Teaching what, when?
Seraph-sama
>You are contradicting yourself above. I assume that it is a type and that
>you are saying that students should NOT be forced to learn all math up to
>basic algebra UNLESS they are to study math to that point. But then
>apparently no further.
>
>Why is basic algebra your break point? Based upon what? Why do they need
>that much? Why do they need basic geometry? Why do they need just about any
>of it? If someone that isn't going to study at the university level doesn't
>need to take math up through basic algabra, then why should someone that is
>going to major in art or English at the university need to take any more.
I chose basic algebra as a break point because that is all the math you need
for general daily situations. Basic geometry might also be needed, maybe in
putting wallpaper on the walls or something. I'll take back then what I said
about basic algebra, and replace "basic algebra" with anything that could be
considered the "break point".
>> There may be some instances where you must learn more; then in that case
>>pick up a book and learn. It's ridiculous how they're teaching kids about
>>logarithms and conic sections when they don't want to, or need to, hear
>about
>>it.
>
>It's also ridiculous how they're teaching kids about world history and
>English literature and basic biology and music when they don't want to, or
>need to, hear about it. Let alone forcing them to participate in gym class.
Yeah, exactly. :)
That's a debate for another forum, though. I don't want my views to start a
riot.
>>If someone wants to major in literature or history, knowing the formula for
>>the polar represenation of an arbitrary complex number isn't going to help
>them
>>at all.
>
>And if someone is going to major in engineering, having read A Midsummer
>Night's Dream isn't going to help them at all.
>
>> At the same time, one should allow kids in midde/high school to take
>>upper level math like abstract algebra if they want to. Advanced math is
>just
>>something that's "there" and whoever wants it should get it.
>
>
>Why don't we just acknowledge that virtually everything that is taught is
>only used directly by a rather small percentage of people. Since we can't
>identify which students need what subjects, we resort to wasting virtually
>all of everyone's time. The rationale has been that students, with very few
>exceptions, simply don't know what it is that they really want to do and so
>we require that they all be exposed to a rather broad sampling of all of
>these areas - in part because most of these subjects enrich people in ways
>that are important, even if it is never used directly. By taking this
>approach the claim is that students are able to knit together more
>sophisticated ways of viewing things and thinking about things and about how
>they interrelate and interact. We have long claimed that the broader and
>deeper a person's base education is, the better prepared they are to
>evaluate their options and decide what it is that they are interested in and
>that they want to focus on.
That's pretty much the view on things based on what I've heard. I have no
intention of going up against the system and changing that though.
>But we should abondon this approach. Stop wasting these poor kids' time. If
>they want to know something, then they should be able to take it if they
>want to. But don't force them to take classes that don't interest them or
>that they have no desire to attend. If a first grader would rather spend all
>day out on the playground, then perhaps it is because their interests are
>leading them in that direction. Why should adults interfer and force the
>child to learn how to spell. If the child eventually wants to learn how to
>read and write, then those classes should be made available to them at that
>time.
I disagree. I think kids should be taught the things the absolutely need to
know to be able to "survive" in our society, two of which are reading and
writing, and another of which is basic mathematics. What I disagree with is
forcing kids to learn out-of-this-world stuff that they don't want to use. If,
on the other hand, they become interested in a subject that requires calculus,
they can go get a calculus book and learn calculus. Or alternatively, they can
take a calculus class. A subject is NEVER "too hard" if one is interested in
it.
P.S. I'm sorry if my views or opinions offend anyone reading them. In
retrospect what I have been saying might be controversial based on my
experience, and I don't intend to make people mad again.
---
Seraph-sama
16/m, so don't call me "sera" or nothin'
http://members.tripod.com/~SeraphSama/class.html
for my research in solving for f in g(x) = f(f(x))
Seraph-sama
>> It's ridiculous how they're teaching kids about
>>
>> > logarithms and conic sections when they don't want to,
>
>generally not be "what do they need" but "what will enrich
>their intellect", and I believe math is included here (obviously,
>things that are necesarry, such as basic traffic laws, must
>also be taught, and also there are many intelluctual pleasures
>besides mathematics).
This sounds like an excellent criterion for teaching, as well as the
complimentary criterion that "they will definitely need the knoweldge to
succeed."
That the algebra 2 classes, as well as the other high-school classes, are
supposed to enhance the intellect, is confusing. If they were, I suppose at
least half of all high school students should be in college.
Unless I'm the only one and I'm a supergenius, which isn't very probable at
all. But I have my own experiences - studying math HAS enhanced my intellect, a
lot, and I think to an extent unnecesarry for purposes other than math. This is
the math like calculus and above. If the other math courses, as "manipulative"
as they are, are intellectually enhancing, than all the power to it. America
needs less dumb people anyway. lol.
Penny314
In the old days schools taught translations from latin and greek
which were severely graded to a high standard.
From this the students learned : accuracy,
elegance, careful reading, mastery of complexity, style and grace. It was
called building mental muscles.
Then, it was decided somehow that
this was a terrible waste of time . After a
while, we had a generation of functional illiterates who could barely read.
The liberal arts became known as
intellectual puff ---at least to science majors who were held to a high
standard of
mathematics.
From mathematics students learned:
accuracy, elegance ,careful reading,mastery of complexity , style and grace.
Then , it was decided somehow that this was a terrible waste of time.
Now ,we have a generation of functional innumerates
who are also functional illiterates.
pennysmith
>But I have my own experiences - studying math HAS enhanced my intellect, a
Absolutely.
Penny314
>And if someone is going to major in engineering, having read A
Midsummer
>>Night's Dream isn't going to help them at all.
In this you are wrong. One reason technical manuals are so poorly written
is that the engineers who write them do not have a mastery of writing complex
material.
In this, Shakespeare can be a corrective.
Personally, I think that young engineers should read Spencer's Fairy Queen. It
is denser than anything by Shakespeare and
is written on five levels of meaning. This would be good intellectual
preparation for college level engineering.
pennysmith
Many of my students cannot read their
college math and physics texts. They have very high math Sat scores and low
english
board scores. Thus ,they are going though
college crippled.
The best predictor of success in college math courses has been found to be the
Sat english score. That is because reading
well and carefully matters a great deal in
college math.
As to gym. Yes, this is a waste of time.
It is true that physical fitness improves intellectual performance. However,
the primary reason for gym is that ,historically,
most students were boys and most people were farmers. The parents wanted the
boys to be fit for harvest. The secondary reason for gym was that the boys were
possible soldiers --- keeping them fit eased the transition to warriorhood.
The ancient greeks celebrated the concept of a fit mind in a fit body.
Since you dont like "pithy Latin and Greek sayings" I won't quote one.
Seraph-sama
>Dear seraph-sama,
> >And if someone is going to major in engineering, having read A
>Midsummer
>>>Night's Dream isn't going to help them at all.
>
>In this you are wrong.
(Psst. I think that was someone else talking. :)
Seraph-sama
>In this you are wrong. One reason technical manuals are so poorly written
>is that the engineers who write them do not have a mastery of writing complex
>material.
> In this, Shakespeare can be a corrective.
>
>Personally, I think that young engineers should read Spencer's Fairy Queen.
>It
>is denser than anything by Shakespeare and
>is written on five levels of meaning. This would be good intellectual
>preparation for college level engineering.
> pennysmith
>
>Many of my students cannot read their
>college math and physics texts. They have very high math Sat scores and low
>english
>board scores. Thus ,they are going though
>college crippled.
It has been my experience that reading the "get-to-the-point" math and science
jargon language is the EASIEST out of all the writing styles. Also, my math on
the PSAT is 99%ile but my English is only 51%ile.
>The best predictor of success in college math courses has been found to be
>the
>Sat english score. That is because reading
>well and carefully matters a great deal in
>college math.
In math books, there is a very homogenous style of exposition. "It is clear
that... from this it follows.. we are tempted to say... let us assume.." You
don't have to be a Shakespeare at all to get used to this.
>As to gym. Yes, this is a waste of time.
>It is true that physical fitness improves intellectual performance. However,
>the primary reason for gym is that ,historically,
>most students were boys and most people were farmers. The parents wanted the
>boys to be fit for harvest. The secondary reason for gym was that the boys
>were
>possible soldiers --- keeping them fit eased the transition to warriorhood.
> The ancient greeks celebrated the concept of a fit mind in a fit body.
>Since you dont like "pithy Latin and Greek sayings" I won't quote one.
---
Seraph-sama
>dear Seraphs-sama,
> In the old days schools taught translations from latin and greek
>which were severely graded to a high standard.
>From this the students learned : accuracy,
>elegance, careful reading, mastery of complexity, style and grace. It was
>called building mental muscles.
I will conform to the analogy of building muscles: Students being introduced to
systematic arithmetic and geometry is like gathering a group of weaklings for a
weight-training program designed to bulk people up. When these weaklings begin
their program, they are expected, say, to curl 20 pounds. Eventually, they
adapt and they can move on; they have built muscle. However, the program
requires not that they move up to 25 or 30, but that they stay at 20, for a
long period of time. They get really lazy after some time, and it becomes hard
to lift more. This is okay to the system because the system requires they be
able to lift 20.
On one hand, this frustrates those who would not need to have been building
those particular muscles at all. On the other hand, it stunts the ability of
potential bodybuilders.
To place the analogy back in explicit terms, I believe math should be
force-taught up to the point where one has all the knowledge he/she needs to
succeed in everyday life. This includes arithmetic, basic algebra, maybe basic
geometry. The only reason math should be force-fed after this is if its purpose
is indirect: to enhance the intellect. The only kind of math that can do this,
especially at puberty when cognitive abilities soar, is the advanced kind.
You write:
>>But I have my own experiences - studying math HAS enhanced my intellect, a
>
>Absolutely.
When I refer to "math" here, I am referring specifically to advanced
mathematics. Starting at algebra and passing through calculus and a course
beyond, perhaps, I have gone through the mathematical stages very many go
through, especially at a young age: undefined concepts are "defined,"
outrageous theorems with these concepts are "proven," and so on. (For example,
in eighth grade (the first year I started studying mathematics seriously) I
recall when I was in a debate with another student on the value of infinity -
infinity. He took the side of "zero" and I took the side of "negative
infinity".) After this, and by means of advanced mathematics only is ones
intellect truly enriched.
However, it is much easier to introduce "intellectual enrichment" classes that
work much more efficiently and directly than forcing the students to learn math
- for example, classes in logic. (Hm. This does sound like a very good class to
teach to the general student in the teenage years. I wonder why it doesn't
exist already.)
Finally, in addition to avoiding the requirements of things like algebra 2 in
high school students when they don't need it, one should also allow the
opportunity for the student with mathematical interests to take courses at the
graduate level. I was infuriated when in my place I heard "No high school
student can take graduate level courses, under any circumstances." I had to go
around talking to lots of people just to take an introductory abstract algebra
class, which was in my own humble opinion quite pathetic. (The talking, not the
class. This class which I am taking right now, rules! One of my first real math
classes.)
I don't expect my opinions to actually be put to use, but I thought I might
say them when I had the chance. :)
William Hale
serap...@aol.com (Seraph-sama) wrote:
> Penny314 wrote:
> >In this you are wrong. One reason technical manuals are so poorly written
> >is that the engineers who write them do not have a mastery of writing complex
> >material.
> > In this, Shakespeare can be a corrective.
> >
> >Personally, I think that young engineers should read Spencer's Fairy Queen.
> >It
> >is denser than anything by Shakespeare and
> >is written on five levels of meaning. This would be good intellectual
> >preparation for college level engineering.
> > pennysmith
> >
> >Many of my students cannot read their
> >college math and physics texts. They have very high math Sat scores and low
> >english
> >board scores. Thus ,they are going though
> >college crippled.
>
> It has been my experience that reading the "get-to-the-point" math and science
> jargon language is the EASIEST out of all the writing styles. Also, my math on
> the PSAT is 99%ile but my English is only 51%ile.
I agree with Penny314 that learning should include more than totally
specializing in one field. There is a series of classic books published
under the title of "Great Books of Western Civilization". Some of these
books deal with mathematics, which you might be interested in reading.
I would also recommend reading other non-mathematical books in this series.
Besides the recommendations of Penny134, I like the dialogs of Plato.
One dialog, Meno, is about mathematics, which you might find interesting
to read. A URL for it is:
http://www.ilt.columbia.edu/academic/digitexts/plato/meno/meno.html
--
Bill Hale
Eric Dew
Seraph-sama <serap...@aol.com> wrote:
>Penny314 wrote:
>>In this you are wrong. One reason technical manuals are so poorly written
>>is that the engineers who write them do not have a mastery of writing complex
>>material.
>> In this, Shakespeare can be a corrective.
>>
>>Personally, I think that young engineers should read Spencer's Fairy Queen.
>>It
>>is denser than anything by Shakespeare and
>>is written on five levels of meaning. This would be good intellectual
>>preparation for college level engineering.
>> pennysmith
>>
>>Many of my students cannot read their
>>college math and physics texts. They have very high math Sat scores and low
>>english
>>board scores. Thus ,they are going though
>>college crippled.
>
>It has been my experience that reading the "get-to-the-point" math and science
>jargon language is the EASIEST out of all the writing styles. Also, my math on
>the PSAT is 99%ile but my English is only 51%ile.
>
>>the
>>Sat english score. That is because reading
>>well and carefully matters a great deal in
>>college math.
>
>In math books, there is a very homogenous style of exposition. "It is clear
>that... from this it follows.. we are tempted to say... let us assume.." You
>don't have to be a Shakespeare at all to get used to this.
>
>>As to gym. Yes, this is a waste of time.
>>It is true that physical fitness improves intellectual performance. However,
>>the primary reason for gym is that ,historically,
>>most students were boys and most people were farmers. The parents wanted the
>>boys to be fit for harvest. The secondary reason for gym was that the boys
>>were
>>possible soldiers --- keeping them fit eased the transition to warriorhood.
>> The ancient greeks celebrated the concept of a fit mind in a fit body.
>>Since you dont like "pithy Latin and Greek sayings" I won't quote one.
I can just imagine Shakespeare is rolling in his grave, realizing that all
his works are now used as a benchmark for educational standards, and not
read or used for enjoyment (which is why he wrote them in the first place).
Please folks, let's return the classics back to the bookstores for reading
enjoyment and teach kids how to effectively communicate using the written
language. There are plenty of programs and classes which explain how to
write effectively. All of them will tell you to write quite unlike the
style used by Shakespeare or Dickens or any other classics authors.
EDEW
John R Ramsden
>
> From mathematics students learned:
> accuracy, elegance ,careful reading,mastery of complexity , style and grace.
> Then , it was decided somehow that this was a terrible waste of time.
> Now ,we have a generation of functional innumerates
> who are also functional illiterates.
> pennysmith
It seems to me that the unspoken assumption behind the motives of
modern educationalists who seek to toss out the demanding studies
that Penny mentioned, and maybe even corporal punishment, is that
kids today couldn't and shouldn't be expected handle it.
In other words these intellectuals, doubtless themselves having
benefited from a "classical" education (even if at the price of
some drudgery and misery), are implicitly saying they are somehow
_better_ than children today. Looking at it in that light, their
breathtaking arrogance is clear.
The trouble is this becomes a self-fulfilling prophesy, in that
as a result of educationalists' obstructive attitudes many kids
will find it much harder, or impossible, to achieve their full
potential.
Apologies if to any teachers reading this it is no more than a
Forrest Gumpish truism, and too obvious to mention, but I think
it should be emphasised.
Cheers
---
John R Ramsden # "No one who has not shared a submarine
# with a camel can claim to have plumbed
(j...@redmink.demon.co.uk) # the depths of human misery."
#
# Ritter von Haske
# "Adventures of a U-boat Commander".
Herman Rubin
John R Ramsden <j...@redmink.demon.co.uk> wrote:
>On 28 Sep 1999 21:15:38 GMT, penn...@aol.com (Penny314) wrote:
>> From mathematics students learned:
>> accuracy, elegance ,careful reading,mastery of complexity , style and grace.
>> Then , it was decided somehow that this was a terrible waste of time.
>> Now ,we have a generation of functional innumerates
>> who are also functional illiterates.
>> pennysmith
>It seems to me that the unspoken assumption behind the motives of
>modern educationalists who seek to toss out the demanding studies
>that Penny mentioned, and maybe even corporal punishment, is that
>kids today couldn't and shouldn't be expected handle it.
>In other words these intellectuals, doubtless themselves having
>benefited from a "classical" education (even if at the price of
>some drudgery and misery), are implicitly saying they are somehow
>_better_ than children today. Looking at it in that light, their
>breathtaking arrogance is clear.
Are they intellectuals? This can be questioned; they do not
seem to consider that gifted children should get a better
education than the somewhat subnormal ones; they are still
pushing age grouping over more learning for the bright.
Few of the educationists are moderately well versed in what
mathematics is. Unless they went into mathematics education,
the most real mathematics they were likely to have seen was
the old Euclidean geometry, and this is one of the things
they cannot see as important. To them, mathematics is the
set of routine tools to calculate answers.
The older ones at least had that. Most of the newer ones
have not had that, nor do they understand the idea of formal
arguments; they are primarily literary philosophers.
Even the ones involved in teaching mathematics do not know
anything about the foundations of mathematics. Notice that
the proposed NCTM standards never gets into formal proofs,
except for Euclidean geometry, if it is given.
>The trouble is this becomes a self-fulfilling prophesy, in that
>as a result of educationalists' obstructive attitudes many kids
>will find it much harder, or impossible, to achieve their full
>potential.
As I said, the obstructive attitudes are not limited to
mathematics. Most of those going to college do not have
grammar, either, unless they have some old-fashioned
teachers for foreign languages. The educationists'
attitude is that, if it is not currently relevant, it
should not be taught. Also, they cannot consider teaching
abstract concepts directly, but only as generalizations
of special cases.
>Apologies if to any teachers reading this it is no more than a
>Forrest Gumpish truism, and too obvious to mention, but I think
>it should be emphasised.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
dann...@here.com
Rubin) wrote:
>Few of the educationists are moderately well versed in what
>mathematics is. Unless they went into mathematics education,
>the most real mathematics they were likely to have seen was
>the old Euclidean geometry, and this is one of the things
>they cannot see as important. To them, mathematics is the
>set of routine tools to calculate answers.
>
This is the biggest pile of unsubstantiated crap I've ever witnessed.
There were some, but only the smallest handful of teachers whom I met
whom I would call incompetent. Your own competence is yet to be
established. However, if this is an example, you fail. You have no
real figures to back up your claim, and will not be getting any until
you become a teacher youself and see the reality rather than your own
myth.
Such claims as this make people read James Harris' feeble attempts at
fame.
Dan.
Curtis James
Herman Rubin <hru...@odds.stat.purdue.edu> wrote in message > Even the ones
involved in teaching mathematics do not know
> anything about the foundations of mathematics. Notice that
> the proposed NCTM standards never gets into formal proofs,
> except for Euclidean geometry, if it is given.
>
pp. 143-145 and pp. 157-160, Curriculum and Evaluation Standards for School
Mathematics, NCTM, 1989
pp. 316-318, Principles and Standards for School Mathematics: Discussion
Draft, NCTM, 1998 (draft)
There are other places that reasoning and proof are mentioned and NCTM has
other publications that discuss the role of proof. Often, the Mathematics
Teacher (NCTM high school journal) publishes articles about proof and their
roles. I believe an entire yearbook (a yearly publication focusing on
specific areas of mathematics) was devoted to reasoning and proof.
By the way, proof by mathematical induction is also mentioned in the above
cited pages.
I would love to discuss whether or not you think the Standards says enough
about the role of proof and reasoning. I would assume you do not think so,
but by your statements, it is not clear whether you have appropriate
knowledge of the Standards to discuss them.
John R Ramsden
wrote:
>In article <37f62496...@news.demon.co.uk>,
>John R Ramsden <j...@redmink.demon.co.uk> wrote:
>>On 28 Sep 1999 21:15:38 GMT, penn...@aol.com (Penny314) wrote:
>
>>> From mathematics students learned:
>>> accuracy, elegance ,careful reading,mastery of complexity , style and grace.
>>> Then , it was decided somehow that this was a terrible waste of time.
>>> Now ,we have a generation of functional innumerates
>>> who are also functional illiterates.
>>> pennysmith
>
>>It seems to me that the unspoken assumption behind the motives of
>>modern educationalists who seek to toss out the demanding studies
>>that Penny mentioned, and maybe even corporal punishment, is that
>>kids today couldn't and shouldn't be expected handle it.
>
>>In other words these intellectuals, doubtless themselves having
>>benefited from a "classical" education (even if at the price of
>>some drudgery and misery), are implicitly saying they are somehow
>>_better_ than children today. Looking at it in that light, their
>>breathtaking arrogance is clear.
>
> Are they intellectuals? This can be questioned;
I was using that term in a disparaging sense, to indicate someone
who is more concerned to be seen as being up to speed on all the
latest fads and theories, and as a natural consequence is likely
to despise and/or loath anything that preceded them. It's rather
like people who mainly eat out in order to be seen at the most
fashionable restaurants, rather than to enjoy a good meal and
conversation.
Obviously one can go to the other extreme and be a reactionary
stick in the mud, and it would be foolish to ignore new ideas
or adapt old ones. But to an intellectual, as I meant the word,
novelty far outranks intrinsic merit.
John R Ramsden
>On 2 Oct 1999 20:59:27 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>
>>Few of the educationists are moderately well versed in what
>>mathematics is. Unless they went into mathematics education,
>>the most real mathematics they were likely to have seen was
>>the old Euclidean geometry, and this is one of the things
>>they cannot see as important. To them, mathematics is the
>>set of routine tools to calculate answers.
>
> This is the biggest pile of unsubstantiated crap I've ever witnessed.
> There were some, but only the smallest handful of teachers whom I met
> whom I would call incompetent.
Danny [?], I was using the word "educationalist" to mean someone who trains
teachers, like a strategist at a military academy as opposed to a "grunt"
at the front line. (Not a very flattering description, but many teachers
these days might consider it quite apt. What was it they said about WW1
soldiers - "lions led by donkeys" ?-)
I can't speak for Herman, but presumably he inferred that meaning and was
using the word in the same sense.
John R Ramsden
wrote:
>
> Obviously one can go to the other extreme and be a reactionary
> stick in the mud, and it would be foolish to ignore new ideas
> or adapt old ones. But to an intellectual, as I meant the word,
> novelty far outranks intrinsic merit.
I meant "or _not_ adapt old ones".
Herman Rubin
Curtis James <jam...@postnet.com> wrote:
The use of clear concepts and proofs need to come before drill
on computation. Variables, not just for mathematical entities,
need to come with beginning reading, and all at once; it is
harder to use them for this, and then for that, etc.
Proof should come quite early. The full restricted predicate
calculus HAS been taught to above average fifth graders, and
I believe that, with minor modifications, it can be taught in
essentially the same way to third graders. Mathematical induction
should come much earlier, even in discussing what integers are
to first graders.
Sound mathematics in the NCTM program is too little, in that
the impression is given that proof is overly restricted and
not that basic, and too late, in that it is not started and
stressed in the primary grades.
I have read the 1989 document, and the emphasis on what I
consider to be the trivia, manipulations that add nothing
to insight, and not presenting the simple general concept
rather than more complicated special cases to be quite
damaging. For probability and statistics, for example,
someone who goes through this, or any other computational
approach, will have great difficulty in understanding the
concepts later.
Curtis James
>
> The use of clear concepts and proofs need to come before drill
> on computation. Variables, not just for mathematical entities,
> need to come with beginning reading, and all at once; it is
> harder to use them for this, and then for that, etc.
>
> Proof should come quite early.
as professional mathematicians would create, a high school geometer, or
merely arguments that are thoughtful and convincing? I do believe that
some geometric proofs, such as triangle congruence, can in fact be taught
to, understood by, and developed independently by 5th graders (most of it is
elementary logic and many would be ready for it). They could learn to
develop 2-column proofs, but I really don't consider these anything more
than rudimentary.
The full restricted predicate
> calculus HAS been taught to above average fifth graders, and
> I believe that, with minor modifications, it can be taught in
> essentially the same way to third graders. Mathematical induction
> should come much earlier, even in discussing what integers are
> to first graders.
I am afraid that I don't understand what you mean by "full restricted
predicate calculus". I would also appreciate a reference to this study (or
studies) in which this occurred.
>
> Sound mathematics in the NCTM program is too little, in that
> the impression is given that proof is overly restricted and
> not that basic, and too late, in that it is not started and
> stressed in the primary grades.
Again, depending upon your definition of "proof", I would argue that
"mathematical reasoning" is stressed throughout the grades. Perhaps more
can be said by the Standards about proof and reasoning. I think it will
eventually, as more people who care about mathematics education becomes
involved in the process of deciding what's important. It is often said that
the US mathematics curriculum is a "mile wide and an inch deep." The 1989
document (and the 2000 document, no doubt) will still reflect this. My
general impression of the standards is that it didn't tell me I couldn't
teach proof or that I shouldn't teach proof because it's too difficult for
the masses. I inferred that I should be teaching less specifics and more
generalities of mathematics.
> I have read the 1989 document, and the emphasis on what I
> consider to be the trivia, manipulations that add nothing
> to insight, and not presenting the simple general concept
> rather than more complicated special cases to be quite
> damaging. For probability and statistics, for example,
> someone who goes through this, or any other computational
> approach, will have great difficulty in understanding the
> concepts later.
>
I wonder why we continued to use the ideas of Newton and Leibniz for so long
without the definition of the limit? I think it was because their ideas
were checked by other mathematician numerically. Neither had the modern
definition of the limit but both intuitively understood what a limit was. I
imagine they had performed some "computations" on functions that had limits
do back up their ideas. True, computation alone will not do the trick, ask
Fermat. He was notorious for checking a few numerical examples and then
claiming he had a theorem but didn't have the time to write the proof (FLT
wasn't the only one). But then again, why does the Monty Hall Problem cause
so much consternation among even the mathematically minded? I know plenty
of people who have only been convinced through simulations of the
experiment. I guess what I am saying is that we have to teach understanding
and computation hand in hand. The computation provides concrete examples of
what the concepts say. Of course computation can get in the way of concepts
at times--mathematical rigor can also get in the way. Don't the two
complement each other?
Dr. Michael Albert
> calculus HAS been taught to above average fifth graders, and
> I believe that, with minor modifications, it can be taught in
> essentially the same way to third graders. Mathematical induction
> should come much earlier, even in discussing what integers are
> to first graders.
An interesting idea, but I'm dubious. For example, truth-tables
and the first-order predicate calculus (ie, A imples B and B implies C
in turn imples A implies C) can be made into very much rote
calculation without understanding. I'd be equally dubious
that people learn much if their first introduction to integers
in Peano's axioms.
I will, however, argue that drill and calculation without understanding
is of little use. By understanding, let me give an example. One
can teach distributivity a*(b+c) = a*b + a*c by rote (indeed, you
can get young folks to carry out rather long formal calculations).
You can also prove it from Peano's axioms. But I still say
that the first introdution to this should be a diagram like this:
A A A B B B B B
A A A B B B B B
A A A B B B B B
A A A B B B B B
which "proves" that 4*(3+5) = 4*3 + 4*5. I believe such a diagram
appears in (rats, I can't think of the name, but it's a high school
level algebra book by a very famous Russian mathematician).
As far as proof, most students don't seem aware that mathematical
truth is judged by anything other than the authority of their
teacher or their textbook. While showing proofs is usually
worthwhile (even if students don't "get it" just then) one
really must first get students to realize that 1)they are
capable of deciding truth for themesleves (for example, by
testing special cases computationaly) and 2) that it really
is easy to be fooled into believing perfectly sensible statements
which are simply false.
Best wishes,
Mike
Herman Rubin
Curtis James <jam...@postnet.com> wrote:
>> The use of clear concepts and proofs need to come before drill
>> on computation. Variables, not just for mathematical entities,
>> need to come with beginning reading, and all at once; it is
>> harder to use them for this, and then for that, etc.
>> Proof should come quite early.
>Please define what you constitute a proof to be here. Do you mean a proof
>as professional mathematicians would create, a high school geometer, or
>merely arguments that are thoughtful and convincing?
The last are merely plausibility arguments, not proofs. There are
cases where they should be used.
A proof is a sequence of statements satisfying certain specific
rules. Those in high school geometry are of this type. What is
generally done by a professional mathematician is to create an
argument which is believed to be extendible to such a formal
sequence, although usually the details are omitted. This can
be done in the early grades.
This should be preceded by the use of formal symbolic language,
often called "mathematical notation", in first grade. The use
of this language should not be restricted to mathematical
entities.
I do believe that
>some geometric proofs, such as triangle congruence, can in fact be taught
>to, understood by, and developed independently by 5th graders (most of it is
>elementary logic and many would be ready for it). They could learn to
>develop 2-column proofs, but I really don't consider these anything more
>than rudimentary.
These are far more than rudimentary. In fact, using the method
of proof called natural deduction, which is easier to follow,
proofs are 3-column, the first column being the set of labels
of the premises involved. These proofs could be done even in
first grade if the ordinal approach to the integers is used.
There is a difference between understanding what constitutes
a proof and being adept at producing proofs. The former is
important for all, while the latter only for those with the
ability to do it.
>The full restricted predicate
>> calculus HAS been taught to above average fifth graders, and
>> I believe that, with minor modifications, it can be taught in
>> essentially the same way to third graders. Mathematical induction
>> should come much earlier, even in discussing what integers are
>> to first graders.
>I am afraid that I don't understand what you mean by "full restricted
>predicate calculus". I would also appreciate a reference to this study (or
>studies) in which this occurred.
The fifth grade book is, _First course in mathematical logic_
by Patrick Suppes and Shirley Hill. It is no longer in print;
it does not include existential arguments. One which is, and
in my opinion is just as easy if some of the applications are
omitted, is _Mathematical logic : applications and theory_ by
Jean E. Rubin. This is intended for college students, but
would only need small modifications to be used by elementary
school children.
>> Sound mathematics in the NCTM program is too little, in that
>> the impression is given that proof is overly restricted and
>> not that basic, and too late, in that it is not started and
>> stressed in the primary grades.
>Again, depending upon your definition of "proof", I would argue that
>"mathematical reasoning" is stressed throughout the grades. Perhaps more
>can be said by the Standards about proof and reasoning. I think it will
>eventually, as more people who care about mathematics education becomes
>involved in the process of deciding what's important. It is often said that
>the US mathematics curriculum is a "mile wide and an inch deep." The 1989
>document (and the 2000 document, no doubt) will still reflect this. My
>general impression of the standards is that it didn't tell me I couldn't
>teach proof or that I shouldn't teach proof because it's too difficult for
>the masses. I inferred that I should be teaching less specifics and more
>generalities of mathematics.
See my comments above. The production of proofs is difficult
for many, but the understanding is not. In general, we ask
students to do too much "solving" and not enough formulating
and understanding. One way to put this at a low level is that
it does no good to know how to add if one does not know when.
And if one knows when, if one can get it done otherwise, such
as by using a calculator, does knowing how become that
important? And yet, form kindergarten through calculus and
beyond, the overwhelming emphasis is on the calculations.
With the present curriculum, I would make the mathematics
requirement for college entrance that the student could
formulate long word problems, requiring many variables, at the
high school algebra level, but not asking anything about the
solution. This would have to be remedial now; such an exam
would be failed by most. But if this is present, mathematics
could really be used in courses in other fields.
>> I have read the 1989 document, and the emphasis on what I
>> consider to be the trivia, manipulations that add nothing
>> to insight, and not presenting the simple general concept
>> rather than more complicated special cases to be quite
>> damaging. For probability and statistics, for example,
>> someone who goes through this, or any other computational
>> approach, will have great difficulty in understanding the
>> concepts later.
>I wonder why we continued to use the ideas of Newton and Leibniz for so long
>without the definition of the limit?
Newton's contemporaries had the concept of limit, even though
there was not yet a formal definition. Read how he presented
his laws to them; they would not accept calculus, but very
definitely would accept arguments by limits. The Greeks, at
least after Euclid and Archimedes, had a good understanding
of limit. They understood an integral as a limit, even in
those cases they had no means of calculating it. One of the
contributions of Archimedes was in making the arc length of
a circle, and the surface area of a sphere, plausible by an
appropriate limiting argument. For the area of a circle and
the volume of a sphere, it was already clear.
I think it was because their ideas
>were checked by other mathematician numerically.
There was little numerical checking; there was much checking
of manipulations.
Neither had the modern
>definition of the limit but both intuitively understood what a limit was. I
>imagine they had performed some "computations" on functions that had limits
>do back up their ideas.
This goes back to the Greeks. Even without algebraic notation,
they did a fair amount of this, including providing algorithms,
long before the name of al-Khwarismi was given to it.
True, computation alone will not do the trick, ask
>Fermat. He was notorious for checking a few numerical examples and then
>claiming he had a theorem but didn't have the time to write the proof (FLT
>wasn't the only one).
Did Fermat claim that in any communication to anyone else? His proof
for 4 was later, and many believe that he thought this would work in
general.
But then again, why does the Monty Hall Problem cause
>so much consternation among even the mathematically minded?
Does it? It might among those who do not understand probability.
Now this might be because they are only taught to compute; the
ideas are not fostered by these computations.
I know plenty
>of people who have only been convinced through simulations of the
>experiment.
Why is simulation of this very simple problem necessary? As the
problem is essentially a computational problem, computation is
necessary, but no method of computation here is going to need a
number larger than 6. The problem arises when people use wrong
computations, because they do not understand the question.
I guess what I am saying is that we have to teach understanding
>and computation hand in hand. The computation provides concrete examples of
>what the concepts say.
Examples are useful, if carefully done. They usually are not,
and to avoid problems, the examples should follow the presentation,
not precede it.
Of course computation can get in the way of concepts
>at times--mathematical rigor can also get in the way. Don't the two
>complement each other?
Not always. But concepts are not proofs, and what the users
of mathematics need most are the concepts.
Herman Rubin
Dr. Michael Albert <alb...@esther.rad.tju.edu> wrote:
>> Proof should come quite early. The full restricted predicate
>> calculus HAS been taught to above average fifth graders, and
>> I believe that, with minor modifications, it can be taught in
>> essentially the same way to third graders. Mathematical induction
>> should come much earlier, even in discussing what integers are
>> to first graders.
>An interesting idea, but I'm dubious. For example, truth-tables
>and the first-order predicate calculus (ie, A imples B and B implies C
>in turn imples A implies C) can be made into very much rote
>calculation without understanding.
What you have given is the sentential calculus, not predicate
calculus. That this situation is subject to rote calculation
does not, however, mean that the ideas should not be taught.
The definition of "implies" is going to give quite a few real
problems; that a false proposition implies anything is
counterintuitive, but it is the only form which allows a
consistent approach, as does the problem of the meaning of
disjunction.
The predicate calculus involves quantifiers.
I'd be equally dubious
>that people learn much if their first introduction to integers
>in Peano's axioms.
This has not been tried, to my knowledge. However, all the
Peano approach does is to axiomatize counting. The arithmetic
operations are characterized by their relationship with counting.
>I will, however, argue that drill and calculation without understanding
>is of little use. By understanding, let me give an example. One
>can teach distributivity a*(b+c) = a*b + a*c by rote (indeed, you
>can get young folks to carry out rather long formal calculations).
>You can also prove it from Peano's axioms. But I still say
>that the first introdution to this should be a diagram like this:
> A A A B B B B B
> A A A B B B B B
> A A A B B B B B
> A A A B B B B B
>which "proves" that 4*(3+5) = 4*3 + 4*5.
It verifies it, using the cardinal approach to numbers. And
even this is sloppy; one would need to add tags, as the elements
of a set are distinct. The ordinal and cardinal approaches can
be taught together. The new math approach was strictly
cardinal, and incomplete. How do you know that a set is finite?
At this level, and this will work at any level, it is because
when counting it out, one will stop; this is part of the key
idea of induction.
I believe such a diagram
>appears in (rats, I can't think of the name, but it's a high school
>level algebra book by a very famous Russian mathematician).
>As far as proof, most students don't seem aware that mathematical
>truth is judged by anything other than the authority of their
>teacher or their textbook. While showing proofs is usually
>worthwhile (even if students don't "get it" just then) one
>really must first get students to realize that 1)they are
>capable of deciding truth for themesleves (for example, by
>testing special cases computationaly)
This only allows deciding when something is false.
and 2) that it really >is easy to be fooled into believing
perfectly sensible statements >which are simply false.
It would be necessary to provide lots of examples where
the few calculated cases work, but the general result does
not. In general, counterexamples should be amply provided
to point out limitations.