalize them, and then do the reverse when asked to
> remember. Thus, in math, understanding and useful memorization are inseparably
> intertwined.
You do not have ONE single thing in your brain
that's there from a process other than memorization.
All your understanding is based on what you've memorized,
and on the relationship you've memorized no the parts
you've memorized, either consciously so or subconsciously.
Patrick
Patrick Reany
Ben Golub wrote:
> Dr. Arm wrote:
>
> > But mathematical memorization can also be the equivalent of beating your
> > kids: Works good only for solving previously addressed problems.
>
> Well, I disagree. For instance, a true understanding of the definition can be used
> to prove a wide variety of limits, some of them quite nontrivial. This is very
> distinct from the true inability to solve new problems gleaned from a rote
> memorization of formulas and words, not concepts.
>
> When we commit mathematical concepts to memory by understanding them, we enable
> ourselves to link them later or to use the "tricks" we learn to solve completely
> new problems.
>
> I agree with OP to an extent in that you can't ever do any interesting math unless
> you have a certain critical mass of information conceptually committed to
> memory -- but it must be conceptual.
You can't get anywhere without memorization.
If you want to understand the Pythagorean theorem,
you have to first understand its parts. But before
you can do that you have to memorize the parts.
Then, to form relationships on the parts of the theorem
in your mind, you have to have the parts memorized
at least for short term use. If you are trying to
understand the relationship of a right angle to the
hypotenuse of a triangle and you've FORGOTTEN
what a triangle is then go home and take your failing
grade. You haven't a chance. Something has to
already be in memory to use for the mental play things
in the "exploring" process, as people loved to
euphemize it. I just don't play these damn word games.
The subject is too important to me. No pain of
memorization, no gain of understanding!
> (And the more advanced the math you want to
> do, the greater the critical mass, of course). Otherwise, your garden variety
> amateur mathematician (read: crank or JSH) really could prove FLT (AAAH! AAH!)
>
> In addition, to do interesting math, you need a creative mind that can make good
> use of those concepts while seeking out new ones with which to grow its abilities.
> So its an intertwined process.
I am all for understanding difficult math. I have many web
pages on how to understand abstract algebra. I learned
late in life and the hard way to be a memorizer. I wish
I had been told to do this as a child. It's so obvious now
what was so unobvious back then. Each new concept
has to be understood on the basis of the older concepts.
And if those older concepts are NOT in memory for
instant, accurate retrieval, you're sunk. He who hesitates
in class because he could remember what Simpson's
rule is or what Lagrange's theorem is IS LOST!!! Wasted
time that will have to be made up. Inefficient and
discouraging. We end up with math anxiety because
we're too damn lazy to do the memorization as we go
through the class in a timely manner.
So why don't we just do the obvious and memorize?
Because by human nature it seems to hurt to do so. We're
just plain lazy about it, I guess. In the early 1980s I made
a living as a professional math tutor. The one thing I
leaned about the students I tutored that made the
biggest impression on me is this: The A students ALWAYS
memorized everything of importance in the chapter
before we got together; the C and D students never
did this!!!! This seems obvious to me what to
make of it. I just expand on the real use of this
intensive memorization as it relates to the Process
of Understanding deep concepts.
Patrick
Patrick Reany
Bill Taylor wrote:
> Patrick Reany <re...@asu.edu> writes:
>
> |> Memorization comes before understanding.
>
> Blimey, watchit mate!
>
> You're going to cop a load of old cobblers from Herman Rubin for this heresy!!
>
> I would not go quite as far as the other fellow in denouncing memorizing
> anything AT ALL, but there is a grain of truth in what he said.
>
> If you want a quick and dirty epigram to cover the case, I would suggest this:
>
> Manipulation comes before understanding.
> =======================================
>
Manipulation of what?
Patrick
Spaceman
>Manipulation of what?
Manipulation of WHAT is trying to be understood.
Try manipulating an atomic clock.
and find out how wrong physics time travel thories are.
memorization is part of understanding.
It is not all of it.
and manipulation is also part of it.
not all of it.
You need both.
not just the memory
to understand anything.
memory can be wrong if not used correcly
and manipulation allowed.
If I show you a pool that has water.
you will memorize the pool with water.
If you go by memory alone
and somone drains the pool.
you are in trouble
but a manipulator will see
the memory has changed.
you need both.
or you lack undersatanding th object as it is now..
the memeories of a clock being a God
is what drives the phsycis wrongs too.
Try manipulating a few REAL things
before you memorize them ever again.
James M Driscoll Jr
Spaceman
http://www.realspaceman.com
Patrick Reany
Robert Israel wrote:
Every thing you KNOW is in your memory.
Every thing you ever hope to learn that's new
to you right now is built on top of what you already
have in your memory. Every concept that has parts,
such as a math theorem, has to be understood in
terms of the relationships those parts have to each other,
and to do that in your brain requires you to have
those parts memorized accurately. Memorization
comes before understanding.
Memorization is NOT the same as rote. Anyone
who thinks otherwise can define rote memorization
for us right now. (Said the spider to the fly!)
Although I do advocate lots of rote memorization,
I NEVER claim that it is the only way to memorize,
just as jogging isn't the only way to get aerobic
exercise. When people "manipulate" ideas around,
that's just a sneaky way of doing mental repetitions.
Patrick
Patrick Reany
"David C. Ullrich" wrote:
> On 9 Jul 2002 06:53:34 GMT, isr...@math.ubc.ca (Robert Israel) wrote:
>
> >In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
> >
> >>My basic premise is that:
> >
> >> Memorization comes before understanding.
> >
> >"... comes instead of ..." is more like it, in many cases.
> >
> >Once it has the conceptual framework, the mind finds it easy to
> >fit in the individual pieces, because they all fit together neatly
> >and logically. Without that framework, you have a jumble of unrelated
> >facts and techniques that take a lot of effort to memorize. If you
> >understand the basic ideas, the actual amount of memorization needed for
> >the average math or physics course is very small.
>
> Maybe people are talking past each other. The people who say there
> should be less emphasis on memorization, etc, are right.
1) I think we aren't talking past each other.
I think you hit it on the head in what you say
later in your post: A cultural hatred of and denial
of the use of memorization in education.
2) We need both more memorization and less
memorization at the same time. I don't
advocate that we be forced to memorize
tedious stuff that is specialized and can be
looked up in a reference table or graphic at
one's leisure. I remember taking a class on
inorganic chemistry and we were asked to
memorize a very long flow chart of precipitations
of salts. This is stupid memorization, except
perhaps for professionals in that field, which
we students were not and probably not going
to be. But in what you say below makes a
irrefutable argument that we need to encourage
students to memorize all the core facts, definitions,
concepts, and theorems that they encounter
in their classes. He who hesitates, due to not
having a definition or theorem memorized, is lost!
> But I
> suspect maybe we're talking about memorizing different things:
>
> For example when I teach linear algebra one of my big problems
> is getting the kids to believe that they actually do have to
> "memorize" the definitions - I try to use a different word
> because they find the word "memorize" so scandalous. But there's
> no way to figure out that the definition of "basis" is
> "independent spanning set", you simply have to memorize that
> definition.
That's what I already knew to be the truth:
Our culture treats memorization as scandalous!
Look at the loony posts we've seen here.
This is precisely what I'm trying to confront
in this thread. And one reason you teachers
have such a hard time getting your students to
memorize is because of this cultural crap value
system that tells them that they shouldn't have to
memorize anything. The posters on this thread
are largely a part of the problem and not part
of the solution. It's time to accept that all successful
intellectual people are memorizers. The people who
think they don't memorize have just invented
a crafty definition of "memorization" that supports
their loony politically correct dogma of denial.
> Of course you don't have to memorize the definition verbatim,
> but you _do_ have to memorize in some sense exactly what the
> definition _says_. Dealing with students who have essentially
> no experience manipulating abstract concepts, it seems that
> suggesting they memorize the definitions verbatim at first
> is the easiest way to get them to memorize exactly what the
> definition says. Gets frustrating - they have no idea that
> if they want to show something is a basis they need to
> show that it's independent and show that it spans, when
> if they'd just concede and memorize the definition then
> they _would_ know that this is what is required.
Precisely. And obvious, isn't it. So where does
this loony anti-memorization dogma come from?
I think it came from two sources:
1) from some of our own experiences of being
forced to memorize tedious and pointless stuff
(something I NEVER advocate!)
2) from the educational philosophy that memorization
of facts ALONE can lead people to intellectual
proficiency. Of course it can't. Memorization of
central concepts and definitions and theorems
is the beginning of building one's own understanding
of a subject. It is necessary but NOT sufficient
in itself! Understanding is the PROCESS of
constructing relationships among parts. If you
don't have the parts memorized, how can you
efficiently construct the mental relationships?
You can't!
> What they learn instead is garbled and/or wrong versions
> of the definitions. If they _knew_ the definitions then
> they could eventually see how the concepts fit together,
> which is of course the goal. But it's impossible for them
> to see how the concepts fit together, because with their
> garbled versions of the definitions they _don't_ fit
> together.
Correct: Memorization comes before understanding.
Or in this case, before problem solving, a closely
related concept to understanding.
> >I always tried to
> >memorize as little as possible - as long as I could readily reconstruct
> >some idea when needed, there was no point in memorizing it. In addition,
> >this type of attitude makes it easier to tackle new problems that don't
> >fit a memorized pattern.
Denial by convenient definition of "memorize." Our
culture is in denial of memorization. Why?
> Of course. But you _did_ memorize _something_ - in my experience
> with kids these days they don't believe that they should have
> to memorize _anything_, and that attitude makes it much _harder_
> to tackle new problems. The reason you found you could tackle
> new problems is you _did_ believe that it was necessary to
> know exactly what the words meant. Making it easier to learn
> new things is exactly why I wish I could convince them that
> it's necessary to memorize _some_ things - when I say that
> memorization is much more important than a lot of people
> admit these days I'm certainly not saying that they should
> memorize the sort of thing I suspect you're alluding to as
> things you didn't memorize, just the opposite; the point
> to memorizing exactly what things mean is to allow them
> to figure things out without memorizing the answers to
> millions of problems.
>
> >Robert Israel isr...@math.ubc.ca
> >Department of Mathematics http://www.math.ubc.ca/~israel
> >University of British Columbia
> >Vancouver, BC, Canada V6T 1Z2
>
> David C. Ullrich
As Ullrich points out, this cultural anti-memorization
attitude that has dominated our students is hurting
them learning mathematics and -- I claim -- physics
as well.
Patrick
Ben Golub
> It's either memorized or its not. There is no "some element"
> about it. Why the equivocation about this?
I think we're more or less talking about the same thing. Look, I've never had any
trouble memorizing math or anything else, so I'm not one of those silly people who
try to downplay knowing facts in favor of "synthesizing ideas" or whatever crap
like that. No question, BEING ABLE TO REMEMBER IMPORTANT FACTS IS IMPORTANT!
But there are VERY different ways to get there which make a huge difference in how
effectively you study math.
I know a lot of students who will sit and memorize formulas and worked problems
for hours and still have trouble with even a slight conceptual deviation from the
known case. ( "But we didn't do that example, professor!" ) One could make the
argument that they have a much more encyclopedic memorized library of knowledge
than I do, but because it's memorized in the wrong way, they can't use it for any
real math, and their grades reflect that.
On the other hand, when better students memorize, they memorize concepts. For
instance, I'll look at the page and temporarily commit it to memory, (sort of like
putting arguments on a stack) but it'll be gone in ten minutes when I have the
much more useful knowledge of what it means and how to use it. If I need to write
down a theorem, it will not be words that I remember, but the idea of how a
theorem works.
There is certainly no surer way to screw yourself than by thinking that as long as
you memorize enough examples, you will be fine. A close second is thinking that
you don't need to remember anything, since you can just think it up when the time
comes. The memorization needs to exist, but it must be a correct and useful kind.
Ben
Neil W Rickert
>Neil W Rickert wrote:
>> Patrick Reany <re...@asu.edu> writes:
>> >Daniel McLaury wrote:
>> >> Nothing could possibly be farther from the truth.
>> >I take it you read nothing on my website.
>> I take it that he probably read a lot of your web site and
>> sincerely disagreed.
>> >> Throughout my
>> >> entire higher-math career, since I started Calculus my Sophomore year
>> >> of highschool, I have made it a point *never* to memorize anything.
>> >So, you're saying that you have nothing of
>> >math content in your head? Then you must
>> >know absolutely NO math.
>> Know, that is not what he said at all. It is dishonest of you to
>> misrepresent what was said.
>It is not dishonest to get the poster to define his
>or her terms.
You misrepresented what he said, and you did so in a public forum.
That's dishonest. There are honest ways of probing for the intended
meaning.
>> >> The sole meaning of mathematics, and life itself (Math is life;
>> >> everything else is just details...) is understanding and proof.
>> >How can one understand a concept C if one has NOTHING
>> >at all in one's memory related to that concept?
>> You have it completely backwards. One cannot have a memory of
>> what one does not understand.
>You seem to have no idea how the human mind
>accomplishes "understanding." It starts off as
>bootstrapping: One memorizes and then
>builds on top of that. [snip]
Hogwash.
Perhaps you have great ideas about how one accomplishes
understanding, but they are great only in the extent in which they
are mistaken.
>> Mathematicians are concerned with relations between infinitudes of
>> things. If they had memorize all of those uncountably many things as
>> prerequisite to finding relations, they could never get started.
>Obviously no one is supposed to memorize
>an infinite number of things. Where did I ever
>say one should?
You said "If you don't have the parts memorized, you have no hope of
finding the relationships among them."
My comment about infinitudes was to demonstrate that your reasoning
was faulty.
Patrick Reany
Neil W Rickert wrote:
Patrick Reany <re...@asu.edu> writes:Daniel McLaury wrote:
[snip]
How can one understand a concept C if one has NOTHING
at all in one's memory related to that concept?
You have it completely backwards. One cannot have a memory of
what one does not understand.
me and memorize the statement of any theorem
at all, even though I would have NO understand
of what that theorem means. But I'm NOT saying
that this is the way to learn new math. That would be
to start on a good foundation of memorized stuff, take
the next theorem or concept to be understood, break
it up into parts, memorize the parts, and construct
relationships among those parts: That is the process of
coming to understand something.
In the case of mathematics, the understanding is usually sufficient and
there is no need to memorize.
Patrick
Ben Golub
> The posters on this thread
> are largely a part of the problem and not part
> of the solution. It's time to accept that all successful
> intellectual people are memorizers. The people who
> think they don't memorize have just invented
> a crafty definition of "memorization" that supports
> their loony politically correct dogma of denial.
Patrick, first of all, as a matter of style, don't argue ad hominem. You started
this thread to provoke a discussion, and now you are calling the people that
courteously answered your call "part of the problem." Call their arguments invalid
if you choose, but don't insult them as loony or problematic in themselves.
Secondly, your dogma of "memorization before understanding" has the potential to
cause as much trouble as the opposite "memorization is old-style garbage." I know
a lot... a LOT... of people who will never be successful at a hard science who
have convinced themselves that memorization will lead them to success. It won't!
Educators MUST tell people that if their memorization must be a dynamic process
where they constanly build on a memorized fact with a layer of understanding, like
a cake with cream. Otherwise, the result of all their pain will be dry and
impalatable.
I agree that all successful intellectuals are capable of strong fact-retention,
but don't harp on that as if you've found some sort of new gem. It's well-known,
and the key is finding an edcuation process that makes it easier for more people
to memorize WELL -- that is, not a process of torture, but almost a fun process
where you , (1) memorize something, (2) make it your own, and (3) don't think
about it as external to yourself anymore. Most people miss (2) and thus can't do
(3), which makes true learning impossible for them.
Neil W Rickert
>Manipulation of what?
If you were thinking like a mathematician, you wouldn't need to ask
that question. And you wouldn't be making mistaken claims about the
role of memorization.
Patrick Reany
Ben Golub wrote:
> Patrick Reany wrote:
>
> > It's either memorized or its not. There is no "some element"
> > about it. Why the equivocation about this?
>
> I think we're more or less talking about the same thing. Look, I've never had any
> trouble memorizing math or anything else, so I'm not one of those silly people who
> try to downplay knowing facts in favor of "synthesizing ideas" or whatever crap
> like that. No question, BEING ABLE TO REMEMBER IMPORTANT FACTS IS IMPORTANT!
>
> But there are VERY different ways to get there which make a huge difference in how
> effectively you study math.
>
> I know a lot of students who will sit and memorize formulas and worked problems
> for hours and still have trouble with even a slight conceptual deviation from the
> known case. ( "But we didn't do that example, professor!" ) One could make the
> argument that they have a much more encyclopedic memorized library of knowledge
> than I do, but because it's memorized in the wrong way, they can't use it for any
> real math, and their grades reflect that.
>
> On the other hand, when better students memorize, they memorize concepts. For
> instance, I'll look at the page and temporarily commit it to memory, (sort of like
> putting arguments on a stack) but it'll be gone in ten minutes when I have the
> much more useful knowledge of what it means and how to use it. If I need to write
> down a theorem, it will not be words that I remember, but the idea of how a
> theorem works.
I don't know how to memorize a concept for effective
use -- as you seem to be suggesting -- unless that
concept first is understood by the student. How do
people come to understand the deep concepts of math?
I claim by going thru a PROCESS that begins with
the identification of the primitive parts of that concept.
I explain below.
> There is certainly no surer way to screw yourself than by thinking that as long as
> you memorize enough examples, you will be fine. A close second is thinking that
> you don't need to remember anything, since you can just think it up when the time
> comes. The memorization needs to exist, but it must be a correct and useful kind.
>
> Ben
Memorizing example and formulas is crucial. But
if one stops there it's like bating the hook of
a fishing pole but never putting the pole in the
water. You'll get no fish! One also has to get good
at problem solving and that means understanding the
concepts, as I have said from the start. But to do
that one has to start somewhere. And that starting
point (of departure) is with memorizing the parts
of the concept. Then one constructs relationships
among those parts. If that construction works
to solve the right problems, that is an objective
means of deciding that the concept has been
understood.
You cannot think about anything unless those
things you are thinking about are at least in
your short term memory. If the building blocks
you need to think about math concepts are required
over and over, then the sooner those concept
are memorized and understood the better.
Patrick
David C. Ullrich
wrote:
>
>
>"David C. Ullrich" wrote:
>
>> On 9 Jul 2002 06:53:34 GMT, isr...@math.ubc.ca (Robert Israel) wrote:
>>
>> >In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>> >
>> >>My basic premise is that:
>> >
>> >> Memorization comes before understanding.
>> >
>> >"... comes instead of ..." is more like it, in many cases.
>> >
>> >Once it has the conceptual framework, the mind finds it easy to
>> >fit in the individual pieces, because they all fit together neatly
>> >and logically. Without that framework, you have a jumble of unrelated
>> >facts and techniques that take a lot of effort to memorize. If you
>> >understand the basic ideas, the actual amount of memorization needed for
>> >the average math or physics course is very small.
>>
>> Maybe people are talking past each other. The people who say there
>> should be less emphasis on memorization, etc, are right.
>
>1) I think we aren't talking past each other.
>[...] The posters on this thread
>are largely a part of the problem and not part
>of the solution. It's time to accept that all successful
>intellectual people are memorizers. The people who
>think they don't memorize have just invented
>a crafty definition of "memorization" that supports
>their loony politically correct dogma of denial.
Um. I don't think it's so much a crafty definition that
supports their loony dogma as just an example of the fact
that people often mean slightly different things by a
word in a given context. Israel says he hardly ever
memorized anything and doesn't think it's that important,
I say that kids were willing to memorize more they'd
have a chance - I really don't think that the two of
us disagree on what a student actually needs to do to
be successful, we're just using words in different
ways. Although he says he didn't memorize so much
I _know_ he memorized a lot more of the things that
I think need to be memorized than my students do
(and the things that I suspect are the sorts of
things he didn't memorize are not the things that
I'm saying students need to memorize, in fact I
agree that they'd be better off doing less memorization
of that sort of thing.)
[...]
>
>> >I always tried to
>> >memorize as little as possible - as long as I could readily reconstruct
>> >some idea when needed, there was no point in memorizing it. In addition,
>> >this type of attitude makes it easier to tackle new problems that don't
>> >fit a memorized pattern.
>
>Denial by convenient definition of "memorize."
No, or at least I don't think so.
David C. Ullrich
Herman Rubin
>I want to start a conversation about the role
>of memorization in the learning of math and
>physics. I did this sometime ago and found
>the discussion interesting.
>My basic premise is that:
> Memorization comes before understanding.
This is quite false. Memorization hinders understanding.
The person who has memorized the methods of arithmetic
is in a worse position to understand what the operations
mean than the one who starts from the concepts. Concepts
are not learned by memorizing the words.
The classical "Euclid" course never started with much
memorization. Only a few axioms and rules were even
presented, and these were learned by usage, not by
memorization as such. The same holds for any other
mathematics; it is not computation.
It ain't what you don't know that hurts you;
It's what you know that ain't so.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Jesse F. Hughes
> It's time to accept that all successful intellectual people are
> memorizers. The people who think they don't memorize have just
> invented a crafty definition of "memorization" that supports their
> loony politically correct dogma of denial.
"All successful intellectual people are memorizers."
"Everyone who denies this is dogmatic."
(As far as that goes, what the heck is "politically correct" about
this dogma?)
Note: I'm not taking one side or the other in this debate. I'm just
tickled by the above argument.
--
Jesse Hughes
"We will run this with the same kind of openness that we've run
Windows." Steve Ballmer, speaking about MS's new ".Net" project.
Patrick Reany
Neil W Rickert wrote:
I knew you wouldn't give a straight answer
to a straight question. Or is all you want
demagoguery and platitudes? I repeat
obfuscater: Manipulation of what?
Patrick
Patrick Reany
Herman Rubin wrote:
> In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>
> >I want to start a conversation about the role
> >of memorization in the learning of math and
> >physics. I did this sometime ago and found
> >the discussion interesting.
>
> >My basic premise is that:
>
> > Memorization comes before understanding.
>
> This is quite false. Memorization hinders understanding.
Tell us all what you think understanding
consists of.
Patrick
Patrick Reany
"Jesse F. Hughes" wrote:
> Patrick Reany <re...@asu.edu> writes:
>
> > It's time to accept that all successful intellectual people are
> > memorizers. The people who think they don't memorize have just
> > invented a crafty definition of "memorization" that supports their
> > loony politically correct dogma of denial.
>
> "All successful intellectual people are memorizers."
>
> "Everyone who denies this is dogmatic."
>
> (As far as that goes, what the heck is "politically correct" about
> this dogma?)
The term "politically correct" is a metaphor for a
sub-dogma based on irrational doctrines held by
the masses to keep in line with some overarching
"politically motivated super-dogma." I am NOT on
the side of the masses on this one. I learned my lesson.
Now I challenge this false dogma. Let its defenders for
once actually explain what they mean by "memorize"
and "understand." Or must I do all the work in this
debate? Not one person has bothered to define
either of these terms when I asked them to do so.
You mentioned denials. Denials are easy. Clearly
explaining what's wrong with my claims is
apparently too hard to my detractors to get
around to doing. So who are the dogmatists?
Patrick
Ben Golub
> sub-dogma based on irrational doctrines held by
> the masses to keep in line with some overarching
> "politically motivated super-dogma."
What the heck is this crap? "Politically correct" is not a metaphor, first of all.
Also what is a sub-dogma? What is the super-dogma? What is the political
motivation? Toward what political end are these people working?
Let's not get absurd now.
Neil W Rickert
>>>How can one understand a concept C if one has NOTHING
>>>at all in one's memory related to that concept?
>>You have it completely backwards. One cannot have a memory of
>>what one does not understand.
>I can pick up a math book on a topic unfamiliar to
>me and memorize the statement of any theorem
>at all, even though I would have NO understand
>of what that theorem means. But I'm NOT saying
>that this is the way to learn new math.
Good. So far we agree.
> That would be
>to start on a good foundation of memorized stuff,
That's where we disagree. Memorized stuff does not a good foundation
make.
>>In the case of mathematics, the understanding is usually sufficient and
>>there is no need to memorize.
>Define what you mean by "memorize."
I seem to be using it in the standard way. I doubt that it is any
more definable than the rest of the words in the dictionary.
Dr Arm®
>
> Dr. Arm wrote:
>
> > But mathematical memorization can also be the equivalent of beating your
> > kids: Works good only for solving previously addressed problems.
>
> Well, I disagree. For instance, a true understanding of the definition can be used
> to prove a wide variety of limits,
some of them quite nontrivial.
Understanding is not memorization. Argue your statement of disagreement
before you begin another arguement.
Neil W Rickert
>> >> Patrick Reany <re...@asu.edu> writes:
>> >> |> Memorization comes before understanding.
>> >> Blimey, watchit mate!
>> >Manipulation of what?
It is not a straight question.
Mathematics is a very broad subject. What is manipulated varies
greatly, depending on what is to be understood.
You apparently have a particular folk-theory of what it takes to
master mathematics. You are making arguments based on that
folk-theory. It seems, from the reaction you are getting, that most
of the respondants sense that your folk-theory does not fit very well
with how they actually do mathematics.
Cognitive science is not currently at the point where it can give
clear accounts of what is involved in understanding mathematics. So
at present this will have to remain a question where definitive
answers are not available.
Herman Rubin
>Mathematical memorization in its proper form is a type of understanding. Take for
>example something very simple (though very beautiful), the definition of a limit.
>* * *
>Let f(x) be defined for all x in some open interval containing the number a, with
>the possible exception that f(x) need not be defined at a. We will write
>lim_(x->a) f(x) = L
>if for any number epsilon > 0 there exists a number b > 0 such that
>|f(x) - L| < epsilon if 0 < |x-a| < b
>* * *
This is not the only way to teach limits; it is the usual cute way.
>Okay. Now, that's a bunch of words and symbols at the surface level. The
>definition could be memorized by somebody with no mathematical knowledge
>whatsoever as a series of English words, or by somebody with no knowledge of
>English or math as a series of sounds (let eff of ex bee dee find, etc.) Of
>course, such a memorization is useless and not even worth discussing.
>Nevertheless, anybody who wants to do calculus at any real level needs to know
>this definition intimately. No question, he should be able to write it on a
>blackboard without pausing, and there is some element of memory in that. But it
>shouldn't be the kind of trivial memory described above, which is just mindless.
Limits can be well understood without memorizing those words,
or a logical equivalent of them. Using a special case typically
obscures the concept, and learning limit as above is an example
of this. It is also the case that someone can learn, and apply,
a concept without being able to give a rote statement of it.
Herman Rubin
>|> Memorization comes before understanding.
>Blimey, watchit mate!
>You're going to cop a load of old cobblers from Herman Rubin for this heresy!!
>I would not go quite as far as the other fellow in denouncing memorizing
>anything AT ALL, but there is a grain of truth in what he said.
>If you want a quick and dirty epigram to cover the case, I would suggest this:
> Manipulation comes before understanding.
> =======================================
>Oh dammit - now I'm going to cop it from old Hermy as well! Oh well.
It is worse than pulling teeth to teach probability or
statistics concepts to those who have learned to calculate
probabilities or to calculate statistical procedures.
The user of probability and statistics needs to understand
the concepts, but little manipulation is needed, and it is
often the case that the manipulations in the trivial cases
which can be readily done do not help.
>Eggshelly, my epigram isn't quite fully correct, there CAN be understanding
>without manipulation, but rarely, and it is likely to be a more complete grok
>afterwards. Until you've done a few manipulations, usually called "examples",
>your understanding is likely to be patchy at best.
>As for the "no-memorizing" part of the rule - I would soften it to this:-
>"Don't memorize anything until you have redone the deduction of it for yourself."
This fails in both directions. Someone understanding concepts
can use a concept, even if it requires memorization, and be
unable to deduce it. I doubt that anyone needs to be able to
carry out a proof of Fermat's Last Theorem to use it, or to
be able to prove the Hahn-Banach Theorem to follow (NOT to redo)
an argument that self-consistent behavior must be Bayesian.
>Only then will you fully grok what it is you're memorizing, and attempting
>to understand. Then, after you have achieved this step, you can go ahead
>and memorize as much of it as you think fit. That'll then be a big help.
Patrick Reany
Neil W Rickert wrote:
> Patrick Reany <re...@asu.edu> writes:
> >Neil W Rickert wrote:
> >>Patrick Reany <re...@asu.edu> writes:
>
> >>>How can one understand a concept C if one has NOTHING
> >>>at all in one's memory related to that concept?
>
> >>You have it completely backwards. One cannot have a memory of
> >>what one does not understand.
>
> >I can pick up a math book on a topic unfamiliar to
> >me and memorize the statement of any theorem
> >at all, even though I would have NO understand
> >of what that theorem means. But I'm NOT saying
> >that this is the way to learn new math.
>
> Good. So far we agree.
>
> > That would be
> >to start on a good foundation of memorized stuff,
>
> That's where we disagree. Memorized stuff does not a good foundation
> make.
>
> >>In the case of mathematics, the understanding is usually sufficient and
> >>there is no need to memorize.
>
> >Define what you mean by "memorize."
>
> I seem to be using it in the standard way.
Then DEFINE it. Or are platitudes all you
want to commit to? You call yourself a
rational debater? Ha!
> I doubt that it is any
> more definable than the rest of the words in the dictionary.
Cop out, as usual by my *&%*& detractors.
Memorization of a set of objective knowledge
(knowledge that is conventionally testable by standard
tests, the likes of which are given in classrooms
all over the world) is ANY process whereby
a person is able at the end of the process
to retrieve this knowledge quickly and
accurately as a mental representation of this
knowledge, and then effectively use this
knowledge correctly on standardized tests.
Naturally, we all believe that we know things which
are not under the usual meaning of objective
knowledge, but I am only compelled to define
-- for purposes of the present discussion --
"memorization" and "memory" for purposes of dealing
with objective knowledge as it can be defined in the
context of formal education.
If a set of objective knowledge has been
memorized, it is said to "reside in one's memory."
If a set of objective knowledge resides in one's
memory, it is said to "memorized."
If you KNOW anything at all, it is memorized
by you. It is in your memory. One learns new
things based on what one already has in memory.
How can YOU claim to know or understand
something that isn't in your memory?
If I tell you that the Pythagorean theorem is about a
relationship among lengths of the sides of a right
triangle and the length of the hypotenuse of the
triangle, and you don't have in your memory what
a right triangle is or what a hypotenuse is, you aren't
going to understand me. The concept of a right
triangle must be memorized first, before it can be
used meaningfully in a larger context. Memorization
comes before understanding.
Memorization is necessary but NOT sufficient
for understanding.
Patrick
Patrick Reany
"Dr Arm®" wrote:
Memorization is necessary but NOT sufficient
for understanding. Don't believe me? Then
provide a counterexample!
Patrick
Herman Rubin
>Herman Rubin wrote:
>> > Memorization comes before understanding.
>Patrick
Conceptual understanding occurs when the "light bulb goes
on". Something happens mentally, but it is not the ability
to recall any identifiable pieces of information.
Much of what is taught is not understanding. Few of us are
capable of realizing that we do not understand, even though
we can do the problems and prove the theorems. I do not
think understanding can be better explained than creativity
can be taught.
Patrick Reany
Neil W Rickert wrote:
Where did I claim that my thread is about
"mastery" of mathematics? I do claim that
for every concept that one wants to understand
in mathematics there is always something
that has to be memorized first, at least
just to get the statement of the concept into
short-term memory, or do you claim that even
that is unnecessary?
> You are making arguments based on that
> folk-theory. It seems, from the reaction you are getting, that most
> of the respondants sense that your folk-theory does not fit very well
> with how they actually do mathematics.
All I get out of the sum total of demagoguery
of my respondents is that they refuse to define
the terms I ask them to define, and they also
refuse to give counterexamples. And last time
I checked this is incompetence and cowardliness
in a rational debate.
> Cognitive science is not currently at the point where it can give
> clear accounts of what is involved in understanding mathematics. So
> at present this will have to remain a question where definitive
> answers are not available.
Then just give a counterexample of even one
mathematical concept that can be understood
without any prerequisite memorization.
I don't give a damn, Neil, what you personally
believe. But if you're going to debate me, do it
by rational rules and NOT by demagoguery. The
rules are quite simple: There are only two terms that
have to be defined: "memorization" and "understanding."
And at least one counterexample that must be given
by your side. Is that damn plain enough for you? I'm out
here alone trying to get anyone to actually stand up like
a decent human being and freely debate me
rationally on the merits. So either do that or stop
wasting my time! All I've gotten so far is a bunch
of claims that I'm wrong, which constitutes no
debate at all, yet this alone seems to impress YOU.
If any side has the claim to being a "folk-theory"
it's YOUR side. I'm here to challenge your
absurd "folk-theory" of the understanding
of math and physics concepts. When the hell is
YOUR side going to debate me by the rules?
Patrick
W. Edwin Clark
Herman Rubin wrote:
>
> In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>
> >My basic premise is that:
>
> > Memorization comes before understanding.
>
> This is quite false. Memorization hinders understanding.
Maybe you would admit that for certain courses some
memorization is necessary?
Suppose you are teaching a beginning course in, say,
abstract algebra or number theory. Call it what you may,
but there are lots of definitions and theorems that one
must KNOW in order to be able to do the kind of things
that such students should be able to do.
Wouldn't you, for example, expect a student to be able
to prove that if two groups are isomorphic and one is
cyclic then the other is also?
Now tell me how someone can do this who doesn't know
the definition of a group, a cyclic group and what it
means to say that a group is isomorphic to another group
----and a few other things?
And how would they "know" these things if they didn't
"memorize" them?
Some people have very good memories. They are told ONCE
these definitions and they never forget them. I guess these
are the people like Robert who never have to "memorize".
Others must make a significant effort to absorb definitions.
[Of course, some refuse to learn them and some make up
their own.]
When I teach such courses I try to FORCE the students to learn
to state the definitions precisely by giving short weekly
quizzes on definitions. For the less gifted this means I
force them to "memorize" the definitions. Is this bad?
Many years ago when I taught abstract algebra to Caltech
sophmores I didn't have to resort to such strategies,
but I don't know of another way to deal with students
at my university where (unlike at Lake Wobegon) all students
are not above average.
I recall that Marvin Minsky had a theory of "frames"
to explain how people learn things -- especially new concepts.
It was something like this: Each of us has a bunch of frames
in our head. Frames for words like "tree", "chair", "senator",
etc. When we come across a new concept we attempt first to
fit it to one of the frames we have. If we don't have one, we
have to construct a new frame. This can be painful sometimes.
The frame for the mathematical "group" it seems would be some
kind of network with various nodes. One node might be the
definition of a group. Other nodes would have examples of groups.
Other nodes would have related concepts. Of course, we
should help our students form these new "frames". They can start
building the frame by memorizing a few definitions. As the frame
grows it is easier to flesh it out. But it has to start somewhere.
So I say, with apologies to Tom Lehrer:
Memorize, memorize, let no definition escape your eyes.
--Edwin Clark
Daniel McLaury
>
> >I take it you read nothing on my website.
>
> I take it that he probably read a lot of your web site and
> sincerely disagreed.
>
all. If there's something other than what I think there is, somebody
tell me.
Daniel McLaury
saying that not everything can be proven? That there must be some
axioms to any system which cannot really be "discovered," since they
themselves define the system? If so, I agree. It has been obvious
from the time of Euclid that there can be no math without axioms.
However, *everything* other than the axioms should be derived, just as
someone before did it. The fact that people do not subscribe to this,
I believe, contributes to their inability to comprehend basic (and
not-so-basic) math.
Ben Golub
> This is not the only way to teach limits; it is the usual cute way.
Believe me, I know :-). I've been through the AP Calc program some years ago where
the prevailing wind with my teacher was to teach limits using a TI-83 graphing
calculator and say -- look, if lim_(x -> 2) f(x) = 5, that means that if you put
in x=1.9999 you'll probably get an f(x) that's very close to 5! Isn't that pretty?
But no rigorous understanding was ever laid of how to prove it mathematically. The
same thing went for continuity ("You can draw it without taking your pencil off
the paper.")
I think the definition of a limit though is what most mathematicians would prefer
to stick with... (e.g. try formulating a solid proof based on limits without it).
For students, probably the best is to get a good ballpark idea of what it means
FIRST and then see how exactly the mathematical definition expresses the concept,
so that the student can appreciate it.
Ben
Herman Rubin <hru...@odds.stat.purdue.edu> wrote in message
news:agfk8p$26...@odds.stat.purdue.edu...
Ben Golub
used
> > to prove a wide variety of limits,
> some of them quite nontrivial.
>
> Understanding is not memorization. Argue your statement of disagreement
> before you begin another arguement.
Sorry, I was unclear in terms. I intentionally said understanding, because the
only useful memorization in my view is that which immediately conduces to
understanding. So once you've memorized briefly and understood a concept
long-term, you can apply it to more difficult problems. But to agree with OP
Patrick to some extent, you can't get an understanding before a short term
memorization of the fact at issue. Sort of like putting arguments on a stack for
us programmers.
Ben
Dr Arm® <Genuin...@Yahoo.com> wrote in message news:3D2B52...@Yahoo.com...
> Ben Golub wrote:
> >
> > Dr. Arm wrote:
> >
> > > But mathematical memorization can also be the equivalent of beating your
> > > kids: Works good only for solving previously addressed problems.
> >
-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
Patrick Reany
Herman Rubin wrote:
> In article <3D2B4366...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>
> >Herman Rubin wrote:
>
> >> In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>
> >> >I want to start a conversation about the role
> >> >of memorization in the learning of math and
> >> >physics. I did this sometime ago and found
> >> >the discussion interesting.
>
> >> >My basic premise is that:
>
> >> > Memorization comes before understanding.
>
> >> This is quite false. Memorization hinders understanding.
>
> >Tell us all what you think understanding
> >consists of.
>
> >Patrick
>
> Conceptual understanding occurs when the "light bulb goes
> on".
Gee, that's rational and clear. Just because it's hard to
to have a complete and detailed model of the concept
of understanding, doesn't mean that we can't say anything
definitive about it.
> Something happens mentally, but it is not the ability
> to recall any identifiable pieces of information.
I NEVER said it was. The ability to recall identifiable pieces
of information relating to the parts of a math concept is
a necessary but NOT sufficient part of having understanding
of the concept.
Patrick
Daniel McLaury
> On 9 Jul 2002 06:53:34 GMT, isr...@math.ubc.ca (Robert Israel) wrote:
>
> >In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
> >
> >>My basic premise is that:
>
> >> Memorization comes before understanding.
> >
> >
> >Once it has the conceptual framework, the mind finds it easy to
> >fit in the individual pieces, because they all fit together neatly
> >and logically. Without that framework, you have a jumble of unrelated
> >facts and techniques that take a lot of effort to memorize. If you
> >understand the basic ideas, the actual amount of memorization needed for
> >the average math or physics course is very small.
>
> Maybe people are talking past each other. The people who say there
> suspect maybe we're talking about memorizing different things:
>
> For example when I teach linear algebra one of my big problems
> is getting the kids to believe that they actually do have to
> "memorize" the definitions - I try to use a different word
> because they find the word "memorize" so scandalous. But there's
> no way to figure out that the definition of "basis" is
> "independent spanning set", you simply have to memorize that
> definition.
Of course, technically you would have to "memorize" this definition.
But the concept of a basis is *so intuitive* that you scarcely have to
work at it. Even when you are dealing with completely abstract
concepts that do not connect at any point to reality (see the ongoing
discussion about a "subaddition function" under the title of The
Wierdest Math Problem Ever on sci.math), there is nothing that is
gonna be definied UNLESS it makes sense to do so. So, if you
understand the reasoning behind a definition, you don't need to
memorize the text of the definition.
> Of course you don't have to memorize the definition verbatim,
> but you _do_ have to memorize in some sense exactly what the
> definition _says_. Dealing with students who have essentially
> no experience manipulating abstract concepts, it seems that
> suggesting they memorize the definitions verbatim at first
> is the easiest way to get them to memorize exactly what the
> definition says. Gets frustrating - they have no idea that
> if they want to show something is a basis they need to
> show that it's independent and show that it spans, when
> if they'd just concede and memorize the definition then
> they _would_ know that this is what is required.
But do the students, who now know what a basis is, realize the
consequences of what it is to be a basis -- that a space with an
n-element basis has no basis with k elements unless k = n? If they do
not realize such concepts themselves due to a lack of feeling for the
concept itself, concepts such as the dimension of a space will come as
shock to them.
> What they learn instead is garbled and/or wrong versions
> of the definitions. If they _knew_ the definitions then
> they could eventually see how the concepts fit together,
> which is of course the goal. But it's impossible for them
> to see how the concepts fit together, because with their
> garbled versions of the definitions they _don't_ fit
> together.
> >I always tried to
> >memorize as little as possible - as long as I could readily reconstruct
> >some idea when needed, there was no point in memorizing it. In addition,
> >this type of attitude makes it easier to tackle new problems that don't
> >fit a memorized pattern.
>
> Of course. But you _did_ memorize _something_ - in my experience
> with kids these days they don't believe that they should have
> to memorize _anything_, and that attitude makes it much _harder_
> to tackle new problems. The reason you found you could tackle
> new problems is you _did_ believe that it was necessary to
> know exactly what the words meant. Making it easier to learn
> new things is exactly why I wish I could convince them that
> it's necessary to memorize _some_ things - when I say that
> memorization is much more important than a lot of people
> admit these days I'm certainly not saying that they should
> memorize the sort of thing I suspect you're alluding to as
> things you didn't memorize, just the opposite; the point
> to memorizing exactly what things mean is to allow them
> to figure things out without memorizing the answers to
> millions of problems.
>
> >Robert Israel isr...@math.ubc.ca
> >Department of Mathematics http://www.math.ubc.ca/~israel
> >University of British Columbia
> >Vancouver, BC, Canada V6T 1Z2
>
>
> David C. Ullrich
Herman Rubin
>Ben Golub wrote:
>> Dr. Arm wrote:
>> > But mathematical memorization can also be the equivalent of beating your
>> > kids: Works good only for solving previously addressed problems.
>> Well, I disagree. For instance, a true understanding of the definition can be used
>> to prove a wide variety of limits,
>some of them quite nontrivial.
It also excludes an even wider variety. The more general
versions are easier to understand.
Many years ago, a student (who did get a PhD, under standards
at least as strong as now) told me that the biggest problem
he had in topology was previous metric space topology.
>Understanding is not memorization. Argue your statement of disagreement
>before you begin another arguement.
Neil W Rickert
>> >Define what you mean by "memorize."
>> I seem to be using it in the standard way.
>Then DEFINE it. Or are platitudes all you
>want to commit to? You call yourself a
>rational debater? Ha!
>> I doubt that it is any
>> more definable than the rest of the words in the dictionary.
>Cop out, as usual by my *&%*& detractors.
> Memorization of a set of objective knowledge
> (knowledge that is conventionally testable by standard
> tests, the likes of which are given in classrooms
> all over the world) is ANY process whereby
> a person is able at the end of the process
> to retrieve this knowledge quickly and
> accurately as a mental representation of this
> knowledge, and then effectively use this
> knowledge correctly on standardized tests.
There is a whole discipline of cognitive science out there. Just
about all of the cognitive scientists would view the above as naive
and simplistic.
But if you disagree, I suggest you try to publish your theory of
understanding in a respectable cognitive science journal.
nick
> Patrick Reany wrote:
>
> > The posters on this thread
> > are largely a part of the problem and not part
> > intellectual people are memorizers. The people who
> > think they don't memorize have just invented
> > a crafty definition of "memorization" that supports
> > their loony politically correct dogma of denial.
>
> this thread to provoke a discussion, and now you are calling the people that
> courteously answered your call "part of the problem." Call their arguments invalid
> if you choose, but don't insult them as loony or problematic in themselves.
>
> Secondly, your dogma of "memorization before understanding" has the potential to
> cause as much trouble as the opposite "memorization is old-style garbage." I know
> a lot... a LOT... of people who will never be successful at a hard science who
> have convinced themselves that memorization will lead them to success. It won't!
> Educators MUST tell people that if their memorization must be a dynamic process
> where they constanly build on a memorized fact with a layer of understanding, like
> a cake with cream. Otherwise, the result of all their pain will be dry and
> impalatable.
>
> I agree that all successful intellectuals are capable of strong fact-retention,
Which demands the obvious, supposedly generally
accepted, but probably unwelcome observation
that correlation does not even _imply_
causality, much less prove it. Discussions of
educational strategies ususally become even more
warm and fuzzy than those about causality in
physics. At the risk of evoking a resounding
silence, I will use the e-word, and suggest that
Patrick describe an experiment to test his
hypothesis. It would be necessary to demonstrate
that what he requires of students creates an
outcome independent of the mental
characteristics of his students at the start of
the experiment.
> but don't harp on that as if you've found some sort of new gem. It's well-known,
> and the key is finding an edcuation process that makes it easier for more people
> to memorize WELL -- that is, not a process of torture, but almost a fun process
> where you , (1) memorize something, (2) make it your own, and (3) don't think
> about it as external to yourself anymore. Most people miss (2) and thus can't do
> (3), which makes true learning impossible for them.
>
Yes, but perhaps for some people (2) comes (and should) before (1). Very
slightly, and definitely not simultaneously.
Nick
> Ben
Neil W Rickert
>> >I knew you wouldn't give a straight answer
>> >to a straight question.
>> It is not a straight question.
>> Mathematics is a very broad subject. What is manipulated varies
>> greatly, depending on what is to be understood.
>> You apparently have a particular folk-theory of what it takes to
>> master mathematics.
>Where did I claim that my thread is about
>"mastery" of mathematics?
You did talk about understanding of mathematics, which is pretty much
the same thing.
> I do claim that
>for every concept that one wants to understand
>in mathematics there is always something
>that has to be memorized first, at least
>just to get the statement of the concept into
>short-term memory, or do you claim that even
>that is unnecessary?
Memorization is not necessary to get statements into short-term
memory.
>> You are making arguments based on that
>> folk-theory. It seems, from the reaction you are getting, that most
>> of the respondants sense that your folk-theory does not fit very well
>> with how they actually do mathematics.
>All I get out of the sum total of demagoguery
>of my respondents is that they refuse to define
>the terms I ask them to define, and they also
Perhaps they are not so naive as to believe that the terms are
definable.
>I don't give a damn, Neil, what you personally
>believe. But if you're going to debate me, do it
>by rational rules and NOT by demagoguery. The
>rules are quite simple: There are only two terms that
>have to be defined: "memorization" and "understanding."
If you ever come up with a clear and precise definition of
"understanding," then publish it in one of the cognitive science
journals. Maybe you can also use that definition to solve the
outstanding problems in AI.
Dave Rusin
W. Edwin Clark <ecl...@math.usf.edu> wrote:
>Many years ago when I taught abstract algebra to Caltech
>sophmores I didn't have to resort to such strategies,
>but I don't know of another way to deal with students
>at my university where (unlike at Lake Wobegon) all students
>are not above average.
Gee, that's too bad. Around here the worst I have to deal with
is that not all the students are above average!
>Memorize, memorize, let no definition escape your eyes.
I'll second the call of those who ask for a certain amount of
memorization of definitions. When you see what's "really going on",
then you can simply state on the spot what it means e.g. for a
group to be cyclic. But most students first attempting to deal with
this concept have little to go on -- few familiar examples,
no mental pictures, only a few theorems -- and are well served
by having a precise definition handy. Often the simplest
things to be proven amount merely to writing down the definitions
and studying the syntax of the sentences. ("Let's, the phrase
is '... if there exists a g such that...' so I guess I had
better find a 'g' ...")
I would say the need for memorizing definitions would have to go
up in a setting in which the instructor prefers to go straight for
the most general and flexible concepts rather than sully the
students' minds with special examples drawn from more rigid categories.
The students have to have _something_ to build on!
For proof of the necessity of having really learned/memorized/internalized
the definitions, I suggest only that people look at the kinds of
conversations which turn up in this newsgroup when non-math-trained
people attempt to articulate ideas using unclear concepts and
incorrectly-mimicked terminology. Math is, like, all about precision, y'know?
dave
Gregory L. Hansen
>of memorization in the learning of math and
>physics. I did this sometime ago and found
>the discussion interesting.
>
>
> Memorization comes before understanding.
>
>one has to have a good definition of what it
>mean to "understand" something, right. Also,
>is wouldn't hurt to have a definition of what
>memorization means, either.
Also the scope of memorization. You couldn't even *read* if you can't
remember what the words mean! You can't solve an algebraic equation if
you can't remember that you were supposed to solve the equation.
But the whole point of learning mathematics is to understand it, and how
to apply it. If you don't understand it you've wasted your time, no
matter how much of it you remember. And the better you understand it the
less you need to remember, because you can derive things as needed. I've
done that often enough.
I guess the question to ask is, what does it matter? Will it affect the
way you teach a class?
--
"For every problem there is a solution which is simple, clean and wrong. "
-- Henry Louis Mencken
Patrick Reany
Neil W Rickert wrote:
> Patrick Reany <re...@asu.edu> writes:
> >Neil W Rickert wrote:
> >> Patrick Reany <re...@asu.edu> writes:
>
> >> >I knew you wouldn't give a straight answer
> >> >to a straight question.
>
> >> It is not a straight question.
>
> >> Mathematics is a very broad subject. What is manipulated varies
> >> greatly, depending on what is to be understood.
>
> >> You apparently have a particular folk-theory of what it takes to
> >> master mathematics.
>
> >Where did I claim that my thread is about
> >"mastery" of mathematics?
>
> You did talk about understanding of mathematics, which is pretty much
> the same thing.
>
> > I do claim that
> >for every concept that one wants to understand
> >in mathematics there is always something
> >that has to be memorized first, at least
> >just to get the statement of the concept into
> >short-term memory, or do you claim that even
> >that is unnecessary?
>
> Memorization is not necessary to get statements into short-term
> memory.
To be "in short-term memory" is to be memorized.
To be memorized is simply to be in your brain for
instant, accurate retrieval. You can't claim to have
a rational argument founded on vague gut feelings.
> >> You are making arguments based on that
> >> folk-theory. It seems, from the reaction you are getting, that most
> >> of the respondants sense that your folk-theory does not fit very well
> >> with how they actually do mathematics.
>
> >All I get out of the sum total of demagoguery
> >of my respondents is that they refuse to define
> >the terms I ask them to define, and they also
>
> Perhaps they are not so naive as to believe that the terms are
> definable.
Then what the hell are they doing arguing with me so
dogmatically? Anyone who can't define these
terms has no business arguing with me in the first
place.
I have stated my definition of understanding math
concepts on my website and on this thread
many times already. Understanding of a concept
is any process that builds relationships among the
parts of a concept that hold the parts as a single
meaningful object to the mind. I admit that this
definition is not particular strong on what "understanding"
is, but that's not my objective. My objective is to
explain the "how" of achieving understanding. The
objective proof of having attained "understanding" of
a concept is the ability to solve problems that require
the concept. What's the point of fancying it up
for some journal? It's too simple for that.
Patrick
W. Edwin Clark
Dave Rusin wrote:
> For proof of the necessity of having really learned/memorized/internalized
> the definitions, I suggest only that people look at the kinds of
> conversations which turn up in this newsgroup when non-math-trained
> people attempt to articulate ideas using unclear concepts and
> incorrectly-mimicked terminology. Math is, like, all about precision, y'know?
>
Exactly! My student often accuse me of "nit-picking". I explain
that that is exactly what mathematics is all about. In fact, it is
the science of "nitpicking"!
Edwin
William Hale
[cut]
> The objective proof of having attained "understanding" of
> a concept is the ability to solve problems that require
> the concept.
This is not correct.
-- Bill Hale
Patrick Reany
William Hale wrote:
Gee, don't ya just love these hyper-terse replies
that claim something but explain nothing at all.
Why don't you elaborate. Does anyone at all who
argues with me ever elaborate to say more
that mushy dogma? So far the answer is NO!
Patrick
Gregory L. Hansen
Dr Arm®
>
> Also the scope of memorization. You couldn't even *read* if you can't
> remember what the words mean! You can't solve an algebraic equation if
> you can't remember that you were supposed to solve the equation.
You;ve streached the concept of memorization until it covers everytihig
you wre attempting to show,
Or did you forget the original premise?
da
Herman Rubin
>Neil W Rickert wrote:
If you had been reading this newsgroup before, or the
education newsgroup, you would know that many of us
do not believe that such a test should ever be given,
especially not in mathematics.
A decent mathematics problem should be a multipart
problem, where the understanding of the problem is
the first task, then having an idea how to go about
attacking the problem, then attacking it, and only
at the end attempting to get the answer. This must
be done with NO HINTS, such as possible answers.
What you are asking for is the least important part
of just about anything.
You put importance on this obstacle to understanding,
which can be done by mindless automata. Those who
have been taught this way seem to find it very hard
to understand.
Herman Rubin
>Herman Rubin wrote:
>> In article <3D29CB81...@asu.edu>, Patrick Reany <re...@asu.edu> wrote:
>> >My basic premise is that:
>> > Memorization comes before understanding.
>> This is quite false. Memorization hinders understanding.
>Maybe you would admit that for certain courses some
>memorization is necessary?
>Suppose you are teaching a beginning course in, say,
>abstract algebra or number theory. Call it what you may,
>but there are lots of definitions and theorems that one
>must KNOW in order to be able to do the kind of things
>that such students should be able to do.
Not many are needed, and very definite I do not want
students to memorize a particular characterization as
a definition, especially when there are others.
>Wouldn't you, for example, expect a student to be able
>to prove that if two groups are isomorphic and one is
>cyclic then the other is also?
>Now tell me how someone can do this who doesn't know
>the definition of a group, a cyclic group and what it
>means to say that a group is isomorphic to another group
>----and a few other things?
What is the definition of a group? This is not just
a rhetorical question; there are several quite different
ones. Groups form an equational class, as they satisfy
the conditions of Birkhoff's theorem characterizing them;
I have seen many textbooks defining a group, not one of
them using equations, although it can be done. Each
"definition" uses enough properties to characterize them,
and should be called characterizations.
Definitions can be looked up when needed.
>And how would they "know" these things if they didn't
>"memorize" them?
Do they have to memorize them? A student given the
problem can look up what is a cyclic group. And since
cyclic groups are abelian, what if they used Tarski's
nice characterization of an abelian group as a set
with an operation "-" such that c=a-(b-(c-(a-b))) ?
>Some people have very good memories. They are told ONCE
>these definitions and they never forget them. I guess these
>are the people like Robert who never have to "memorize".
>Others must make a significant effort to absorb definitions.
>[Of course, some refuse to learn them and some make up
>their own.]
>When I teach such courses I try to FORCE the students to learn
>to state the definitions precisely by giving short weekly
>quizzes on definitions. For the less gifted this means I
>force them to "memorize" the definitions. Is this bad?
Yes. The definition displaces the concept. The concept
of a group is not just something which satisfies a
characterization; this is a way that one can show that
something is a group. It has lots of properties, not
the few in the characterization, and no one particular
set is "key".
>Many years ago when I taught abstract algebra to Caltech
>sophmores I didn't have to resort to such strategies,
>but I don't know of another way to deal with students
>at my university where (unlike at Lake Wobegon) all students
>are not above average.
Make them think, not memorize.
The World Wide Wade
"W. Edwin Clark" <ecl...@math.usf.edu> wrote:
> Exactly! My student often accuse me of "nit-picking". I explain
> that that is exactly what mathematics is all about. In fact, it is
> the science of "nitpicking"!
Very amusing. Now please tell me you're joking.
--WWW.
Bill Taylor
|> I don't give a damn, Neil, what you personally believe.
And we don't really give a damn about what is making you behave so abusively
and anal-compulsively about this issue, Patrick!
Neil is a highly respected regular here, and has made excellent responses
to your queries. I suggest you read them again with a clearer mind, more open
to improvement than polemics of your own. It is not our fault if you cannot
understand what everyone else seems to be having little trouble with!
|> But if you're going to debate me, do it
|> by rational rules and NOT by demagoguery.
This is a typical comment by someone being irrational in his own particular
way that he feels safe with.
|> The rules are quite simple:
YOUR alleged rules are FAR too simple(-minded) to be much use for anything.
|> There are only two terms that
|> have to be defined: "memorization" and "understanding."
This is utter rubbish on several counts. An obsessive demand for "definitions"
at all costs, in everyday debate, betokens a complete failure to understand
the main intent of such debate (which is enlightenment rather than debating
points), and a refusal to listen to anything that would help achieve that end.
|> And at least one counterexample that must be given by your side.
This is near-meaningless gibberish in the current context.
|> Is that damn plain enough for you?
What is plain is that you are an undergraduate-minded twit with little
understanding of what the rest of us are talking about.
|> I'm out here alone
Oops - I apologize - you *have* understood at least one vital point.
|> trying to get anyone to actually stand up like
|> a decent human being and freely debate me rationally on the merits.
What a pathetic, self-serving, whingeing, petulant piece of twaddle!
|> So either do that or stop wasting my time!
You're wasting your own time. And our patience, which is almost gone.
|> I'm here to challenge your absurd "folk-theory" of the understanding
|> of math and physics concepts.
What you ARE here for is very unclear, but it doesn't seem to be effective.
|> When the hell is YOUR side going to debate me by the rules?
When are you going to stop blethering about alleged rules, stop shouting,
start listening, grow up and learn something? Good advice and wisdom
from many of your betters is being cast before you, and you are rooting
around trampling it in the muddy inadequacy of your own intellect.
Now grow up or piss off!
------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
I'm a little crackpot short and stout.
Don't ask my handle, let me spout.
When I get all steamed up then I shout,
Snip a poster - flame the lout.
------------------------------------------------------------------------------
James Hunter
Neil W Rickert wrote:
> Patrick Reany <re...@asu.edu> writes:
> >Bill Taylor wrote:
>
> >> Patrick Reany <re...@asu.edu> writes:
> >>
> >> |> Memorization comes before understanding.
> >>
> >> Blimey, watchit mate!
> >>
> >> You're going to cop a load of old cobblers from Herman Rubin for this heresy!!
> >>
> >> I would not go quite as far as the other fellow in denouncing memorizing
> >> anything AT ALL, but there is a grain of truth in what he said.
> >>
> >> If you want a quick and dirty epigram to cover the case, I would suggest this:
> >>
> >> Manipulation comes before understanding.
> >> =======================================
> >>
>
> >Manipulation of what?
>
> If you were thinking like a mathematician, you wouldn't need to ask
> that question. And you wouldn't be making mistaken claims about the
> role of memorization.
You can't think like a born-again, newer-than-new-metamath
revivalist dork with this one, since we have to remind
the demented that you don't the axiom of choice
available MORONS.
Jesse F. Hughes
> "Jesse F. Hughes" wrote:
>
> > Patrick Reany <re...@asu.edu> writes:
> >
> > > It's time to accept that all successful intellectual people are
> > > memorizers. The people who think they don't memorize have just
> > > invented a crafty definition of "memorization" that supports their
> > > loony politically correct dogma of denial.
> >
> > "All successful intellectual people are memorizers."
> >
> > "Everyone who denies this is dogmatic."
> >
> > (As far as that goes, what the heck is "politically correct" about
> > this dogma?)
>
> The term "politically correct" is a metaphor for a
> sub-dogma based on irrational doctrines held by
> the masses to keep in line with some overarching
> "politically motivated super-dogma."
Damn commies. Trying to keep the sheeple down by denying them the
power of memorization. Diabolical!
I am NOT on
> You mentioned denials. Denials are easy. Clearly
> explaining what's wrong with my claims is
> apparently too hard to my detractors to get
> around to doing. So who are the dogmatists?
No doubt, your detractors are following the sub-dogma in service to
the super-dogma. But, hit it with a bit of your kryptonite-dogma and
all will be well in the end.
--
Jesse Hughes
"Still I think [Wiles's] story is kind of suspicious, but hey, truth is
stranger than fiction."
-- James Harris, colleague of Andrew Wiles
Rune Allnor
> You cannot think about anything unless those
> things you are thinking about are at least in
> your short term memory.
> Patrick
This statement makes me wonder if this whole thread is a
clever troll... I think it would be safe to assume that
any student is capable to keep a statement he has heard
or read in his short term memory sufficiently long to
contemplate its contents.
Whether he actually does contemplate such information, and
how he does so, is a completely different matter...
Rune
Mark Fergerson
>
> I want to start a conversation about the role
> of memorization in the learning of math and
> physics. I did this sometime ago and found
> the discussion interesting.
>
>
> Memorization comes before understanding.
In the third grade, we were introduced
to multiplication. We learned that it
was just repeated addition, and that
there were some simple rules, but the
teacher eventually defaulted to rote
memorization (entire class chanting "six
times seven is forty-two" and so on).
Ever seen something called a "Pee-chee
folder"? It's a thick paper jobbie
designed to hold school paper work, and
has all sorts of neat stuff printed on
it, like a _multiplication table_ that
goes up to 12x12. One glance, and I had
an "AHA moment". The next couple weeks
of "six times seven" were a horrible
torture for me.
So IMNSHO yes, you have to memorize
the parts, and the rules, but not all
the results.
> But to fully appreciate what this means,
> one has to have a good definition of what it
> mean to "understand" something, right. Also,
> is wouldn't hurt to have a definition of what
> memorization means, either.
Well, let's all memorize this, shall
we, and see if it helps?
> Now, I want to counter the obvious charge
> that all I'm advocating is for students to
> memorize their textbooks word-for-word.
> This I do NOT do. And I'm sure that even
> my strongest critiques will agree that some
> memorization of new material must be
> accomplished somehow, if conceptual
> understanding is to be achieved. So, the
> question is, "How to find the happy
> balance?"
Well, to begin with, your "critiques"
are what you generate when you examine
your own work, whereas your "critics"
are those persons other than yourself
that criticize your work. Ahem.
Mark L. Fergerson
Brian Chandler
[snip]
> Memorization is necessary but NOT sufficient
> for understanding. Don't believe me? Then
> provide a counterexample!
I certainly don't understand the point of this discussion, unless it
is on the same general level as "no gain without pain", and I don't
see quite what you're getting at with this "Memorization" word (which
will mislead at least some into thinking you mean 'rote learning').
But putting all that aside, consider proofs that are totally visual.
In the proof below - what can be said to have been "memorized" in
order to understand the proof?
(Roger Penrose has an interesting essay on this kind of thing in "What
is intelligence?" edited by Jean Khalfa - perhaps out of print: try
http://dogbert.abebooks.com/abe/BookSearch?sby=key&stext=intelligence+khalfa&image.x=0&image.y=0&ph=2
)
.o0o._.o0o._.o0o._.o0o._.o0o._.o0o._.o0o.
Excerpt from http://imaginatorium.org/stuff/bubbles.htm (My page, but
of course the proof is not original)
PROBLEM:
In a plane (i.e. on a table napkin) draw three circles all of
different sizes (they may overlap, but may not be completely inside
each other). Now for each pair of circles, draw the two tangents that
touch both on the same side (i.e. place two chopsticks to sandwich
each pair of circles). Each of these pairs of lines (chopsticks) meets
on a point (because the circles aren't the same size). Prove that the
three points at which the three pairs of chopsticks intersect all lie
on a straight line.
PROOF:
OK, imagine that on each circle on the napkin stands a hemispherical
bubble of the same radius (and of rather strong material as bubbles
go). Also imagine that the table has a mirror surface, so you appear
to see three spheres floating in space with the knives and forks. Now
imagine that for each pair of spheres, the pair of touching chopsticks
is replaced by a carefully cut piece of paper from the inside of the
menu, curved around the bubbles. Now you see a floating cone, and
obviously the vertex (er, "point") of the cone is at the same place
where the chopsticks used to intersect. Almost there: we have to show
that the vertices of the three cones lie on a straight line. So we
take the stiff (flat) outer cover of the menu, and lay it on top of
the three bubbles. It isn't hard to see that it therefore lies neatly
on top of the three cones, and therefore intersects the table at each
of the three vertices. But both the menu and the table are planes, and
two planes intersect in a straight line. Therefore the three vertices
lie on a straight line. Therefore the two-dimensional problem follows
trivially, as a subset of the three-dimensional one.
.o0o._.o0o._.o0o._.o0o._.o0o._.o0o._.o0o.
Brian Chandler
----------------
geo://Sano.Japan.Planet_3
http://imaginatorium.org/
Nico Benschop
> >> having attained "understanding" of a concept [2]
> >> is the ability to solve problems that require the concept. [3]
> >
> >This is not correct.
>
> T'is.
Indeed, but why? Can you elaborate? Your 'yes' and my 'yes' to
agreeing with [3] as charactristic of [2] may be different...
What I mean is: your 'understanding' may still be different
from my 'understanding', although on the surface we seem to agree.
In other words, what we observe is always filtered by something
like the sum-total of our previous experiences, which filter-bank
is likely to be quite different for you and me. Of course, we
both 'know' enough of the English language, we both are raised
in a Western society, at schools that roughly aim for similar
values to be 'programmed' into our little brains, etc. etc.
These unmentioned assumptions form the basis for our communication,
and quite some 'memory' is involved in such 'filter-bank'. I'm
just reading "The Wholeness of Nature" (1996, UK) by Henri Bortoft
(philosofer) http://www.dialogonleadership.org/Bortoft-1999cp.html
"In 1972 I came to know Goethe's work just by accident…
Goethe's point was to develop a different kind of seeing,
a seeing that strives from the whole to the parts.
That was very close to Bohm's hologram…"
In essence: there is no 'objective' observation of 'facts',
we always interprete (often unconsciously) what we see / read,
hence filter it, while putting it in our 'memory' with some
label as to its context and value. The concept of TRUTH is
also very much filtered personally, and dependent on context.
E.g.: Goethe did not understand, or rather disagreed, with
Newton's theory of colours (being purely numerical, quantitative)
while he (Goethe), as artist, was missing something very essential
namely the 'quality' of colours and their value in our visual
image of the world we live in, and the physiology of our eyes,
yielding a 'colour-in-contex' of other colours and shapes
(re: a 'random' collection of spots (a la Rohrschach) that, after
some observation time, many people agree to represent a giraffe)
So, as to 'memory' and 'understanding': the terms into which
you provide a further decomposition / explanation (or rather
'embedding' into a larger context of personal experiences)
will probably depend on _your_ context & expectations.
For me, 'memory' in this context would correlate with
'association' and 'resonance' of certain 'features'
(re feature extraction in OCR: optical character recognition)
while 'understanding' has to do with 'able-to-apply' in
a 'relevant' context (recognized as such by my filterbank;-)
(some bells went ringing...)
Towards more math-like terms:
understand => similar => resonance => analogy => (iso)morphism
In 'fact': the two concepts of memory and understanding are
not so clearly separated in my mind, really...
(in my early schooldays I hated learning 'by rote' the
words & rules of the several foreign langages that we
in the Netherlands must learn: French, German & English.
So it was a relief to me that e.g. all those trig formula's
could be derived easily from very few basic ones; hence my
preference for a 'generative' view of things & FSM's &c :-)
[*] countable = A* and uncountable = B*, with |A|=1, |B|>1.
-- NB - http://www.iae.nl/users/benschop/math-use.htm
http://www.iae.nl/users/benschop/cantor.htm [*]
Patrick Reany
Rune Allnor wrote:
People must really hate to memorize, by rote or by
any other method, because even though people
obviously have things in their memory (as evidenced
by the fact that they can remember words and
syntax to write replies to me) they still deny
that what they have in their memory has been
memorized. Ludicrous. I have no time to troll
around. I have a simple epigram, which used to be
utterly obscure to me but now seems obvious:
Memorization comes before understanding.
Yes, the obvious implication of this is that I
should make a determined effort to memorize
as much as I can stand to master every subject
I care to. Sometimes understanding is difficult
and impossible to accomplish as an algorithmic
process. But memorization of the parts to a
difficult concept is straightforward and algorithmic.
It's the one parts of the understanding process
that is unmysterious.
The pianist must practice scales, the pitcher
must practice his pitch, so too the intellectual
must fill his or her mind with memorized
knowledge -- in long-term memory, of course.
It's only laziness that keeps us from doing this
so we make up silly reasons NOT to do it.
Is this then really about ROTE memorization.
Partly. Unless you're a see-once-remember-
forever type. I don't care how one gets knowledge
in memory, just so long as it gets done, sooner
rather than later.
When I was a professional math tutor, I
noticed something which, at the time, amazed
me. All my A-students always had all the
central concepts and theorems memorized
before we met. The C and D students
never did. This was NOT due to any
prompting on my part. It's just a "habit" of
effective, good students! Were these A-
students just naturally good at memorizing,
or did they just learn on their own that
memorization comes before understanding,
and just decided that memorization by rote
is just something good students do to get
good grades? I don't know. But the result
was the same. The chance to understand a
concept is made better by preparation. It's
a case of chance favoring the prepared mind.
Patrick
People often overlook the obvious.
-- Dr. Who
Patrick Reany
Mark Fergerson wrote:
I just might!
> > Now, I want to counter the obvious charge
> > that all I'm advocating is for students to
> > memorize their textbooks word-for-word.
> > This I do NOT do. And I'm sure that even
> > my strongest critiques will agree that some
> > memorization of new material must be
> > accomplished somehow, if conceptual
> > understanding is to be achieved. So, the
> > question is, "How to find the happy
> > balance?"
>
> Well, to begin with, your "critiques"
> are what you generate when you examine
> your own work, whereas your "critics"
> are those persons other than yourself
> that criticize your work. Ahem.
>
> Mark L. Fergerson
I have yet to find anyone who will debate
by rational means, though I suppose the
opposition thinks they are. Do you people
KNOW how to debate rationally? I
made a simple claim. I got claims back
that my statement is wrong. FINE! So, if it's
wrong, you smart people who don't "memorize"
should be able to show it's wrong by defining
the terms I use and providing counterexamples.
(You people know. Counterexamples, right?
Hello! That's right, same as in math itself. OK.)
I am quite willing to accept defeat if this
counterexample of your side is devastating to
my claim. But so far no one has even tried to
do this. You'd think that mathematicians would
make good debaters. Learn something new
every day.
Well, what can I expect of people who think
that "memorization" rots the brain.
You know, I really expected a better show
of it from mathematicians.
Patrick
Gregory L. Hansen
Nico Benschop <n.ben...@chello.nl> wrote:
>glha...@steel.ucs.indiana.edu (Gregory L. Hansen) wrote:
>>
>> William Hale <ha...@tulane.edu> wrote:
>> > Patrick Reany <re...@asu.edu> wrote:
>> > [cut]
>> >> The objective proof of [1]
>> >> having attained "understanding" of a concept [2]
>> >> is the ability to solve problems that require the concept. [3]
>> >
>> >This is not correct.
>>
>> T'is.
>
>Indeed, but why? Can you elaborate? Your 'yes' and my 'yes' to
> agreeing with [3] as charactristic of [2] may be different...
>
>What I mean is: your 'understanding' may still be different
>from my 'understanding', although on the surface we seem to agree.
Mainly I was responding to a pitifully curt reply with an even more
pitifully curt reply. Going back and forth saying "T'is", "T'aint'"
accomplishes nothing.
But you can't say you really understand a concept unless you can teach it
to another person, and discuss it in novel ways. Not all concepts are
mathematical theorems that you solve problems with, but you should still
be able to relate it to things that you've never seen it related to. In
the case of math, that usually means solving problems.
How well can you say you understand vectors if you can't solve one of
those "A person walks 10 meters east and 20 meters northeast, what total
distance has he traveled?" problems? The whole point of a vector is to
apply it to that sort of problem with magnitudes and directions. Though
if you're a little confused about a Fourier transform as a state space
vector, that's a more abstract use, and we don't expect the freshman to
crank out Fourier transforms, anyway. There are degrees of understanding.
Gregory L. Hansen
>Gregory L. Hansen wrote:
>>
>> Also the scope of memorization. You couldn't even *read* if you can't
>> remember what the words mean! You can't solve an algebraic equation if
>> you can't remember that you were supposed to solve the equation.
>
>You;ve streached the concept of memorization until it covers everytihig
>you wre attempting to show,
Exactly. As above, it's trivially easy to show that memorization comes
before understanding. But that's not very useful in figuring out how to
teach a class, and it's probably not what Patrick had in mind. The
question seemed a little vague to me.
Nico Benschop
>
> In article <caeab7cb.02071...@posting.google.com>,
> Nico Benschop <n.ben...@chello.nl> wrote:
> >glha...@steel.ucs.indiana.edu (Gregory L. Hansen) wrote:
> >>
> >> William Hale <ha...@tulane.edu> wrote:
> >> > Patrick Reany <re...@asu.edu> wrote:
> >> > [cut]
> >> >> The objective proof of [1]
> >> >> having attained "understanding" of a concept [2]
> >> >> is the ability to solve problems that require the concept. [3]
> >> >
> >>
> >> T'is.
> >
> >Indeed, but why? Can you elaborate? Your 'yes' and my 'yes' to
> > agreeing with [3] as charactristic of [2] may be different...
> >
> >What I mean is: your 'understanding' may still be different
> >from my 'understanding', although on the surface we seem to agree.
>
> Mainly I was responding to a pitifully curt reply with an even more
> pitifully curt reply. Going back and forth saying "T'is", "T'aint'"
> accomplishes nothing.
Precisely.
And I took your terse remark indeed as critique on reasonless [*].
(in Dutch we refer to that style as "welles, nietes".)
> But you can't say you really understand a concept unless you
> can teach it to another person, and discuss it in novel ways.
Good point.
Richard Feynman on why spin one-half particles obey Fermi-Dirac
statistics:
"I couldn't do it. I couldn't reduce it to the freshman level.
That means we don't really understand it. "
-- He also expected simplicity at the bottom of things:
"It always bothers me that, according to the laws
as we understand them today, it takes a computing
machine an infinite number of logical operations
to figure out what goes on in no mater how tiny a
region of space, and no matter how tiny a region
of time. How can all that be going on in that
tiny space? Why should it take an infinite
amount of logic to figure out what one tiny piece
of space/time is going to do? So I have often
made the hypothesis that ultimately physics will
not require a mathematical statement, that in the
end the machinery will be revealed, and the laws
will turn out to be simple..." -- Feynman
-- NB
Ben Golub
> must practice his pitch, so too the intellectual
> must fill his or her mind with memorized
> knowledge -- in long-term memory, of course.
Patrick, you are trolling about. The third part of this silly analogy doesn't fit!
The correct continuation would be "... and the mathematician must PRACTICE DOING
MATHEMATICS!" How he does that and whether it follows from your creed of
"memorization before understanding" is another matter entirely. But doing science
and "filling one's head with memorized knowledge" is not the same thing.
I think your purpose on this thread is to clobber everybody with your theory and
refuse to make even small allowances... well, in that sort of discussion, there's
a great thing: you ALWAYS win when people realize that trying to discuss the issue
is pointless. So take your victories as they come.
Ben Golub
> the super-dogma. But, hit it with a bit of your kryptonite-dogma and
> all will be well in the end.
I think my karma just ran over Patrick's dogma.
nick
> "Jesse F. Hughes" wrote:
>
> > Patrick Reany <re...@asu.edu> writes:
> >
> > > It's time to accept that all successful intellectual people are
> > > memorizers. The people who think they don't memorize have just
> > > invented a crafty definition of "memorization" that supports their
> > > loony politically correct dogma of denial.
> >
> > "All successful intellectual people are memorizers."
> >
> > "Everyone who denies this is dogmatic."
> >
> > (As far as that goes, what the heck is "politically correct" about
> > this dogma?)
>
> The term "politically correct" is a metaphor for a
> sub-dogma based on irrational doctrines held by
> the masses to keep in line with some overarching
> the side of the masses on this one. I learned my lesson.
> Now I challenge this false dogma. Let its defenders for
> once actually explain what they mean by "memorize"
> and "understand." Or must I do all the work in this
> debate? Not one person has bothered to define
> either of these terms when I asked them to do so.
>
> You mentioned denials. Denials are easy. Clearly
> explaining what's wrong with my claims is
> apparently too hard to my detractors to get
> around to doing. So who are the dogmatists?
>
-You don't explain how you differentiate outcomes
of your process from pre-existing conditions.
-You don't explain how you test the outcome
'understanding' in isolation from the outcome
'having memorized'.
nick
> Patrick
nick
Ben Golub
> before understanding. But that's not very useful in figuring out how to
> teach a class, and it's probably not what Patrick had in mind. The
> question seemed a little vague to me.
I agree. "You're uttering English words, so you've been memorizing" is absurd,
though true on Patrick's level. He's been accusing everybody of being unable to
have a reasoned debate, but he himself is the one who has stuck to his idea like
some sort of revelation and refused a discussion of its practical merits.
Ben Golub
> You know, I really expected a better show
> of it from mathematicians.
Go talk to JSH over there. He's been repressed by us loonies for... oh... seven
years now.
Look, I'll try one time and I have a feeling that your narrowness in this debate
will continue and you will be unable to grasp the point of what a lot of the
posters on this theread are saying... but here we go.
To memorize: to make a continuous effort to FORCE certain facts into long-term
memory, without necessarily pausing to make those facts comprehensible or
palatable to oneself.
In that case, memorization can hinder understanding because the memorizer will sit
around swallowing things that he does not work to digest. If he digested them,
there would be no need to force them down because they would go smoothly (anybody
who has "gone over the hump" of a difficult math concept and then said, "what's so
bad about that?" knows what I'm talking about.) So a lot of successful
intellectuals have agreed that the way they learn best is to briefly think on it
(which they don't believe is memorizing), and then internalize it -- which is
distinct from forcing it into long term memory ("memorizing") because it happens
naturally.
You will say: memorization is simply being able to recall. Memorization INCLUDES
putting things in your short term memory. If you "know" what English words mean,
you've memorized them, and so forth.
As others have pointed out, this is a very primitive cognitive theory. When you
read a book and remember the plot, or watch a fun movie and tell a friend later,
or remember what your friend's face looks like, you indeed have committed all
those things to memory... but almost all cognitive scientists and indeed most
people will say that you never "memorized" the book or the movie or the face. Many
of us posters believe that math works best when you learn to think of it as a book
or a movie or a face -- something you internalize naturally by contemplating it
and remembering them in a form that works for you.
The most interesting issues in cognitive science happen when we start to think,
where is the border between forced learning and natural learning? No child
"memorizes" motor skills like walking or English words. How do we know? Studies
show different areas of the brain working in different ways when the two happen.
When shown a fun video game and asked to absorb the rules in order to play, the
child's "natural" areas show increased activity -- the same ones as those that
work when new words are learned in everyday conversation. This child wants to
learn because it's fun and he knows why he wants to learn it, so there's no
"memorization effort" going on. On the other hand, when shown a list of words and
asked to memorize by rote, totally different areas tend to fire -- and
incidentally, the learning is significantly slower. I read this particular
information in a cognitive science paper a few years ago -- it was background in a
study on memorization and dyslexia. Incidentally, I made no effort to memorize all
that, it just came back naturally.
The point is, as people have said above, you can stretch your silly definition to
encompass any aspect of thinking you want, and that won't get us anywhere becuase
you are ignoring the subtleties of real cognitive science that actually help
people learn better. That's the counterexample -- that you are unwilling to
consider that "memorization" according to most intelligent people is significantly
narrower than for you. And no matter how much you yell that "anything is
memorization," you won't get anything accomplished.
Are we clearer now?
Ben
Patrick Reany <re...@asu.edu> wrote in message news:3D2C2A2B...@asu.edu...
>
> Patrick
Patrick Reany
Ben Golub wrote:
> > The pianist must practice scales, the pitcher
> > must practice his pitch, so too the intellectual
> > must fill his or her mind with memorized
> > knowledge -- in long-term memory, of course.
>
> Patrick, you are trolling about.
I refuse to accept that my attempt to provoke
people to provide rational arguments against me
is "trolling." This is yet another obfuscation of
the whole argument. So you're faulting me for
having no patience with people who counter
my arguments with rhetoric rather than real
debate. If I'm wrong prove it with rational
argumentation. You mathematicians are supposed
to KNOW how to do this. Quit making up
excuses for people who break the rules of
debate. If you were contesting a proffered
math theorem I might propose you'd present
either a logical flaw or a counterexample. That's
precisely what's needed here too. And you
KNOW it! If you people can't provide this
kind of proof against me, then just say "I think
you are wrong," and leave it at that. But don't
pretend that that adds any points to your resume.
> The third part of this silly analogy doesn't fit!
> The correct continuation would be "... and the mathematician must PRACTICE DOING
> MATHEMATICS!" How he does that and whether it follows from your creed of
> "memorization before understanding" is another matter entirely. But doing science
> and "filling one's head with memorized knowledge" is not the same thing.
Yes, practice doing mathematics. Don't you ever
get rusty on anything you've ever learned?
Just how perfect are you, anyway? Filling your
head with math facts increases the likelihood of
making an important connection. Years ago I found
a solution to the cubic equation using a Clifford
algebra. I did this while playing around with
cube roots of hyperintegers. In the process I
noticed that I got a special version of the cubic
equation with roots provided. The special form
is called the "reduced cubic." I had just memorized
the year before that the "reduced cubic" is just
as general as the full cubic. If I had not memorized
this fact, I might have just dismissed this special
cubic as a mere uninteresting special case. Chance
favors the prepared mind. The more you have
accurately stored in your memory the better
chance you have of "seeing" connections that might
otherwise would go unnoticed or appreciated. This
is just obvious commonsense, right?
A couple years ago I responded to a poster on
sci.math who asked for help to prove
a theorem in group theory. I thought about it
for a few second and was able to link together
three diverse theorems into a complete proof --
something I surely could never have done so
efficiently, if at all, if I had not just finished
memorizing the statements of all the main
theorems of group theory at that level. It's
so obvious why people can't do proofs if they
haven't bothered to memorize the statements
of the main theorems of their subject. This
ain't rocket science here -- just commonsense.
Chance favors the prepared mind. Memorize
all the main theorems of every math subject
you take!
> I think your purpose on this thread is to clobber everybody with your theory and
> refuse to make even small allowances..
Nobody has to read my post in this thread.
They're here of their own volition, so get off it.
> . well, in that sort of discussion, there's
> a great thing: you ALWAYS win when people realize that trying to discuss the issue
> is pointless. So take your victories as they come.
>
> B
I'm just tired of watching people give themselves points
for purely rhetorical arguments. Let's get down to real
argumentation. Just provide counterexamples to my epigram.
No wonder the teacher can't teach and the student
can't learn math concepts. Neither one seems to have
the slightest idea of what "understanding a concept in
math" means. OK, maybe I'm wrong. So what's your
correct explanation of how to best "understand a
concept in math"? This question is ultimately the
point of this thread. What do you have contribute
to this important question in a substantial way?
Patrick
W. Edwin Clark
Herman Rubin wrote:
>
> What is the definition of a group? This is not just
> a rhetorical question; there are several quite different
> ones. Groups form an equational class, as they satisfy
> the conditions of Birkhoff's theorem characterizing them;
Well, one must define a group before one can prove that
groups form a variety (= equational class). Birkhoff's
Theorem characterizes varieties -- not the variety of groups.
And you must define a universal algebra and lots of
other things to even state Birkhoff's Theorem. Certainly
Birkhoff's Theorem is not useful in developing group
theory.
Here are four ways to define a group:
A group is a set together with a binary operation (x,y)->xy,
a unary operation x->x', and a nullary operation {}->e
satisfying any one of the following sets of
identities:
1. x(yz) = (xy)z, ex = xe = x, x'x = xx' = e (Common definition)
2. x(yz) = (xy)z, ex = x, x'x = e (Another common definition)
3. x(yz) = (xy)z, xx'x = x (and x->x' is a bijection)
4. (w((x'w)'z))((yz)'y) = x
Only a masochist would use any other than 1 in
an elementary course. Maybe 2 might be stated as an
exercise. [BTW 4 is the shortest single identity in these
operations capable of defining groups--but useless
for human understanding.]
> I have seen many textbooks defining a group, not one of
> them using equations, although it can be done.
I don't recall seeing a definition of a group that doesn't
use equations! I would be very interested in seeing a definiton
of a group that doesn't use equations.
> Each
> "definition" uses enough properties to characterize them,
> and should be called characterizations.
>
> Definitions can be looked up when needed.
There are a large number of definitons that one must
know to be able to do abstract algebra. Suppose a proof
requires 20 definitions. Someone who has to look up
all these has a low probability of being able to find
the proof.
> Do they have to memorize them? A student given the
> problem can look up what is a cyclic group. And since
> cyclic groups are abelian, what if they used Tarski's
> nice characterization of an abelian group as a set
> with an operation "-" such that c=a-(b-(c-(a-b))) ?
It would be fascinating to see a proof that a group isomorphic
to a cyclic group is also cyclic using this characterization
of abelian groups. Such characterizations are fun
for logicians to play with but are useless for understanding
what an abelian group is.
I think it is very important to distinguish between the
definition of something and a characterization of it.
In fact I think that the confusion of the two is serious
roadblock for students.
For example, there is the definiton of semi-simple Artinian
ring and there is the "characterization" of such as the product
of a finite number of full matrix rings over division rings.
I don't think one would want to use the latter as a
substitute for the former.
To me a characteriztion of something is a theorem. Students
need to know the difference between a definition and a
theorem. Otherwise mathematics can become very confusing.
Part of the art of doing mathematics is the designing of
useful definitions.
This is one of the points made by Sachs and Wu in their
book General Relativity for Mathematicians. They note that
the main difficulty in writing the book was coming up
with the right definitions. They point out that the
way mathematicians and physicists use definitions is
quite different. What physicists call theorems in some
cases mathematicians call definitions. --but I guess that's
another topic...
--Edwin Clark
Daniel Grubb
>Of course, technically you would have to "memorize" this definition.
>But the concept of a basis is *so intuitive* that you scarcely have to
>work at it. Even when you are dealing with completely abstract
>concepts that do not connect at any point to reality (see the ongoing
>discussion about a "subaddition function" under the title of The
>Wierdest Math Problem Ever on sci.math), there is nothing that is
>gonna be definied UNLESS it makes sense to do so. So, if you
>understand the reasoning behind a definition, you don't need to
>memorize the text of the definition.
I very much disagree. Having the basic intuition is important, but it
is also important to memorize or at least be very familiar with the
actual definition. Sometimes the intuition just isn't enough. The
intuition of a curve is one we have, but if you then start talking about
space filling curves, much of that intuition goes out the window,
at least at first. If you go further and start talking about simple
closed curves with non-zero area, you had better know the definitions
of all the concepts involved and be willing to change your intuition to
fit those definitions.
>But do the students, who now know what a basis is, realize the
>consequences of what it is to be a basis -- that a space with an
>n-element basis has no basis with k elements unless k = n? If they do
>not realize such concepts themselves due to a lack of feeling for the
>concept itself, concepts such as the dimension of a space will come as
>shock to them.
Considering that this is something that you should *prove* from
the definitions, I don't think that they need have that intuition
*at first*. However, it should get integrated into their intuition
after the result is shown. I would think that the initial intuition is
'a basis is something like the standard basis in R^n'. We then define
the concept and see how much that definition corresponds to our initial
intuition. If it does, all is fine and good. If it doesn't we may still
have defined an important concept.
I guess what I am trying to say is that the definitions are originally
given to fit an intuition, but that as results are proved that intuition
changes to fit the definitions. If need be, we will define *other* concepts
for places where the intuitions didn't fit. It is an interactive
process, but knowing the specific definitions is crucial for
developing correct intuitions. This means that those specific
definitions must be memorized at some point.
---Dan Grubb
X
>>>must practice his pitch, so too the intellectual
>>>must fill his or her mind with memorized
>>>knowledge -- in long-term memory, of course.
Umm... I play piano in a very special manner, improvising my own very
expressive and filmic creations, and I attribute this to not being
formally educated in the "correct" way to play piano, which murders the
creative process. I've never done scales. The best potential physicist
would be advised to avoid the formal sciences completely, save for
possibly the easiest courses in grammerschool and highschool, and learn
first how to think and reason about anything, the cause and effect of
anything, which they could apply in any potential direction and not just
in one branch of the sciences.
-X
Patrick Reany
nick wrote:
1) I have stated my epigram with conviction, that's true.
But my opposition has also stated its condemnation
of my epigram with conviction. So, why is the onus to
give explanations ONLY on my shoulders? Why
do you make such prejudicial complaints. Ask them
to explain what the hell they mean.
2) I have given some examples in a recent reply to
to Ben Golub. Maybe they will help to clarify
some of the issues.
As for your complaints above, I can address
them better if you rephrase them. They are
unclear to me. A lot of territory has been covered
since my first post, so some specificity would help.
Are we finally getting somewhere? Do you accept
that my detractors have behaved themselves properly
so far? Would you hold up this thread as an example
of how mathematicians should engage in rational debate
to debunk what they believe is a false doctrine? Mind
you I knew from the start that I'd get this rhetoric
crap, but it's a damn waste of everybody's time. It's
long past time to get to specifics.
The stakes are high: How should teachers teach
math to their students? What learning advise should
they give their students? How should students try
to understand deep math concepts, with or without
advice from their teachers? You know, I have only
a BS in math so I am no expert on math. But I
do know one thing: No math teacher ever gave any
systematic advice to any class I was ever in on
how to understand math concepts or succeed in a
math class. This is abominable to me. I learned
the hard way, but other students could be told how
to do this.
Patrick
Herman Rubin
W. Edwin Clark <ecl...@math.usf.edu> wrote:
>Herman Rubin wrote:
>> What is the definition of a group? This is not just
>> a rhetorical question; there are several quite different
>> ones. Groups form an equational class, as they satisfy
>> the conditions of Birkhoff's theorem characterizing them;
>Well, one must define a group before one can prove that
>groups form a variety (= equational class). Birkhoff's
>Theorem characterizes varieties -- not the variety of groups.
>And you must define a universal algebra and lots of
>other things to even state Birkhoff's Theorem. Certainly
>Birkhoff's Theorem is not useful in developing group
>theory.
>Here are four ways to define a group:
There cannot be four ways to DEFINE anything; there can
only be one. These are ways to characterize a group.
>A group is a set together with a binary operation (x,y)->xy,
>a unary operation x->x', and a nullary operation {}->e
>satisfying any one of the following sets of
>identities:
>1. x(yz) = (xy)z, ex = xe = x, x'x = xx' = e (Common definition)
A more common version has only the binary operation,
and the existence of an element e and an element x'.
The uniqueness is proved.
>2. x(yz) = (xy)z, ex = x, x'x = e (Another common definition)
>3. x(yz) = (xy)z, xx'x = x (and x->x' is a bijection)
>4. (w((x'w)'z))((yz)'y) = x
>Only a masochist would use any other than 1 in
>an elementary course. Maybe 2 might be stated as an
>exercise.
Many textbooks use essentially 2. However, the book I
learned it from, by A. A. Albert, used the associative law
and the existence of (unique) solutions of ab=c for a given
b and c, and for b given a and c.
I do not consider this at all masochistic, and it has the
added feature that one can discuss quasigroups by just
dropping the associative law.
[BTW 4 is the shortest single identity in these
>operations capable of defining groups--but useless
>for human understanding.]
One could also start with the operation corresponding
to xy' in your notation, and do everything with that.
>> I have seen many textbooks defining a group, not one of
>> them using equations, although it can be done.
>I don't recall seeing a definition of a group that doesn't
>use equations! I would be very interested in seeing a definiton
>of a group that doesn't use equations.
I do not recall seeing one assuming the unary and the
nullary operations above.
>> Each
>> "definition" uses enough properties to characterize them,
>> and should be called characterizations.
>> Definitions can be looked up when needed.
>There are a large number of definitons that one must
>know to be able to do abstract algebra. Suppose a proof
>requires 20 definitions. Someone who has to look up
>all these has a low probability of being able to find
>the proof.
>> Do they have to memorize them? A student given the
>> problem can look up what is a cyclic group. And since
>> cyclic groups are abelian, what if they used Tarski's
>> nice characterization of an abelian group as a set
>> with an operation "-" such that c=a-(b-(c-(a-b))) ?
>It would be fascinating to see a proof that a group isomorphic
>to a cyclic group is also cyclic using this characterization
>of abelian groups. Such characterizations are fun
>for logicians to play with but are useless for understanding
>what an abelian group is.
>I think it is very important to distinguish between the
>definition of something and a characterization of it.
>In fact I think that the confusion of the two is serious
>roadblock for students.
For there to be a "definition", it would have to be universally
agreed upon.
>For example, there is the definiton of semi-simple Artinian
>ring and there is the "characterization" of such as the product
>of a finite number of full matrix rings over division rings.
>I don't think one would want to use the latter as a
>substitute for the former.
>To me a characteriztion of something is a theorem. Students
>need to know the difference between a definition and a
>theorem. Otherwise mathematics can become very confusing.
>Part of the art of doing mathematics is the designing of
>useful definitions.
>This is one of the points made by Sachs and Wu in their
>book General Relativity for Mathematicians. They note that
>the main difficulty in writing the book was coming up
>with the right definitions. They point out that the
>way mathematicians and physicists use definitions is
>quite different. What physicists call theorems in some
>cases mathematicians call definitions. --but I guess that's
>another topic...
>--Edwin Clark
Patrick Reany
Ben Golub wrote:
> > Exactly. As above, it's trivially easy to show that memorization comes
> > before understanding. But that's not very useful in figuring out how to
> > teach a class, and it's probably not what Patrick had in mind. The
> > question seemed a little vague to me.
>
> I agree. "You're uttering English words, so you've been memorizing" is absurd,
It's NOT absurd. It's the truth. It's the foundation
of getting people to appreciate that problem solving
is based on having lots of memorized information
in one's mind for instant, accurate retrieval. I don't
know how the mind does this, but it does. I'm
dealing mostly with attitude problems here, and
a form of political correctness. (Don't ask me
to explain this, please!)
>
> though true on Patrick's level. He's been accusing everybody of being unable to
> have a reasoned debate, but he himself is the one who has stuck to his idea like
> some sort of revelation and refused a discussion of its practical merits.
I have not yet been asked for specifics! I
wanted to get to specifics after the first set
of replies. Granted, from my experience I had the
last time I did this I knew that was pretty unlikely
to happen. What I got instead of requests for
specifics was the expected loony denunciations
of the obvious. I got the "memorization is harmful
to understanding" quip. This was a new twist to the
debunking I received last time. Nobody bothered
to just asked me what I meant; they just mindlessly
condemned. So they got the response from me they
deserved. I expect a whole lot better of
mathematicians than what I got. So should you!
I'll accept that "memorization is harmful to
understanding" in the same way that "water is
harmful to health." Is it the case that you take
a mathematician out of the realm of pure mathematics
and he or she looses their ability to think logically?
Hasn't even one of them studied what a logical
fallacy is?
But your charge that I gave no more amplification to
the meaning of what I promote here is untrue. I have
defined both "memorization" and "understanding,"
something that no one else in this thread has bothered
to do. I have presented models of the process of
understanding. I have given many anecdotal cases that
support my thesis. So I HAVE indeed given more specifics!
>
>
> Ben
>
> -----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
> http://www.newsfeeds.com - The #1 Newsgroup Service in the World!
> -----== Over 80,000 Newsgroups - 16 Different Servers! =-----
Perhaps some help in getting to specifics
will be found for you in a couple recent
replies I gave to Nick and to you in another
place.
Patrick
Ben Golub
> defined both "memorization" and "understanding,"
> something that no one else in this thread has bothered
> to do.
See my post below (10:25 AM eastern time). I define "memorization" there, though I
suspect not to your liking.
Randy Poe
>
> >>>The pianist must practice scales, the pitcher
> >>>must practice his pitch, so too the intellectual
> >>>must fill his or her mind with memorized
> >>>knowledge -- in long-term memory, of course.
>
> Umm... I play piano in a very special manner, improvising my own very
> expressive and filmic creations, and I attribute this to not being
> formally educated in the "correct" way to play piano, which murders the
> creative process. I've never done scales.
And if you want to create a work with scales in it, what
then?
Or arpeggios?
Or a rapid series of descending octaves?
Or counter rhythms?
Would you still say that the technical ability to play what
you hear in your mind "murders" the creative process, in
the way that being able to type murders the process of
writing a novel?
- Randy
Herman Rubin
>nick wrote:
>> Patrick Reany <re...@asu.edu> wrote in message news:<3D2B47B8...@asu.edu>...
>> > "Jesse F. Hughes" wrote:
>> > > Patrick Reany <re...@asu.edu> writes:
....................
>The stakes are high: How should teachers teach
>math to their students?
Teach concepts. Do not worry at all about speed in
getting results. If a student knows how to DERIVE
sums and products from first principle, no matter
how slowly, is highly essential; memorizing how to
calculate them is useful, but if the concepts are
not understood, it can prevent understanding.
What learning advise should
>they give their students? How should students try
>to understand deep math concepts, with or without
>advice from their teachers?
In this case, if the concepts are not explicitly
presented, they have essentially no chance to learn
them. Modern mathematics started in the 16th century,
but it was not until the 19th that there was other
than a hopeful intuitive understanding. It is not
that the explicit concepts are hard to understand;
just that they are hard to extract.
You know, I have only
>a BS in math so I am no expert on math.
The conceptual understanding of mathematics I am
advocating can be taught in elementary school;
it has been. However, it is by no means clear
that someone with a BS in math is in a better
position to understand the concepts than a small
child. The classical "Euclid" course was an
ancient attempt to approximate this, and was
moderately successful.
But I
>do know one thing: No math teacher ever gave any
>systematic advice to any class I was ever in on
>how to understand math concepts or succeed in a
>math class. This is abominable to me. I learned
>the hard way, but other students could be told how
>to do this.
I do not think anyone knows how to teach concepts.
It has been shown that drill after a concept has
been learned is of little, if any, value, and that
learning one concept makes it harder to learn a
related one. Reasonable drill before a concept has
been learned seems to benefit learning for some.
Also, going from general to special seems to be less
difficult than the other way. Generalization from
a set of examples which have more in common than the
general concept is a problem to be avoided.
Jesse F. Hughes
[More passionate silliness]
Really, I think it's bad form to recycle old JSH subject lines unless
you have squared the circle or something similarly impressive.
--
Jesse Hughes
"A factor is simply something that multiplies against another factor
to produce a 'product'." -- James Harris offers a definition.