What's wrong with education and what is being done to change it?
Herman Rubin
In article <319e9fc9.16890041@news>, Starfollower <ck...@cyberg8t.com> wrote:
>mac...@alaska.net (Kim Mackey) wrote:
>>In article <4ng0f3$1p...@b.stat.purdue.edu>, hru...@b.stat.purdue.edu says...
>>>This topic is here again. It is useful to be able to do arithmetic,
>>>but as for understanding, it is not clear that it helps.
>>Not clear to whom? As a high school geometry teacher I see very few high
>>quality students who lack arithmetic ability. But I see lots of low quality
>>students who _do_ lack arithmetic ability. Lack of skills gets students hung
>>up on parts of problems that their peers rip right through without a pause.
>>Their learning is affected, their self-esteem is affected, and ultimately,
>>what they can accomplish in life is affected.
>Just another math teacher wanting to add her $0.02:
>I must agree with what Kim writes above. I have had the same experience many
>times with my own students.
I sense a lot of confusion here. A good part of it is due to the fact
that, while mathematical concepts can easily be taught even before
arithmetic, this is not done, and in fact, the important ones are
pretty much ignored. Also, when algebraic concepts are introduced,
the early emphasis is almost entirely on solution, and the grading
is mainly on solution, so that the one without arithmetic ability will
come out quite poorly.
It is also the case that getting numerical answers quickly for someone
without arithmetic ability is impossible, and that having a good chance
to get the answers at all for such will require calculational assistance.
So the present system is quite effective at keeping those without fairly
good arithmetic ability out of real mathematics, which of course reinforces
the claim.
To test this in high school geometry, it would be necessary to go to the
approach of Euclid and others for over 2000 years, and teach the classical
geometry course which had almost no arithmetic in it. Or even better, a
good course in logic, as can be given in elementary school, and a development
of the foundations of mathematics, which does not involve knowledge of
arithmetic. This was the main function of the old geometry course, whether
or not it was realized.
I can do arithmetic quickly and accurately, and I deliberately developed
the scope of this beyond what was taught. But as soon as I learned
mathematics, I realized that this was not mathematics, but merely a
useful tool.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
Herman Rubin
In article <4nm536$1s...@thor.cmp.ilstu.edu>,
Thomas W. Cowdery <twc...@rs6000.cmp.ilstu.edu> wrote:
>>This is a place to be MORE careful. This is probability; the decay
>>will NOT be exponential, even when there are large numbers, except
>>with small probability. One needs the appropriate concepts, and in
>>mathematics, the concepts are completely precise.
>I won't argue that fact that the M&M experiment isn't an exact case of
>exponential decay, but it does model the concept of a half-life [until
>you get to that last M&M and have to fight over it ;-) ].
The effect takes place much earlier. With 8, assuming that they are
equally likely to end M up as down, the probability of exactly 4 coming
up head is exactly .2734375. For 2, it is .5. For 100, it is approximately
.04.
What can you learn from the M&M experiment? Not mathematics, but what
can be called "statistical physics". Mathematical ideas may have been
obtained initially by people who were brilliant and lucky who observed
properties of the "real world", but they are not properties of the real
world. We use mathematics to model the real world, not the converse.
>>>She will never comprehend the M&M exercise because she can't see the
>>>relationships due to her arithmetic deficiency.
>>It is nnot a problem of arithmetic. If she understood the operations,
>>and could do them on a calculator, it would make no difference.
>But she *doesn't* understand the operations because all she can do is
>punch them in on a calculator. She can't *see* fifteen being one
>quarter of 60. I suspect that she knows that 4 quarters equals one
>dollar, but only by rote. She doesn't see them as a fraction of a
>whole dollar.
Has anyone ever attempted to teach her the properties of the integers,
rational numbers, and real numbers? I believe that it could have been
done, and if she had learned those, she would still be no better at
arithmetic, but she would understand what it means. As for understanding,
it is no more important to see fifteen being one quarted of 60 than to see
that 43 is one 79th of 3397.
Kids with better arithmetic skills also have better
>number sense.
The impetus for developing the new math came from the observation that
children with excellent arithmetic skills often did not have the
slightest understanding of anything except that. Also, they could
no longer learn the concepts without great difficulty, because their
success at arithmetic had convinced them that they understood.
They don't have to resort to punching buttons or long
>division to see relationships. If we talk about 7 being about 25% of
>30, they can understand that because they know 7 is exactly a fourth
>of 28. But the girl that I was talking about wouldn't be able to
>think that way because she couldn't even figure how many times 7 goes
>into 28.
As I said, this is a convenience.
And she certainly couldn't handle anything more abstract.
Has anybody ever considered trying to teach the concepts to her without
tying them up with arithmetic manipulations? From your comments, it
seems you would have to relearn them yourself to attempt this.
Unlearning is possibly the hardest academic activity, so the need for it
should be avoided. Teaching arithmetic first, and building mathematics
on it when it is not needed, is a highly effective way of keeping those
without arithmetic skills from understanding mathematical concepts.
>>The teachers who taught her fractions and decimals also only knew the
>>manipulations. Even if they were not good, after years of teaching,
>>the answers would become overly routine. So the concepts would be
>>again swept under the computational rug.
>Perhaps. But I think the bigger problem is that she has used a
>calculator to avoid having to think about the arithmetic at all.
>Whatever is in the display must be right. She doesn't need to think
>about how it got there.
>As you have rightly pointed out in the past, we learn better when we
>are young. I agree with that, but I don't believe that young children
>are capable of learning the same types of things that older children
>and adults are capable of learning. I believe that there is a
>physical maturity factor involved that you apparently do not believe
>in. I also believe that some things must be learned young if a
>significant level of mastery can ever occur. I believe that
>arithmetic is one of those things that must be learned young. The
>girl in my example has missed much of her window of opportunity to
>learn arithmetic well. That deficiencey will also cripple her ability
>to learn higher concepts.
You have provided no evidence of that. Persistence in making heavy use
of arithmetic in the teaching of mathematics, even if much of it is
subconscious, will prevent that. It would not be hard to separate
out the arithmetic involved in calculation from an algebra course.
I do not know of any approach to teaching logic, including the entire
restricted predicate calculus, which involves arithmetic at all, except
for numbering statements and knowing the order in which they occur.
Euclidean geometry does not involve arithmetic at all, and we should
go back to teaching THAT in high school instead of what is now taught
in most places.
Thomas W. Cowdery
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
<snip>
>Has anyone ever attempted to teach her the properties of the integers,
>rational numbers, and real numbers? I believe that it could have been
>done, and if she had learned those, she would still be no better at
>arithmetic, but she would understand what it means.
Yes, I spent a fair amount of time on this. But without much in the
way of preexisting arithmetic skills, those properties are nonsensical
to her and kids like her. *You* may be able to comprehend the
abstract before the concrete, but *most* people cannot.
> As for understanding,
>it is no more important to see fifteen being one quarted of 60 than to see
>that 43 is one 79th of 3397.
Both are important as they represent a relationship, albeit a simple
one. The former should be fairly easy to see and to visualize.
Anyone who can tell time using an analog clock ought to be able to
understand it (with *ought to* being the operative phrase here).
Someone who can comprehend the first one, can better understand the
second even if they cannot visualize one 79th of 3397.
> Kids with better arithmetic skills also have better
>>number sense.
>The impetus for developing the new math came from the observation that
>children with excellent arithmetic skills often did not have the
>slightest understanding of anything except that. Also, they could
>no longer learn the concepts without great difficulty, because their
>success at arithmetic had convinced them that they understood.
Well 'new math' was an abject failure. I know that you blame the
teachers, but I tend to think it was the approach. What I know of it,
which comes more from reading about it and from your descriptions of
it in this forum, than from personal experience, strikes me as poor
pedagogy.
> They don't have to resort to punching buttons or long
>>division to see relationships. If we talk about 7 being about 25% of
>>30, they can understand that because they know 7 is exactly a fourth
>>of 28. But the girl that I was talking about wouldn't be able to
>>think that way because she couldn't even figure how many times 7 goes
>>into 28.
>As I said, this is a convenience.
No, it is a level of understanding that she has not reached. Granted,
by itself it isn't a major concept, but every journey starts with a
single step. She, and others like her, keep tripping over this one.
> And she certainly couldn't handle anything more abstract.
>Has anybody ever considered trying to teach the concepts to her without
>tying them up with arithmetic manipulations? From your comments, it
>seems you would have to relearn them yourself to attempt this.
Perhaps I would. I certainly cannot see how that would be done
effectively. The few times that I have seen attempts to do that with
the types of things that I would be teaching, the explanations seemed
to muddy the water rather than clarify it. I would enjoy seeing what
you would consider good approaches to teaching some Algebra 1 topics
that wouldn't have any tie-in to arithmetic. If you are game, I'll
pick a couple of topics from the Algebra 1 curriculum and you can
suggest a method of instruction.
<snip>
>>As you have rightly pointed out in the past, we learn better when we
>>are young. I agree with that, but I don't believe that young children
>>are capable of learning the same types of things that older children
>>and adults are capable of learning. I believe that there is a
>>physical maturity factor involved that you apparently do not believe
>>in. I also believe that some things must be learned young if a
>>significant level of mastery can ever occur. I believe that
>>arithmetic is one of those things that must be learned young. The
>>girl in my example has missed much of her window of opportunity to
>>learn arithmetic well. That deficiencey will also cripple her ability
>>to learn higher concepts.
>You have provided no evidence of that.
Some of it is based solely on my own observations as a teacher, and I
concede that I have no scientific proof. Some, I believe is related
to Piaget's theories, which I know that you do not put as much stock
in as I do. But I think that his general ideas were correct, even if
he didn't have all of the details down.
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Thomas W. Cowdery When I die, I want to go peacefully in
twc...@rs6000.cmp.ilstu.edu my sleep like my grandfather did.
tcow...@dave-world.net Not screaming like the passengers in his car.
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Thomas W. Cowdery
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>I sense a lot of confusion here. A good part of it is due to the fact
>that, while mathematical concepts can easily be taught even before
>arithmetic, this is not done, and in fact, the important ones are
>pretty much ignored. Also, when algebraic concepts are introduced,
>the early emphasis is almost entirely on solution, and the grading
>is mainly on solution, so that the one without arithmetic ability will
>come out quite poorly.
>It is also the case that getting numerical answers quickly for someone
>without arithmetic ability is impossible, and that having a good chance
>to get the answers at all for such will require calculational assistance.
>So the present system is quite effective at keeping those without fairly
>good arithmetic ability out of real mathematics, which of course reinforces
>the claim.
What I have seen over the years is that the kids who have the poorest
arithmetic skills, also have the least understanding of the concepts
behind what they are being taught. For example, in a recent chapter
on solving systems of equations, one question on the test asked the
students to draw graphs of the lines to represent the system and to
identify the solution from the graph. The kids that I would identify
as being the most calculator-dependent were also the kids who were
most likely to draw one line and write down an 'answer'. They
obviously didn't understand that the solution was the intersection of
the two lines, since they didn't bother to draw the second line. The
kids who I would identify as being stronger arithmetically, may not
have always gotten the right answer, but they at least had two lines
on their graph! I grant you that there are other factors at work
here. Some of the former group still haven't figured out how to draw
a line by any method - slope-intercept, x & y intercepts, plug & chug
(no they don't have graphing calculators, I'll use those *after* they
show me that they can draw a graph on their own), but they got one
line on the paper. Even if it was wrong, the fact that they didn't
have a second shows a lack of understanding.
>To test this in high school geometry, it would be necessary to go to the
>approach of Euclid and others for over 2000 years, and teach the classical
>geometry course which had almost no arithmetic in it. Or even better, a
>good course in logic, as can be given in elementary school, and a development
>of the foundations of mathematics, which does not involve knowledge of
>arithmetic. This was the main function of the old geometry course, whether
>or not it was realized.
When I taught geometry a few years ago, the kids asked me if they
could use a calculator. I said 'sure'. About halfway through the
first semester, one of the kids piped up "I know why you said it was
OK to use calculators, there isn't anything to use them for!". He was
correct. At that point it was all logic, postulates, theorems and
proofs. They hadn't seen a numeral used for anything except numbering
the questions. When we got to second semester, however, and started
doing more problems that involved arithmetic, such as area, volume and
surface area problems, then any arithmetic deficiencies would have had
an opportunity to show up. I saw no difference in performance. The
kids who did poorly first semester, in general, did poorly second
semester and vice versa. I noticed that the kids who did the best,
used their calculators the least. One of my students was a Japanese
boy. He didn't even bring one to class, and aced every test even
though he had to translate the directions to Japanese using his
Japanese-English dictionary. I realize that this still isn't
scientific proof, but it is part of the reason that I believe as I do.
David K. Davis
Herman Rubin (hru...@b.stat.purdue.edu) wrote:
: In article <319e9fc9.16890041@news>, Starfollower <ck...@cyberg8t.com> wrote:
: >mac...@alaska.net (Kim Mackey) wrote:
: >>In article <4ng0f3$1p...@b.stat.purdue.edu>, hru...@b.stat.purdue.edu says...
: >>>This topic is here again. It is useful to be able to do arithmetic,
: >>>but as for understanding, it is not clear that it helps.
: >>Not clear to whom? As a high school geometry teacher I see very few high
: >>quality students who lack arithmetic ability. But I see lots of low quality
: >>students who _do_ lack arithmetic ability. Lack of skills gets students hung
: >>up on parts of problems that their peers rip right through without a pause.
: >>Their learning is affected, their self-esteem is affected, and ultimately,
: >>what they can accomplish in life is affected.
: >Just another math teacher wanting to add her $0.02:
:
: >I must agree with what Kim writes above. I have had the same experience many
: >times with my own students.
: I sense a lot of confusion here. A good part of it is due to the fact
: that, while mathematical concepts can easily be taught even before
: arithmetic, this is not done, and in fact, the important ones are
: pretty much ignored. Also, when algebraic concepts are introduced,
: the early emphasis is almost entirely on solution, and the grading
: is mainly on solution, so that the one without arithmetic ability will
: come out quite poorly.
: It is also the case that getting numerical answers quickly for someone
: without arithmetic ability is impossible, and that having a good chance
: to get the answers at all for such will require calculational assistance.
: So the present system is quite effective at keeping those without fairly
: good arithmetic ability out of real mathematics, which of course reinforces
: the claim.
: To test this in high school geometry, it would be necessary to go to the
: approach of Euclid and others for over 2000 years, and teach the classical
: geometry course which had almost no arithmetic in it. Or even better, a
: good course in logic, as can be given in elementary school, and a development
: of the foundations of mathematics, which does not involve knowledge of
: arithmetic. This was the main function of the old geometry course, whether
: or not it was realized.
: I can do arithmetic quickly and accurately, and I deliberately developed
: the scope of this beyond what was taught. But as soon as I learned
: mathematics, I realized that this was not mathematics, but merely a
: useful tool.
I'm totally unable to understand this argument, and my experience entirely
supports those on the other side. Arithmetic is the first and most familiar
mathematical system students are exposed to. All kinds of other algebraic
structures are modelled after or abtracted from arithmetic - groups, vector
spaces, Banach and Hilbert spaces. All these abstractions are hopelessly
beyond someone who doesn't have a good feel for arithmetic, a feel that comes
only with the ability to perform operations. It's ridiculous to see kids
pulling out calculators to do .01 x 100.
You build all these other structures, all of which also involve calculation,
on modifying this or that feature of arithmetic. Logic also involves
calculation at some level. Is geometry important too? Yes! All these abstract
spaces are meaningless without a solid geometric intuition underneath.
In no way am I against understanding or teaching it. But technique is vital
and it starts with arithmetic. Sugar coat it if need be, but one way or another
arithmetic ability needs to be raised considerably above what it is now. I
don't think there's such a thing as a theoretical violinist - that is,
someone who "understands" the violin but can't play it. It's the same with
mathematics - if you don't "play" arithmetic I don't see how you're going
to play or understand the more advance instruments.
Neglect of technique is NOT the answer to rote learning. Very little should
be memorized in mathematics, I agree, but there has to be a whole lot of
doing, and the doing ought to be such as to make arithmetic second nature.
I strongly suspect that means denying the use of the calculator up to
some fairly late grade.
-Dave D.
Herman Rubin
In article <Droqv...@spcuna.spc.edu>,
David K. Davis <dav...@spcunb.spc.edu> wrote:
>Herman Rubin (hru...@b.stat.purdue.edu) wrote:
>: In article <319e9fc9.16890041@news>, Starfollower <ck...@cyberg8t.com> wrote:
>: >mac...@alaska.net (Kim Mackey) wrote:
>: >>In article <4ng0f3$1p...@b.stat.purdue.edu>, hru...@b.stat.purdue.edu says...
...................
Arithmetic is NOT the mathematical system; it has nothing to do with
mathematics, but is merely a mechanical means of deriving answers.
If you mean that integers are the first mathematical system people are
exposed to, you are correct. But whether the number we write as 13 is
so written, or is written as 1101 as a binary number, or 111 in base 3,
or xiii in Roman numerals, or jc in a version of the Greek number system
converted to Roman letters, or ''''''''''''' by just making tick marks,
the number meant is exactly the same. How a number is represented has
little to do with the concepts of number; the tick marks get much closer
to the essence than all of the others.
Teaching the ideas of arithmetic carefully and axiomatically can be
done at the first grade level with little problem; what it states is
that the tick mark approach is a correct way to proceed. Now a child
may even ask if there is a better way; this should be the motivation
to teach the mechanics. And all of this can, and should, be done
using algebraic notation, as a means of clear and precise communication.
As for the specific example you gave, Archimedes would not have been
able to see it as you wrote it, because decimals had not been invented.
And Archimedes would not have been able to multiply by 100 by adding
two 0's, nor would Diophantus even after he invented variables. The
current means of arithmetic manipulation are just that, and do not
give a feel for anything other than those manipulations. The important
parts do not involve the manipulations.
>You build all these other structures, all of which also involve calculation,
>on modifying this or that feature of arithmetic. Logic also involves
>calculation at some level. Is geometry important too? Yes! All these abstract
>spaces are meaningless without a solid geometric intuition underneath.
They modify properties of other systems, but not of the arithmetic which
is taught. When an algebraist talks about the arithmetic of an algebra,
the precise notation is not mentioned, as it does not matter. And one
can do quite well in geometry without "geometric intuition." Try the
7-element projective plane, or any other finite projective plane. These
have practical uses.
Logic no more involves calculation than writing text in a language does.
The notion of variable is purely linguistic; introducing it for numbers
first makes it harder to learn the general concept later.
>In no way am I against understanding or teaching it. But technique is vital
>and it starts with arithmetic. Sugar coat it if need be, but one way or another
>arithmetic ability needs to be raised considerably above what it is now. I
>don't think there's such a thing as a theoretical violinist - that is,
>someone who "understands" the violin but can't play it.
How many instruments do composers play? How many people have not become
composers because they played no instruments? And even more so, how many
musicians could make a violin or a piano?
It's the same with
>mathematics - if you don't "play" arithmetic I don't see how you're going
>to play or understand the more advance instruments.
So which arithmetic do you play? Do you use the Babylonian system, alternating
between base 10 and base 6? Do you use the Egyptian system, using repeated
numbers of characters for a given power of 10, or the Roman more efficient
version of that? Do you use the Greek system, or something like that, in
which nine characters are used for the digits in each place? Do you use base
2, where the addition and multiplication tables are trivial?
And you can do all these other branches, including numerical analysis, if
you can call an arithmetic "player" to assist you. It is only time that
is lost by this.
>Neglect of technique is NOT the answer to rote learning. Very little should
>be memorized in mathematics, I agree, but there has to be a whole lot of
>doing, and the doing ought to be such as to make arithmetic second nature.
>I strongly suspect that means denying the use of the calculator up to
>some fairly late grade.
The use of the calculator to do arithmetic is not the problem. The problem
is that we teach the mechanics of arithmetic without any undersanding of
what it means. This understanding CAN be taught with essentially no
mechanics. What is the point of finding that the answer is 382 if the
student does not have any idea of what that means?
David K. Davis
Herman Rubin (hru...@b.stat.purdue.edu) wrote:
<lots snipped>
: >I'm totally unable to understand this argument, and my experience entirely
: >supports those on the other side. Arithmetic is the first and most familiar
: >mathematical system students are exposed to. All kinds of other algebraic
: >structures are modelled after or abtracted from arithmetic - groups, vector
: >spaces, Banach and Hilbert spaces. All these abstractions are hopelessly
: >beyond someone who doesn't have a good feel for arithmetic, a feel that comes
: >only with the ability to perform operations. It's ridiculous to see kids
: >pulling out calculators to do .01 x 100.
: Arithmetic is NOT the mathematical system; it has nothing to do with
: mathematics, but is merely a mechanical means of deriving answers.
: If you mean that integers are the first mathematical system people are
: exposed to, you are correct. But whether the number we write as 13 is
: so written, or is written as 1101 as a binary number, or 111 in base 3,
: or xiii in Roman numerals, or jc in a version of the Greek number system
: converted to Roman letters, or ''''''''''''' by just making tick marks,
: the number meant is exactly the same. How a number is represented has
: little to do with the concepts of number; the tick marks get much closer
: to the essence than all of the others.
: Teaching the ideas of arithmetic carefully and axiomatically can be
: done at the first grade level with little problem; what it states is
: that the tick mark approach is a correct way to proceed. Now a child
: may even ask if there is a better way; this should be the motivation
: to teach the mechanics. And all of this can, and should, be done
: using algebraic notation, as a means of clear and precise communication.
This is ridiculous. Meaning is fine, sticks are fine, and so on. But in
first grade kids need to start on the addition tables, and later the times
tables, just as I did in the 40s - whatever else is thrown in. The numbers
(and basic operations on them) must be part of us, must become part of us,
and early. Abstraction is built on the concrete. Without that first
internalized system, there's nothing to build abstractions on.
: As for the specific example you gave, Archimedes would not have been
: able to see it as you wrote it, because decimals had not been invented.
: And Archimedes would not have been able to multiply by 100 by adding
: two 0's, nor would Diophantus even after he invented variables. The
: current means of arithmetic manipulation are just that, and do not
: give a feel for anything other than those manipulations. The important
: parts do not involve the manipulations.
There's a lot of stuff I learned early that only later I saw in a different
deeper light. Right now I'm trying to bone up on function spaces, Banach
and Hilbert spaces. Book after book goes - theorem, lemma, theorem, def,
lemma, def, thm, etc. What a struggle to learn what it's all about! It's
only when I started reading some stuff by physicists that I started getting
a clue. I didn't understand what the abstraction was from - from the
differential and integral equations and stuff that preoccupied earlier
mathematicians. And because I lack that background, it's all very much
up in the air, even when I "understand" it. On the other hand, some of the
math I sweated over 35 years ago, meaning manipulated every which way -
logic and set theory, is still very easy for me to picture and do.
: >You build all these other structures, all of which also involve calculation,
: >on modifying this or that feature of arithmetic. Logic also involves
: >calculation at some level. Is geometry important too? Yes! All these abstract
: >spaces are meaningless without a solid geometric intuition underneath.
: They modify properties of other systems, but not of the arithmetic which
: is taught. When an algebraist talks about the arithmetic of an algebra,
: the precise notation is not mentioned, as it does not matter. And one
: can do quite well in geometry without "geometric intuition." Try the
: 7-element projective plane, or any other finite projective plane. These
: have practical uses.
You're quibbling. Groups drop the requirement of commutativity, combinators
the requirement of associativity, etc. There may well be pieces of geometry
which don't require geometric intuition, but its ridiculous to start
talking about vector spaces and beyond without any geometrical intuition.
Its ridiculous to talk about linear maps and not do some multiplication of
vectors by matrices - yes, by hand even.
: Logic no more involves calculation than writing text in a language does.
: The notion of variable is purely linguistic; introducing it for numbers
: first makes it harder to learn the general concept later.
Logic can involve a long and intricate series of steps. Following those,
and creating those steps, involves not just insight, but a well exercised
insight - and technique! It's a facility developed thru exercise. Chess is
logic. Go up against someone - a 7 year old - who has technique developed
through practice. They'll wipe the floor with you (and me). Their insight
comes through hard work, lots of games, talent, and coaching - but never
from just "understanding the game". Math is the same in that regard.
You've got to get into shape for it, and there are crucial pieces that should
be there. We abstract from what's inside us. In so many cases now, there's
nothing there to abstract from.
: >In no way am I against understanding or teaching it. But technique is vital
: >and it starts with arithmetic. Sugar coat it if need be, but one way or another
: >arithmetic ability needs to be raised considerably above what it is now. I
: >don't think there's such a thing as a theoretical violinist - that is,
: >someone who "understands" the violin but can't play it.
: How many instruments do composers play? How many people have not become
: composers because they played no instruments? And even more so, how many
: musicians could make a violin or a piano?
: It's the same with
: >mathematics - if you don't "play" arithmetic I don't see how you're going
: >to play or understand the more advance instruments.
: So which arithmetic do you play? Do you use the Babylonian system, alternating
: between base 10 and base 6? Do you use the Egyptian system, using repeated
: numbers of characters for a given power of 10, or the Roman more efficient
: version of that? Do you use the Greek system, or something like that, in
: which nine characters are used for the digits in each place? Do you use base
: 2, where the addition and multiplication tables are trivial?
: And you can do all these other branches, including numerical analysis, if
: you can call an arithmetic "player" to assist you. It is only time that
: is lost by this.
It's not lost time. It's called building the foundation. Learn arithmetic
in the decimal system, internalize it. Learn how it works later. It'll
be available to you anytime you want to think about it. Not for a lot
of todays kids, though.
: >Neglect of technique is NOT the answer to rote learning. Very little should
: >be memorized in mathematics, I agree, but there has to be a whole lot of
: >doing, and the doing ought to be such as to make arithmetic second nature.
: >I strongly suspect that means denying the use of the calculator up to
: >some fairly late grade.
: The use of the calculator to do arithmetic is not the problem. The problem
: is that we teach the mechanics of arithmetic without any undersanding of
: what it means. This understanding CAN be taught with essentially no
: mechanics. What is the point of finding that the answer is 382 if the
: student does not have any idea of what that means?
I've just never seen an instance. What you advocate is a wild and dangerous
experiment, just like the new math crap back in the 70s. I KNOW that the way
I learned back in the 40s and 50s, despite its drawbacks, works. There was
an excessive reliance on rote. But it worked and didn't do irreparable
damage to me or others.
Later on, after having majored in set theory and logic in college, I saw
(from afar) the new math hit the schools, in the early 70s I believe. As
much as I had been fascinated by the hierarchy of infinities (cardinal
and ordinal), the introduction of premature and unprepared for abstraction
was the first assault against math teaching. It was result of a major mis-
conception among educators and maybe some mathematicians. Set theory
was an answer to a problem in mathematics. At a certain level of
abstraction, it provides a unifying approach and imagery for most
of mathematics. But it didn't solve a problem for 3rd graders or
3rd grade teachers. Rather it was a disastrous detour from learning
arithmetic and the rudiments of geometry.
The reliance on computers and calculators is compounding this disaster.
It's the same thing but from the opposite end. New math says study the ideas
behind it all, that'll make your calculations more fun and less onerous.
The calculators and computers say - now you don't need to do the drudgery
at all, just study ideas, the whys, (but don't call it new math).
But I don't believe any of it. Abstraction is dead without the concrete.
Just do it - I'll show you why later. Why won't you show me now? Because
you won't understand it now, or at least won't see the importance. I'm
not a Piagetan necessarily, but there are stages. I see it with my kids,
myself, everywhere. Yours seems to be a perhaps extreme version of a
very prevalent philosophy these days. But I really think the results
are proving and will prove to be disastrous.
-Dave D.
Tony2back
In article <DrqoJ...@spcuna.spc.edu>, dav...@spcunb.spc.edu (David K.
Davis) writes:
>Herman Rubin (hru...@b.stat.purdue.edu) wrote:
>
>: Arithmetic is NOT the mathematical system; it has nothing to do with
>: mathematics, but is merely a mechanical means of deriving answers.
>
>: Teaching the ideas of arithmetic carefully and axiomatically can be
>: done at the first grade level with little problem; what it states is
>: that the tick mark approach is a correct way to proceed. Now a child
>: may even ask if there is a better way; this should be the motivation
>: to teach the mechanics. And all of this can, and should, be done
>: using algebraic notation, as a means of clear and precise
communication.
>
>This is ridiculous. Meaning is fine, sticks are fine, and so on. But in
>first grade kids need to start on the addition tables, and later the
times
>tables, just as I did in the 40s - whatever else is thrown in. The
numbers
>(and basic operations on them) must be part of us, must become part of
us,
>and early. Abstraction is built on the concrete. Without that first
>internalized system, there's nothing to build abstractions on.
>
>
I fear we are entering the realm of the sterile religious disputations of
the sixteenth and seventeenth centuries. Both points of view are correct
in context. The mathematicians of the next generation will need Herman's
preferred type of education, the average citizen, on the other hand, with
little interest in mathematics apart from adding up a shopping list, would
not make the intellectual investment required to understand the rules that
he applies. I think it is safe to assume that most pupils fall into the
second category, and start their education on that basis. What the
education system must however do, (and at the moment it is NOT doing) is
to recognize early those in the first category and be able to give them
the much deeper training that they will require. We are able to do this
with sporting prodigies, but not it seems with intellectual ones.
Anthony Hugh Back
Herman Rubin
In article <DrqoJ...@spcuna.spc.edu>,
................
>: If you mean that integers are the first mathematical system people are
>: exposed to, you are correct. But whether the number we write as 13 is
>: so written, or is written as 1101 as a binary number, or 111 in base 3,
>: or xiii in Roman numerals, or jc in a version of the Greek number system
>: converted to Roman letters, or ''''''''''''' by just making tick marks,
>: the number meant is exactly the same. How a number is represented has
>: little to do with the concepts of number; the tick marks get much closer
>: to the essence than all of the others.
>: Teaching the ideas of arithmetic carefully and axiomatically can be
>: done at the first grade level with little problem; what it states is
>: that the tick mark approach is a correct way to proceed. Now a child
>: may even ask if there is a better way; this should be the motivation
>: to teach the mechanics. And all of this can, and should, be done
>: using algebraic notation, as a means of clear and precise communication.
>This is ridiculous. Meaning is fine, sticks are fine, and so on. But in
>first grade kids need to start on the addition tables, and later the times
>tables, just as I did in the 40s - whatever else is thrown in. The numbers
>(and basic operations on them) must be part of us, must become part of us,
>and early. Abstraction is built on the concrete. Without that first
>internalized system, there's nothing to build abstractions on.
Sticks are a way to get the ideas across, but not the best. The idea of
number is important, but not of a particular representation. With the
way the Egyptians wrote numbers, would they have addition tables to learn?
They used symbols for each power of ten, and repeated as needed. What about
the Roman numerals? One might get some practice at recognizing forms, but
the ideas are the same. The abacus embodies these. And the Babylonians
would have you memorize their procedures for base 60, really alternating
base 10 and base 6. But none of this has anything to do with understanding
numbers. The Egyptians and Babylonians and Greeks did arithmetic.
..................
>There's a lot of stuff I learned early that only later I saw in a different
>deeper light. Right now I'm trying to bone up on function spaces, Banach
>and Hilbert spaces. Book after book goes - theorem, lemma, theorem, def,
>lemma, def, thm, etc. What a struggle to learn what it's all about!
I am quit familiar with the almost total lack of presenting concepts.
Part of the problem is that the concepts are not that well known. However,
Newton presented his ideas to his contemporaries who had not progressed
beyond "analytic geometry" by using limits. Even if the concept was not
formulated (this took almost two more centuries) they understood it.
The students who supposedly learn about it in calculus do not understand
limits, or anything else except to compute. They do not understand what
a definition or a theorem or a proof is. BTW, I would use characterization
rather than definition; there can only be one definition, but there can
be many characterizations of the same type of object.
One can teach concepts. But to do this often requires abandoning where
they came from, historically. For the concepts of measure and integration,
everything is discrete or limits of discrete. But it was not until about
a century ago that it started to be recognized that length and counting
were both examples of measures, and that measure and integration is
discrete or the limit of discrete. It is not often taught that way now.
While the words were not used, integrals with respect to discrete
measures go back almost 5000 years, and the measures themselves are
even older.
..................
>You're quibbling. Groups drop the requirement of commutativity, combinators
>the requirement of associativity, etc. There may well be pieces of geometry
>which don't require geometric intuition, but its ridiculous to start
>talking about vector spaces and beyond without any geometrical intuition.
>Its ridiculous to talk about linear maps and not do some multiplication of
>vectors by matrices - yes, by hand even.
Research operates this way, but pedagogy should not. By proceding in this
manner, far too much of the time is spent in the most painful part, unlearning.
Examples should be given, but the examples should have nothing in common
which the student can recognize but the general concept.
As for doing things by hand, this may help to get an understanding IF
the person doing it is adept at arithmetic manipulations. What you are
saying is that nobody else should learn it, that you do not believe in
letting them see the ideas except through manipulation.
>: Logic no more involves calculation than writing text in a language does.
>: The notion of variable is purely linguistic; introducing it for numbers
>: first makes it harder to learn the general concept later.
>Logic can involve a long and intricate series of steps. Following those,
>and creating those steps, involves not just insight, but a well exercised
>insight - and technique!
Following the steps does not involve insight; a computer can follow a
formal proof. Creating them is an art form, and art forms cannot be
taught, but only encouraged.
.................
>: How many instruments do composers play? How many people have not become
>: composers because they played no instruments? And even more so, how many
>: musicians could make a violin or a piano?
>: It's the same with
>: >mathematics - if you don't "play" arithmetic I don't see how you're going
>: >to play or understand the more advance instruments.
>: So which arithmetic do you play? Do you use the Babylonian system, alternating
>: between base 10 and base 6? Do you use the Egyptian system, using repeated
>: numbers of characters for a given power of 10, or the Roman more efficient
>: version of that? Do you use the Greek system, or something like that, in
>: which nine characters are used for the digits in each place? Do you use base
>: 2, where the addition and multiplication tables are trivial?
>: And you can do all these other branches, including numerical analysis, if
>: you can call an arithmetic "player" to assist you. It is only time that
>: is lost by this.
>It's not lost time. It's called building the foundation. Learn arithmetic
>in the decimal system, internalize it. Learn how it works later. It'll
>be available to you anytime you want to think about it. Not for a lot
>of todays kids, though.
If you are willing to reject those who cannot understand the concepts in
spite of this, this is a way to proceed. But if the current miseducational
system is used, in which it is required that the great bulk of students
pass a course, it merely keeps everyone from learning more than trivia.
...................
>I've just never seen an instance. What you advocate is a wild and dangerous
>experiment, just like the new math crap back in the 70s. I KNOW that the way
>I learned back in the 40s and 50s, despite its drawbacks, works. There was
>an excessive reliance on rote. But it worked and didn't do irreparable
>damage to me or others.
The new math was introduced in the 50s. I did not believe then that it was
the best way; the cardinal concept, while important, is extremely difficult
to develop. What it became when the attempt was made to adjust it to what
the teachers could handle was an atrocity.
>Later on, after having majored in set theory and logic in college, I saw
>(from afar) the new math hit the schools, in the early 70s I believe. As
>much as I had been fascinated by the hierarchy of infinities (cardinal
>and ordinal), the introduction of premature and unprepared for abstraction
>was the first assault against math teaching.
As I said, the attempt to develop from the cardinal approach could only be
handled by mathematicians. The teachers, who had been brought up as you
suggest, could no longer teach what the children could learn. Now either
there was a strong adverse selection against those who could understand
mathematics, or the teachers had lost the ability to understand mathematical
concepts almost completely. The sensible strategy, to remove these teachers
from the teaching of mathematics, was not strongly considered.
It was result of a major mis-
>conception among educators and maybe some mathematicians. Set theory
>was an answer to a problem in mathematics. At a certain level of
>abstraction, it provides a unifying approach and imagery for most
>of mathematics. But it didn't solve a problem for 3rd graders or
>3rd grade teachers. Rather it was a disastrous detour from learning
>arithmetic and the rudiments of geometry.
Set theory has never been taught in third grade. Calling set algebra by
this name is a misnomer. But I am not convinced that third graders would
have more difficulty with it than most third year college students. I am
not convinced the other way, either. But unless the college students have
had a course where formal proofs were required, they have been no better
prepared than third graders.
>The reliance on computers and calculators is compounding this disaster.
>It's the same thing but from the opposite end. New math says study the ideas
>behind it all, that'll make your calculations more fun and less onerous.
>The calculators and computers say - now you don't need to do the drudgery
>at all, just study ideas, the whys, (but don't call it new math).
The present teachers do not know the why's themselves. Most of the
present high school teachers do not, either.
james dolan
david k. davis writes:
-I'm totally unable to understand this argument, and my experience entirely
-supports those on the other side. Arithmetic is the first and most familiar
-mathematical system students are exposed to. All kinds of other algebraic
-structures are modelled after or abtracted from arithmetic - groups, vector
-spaces, Banach and Hilbert spaces.
not really. trying to learn group theory as an abstraction of
arithmetic is a really lousy and misleading way of trying to learn
group theory.
cameron
Quotations were getting excessive, so I've paraphrased a bit.
Herman Rubin (hru...@b.stat.purdue.edu) wrote:
>>: [ Arithmetic is merely mechanics, not mathematics, and should be
>>: taught only incidentally to an axiomatic presentation of the
>>: underlying mathematics. ]
David K. Davis (dav...@spcunb.spc.edu) wrote:
>> [ Kids need to learn concrete manipulations first, such as addition
>> and multiplication tables, because abstraction is built on the
>> concrete, and without internalizing concrete manipulations,
>> kids have no basis on which to build abstractions. ]
Anthony Hugh Back (tony...@aol.com) wrote:
> I fear we are entering the realm of the sterile religious disputations of
> the sixteenth and seventeenth centuries. Both points of view are correct
> in context. The mathematicians of the next generation will need Herman's
> preferred type of education, the average citizen, on the other hand, with
> little interest in mathematics apart from adding up a shopping list, would
> not make the intellectual investment required to understand the rules that
> he applies. I think it is safe to assume that most pupils fall into the
> second category, and start their education on that basis. What the
> education system must however do, (and at the moment it is NOT doing) is
> to recognize early those in the first category and be able to give them
> the much deeper training that they will require. We are able to do this
> with sporting prodigies, but not it seems with intellectual ones.
I like very much Anthony's comparison between our treatment of athletic
and intellectual prodigies, and I find it apt. He's right that we insist
on early recognition, reward and encouragement for physical prowess, but
not for mental ability. I have never heard it said of a promising third
grade little leaguer or soccer star that "he's just fine taking phys. ed.
with the other kids; he doesn't need an extra program -- the more advanced
kids in the phys. ed. class will exercise and develop coordination naturally,
on their own, whether we train them or not, so we should concentrate our
efforts on the inept and uncoordinated ones."
I'm a little confused by this statement, however:
> [T]he average citizen, [...] with little interest in mathematics apart
> from adding up a shopping list, would not make the intellectual investment
> required to understand the rules that he applies. I think it is safe to
> assume that most pupils fall into the second category, and start their
> education on that basis.
Who is the "average citizen" in question? I get the feeling that he's
an adult: he adds up shopping lists, and decides for himself when and
whether he will make an intellectual investment in understanding something.
But in the next breath we use our model of the "average citizen" to define
a category into which "most pupils" fall, suggesting that we're talking
about children. Which is it?
I think this is more than just a quibble. I think that if we set the
curriculum we use in teaching our kids by appealing to the standard
set by today's adults, we will obtain a result that is (at best) no
better than the current (deplorable) situation.
I don't think either Herman or David is suggesting that we round up
mathematically illiterate adults and coerce them into studying an
axiomatic treatment of arithmetic. Herman is suggesting that we present
axiomatics to first-graders, and David seems to be suggesting not that
we shouldn't teach axiomatics, but that we should teach arithmetic first
and then use that as a basis upon which to develop a deeper understanding.
I must admit I lean more toward David's view than Herman's on this matter,
but in any case I don't see that either one of them is talking about adults.
We're talking about children, who are enrolled in a compulsory curriculum.
It's our job to figure out what our students need to know, and to teach
it to them. I don't see any room for us to get off the hook for that
responsibility by saying "most of today's adults don't care about this,
and most pupils of today will turn out tomorrow to be like the adults of
today, so we'll let that be our guide for defining the majority curriculum."
If we do that, it'll be a self-fulfilling prophecy: we *will* perpetuate
the low level of mathematical awareness and competency that now prevails.
Locating the mathematically gifted few and placing them in enriched
programs (comparable to special athletic training for pint-sized baseball
players, ice skaters, and tennis stars) is a good idea, but it is not
enough. Skimming off the cream doesn't remove the need to improve the
quality of the milk. (Please forgive the abrupt shift in metaphor!)
To return to the comparison with athletic programs, how many of us
remember the President's Council on Physical Fitness, and all those
dreadful sit-ups and push-ups and chin-ups and 50-yard and 600-yard
dashes that we had to do twice a year to improve the national state
of physical fitness? It didn't work, of course -- there are more
couch potatoes now than ever -- but the point is that we recognized
a national deficiency and implemented a nationwide program to raise
*everybody*'s level of fitness. We did not aim to improve the skills
of the most fit by making them even more fit; that was not the point.
The idea was to raise the average and the median, not just to make
the extremes more extreme. Why can't we take that view toward improving
mathematical fitness too? (Only this time hopefully we'll come up with
a program that actually works.)
And by the by, I don't think the average citizen cares two shakes about
adding up shopping lists. The computerized cash registers in the checkout
lanes do that quite nicely, thank you very much. Speaking for myself,
I can say that the most challenging "real life" mathematical problems
I have lately had to deal with are much more serious:
* figuring out a break-even point for what fraction of my income
to put in a 401(k), and
* figuring out how soon it becomes worth it to pay an extra $500 or
$1000 up front on a mortgage in order to buy down the interest
rate a quarter or a half a point, and whether I'm likely to own
my house that long, and
* similarly, figuring out how much earlier I'll pay off that mortgage
if I can pay an extra $50 or $100 a month direct to principal, and
also figuring out whether paying off the mortage early will actually
*hurt* me in the long run (by making me ineligible for tax credits
for mortgage interest!), and
* figuring out whether a health insurance plan with a $15 biweekly
premium and a 20% co-pay is better than a plan with a $75 premium
and a $10 co-pay, given that I have two children under the age of two
who visit the doctor an average of eight times a year, and who've
each been to the emergency room once already (there's a separate
deductible for ER visits in one of the plans), and
* figuring out how the hell I'm going to send either one of them to
college (under any assumptions you care to make!)
And on a less personal, more political note:
* figuring out how much money our society "ought" to spend on HIV/AIDS
research, given that the number of people dying of AIDS, or even the
number of people infected with HIV, is a tiny fraction of those dying
of heart disease, but given also that the rate of growth of HIV infection
is very much greater than the rate of growth of heart disease
(note: one point concerns f(x), while the other concerns f'(x)!),
* figuring out whether the harm to the economy of putting lots of tobacco
farmers in North Carolina out of work would really be greater than the
combined cost of subsidizing tobacco farmers and paying for medical care
for people with lung cancer and emphysema,
* figuring out whether the money saved today by eliminating Head Start
programs isn't really more than used up by the increased cost ten or
fifteen years from now of paying for welfare and (often) incarceration
of unemployable high-school dropouts (hint: this one's a no-brainer)
Etc., etc., etc.
These are all very real problems, and the "average citizen" is often
woefully unprepared even to understand their true dimensions, let alone
evaluate critically the pros and cons and come to a well-reasoned
conclusion. (Witness the fact that Ronald Reagan could state that
75% of air pollution is CAUSED BY TREES, and not only not be laughed
out of the political arena, but get elected president twice!)
No, Anthony, I really do not think that simply skimming off the prodigies
and placing them in special programs is enough -- not if we are going to
continue to leave the remaining "average citizens" with mathematical
skills that are only suitable for adding up cash register tapes.
I think we are all headed for disaster, in our personal lives and in
our public policy deliberations, if we settle for that.
But I do agree that it would be nice to see, just once, ANY academic
discipline -- mathematics, history, english lit, or what have you --
get the same kind of institutional and community support that we
give to soccer and gymnastics -- let alone (gasp!) football.
I mean not only money for equipment and special training programs,
but also the societal support: recognition and reward for achievement,
admiration rather than contempt from peers, organized clubs of parents
who coach teams in friendly competition in order to improve the skills
of all while making the activity fun rather than a chore, etc. etc. etc.
(Yes, I was an uncoordinated, physically inept, nerdy math prodigy.
Does it show?)
--Cameron Smith
cam...@dnaco.net
P.S. This is posted to the newsgroups listed in the Followup-To: header,
but the only one of those I read is k12.ed.math, and my ISP's feed
for that one seems to be spotty, so I may or may not see any responses.
kc7cc
You people use too many words ! We must , at an early age , identify
left and right brained students and allow them to find their niche.
But most important is ...you can't allow gov't to have anything to do
with education ! Our system is crap ! we can't get rid of the bad
teachers, we can't hire this or that very good teacher cause they don't
have the right degree , etc etc . A private run system will hire
teachers who can teach ( never mind the sheepskin) and pay them
according to performance ( when have you heard of a gov't sys' doing
that ?) . A private school will within months , go bankrupt if it does
not perform , a gov't school ?? ha ha taxes will keep it running for
years. A voucher system allows you to move to a better school in
minutes . We have 10 million good teachers in the USA and not one has
a college degree !! And they of course aren't employed !
Alberto C Moreira
I don't know. Maybe once you go into group theory proper - that is,
once you start specializing - it might be a good idea to abstract
from arithmetics. But if you're looking a a group as a component
in a hierarchy of algebraic structures, you can't avoid the fact
that things are deeply interwoven with our number system. We
define "elements" (read numbers) and an operation (read sum, or
multiply), an "inverse" and an "identity" (directly derived from
our number experience). We define order of a group. We define
finiteness. We define the concept of Abelian group based on our
experience with elementary arithmetic operations. We then go
study a number of individual groups with 3, 4, more elements.
We simplify the concept into monoids and semigroups, and use
them as basic objects to model rings and fields - that is,
our integers, reals, complex. We then build vector spaces; vectors
aren't but sets of numbers, up to isomorphism.
I believe that's the point: up to isomorphism, a lot of these
abstract constructs aren't but number structures in disguise.
The first concrete usage of a Hilbert space, as far as I can
remember, is in Quantum Computing; which basically boils down
to numbers and arithmetic again. The whole buildup of logic,
set theory, algebraic structures, measure theory, topology,
analysis, isn't but a road to formally establish properties
of our number system and the operations on them.
It's only when you start working with computers, specially on
functional programming, that numbers sort of become secondary
and play second fiddle to the concept of function. Even then,
Church numerals give you a set of functions that behave like
our numbers; and we're back were we started, only now we have
a much more comprehensive set of tools.
But, even then, it's never too far from arithmetic...
Alberto.
Brian M. Scott
>: They modify properties of other systems, but not of the arithmetic which
>: is taught. When an algebraist talks about the arithmetic of an algebra,
>: the precise notation is not mentioned, as it does not matter. And one
>: can do quite well in geometry without "geometric intuition." Try the
>: 7-element projective plane, or any other finite projective plane. These
>: have practical uses.
>You're quibbling. Groups drop the requirement of commutativity, combinators
>the requirement of associativity, etc. There may well be pieces of geometry
>which don't require geometric intuition, but its ridiculous to start
>talking about vector spaces and beyond without any geometrical intuition.
>Its ridiculous to talk about linear maps and not do some multiplication of
>vectors by matrices - yes, by hand even.
Speaking as a topologist, I find it ridiculous to insist that geometrical
intuition is required in order to understand any kind of space, vector or
otherwise. And Herman is hardly quibbling: most of what is called
'geometry' is rather far removed from ordinary 'geometric intuition'.
In fact, that intuition can get in the way. That's one of the drawbacks
to the common approach to topology via metric spaces, or worse yet, the
Euclidean spaces: students develop intuitions that won't serve them at
all well in more general settings.
> We abstract from what's inside us.
But we can be presented someone else's abstraction as a starting point. I
never properly understood continuity until I encountered it in the setting
of general topological spaces, where - finally! - it wasn't cluttered up
with inessential details. Epsilons and deltas were then an obvious
application of the underlying abstract notion.
>I've just never seen an instance. What you advocate is a wild and dangerous
>experiment, just like the new math crap back in the 70s.
The 70s?! The so-called 'new math' was part of the response to Sputnik.
>But I don't believe any of it. Abstraction is dead without the concrete.
>Just do it - I'll show you why later. Why won't you show me now? Because
>you won't understand it now, or at least won't see the importance.
Many of my adult students were damaged by just this sort of response in
grade school and high school. Yes, I think that there is value in
learning to compute with some facility. But students who need to know
'why' - or at least to crack their heads on the ideas - are not well
served by rote training. (I doubt that anyone is really very well served
by it.)
> Yours seems to be a perhaps extreme version of a
>very prevalent philosophy these days.
Hardly! Herman's views aren't in line with either of the popular views
nowadays, which can be summarized as back-to-basics and problem-solving.
Neither are mine; reading most published discussion is like reading a
report by the blind men who encountered the elephant. (I think that
Herman's overoptimistic, but I'd love to be proved wrong.)
Brian M. Scott
Donna Mettler
musician on any instrument but their primary-I understand enough about the
violin (your example) to be able to teach it-how to hold it, how to generate
a decent tone, vibrato, hand/finger positions, etc. However, I really can't
play it passibly. Nevertheless, I can teach it by being able to generalize,
using this specific knowledge, because I can play other instruments, and
understand them. I suspect that if students really understand mathematical
operations-with or without notation, then they can learn higher level
ones, but that the basics have to come first. One of the main focii of
the NCTM math is teaching arithmetic operations, but at the motor and
symbolic level. I don't think you'd find any math teacher trained under
NCTM standards who would advocate teaching higher level concepts before basic
ones, but they may not agree on what level concepts fall under. 3+x=6, in
an NCTM classroom, is considered to be on the same level as 3+3=x-this may
be where the confusion on algebra comes from.
D2M
Donna DeVore Mettler
dmet...@math.ttu.edu
http://www.math.ttu.edu/~dmettler/
Math Education Page-http://www.math.ttu.edu/~dmettler/title.html
Musician, Preschool Teacher, and Education Grad Student
All Children deserve a SPECIAL education!
Herman Rubin
In article <4ntthc$2...@csu-b.csuohio.edu>,
Brian M. Scott <sc...@math.csuohio.edu> wrote:
>In article <DrqoJ...@spcuna.spc.edu>, dav...@spcunb.spc.edu
>(David K. Davis) responds to Herman Rubin, who said:
................
>>I've just never seen an instance. What you advocate is a wild and dangerous
>>experiment, just like the new math crap back in the 70s.
>The 70s?! The so-called 'new math' was part of the response to Sputnik.
The "new math" development was started right after WWII, by those who
observed that learning to do arithmetic, even well, very often was
accompanied by almost total lack of understanding. It was developed
and tested on large numbers of children before being introduced to the
general educational system. This large-scale introduction was started
before Sputnik, although Sputnik may have provided the impetus for more
general use.
The early materials (essentially before 1960) were far superior to the
later ones, produced in an unsuccessful attempt to counter the inability
of the teachers to learn the concepts which the children could handle.
David Ullrich
kc7cc wrote:
>
> You people use too many words ! We must , at an early age , identify
> left and right brained students and allow them to find their niche.
Aargh. What about the idea of trying to teach them all to
use both sides of their brains?
David "Sorry About All Those Words" Ullrich
Brian M. Scott
In article <4nuls7$19...@b.stat.purdue.edu>, hru...@b.stat.purdue.edu (Herman
Rubin) says:
>The "new math" development was started right after WWII, by those who
>observed that learning to do arithmetic, even well, very often was
>accompanied by almost total lack of understanding. It was developed
>and tested on large numbers of children before being introduced to the
>general educational system. This large-scale introduction was started
>before Sputnik, although Sputnik may have provided the impetus for more
>general use.
I don't know when it actually started - I'll take your word for it - but
on further consideration I realize that I should have known that it
pre-dated Sputnik: my father was already involved with the CBA high school
chemistry project by 1960, and I remember seeing the Illinois and SMSG
material at about the same time. (I think that Sputnik did give it
considerable impetus, however.)
Brian M. Scott
ed...@netcom.com
In article <Droqv...@spcuna.spc.edu> dav...@spcunb.spc.edu (David K.
Davis) writes:
[snip]
=I'm totally unable to understand this argument, and my experience
entirely
=supports those on the other side. Arithmetic is the first and most
familiar
=mathematical system students are exposed to. All kinds of other algebraic
=structures are modelled after or abtracted from arithmetic - groups,
vector
=spaces, Banach and Hilbert spaces. All these abstractions are hopelessly
=beyond someone who doesn't have a good feel for arithmetic, a feel that
comes
=only with the ability to perform operations. It's ridiculous to see kids
=pulling out calculators to do .01 x 100.
=
What is the difference -- mathematically -- between adding 3 + 4 and
3958917390183.4897952070382 + 845082.29309037917004874? The mechanics are
the same, but for the former, you and I (and a lot of other people) will
just our heads. For the latter, we'd whip out the calculator. Who
decides at what point is the calculator useful and when is the calculator
not? My guess is that it's the person who has to do the problem.
Thus, your claim of ridiculousness for using calculators to multiply 0.01
by 100 is limited in some degree by how the person interprets the
difficulty.
Of course, I do realize that the ``trick'' in the 0.01 X 100 problem is
that 0.01 and 100 are multiplicative inverses, so they should multiply out
to 1 (and not 0.99999 or whatever pentium-based calculator the student
might have). But, if you think about it, the notations we use to denote
our numbers already has a calculator interface built in. We were taught
once that to do the multiplication, move the decimal point to the right by
the number of zeros following the ``1'' of the ``100.'' That abstract
step (abstract in the sense of `why does it work? because it does...') is
very much like the black-boxiness of the calculator.
We must realize that the calculator is here to stay. Let students use it,
even if it's to add 2 + 3. They will, in time, develop an abstraction
from using it and will think differently that the way we do now. Maybe it
might move mathematics into a new interesting direction. The main point,
however, is that they have to use the calculator a lot before they can
grow their own abstraction onto it.
EDEW
ed...@netcom.com
In article <4nnql6$1h...@b.stat.purdue.edu> hru...@b.stat.purdue.edu
(Herman Rubin) writes:
[snip]
=I sense a lot of confusion here. A good part of it is due to the fact
=that, while mathematical concepts can easily be taught even before
=arithmetic, this is not done, and in fact, the important ones are
=pretty much ignored. Also, when algebraic concepts are introduced,
=the early emphasis is almost entirely on solution, and the grading
=is mainly on solution, so that the one without arithmetic ability will
=come out quite poorly.
One mathematical concept that I wish is taught more often and earlier is
the idea that one can quantify an unknown and label it as X (or whatever
is your favorite variable label). Many students have this death grip fear
when they see a variable X, when it is nothing more than short hand for
``this unknown quantity''. Students have trouble understanding what
should be the answer to the question, ``What is X?'' Perhaps we math
educators should rephrase the question. For example, I ask, ``What is
X?'' When I mean, ``What does X represent?'' and I ask the same question
when I mean ``What is the value of X?'' In the former case, the answer
might be ``distance from school to home,'' and in the latter, ``we don't
know yet,'' or ``twice the distance from playground to home.''
Of all things from math, the ability to quantify the unknown and work with
it is most widely useful in all other fields and endeavors. People who
aren't really mathematical in thinking cannot quantify unknowns. If you
ask someone, ``How many shoes will you sell this weekend?'' They can
either give you an answer based on experience (``Oh, I usually sell about
20 pairs.'') or a blanket, ``I don't know.'' They don't use the idea that
the unknown, number of shoes sold, can be equated to other data, like the
number of shoes available, the number of people in the store, etc.
EDEW
ibo...@metz.une.edu.au
In article <DrqoJ...@spcuna.spc.edu> David K. Davis,
dav...@spcunb.spc.edu writes:
>This is ridiculous. Meaning is fine, sticks are fine, and so on.
But in
>first grade kids need to start on the addition tables, and later
the times
>tables, just as I did in the 40s - whatever else is thrown in.
But the addition tables you learnt were already in algebraic form!
You learnt a system of representing numbers by means of well-formed
formulae on an "alphabet" of ten primitives. You learnt tables for
combining the primitives and then rules for deriving the combination
of words of length greater than one bt a more or less complex
algorithm for combining the results obtain from combining certain
pairs of the primitives appearing. At no stage during the algorithm
do you actually worry about the arithmetic significance. You just do
the algebra.
> The numbers
>(and basic operations on them) must be part of us, must become part
of us,
>and early. Abstraction is built on the concrete.
That may be true later in life, but it does not seem to be
true early in life. Children seem to make large leaps of
abstraction very early without any clear relationship to
"concreteness".
d.A.
ibo...@metz.une.edu.au
Subject: When was the new math developed?
From: Herman Rubin, hru...@b.stat.purdue.edu
Date: 22 May 1996 04:13:43 -0500
In article <4nuls7$19...@b.stat.purdue.edu> Herman Rubin,
hru...@b.stat.purdue.edu writes:
>The "new math" development was started right after WWII, by those
who
>observed that learning to do arithmetic, even well, very often was
>accompanied by almost total lack of understanding. It was developed
>and tested on large numbers of children before being introduced to
the
>general educational system. This large-scale introduction was
started
>before Sputnik, although Sputnik may have provided the impetus for
more
>general use.
>
>The early materials (essentially before 1960) were far superior to
the
>later ones, produced in an unsuccessful attempt to counter the
inability
>of the teachers to learn the concepts which the children could
handle.
>
An interesting article in this context is
"Which Subjects in Modern Mathematics and Which Applications in
Modern Mathematics Can Find Place a Place in Programs of Secondary
School Instruction"
by J. Kemeny
L'enseignement mathematiquesVol 10 (1964) pp 152 -- 176.
d.A.
ibo...@metz.une.edu.au
I disagree. I think we abstract to understand and to communicate.
Language is already abstract. To recognise what is being
discussed from the sounds heard is a tour de force of
abstraction. To speak of "yellow" or "table" pre-supposes
sophisticated abstraction.
d.A.
Douglas J. Zare
In article <31A21B...@azstarnet.com>, kc7cc <kc...@azstarnet.com> wrote:
>You people use too many words ! We must , at an early age , identify
>left and right brained students and allow them to find their niche.
I shudder to think of what you could mean by that. It looks similar to the
historical justification of restricting educational opportunities based on
gender.
>But most important is ...you can't allow gov't to have anything to do
>with education !
>[...]
Please go learn some economics.
>A private run system will hire
>teachers who can teach ( never mind the sheepskin) and pay them
>according to performance ( when have you heard of a gov't sys' doing
>that ?) .
>[...]
Do you think private universities emphasize teaching more than public ones?
Douglas Zare
ibo...@metz.une.edu.au
In article <Droqv...@spcuna.spc.edu> David K. Davis,
dav...@spcunb.spc.edu writes:
>I'm totally unable to understand this argument, and my experience
entirely
familiar
algebraic
groups, vector
hopelessly
that comes
kids
You have put the cart before the horse.
For centuries now arithmetic has been taught algebraically,
not vice versa. We no longer teach multiplication of numbers
as calculating the "volume" of rectangular figures.
We use an algebraic representation of numbers and teach
algorithms which depend on an axiomatisation of the
natural numbers to get people to multpily, say
fivehundredandfortyseven by sixhundredandtwentynine.
Doing it arithmetically, as opposed to algebarically
would take many hours rather than a few seconds.
d.A.
ibo...@metz.une.edu.au
David K. Davis
Brian M. Scott (b.s...@bscott.async.csuohio.edu) wrote:
: In article <4nuls7$19...@b.stat.purdue.edu>, hru...@b.stat.purdue.edu (Herman
: Rubin) says:
: >The "new math" development was started right after WWII, by those who
: >observed that learning to do arithmetic, even well, very often was
: >accompanied by almost total lack of understanding. It was developed
: >and tested on large numbers of children before being introduced to the
: >general educational system. This large-scale introduction was started
: >before Sputnik, although Sputnik may have provided the impetus for more
: >general use.
: I don't know when it actually started - I'll take your word for it - but
: on further consideration I realize that I should have known that it
: pre-dated Sputnik: my father was already involved with the CBA high school
: chemistry project by 1960, and I remember seeing the Illinois and SMSG
: material at about the same time. (I think that Sputnik did give it
: considerable impetus, however.)
I don't know the history, but I do know that it was not part of my education
in the 50s (I graduated high school in 1959). All but the last year of my
schooling was in the western suburbs of Chicago - the schools were considered
very good. It was only much later that I remember friends coming to me with
stuff their kids were doing - and which they didn't understand.
I also remember, as a grad student in Cincinnati, teaching logic to high
school teachers one summer. That would have to have been around '64. Presumably
they were gearing up for the 'new math'.
This is all anecdotal, of course, but it's what inclines me to believe that
the 'new math' didn't hit the millions until at earliest the 60s. (I also
remember tracking Sputnik's beep every 90 minutes - Sputnik was a really
big deal back then - '57? ).
-Dave D.
Herman Rubin
In article <DruCH...@spcuna.spc.edu>,
>: In article <4nuls7$19...@b.stat.purdue.edu>, hru...@b.stat.purdue.edu (Herman
>: Rubin) says:
>: >The "new math" development was started right after WWII, by those who
>: >observed that learning to do arithmetic, even well, very often was
>: >accompanied by almost total lack of understanding. It was developed
>: >and tested on large numbers of children before being introduced to the
>: >general educational system. This large-scale introduction was started
>: >before Sputnik, although Sputnik may have provided the impetus for more
>: >general use.
>: I don't know when it actually started - I'll take your word for it - but
>: on further consideration I realize that I should have known that it
>: pre-dated Sputnik: my father was already involved with the CBA high school
>: chemistry project by 1960, and I remember seeing the Illinois and SMSG
>: material at about the same time. (I think that Sputnik did give it
>: considerable impetus, however.)
>I don't know the history, but I do know that it was not part of my education
>in the 50s (I graduated high school in 1959). All but the last year of my
>schooling was in the western suburbs of Chicago - the schools were considered
>very good. It was only much later that I remember friends coming to me with
>stuff their kids were doing - and which they didn't understand.
The new math was intended to be started in the primary grades, and followed
up. For you to have had that version, it would have had to be by 1950.
This is too early. There was followup material, but the original idea
was to introduce it before too much arithmetic had been taught.
>I also remember, as a grad student in Cincinnati, teaching logic to high
>school teachers one summer. That would have to have been around '64. Presumably
>they were gearing up for the 'new math'.
By that time, logic and set algebra had gotten into the high school
algebra courses. This is essentially what remained of the new math.
As this was taught in a manipulative manner, the teachers were able to
manage this to at least some extent.
Albert Y.C. Lai
In article <4nsdn0$h...@newsbf02.news.aol.com>,
tony...@aol.com (Tony2back) wrote:
>The mathematicians of the next generation will need Herman's
>preferred type of education, the average citizen, on the other hand, with
>little interest in mathematics apart from adding up a shopping list, would
>not make the intellectual investment required to understand the rules that
>he applies.
The average citizen, with no interest in literature, should not be
required to read Shakespear, Dickson, or Coles(*), just like they are
not required to read Euclid, Diophantus, Gauss, Spivak.
(*) as in Coles Notes. Or is that Cole's Notes?
--
Albert Y.C. Lai tre...@io.org http://www.io.org/~trebla/
Eric Gindrup
ibo...@metz.une.edu.au wrote:
> In article <DrqoJ...@spcuna.spc.edu> David K. Davis,
> dav...@spcunb.spc.edu writes:[...]
> > The numbers (and basic operations on them) must be part of us, must >
> >become part of us, and early. Abstraction is built on the concrete.
>
> That may be true later in life, but it does not seem to be
> true early in life. Children seem to make large leaps of
> abstraction very early without any clear relationship to
> "concreteness".
>
> d.A.
Largely, children are less concerned about concreteness and grounding.
It takes a few times of being wrong because of poor grounding to instill
a broad desire for concreteness. Children, while still young are able to
make "large leaps of abstraction" for the same reason that ultimately
they'll learn to do arithmetic in their heads and if they become
methematicians, they'll generally lose that ability. It's because of an
overdeveloped sense of caution. Most mathematicians have had much of
their wild, hairy intuition squashed so that only useful intuition will
remain. That is, groundless intuition goes away to be replaced with a
throttled form of intuition. In mathematics, though, this is a good
thing. It allows the mathematician to devote energies to projects for
which the mathematician's restricted intuition still sees hope of
resolution.
The child however doesn't have this caution and will wildly assume the
existence of all kinds of relationships in arithmetic before settling on
repeatably correct ones. Forcing a child to learn how to find grounding
when it is not provided for free will probably result in two styles of
response:
1) The child gives up assuming that math is too hard to do;
2) The child learns how to hypothesize about abstract things and
hypothesizes with sufficient accuracy and precision to pass math.
Learning how to hypothesize is a necessary life skill. It ranks up there
with learning how to use a library and reference materials, reading,
writing, socializing (in the *broad* sense), and self-actualizing.
So, why doesn't this happen in the normal pre-college curriculum? The
answer is a simple choice of economics (probably subconscious) made by
the children responding using style 1) above: devote energies into other
curricula in which it is easier to succeed. There is probably little
wrong with the majority of pre-college math curricula. The problem is in
the system which allows the children to decide NOT to learn math. The
same things happen in the college curricula. Students are given a choice
of major and some choice of electives. Largely, the students choose to
take classes in an economic fashion: which
classes/instructors/departments/majors are easy? This is no different in
essence.
The most effective solution to this problem is NOT to try to construct a
golden road to mathematics in college or pre-college, but to make all
roads equally difficult. Force the children to learn the basic skills by
allowing no alternative. This solution is unpopular because it costs
more.
I suppose we're getting what we pay for.
-- Eric Gindrup! gin...@okway.okstate.edu
M. A. Baker
There have been several generations of "new math." The first curriculum
I had was School Mathematics Study Group (SMSG) in 1962. I beleave this
was part of the post-Sputnick push which also included Physical Sciene
Study Committee (PSSC) Physics.
cameron
(In quoting, I have snipped and reformatted in order to make clear
what points I'm responding to.)
Herman Rubin (hru...@b.stat.purdue.edu) wrote:
> [...] the cardinal concept, while important, is extremely difficult
> to develop. What it became when the attempt was made to adjust it to what
> the teachers could handle was an atrocity. [...]
> [...] the attempt to develop from the cardinal approach could only be
> handled by mathematicians. The teachers, who had been brought up as you
> suggest, could no longer teach what the children could learn.
I'm not convinced that any but a few of the children could learn it
either, but I do agree that when taught to children by teachers who
didn't understand what they were teaching, it failed. I don't know
that I would call it an "atrocity", but it was certainly an unsuccessful
strategy for teaching mathematics to elementary school students.
> Now either
> there was a strong adverse selection against those who could
> understand mathematics,
> or
> the teachers had lost the ability to understand mathematical
> concepts almost completely.
Which do you think it was, Herman? I get the idea that you were assuming
it was the latter, whereas I am 100% convinced it was the former.
That is, I claim that few teachers *lost* any mathematical ability;
most just never had it to begin with, and in fact, that's how they got
to be teachers.
NOW BEFORE YOU FLAME ME, PLEASE NOTICE:
I am NOT NOT NOT saying that all primary and secondary school teachers
are mathematically incompetent!
But I *am* saying that in my experience
1) anyone who shows evidence of mathematical competence is steered
*away* from teaching, and
2) many people who show evidence of mathematical incompetence are
steered *into* teacing, and
3) the incompetent are steered into teaching not *in spite of* but
*because of* their incompetence.
As an undergraduate math major, I knew gifted math students of whom
other students AND FACULTY said "gee, it's too bad about him -- he's
so talented, but he's going into math education." I know of one major
midwest university math department where a hiring plan for the next
several years is being formulated. Thirty hires are expected in a
department of about 80-100, and the idea was proposed that *one position*
be set aside for a math education professorship, to replace the one
math ed professor the department had previously had, who had retired.
The idea was ridiculed in the faculty meeting as a waste of resources,
and dismissed. I have additional anecdotal evidence, gleaned from
experiences as a teaching assistant in a "math for education majors"
class, and from talking to math ed. and el.ed. students, both before
and after they secured jobs. I have heard comments like "I sort of
like math, but I couldn't do well enough in math classes to be a math
major, so I was told to go into math education instead." Especially
among the elementary ed. majors, I encountered great resistance to our
expectations that they actually LEARN SOME MATH; they seemed to want
to turn our "math for ed majors" class into a "teaching methods for
elementary education" class. I had to stress over and over that our
class was NOT a methods class, and we were not there to discuss
"manipulatives" and other teaching strategies, but to do actual proofs
and solve actual math problems involving both theory and computation.
I do know that some mathematically talented persons survive this process
and do go on to become quite good teachers of mathematics. I had a few
such teachers myself, and I credit them for my own decision to pursue
math at the undergraduate and graduate levels. But I also claim that
there are strong selection forces operating to channel as many such
people as possible into other careers, and to dump mathematically
incompetent persons into education, where their incompetence "won't
hurt anything."
(And, for what it's worth, my default assumption about the teachers
who participate in forums like k12.ed.math is that they are the
better ones. Doing this is an extra effort that you don't have to
make, but do anyway, and that to me is prima facie evidence that
you are not the kinds of math teachers that most of the math ed
students I encountered surely must have become.)
> The sensible strategy, to remove these teachers from the teaching of
> mathematics, was not strongly considered.
Given the low esteem assigned to the teaching profession, and to
elementary education in particular, who would you have replaced them
with? Where were you going to get a lot of more mathematically
competent persons to fill low-paid, low-prestige jobs vacated by
the departing teachers?
> Set theory has never been taught in third grade. Calling set algebra by
> this name is a misnomer. But I am not convinced that third graders would
> have more difficulty with it than most third year college students. I am
> not convinced the other way, either. But unless the college students have
> had a course where formal proofs were required, they have been no better
> prepared than third graders.
Even having such a course doesn't help. I know, because I helped teach
one. I had a lot of students frankly admit to me that that course was
(rightly) reputed to be one of the most difficult they had to take, so
they entered it with expectations of getting a C or D (they couldn't risk
an E, because passing it was a requirement for graduation in their major,
but a D was OK), and that since they had an A or B average overall, a low
grade in my course would in no way inhibit them from pursuing their career
plans. Now how much math do you think I managed to teach to students
like that? They expected to do poorly whether they made an effort or not,
and they knew that doing poorly wouldn't hurt them in any way, so how much
effort do you think they made?
I'll go further: I encountered a number of students (and for that matter,
continue to encounter a number of people in the population generally)
who wear mathematical incompetence as a sort of a badge of honor.
People who say things like "I don't really *get* math, but then I'm
more of a creative, right-brain sort of person (smirk)." People who
would think it scandalous if someone said "I never read books or novels"
or "I hate music; it annoys me and I don't understand it" are perfectly
comfortable with saying "I always hated math and was never any good at it."
This is acceptable in our culture; I'm not sure why.
I don't know what to do about the problems I've sketched here, but
Herman's comment prompted me to relate my impressions. I'd be interested
in hearing how they compare to other people's experiences.
--Cameron Smith
cam...@dnaco.net
Herman Rubin
In article <4o2cku$s...@sisko.dnaco.net>,
cameron <cam...@sisko.dnaco.net> wrote:
>(In quoting, I have snipped and reformatted in order to make clear
>what points I'm responding to.)
>Herman Rubin (hru...@b.stat.purdue.edu) wrote:
>> [...] the cardinal concept, while important, is extremely difficult
>> to develop. What it became when the attempt was made to adjust it to what
>> the teachers could handle was an atrocity. [...]
>> [...] the attempt to develop from the cardinal approach could only be
>> handled by mathematicians. The teachers, who had been brought up as you
>> suggest, could no longer teach what the children could learn.
>I'm not convinced that any but a few of the children could learn it
>either, but I do agree that when taught to children by teachers who
>didn't understand what they were teaching, it failed. I don't know
>that I would call it an "atrocity", but it was certainly an unsuccessful
>strategy for teaching mathematics to elementary school students.
Considering the amount of testing before it was made available for
general use, I must disagree with this.
>> Now either
>> there was a strong adverse selection against those who could
>> understand mathematics,
>> or
>> the teachers had lost the ability to understand mathematical
>> concepts almost completely.
>Which do you think it was, Herman? I get the idea that you were assuming
>it was the latter, whereas I am 100% convinced it was the former.
>That is, I claim that few teachers *lost* any mathematical ability;
>most just never had it to begin with, and in fact, that's how they got
>to be teachers.
>NOW BEFORE YOU FLAME ME, PLEASE NOTICE:
>I am NOT NOT NOT saying that all primary and secondary school teachers
>are mathematically incompetent!
Most of the teachers involved had gone through before the dumbing down had
taken place. Nobody expected that the teachers did not have the ability.
However, we must remember that the impetus for the program was the observation
that many, if not most, children came out of arithmetic without the concepts,
even if they were good at it.
....................
H. Jurjus
In Article <4o09b8$a...@news.next.com> "ed...@netcom.com" says:
> Of all things from math, the ability to quantify the unknown and work with
> it is most widely useful in all other fields and endeavors. People who
????
> aren't really mathematical in thinking cannot quantify unknowns. If you
> ask someone, ``How many shoes will you sell this weekend?'' They can
> either give you an answer based on experience (``Oh, I usually sell about
> 20 pairs.'') or a blanket, ``I don't know.'' They don't use the idea that
> the unknown, number of shoes sold, can be equated to other data, like the
Maybe they are able to, but they (rightly) think it would not be useful ?
Another silly question: can you give me an example of *quantification*
over the unknown you mention above ("number of shoes sold") ?
H.Jurjus
John Spiller
It seems to me that it is the generation raised on "New Math"
that so strongly feels that the way they were taught is not good enough
and needs revision. Perhaps that should tell us something:-)
Alberto C Moreira
dav...@spcunb.spc.edu (David K. Davis) wrote:
[some deleted...]
>There's a lot of stuff I learned early that only later I saw in a different
>deeper light. Right now I'm trying to bone up on function spaces, Banach
>and Hilbert spaces. Book after book goes - theorem, lemma, theorem, def,
>lemma, def, thm, etc. What a struggle to learn what it's all about!
This is exactly the point, and a sticky one, too.
A lot of people have this same problem: they get to something like
Hilbert spaces without the corresponding capacity of abstracting concept
from formalism, and unwilling to accept that the only way to really learn
mathematics is to be able to do that. Building the concept is as important
as learning it; I differ with Herman here, in that I find that "concept",
whatever it means, must emanate from the formalism. After all, mathematics
is the science of formalizing inference systems. The sequence, theorem,
lemma, theorem, def, lemma, def, etc., is a natural one; in mathematics
we start from formal definitions, and infer conclusions from premises.
It's not that we use concepts to generate manipulation, but that we learn
concepts by manipulating chains of inference. The concept is a product,
not a prerequisite, and emanates from inference.
>It's only when I started reading some stuff by physicists that I started getting
>a clue. I didn't understand what the abstraction was from - from the
>differential and integral equations and stuff that preoccupied earlier
>mathematicians. And because I lack that background, it's all very much
>up in the air, even when I "understand" it. On the other hand, some of the
>math I sweated over 35 years ago, meaning manipulated every which way -
>logic and set theory, is still very easy for me to picture and do.
If you took the same mathematical machinery used in physics and decided to
apply it so something else, say, computer science, you'd find that your
mental images would change dramatically. Yet, it's the same mathematics;
the definitions, theorems, lemmas, inferences, within the formal system
are the same. What's important is neither manipulation alone nor applicative
concept alone, but to be able to build a formal logic system and walk from
definitions to lemmas to theorems with ease, and so build a mass of abstract
knowledge that can be turned efficiently into use at the user's command:
be it in physics, computer science or any other application.
>You're quibbling. Groups drop the requirement of commutativity, combinators
>the requirement of associativity, etc. There may well be pieces of geometry
>which don't require geometric intuition, but its ridiculous to start
>talking about vector spaces and beyond without any geometrical intuition.
>Its ridiculous to talk about linear maps and not do some multiplication of
>vectors by matrices - yes, by hand even.
It is certainly not ridiculous to talk about abstract entities when we can't
have a sensible intuition anymore. Just go beyond three dimensions and you'll
be needing a fair amount of abstraction. When you're talking about
"multiplication" of vectors by matrices, you must accept that what you
define by "multiplication" may not necessarily be the traditional arithmetic
operation; your "vectors" may have more than three dimensions, and your
field may not be one of numbers at all; the important thing is that, while
such concepts as vectors and maps apply easily to normal "intuitive"
situations, they can be used well beyond. And this is exactly what's needed,
a machinery that can be stretched into a lot of situations, many of them
far beyond one's intuition. We don't need mathematics to confirm our intuition;
we rather need it to help us when intuition fails.
>Logic can involve a long and intricate series of steps. Following those,
>and creating those steps, involves not just insight, but a well exercised
>insight - and technique! It's a facility developed thru exercise.
Mathematical logic is a science, where every entity is defined and every step
must be taken in accordance with a very basic set of rules of inference.
>Chess is logic. Go up against someone - a 7 year old - who has technique >developed through practice. They'll wipe the floor with y=
ou (and me). Their >insight comes through hard work, lots of games, talent, and coaching - but never
>from just "understanding the game".
I played a fair amount of chess, and it certainly doesn't look like logic. It's
more like vision; imagination; creation. Yes, it needs a lot of hard work,
and talent; but not always coaching, and not always lots of games. The greatest
chess players seem to have reached dizzying heights before they even matured
physically or intellectually.
>Math is the same in that regard.
>You've got to get into shape for it, and there are crucial pieces that should
>be there. We abstract from what's inside us. In so many cases now, there's
>nothing there to abstract from.
To an extent you're right, but I daresay it's a bit in the opposite direction
of which you're trying to get. Mathematics seems to be like chess exactly in
that creative, non-logical part, where heuristics and creation reign supreme.
>It's not lost time. It's called building the foundation. Learn arithmetic
>in the decimal system, internalize it. Learn how it works later. It'll
>be available to you anytime you want to think about it. Not for a lot
>of todays kids, though.
I agree with this. Knowing arithmetic is important for a number of reasons,
one of them being the exercising of our capacity of exact inference.
Mathematical thought requires precision, which is not usually exercised
by many other disciplines.
>I've just never seen an instance. What you advocate is a wild and dangerous
>experiment, just like the new math crap back in the 70s. I KNOW that the way
>I learned back in the 40s and 50s, despite its drawbacks, works. There was
>an excessive reliance on rote. But it worked and didn't do irreparable
>damage to me or others.
>Later on, after having majored in set theory and logic in college, I saw
>(from afar) the new math hit the schools, in the early 70s I believe. As
>much as I had been fascinated by the hierarchy of infinities (cardinal
>and ordinal), the introduction of premature and unprepared for abstraction
>was the first assault against math teaching. It was result of a major mis-
>conception among educators and maybe some mathematicians. Set theory
>was an answer to a problem in mathematics. At a certain level of
>abstraction, it provides a unifying approach and imagery for most
>of mathematics. But it didn't solve a problem for 3rd graders or
>3rd grade teachers. Rather it was a disastrous detour from learning
>arithmetic and the rudiments of geometry.
But, pray, what is the problem for 3rd graders and their teachers ? I would
contend that arithmetic, although necessary, isn't mathematics. While it is
sort of up to debate what a 3rd grader should be taught as far as math goes,
if you take a 5th or 6th grader there's a lot less doubt in my mind: he/she
should be doing algebra, and perhaps set theory and logic with it. By that
age, the numerical phase should have been long gone, at least in the
mathematics class.
>The reliance on computers and calculators is compounding this disaster.
>It's the same thing but from the opposite end. New math says study the ideas
>behind it all, that'll make your calculations more fun and less onerous.
>The calculators and computers say - now you don't need to do the drudgery
>at all, just study ideas, the whys, (but don't call it new math).
I do not agree with either calculators or computers in a math class, if
nothing else because precious little mathematical teaching should be
numerical. And precisely because computers exist, mathematical skills
must now emphasize non-numerical application much more than numerical.
It's not that computations are drudgery or onerous, it's just that
they're irrelevant. There's more important things to be learned, some
of which require at least as much mechanical skill, and maybe more.
>But I don't believe any of it. Abstraction is dead without the concrete.
>Just do it - I'll show you why later. Why won't you show me now? Because
>you won't understand it now, or at least won't see the importance. I'm
>not a Piagetan necessarily, but there are stages. I see it with my kids,
>myself, everywhere. Yours seems to be a perhaps extreme version of a
>very prevalent philosophy these days. But I really think the results
>are proving and will prove to be disastrous.
I look at my own kids, who are now in college. I don't believe in Piaget
anymore than I believe that Euler or Cauchy had the answers to how to
model recursive types in object oriented computer languages. Abstraction
must be built without the concrete, because we need math exactly to
take us where the concrete cannot. The concrete today belongs to the
computer, and we usually don't handle it: not even computer programmers
do. If we keep emphasizing 19th century skills, we get 19th century
people: they won't be able to compete with the machine, let alone
command it.
Alberto.
Thomas W. Cowdery
I don't know whether to laugh or to cry. The sad thing is that you
really believe this nonsense. Well, I spent 15 years in the private
sector before becoming a teacher, and I know better. The private
sector isn't any more efficient than education is. In fact, corporate
America could take a few lessons from our public schools. They
couldn't run the schools any better than they are run. In fact, they
couldn't do as good a job, which is why the cities that contracted
private enterprise to run their schools are ready to give up and go
back to the old system.
A private run system would just hire hacks that need a job because no
self-respecting teacher would want anything to do with them. There
isn't any way to tie a teacher's compensation to their performance
because *they* aren't the ones taking the tests! You can lead a horse
to water, but you can't make him drink, and you can teach a lesson,
but you can't make someone learn.
That is part of the problem now. There aren't enough teachers to go
around, and the money isn't good enough to attract and retain the
young people coming in. Virtually every person that I know that quit
education cited the lack of respect, lack of money, and the c*** they
had to deal with from students and parents. As one put it, "you can't
pay me enough to do this anymore".
The salaries that exist aren't enough to attract and retain enough
quality people. I've seen so many 'new teachers' come an go in the
last nine years that we could have replaced our entire faculty. Yet,
with 9 years seniority, I'm still in the bottom half of the list. The
majority of new teacher's can't cut it, and/or won't do it for the
money they are being paid. The ones that are here are the ones who
care enough about education that we tough it out in spite of all of
the c*** that we put up with.
kc7cc <kc...@azstarnet.com> wrote:
>You people use too many words ! We must , at an early age , identify
>left and right brained students and allow them to find their niche.
>But most important is ...you can't allow gov't to have anything to do
>with education ! Our system is crap ! we can't get rid of the bad
>teachers, we can't hire this or that very good teacher cause they don't
>have the right degree , etc etc . A private run system will hire
>teachers who can teach ( never mind the sheepskin) and pay them
>according to performance ( when have you heard of a gov't sys' doing
>that ?) . A private school will within months , go bankrupt if it does
>not perform , a gov't school ?? ha ha taxes will keep it running for
>years. A voucher system allows you to move to a better school in
>minutes . We have 10 million good teachers in the USA and not one has
>a college degree !! And they of course aren't employed !
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Thomas W. Cowdery When I die, I want to go peacefully in
twc...@rs6000.cmp.ilstu.edu my sleep like my grandfather did.
tcow...@dave-world.net Not screaming like the passengers in his car.
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H. Jurjus
In Article <31A4CE...@okway.okstate.edu> "Eric Gindrup <gin...@okway.okstate.edu>" says:
> Learning how to hypothesize is a necessary life skill.
(That's what you call 'hypothesizing', I guess ?)
One actually starts really to learn mathematics when one studies, e.g. Euclidean
geometry starting from *axioms*. Yes, it is possible for children
(13, 14 years old, maybe even younger) to understand such an approach,
if it is well taught.
Learning math is not like learning, say, French.
2) The kind of hypothesizing necessary in life has *nothing* to do with
the mentality to do mathematics. On the contrary. Applying the mathematical
mentality in real life may be disastrous: in mathematics a problem stops
being a problem whenever one can prove it has no solution. In life,
that is when they become real problems.
(Yes, there are mathematicians who make that error, for example in their
marriage.)
It ranks up there
> with learning how to use a library and reference materials, reading,
> writing, socializing (in the *broad* sense), and self-actualizing.
Starfollower
ed...@netcom.com wrote:
>
>What is the difference -- mathematically -- between adding 3 + 4 and
>3958917390183.4897952070382 + 845082.29309037917004874? The mechanics are
>the same, but for the former, you and I (and a lot of other people) will
>just our heads. For the latter, we'd whip out the calculator. Who
>decides at what point is the calculator useful and when is the calculator
>not? My guess is that it's the person who has to do the problem.
...<deletions>...
>We must realize that the calculator is here to stay. Let students use it,
>even if it's to add 2 + 3. They will, in time, develop an abstraction
>from using it and will think differently that the way we do now. Maybe it
>might move mathematics into a new interesting direction. The main point,
>however, is that they have to use the calculator a lot before they can
>grow their own abstraction onto it.
>
It is true that the calculator is here to stay, but that is a pretty poor reason
for allowing ulimited and indiscriminant use of the tool in the classroom. In my
experience, students who are allowed such use do not develop abstractions
because there is no impetus for them to do so. Students naturally use as little
effort and thought as they can get away with in any situation, and the emphasis
in our school system is :-( on good grades. When students find they can do the
math with a calculator, get the right answers, get good grades, and have
absolutely no idea what they have just done, they are only too happy to do so.
How many students have you taught under your proposed method of unlimited
calculator use and for how long, and did you follow their "math career" into
high school and beyond to find out how they were doing?
Sheila King
Herman Rubin
In article <1996052411...@pi0220.kub.nl>,
H. Jurjus <jur...@kub.nl> wrote:
>In Article <31A4CE...@okway.okstate.edu> "Eric Gindrup <gin...@okway.okstate.edu>" says:
>> Learning how to hypothesize is a necessary life skill.
>1) Mathematics isn't necessarily learned by trial and error.
>(That's what you call 'hypothesizing', I guess ?)
This is needed in doing mathematical research. One cannot, however, teach
this to those who do not have the ability.
>One actually starts really to learn mathematics when one studies, e.g. Euclidean
>geometry starting from *axioms*. Yes, it is possible for children
>(13, 14 years old, maybe even younger) to understand such an approach,
>if it is well taught.
>Learning math is not like learning, say, French.
What ins involved in learning mathematics is exactly as you state.
As for the age involved, I believe it could be started for most children
by age 6; I have advocated that arithmetic be so taught. It is possible
to teach the concepts and the structure without having the student able
to PROVE the theorems. It is quite possible that we may find that far
more of them have that ability if it is started earlier.
But even if it is desired to have the students prove some of them, it
can certainly be done by age 10; I can state this because people who
understand what is involved have done so to children believed to be
representative of the upper half of fifth graders. I believe that
with material involving somewhat less difficult vocabulary, it can
be done for a larger proportion at least two years earlier.
James F. Epperson
On Sat, 25 May 1996, Michael Greene wrote:
> twc...@rs6000.cmp.ilstu.edu (Thomas W. Cowdery) wrote:
>
> >There
> >isn't any way to tie a teacher's compensation to their performance
> >because *they* aren't the ones taking the tests! You can lead a horse
> >to water, but you can't make him drink, and you can teach a lesson,
> >but you can't make someone learn.
>
> That isn't clear to me. It's like saying an engineer's proficiency
> isn't gauged by how well the rocket she designs performs.
>
> It's true a teacher is not in direct control of the child but a
> teacher's ability to convey information and motivate children is being
> gauged by the test. It's not a perfect measure but it's better than no
> measure.
It is a very =imperfect= and =counterproductive= measure. In the first
place, if I am given a class of underprepared students for Calc I, then
the odds are very low that they will do well on a standardized exit
exam. If I am a good teacher -- and I think I am -- I will be able
to bring them along, but the poor initial preparation will almost
certainly be reflected in lower scores on the exit exam. Thus I am
punished for the student's poor preparation, which might be the fault of
the previous teacher or the students' advisors.
Tying my compensation to an exit exam will also inevitably transform
teaching into coaching, as in "here is how to do this kind of problem
that is on the exam," instead of actually teaching concepts and ideas.
And the rocket analogy is very poor. Stainless steel has no free will
-- if the design is right, and the construction correct, the rocket will
fly. On the other hand, students have free will and they also have "bad
days." I don't want my performance evaluation to depend on someone
else's performance. Football coaches have to deal with that, but they
are paid a damn sight more than I am!
[snips]
> OTOH, there are some teachers in my son's school district that don't
> have my respect. One of them mistaught my son arithmetic and the other
> one repeatedly sent home spelling lists with misspelled words on it.
> Those two teachers reflect poorly on an administration that fails to
> detect the problem on its own or rectify it when its brought to their
> attention. Before we raise salaries to draw better staff, I want to
> see mechanisms in place to identify non-performing staff and
> administrators.
But this is a chicken-and-egg issue. Because salaries are so low, smart
and qualified people are not interested in teaching in public schools, so
their places are taken by the less smart and less qualified. I remember
a young lady in a linear algebra class I taught at the University of
Georgia. She couldn't speak a grammatically correct sentence to save her
soul, yet she was going to be a public school mathematics teacher!
We have to increase the salaries in order to attract decent people.
Other changes can be made as well, like some kind of competancy testing
to weed out the deadwood. But I think that we have to face the fact that
one cost of having so disrespected the teaching profession for so long is
that it will take time to get the deadwood out, by natural attrition.
> The vast majority of teachers I've dealt with over the years have been
> excellent teachers. I can't say the same about the principals I've
> seen come and go. Any test that's applied to the teachers should also
> be applied to the administrators because they set the course for a
> school.
Now this I can agree with.
Jim Epperson | I would like to see truthful
Department of Mathematical Sciences | history written -- US Grant
University of Alabama in Huntsville +-------------------------------------
eppe...@math.uah.edu URL: http://www.math.uah.edu/~epperson
URL: http://members.aol.com/jfepperson
Herman Rubin
In article <4o4of3$c...@news.redshift.com>,
>>There
>>isn't any way to tie a teacher's compensation to their performance
>>because *they* aren't the ones taking the tests! You can lead a horse
>>to water, but you can't make him drink, and you can teach a lesson,
>>but you can't make someone learn.
>That isn't clear to me. It's like saying an engineer's proficiency
>isn't gauged by how well the rocket she designs performs.
The quality of the materials in the rocket are much more controlled
than those of the children being taught in a class. And often the
best designed machines fail because of poor assembly and poor operation.
The failure to produce (yet) fusion power plants does not mean that
those designing them are incompetent.
>It's true a teacher is not in direct control of the child but a
>teacher's ability to convey information and motivate children is being
>gauged by the test. It's not a perfect measure but it's better than no
>measure.
Suppose a teacher does a first-class job in teaching a foreign
language to children who have an understanding of phonetic reading
and English grammar. Would you expect that teacher to do well in
teaching that language to those who cannot pronounce words they have
not seen, and who cannot recognize the various parts of speech?
On the other hand, those who teach a foreign language by the oral-aural
method will do fairly well with that group, but will not accomplish
much more with the ones who understand letters and grammar.
Likewise, the teacher who drills arithmetic manipulation will get
children who have difficulty in seeing the concepts to learn that better
than those emphasizing the concepts. But from the evidence, those
capable of learning the concepts if taught them early may never get
them later.
And a teacher who has a class of children form a group which stresses
learning will do better than one whose class comes from those which
do not place much value on it.
We can assess what the teacher knows, and we can see how the teacher
teaches. We can look at the examinations, if the teacher has the
choice of how to examine.
But different teachers may very well be appropriate for different
students, just as different students should take different curricula
at different rates starting from the beginning.
>My sense is that if the school's administration and staff were
>evaluated in part by standardized test scores, a lot of the subjective
>arguments would go away. The school principals and superintendents
>would be testing pedagogical changes before jumping in with both feet.
This is likely to result in stressing the teaching of trivia. This is
all that is tested well by "objective" tests. Universities use SATs
because there is nothing else which is available and reliable; these
are more a measure of ability than knowledge.
.............
>The vast majority of teachers I've dealt with over the years have been
>excellent teachers. I can't say the same about the principals I've
>seen come and go. Any test that's applied to the teachers should also
>be applied to the administrators because they set the course for a
>school.
A few of the teachers my children had were excellent. Some were good.
The majority were at best tolerable. A fair number did not know what
they were teaching.
Kevin Anthony Scaldeferri
>One mathematical concept that I wish is taught more often and earlier is
>the idea that one can quantify an unknown and label it as X (or whatever
>is your favorite variable label). Many students have this death grip fear
>when they see a variable X, when it is nothing more than short hand for
>``this unknown quantity''. Students have trouble understanding what
>should be the answer to the question, ``What is X?'' Perhaps we math
>educators should rephrase the question. For example, I ask, ``What is
>X?'' When I mean, ``What does X represent?'' and I ask the same question
>when I mean ``What is the value of X?'' In the former case, the answer
>might be ``distance from school to home,'' and in the latter, ``we don't
>know yet,'' or ``twice the distance from playground to home.''
>
I've always found it a bit baffling as to why so many people have this
problem. People who have no difficulty at all understanding
linguistic variables (ie pronouns) shudder at the sight of a
mathematical variable.
>Of all things from math, the ability to quantify the unknown and work with
>it is most widely useful in all other fields and endeavors. People who
>aren't really mathematical in thinking cannot quantify unknowns. If you
>ask someone, ``How many shoes will you sell this weekend?'' They can
>either give you an answer based on experience (``Oh, I usually sell about
>20 pairs.'') or a blanket, ``I don't know.'' They don't use the idea that
>the unknown, number of shoes sold, can be equated to other data, like the
>number of shoes available, the number of people in the store, etc.
>
I don't think that this is really someting that is limited to 'people
who aren't really mathematical'. For example, the physics society
here sells coffee and donuts in the lobby of the physics bulding to
raise money. Yet most of the workers are oblivious to the factor that
control sales. I have walked in on a rainy morning and pronounced, "What a
miserable day. Sales are going to be great." and found that most
people couldn't see the correlation between the weather and the amount
of coffee sold.
Michael Greene
twc...@rs6000.cmp.ilstu.edu (Thomas W. Cowdery) wrote:
>There
>isn't any way to tie a teacher's compensation to their performance
>because *they* aren't the ones taking the tests! You can lead a horse
>to water, but you can't make him drink, and you can teach a lesson,
>but you can't make someone learn.
That isn't clear to me. It's like saying an engineer's proficiency
isn't gauged by how well the rocket she designs performs.
It's true a teacher is not in direct control of the child but a
teacher's ability to convey information and motivate children is being
gauged by the test. It's not a perfect measure but it's better than no
measure.
My sense is that if the school's administration and staff were
evaluated in part by standardized test scores, a lot of the subjective
arguments would go away. The school principals and superintendents
would be testing pedagogical changes before jumping in with both feet.
> Virtually every person that I know that quit
>education cited the lack of respect, lack of money, and the c*** they
>had to deal with from students and parents. As one put it, "you can't
>pay me enough to do this anymore".
A teacher simply shouldn't have to deal with a recalcitrant student.
If the student is disrespectful, or worse, the kids should be out of
the classroom.
OTOH, there are some teachers in my son's school district that don't
have my respect. One of them mistaught my son arithmetic and the other
one repeatedly sent home spelling lists with misspelled words on it.
Those two teachers reflect poorly on an administration that fails to
detect the problem on its own or rectify it when its brought to their
attention. Before we raise salaries to draw better staff, I want to
see mechanisms in place to identify non-performing staff and
administrators.
The vast majority of teachers I've dealt with over the years have been
Karen Dee Michalowicz
I always enjoy the discussion on the "new math" because it
demonstrates our lack of knowledge about the history of
mathematics education, particularly in the U.S. NCTM is now in
the process of putting together a monograph on the topic.
The movement which we call "new math" did actually start
germinating in the universities in the 50's. It started
filtering into the classrooms during the 60's and early 70's.
Having been in this field for 34 years, let me assure you the
ideas were super. The problem was that the "inservice"
component was either not there or not considered essential.
The new standards, which reflect many of the "new math" ideas,
have stressed the importance of inservice.
I now use many of the materials I learned about in the 60's
when I worked on my first masters. Many manipulatives were
"home made" with little expense. My luck was that I went to
graduate school when the "new math" movement was gaining
momentum. Again, its problem was coordination with teachers.
There was virtually no instruction for teachers in the field.
Now, if you think that this was the first "new math" movement,
you will be surprised. There was a movement in the early
l800's to revise textbooks and change pedagogy. It seems if
you go through old arithmetics from 100-200 years ago, you will
notice significant changes about every 30 or 40 years. I have
a super book from the turn of this century that sounds like it
was written today. My mentor, Mary Everest Boole, b.1832
d.1916, wrote books that were 80 years ahead of her time. Her
books speak to groups, journals, hands-on activities, sensory
education, historical connections, real-world problems.
Before we condemn a particular trend or movement, it might be
good to have some experience with it as a teacher. Or, before
we put something down, we might look at the historical
perspectives. My opinions on the movements in math ed during
my tenure as a teacher are a result of teaching 34 years, in
and out, with an average of 35 students. (My first years I had
in the upper 40s.) The secret is taking the best from
everything and celebrating it in ones teaching.
---
Cheers!
Karen Dee
Presidential Awardee Mathematics,'94
Karen Dee Michalowicz Adjunct Faculty
Upper School Mathematics Chair George Mason University
The Langley School Fairfax, VA
1411 Balls Hill Rd, McLean, VA
22012 USA
703-356-1920(w) E-Mail: kmic...@pen.k12.va.us
Fax: (703) 790-9712 --or-- Kar...@aol.com
Thomas W. Cowdery
ed...@netcom.com wrote:
>We must realize that the calculator is here to stay. Let students use it,
>even if it's to add 2 + 3. They will, in time, develop an abstraction
>from using it and will think differently that the way we do now. Maybe it
>might move mathematics into a new interesting direction. The main point,
>however, is that they have to use the calculator a lot before they can
>grow their own abstraction onto it.
You are missing the whole point. Using the calculator, when they
can't do the arithmetic themselves, PREVENTS them from developing that
abstraction. The calculator just presents the answers. To see the
patterns behind the answers you need a degree of 'number sense' that
comes from working with numbers. But you don't get that from using
the calculator, only by doing the arithmetic mentally or manually.
That forces you to think about what is happening in a way that
punching buttons cannot duplicate. Granted, some will never develop
that 'number sense' even if they do the work themselves. But they
won't get it from the calculator use either.
Thomas W. Cowdery
<ibo...@metz.une.edu.au> wrote:
>In article <DrqoJ...@spcuna.spc.edu> David K. Davis,
>dav...@spcunb.spc.edu writes:
>>... We abstract from what's inside us.
>I disagree. I think we abstract to understand and to communicate.
>Language is already abstract. To recognise what is being
>discussed from the sounds heard is a tour de force of
>abstraction. To speak of "yellow" or "table" pre-supposes
>sophisticated abstraction.
Nonsense! Both 'table' and 'yellow' are very concrete concepts. The
former is a word that represents a physical object. The latter is a
color that any person of normal eyesight can distinguish. A child's
language skills have very little abstraction in them. Why? Because
the child (prodigies excepted) doesn't have the capability to conceive
much that is abstract. Even 'love', which can be about as abstract as
anything, has tangible effects that a child can identify as 'love'.
Mike Kent
Not everyone taught the new math by rote. I was fortunate
enough to be taught geometry by an extremely gifted teacher
in 1961-62 -- though "taught" is not exactly the right word,
"encouraged and assisted in learning" is a lot closer as the
course was conducted Moore-style after the first few weeks.
The course notes were essentially Ed Moise's book "Geometry
from an Advanced Standpoint" sans proofs, and it was covered
in about 24 weeks. there was time enough left over for a good
intro to analytic geometry and trig. The next year another
gifted teacher gave us an "Algebra II" course that introduced
groups, rings (including euclidean rings) and fields, elementary
linear algebra, and some number theory. Finally, as a senior, I
had a third gifted teacher who taught both the computational
"applied" parts of calculus and what I later learned was a
rigourous introduction to real analysis.
In all these courses the emphasis was on developing understanding
and mathematical maturity, but mastering the computational
skills to be able apply this understanding was not neglected.
There was (and is) nothing wrong with the new math. It's a lot
better than hoping that rote and practice will somehow, perhaps
by magic, lead to real understanding. HOWEVER, it does require
that the person teaching knows mathematics, enjoys doing and
teaching math, and communicates this to the kids.
Such people are hard to find, and then we underpay them ..
ibo...@metz.une.edu.au
Subject: Re: What's wrong with education and what is being done to
change it?
From: Starfollower, ck...@cyberg8t.com
Date: Fri, 24 May 1996 15:30:07 GMT
>It is true that the calculator is here to stay, but that is a
pretty poor reason
>for allowing ulimited and indiscriminant use of the tool in the
classroom.
I agree that the ubiquity of calculators is no argument for their
use in classrooms.
After all, prositution is also here to stay .....
d.A.
Thomas W. Cowdery
mgr...@redshift.com (Michael Greene) wrote:
>twc...@rs6000.cmp.ilstu.edu (Thomas W. Cowdery) wrote:
>>There
>>isn't any way to tie a teacher's compensation to their performance
>>because *they* aren't the ones taking the tests! You can lead a horse
>>to water, but you can't make him drink, and you can teach a lesson,
>>but you can't make someone learn.
>That isn't clear to me. It's like saying an engineer's proficiency
>isn't gauged by how well the rocket she designs performs.
The engineer can control virtually every aspect of whatever they
design. They can use only those components that meet their criteria
and reject those that do not. They can specify the conditions that
the rocket (or whatever) is tested under.
A manufacturer has similar abilities to control what they produce. I
recently saw a video on the quality control system used by Frito-Lay
in making their potatoe chips. They would sample potatoes before
accepting delivery. If the sample didn't meet their expectations,
they didn't buy it (and presumably some other company did). They had
an ongoing system to test that the chips were not overcooked, did not
have too much or too little salt, and didn't suffer too much breakage.
If any of those problems cropped up, the entire batch might be
discarded (or sold as a generic or store brand).
By contrast, a teacher have virtually no control of the 'raw
materials' that they have to work with. We certainly do not get to
pick and reject our students. If a class does poorly, we don't
discard the batch and start over. There is absolutely no comparison.
>It's true a teacher is not in direct control of the child but a
>teacher's ability to convey information and motivate children is being
>gauged by the test. It's not a perfect measure but it's better than no
>measure.
It is no measure at all. The teacher is but one influence out of many
that children have. And with the lack of respect for teachers that
is prevelent in our society, and influence that we might have is
further diluted. The best teachers will not be able to motivate
certain students. Even the worst teacher will motivate some that
nobody else has been able to reach. Then it becomes the luck of the
draw, whether you get the ones that you can reach or not.
Every teacher that I know that has taught multiple sections of the
same course has at least one story of the class that was significantly
above or below all of the others, in spite of having the same teacher,
teaching the same lessons, and giving the same tests. It happens
every year for me, since I usually teach multiple sections of the same
course. I've seen times when there was as much as a 15% difference in
the average grades, with no explanation that I could see other than
the mix of kids in the respective classes.
>My sense is that if the school's administration and staff were
>evaluated in part by standardized test scores, a lot of the subjective
>arguments would go away. The school principals and superintendents
>would be testing pedagogical changes before jumping in with both feet.
Again, you run into the problems of demographics. If my school is
going to be compared to a school in a poverty stricken area of the
country, like an inner-city school, or the hills and hollers of West
Virginia, then we will look very good in the standardized test scores.
But if we are compared to many of the schools from the wealthy
suburbs, then we won't look so good. Socio-economic status is one of
the strongest predictors of academic success. Even within my school
district, the elementary school in the 'poor' part of town has test
scores that are significantly lower than the other elementary schools.
All of the elementary schools use the same curriculum. And enough
teachers switch schools from year to year, with no effect on the
overall test scores, that it is ludicrous to think that the teachers
in that school are all that much worse than the teachers in the other
schools. It isn't the teachers. It is the kids.
>> Virtually every person that I know that quit
>>education cited the lack of respect, lack of money, and the c*** they
>>had to deal with from students and parents. As one put it, "you can't
>>pay me enough to do this anymore".
>A teacher simply shouldn't have to deal with a recalcitrant student.
>If the student is disrespectful, or worse, the kids should be out of
>the classroom.
And what do you do with them? That is easier said than done; not to
mention illegal in most states.
> OTOH, there are some teachers in my son's school district that don't
>have my respect. One of them mistaught my son arithmetic and the other
>one repeatedly sent home spelling lists with misspelled words on it.
>Those two teachers reflect poorly on an administration that fails to
>detect the problem on its own or rectify it when its brought to their
>attention. Before we raise salaries to draw better staff, I want to
>see mechanisms in place to identify non-performing staff and
>administrators.
I understand where you are coming from here. But you are putting the
cart before the horse. If you cannot attract the better staff, you'd
better hold on to what you have. Many schools in the large cities
simply cannot find enough teachers in some fields, so they fill the
classroom with whoever they can get. It shouldn't happen, but it
isn't unique to education. I see it regularly in private business.
Besides, even people who aren't teachers make mistakes.
>The vast majority of teachers I've dealt with over the years have been
>excellent teachers. I can't say the same about the principals I've
>seen come and go. Any test that's applied to the teachers should also
>be applied to the administrators because they set the course for a
>school.
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Brian M. Scott
In article <4o5rf2$l...@thor.cmp.ilstu.edu>, twc...@rs6000.cmp.ilstu.edu
(Thomas W. Cowdery) says:
><ibo...@metz.une.edu.au> wrote:
[snip]
>>Language is already abstract. To recognise what is being
>>discussed from the sounds heard is a tour de force of
>>abstraction. To speak of "yellow" or "table" pre-supposes
>>sophisticated abstraction.
>Nonsense! Both 'table' and 'yellow' are very concrete concepts. The
>former is a word that represents a physical object. The latter is a
>color that any person of normal eyesight can distinguish.
A particular table is a concrete object. The concept of 'table', on
the other hand, is a rather sophisticated abstraction. (Consider how
difficult it is to formulate a comprehensive and accurate definition
of the term.) 'Yellow' is even more obviously an abstraction: many
languages don't even have a term for it! I think that you've been
misled by familiarity with the concepts into seriously underestimating
how far they've been abstracted from perception.
Brian M. Scott
David K. Davis
Alberto C Moreira (alb...@moreira.mv.com) wrote:
< much deleted >
: dav...@spcunb.spc.edu (David K. Davis) wrote:
: [some deleted...]
: >You're quibbling. Groups drop the requirement of commutativity, combinators
: >the requirement of associativity, etc. There may well be pieces of geometry
: >which don't require geometric intuition, but its ridiculous to start
: >talking about vector spaces and beyond without any geometrical intuition.
: >Its ridiculous to talk about linear maps and not do some multiplication of
: >vectors by matrices - yes, by hand even.
: It is certainly not ridiculous to talk about abstract entities when we can't
: have a sensible intuition anymore. Just go beyond three dimensions and you'll
: be needing a fair amount of abstraction. When you're talking about
: "multiplication" of vectors by matrices, you must accept that what you
: define by "multiplication" may not necessarily be the traditional arithmetic
: operation; your "vectors" may have more than three dimensions, and your
: field may not be one of numbers at all; the important thing is that, while
: such concepts as vectors and maps apply easily to normal "intuitive"
: situations, they can be used well beyond. And this is exactly what's needed,
: a machinery that can be stretched into a lot of situations, many of them
: far beyond one's intuition. We don't need mathematics to confirm our intuition;
: we rather need it to help us when intuition fails.
My position is that mathematics helps us extend our intuition - it doesn't
obviate the need for it at any level. 2 and 3 dimensional spaces extend
our intuition (and knowledge) about the real line, plus unifying this with
some of our geometrical ideas and intuitions. n dimensional space takes us
beyond these intuitions, but does not at all obviate the need for them.
Those intuitions guide us in exploring what's still the same in n dimensions,
what carries over, what doesn't. And the same applies in moving to complex
and infinite dimensional spaces. But just as the 2 and 3 dimensional real
spaces tie togther a lot elementary geometric knowledge, and as the n
dimensional spaces tie together a lot of knowledge about linear equations,
the inf dimensional space tie toghter a lot of knowledge about classical
diff eq and integral eqs - fourier series, etc. (Never mind if I'm wrong
on this or that detail.)
My point is that our abstractions, ultimately, are historically conditioned.
We do not, now can we, just arbitrarily pick a bunch of axioms and start
exploring the consequences. We abstract on the basis of what precedes.
Of course, we often say, what happens if we loosen this requirement, etc.
Something went wrong in the 50s and 60s, maybe even before. I have a lot
of books on my shelves from that period that just go like a bat out of
hell - def,def,lemma,thm,lemma,def,thm, etc. The exercises might condescend
to dropping a hint or two as to where it all came from. But there is
precious little motivation, precious little acknowledgment of the actual
historical route of development, of what was abstracted from.
Abstraction is historical, it is layered, it is relative, it has a
context. I don't deny that there has a been a great deal of exploration
of variants of historically presented abstract systems, explorations
perhaps motivated only by a desire to fill in the holes. But the number
of abstract systems is vast, and even the variants which are explored for
their own sakes get explored because the are contiguous to something
historically conditioned. And those that get a lot of interest will
be those that are fruitful in unifying other stuff, i.e. turn out to
be abstractions of pre-existing material after all.
I think that mathematics has gone a little bit (well, quite a bit really)
nuts with some of this, and tries too pretend that this is not so. There
is a book by R. Smullyan, TO MOCK A MOCKINGBIRD, which presents combinators
franky as an amusement, as a source of puzzles, an abstract game. He does
give a hint later as to some of the "applications" and history, but not
much. It's entertainment and is so presented. But the books I'm talking about
are dead serious, and deadly dry, and dead. So much suffering could be
relieved so easily, but they do not, they will not compromise their
abstractions with mention of where it all came from.
Bottom line, pedagogically, teaching or learning, we can't just skip the
historical roots of math, we can't just proceed to the top levels of
abstraction, present this top level the to students, and neglect all the
layers of suffering that went before. And in the basement there needs
to be an arithmetic intuition that is based on considerable exercise
as well as a geometric intuition also based on considerable exercise.
... <a lot i agree with>
: I look at my own kids, who are now in college. I don't believe in Piaget
: anymore than I believe that Euler or Cauchy had the answers to how to
: model recursive types in object oriented computer languages. Abstraction
===========
: must be built without the concrete, because we need math exactly to
==================================
: take us where the concrete cannot. The concrete today belongs to the
: computer, and we usually don't handle it: not even computer programmers
: do. If we keep emphasizing 19th century skills, we get 19th century
: people: they won't be able to compete with the machine, let alone
: command it.
==== is mine
That's where I disagree. Abstraction is relative, and the direction of
movement is from the concrete to the abstract, but there are layers. The
computer implements abstractions - data types and algorithms. These
abstractions are, it's true, but a small subset of the abstraction that
mathematics in its whole deals with.
I do agree with the importance of abstraction - that's the power of
mathematics. But abstraction can be undermined in TWO ways, not just one.
It can be undermined by refusing to recognize its usefulness, by refusing
to move beyond the concrete, by refusing to recognize the legitimacy of
purely abstract explorations of mathematical objects.
But it can also be undermined by forgetting the relativity of abstraction,
by forgetting that abstraction is abstraction from something, that
abstraction is historical, by forgetting - please forgive me - the dialectical
interaction of the abstract and the concrete. Abstraction can be undermined by
our getting lost in abstraction, by forgetting that we have fingers and that
we move in space and that our loftiest abstractions trace back to these
simple facts.
-Dave D.
Herman Rubin
In article <4o5rf2$l...@thor.cmp.ilstu.edu>,
>>In article <DrqoJ...@spcuna.spc.edu> David K. Davis,
>>dav...@spcunb.spc.edu writes:
>>>... We abstract from what's inside us.
>>I disagree. I think we abstract to understand and to communicate.
>>Language is already abstract. To recognise what is being
>>discussed from the sounds heard is a tour de force of
>>abstraction. To speak of "yellow" or "table" pre-supposes
>>sophisticated abstraction.
>Nonsense! Both 'table' and 'yellow' are very concrete concepts. The
>former is a word that represents a physical object.
^^^^^^^^^^
THIS is a good part of abstraction. Mathematical communication about
the real world is the ability to use formal mathematical systems to
represent other types of entities.
But for this particular term, try to distinguish between table, counter,
desk, and platform. I am not sure that I could do it in a clear manner.
The latter is a
>color that any person of normal eyesight can distinguish. A child's
>language skills have very little abstraction in them.
Try explaining the limitations of "yellow". They are quite unclear.
It is now known that children acquire an understanding of grammatical
structure with little vocabulary.
Why? Because
>the child (prodigies excepted) doesn't have the capability to conceive
>much that is abstract. Even 'love', which can be about as abstract as
>anything, has tangible effects that a child can identify as 'love'.
Now consider the abstract ideas in mathematics. These are more abstract,
so they can be more easily communicated. Only a small amount of initial
knowledge is needed to be able to provide a characterization of other
concepts.
Herman Rubin
In article <Dry38...@spcuna.spc.edu>,
David K. Davis <dav...@spcunb.spc.edu> wrote:
>Alberto C Moreira (alb...@moreira.mv.com) wrote:
>< much deleted >
>: dav...@spcunb.spc.edu (David K. Davis) wrote:
>: [some deleted...]
>My position is that mathematics helps us extend our intuition - it doesn't
>obviate the need for it at any level. 2 and 3 dimensional spaces extend
>our intuition (and knowledge) about the real line, plus unifying this with
>some of our geometrical ideas and intuitions. n dimensional space takes us
>beyond these intuitions, but does not at all obviate the need for them.
If one is dealing with linear spaces over other fields, one had better
get rid of a lot of the so-called intuition.
...............
>My point is that our abstractions, ultimately, are historically conditioned.
>We do not, now can we, just arbitrarily pick a bunch of axioms and start
>exploring the consequences. We abstract on the basis of what precedes.
This is how mankind arrives at intuitions. That does not mean that this
is the way an individual should arrive at them. Unlearning is very
difficult, and very simple ways of looking at problems are extremely
hard to learn if one knows too much which has to be discarded.
Measure and integration are not that difficult provided that one abandons
the misconceptions coming from the domain being the real numbers or
Euclidean space. Probability suffered from this in the 18th and 19th
centuries, before general measure theory, when simplicity sets in. But
we still teach too much with densities, causing major confusion in
the understanding of probability and expectation.
A large number of mathematicians have great difficulty in set theory.
Part of this comes from the fact that they have an intuitive idea of
what it means for something to be a set, which must be dropped.
Alberto C Moreira
twc...@rs6000.cmp.ilstu.edu (Thomas W. Cowdery) wrote:
>Nonsense! Both 'table' and 'yellow' are very concrete concepts. The
>former is a word that represents a physical object. The latter is a
>color that any person of normal eyesight can distinguish.
I can think of several things I'd call a "table", and some are more
abstract than others. Even when we're talking about the ordinary
furniture object, what really stands in the mind for "table" is an
abstract idea that typically won't match any real table in the world.
With "yellow", it's even harder. A wide variety of colors correspond
to "yellow", and "yellow" is a perception that varies with the
ambient light. A "red" ball will look "white" when seen with a "red"
ambient light. It's only when we abstract from the sensation and go
to the mathematical model - including frequency, wavelength, hue,
saturation, intensity, chromacity, RGB, CMY or whatever, that we
can really say what "yellow" is. Obvious things usually aren't,
and the same happens to "concrete": the more intently one looks at
a concept, the less concrete it becomes.
>A child's
>language skills have very little abstraction in them. Why? Because
>the child (prodigies excepted) doesn't have the capability to conceive
>much that is abstract. Even 'love', which can be about as abstract as
>anything, has tangible effects that a child can identify as 'love'.
It's exactly the opposite way: children have a powerful capacity of
abstraction, and that unspoiled by adult concerns. A children can
feel "love" all right, what he/she cannot do is to fathom the
arbitrary meaning we give to the word itself. Because even the
spoken language is somewhat of an appendage, a front end to something
much more intuitive and basic. We - even children - can handle the
abstract very well, it's only when we have to classify it and
pigeonhole it in words that children are in disavantage.
Alberto.
Alberto C Moreira
kmic...@pen.k12.va.us (Karen Dee Michalowicz) wrote:
>I always enjoy the discussion on the "new math" because it
>demonstrates our lack of knowledge about the history of
>mathematics education, particularly in the U.S. NCTM is now in
>the process of putting together a monograph on the topic.
>
>The movement which we call "new math" did actually start
>germinating in the universities in the 50's. It started
>filtering into the classrooms during the 60's and early 70's.
>Having been in this field for 34 years, let me assure you the
>ideas were super. The problem was that the "inservice"
>component was either not there or not considered essential.
>The new standards, which reflect many of the "new math" ideas,
>have stressed the importance of inservice.
I don't know about the history of "new math" in this country;
I wasn't here then. But the statement that the NCTM standard
somehow reflects ideas that have to do with what I - and
other science professionals - mean by "new math" is plainly
not true.
The tenet behind any sort of "new math" is that mathematics
learning should be approached from its formal basis: logic,
set theory, algebraic structures. That numeric application
is far less important than non-numeric. That structure is
more important than computation. That the fundamental thing
in mathematics isn't "find the result" but "build the object".
That mathematics isn't about analysis and step by step
discovery, but about synthesis; not about understanding
concepts, but about building them from previously known ones,
through formally sound inference rules. Mathematics isn't
something to be taken apart, but a kind of Lego that must be
used to build things from scratch.
I read the NCTM materials over and over, and I see nothing
I'd myself classify as "new math".
>I now use many of the materials I learned about in the 60's
>when I worked on my first masters. Many manipulatives were
>"home made" with little expense. My luck was that I went to
>graduate school when the "new math" movement was gaining
>momentum. Again, its problem was coordination with teachers.
>There was virtually no instruction for teachers in the field.
One can't just "instruct" teachers in this. It takes years of
study to go through mathematical logic, set theory, algebraic
structures, measures, topology, integration, and other relevant
material. You can't just take any teacher and say "do it!",
nor even "train" them; a whole new educational framework is
needed as far as the teacher is concerned.
>Now, if you think that this was the first "new math" movement,
>you will be surprised. There was a movement in the early
>l800's to revise textbooks and change pedagogy. It seems if
>you go through old arithmetics from 100-200 years ago, you will
>notice significant changes about every 30 or 40 years. I have
>a super book from the turn of this century that sounds like it
>was written today. My mentor, Mary Everest Boole, b.1832
>d.1916, wrote books that were 80 years ahead of her time. Her
>books speak to groups, journals, hands-on activities, sensory
>education, historical connections, real-world problems.
Real world tends to change according to who's speaking about it.
To me - a computer geek - real world involves advanced mathematics
on a daily basis, much of which is non-numeric. I believe that
one of the important things in "new math", or at least in what I
give that label, is that arithmetic is quite irrelevant. Who
cares how much is 5 + 7, what's important is the structure of the
"+" operation, its domain and range, its properties, and the fact
that I - or any other student - can redefine "+" or overload it
to mean anything we need it to mean. In other words, the transition
from manipulating objects to manipulating meta-objects is what,
in my opinion, qualifies as "new math".
>Before we condemn a particular trend or movement, it might be
>good to have some experience with it as a teacher. Or, before
>we put something down, we might look at the historical
>perspectives. My opinions on the movements in math ed during
>my tenure as a teacher are a result of teaching 34 years, in
>and out, with an average of 35 students. (My first years I had
>in the upper 40s.) The secret is taking the best from
>everything and celebrating it in ones teaching.
When one's upstream from such trends and see how much damage
is being done by them, one can't avoid but disagree strongly.
To me, the definition of "real life" as used by today's
education establishment is utterly inadequate, and excludes most
of the areas where mathematical knowledge if of paramount
importance. So, ok, if you exclude every advanced application of
mathematics, what remains can pretty much be handled by any
half-pint theory or trend; we can afford to use easy-street
methods because the objectives have been so diluted that anything
works.
It would help every HS teacher to take a couple of years off and
go teach mathematics at college undergrad - or even graduate -
level. That's the only way to gauge how much mathematics is
needed, and of what kind. But anybody who comes and says that
my "real life" mathematical needs is fully satisfied by today's
HS teaching, immediate loses credibility as far as I'm concerned.
Alberto.
Herman Rubin
In article <4o5r5i$l...@thor.cmp.ilstu.edu>,
>>We must realize that the calculator is here to stay. Let students use it,
>>even if it's to add 2 + 3. They will, in time, develop an abstraction
>>from using it and will think differently that the way we do now. Maybe it
>>might move mathematics into a new interesting direction. The main point,
>>however, is that they have to use the calculator a lot before they can
>>grow their own abstraction onto it.
>You are missing the whole point. Using the calculator, when they
>can't do the arithmetic themselves, PREVENTS them from developing that
>abstraction. The calculator just presents the answers. To see the
>patterns behind the answers you need a degree of 'number sense' that
>comes from working with numbers.
I do not see that any more number sense is acquired by memorizing
addition and multiplication tables, and learning the rules for
multi-digit arithmetic.
It might help if the child started with the development of the
operations from counting, and had to produce the tables from
scratch. But this requires that one start with counting, and
characterize addition and multiplication in terms of that process,
which IS what I am advocating.
Otherwise, a number sense can be just as easily produced by using
the calculator to produce answers as to develop a calculator in
the brain doing the same thing.
But you don't get that from using
>the calculator, only by doing the arithmetic mentally or manually.
>That forces you to think about what is happening in a way that
>punching buttons cannot duplicate.
If we want to just achieve this, I would suggest a computer program
which does not just print out the answer, but prints out the steps.
Granted, some will never develop
>that 'number sense' even if they do the work themselves. But they
>won't get it from the calculator use either.
But it can be taught, if we teach the concepts. I suggest starting
with the ordinal, because it is self-contained. The cardinal concepts
should also be taught.
kc7cc
kc7cc wrote: You people use too many words ! We must , at an early age
, identify left and right brained students and allow them to find
their niche.
David: Aargh. What about the idea of trying to teach them all to
use both sides of their brains?
kc7cc: Costs us too much . We need to find their natural abilities
and educate them in that specialty. But 1st they must be hungry .
But most sheepskin mentality gov't teachers don't want that cause
it's means fewer teachers paychecks ( similar to cops wanting to
maintain drug laws so more cops can be employed). Hunger means they
learn faster and therefore need less teacher/hours .
This nonsense that "I have a piece of paper that says i can teach"
has got to go ! There's 10 million teachers in U.S. that don't have
a coll' deg' and in a competition they would teach same subject,
in fewer hours !
Milton Fredmans voucher idea is genius ! The secound you think your
kid isn't getting taught , you move to a different school ! Which
means that the school goes bankrupt quickly . The gov't will try to
"SAVE" that school cause it's a "GOOD" school ! But the fact it has
no students proves it aint teaching !! Private interprize is the only
way to educate .
How about the problem with propganda when gov runs the schools ?
Private schools are watching the bottom line , and anything that reduces
the bottom line isn't taught !! Like "SAY NO TO DRUGS"
Alberto C Moreira
dav...@spcunb.spc.edu (David K. Davis) wrote:
>My position is that mathematics helps us extend our intuition - it doesn't
>obviate the need for it at any level. 2 and 3 dimensional spaces extend
>our intuition (and knowledge) about the real line, plus unifying this with
>some of our geometrical ideas and intuitions. n dimensional space takes us
>beyond these intuitions, but does not at all obviate the need for them.
There cannot be any intuition in much of what's covered by mathematics. It
is actually the other way around: our intuition often leads us into false
paths and erroneous concepts that can get very hard to eradicate. Our
intuition falls so often flat on its face, that we can't use it; that's
why the turn of the century saw the emergence of formal systems. We use
mathematics to model intuitive things, not the other way around. And
when they clash, we dump intuition and stick to the formalism.
>Those intuitions guide us in exploring what's still the same in n dimensions,
>what carries over, what doesn't. And the same applies in moving to complex
>and infinite dimensional spaces. But just as the 2 and 3 dimensional real
>spaces tie togther a lot elementary geometric knowledge, and as the n
>dimensional spaces tie together a lot of knowledge about linear equations,
>the inf dimensional space tie toghter a lot of knowledge about classical
>diff eq and integral eqs - fourier series, etc. (Never mind if I'm wrong
>on this or that detail.)
It's not that n-dimensional spaces tie together a lot of mathematical
knowledge, it's exactly the reverse process: there's a lot of mathematical
knowledge that can be applied to model our intuition about multi-dimensional
spaces. Mathematics doesn't need intuition, but intuition needs mathematics.
Besides, there are spaces and spaces; many of which can be quite beyond
our intuition (take, for example, an infinite-dimensional space). Actually,
the same knowledge about multi-dimensional intuitive spaces can be used in
far less intuitive circumstances, which would be totally outside our
reach weren't for the generality of mathematical formalism and its total
independence from intuition.
>My point is that our abstractions, ultimately, are historically conditioned.
>We do not, now can we, just arbitrarily pick a bunch of axioms and start
>exploring the consequences. We abstract on the basis of what precedes.
>Of course, we often say, what happens if we loosen this requirement, etc.
Yet that's exactly what the so-called "modern" mathematics does, and that's
the only way to approach mathematics that holds widespread validity. Yes,
we pick up a bunch of axioms and develop a theory by formalizing not only
the axioms, but the allowed rules of inference as well. That's how, for
example, we derive functional programming from the basic Lambda Calculus.
Also, it used to be the case that the only reality available to us was the
intuitive one. Today, with computers at every desk, alternative realities
aren't but a few keystrokes away. There's no need to stick to intuition,
and all the incentive to go search for greener pastures.
>Something went wrong in the 50s and 60s, maybe even before. I have a lot
>of books on my shelves from that period that just go like a bat out of
>hell - def,def,lemma,thm,lemma,def,thm, etc. The exercises might condescend
>to dropping a hint or two as to where it all came from. But there is
>precious little motivation, precious little acknowledgment of the actual
>historical route of development, of what was abstracted from.
Yet this is exactly what mathematical knowledge is about. Mathematics is about
axioms, rules of inference, lemmas, theorems. Concepts emerge from chains of
inference. Mathematical objects are built from more elementary ones. Theories
aren't "right" or "wrong", but "consistent" or "complete"; a theory that
closely follows the objective world can be deemed "adequate", but if it doesn't
it's not necessarily "wrong".
The motivation of mathematics is is universal applicability. If I have, say,
a pair of vectors, the same vector product can be used in computer graphics,
quantum mechanics, fluid flow, electrical field modelling. If I have a function,
I can build the whole universe with that one concept; I don't even need numbers,
I can build numbers out of functions. The same mathematics that allows me to
multiply two numbers allows me to multiply two abstract objects, as long as I
properly overload my multiplication; the whole field of object oriented programming
is based on this. The same set theory that provides me with functions, continuity
and differential calculus, also gives me models of programming recursion and of
abstract computing machines.
The march from "concrete" to "abstract" didn't quite take place; what happened was
that the "concrete" left far too many things unexplained, and people had to devise
much better ways of building up mathematics. It's not that "numbers led to
abstraction", but that numbers were so ill-defined that people had to raze the
old ways out and restart fresh; so, today, what we call number, although keeping
all of the traditional properties, has precious little to do with what people used
to call number before, say, Cantor came to be. If you take, say, Russell's
definition of number that says that (f.ex.) "the number seven is the set comprised
of its six predecessors plus the set of those predecessors", you'll agree with
me that there's nothing intuitive in this. Even worse, look at Church numerals
and tell me if intuition plays any part; in both cases, numbers haven't been
built to be intuitive, but to satisfy Peano's axioms.
>Abstraction is historical, it is layered, it is relative, it has a
>context. I don't deny that there has a been a great deal of exploration
>of variants of historically presented abstract systems, explorations
>perhaps motivated only by a desire to fill in the holes. But the number
>of abstract systems is vast, and even the variants which are explored for
>their own sakes get explored because the are contiguous to something
>historically conditioned. And those that get a lot of interest will
>be those that are fruitful in unifying other stuff, i.e. turn out to
>be abstractions of pre-existing material after all.
There are a few concepts that needed formalism far beyond what classical
mathematics could deliver. That's why abstract systems came to be. But
once we have them, people found out that the old vision is no longer
necessary. There's a discontinuity in the history of mathematics; the
evolution of the old, intuitive approach was almost dropped, and the
modern, formal systems way took over.
In many cases, abstract systems stay dormant for a long time, until someone
bumps into something that reawakens them. Look at the Lambda Calculus, for
example, it's now a cornerstone of good computer programming. Tarski published
his Fixed Point Theorem paper in 1955; it took the modern computer age to
find its dramatic use in modelling recursion. If we only look at what we can
reach with intuition, we don't need mathematics; and like Tolkien said, seeds
lay dormant for a long time, and germinate often in places and times unlooked
for.
>I think that mathematics has gone a little bit (well, quite a bit really)
>nuts with some of this, and tries too pretend that this is not so. There
>is a book by R. Smullyan, TO MOCK A MOCKINGBIRD, which presents combinators
>franky as an amusement, as a source of puzzles, an abstract game. He does
>give a hint later as to some of the "applications" and history, but not
>much. It's entertainment and is so presented. But the books I'm talking about
>are dead serious, and deadly dry, and dead. So much suffering could be
>relieved so easily, but they do not, they will not compromise their
>abstractions with mention of where it all came from.
"Fun" is not a prerequisite for science. Neither it is for knowledge. But the
sort of things that go behind scientific papers is neither dry nor dead; what
they do require, though, is knowledge, effort and time to be digested. When
it comes to mathematical learning, there's no easy street.
Lambda Calculus combinators, by the way, are worth more than puzzles: we
can actually build abstract - or concrete - computing machines whose
instruction sets are made up exclusively of combinators.
>Bottom line, pedagogically, teaching or learning, we can't just skip the
>historical roots of math, we can't just proceed to the top levels of
>abstraction, present this top level the to students, and neglect all the
>layers of suffering that went before. And in the basement there needs
>to be an arithmetic intuition that is based on considerable exercise
>as well as a geometric intuition also based on considerable exercise.
I don't agree with this at all. The "roots" of mathematics can just as much
be forgotten, at least most that went on before 1870. Arithmetic intuition
today is a slave of something bigger, which is how to represent it inside
a computer. Geometric intuition became a side-effect of algebraic vision
since Descartes invented Analytic Geometry. And today with computers, what
we call "computational geometry" and "affine geometry" is much more important
than the traditional geometry as it is taught in high school. A healthy dose
of Beziers and Splines, for example, is much more important than developing
traditional geometric intuition.
>That's where I disagree. Abstraction is relative, and the direction of
>movement is from the concrete to the abstract, but there are layers. The
>computer implements abstractions - data types and algorithms. These
>abstractions are, it's true, but a small subset of the abstraction that
>mathematics in its whole deals with.
The direction must be from the abstract to the concrete. If I know how to
integrate and I can model a problem with integrals, I can solve the
problem. But if I use the problem to learn integrals, I won't be able to
use integrals in a field that's markedly different from the problem at
hand. Going even further: an integral, an inner product of vectors,
they're just two particular cases of one structure: a sum of products.
If I know that, and I have my basic abstract skills in place, I can
twist the concept into any area of application I care to; it doesn't
matter, for example, if I face a finite sum in a computer algorithms
course, or an integral in elementary physics, or a surface or volume
integral in fluid flow, or a scalar product defined as an integral over
an n-dimensional space in quantum mechanics: it's exactly the same
concept, and I need to learn zero additional mathematics to go from one
to the other. But if somebody imprints in my mind that an integral is
the area under a curve, I'll have a heck of a problem trying to
comprehend how the same concept is used in Quantum Mechanics, or how
a discrete sum is exactly the same as an integral, or that there's no
essential difference between a continuous Fourier Transform and a
discrete one. Furthermore, me the computer person can say that who
cares how one defines "sum" and "product": as long as these two
operations satisfy certain basic structural requirements, I can
replace them with a wide range of other operations or even replace
the vector space with something else, and still the mathematics is
exactly the same.
>I do agree with the importance of abstraction - that's the power of
>mathematics. But abstraction can be undermined in TWO ways, not just one.
>It can be undermined by refusing to recognize its usefulness, by refusing
>to move beyond the concrete, by refusing to recognize the legitimacy of
>purely abstract explorations of mathematical objects.
>But it can also be undermined by forgetting the relativity of abstraction,
>by forgetting that abstraction is abstraction from something, that
>abstraction is historical, by forgetting - please forgive me - the dialectical
>interaction of the abstract and the concrete. Abstraction can be undermined by
>our getting lost in abstraction, by forgetting that we have fingers and that
>we move in space and that our loftiest abstractions trace back to these
>simple facts.
No, this was maybe in the 19th century, but not today. Today the path is exactly
the opposite: we start at the abstract, we learn the abstract, and then we
apply it. We learn the mathematics, and let the physics, chemistry, computer
science professors to do the applied part. There's so much math to be learned,
and so little time, that we can't afford to do it the old way.
Alberto.
Karen Dee Michalowicz
Alberto C Moreira (Alberto C Moreira ) writes:
>
> I don't know about the history of "new math" in this country;
> I wasn't here then. But the statement that the NCTM standard
> somehow reflects ideas that have to do with what I - and
> other science professionals - mean by "new math" is plainly
> not true.
>
If you dispute a concept, or in this case a movement, it is
expected that you have some information about what you are
criticizing. As Alberto says above, he doesn't know about the
"new math" (movement) about which I wrote. Therefore, since I
was equating the standards to this movement, about which he
states he knows nothing, he logically shouldn't dispute what I
said.
I welcome any positive or negative discussion of my comments
based on knowledge.
>
---
Cheers!
Karen Dee
Math History Lives!
Thomas W. Cowdery
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>>You are missing the whole point. Using the calculator, when they
>>can't do the arithmetic themselves, PREVENTS them from developing that
>>abstraction. The calculator just presents the answers. To see the
>>patterns behind the answers you need a degree of 'number sense' that
>>comes from working with numbers.
>I do not see that any more number sense is acquired by memorizing
>addition and multiplication tables, and learning the rules for
>multi-digit arithmetic.
I do. In fact, I see a noticeable difference in my students when I
just make them do the arithmetic by hand for the first semester of
algebra. And I know that they are using the calculator at home, when
I'm not around. I can only speculate what kind of difference that
would be possible if they didn't get hooked on the darn things in the
first place. Breaking the calculator addiction is harder than
avoiding it.
>It might help if the child started with the development of the
>operations from counting, and had to produce the tables from
>scratch. But this requires that one start with counting, and
>characterize addition and multiplication in terms of that process,
>which IS what I am advocating.
Well, we are in agreement here. I know that is primarily how I was
taught. Naturally, there was drill and memorization too. But sheer
repitition is irreplaceable. Recently, while teaching an introductory
lesson on square roots, I had the kids create a table of squares from
1^2 to 25^2. The kids asked me if I knew all of them by heart.
Naturally I did, but I explained to them that I hadn't memorized all
of them. I memorized the lower ones as a child while learning my
times tables. The others I had just used enough over the years that I
now knew them. That still isn't 'number sense', but it helps lead to
it. The person who lacks that, has one more impediment to overcome
when trying to develop 'number sense'.
>Otherwise, a number sense can be just as easily produced by using
>the calculator to produce answers as to develop a calculator in
>the brain doing the same thing.
Nonsense. Using a calculator requires little if any thought. Number
sense cannot be developed or acquired without thought.
> But you don't get that from using
>>the calculator, only by doing the arithmetic mentally or manually.
>>That forces you to think about what is happening in a way that
>>punching buttons cannot duplicate.
>If we want to just achieve this, I would suggest a computer program
>which does not just print out the answer, but prints out the steps.
But the person still doesn't have to think. Doing the mental or
manual arithmetic requires the brain to act and react. Those mental
gymnastics are what lead to learning and understanding. Having a
calculator or computer program do the work for you isn't the same.
*Writing* the computer program that shows the steps will do the job,
but you'd need to understand the manual process first to write the
program.
> Granted, some will never develop
>>that 'number sense' even if they do the work themselves. But they
>>won't get it from the calculator use either.
>But it can be taught, if we teach the concepts. I suggest starting
>with the ordinal, because it is self-contained. The cardinal concepts
>should also be taught.
No it cannot be taught. It can be learned, but it cannot be taught.
All a teacher can do is provide the learning experiences necessary for
that understanding to grow, much like a farmer provides the means for
a seed to grow, but doesn't really make it grow.
Herman Rubin
In article <4o5r9b$m95$1...@mhafc.production.compuserve.com>,
I have little to comment on this, except to stress that:
Concepts CAN be taught, and earlier than most think,
by teaching them, and not just the rote.
For someone to teach conceptual material, on has
to understand it. I would be surprised if most high school
mathematics teachers know what these students were expected
to learn.
At lower levels, the problem is far worse. The
proportion of elementary teachers who can teach conceptual
material is very low. Those who can learn it can manage
moderately well if well written material stresses it, even
if they have not learned it before. But most can not.
This problem is not going to be sensibly addressed by taking
the attitude that what the teachers cannot teach should not
be taught; this makes it harder for children to understand,
and perpetuates the problem.
Brian M. Scott
In article <Ds0JM...@pen.k12.va.us>, kmic...@pen.k12.va.us
(Karen Dee Michalowicz) responds to Alberto C Moreira, who had written:
I don't know about the history of "new math" in this country;
I wasn't here then. But the statement that the NCTM standard
somehow reflects ideas that have to do with what I - and
other science professionals - mean by "new math" is plainly
not true.
>If you dispute a concept, or in this case a movement, it is
>expected that you have some information about what you are
>criticizing. As Alberto says above, he doesn't know about the
>"new math" (movement) about which I wrote. Therefore, since I
>was equating the standards to this movement, about which he
>states he knows nothing, he logically shouldn't dispute what I
>said.
He didn't. He disputed 'the statement that the NCTM standard somehow
reflects ideas that have to do with what [he] - and other science
professionals - mean by "new math"'. From this you may fairly safely
infer that what he means by 'new math' is different from what you mean
by the term, that's all.
Brian M. Scott
Kevin Anthony Scaldeferri
In article <4o5rf2$l...@thor.cmp.ilstu.edu>,
>
>>In article <DrqoJ...@spcuna.spc.edu> David K. Davis,
>>dav...@spcunb.spc.edu writes:
>>>... We abstract from what's inside us.
>
>>I disagree. I think we abstract to understand and to communicate.
>
>>Language is already abstract. To recognise what is being
>>discussed from the sounds heard is a tour de force of
>>abstraction. To speak of "yellow" or "table" pre-supposes
>>sophisticated abstraction.
>
>former is a word that represents a physical object. The latter is a
>color that any person of normal eyesight can distinguish.
You are both right, but in different ways. Take 'yellow'. There is a
concrete notion of yellow and an abstract notion of yellow. To teach
a child the concrete notion, point at things that are yellow. To teach
the abstract notion, you need electricity and magnetism, Maxwell's
equations, waves, wavelengths, all in all, several years before you
can hope to 'understand' yellow. And I ask you, which one will be
more useful to most students?
Kevin
Herman Rubin
In article <4o9u5h$1p...@thor.cmp.ilstu.edu>,
>>>You are missing the whole point. Using the calculator, when they
>>>can't do the arithmetic themselves, PREVENTS them from developing that
>>>abstraction. The calculator just presents the answers. To see the
>>>patterns behind the answers you need a degree of 'number sense' that
>>>comes from working with numbers.
Are they getting the abstraction now? I would instead suggest that
they be asked to formulate equations, relations, etc., numerical and
non-numerical, using lots of variables, without doing such things as
using x, y, etc., for "unknowns" and a, b, etc., for "parameters".
Avoid having them formulate in one variable problems to be solved,
unless there is only one variable involved.
Initially, have them list the steps they are doing. You may point
out that later they will combine steps, but to evaluate their algebra,
you need to see their algebraic thinking, and not the solution, which
might be obtained by educated guessing.
>>I do not see that any more number sense is acquired by memorizing
>>addition and multiplication tables, and learning the rules for
>>multi-digit arithmetic.
>I do. In fact, I see a noticeable difference in my students when I
>just make them do the arithmetic by hand for the first semester of
>algebra. And I know that they are using the calculator at home, when
>I'm not around. I can only speculate what kind of difference that
>would be possible if they didn't get hooked on the darn things in the
>first place. Breaking the calculator addiction is harder than
>avoiding it.
And maybe that "number sense" represents itself in educated guessing,
which defeats the purpose of the algebra course. I do not mean that
educated guessing is not useful, but if it prevents learning the
concepts, it is not good.
>>It might help if the child started with the development of the
>>operations from counting, and had to produce the tables from
>>scratch. But this requires that one start with counting, and
>>characterize addition and multiplication in terms of that process,
>>which IS what I am advocating.
>Well, we are in agreement here. I know that is primarily how I was
>taught. Naturally, there was drill and memorization too. But sheer
>repitition is irreplaceable. Recently, while teaching an introductory
>lesson on square roots, I had the kids create a table of squares from
>1^2 to 25^2. The kids asked me if I knew all of them by heart.
>Naturally I did, but I explained to them that I hadn't memorized all
>of them. I memorized the lower ones as a child while learning my
>times tables. The others I had just used enough over the years that I
>now knew them. That still isn't 'number sense', but it helps lead to
>it. The person who lacks that, has one more impediment to overcome
>when trying to develop 'number sense'.
Again, what does "number sense" have to do with algebra? And what is
"number sense"? Proficiency in base 10 arithmetic certainly is not that.
>>Otherwise, a number sense can be just as easily produced by using
>>the calculator to produce answers as to develop a calculator in
>>the brain doing the same thing.
>Nonsense. Using a calculator requires little if any thought. Number
>sense cannot be developed or acquired without thought.
>> But you don't get that from using
>>>the calculator, only by doing the arithmetic mentally or manually.
>>>That forces you to think about what is happening in a way that
>>>punching buttons cannot duplicate.
>>If we want to just achieve this, I would suggest a computer program
>>which does not just print out the answer, but prints out the steps.
>But the person still doesn't have to think. Doing the mental or
>manual arithmetic requires the brain to act and react. Those mental
>gymnastics are what lead to learning and understanding. Having a
>calculator or computer program do the work for you isn't the same.
>*Writing* the computer program that shows the steps will do the job,
>but you'd need to understand the manual process first to write the
>program.
In that case, have the student tell the computer precisely what steps
to take. You are assuming that having the tables memorized will produce
this sense, but I can see no reason for it. And while what I do is
facilitated by using arithmetic, that of most of my colleagues is not,
and the arithmetic is still just a tool.
>> Granted, some will never develop
>>>that 'number sense' even if they do the work themselves. But they
>>>won't get it from the calculator use either.
>>But it can be taught, if we teach the concepts. I suggest starting
>>with the ordinal, because it is self-contained. The cardinal concepts
>>should also be taught.
>No it cannot be taught. It can be learned, but it cannot be taught.
>All a teacher can do is provide the learning experiences necessary for
>that understanding to grow, much like a farmer provides the means for
>a seed to grow, but doesn't really make it grow.
Herman Rubin
In article <4oa2r3$4...@perl.eng.umd.edu>,
Kevin Anthony Scaldeferri <cool...@Glue.umd.edu> wrote:
>In article <4o5rf2$l...@thor.cmp.ilstu.edu>,>><ibo...@metz.une.edu.au> wrote:
There is no such thing as a concrete notion. Any notion is formed in
the brain, and at this time, we have little idea of how this is done.
When we communicate, it is at best brain to brain. To clarify the
communication, it is put into a restricted symbolism. As we have
found, it is much easier to communicate with a restricted "digital"
character set. While we lose bandwidth, we gain clarity.
Is "yellow" a concept"? How does one decide if something is yellow?
What are the boundaries of yellow? Purdue's colors are black and gold.
Is the gold color "yellow"? Is yellow the color of an object? No, for
if an object is viewed in sodium light, it will look yellow or black,
even if red or orange or green.
Ted Hirsch
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>I have little to comment on this, except to stress that:
>
> Concepts CAN be taught, and earlier than most think,
>by teaching them, and not just the rote.
>
> For someone to teach conceptual material, on has
>to understand it. I would be surprised if most high school
>mathematics teachers know what these students were expected
>to learn.
>
> At lower levels, the problem is far worse. The
>proportion of elementary teachers who can teach conceptual
>material is very low. Those who can learn it can manage
>moderately well if well written material stresses it, even
>if they have not learned it before. But most can not.
>
>This problem is not going to be sensibly addressed by taking
>the attitude that what the teachers cannot teach should not
>be taught; this makes it harder for children to understand,
>and perpetuates the problem.
>--
Herman,
Can you suggest any starting points (books, journals, etc.)
that an interested elementary teacher can go to, assuming
he has a strong conceptual understanding of mathematics?
You may assume that this teacher has been through the
standard compliment of arithmetic and mathematics courses,
including calculus. This teacher was bored by the repetitive
and lengthy calculations--but not the concepts--presented.
This teacher has already read some works in this area,
such as "About Teaching Mathematics" by Marilyn Burns.
Please suggest others.
Note to others: I'm not interested in anyone's demagoguery.
If you have nothing to add to the reading list, just keep
your mouth closed. Saxon people, especially, this means you.
| Long Island, New York http://www.nyiq.net/~ted
_|_ _ __| "There is no expedient a man will not go to avoid
| |/ / | the real labor of thinking." -Thomas Alva Edison
|_/|__/\_/|_/ Disclaimer: Everyone shares my views. So there.
Herman Rubin
>Herman,
This is a somewhat rambling account, but I do not know of a real
organized way of doing it from existing books.
Also, I will have to take the position that the strong conceptual
understanding may not be there. This is not an insult to the
individual; I do not expect you to know what you do not even know
exists, or which you have been poorly taught. I would like to
say that most textbooks convey the concepts, but I must say quite
the contrary.
The situation is not the best, as there has been major resistance
to the teaching of concepts for some time. But there are some old
materials, some of which can even be used for teaching children.
For those teaching mathematics above the elementary level, I
consider the ability to do all of this essential.
For logic and the use of variables, the book _Introduction to Logic_
by Suppes, which is intended for college freshment, but which has
even been used with good 6th graders. The third chapter, on
variables, is good and quite readable. It would be difficult
for children to use without help. The book, _First Course in
Mathematical Logic_, by Suppes and Hill, is intended for the
upper half of 5th graders; in some places, I find the supposedly
more advanced book actually easier. The first part is the part
on logic; I have not looked at it enough recently to comment on
the use of the later parts of it. The part on existential
proof in Suppes is something I would not recommend. There are
other books on this. My children learned from these as early
as they were capable of attacking it, but I am inclined to
doubt that it could have been done at that age without my help.
A good really old-fashioned algebra book would not be bad.
But almost all of them emphasize solution, and also ask for
formulation in few variables. The latter should be resisted
for beginners.
These can be adapted for children. I have recommended that
the idea of variables as pronouns be started very early, and
that everything be done using that as NOTATION. Some have
complained, for example, about the lack of clarity in Euclid's
statements of propositions. They are right; what is lacking
is the necessary notation. In principle, the notation is not
needed, but it is very difficult to express precise information
clearly without such.
The similarity of our multi-place numerical notation with polynomials
is evident to someone who has access to that notation. it almost
trivializes the arithmetic of multi-place numbers. It also makes
it easier to see that the arithmetic operations are only means of
obtaining the answers, and not the mathematics of the basic
arithmetic operations. The algebra "lays out" the arithmetic,
but does not need it, except to solve numerical problems.
For the understanding of the structure of the various number
systems, _Foundations of Analysis_ by Landau is a small concise
book. It is actually too concise, as he wanted to do it for
those who had some understanding with the minimum of difficulty;
the order of development is not universal, and 0 is deliberately
not introduced until late. The initial part, on the development
of the positive integers, is almost readable by sufficiently
literate children. There are some errors, but they are avoided
by having addition and multiplication characterized, rather than
defined. I believe that everyone who teaches mathematics at any
level should be required at least to be able to handle this
development. Anyone who had difficutly with the concept of
induction does not understand the integers.
The development of the positive rationals is also good. Fractions
are much more easily understood using simple formalism than trying
to invoke intuition.
The idea of infinite decimals is usually taught in elementary school;
this brings in the completeness of the real numbers. Whether this is
done as in Landau (short but not necessarily sweet) or in other ways,
it is important. There are no "holes" in the number line.
Once can find books, including books for teachers of mathematics, which
have much of this in a more concise form, but beware that it is the
concepts which are taught, and not just definitions, theorems, and proofs.
I suggest also that one avoid too much use of definitions, especially
"cute" ones. I suggest that characterizations be used instead. The
difference is that something can have only one definition, but as many
characterizations as are convenient. If they are valid characterizations,
they are all equivalent.
Terry Moore
In article <Dry38...@spcuna.spc.edu>, dav...@spcunb.spc.edu (David K.
Davis) wrote:
> Bottom line, pedagogically, teaching or learning, we can't just skip the
> historical roots of math, we can't just proceed to the top levels of
> abstraction, present this top level the to students, and neglect all the
> layers of suffering that went before. And in the basement there needs
> to be an arithmetic intuition that is based on considerable exercise
> as well as a geometric intuition also based on considerable exercise.
This is exactly where the source of disagreement lies. Some may argue
that the motivation can be misleading because the motivating concepts
have special features omitted from the abstract system. Others counter
this by giving many different motivations.
But this is much less a source of disagreement than that over
arithmetical exercises. Certainly it is important to have arithmetic
intuition. The question is whether algorithms help develop this.
How much intuition can one develop while struggling with the
details of a particular algorithm? (And the usual algorithm is
not very efficient either). Before calculators we had no choice,
but now we do. Are we going to continue to teach things that
machines can do better, or encourage the ability to solve
problems while leaving the trivial to machines?
BTW, it is well known that the things we do the most easily
are very complicated, while simple things we can only do
with great difficulty. For example we soon learn to recognise
faces, but to make computers do so is cutting edge research.
OTOH computers could do arithmetic from the beginning,
but it takes years to train students to do it, and some never learn.
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan
ibo...@metz.une.edu.au
Subject: Re: When was the new math developed?
From: Karen Dee Michalowicz, kmic...@pen.k12.va.us
Date: Sun, 26 May 1996 12:49:41 GMT
In article <Ds0JM...@pen.k12.va.us> Karen Dee Michalowicz,
>> I wasn't here then. But the statement that the NCTM standard
>> somehow reflects ideas that have to do with what I - and
>> other science professionals - mean by "new math" is plainly
>> not true.
>>
>
>If you dispute a concept, or in this case a movement, it is
>expected that you have some information about what you are
>criticizing. As Alberto says above, he doesn't know about the
>"new math" (movement) about which I wrote.
If you criticise a contribution, do at least read what it contains.
The previous cntributor did *not* write that 'he doesn't know about
the
"new math" (movement)', he wrote that he doesn't know about its
*history*.
That does not preclude knowing its content or philosophy.
After all, how many teachers of mathematics know the history of
their subject?
> Therefore, since I
>was equating the standards to this movement, about which he
>states he knows nothing, he logically shouldn't dispute what I
>said.
>
>I welcome any positive or negative discussion of my comments
>based on knowledge.
There has been nothing in his postings to suggest that
Alberto C Moreira is unaware of either the aims and content
of "new maths" or the current state of (mathematical)
"education".
d.A.
David K. Davis
Rarely does one find a post that so perfectly, so directly, and so clearly
expresses an idea that one opposes as Alberto has done for me here. It
takes my breath away, because, to me, so plainly put it is so evidently and
so clearly wrong. Why? Because the picture it paints of mathematics is one
of impossible purity and abstraction - and his argument could not possibly
convince anyone - EXCEPT - I realize at last - a mathematician. And a
mathematician is invulnerable to any argument against it because at a
certain level of abstraction, abstraction sucks up the very earth beneath it,
the earth which gave birth to it, and lifts off into the sky, nary a trace
of it origins and sustenance visible to the naked eye.
But I shall try, I shall try to a revive a remembrance of that earth.
Alberto C Moreira (alb...@moreira.mv.com) wrote:
NOR (type in my original)
: >exploring the consequences. We abstract on the basis of what precedes.
: >Of course, we often say, what happens if we loosen this requirement, etc.
: Yet that's exactly what the so-called "modern" mathematics does, and that's
: the only way to approach mathematics that holds widespread validity. Yes,
: we pick up a bunch of axioms and develop a theory by formalizing not only
: the axioms, but the allowed rules of inference as well. That's how, for
: example, we derive functional programming from the basic Lambda Calculus.
But I really don't believe you can make the case that we just pick a bunch
of axioms and start doing deductions. The axioms for groups, for Hilbert
space, for Lamda Calculus, these didn't just drop out of the sky. They
were arrived at after the fact - they formalized prior concepts. The
Lambda Calculus was an abstract study of substitution. These particular
formal systems are interesting because of a lot of history benind them -
they condense a lot of history. Where do we just sit down and start
writing down axioms with no prior motiviation, no prior concerns? Yes,
sometimes having an axiom system, we may vary this or that axiom to see
where it takes us. Even today, mathematics is not just randomly generating
axioms sytems and giving them a spin. Who does that?
: Also, it used to be the case that the only reality available to us was the
: intuitive one. Today, with computers at every desk, alternative realities
: aren't but a few keystrokes away. There's no need to stick to intuition,
: and all the incentive to go search for greener pastures.
I dare say that a great deal of effort has been spent in connecting the
computer with our prior intuitions. We are unable to use them without that
connection. And they have extended our intuition - our intuition is not a
fixed thing. There ARE objects that can't be manipulated with pen and paper,
fractals and all kinds of other stuff to wit.
: >Something went wrong in the 50s and 60s, maybe even before. I have a lot
: >of books on my shelves from that period that just go like a bat out of
: >hell - def,def,lemma,thm,lemma,def,thm, etc. The exercises might condescend
: >to dropping a hint or two as to where it all came from. But there is
: >precious little motivation, precious little acknowledgment of the actual
: >historical route of development, of what was abstracted from.
: Yet this is exactly what mathematical knowledge is about. Mathematics is about
: axioms, rules of inference, lemmas, theorems. Concepts emerge from chains of
: inference. Mathematical objects are built from more elementary ones. Theories
: aren't "right" or "wrong", but "consistent" or "complete"; a theory that
: closely follows the objective world can be deemed "adequate", but if it doesn't
: it's not necessarily "wrong".
The formalization is of concepts. The concepts don't grow out of chains
of inference. The chains of inference are bridges we build from both ends,
from the axioms to the propostion. We use all kinds of things to guide
in finding those chains. It's the last thing we do. We don't wade out from
the axioms, adding a link in our chain at each step, seeing what we'll get
next - wow! so x^n + y^n = z^n for n > 2 - boy, am I lucky.
: The motivation of mathematics is is universal applicability. If I have, say,
: a pair of vectors, the same vector product can be used in computer graphics,
: quantum mechanics, fluid flow, electrical field modelling. If I have a function,
: I can build the whole universe with that one concept; I don't even need numbers,
: I can build numbers out of functions. The same mathematics that allows me to
: multiply two numbers allows me to multiply two abstract objects, as long as I
: properly overload my multiplication; the whole field of object oriented programming
: is based on this. The same set theory that provides me with functions, continuity
: and differential calculus, also gives me models of programming recursion and of
: abstract computing machines.
: The march from "concrete" to "abstract" didn't quite take place; what happened was
: that the "concrete" left far too many things unexplained, and people had to devise
: much better ways of building up mathematics. It's not that "numbers led to
: abstraction", but that numbers were so ill-defined that people had to raze the
: old ways out and restart fresh; so, today, what we call number, although keeping
: all of the traditional properties, has precious little to do with what people used
: to call number before, say, Cantor came to be. If you take, say, Russell's
: definition of number that says that (f.ex.) "the number seven is the set comprised
: of its six predecessors plus the set of those predecessors", you'll agree with
: me that there's nothing intuitive in this. Even worse, look at Church numerals
: and tell me if intuition plays any part; in both cases, numbers haven't been
: built to be intuitive, but to satisfy Peano's axioms.
Without question, the program of formalization led us to go back and look
at old material in a different light. It's fun to go back and do all those
reconstructions. But the various pieces of mathematics were quite real
and quite vital prior to those reconstructions, and we do not think about
Peano too much when doing number theory or whatever else we're doing.
: >Abstraction is historical, it is layered, it is relative, it has a
: >context. I don't deny that there has a been a great deal of exploration
: >of variants of historically presented abstract systems, explorations
: >perhaps motivated only by a desire to fill in the holes. But the number
: >of abstract systems is vast, and even the variants which are explored for
: >their own sakes get explored because the are contiguous to something
: >historically conditioned. And those that get a lot of interest will
: >be those that are fruitful in unifying other stuff, i.e. turn out to
: >be abstractions of pre-existing material after all.
: There are a few concepts that needed formalism far beyond what classical
: mathematics could deliver. That's why abstract systems came to be. But
: once we have them, people found out that the old vision is no longer
: necessary. There's a discontinuity in the history of mathematics; the
: evolution of the old, intuitive approach was almost dropped, and the
: modern, formal systems way took over.
As a matter of fact, there's truth to what you say. It leaves open
the question of desirability.
: In many cases, abstract systems stay dormant for a long time, until someone
: bumps into something that reawakens them. Look at the Lambda Calculus, for
: example, it's now a cornerstone of good computer programming. Tarski published
: his Fixed Point Theorem paper in 1955; it took the modern computer age to
: find its dramatic use in modelling recursion. If we only look at what we can
: reach with intuition, we don't need mathematics; and like Tolkien said, seeds
: lay dormant for a long time, and germinate often in places and times unlooked
: for.
Again, intuition is not a fixed thing, but something that develops and
matures with training and exercise.
: >I think that mathematics has gone a little bit (well, quite a bit really)
: >nuts with some of this, and tries too pretend that this is not so. There
: >is a book by R. Smullyan, TO MOCK A MOCKINGBIRD, which presents combinators
: >franky as an amusement, as a source of puzzles, an abstract game. He does
: >give a hint later as to some of the "applications" and history, but not
: >much. It's entertainment and is so presented. But the books I'm talking about
: >are dead serious, and deadly dry, and dead. So much suffering could be
: >relieved so easily, but they do not, they will not compromise their
: >abstractions with mention of where it all came from.
: "Fun" is not a prerequisite for science. Neither it is for knowledge. But the
: sort of things that go behind scientific papers is neither dry nor dead; what
: they do require, though, is knowledge, effort and time to be digested. When
: it comes to mathematical learning, there's no easy street.
Fun IS A prerequisite for science, mathematics in particular. No one in their
right mind would do this for any other reason (unless maybe something closely
related - like ego, ambition, burning curiosity). Money, health, character-
building? I don't think so. Effort and time are not forthcoming without
one of the varieties of fun I mentioned above. And there are few forms of
fun that don't put you thru a certain amount of shit in pursuit thereof -
if that's what you mean.
: Lambda Calculus combinators, by the way, are worth more than puzzles: we
: can actually build abstract - or concrete - computing machines whose
: instruction sets are made up exclusively of combinators.
I know - I played with combinators for a year or so. And I gradually
developed a little bit of a feel for them, thru playing with them. As
well as a way to picture them sort of (function applications). And I got
help in that by looking up a little of the history.
: >Bottom line, pedagogically, teaching or learning, we can't just skip the
I don't believe we learn it. I believe we go through it, or it goes through
us, but I'm not sure we learn it. To me, if I see a solution, I want to
know what the problem was. What good is a solution without a problem?
The solution is meaningless to me without the problem. What does this
or that definition capture? What does this axiom system formalize? What
prior body of stuff does it tidy up?
I once had an older friend who was in the hospital - something had gone
wrong. She was speaking gramatically and syntactically correct sentences.
But there was no meaning, or none that I could get. It was gibberish. There
was a disconnection of the sentence generating mechanism from reality.
The sentences were flowing, but the connection was broken. I suppose if
I had waited long enough, some pearl or another might have dropped from
her lips. But had I waited that long, I don't know that I would have been
connected enough myself to recognize it. We can't do mathematics this way.
Formalization is not an end unto itself. It's a consisteency check, yes,
very important. Abstraction is not and end unto itself. (Abstraction, BTW,
is not the same as formalization.) Abstraction doesn't exist without the
concrete (relatively). Abstraction unites diverse concepts and fields.
When it does, it's good. When it doesn't, its useless.
It's an illusion to think that we have freed ourselves from history,
from the concrete, from applications, and so on. A lot of mathematics
is abstracted from other parts of mathematics, but at some point the
outer circle connects with reality and history in one way or another.
Take any modern mathematical concept and trace its roots back, layer
by layer. It can be done. There's none that just hangs there in midair.
(SOME of this tracing back might be useful padagogically)
Can one learn modern mathematics, formally, abstractly, sullenly I am
tempted to say - gobble it down, chew it a little, and take it back and
apply it to the less lofty parts of mathemtics and even reality without any
particular trouble? Or can it only be inflicted on yet another generation?
-Dave D.
Karen Dee Michalowicz
Help, I must be reading these postings differently than some
others.
When I made my posting earlier I equated the "New Math"
movement of the 60's and 70's with the present movement
influenced by the NCTM Standards (incidently issued in l989).
I used my many years of experience and my informed readings to
write my posting. My discussion was taken out of this
context. I repeat I would be glad to discuss
the New Math movement of the 60's
and 70's with what is being encouraged by the National Council
of Teachers of Mathematics today.
However, it does appear some people enjoy the exercise of
comparing apples and oranges.
Anyone may discuss any topic they choose. They may title the
topic anything they choose. However, when one discusses
something specific it is expected that there is some knowledge
about the topic. Alberto gave his meaning of "new math".
That's fine and much appreciated. Again, this was not what I
was discussing.
Starfollower
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>In article <4o5r5i$l...@thor.cmp.ilstu.edu>,
>Thomas W. Cowdery <twc...@rs6000.cmp.ilstu.edu> wrote:
>>ed...@netcom.com wrote:
>>>We must realize that the calculator is here to stay. Let students use it,
...<snipped>...
>>> is that they have to use the calculator a lot before they can
>>>grow their own abstraction onto it.
>
>>You are missing the whole point. Using the calculator, when they
>>can't do the arithmetic themselves, PREVENTS them from developing that
>>abstraction.
...<snipped>...
>
>I do not see that any more number sense is acquired by memorizing
>addition and multiplication tables, and learning the rules for
>multi-digit arithmetic.
>
In theory, I think it is possible for a child to memorize addition, etc.. facts
and learn algorithms for computing multi-digit products, etc.. and still acquire
no number sense at all. You keep telling us that having students do manual or
mental computations (I should stress the word computations) will not give them
number sense. I would agree with that statment if we inserted the word
"necessarily" after the word "not". A student could learn to do computations and
not develop number sense. But many students DO develop number sense through such
practice, despite your protestations.
>It might help if the child started with the development of the
>operations from counting, and had to produce the tables from
>scratch. But this requires that one start with counting, and
>characterize addition and multiplication in terms of that process,
>which IS what I am advocating.
FWIW, this is how my two children were introduced to operations (ages 7 and 10),
and when I read educational catalogs and journals, it seems to me to be the way
that addition and subtraction are currently introduced. Even multiplication was
introduced through counting. For instance, 3x5 was characterized as adding up 3
groups of 5 objects. You are always bringing this up, about addition, etc. being
developed from the cardinal sense, and starting with counting. I simply cannot
understand your protestations, since this is the only way I've ever seen it
presented in the lower elementary grades. (Kindergarten & first grade)
>
>Otherwise, a number sense can be just as easily produced by using
>the calculator to produce answers as to develop a calculator in
>the brain doing the same thing.
Students can use the calculator without thinking. While it would be possible to
be some absolutely automated calculating machine, and simply crank out memorized
answers and never engage ones brain, it seems to me it would be difficult to
compute sums and products without a calculator and NEVER have a thought about
it. It is these thoughts during computation that can foster some number sense.
OTOH, I have seen students use calculators without a thought in their head. They
reach for a calculator to do 32 divided by 2. :-(
Sheila King
Herman Rubin
In article <Ds1wA...@spcuna.spc.edu>,
This is not quite the case; we dump our old intuitions, and develop new
intuitions from the formalism. But we have learned not to trust even
these; I have proved theorems by methods which are only intuitive as
methods of proof, where the theorems would be highly unintuitive.
We have these in non-mathematical systems as well. Consider written
language; it developed from pictographs, to formalized pictorial
systems, to syllabaries and related sytems, to consonantal alphabets,
to the present alphabetic method. With a few notable exceptions, all
languages are now written in this latter system. Is there any point
in going through all the "intuitive" stages in the development of
writing before teaching children to read using alphabetic means?
For the development of speech, we do not even know the stages. Early
languages have quite elaborate grammars. Do we attempt to go back to
proto-Indo-European in teaching English?
>: >Those intuitions guide us in exploring what's still the same in n dimensions,
>: >what carries over, what doesn't. And the same applies in moving to complex
>: >and infinite dimensional spaces. But just as the 2 and 3 dimensional real
>: >spaces tie togther a lot elementary geometric knowledge, and as the n
>: >dimensional spaces tie together a lot of knowledge about linear equations,
>: >the inf dimensional space tie toghter a lot of knowledge about classical
>: >diff eq and integral eqs - fourier series, etc. (Never mind if I'm wrong
>: >on this or that detail.)
>: It's not that n-dimensional spaces tie together a lot of mathematical
>: knowledge, it's exactly the reverse process: there's a lot of mathematical
>: knowledge that can be applied to model our intuition about multi-dimensional
>: spaces. Mathematics doesn't need intuition, but intuition needs mathematics.
This is precisely the point which should be stressed. Mathematics, and
logic, forms the formal system in which we can model the world, and if the
model is any good, what we get from it is far more accurate than what we
get from our intuition.
Intuition is still used in coming up with new ideas. But these ideas are
checked by applying formal reasoning. A control mechanism built from
intuitive ideas may go into wild oscillations; we check it out. An
intuitive method for solving equations may go off to infinity; we use
careful analysis first. It is when we use too much intuition that we
run into problems, in engineering, medicine, war, and politics.
Some did, and some did not. It is hard to say if anything just "dropped out
of the sky". But a lot of mathematics is close to that; a formalism which
happens to give answers in simple situations is not pushed to the limit,
with no understanding of it except as formalism. Sometimes it gives nothing
new, sometimes lots. Sometimes the formalism is sufficiently incomplete,
and we cannot do anything with it until we manage to find out what is really
meant.
It is often the case that the developed axiom system makes it possible to
understand what went before more easily by discarding the old intuition.
......................
Herman Rubin
In article <31a9c511.1254487@news>, Starfollower <ck...@cyberg8t.com> wrote:
>hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>>In article <4o5r5i$l...@thor.cmp.ilstu.edu>,
>>Thomas W. Cowdery <twc...@rs6000.cmp.ilstu.edu> wrote:
>>>ed...@netcom.com wrote:
.................
>>I do not see that any more number sense is acquired by memorizing
>>addition and multiplication tables, and learning the rules for
>>multi-digit arithmetic.
>In theory, I think it is possible for a child to memorize addition, etc.. facts
>and learn algorithms for computing multi-digit products, etc.. and still acquire
>no number sense at all. You keep telling us that having students do manual or
>mental computations (I should stress the word computations) will not give them
>number sense. I would agree with that statment if we inserted the word
>"necessarily" after the word "not". A student could learn to do computations and
>not develop number sense. But many students DO develop number sense through such
>practice, despite your protestations.
Many develop number sense DESPITE this. The motivation for introducing the
"new math" was that far too many had no idea of what they were doing.
>>It might help if the child started with the development of the
>>operations from counting, and had to produce the tables from
>>scratch. But this requires that one start with counting, and
>>characterize addition and multiplication in terms of that process,
>>which IS what I am advocating.
>FWIW, this is how my two children were introduced to operations (ages 7 and 10),
>and when I read educational catalogs and journals, it seems to me to be the way
>that addition and subtraction are currently introduced. Even multiplication was
>introduced through counting. For instance, 3x5 was characterized as adding up 3
>groups of 5 objects. You are always bringing this up, about addition, etc. being
>developed from the cardinal sense, and starting with counting. I simply cannot
>understand your protestations, since this is the only way I've ever seen it
>presented in the lower elementary grades. (Kindergarten & first grade)
The most common method, and I think the biggest mistake of the new math,
was to introduce it as union or Cartesian product, rather than counting.
Counting does not need objects. To get 5x3 (I have changed the order so
that the 3 groups idea would still be present from the counting approach),
it would proceed 5x1 = 5; 5x2 = 5x(1+1) = 5x1 + 5; 5x3 = 5x(2+1) = 5x2 + 5.
Taking 3 groups of 5 objects and seeing how many are present is not building
it up from counting; the cardinal concept is only apparently simpler, and
is inadequate by itself for a rigorous development.
>>Otherwise, a number sense can be just as easily produced by using
>>the calculator to produce answers as to develop a calculator in
>>the brain doing the same thing.
>Students can use the calculator without thinking. While it would be possible to
>be some absolutely automated calculating machine, and simply crank out memorized
>answers and never engage ones brain, it seems to me it would be difficult to
>compute sums and products without a calculator and NEVER have a thought about
>it.
Again I must point out that it was the observation that exactly this did
happen which led to the new math program.
It is these thoughts during computation that can foster some number sense.
>OTOH, I have seen students use calculators without a thought in their head. They
>reach for a calculator to do 32 divided by 2. :-(
Okay, now how does someone with "number sense" do 32 divided by 2? If it
is not memorized, it will be done exactly by the division method as usually
taught formally to do short division. Even if the trick to start out with
30/2 = (3-1)/2x10 +5 is used, this is a memorized trick, and carries no
sense about what the numbers mean. The buildup of strategies to do
arithmetic provides no understanding of numbers, but just computational
devices. Now if this is done in base 16, 20 divided by 2 uses a little
more number sense. But if the numeral string 32 refers to a number in
base 16, the answer is 19.
The string of numerals, etc., REPRESENTS a number. Number sense is about
numbers, not these strings.
Frank Rodgers
Ted Hirsch <t...@nyiq.net> wrote:
>Can you suggest any starting points (books, journals, etc.)
>that an interested elementary teacher can go to, assuming
>he has a strong conceptual understanding of mathematics?
Herman Rubin <hru...@b.stat.purdue.edu> responded:
>The situation is not the best, as there has been major resistance
>to the teaching of concepts for some time. But there are some old
>materials, some of which can even be used for teaching children.
>For those teaching mathematics above the elementary level, I
>consider the ability to do all of this essential.
I, too, have been bothered by the lack of available material which
emphasizes mathematical concepts. So I have just recently
completed a HyperCard stack which attempts to remedy this
deficiency vis-a-vis elementary algebra. It is available for
downloading from Info-mac at
ftp://sumex-aim.stanford.edu/info-mac/edu/algebra-tutor-11-hc.h
qx
as well as all other Info-mac mirror sites.
It treats numerals as proper nouns, numerical expressions as
nominal expressions, variables as pronouns, algebraic expressions
as pronominal expressions, equations as sentences about synonyms
and the importance of identities and the equivalence relationship in
algebra. The topics covered include sets, systems of equations,
irrational numbers, the quadratic formula, graphs of conic sections
and rational algebraic expressions. It attempts to impart an
appreciation for the process by which our number system evolved
from counting to rational numbers and the difficulties that process
encountered with the discovery of the need for irrational numbers
and the later introduction of zero into the number system. It does
not treat the real number system for it is my feeling that a
meaningful treatment of that topic should be delayed until the
student has acquired a good deal more sophistication in later course
work.
I would be interested in hearing the opinions of this effort from
those of you who are following this thread. My follow-on project
will be a stack dedicated to mathematical logic.
Frank Rodgers frod...@olympus.net
Duncan Napier
In article <4o4of3$c...@news.redshift.com> mgr...@redshift.com (Michael Greene) writes:
>twc...@rs6000.cmp.ilstu.edu (Thomas W. Cowdery) wrote:
>
>>isn't any way to tie a teacher's compensation to their performance
>>because *they* aren't the ones taking the tests! You can lead a horse
>>to water, but you can't make him drink, and you can teach a lesson,
>>but you can't make someone learn.
>
>That isn't clear to me. It's like saying an engineer's proficiency
>isn't gauged by how well the rocket she designs performs.
>
It isn't that simple, even in rocket design. Poor realization of a good design
could send the rocket hurtling back to earth (eg. choice of materials,
bureaucratic interferences, etc. etc.). My point is that there are many other
factors involved.
>It's true a teacher is not in direct control of the child but a
>teacher's ability to convey information and motivate children is being
>gauged by the test. It's not a perfect measure but it's better than no
>measure.
>
I've taken university courses where more than half the class was failed
(of course, the grades were adjusted to satisfy the Registrar). Would you argue
that those who failed had failed themselves? Doesn't this indicate a failure
of some kind on the TEACHER's part (ie to communicate effectively)? Does it
also indicate failure on the university's part for setting standards
and then renegging on them?
I don't have the answers ...
Duncan Napier.
University of British Columbia.
Herman Rubin
In article <4ocoar$6...@nntp.ucs.ubc.ca>,
Duncan Napier <nap...@theory.chem.ubc.ca> wrote:
>In article <4o4of3$c...@news.redshift.com> mgr...@redshift.com (Michael Greene) writes:
>>twc...@rs6000.cmp.ilstu.edu (Thomas W. Cowdery) wrote:
>>>There
>>>isn't any way to tie a teacher's compensation to their performance
>>>because *they* aren't the ones taking the tests! You can lead a horse
>>>to water, but you can't make him drink, and you can teach a lesson,
>>>but you can't make someone learn.
>>That isn't clear to me. It's like saying an engineer's proficiency
>>isn't gauged by how well the rocket she designs performs.
>It isn't that simple, even in rocket design. Poor realization of a good design
> could send the rocket hurtling back to earth (eg. choice of materials,
> bureaucratic interferences, etc. etc.). My point is that there are many other
> factors involved.
>>It's true a teacher is not in direct control of the child but a
>>teacher's ability to convey information and motivate children is being
>>gauged by the test. It's not a perfect measure but it's better than no
>>measure.
No teacher, however good, could teach a course in English literature to
many in a class of students who are not of genius level who cannot read
English before taking the class.
>I've taken university courses where more than half the class was failed
> (of course, the grades were adjusted to satisfy the Registrar). Would you argue
> that those who failed had failed themselves? Doesn't this indicate a failure
> of some kind on the TEACHER's part (ie to communicate effectively)? Does it
> also indicate failure on the university's part for setting standards
> and then renegging on them?
>I don't have the answers ...
Something like this happened to me quite some time ago. If you count
D's as failures, it is much more common. There was a probability course
which a department decided would be good for its juniors, and so they
put lots of them in. The time I taught a section, there were 6 sections,
with 5 different teachers. Now the students taking this class could be
put in four categories: strong math majors, who should not have been there,
about 1/6; weak majors in mathematics education, about 1/6; students in this
major, about 1/3; and students in everything else, about 1/3.
The first and last groups did well. The others did not, with D being
the dominant grade. Now the math ed students may have been hopeless,
but the department who sent their majors there, seeing the types of
errors their students made, were far more inclined to blame the complete
obstinacy of their students to learn what was not immediately "relevant".
One of their professors, in a faculty discussion between the two
departments about the course, stated:
I do not see how any conceivable combination of
instructor and textbook could lead to such bad results
without the active cooperation of the students.
There are few fundamentalist creationists to whom one can communicate
the ideas of evolution. Communicating mathematical thinking to someone
who is utterly convinced that mathematics means plugging into formulas
may be almost as hard. The teacher might even be a very effective
communicator of mathematical ideas, but if the student comes in needing
several years of remedial work, what can the teacher do in that one term?
The approach taken by the educationists, to teach to the lowest common
denominator, was always a mistake. I could hardly take the time in
which I am expected to teach probability and teach the basic algebraic
ideas, which the students do not have much familiarity with, instead.
ibokor
David K. Davis (dav...@spcunb.spc.edu) wrote:
:
: My position is that mathematics helps us extend our intuition -
Then you should look more closely at the hstory of western
cuture at least. It was often mathematics --- used in its
broadest sense --- which showed that our collective intuition
is flawed: Zeno's paradoxes, the existence of irrational
numbers, the paradoxes of set theory spring readily to mind.
: it doesn't
: obviate the need for it at any level. 2 and 3 dimensional spaces extend
: our intuition (and knowledge) about the real line, plus unifying this with
: some of our geometrical ideas and intuitions.
Regarding the real numbers at all is anything but intuitive!
It is customary since Descartes at least, but it is really
only since about the middle of the last century that there is
a firm basis for this.
I suggest that "intuition" would not take us much further than
rational numbers and would have diffulty coping with the
subtleties of infinite sets.
:
: My point is that our abstractions, ultimately, are historically conditioned.
: We do not, now can we, just arbitrarily pick a bunch of axioms and start
: exploring the consequences. We abstract on the basis of what precedes.
: Of course, we often say, what happens if we loosen this requirement, etc.A
But our "intuition" is informed by the centuries of abstraction even more
than the abstraction is based on "intuition". The fact that you view
the real numbers in terms of a straigt line is in part consequence
of abstract theory. It is difficult to see how one could arrive
at the real numbers by intuition alone, without substantial abstract
development.
d.A.
Brian M. Scott
In article <4o9u5h$1p...@thor.cmp.ilstu.edu>, twc...@rs6000.cmp.ilstu.edu
(Thomas W. Cowdery) says:
[snip]
[Herman had said:]
>>Otherwise, a number sense can be just as easily produced by using
>>the calculator to produce answers as to develop a calculator in
>>the brain doing the same thing.
>Nonsense. Using a calculator requires little if any thought. Number
>sense cannot be developed or acquired without thought.
Like you, I have observed a correlation between ability to carry out
basic computations and ability to understand mathematics. I am not
certain whether there is a cause and effect relationship, but I'm
much more willing to believe in one than Herman is. Nevertheless I
must disagree wholeheartedly with your claim that using a calculator
requires little thought. It is precisely because using a calculator
(usefully) *does* require thought that many of my students get little
or no benefit from theirs. They don't understand the mathematics -
not the computation, but the mathematics - well enough to know what
to do with the calculator. They have tried to substitute it for
thought, and of course they have failed.
Brian M. Scott
Brian M. Scott
In article <Ds1wA...@spcuna.spc.edu>, dav...@spcunb.spc.edu (David K.
Davis) says:
[much snipped]
>The formalization is of concepts. The concepts don't grow out of chains
>of inference. The chains of inference are bridges we build from both ends,
>from the axioms to the propostion.
Apparently you've never encountered the phenomenon of the 'proof in
search of a theorem'. Some concepts *do* grow out of chains of
reasoning. It is by no means unknown to define a class of objects
essentially as those for which a certain proof, originally intended
to apply to a narrower, previously defined class, works; and if this
new class proves sufficiently interesting, it may survive as a 'living'
mathematical concept.
> To me, if I see a solution, I want to
>know what the problem was. What good is a solution without a problem?
>The solution is meaningless to me without the problem. What does this
>or that definition capture? What does this axiom system formalize? What
>prior body of stuff does it tidy up?
This can be valuable information, but not everyone cares. Some find
in the elegance of the solution or the (to them) intrinsic interest of
the defined concept sufficient reason to study them. This is a matter
of taste.
>It's an illusion to think that we have freed ourselves from history,
>from the concrete, from applications, and so on. A lot of mathematics
>is abstracted from other parts of mathematics, but at some point the
>outer circle connects with reality and history in one way or another.
This is a truism, a banality. There is much fascinating mathematics
that is so far removed from physical reality as to have no meaningful
connection therewith.
Brian M. Scott
Thomas W. Cowdery
b.s...@bscott.async.csuohio.edu (Brian M. Scott) wrote:
<snipped>
>It is precisely because using a calculator
>(usefully) *does* require thought that many of my students get little
>or no benefit from theirs. They don't understand the mathematics -
>not the computation, but the mathematics - well enough to know what
>to do with the calculator. They have tried to substitute it for
>thought, and of course they have failed.
I would still contend that using the calculator requires little, if
any, thought. However, knowing what to use it for, and what to enter
in, in what order etc., that requires thought. Now, at the level that
I teach, the calculator isn't used for much else except arithmetic. I
do a little with the graphing calculators, but even if I used them
extensively, that aspect of their use would only show up once in a
while. Until the second semester of Algebra 2/Trig, when the log, x^y
and trig function keys come into play, a student in my class could get
by with one of those credit card calculators, or without a calculator
at all. I would prefer the latter.
Thomas W. Cowdery
<snip>
>Okay, now how does someone with "number sense" do 32 divided by 2? If it
>is not memorized, it will be done exactly by the division method as usually
>taught formally to do short division. Even if the trick to start out with
>30/2 = (3-1)/2x10 +5 is used, this is a memorized trick, and carries no
>sense about what the numbers mean.
Another possibility is that they may realize that (in base 10) any
number ending in 2 is evenly divisible by 2, and the result when it is
divided by 2 must end in either 1 or 6. Then they can easily
eliminate 11 (which they can mentally multiply by 2 and get 22) and 21
(which gives 42). Since 11 produces a product that is too small, and
21 produces one that is too large, then the correct answer must be 16.
This is probably more work than is necessary for this problem; but
I've used that style of thinking on larger problems. And that is
getting a little closer to what I would call 'number sense'. The next
step in the evolution of the 'number sense' might be less reliant on
the quirks of base 10, and more abstract, though I confess that I
would be hard put to define exactly what 'number sense' is. Like one
of the Supreme Court justices said about pornography, I can't define
it, but I know it when I see it. ;-)
>The buildup of strategies to do
>arithmetic provides no understanding of numbers, but just computational
>devices. Now if this is done in base 16, 20 divided by 2 uses a little
>more number sense. But if the numeral string 32 refers to a number in
>base 16, the answer is 19.
>The string of numerals, etc., REPRESENTS a number. Number sense is about
>numbers, not these strings.
True, but working with those strings of numerals can lead to gaining
the number sense. Punching buttons cannot.
Thomas W. Cowdery
>There is no such thing as a concrete notion. Any notion is formed in
>the brain, and at this time, we have little idea of how this is done.
>When we communicate, it is at best brain to brain. To clarify the
>communication, it is put into a restricted symbolism. As we have
>found, it is much easier to communicate with a restricted "digital"
>character set. While we lose bandwidth, we gain clarity.
>Is "yellow" a concept"? How does one decide if something is yellow?
>What are the boundaries of yellow? Purdue's colors are black and gold.
>Is the gold color "yellow"? Is yellow the color of an object? No, for
>if an object is viewed in sodium light, it will look yellow or black,
>even if red or orange or green.
You are looking at this from an adult's perspective; and a highly
educated adult at that. Kids don't think that way because they aren't
capable of that level of thought yet.
Recently the Crayola company had to make a change in the formulation
of a new crayon product. They had added food scents to a line of
crayons and kids were eating them. To a small child, if something
smells like a banana (or lemon, I don't know what 'flavor' they made
yellow) and is the same color as that particular fruit, then they will
think it must taste like it too. They ended up putting some other
scents in the crayons; floral, if I recall. The abstraction that the
scents were not related to edibility was more than most kids could
grasp. Let's face it, kids at that age are apt to put anything in
their mouth, and making a yellow crayon smell like a banana or lemon
was too much temptation.
Thomas W. Cowdery
nap...@theory.chem.ubc.ca (Duncan Napier) wrote:
<snip>
>I've taken university courses where more than half the class was failed
> (of course, the grades were adjusted to satisfy the Registrar). Would you argue
> that those who failed had failed themselves? Doesn't this indicate a failure
> of some kind on the TEACHER's part (ie to communicate effectively)? Does it
> also indicate failure on the university's part for setting standards
> and then renegging on them?
I would say that either explanation is a possibility. I do not
believe that half the class failing is prima facia evidence of poor
teaching. In my alma mater, many of the 100 level courses are
*expected* to have a failure rate that high. I took several of them
and earned "A's" in all of them. The courses aren't that hard.
Especially compared to higher level courses. One was so easy that
after the first of four tests, I skipped 75% of the lectures and still
ended up with a 98% average. But about half of the class failed. I
can't imagine putting in less effort that I did in that class. (For
the record, being that lazy isn't something I'm proud of, but I was
working 40 hours a week on 3rd shift while carrying 17 hours during
the daytime that semester. Being able to skip the class and still
keep up got me through.)
Another one that I took was a remedial (for college) math course that
I enrolled in because I'd been out of school for 14 years and wanted a
'refresher' before I got into the courses that counted toward the math
major. I thought that the professor did an excellent job and was
stunned to find out how poorly the rest of the class was doing. I
didn't speak to many of my classmates. One that I did speak to was an
adult student, like myself, that I had once worked with before going
back to school. He had to pass the class (having failed it once
already) to stay in the university. I tried to help him a little. He
just didn't 'get' math. I don't think the problem was the professor's
teaching as much as his lack of ability. Thankfully, he squeaked out
a 'D' and was able to finish his coursework. He is a safety inspector
for the fire department. I don't know how much algebra he uses, but I
hope it isn't much, because I know he can't do much!
Alberto C Moreira
>(Karen Dee Michalowicz) responds to Alberto C Moreira, who had written:
>
> I wasn't here then. But the statement that the NCTM standard
> somehow reflects ideas that have to do with what I - and
> other science professionals - mean by "new math" is plainly
> not true.
>
>>If you dispute a concept, or in this case a movement, it is
>>expected that you have some information about what you are
>>criticizing. As Alberto says above, he doesn't know about the
>>was equating the standards to this movement, about which he
>>states he knows nothing, he logically shouldn't dispute what I
>>said.
>
>reflects ideas that have to do with what [he] - and other science
>professionals - mean by "new math"'. From this you may fairly safely
>infer that what he means by 'new math' is different from what you mean
>by the term, that's all.
Exactly. What I mean by "new math" is the sort of math I learned
in college between 1965 and 1967: the journey from mathematical
logic to analysis through set theory and formal systems. I've read
plenty of NCTM material, and I see nothing of that sort there.
Alberto.
Alberto C Moreira
Ted Hirsch <t...@nyiq.net> wrote:
>Can you suggest any starting points (books, journals, etc.)
>that an interested elementary teacher can go to, assuming
>he has a strong conceptual understanding of mathematics?
>You may assume that this teacher has been through the
>standard compliment of arithmetic and mathematics courses,
>including calculus. This teacher was bored by the repetitive
>and lengthy calculations--but not the concepts--presented.
>
>This teacher has already read some works in this area,
>such as "About Teaching Mathematics" by Marilyn Burns.
>Please suggest others.
>
>Note to others: I'm not interested in anyone's demagoguery.
>If you have nothing to add to the reading list, just keep
>your mouth closed. Saxon people, especially, this means you.
I'll ignore the warning and stick my neck out. BTW, I'm not a
"Saxon people".
Curry, "Foundations of Mathematical Logic", Dover
Halmos, "Naive Set Theory", Springer Verlag
Suppes, "Axiomatic Set Theory", Dover
Russell, "Introduction to Mathematical Philosophy", Dover
Russell, "Principles of Mathematics", Norton
Landin, "An Introduction to Algebraic Structures", Dover
Rosenlicht, "Introduction to Analysis", Dover
Munkres, "Topology", Prentice-Hall
Hardy and Wright, "An Introduction to the Theory of Numbers",
Oxford
This should keep you busy for a while. I may have omitted some, but
these map the road from logic to calculus, which I believe every
teacher of mathematics should be familiar with.
Alberto.
Ted Hirsch
Herman,
Thanks for the reading list. I appreciate the time you took to
write it out, including explanations of what you thought was
valuable for the teacher.
:-)
Alberto C Moreira
hru...@b.stat.purdue.edu (Herman Rubin) wrote:
>In article <4ocoar$6...@nntp.ucs.ubc.ca>,
>Duncan Napier <nap...@theory.chem.ubc.ca> wrote:
[some deleted...]
>>I've taken university courses where more than half the class was failed
>> (of course, the grades were adjusted to satisfy the Registrar). Would you argue
>> that those who failed had failed themselves? Doesn't this indicate a failure
>> of some kind on the TEACHER's part (ie to communicate effectively)? Does it
>> also indicate failure on the university's part for setting standards
>> and then renegging on them?
>
>>I don't have the answers ...
>
>Something like this happened to me quite some time ago. If you count
>D's as failures, it is much more common. There was a probability course
>which a department decided would be good for its juniors, and so they
>put lots of them in. The time I taught a section, there were 6 sections,
>with 5 different teachers. Now the students taking this class could be
>put in four categories: strong math majors, who should not have been there,
>about 1/6; weak majors in mathematics education, about 1/6; students in this
>major, about 1/3; and students in everything else, about 1/3.
It happens to me all the time, to get students in class who don't have the
minimum prerequisite to follow the course. The more advanced the student's
stage and the more important the course is, the more likely it is that
some percentage of the students shouldn't be there. And for those, there's
little the teacher can do. If I'm teaching Abstract Machines, for example,
I don't have time to teach set theory; and some may fail because of their
lack of knowledge in those prerequisite areas.
Alberto.
H. Jurjus
In Article <4od5gp$6...@grivel.une.edu.au> "ibo...@metz.une.edu.au (ibokor)" says:
> David K. Davis (dav...@spcunb.spc.edu) wrote:
> :
> : My position is that mathematics helps us extend our intuition -
>
> Then you should look more closely at the hstory of western
> cuture at least. It was often mathematics --- used in its
> broadest sense --- which showed that our collective intuition
> is flawed: Zeno's paradoxes, the existence of irrational
> numbers, the paradoxes of set theory spring readily to mind.
In the same way, Russells paradox is a problem of formalizing stupidly,
not of the intuitive idea of a set. The very fact that people immediately
looked for some other formalization illustrates the fact that there *was*
some fundamental intuition to be formalized.
> Regarding the real numbers at all is anything but intuitive!
so long to find a convincing formal elaboration of the intuitions doesn't mean
that the intuitions were not fundamental. On the contrary: the only
reason to appreciate, say, Dedekind's work is that it captures the intuitions
better.
> It is customary since Descartes at least, but it is really
> only since about the middle of the last century that there is
> a firm basis for this.
>
> I suggest that "intuition" would not take us much further than
> rational numbers and would have diffulty coping with the
> subtleties of infinite sets.
mathematics in the first place.
H.Jurjus
H. Jurjus
In Article <4ocidg$31...@b.stat.purdue.edu> "hru...@b.stat.purdue.edu (Herman Rubin)" says:
>
> Some did, and some did not. It is hard to say if anything just "dropped out
> of the sky". But a lot of mathematics is close to that; a formalism which
> happens to give answers in simple situations is not pushed to the limit,
> with no understanding of it except as formalism.
And in some way, one might say: the mathematics starts as soon as the
intuitive concepts have been formalized.
This doesn't mean that studying some formal systems does not have
as a primary purpose the investigations of the intuitions behind them
and, perhaps, the improvement of such intuitions.
It is only the outside appearance (the 'official mathematics') that is purely
formal.
And the art of formalizing intuitions is an important skill in mathematics,
don't you think ? (Note: I don't claim it is the only one.)
H.Jurjus
WJ Bland
On 26 May 1996, Kevin Anthony Scaldeferri wrote:
> You are both right, but in different ways. Take 'yellow'. There is a
> concrete notion of yellow and an abstract notion of yellow. To teach
> a child the concrete notion, point at things that are yellow. To teach
> the abstract notion, you need electricity and magnetism, Maxwell's
> equations, waves, wavelengths, all in all, several years before you
> can hope to 'understand' yellow. And I ask you, which one will be
> more useful to most students?
>
> Kevin
No! Still not right. Your "concrete" notion of yellow is still not really
concrete. You can point to a table and say "that's a table", but when you
point to something and say "that's yellow" you are being abstract.
The notion of "yellow" *is* abstract, because it doesn't exist by itself;
it is only a property of some things.
There is no concrete notion of yellow; just abstract and more abstract.
Bill.
-------
/'I thought dwarfs didn't believe in devils and demons and stuff like that.' \
\'That's true, but ... we're not sure if they know.' Terry Pratchett - M.A.A/
Bill's home page is at http://www.ccc.nottingham.ac.uk/~pmykwjb/bil.html
rdsil...@qed.com
In article <4o7i6r$32...@b.stat.purdue.edu>, <hru...@b.stat.purdue.edu> writes:
> >>Language is already abstract. To recognise what is being
> >>discussed from the sounds heard is a tour de force of
> >>abstraction. To speak of "yellow" or "table" pre-supposes
> >>sophisticated abstraction.
>
> >Nonsense! Both 'table' and 'yellow' are very concrete concepts. The
> >former is a word that represents a physical object.
> ^^^^^^^^^^
>
> THIS is a good part of abstraction. Mathematical communication about
> the real world is the ability to use formal mathematical systems to
> represent other types of entities.
>
Let me also add that the meaning of "table" is CONTEXT DEPENDENT, and
does not necessarily mean a kind of furniture. There are also tables
of numbers for example. Therefore, your assertion is not correct that
"table" represents a physical object. It *can* represent a physical object,
but does not have to.
Furthermore, the word "yellow" is also context dependent. It can refer to
a color, or it can refer to cowardice.
Words, in themselves, represent abstractions which can have *multiple*
mappings. They need not be concrete at all.
cameron
Attribution key:
"DKD" is dav...@spcunb.spc.edu (David K. Davis), in article
<Ds1wA...@spcuna.spc.edu>
"BMS" is Brian M. Scott (b.s...@bscott.async.csuohio.edu), in article
<4odb2p$i...@csu-b.csuohio.edu>
DKD> To me, if I see a solution, I want to
DKD> know what the problem was. What good is a solution without a problem?
DKD> The solution is meaningless to me without the problem. What does this
DKD> or that definition capture? What does this axiom system formalize? What
DKD> prior body of stuff does it tidy up?
BMS> This can be valuable information, but not everyone cares. Some find
BMS> in the elegance of the solution or the (to them) intrinsic interest of
BMS> the defined concept sufficient reason to study them. This is a matter
BMS> of taste.
It's hard to tell, since this thread is cross-posted between some
newsgroups that are specifically devoted to discussions of math education
and at least one that is not, but I *think* this was supposed to be a
discussion of motivation in mathematics education. (Witness the subject
line: "...was 'What's wrong with education...'".) I don't think the
cultivated taste of the mathematical aesthete is what we ought to be
using as the model by which we plan our curruculum. Even those who *are*
able to appreciate the austere beauty of elegant chains of inference are
generally not actually *repelled* by learning that a particular
mathematical concept arose in the process of solving a problem that was
not itself essentially mathematical. (For example, Newton developed much
of his calculus in the course of solving problems of celestial mechanics,
but I don't know any mathematicians who abhor calculus because of that.)
So what is gained by the observation that "not everyone" needs to know
what problem a concept helps solve? I claim that most students *do*
need to have, or at least benefit greatly from having, a motivation
for the concepts they are asked to learn.
(ASIDE: Lest anyone misconstrue that last sentence, let me stress that
I mean "motivation", not "incentive". I am talking about putting
ideas in their proper historical contexts and showing how the ideas
developed, so that someone who wants to learn the ideas has a mental
framework to which to attach them. I am not talking about enticements
to induce students who have no desire to learn to do so anyway.)
Or have pedagogical considerations been abandoned altogether, and are
we now merely engaged in a philosophical discussion about the intrinsic
value of mathematics as an activity for its own sake versus its role
as a tool in the study of non-mathematical problems? If so, then I do
agree that mathematics done this way has as much value as sculpture or
dance or any other art form, but maybe we should drop misc.education and
k12.ed.math from the Newsgroups: line in favor of some sci.philosophy group.
DKD> It's an illusion to think that we have freed ourselves from history,
DKD> from the concrete, from applications, and so on. A lot of mathematics
DKD> is abstracted from other parts of mathematics, but at some point the
DKD> outer circle connects with reality and history in one way or another.
BMS> This is a truism, a banality. There is much fascinating mathematics
BMS> that is so far removed from physical reality as to have no meaningful
BMS> connection therewith.
I find this strongly reminiscent of Hardy's "Mathematician's Apology".
I read it as an undergraduate, so my recollection is not crystal clear, but
it seems to me that Hardy took a smug pleasure in his confidence that number
theory is so pure a pursuit that it will never be useful for anything.
One can only imagine how dismayed he would have been to see, before the
century was out, number theory (as manifested in RSA public-key algorithms
and similar techniques) become the cornerstone of the foundation upon
which the next generation of electronic commerce systems are being built.
Even when (as in this case) an idea did not historically arise in response
to a particular practical problem, if the idea proves crucial to the solution
of that problem, and the problem's practical importance is great enough,
then that problem may become the justification for the idea's continued
interest to the majority of the people who study it. So I think that not
too much should be made of the fact that some concepts of interest to the
mathematician have not yet acquired non-mathematical interpretations.
In any case, this seems to have little relevance to questions of pedagogy.
--Cameron Smith
cam...@dnaco.net
Beth Kevles
John Heilbron (recently of UC Berkeley) is coming out soon with a
geometry textbook that may be of interest to readers of this thread.
It approaches geometry through the history of the problems that
motivated the development of the field. Examples run through history,
starting (I think) with the ancient Greeks.
I haven't seen the book yet, but it promises to be very interesting. I
understand that it is not an innovative curriculum in other senses. The
text, by the way, is designed for undergraduates.
John Heilbron's field is the history of science and technology. I'm
looking forward to reading his geometry textbook.
--Beth Kevles
kev...@mit.edu
ibo...@metz.une.edu.au
Karen Dee Michalowicz (kmic...@pen.k12.va.us) wrote:
: (<ibo...@metz.une.edu.au>) writes:
: > From: Karen Dee Michalowicz, kmic...@pen.k12.va.us
: > Date: Sun, 26 May 1996 12:49:41 GMT
: > In article <Ds0JM...@pen.k12.va.us> Karen Dee Michalowicz,
: > kmic...@pen.k12.va.us writes:
: > >Alberto C Moreira (Alberto C Moreira ) writes:
: > >>
: > >> I don't know about the history of "new math" in this country;
: > >> I wasn't here then. But the statement that the NCTM standard
: > >> somehow reflects ideas that have to do with what I - and
: > >> other science professionals - mean by "new math" is plainly
: > >> not true.
: > >>
: > >
: > >If you dispute a concept, or in this case a movement, it is
: > >expected that you have some information about what you are
: > >criticizing. As Alberto says above, he doesn't know about the
: > >"new math" (movement) about which I wrote.
: >
: > If you criticise a contribution, do at least read what it
contains.
: >
: > The previous cntributor did *not* write that 'he doesn't know
about
: > the "new math" (movement)', he wrote that he doesn't know about
its
: > *history*.
: > That does not preclude knowing its content or philosophy.
: >
: > After all, how many teachers of mathematics know the history of
: > their subject?
: >
[snip]
: Help, I must be reading these postings differently than some
: others.
: When I made my posting earlier I equated the "New Math"
: movement of the 60's and 70's with the present movement
: influenced by the NCTM Standards (incidently issued in l989).
: I used my many years of experience and my informed readings to
: write my posting.
As Herman Rubin has pointed out, the "New Maths" movement is
substantially older than that. It commenced shortly after
World War II and was much and planned in the 1950's. The
article by J. Kemeny I referred to in an earlier posting
was a paper delivered at an international meeting summarising
the situation as reported by an international committee on
what should/could be taught in high school mathematics.
Amongst the points made was the one that a major difficulty
faced in the USA is that there the teachers are too poorly
qualified to teach such mathematics. This is contrasted to
the situation in some other countries, such as France,
where it is not uncommon for people with dctirates in mathematics
to teach in high schools and propsepective teachers of high
school mathematics are expected to have strong background
in mathematics.
So in your many years of personal experience and informed readings
you seem to have overlooked some relevant features/facts
about the "New Maths" movement and you criticise Alberto Moreira
for pointing out that the content and aim of the "New Maths"
and the NCTM Standards are very different rather than very similar.
As to the last point, I cannot comment, since I have not seen a
copy of the NCTM standards, but it is striking that so many
of the postings here critical of the "New Maths" have displayed
little comprehension of its aims and purposes.
d.A.
ibo...@metz.une.edu.au
Brian M. Scott
[some snipping throughout]
Attribution key:
"DKD" is dav...@spcunb.spc.edu (David K. Davis) in article
<Ds1wA...@spcuna.spc.edu>.
"BMS" is Brian M. Scott (b.s...@bscott.async.csuohio.edu) in article
<4odb2p$i...@csu-b.csuohio.edu>.
"CS" is Cameron Smith (cam...@sisko.dnaco.net) in article
<4of817$c...@sisko.dnaco.net>.
DKD> To me, if I see a solution, I want to
DKD> know what the problem was. What good is a solution without a problem?
DKD> The solution is meaningless to me without the problem. What does this
DKD> or that definition capture? What does this axiom system formalize? What
DKD> prior body of stuff does it tidy up?
BMS> This can be valuable information, but not everyone cares. Some find
BMS> in the elegance of the solution or the (to them) intrinsic interest of
BMS> the defined concept sufficient reason to study them. This is a matter
BMS> of taste.
CS>It's hard to tell, since this thread is cross-posted between some
CS>newsgroups that are specifically devoted to discussions of math education
CS>and at least one that is not, but I *think* this was supposed to be a
CS>discussion of motivation in mathematics education. (Witness the subject
CS>line: "...was 'What's wrong with education...'".) I don't think the
CS>cultivated taste of the mathematical aesthete is what we ought to be
CS>using as the model by which we plan our curruculum.
Fair enough; but neither should we pretend that mathematics exists
solely for its applications.
CS> I claim that most students *do*
CS>need to have, or at least benefit greatly from having, a motivation
CS>for the concepts they are asked to learn.
I suspect that you're right. And the motivations are a motley lot.
I have taken students who were scared to death of mathematics and
motivated them by finding a way for them to see something of the
aesthetics, or by finding a mathematical challenge that appealed to
them. I was generally motivated by the aesthetics. DKD is evidently
motivated chiefly by the utility. Insisting on a one-motivation-fits-all
approach is foolish, and that's what he seemed to me to be doing.
DKD> It's an illusion to think that we have freed ourselves from history,
DKD> from the concrete, from applications, and so on. A lot of mathematics
DKD> is abstracted from other parts of mathematics, but at some point the
DKD> outer circle connects with reality and history in one way or another.
BMS> This is a truism, a banality. There is much fascinating mathematics
BMS> that is so far removed from physical reality as to have no meaningful
BMS> connection therewith.
CS>I find this strongly reminiscent of Hardy's "Mathematician's Apology".
CS>I read it as an undergraduate, so my recollection is not crystal clear, but
CS>it seems to me that Hardy took a smug pleasure in his confidence that number
CS>theory is so pure a pursuit that it will never be useful for anything.
He did, though I'm not sure that 'smug' is quite right.
CS>One can only imagine how dismayed he would have been to see, before the
CS>century was out, number theory (as manifested in RSA public-key algorithms
CS>and similar techniques) become the cornerstone of the foundation upon
CS>which the next generation of electronic commerce systems are being built.
I, on the other hand, am fascinated by yet another demonstration of the
unreasonable effectiveness of mathematics.
CS> So I think that not
CS>too much should be made of the fact that some concepts of interest to the
CS>mathematician have not yet acquired non-mathematical interpretations.
CS>In any case, this seems to have little relevance to questions of pedagogy.
That depends on whether one's purpose is solely to teach specific
mathematical ideas, or whether one wishes as well to convey something
of the nature of mathematics as a whole. In at least one of the courses
that I regularly teach, the latter is as important as the former. Even
in the K-12 grades I see no good reason to ignore the matter altogether,
and K-12 *teachers* most certainly ought to have some basic idea of the
aesthetic as well as the practical side of mathematics.
Brian M. Scott
Brian M. Scott
In article <4odpuk$k...@thor.cmp.ilstu.edu>, twc...@rs6000.cmp.ilstu.edu
(Thomas W. Cowdery) says:
Herman said:
There is no such thing as a concrete notion. Any notion is formed
in the brain, and at this time, we have little idea of how this is
done. When we communicate, it is at best brain to brain. To
clarify the communication, it is put into a restricted symbolism.
As we have found, it is much easier to communicate with a restricted
"digital" character set. While we lose bandwidth, we gain clarity.
Is "yellow" a concept"? How does one decide if something is yellow?
What are the boundaries of yellow? Purdue's colors are black and
gold. Is the gold color "yellow"? Is yellow the color of an
object? No, for if an object is viewed in sodium light, it will
look yellow or black, even if red or orange or green.
Thomas responded:
You are looking at this from an adult's perspective; and a highly
educated adult at that. Kids don't think that way because they
aren't capable of that level of thought yet.
Recently the Crayola company had to make a change in the formulation
of a new crayon product. They had added food scents to a line of
crayons and kids were eating them. To a small child, if something
smells like a banana (or lemon, I don't know what 'flavor' they made
yellow) and is the same color as that particular fruit, then they
will think it must taste like it too. They ended up putting some
other scents in the crayons; floral, if I recall. The abstraction
that the scents were not related to edibility was more than most
kids could grasp. Let's face it, kids at that age are apt to put
anything in their mouth, and making a yellow crayon smell like a
banana or lemon was too much temptation.
This example doesn't really help your case, I'm afraid. It shows that
the kids had successfully abstracted the association between food scents
and food. Associating the scent of, let us say, banana - perhaps in
conjunction with the color yellow - with the edible fruit involves an
abstraction from experience.
Brian M. Scott