rrwood@contact.uucp (roy wood) (04/21/91)
I'm collecting a list of subtle math questions designed to stump high-school mathematics teachers. For example, the question "why, when you are dividing by a fraction, do you invert it and multiply?" is typical of the sort of thing I'm interested in. The idea is not to focus on an extremely difficult or obscure mathematical topic, but to come up with a question that relates to a simple high-school level topic, seems innocent, and hopefully questions something no-one ever bothers to ask about. If you have a good question along this line, please e-mail it to me. I'll post a summary of the responses to these groups in a week or two. Thanks, -Roy Wood (rrwood@contact.uucp)
ghot@ms.uky.edu (Allan Adler) (04/22/91)
Roy Wood (rrwood@contact.uucp) solicits questions designed to stump high school math teachers. In view of the public discussion by some politicians, such as George Bush and, in Kentucky, Martha Wilkinson (wife of the current Governor and now running for the office for which he is consitituionally forbidden from seeking a second term) of competency testing for teachers in public schools, it is reasonable to ask: for what purpose will these questions be used ? For competency testing of teachers (i.e., the development of a product to be sold to politicians who probably could not pass such a test either) or as course materials for people planning to become teachers ? The level of the questions suggested by Roy Wood by way of example also raise some questions. Are high school math teachers going to be expected to understand fractions but no higher level ? Are high school math teachers going to be teaching fractions (which are taught over and over again for years prior to high school) but no higher level ? Another question is this: apart from the purpose which these questions are expected to serve, what exactly are these tests supposed to measure ? For example, while it is undoubtedly desirable for a teacher to know the answers to such questions, the answer that is correct from the standpoint of the person grading the test may not be the answer that the naive student who asks the disconcerting question needs to hear. Allan Adler ghot@ms.uky.edu
ghot@ms.uky.edu (Allan Adler) (04/22/91)
In my reply to Roy Wood's posting, I misstated George Bush's position regarding testing. I was under the impression that he wanted competency testing for teachers, but I am unable to confirm this and was probably mistaken. I also mentioned Martha Wilkinson, gubernatorial candidate in Kentucky and wife of the current governor who is not allowed to succeed himself, as advocating teacher competency testing. That is an accurate description of her position. Bush has announced that he plans to become computer literate, so for all I know he may be reading this. :-) Allan Adler ghot@ms.uky.edu
mjo@ttardis.UUCP (Mike O'Connor) (04/22/91)
In article <1991Apr21.194019.352@ms.uky.edu>, ghot@ms.uky.edu (Allan Adler) writes: >schools, it is reasonable to ask: for what purpose will these questions >be used ? For competency testing of teachers (i.e., the development of a >product to be sold to politicians who probably could not pass such a test >either) or as course materials for people planning to become teachers ? Whoa... sounds like you're getting a bit defensive there. Why don't you ASK the poster what his/her point was rather than make all sorts of presumptions and insinuations? >The level of the questions suggested by Roy Wood by way of example also >raise some questions. Are high school math teachers going to be expected If I remember right, he gave only one question as an example. It seems to me that you're reading too much into all of this. ...Mike Phone: TTARDIS Public Access Unix -- (313) 350-2585 Internet: mjo%ttardis@uunet.uu.net UUCP ("domain"): mjo@ttardis.UUCP UUCP (bang): ...!uunet!sharkey!cfctech!ttardis!mjo
rrwood@contact.uucp (roy wood) (04/22/91)
Actually, as the original poster of the "subtle math questions" article, I'd like to point out that I have no "hidden agenda" for the use of these questions. The worst use I have for these questions is to try and stump my friend and Math Department Head. Actually, he'd probably enjoy nothing better than to be stumped, so I really appreciate the questions I've already received. As I said, I'll post a summary for you all..... -Roy Wood
foster@ted.cs.uidaho.edu (04/23/91)
Normally, this posting would have been a private response. But I have a VERY GOOD reason for proposing that we ALL see such a list of questions. Math education in this country is very poor. In part, this is because teachers at lower levels are either not good at math or do not pursue math very deeply. I do not mean this perjoratively. They have little incentive to be good at math. I conclude that it is up to US, we favored few, to tell the students what math is and why it's interesting. YES, I am proposing we volunteer some time in the local schools. One great way to do a one-class talk on math is to ask some little questions which should bug the heck out of a student who really wants to master math. Then have a discussion about the problem. The "Subtle Math Questions" would be great to use in this way. Note that even the most ardent of us are probably only going to donate a day or two every now and then. So we can't expect to actually TEACH much. But we can teach the student to ask critical questions and, more importantly, to discuss and think about the answers. James
ghot@ms.uky.edu (Allan Adler) (04/23/91)
Now that Roy Wood has explained that he has no hidden agenda, I would
like to contribute some "subtle" questions off the top of my head.
(1) Any positive real number can be represented as an infinite
decimal (e.g.3.14159265358979323846...), possibly ending in
all zeroes or all ones. We teach students how to add decimals.
How do we add positive real numbers represented as infinitely
long decimals ? How do we subtract or multiply or divide them ?
(2) We routinely allow students to use calculators. We do not
normally teach them how to know how much confidence they
can have in the answer the calculator gives. Of course,
that depends to some extent on the calculator and on the
problem it is given.
(a) What are some simple tests we can give to a calculator to
determine the nature of the errors it will give us ?
(b) Take a calculator, take the square root of 2, square the
answer, take the square root of the answer, square the result,
and repeat this a dozen times or more. Explain to your
weakest student why this is happening and how much confidence
this student should have in the device he/she is using in
view of this.
(3) You will need a Friden desl calculator for this: how many
interesting rhythms can one play on this device ? How many
can one play on a modern calculator ?
(4) Is i greater than 0 or less than 0 ? (i is the square root of -1).
(5) Galileo gives constructions for regular pentagons and regular
7-gons somewhere in his collected works. How accurate are his
constructions ? (Yes, look them up. That's where I found them.)
(6) What are the last 4 digits of 5 to the 7777th power ?
(YOu are not allowed to use a calculator. Anyone who uses a
calculator will be expelled, their reputation tarnished, their
future ruined and their children left to fend for themselves
in a cold and hostile world.)
(7) Are any of the telephone numbers (7 digits, or 10 with the area
code) at your school perfect cubes ?
(8) Once I was in the Science Center at Harvard on the 5th floor and
passed someone who was frantically trying to get into the men's
room but did not know the combination. Figuring that at Harvard
one could expect someone to figure it out with a little hint,
I told the person that the number is the sum of the cubes of
its digits and walked away. Question: how many solutions would
a person have to try before finding the right combination, in
the worst case ?
(9) We can reduce the fraction 95/19 to lowest terms by cancelling
the 9's, right? When is it safe to use this rule ?
(10) When we teach children to reduce fractions to lowest terms,
we teach them to do it by factoring. We often teach them to
factor by trying to divide by primes. We teach them to decide
whether a number is prime by telling them that it is divisible
only by itself and 1 (which, naively means that we have to try
all numbers less than the number), presumably because they
are not scheduled to learn square roots for several years.
Question: Why don't we teach them to use the Euclidean algorithm
to reduce fractions to lowest terms ?
(11) True or false: x^2-x+41 is always prime ? This is a good exercise
because lots of students ignore general statements and guess
the general rules based on examples. This example shows that
statements can be false in spite of "overwhelming" numberical
evidence.
Please don't send me the answers. I already know them.
Allan Adler
ghot@ms.uky.edughot@ms.uky.edu (Allan Adler) (04/23/91)
Instead of "all ones", please read "all nines". Allan Adler ghot@ms.uky.edu
ndanger@lightning.Berkeley.EDU (Norman Danner) (04/23/91)
In article <1991Apr22.221923.2370@groucho> foster@ted.cs.uidaho.edu writes: ... >I conclude that it is up to US, we favored few, to tell the students what math >is and why it's interesting. YES, I am proposing we volunteer some time in >the local schools. ... Hear, hear!! ------------------------------------------------------------------------ norman |"It must be admitted that even among "The guy with the hair." | intellectuals there are some really ndanger@plasma1.ssl.berkeley.edu| intelligent people." ndanger@ocf.berkeley.edu |-M. Bulgakov, _The Master & Margaritta_ ------------------------------------------------------------------------
ghot@ms.uky.edu (Allan Adler) (04/23/91)
The laudable suggestion has been made that we volunteer some time in the local schools. Presumably the term "we" refers to people who are not already working in the local schools. I think there might be some value in trying to articulate what exactly "we" might do when we go there to donate our time ? "We" don't all have to do the same thing and in fact "we" might find it useful to draw up a list of the things "we" might do, just in case any one of "us" is short on ideas. The first thing "we" should do is talk to "them". I think "they" might have some information that might be useful to "us", if not the other way around, and in addition "our" impressions of "them" might also be stimulating. I have not defined the term "them". I'm sure "they" have their own definition of "them" which might not mean "us", and we mgiht also talk to those that "they", at various times, refer to as "them". The second thing that "we", who do not work in the local schools, should do is to draw up a list of necessary and sufficient conditions under which "we" would be willing to abandon our separate status and work in the local schools. In making up this list, "we" should not be swayed by our impressions of what is possible. The purpose of the list is as much to present an alternative picture of local education, since there seems to be a real need for one. The third thing that "we" should do is to make some simple computations, based on the list of necessary and sufficient conditions, of pertinent figures related to funding this alternative picture: how many students, how many working hours, how many students and how many hours does a teacher have to teach, how many teachers does that require, how much do they have to be paid, how much equipment is required ancillary to various approaches to teaching (such as computers or laboratories in physical sciences or in design of sculptures or machines), how many more books and which books and at what cost, and what will it cost to guarantee us the time and f flexibility and resources for our own scholarship, and so forth ? Then "we" should bring this list (including signatures) to the attention of politicians, media and other bodies concerned with the reform of education and point out that there is an alternative to whatever they may have been planning on. Finally, "we" should pause and wonder why it is that "we" think that the working and educational environment which "we" would insist on for ourselves is not necessary unless "we" happen to be working there. "We" will feel a little bit better making a charitable donation of our time to the local schools, but "we" cannot seriously expect by such means to bridge the gap between what the schools are and what they ought to be. Allan Adler ghot@ms.uky.edu
jimh@welch.jhu.edu (Jim Hofmann) (04/23/91)
In article <1991Apr22.221923.2370@groucho> foster@ted.cs.uidaho.edu writes: >Normally, this posting would have been a private response. But I have a VERY >GOOD reason for proposing that we ALL see such a list of questions. > >Math education in this country is very poor. In part, this is because >teachers at lower levels are either not good at math or do not pursue >math very deeply. I do not mean this perjoratively. They have little >incentive to be good at math. > >I conclude that it is up to US, we favored few, to tell the students what math >is and why it's interesting. YES, I am proposing we volunteer some time in >the local schools. > >One great way to do a one-class talk on math is to ask some little questions >which should bug the heck out of a student who really wants to master math. >Then have a discussion about the problem. The "Subtle Math Questions" >would be great to use in this way. > >Note that even the most ardent of us are probably only going to donate a day or >two every now and then. So we can't expect to actually TEACH much. But we >can teach the student to ask critical questions and, more importantly, to >discuss and think about the answers. > >James Excellent Idea! Career day is a good time to start or fine a school with a math fair. At the fair, you'll see where their interest lies and build off that. Added thought, tutoring. Have undergrads do some one-on-one tutoring. If thereis one area that all teachers agree with is the lack of individual help to the students who really need it. The undergrads will benifit in 2 ways. They will find out how much they do know and they will see what it is like on the other side as a teacher. The students will benifit from the help. Sometimes it only takes a sentence or two and you can save the students hours of frustration. Another benifit for the student is varitity. The see the same math teach ALL year. With a program like tutoring, they will see different views of math and hopefully see someone that likes math. I must defend the math teachers that are out there now. TRUE, there are gym \ and art teachers in math class rooms, but there a some excellent math talent in the system. Unfortunetly, its not only the pay the keeps talent away. Professionalism is missing. Think about it. Hall duty, bus duty, caf. duty, bathroom duty, ect...... Jim
deghare@daisy.waterloo.edu (Dave Hare) (04/23/91)
In article <1991Apr23.124929.2180@welch.jhu.edu> jimh@welchlab.welch.jhu.edu (Jim Hofmann) writes: >Excellent Idea! Career day is a good time to start or fine a school with a >math fair. ^^^^ I suspect that that would be counterproductive :-)
lwallace@javelin.sim.es.com (Raptor) (04/23/91)
I think it would be a great service if you would post the answers to your quiz.
--
Lynn Wallace | I do not represent E&S.
Evans and Sutherland Computer Corp.| Internet: lwallace@javelin.sim.es.com
Salt Lake City, UT 84108 | Compu$erve: 70242,101
Revenge is a dish best not served at all.simon@bowfin.cs.washington.edu (Kevin Simonson) (04/23/91)
In article <1991Apr22.235606.10856@ms.uky.edu> ghot@ms.uky.edu (Allan Adler) writes: = =Now that Roy Wood has explained that he has no hidden agenda, I would =like to contribute some "subtle" questions off the top of my head. = = = =(1) ... =(6) What are the last 4 digits of 5 to the 7777th power ? = (YOu are not allowed to use a calculator. Anyone who uses a = calculator will be expelled, their reputation tarnished, their = future ruined and their children left to fend for themselves = in a cold and hostile world.) Allan, I REALLY didn't use a calculator for this. For all i > 0 5^(4i) mod 10000 = 625. 7777 = 4*1944 + 1, so 5^7777 mod 10000 = 5^(4*1944+1) mod 10000 = 5(5^(4*1944)) mod 10000 = 5*625 = 3125. Somebody with a calculator might want to check me on this. ---Kevin Simonson
mjo@ttardis.UUCP (Mike O'Connor) (04/23/91)
In article <1991Apr22.235606.10856@ms.uky.edu>, ghot@ms.uky.edu (Allan Adler) writes: >(2) We routinely allow students to use calculators. We do not > normally teach them how to know how much confidence they > can have in the answer the calculator gives. Of course, > that depends to some extent on the calculator and on the > problem it is given. > (a) What are some simple tests we can give to a calculator to > determine the nature of the errors it will give us ? Well... on an HP-11 or 15, you can take the cosine of pi/2 and get a number that is not zero. It's rather annoying. ...Mike Phone: TTARDIS Public Access Unix -- (313) 350-2585 Internet: mjo%ttardis@uunet.uu.net UUCP ("domain"): mjo@ttardis.UUCP UUCP (bang): ...!uunet!sharkey!cfctech!ttardis!mjo
mjo@ttardis.UUCP (Mike O'Connor) (04/23/91)
In article <1991Apr23.014114.3603@ms.uky.edu>, ghot@ms.uky.edu (Allan Adler) writes: >The laudable suggestion has been made that we volunteer some time in the >local schools. Presumably the term "we" refers to people who are not already >working in the local schools. > >I think there might be some value in trying to articulate what exactly >"we" might do when we go there to donate our time ? "We" don't all have >to do the same thing and in fact "we" might find it useful to draw up a list >of the things "we" might do, just in case any one of "us" is short on ideas. > >The first thing "we" should do is talk to "them". I think "they" might have etc. I think that "we" all get the point. Why do I get this picture in my head of a college math department swarming on a local high school, ousting the current regime of high school math teachers, and replacing with a brand-new, more highly educated regime ? What I'd really like to see is for "you" to teach these budding HS math teachers better, so "we" don't have to suffer through their miseducation! :) Phone: TTARDIS Public Access Unix -- (313) 350-2585 Internet: mjo%ttardis@uunet.uu.net UUCP ("domain"): mjo@ttardis.UUCP UUCP (bang): ...!uunet!sharkey!cfctech!ttardis!mjo
ewright@convex.com (Edward V. Wright) (04/24/91)
In article <1991Apr22.235606.10856@ms.uky.edu> ghot@ms.uky.edu (Allan Adler) writes: > >(1) Any positive real number can be represented as an infinite > decimal (e.g.3.14159265358979323846...), possibly ending in > all zeroes or all ones. We teach students how to add decimals. An infinitely long decimal that *ends* in zero or one??? This is a trick question, right? >(5) Galileo gives constructions for regular pentagons and regular > 7-gons somewhere in his collected works. How accurate are his > constructions ? (Yes, look them up. That's where I found them.) I'm not sure how many high-school libraries have the collected works of Galileo.
mayne@delta.cs.fsu.edu (William Mayne) (04/24/91)
First: I think one subtle question which should definitely go on the list of things math teachers should know well (but often don't) is one seen recently here (and answered to death): Why is 0!=1? I recall asking my teachers this when I was first exposed to it, and not getting a satisfactory answer. It is actually a pretty interesting question, leads into some real math, and helps show how some things aren't as arbitrary as they seem (contrary to what I was told when I asked this question way back when.) Second: Why are radians used as the preferred measure for angles (in some situations)? Similarly, why is e so important? How are trig tables figured? Granted, the real answer (as I see it) requires going into a little calculus and might be beyond most students, but teachers ought to know it and maybe be able to explain in general terms to students sufficiently advanced to be studying trig. Third: If they still teach how to find square roots by hand, using the algorithm which produces the digits one after another working on two digit chunks of the argument, not the methods in which successive approximation is obvious, teachers should know the justification for it. Fourth: Why isn't 1 considered a prime number? As far as I know (I am not a mathematician) this really is somewhat arbitrary. It makes the definition of a prime arguably easier, or at least the question of the least prime factor of a number more useful, but is this a good reason? Here is a problem, copied from rec.humor, which illustrates possible confusion when people don't remember that 1 is not a prime. Perhaps showing it to students would reinforce that lesson: In article <8098@utacfd5.UUCP> slh@utacfd5.UUCP (Scot Haire) writes: > > One drab day when Perce and Eve were reduced to thumb-twiddling, Perce > suddenly brightened and said, "I'll think of a positive number of 75 or > less. Ask me yes-or-no questions and see how quickly you can guess the > number." Eve, who had never been known to ask an irrelevant question, > plunged in thusly: > > 1. Is it a prime number? > > 2. Is it divisible by 2? > > 3. Is it divisible by 3? > > 4. Is it divisible by 5? > > 5. Is it less than 25? > > The questions are given in the order Eve asked them. After the fifth > question was answered - and not before - Eve had ferreted out Perce's > number. Can you find the number and the answers Perce gave to the five > questions? I'll bet most people who solve this (at least most non-mathematicians) say 15. The first answer I thought of was 1. Verifying that leads to the other possiblity, 49. I don't know if there are more possible answers besides these three. Actually since there are at least three numbers and corresponding answer sets which satisfy the requirements and the question was "Can you find the number" the answer should be "no." How many would accept "no" to a "Can you find..." question on test (assuming someone who knows enough could)? Bill Mayne
ghot@ms.uky.edu (Allan Adler) (04/24/91)
Edward V. Wright points out that most high school libraries do not have the collected works of Galileo. The same is probably true of most libraries accessible to a high school math department head. This points to the need for better libraries and for a zealous concern for keeping editions of great works in print. It does not point to the censorship of reasonable questions. Allan Adler ghot@ms.uky.edu
new@ee.udel.edu (Darren New) (04/24/91)
In article <1991Apr23.144230.14500@mailer.cc.fsu.edu> mayne@cs.fsu.edu writes: >Why isn't 1 considered a prime number? >As far as I know (I am not a mathematician) this really is somewhat >arbitrary. Actually, I ran across a "good" reason in a class a few years ago. I don't really remember what it was, but it was something to do with either order statistics or factorials or discrete probability or something like that; i.e., nothing to do with primes or modulos as such. Maybe somebody out there can really come up with the answer. Unfortunately, I don't have time to find all my old class notes this month... -- Darren -- --- Darren New --- Grad Student --- CIS --- Univ. of Delaware --- ----- Network Protocols, Graphics, Programming Languages, FDTs ----- +=+=+ My time is very valuable, but unfortunately only to me +=+=+ +=+ Nails work better than screws, when both are driven with screwdrivers +=+
tycchow@phoenix.Princeton.EDU (Timothy Yi-chung Chow) (04/24/91)
Two more examples: 1. Is 1/0 equal to infinity? 2. Is 0.9999999... equal to one or infinitely less than one? -- Tim Chow tycchow@phoenix.princeton.edu
henry@zoo.toronto.edu (Henry Spencer) (04/24/91)
In article <2730@ttardis.UUCP> mjo@ttardis.UUCP (Mike O'Connor) writes: >Well... on an HP-11 or 15, you can take the cosine of pi/2 and get >a number that is not zero. It's rather annoying. Did you really expect anything else, from floating-point arithmetic? -- And the bean-counter replied, | Henry Spencer @ U of Toronto Zoology "beans are more important". | henry@zoo.toronto.edu utzoo!henry
balden@wimsey.bc.ca (Bruce Balden) (04/24/91)
What is so "natural" about natural logarithms, considering that their base cannot be conveniently written down exactly? If you write 10^10^10 in ordinary 10-point type, will the answer string around the yard, around the planet, around the solar system, or to the next star? (for mathematical physicists) Name four principles of physics violated in each and every episode of Star Trek. -- ******************************************************************************* * Bruce E. Balden Computer Signal Corporation Canada * * Thaumaturgist 225B Evergreen Drive * * balden@xenophon.wimsey.bc.ca Port Moody, B.C. V3H 1S1 CANADA *
ndallen@contact.uucp (Nigel Allen) (04/24/91)
Since people are discussing high school mathematics education,
I thought the following message might be appropriate.
High school and junior college mathematics teachers:
Are you interested in starting a math club at your school?
If so, you may want to get in touch with Mu Alpha Theta,
the national high school and junior college mathematics club
sponsored by the National Council of Teachers of Mathematics
and the Mathematical Association of America.
It has chapters across the U.S. and Canada.
Mu Alpha Theta publishes a quarterly newsletter and other interesting
publications, and sponsors an annual convention every August.
For more information, contact:
Mu Alpha Theta
601 Elm, Room 423
Norman, Oklahoma 73019 or phone (405) 325-4489 voice.foster@ted.cs.uidaho.edu (04/24/91)
I think I mis-implied something in my posting about volunteerism. I spoke of "we favored few" ironically to mean us working mathematicians. I did not mean anything perjorative about current teachers...though there is little incentive for good math teachers to teach K12 and as a result there are not as many as there should be. Nor did I mean to imply that "we" should drop everything and donate all of our time to K12 education without sacrificed our academic status. Mine was a modest proposal. I had in mind ocassional visits to the local schools to let the students know that mathematics is a live (literally) subject. James
hansz@ruuinf.cs.ruu.nl (Hans Zantema) (04/24/91)
In article <1991Apr23.144230.14500@mailer.cc.fsu.edu>, mayne@delta.cs.fsu.edu (William Mayne) writes: > > Why isn't 1 considered a prime number? > > As far as I know (I am not a mathematician) this really is somewhat > arbitrary. It makes the definition of a prime arguably easier, or > at least the question of the least prime factor of a number more > useful, but is this a good reason? Here is a problem, copied from A main theorem of number theory is the unique factorization: every natural number can be written as a product of prime numbers. This prime number decomposition is unique up to the order. If 1 is considered as a prime number then this uniqueness is not true any more. This is a very good reason for not considering 1 a prime number. Every math teacher should know this. with kind regards, Dr Hans Zantema Department of Computer Science University of Utrecht P.O. Box 80.089 3508 TB Utrecht The Netherlands.
wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys) (04/24/91)
In article <51667@nigel.ee.udel.edu> new@ee.udel.edu (Darren New) writes: >In article <1991Apr23.144230.14500@mailer.cc.fsu.edu> mayne@cs.fsu.edu writes: >>Why isn't 1 considered a prime number? >>As far as I know (I am not a mathematician) this really is somewhat >>arbitrary. > >as such. Maybe somebody out there can really come up with the answer. >Unfortunately, I don't have time to find all my old class notes Prime = not a product of smaller factors. You want your theory to look neat. How would you formulate "every integer has a unique (up to order) decomposition in prime factors" if you could throw in an arbitrary number of factors 1? By the way, for some of the subtle math questions see the list of Frequently Asked Questions that is posted to sci.math every now and then. JWN
pjh@mccc.edu (Pete Holsberg) (04/24/91)
In article <2731@ttardis.UUCP> mjo@ttardis.UUCP (Mike O'Connor) writes: =In article <1991Apr23.014114.3603@ms.uky.edu>, ghot@ms.uky.edu (Allan Adler) writes: =Why do I get this picture in my head of a college math department swarming =on a local high school, ousting the current regime of high school math =teachers, and replacing with a brand-new, more highly educated regime =? = =What I'd really like to see is for "you" to teach these budding HS =math teachers better, so "we" don't have to suffer through their =miseducation! I think you'll find that the majority of primary and secondary school math teachers do not get their math education from a college's math department in "regular" math courses but either from a regular college's math department's special math courses for wannabes, OR from the math departments of teachers colleges!! :-( In either case, the students are not expected to learn much math at all. (My ex-wife is now a HS math teacher and her education matches the "ed major" model implied above.) Pete -- Prof. Peter J. Holsberg Mercer County Community College Voice: 609-586-4800 Engineering Technology, Computers and Math UUCP:...!princeton!mccc!pjh 1200 Old Trenton Road, Trenton, NJ 08690 Internet: pjh@mccc.edu Trenton Computer Festival -- 4/20-21/91
new@ee.udel.edu (Darren New) (04/24/91)
In article <1991Apr23.214223.7549@wimsey.bc.ca> balden@wimsey.bc.ca (Bruce Balden) writes: >(for mathematical physicists) Name four principles of physics violated in each >and every episode of Star Trek. Old or new generation? I think the new generation is quite a bit better than the old when it comes to "realism." Care to enlighten us? -- Darren -- --- Darren New --- Grad Student --- CIS --- Univ. of Delaware --- ----- Network Protocols, Graphics, Programming Languages, FDTs ----- +=+=+ My time is very valuable, but unfortunately only to me +=+=+ +=+ Nails work better than screws, when both are driven with screwdrivers +=+
mjo@ttardis.UUCP (Mike O'Connor) (04/25/91)
In article <1991Apr23.212624.5276@zoo.toronto.edu>, henry@zoo.toronto.edu (Henry Spencer) writes: >In article <2730@ttardis.UUCP> mjo@ttardis.UUCP (Mike O'Connor) writes: >>Well... on an HP-11 or 15, you can take the cosine of pi/2 and get >>a number that is not zero. It's rather annoying. > >Did you really expect anything else, from floating-point arithmetic? I just found it amusing because the problem does not seem to occur on cheaper, "lesser" calculators. Also, because I was burned on it once while in high school. Such is life... ...Mike Phone: TTARDIS Public Access Unix -- (313) 350-2585 Internet: mjo%ttardis@uunet.uu.net UUCP ("domain"): mjo@ttardis.UUCP UUCP (bang): ...!uunet!sharkey!cfctech!ttardis!mjo
ken@aiai.ed.ac.uk (Ken Johnson) (04/25/91)
In article <8641@idunno.Princeton.EDU> tycchow@phoenix.Princeton.EDU (Timothy Yi-chung Chow) writes: >2. Is 0.9999999... equal to one or infinitely less than one? Suppose X=0.9999... Then 10X = 9.99999... Therefore 10X-X = 9.999... - 0.999... => 9X = 9 => X = 1 ps By `infinitely' you probably meant `infinitesimally'. -- Ken Johnson, /// Stamp out ATHLETE'S FOOT! 80 South Bridge, Edinburgh /// Save up to #50,000! Draw your own badge! E-mail ken@aiai.ed.ac.uk /// Welcome to Earth. No-one gets out of here alive 031-650 2756 direct line /// Muslims say: Hands Off Shoplifters
ronerwin@milton.u.washington.edu (04/25/91)
I agree with the tutoring theme. Subtle math isn't the problem, there are many children and adults afraid of math - but math doesn't have to be scary, it's simpler and more logical than English. Many calculators now store formulas and written words. I have such a calculator and it has an excellent memory. But to contradict my weak point, we don't need to rote memorize math - the key to math is the logical processes within the math. So let's not be subtle, let's be very obvious. Ron Erwin ronerwin@cac.washington.edu
suriano@iitmax.iit.edu (candice suriano) (04/25/91)
In article <1991Apr23.235053.6458@groucho> foster@ted.cs.uidaho.edu writes: >I think I mis-implied something in my posting about volunteerism. I spoke of >"we favored few" ironically to mean us working mathematicians. I did not mean >anything perjorative about current teachers...though there is little incentive >for good math teachers to teach K12 and as a result there are not as many >as there should be. Nor did I mean to imply that "we" should drop everything >and donate all of our time to K12 education without sacrificed our academic status. >Mine was a modest proposal. I had in mind ocassional visits to the local schools >to let the students know that mathematics is a live (literally) subject. > >James I applaud your idea of volunteerism, but you may have a tough time getting any school to let you, especially a public elementary school. For example, in Illinois it is illegal for a child to be in the school library without a certified teacher in a certified position being present. My daughter's school lost their librarian. The idea was that some parents could spend an hour or two a month as volunteers. We all saw it as a great way to be involved, help out, and keep our taxed down. No way!! The kids can't be there without their regular classroom teacher or a certified librarian. They have an aide who is a certified librarian but she doesn't count because the aide position is not a certified position!! And they're having trouble finding a new librarian because the school year is almost over. But we do get to volunteer. We shelve books and put the plastic covers on the new ones. That frees the aide to help the teachers who can then bring the kids to the library!! The idead behind this brilliant law is that only people who know something about elementary ed should be teaching the kids. On the one hand it really makes me mad, but when I look at some of the parents who might be teaching my child I'm sort of glad I'm protected this way. Anyway, my point is, before you get too excited about volunteering, you need to check to see what you're allowed to do. (And at my daughter's school it's only clerical) Next I get to help duplicate computer disks :-). Candi
shimeall@taurus.cs.nps.navy.mil (timothy shimeall) (04/25/91)
Most of these questions have focussed on "puzzle-solving" as a measure of math ability. Having the opportunity, I asked Richard Hamming (of Hamming codes, Hamming integral, Hamming integration method and Hamming primes fame...) the question of how to judge a person's math understanding. He indicated that he wouldn't ask them to solve puzzles. Instead, he'd ask them to define the field of mathmatics, the activities mathematicians perform and the skills needed by students of mathematics. The trick is, he isn't looking for any specific response. He feels that if the respondent is able to put together ANY well-reasoned response, that would be sufficient. He makes to cogent point that much of the subtle points of mathematics are open to debate and to change -- its far better that the teachers and students have a view of the broad scope of the field, its reasoning and its use than if they're able to solve a laundry-list of puzzles. Tim -- Tim Shimeall ((408) 646-2509)
eepjm@cc.newcastle.edu.au (04/26/91)
In article <1991Apr23.214223.7549@wimsey.bc.ca>, balden@wimsey.bc.ca (Bruce Balden) writes: > What is so "natural" about natural logarithms, considering that their base > cannot be conveniently written down exactly? Not a very tricky question, given that the answer is standard high-school material. > If you write 10^10^10 in ordinary 10-point type, will the answer string around > the yard, around the planet, around the solar system, or to the next star? (10^10)^10, or 10^(10^10)? It makes a big difference. > (for mathematical physicists) Name four principles of physics violated in each > and every episode of Star Trek. Aha, at last a *very* tricky question. The correct answer is, of course, "I don't know". Any other response will get you labelled as the sort of moron who actually watches Star Trek. (Watches? Watched? Is this still running? It must be 25 years since I last saw an episode of this.) Peter Moylan eepjm@cc.newcastle.edu.au
hrubin@pop.stat.purdue.edu (Herman Rubin) (04/27/91)
In article <1991Apr24.142835.26475@mccc.edu>, pjh@mccc.edu (Pete Holsberg) writes: > In article <2731@ttardis.UUCP> mjo@ttardis.UUCP (Mike O'Connor) writes: > =In article <1991Apr23.014114.3603@ms.uky.edu>, ghot@ms.uky.edu (Allan Adler) writes: > =Why do I get this picture in my head of a college math department swarming > =on a local high school, ousting the current regime of high school math > =teachers, and replacing with a brand-new, more highly educated regime > =What I'd really like to see is for "you" to teach these budding HS > =math teachers better, so "we" don't have to suffer through their > =miseducation! > I think you'll find that the majority of primary and secondary school > math teachers do not get their math education from a college's math > department in "regular" math courses but either from a regular college's > math department's special math courses for wannabes, OR from the math > departments of teachers colleges!! :-( In either case, the students > are not expected to learn much math at all. (My ex-wife is now a HS > math teacher and her education matches the "ed major" model implied above.) The situation is even far worse than this. The "regular" math courses have also declined; even in a good school, the mathematics (or physics, or chemistry, or whatever) department cannot really maintain standards. I believe it can be done, but only by refusing to recognize credits. There are remedial courses, but they are taught on the assumption that the student was unable to learn the subject when taught in high school, rather than the more appropriate assumption that the subject was so badly taught that the situation may even be worse than it it had not been. At Purdue, the mathematics department can legally maintain standards for prospective teachers, but what would happen if it did? Purdue would turn out very few HS teachers, and they would have very little, if any, advantage over those turned out by other schools which do not have standards. To the school superintendant, a C in an honest course on the foundations of analysis is bad, while an A in a course with 1% of the content is good. As long as the current grade-credit system is being used as information to others, this state will continue. I have argued for change, not merely in these groups. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 Phone: (317)494-6054 hrubin@l.cc.purdue.edu (Internet, bitnet) {purdue,pur-ee}!l.cc!hrubin(UUCP)
mjl@cs.rit.edu (Michael J Lutz) (04/30/91)
In article <11415@mentor.cc.purdue.edu>, hrubin@pop.stat.purdue.edu (Herman Rubin) writes: > The situation is even far worse than this. The "regular" math courses have > also declined; even in a good school, the mathematics (or physics, or > chemistry, or whatever) department cannot really maintain standards. Actually, it's even worse than this -- the rigor and challenge of most liberal arts courses is also on the wane. > At Purdue, the mathematics department can legally maintain standards for > prospective teachers, but what would happen if it did? Purdue would turn > out very few HS teachers, and they would have very little, if any, advantage > over those turned out by other schools which do not have standards. To the > school superintendant, a C in an honest course on the foundations of analysis > is bad, while an A in a course with 1% of the content is good. I've tinkered with the idea of grading based on both achievment and difficulty of material -- you know, like diving in the Olympics. Courses would be given ``degrees of difficulty (dod)'' (assigned by faculty in each discipline). Students would receive a standard letter grade for achievement, which would be multiplied by the ``dod'' to arrive at a cumulative difficulty rating. I'd probably use the raw grades to determine academic standing, but the difficulty rating of would let prospective employers or graduate schools know whether the new graduate was a belly-whopper or a Greg Louganis (sp?). --------- Mike Lutz Rochester Institute of Technology Rochester, NY 14623-0887 mjl@cs.rit.edu
blaak@csri.toronto.edu (Raymond Blaak) (04/30/91)
ghot@ms.uky.edu (Allan Adler) writes: >(6) What are the last 4 digits of 5 to the 7777th power ? How about the FIRST 4 digits? Cheers, Ray
kym@bingvaxu.cc.binghamton.edu (R. Kym Horsell) (04/30/91)
In article <1991Apr30.002142.21112@jarvis.csri.toronto.edu> blaak@csri.toronto.edu (Raymond Blaak) writes: >ghot@ms.uky.edu (Allan Adler) writes: >>(6) What are the last 4 digits of 5 to the 7777th power ? > >How about the FIRST 4 digits? > >Cheers, >Ray 7757 -kym
kym@bingvaxu.cc.binghamton.edu (R. Kym Horsell) (04/30/91)
In article <1991Apr30.072900.7073@bingvaxu.cc.binghamton.edu> kym@bingvaxu.cc.binghamton.edu (R. Kym Horsell) writes: >In article <1991Apr30.002142.21112@jarvis.csri.toronto.edu> blaak@csri.toronto.edu (Raymond Blaak) writes: >>ghot@ms.uky.edu (Allan Adler) writes: >>>(6) What are the last 4 digits of 5 to the 7777th power ? >> >>How about the FIRST 4 digits? >> >>Cheers, >>Ray > > >7757 > >-kym Repeatedly divide by 2, starting with 1, and keep first 4 nonzero digits after dp? -kym
kym@bingvaxu.cc.binghamton.edu (R. Kym Horsell) (04/30/91)
In article <1991Apr30.002142.21112@jarvis.csri.toronto.edu> blaak@csri.toronto.edu (Raymond Blaak) writes: >ghot@ms.uky.edu (Allan Adler) writes: >>(6) What are the last 4 digits of 5 to the 7777th power ? > >How about the FIRST 4 digits? > >Cheers, >Ray As I've had several requests for the secret of my posted solution: /* calculate ms 4 digits of 5^7777 */ ipow(ix,iy) { double u=1; double x=ix; while(x>1) x/=10; while (iy) { if(iy&1) { u*=x; while(10*u<1) u*=10; } x*=x; while(10*x<1) x*=10; iy>>=1; } return u*10000; } main() { printf("%d\n",ipow(5,7777)); } -kym
kurtze@plains.NoDak.edu (Douglas Kurtze) (04/30/91)
In article <1991Apr24.142835.26475@mccc.edu> pjh@mccc.edu (Pete Holsberg) writes >I think you'll find that the majority of primary and secondary school >math teachers do not get their math education from a college's math >department in "regular" math courses but either from a regular college's >math department's special math courses for wannabes, OR from the math >departments of teachers colleges!! :-( In either case, the students >are not expected to learn much math at all. (My ex-wife is now a HS >math teacher and her education matches the "ed major" model implied above.) This is a strong argument for mathematicians to get involved in research on how students (at whatever level) learn mathematics, how to present concepts, what misconceptions exist, etc. The results could then feed into the courses for wannabes, who could then learn the mathematics they need AND how to teach it. The latter would, of course, require that they understand thoroughly the material they are about to teach. It's unlikely that they will get that in an education course, without deep involvement of mathematicians. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Doug Kurtze kurtze@plains.NoDak.edu Physics, North Dakota State "Patience is its own reward" -- Flann O'Brien ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
csuwr@warwick.ac.uk (Derek Hunter) (05/01/91)
Sorry to bug you all, but can you restrict this to USA distribution only please? - Derek Hunter
Chris.Holt@newcastle.ac.uk (Chris Holt) (05/01/91)
csuwr@warwick.ac.uk (Derek Hunter) writes: >Sorry to bug you all, but can you restrict this to USA distribution >only please? Why? Do you think we don't have the same problems here? ----------------------------------------------------------------------------- Chris.Holt@newcastle.ac.uk Computing Lab, U of Newcastle upon Tyne, UK ----------------------------------------------------------------------------- "And when they die by thousands why, he laughs like anything." G Chesterton
ljdickey@watmath.waterloo.edu (L.J.Dickey) (05/02/91)
In article <1991Apr24.142835.26475@mccc.edu> pjh@mccc.edu (Pete Holsberg) writes: >I think you'll find that the majority of primary and secondary school >math teachers do not get their math education from a college's math >department in "regular" math courses but either from a regular college's >math department's special math courses for wannabes, OR from the math >departments of teachers colleges!! :-( In either case, the students >are not expected to learn much math at all. (My ex-wife is now a HS >math teacher and her education matches the "ed major" model implied above.) Fortunately, there are a few nice exceptions to these models, and students at Waterloo are some of them. Here, students in the Faculty of Mathematics who are enrolled in our Teaching Option alternate study terms and work terms. During their eight study terms they work on their undergraduate degree in Mathematics, and during their work terms the do supervised teaching. At the end of their five year programme, they have earned a degree called Bachelor of Mathematics, Honours, and the right to attend a one term course at the nearby teacher's college where they get their teaching credentials. This is a far cry from special courses for wannabes. -- Prof L.J. Dickey, Faculty of Mathematics, U of Waterloo, Canada N2L 3G1 Internet: ljdickey@watmath.waterloo.edu UUCP: ljdickey@watmath.UUCP ..!uunet!watmath!ljdickey X.400: ljdickey@watmath.UWaterloo.ca
ssingh@watserv1.waterloo.edu ( Ice ) (05/02/91)
So what are some examples of countries which have good math programs?
--
(1ST HYPERMEDIA .SIG) ; #include <black_rain.h> ; #include <robotron.h>
"Ice" is a UW AI living at: ssingh@watserv1.[u]waterloo.{edu|cdn}/[ca]
"The human race is inefficient and therefore must be destroyed"-Eugene Jarvis
Visual component of .sig: Saito in the cafe doing some slicing in _Black_Rain_ grant@psych.toronto.edu (Stuart Grant) (05/02/91)
>>I think you'll find that the majority of primary and secondary school >>math teachers do not get their math education from a college's math >>department in "regular" math courses but either from a regular college's >>math department's special math courses for wannabes, OR from the math >>departments of teachers colleges!! :-( In either case, the students >>are not expected to learn much math at all. (My ex-wife is now a HS >>math teacher and her education matches the "ed major" model implied above.) I agree that watered down courses in which students are not expected to learn are not much use to anyone. However, I don't think that this is the biggest problem with the math instruction in primary and secondary schools. _Any_ math course taught at a college or university will be at least as sophisticated as what teachers will be teaching in primary and secondary schools. Not knowing how to do differential equations is not the greatest problem math teachers have. Calling the education faculty math courses wimpy, and making math teachers take "regular" math courses is not the answer. The quality of math instruction will improve if teachers are given more training in the teaching of math. Teaching math is difficult, Motivating students and getting across abstract concepts that the students have not used before is, I believe, the greatest difficulty. So, I think math instruction can be best improved not by teaching the teachers more math, but by giving them more teaching skills, including additinal training in how to teach math.
balden@wimsey.bc.ca (Bruce Balden) (05/03/91)
In article <1991May2.133856.8338@psych.toronto.edu> grant@psych.toronto.edu (Stuart Grant) writes: >>>I think you'll find that the majority of primary and secondary school >>>math teachers do not get their math education from a college's math >>>department in "regular" math courses but either from a regular college's >I agree that watered down courses in which students are not expected to learn >are not much use to anyone. However, I don't think that this is >the biggest problem with the math instruction in primary and secondary >schools. _Any_ math course taught at a college or university will be at >least as sophisticated as what teachers will be teaching in primary and >secondary schools. Not knowing how to do differential equations is not >the greatest problem math teachers have. Nevertheless, the good teacher of mathematics will have a deep appreciation of the way mathematics is actually used in the world at large and not just a good understanding of a traditional list of arithmetical and algebraic algorithms and formulas. The student who sees his mathematics teacher as inadequate, not only in the internal mechanics of the subject, but in success in making the subject relevant to the world at large, will correctly reason (YES, students are capable of reasoning) that this person has nothing of importance to tell him. Just as the coach of the football team is normally expected to be a good athlete well beyond the capabilities of the average high-school athlete, so should a high-school or even elementary school mathematics teacher be a source of inspiration. Currently, of course, we cannot attract people with the requisite combination of people and technical skills into the school system, particularly at the lower levels. Of course, the mathematics community itself is not immune to criticism in this regard. Take the college level, at which I have some experience. The "sexy" subject, regarded as the principal goal of a good engineering and science student is Calculus, which, in my experience, is one of the most bizarre and arcane subjects students ever encounter, being obsessed with complex derivative and integral calculations of dubious value. The dreary subject, reserved for "slow" student and non-specialists, is "Finite Mathematics". In my opinion, the topics in this course are far more relevant to the ordinary experiences of people than first year calculus. It is true of course, that if you want to extend these techniques and ideas much further, then you have to drag in a LOT of mathematical machinery, especially linear algebra, but there is no motivation to do so otherwise. Therefore, when the average second year student encounters linear algebra, he finds it a dry, if not extremely difficult subject and quickly forgets everything about the subject twenty minutes after the final exam. I have myself answered many net queries which would be quite unnecessary if these courses had any habit of sinking in. Let's face the basic truth: People in general lose interest in mathematics at an early age because the parts of the subject that they see are INTRINSICALLY uninteresting and unimportant. Even a slow student can figure out that his bank president doesn't know the fine points of long division. -- DISCLAIMER: Opinions expressed are my own, not those of my employer. ******************************************************************************* * Bruce E. Balden Computer Signal Corporation Canada * * Thaumaturgist 225B Evergreen Drive *
kludge@grissom.larc.nasa.gov ( Scott Dorsey) (05/03/91)
In article <1991May2.133856.8338@psych.toronto.edu> grant@psych.toronto.edu (Stuart Grant) writes: >Calling the education faculty math courses wimpy, and making math teachers >take "regular" math courses is not the answer. The quality of math >instruction will improve if teachers are given more training in the >teaching of math. Teaching math is difficult, Motivating students and >getting across abstract concepts that the students have not used before >is, I believe, the greatest difficulty. > >So, I think math instruction can be best improved not by teaching the >teachers more math, but by giving them more teaching skills, including >additinal training in how to teach math. I think that teachers tend to teach math the way they have been taught math. Which means that teaching them properly in the first place and giving them a good example is half the struggle. --scott
grant@psych.toronto.edu (Stuart Grant) (05/03/91)
In article <1991May02.171317.751@wimsey.bc.ca> balden@wimsey.bc.ca (Bruce Balden) writes: >In article <1991May2.133856.8338@psych.toronto.edu> grant@psych.toronto.edu (Stuart Grant) writes: >>>>I think you'll find that the majority of primary and secondary school >>>>math teachers do not get their math education from a college's math >>>>department in "regular" math courses but either from a regular college's >>I agree that watered down courses in which students are not expected to learn >>are not much use to anyone. However, I don't think that this is >>the biggest problem with the math instruction in primary and secondary >>schools. _Any_ math course taught at a college or university will be at >>least as sophisticated as what teachers will be teaching in primary and >>secondary schools. Not knowing how to do differential equations is not >>the greatest problem math teachers have. > >Nevertheless, the good teacher of mathematics will have a deep appreciation >of the way mathematics is actually used in the world at large and not just >a good understanding of a traditional list of arithmetical and algebraic >algorithms and formulas. The student who sees his mathematics teacher as >inadequate, not only in the internal mechanics of the subject, but in >success in making the subject relevant to the world at large, will correctly >reason (YES, students are capable of reasoning) that this person has nothing >of importance to tell him. Just as the coach of the football team is >normally expected to be a good athlete well beyond the capabilities of >the average high-school athlete, so should a high-school or even elementary >school mathematics teacher be a source of inspiration. I agree. I don't see why you begin with "Nevertheless", unless you believe that the use of mathematics is as some sort of way to bludgeon college students :-) The completion of any number of university math courses will not in itself, enable a teacher to motivate students. I don't believe that it is even necessary. Perhaps I should have elaborated when I suggested that math teachers should be given more help in motivating their students. This, I believe, would certainly include being able to show the real world relevance of the topic.
mcramer@watdragon.waterloo.edu (Mert Cramer) (05/03/91)
> Perhaps I should have elaborated when I suggested that > math teachers should be given more help in motivating their students. > This, I believe, would certainly include being able to show the real world > relevance of the topic. The notion that, within the present framework, any change in maths instruction will make a difference is naive. An informative discussion of the development (or not) of maths skills in pre-school children is in a BBC documentary called "Four plus four equals the wings of a bird". Among the points it makes: 1. For most people math is something you do at a desk and has no relivance to life problems. 2. The formal method of teaching math makes the subject which the student encounters which is NOT concrete (numbers apply to anything) hard to visualize. 3. The teaching of math concepts by exploration rather than by lecture is a more effective technique. This film was presented on David Suzuki's "The Nature of Things" program a few years ago. I recorded it but since is was the first use of my VCR it is a bit ragged. If you are in the Waterloo, Ont. area and want borrow the tape let me know. One the major points in the film is that the curiosity about math and numbers is largly destroyed by the usual techniques of promary teaching. You might say that anyone who has an interest in math by the time they get to university has survived in spite of all formal education has tried to do to them. Of course, the university maths education is in exactly the same distructive mold as all that went before so that anyone who survives at the univ. level is either really dedicated and intersested in math or a masochist (or both).
news@cec1.wustl.edu (USENET News System) (05/04/91)
It is not all that hard to liven math for students in grades 9-13. Anyone
can do it; it doesn't even need to be a teacher, just anyone who will talk
to the student a little. If you care about math or science, work up a good
talk for your neighborhood school's math or science club. The teachers in
charge are usually the best ones, and welcome outside speakers.
The Mathematics Association of America still has a program, in most sections,
to arrange such visits, I believe.
If you encounter a youngster who is studying algebra, ask if he or
she has been told why the rules are meaningful-- usually nothing has been
said about this. Then spend a few minutes showing that in clock arithmetic
you have funny things like 10 + 5 = 3 and 3*4= 0, and on the other hand
the base 2 "odd, even" arithmetic *does* satisfy all the rules.
I hope others have more examples of this type of goodie; we ought
to collect them-- they seem to help, now and then.
From: delliott@cec2.wustl.edu (Dave Elliott)
Path: cec2!delliott
David L. Elliott
Dept. of Systems Science and Mathematics
Washington University, St. Louis, MO 63130
delliott@CEC2.WUSTL.EDUbrs@cci632.cci.com (Brian Scherer) (05/04/91)
In article <10060@plains.NoDak.edu> kurtze@plains.NoDak.edu (Douglas Kurtze) writes: >In article <1991Apr24.142835.26475@mccc.edu> pjh@mccc.edu (Pete Holsberg) writes > >>I think you'll find that the majority of primary and secondary school >>math teachers do not get their math education from a college's math >>department in "regular" math courses but either from a regular college's >>math department's special math courses for wannabes, OR from the math >>departments of teachers colleges!! :-( In either case, the students >>are not expected to learn much math at all. (My ex-wife is now a HS >>math teacher and her education matches the "ed major" model implied above.) > >This is a strong argument for mathematicians to get involved in research on how >students (at whatever level) learn mathematics, how to present concepts, what >misconceptions exist, etc. The results could then feed into the courses for >wannabes, who could then learn the mathematics they need AND how to teach it. >The latter would, of course, require that they understand thoroughly the >material they are about to teach. It's unlikely that they will get that in an >education course, without deep involvement of mathematicians. > >^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ >Doug Kurtze kurtze@plains.NoDak.edu >Physics, North Dakota State > >"Patience is its own reward" -- Flann O'Brien >^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ I wish to disagree on the point that math teachers do not learn much. I went to a college that was a teachers college and a science college. The only difference between the major in Math and the major in Math education was areas outside of the math department. Like english etc. The math ed people had to take courses in lesson preperation, and the sych(sp) courses. The math courses were geared to the science arena and you had to have a B or better in them. I would like to make a comment about the so-called teachers that teach at a college. Do you know that most never have to take any educational courses to be able to teach? Many do not know how to write lesson plans, have good examples (worked out ahead of time), and really know how to present the material to the students. As an ex-secondary math teacher, who still teaches on the side, not only in the math arena, but in the computers (micro) and also for the bouy scouts I would guess that the whole area of education needs to be looked at and re-done. Brian Scherer
grant@psych.toronto.edu (Stuart Grant) (05/04/91)
In article <1991May3.124454.12758@watdragon.waterloo.edu> mcramer@watdragon.waterloo.edu (Mert Cramer) writes: >> Perhaps I should have elaborated when I suggested that >> math teachers should be given more help in motivating their students. >> This, I believe, would certainly include being able to show the real world >> relevance of the topic. > >The notion that, within the present framework, any change in maths instruction >will make a difference is naive. An informative discussion of the development >(or not) of maths skills in pre-school children is in a BBC documentary called >"Four plus four equals the wings of a bird". Among the points it makes: >1. For most people math is something you do at a desk and has no relivance to > life problems. Isn't this what I just acknowledged?
david.lloyd-jones@rose.uucp (DAVID LLOYD-JONES) (05/04/91)
Replying to : ssingh@watserv1.waterloo.edu ( Ice ) >Orga: University of Waterloo > >So what are some examples of countries which have good math programs? > My daughters, now 14 and 8, are in the regular Japanese school system, and the older one was able at the age of seven to tell me that her mean time to math error was about three weeks. Shortly after, in fourth grade, she explained to me that the way you solved equations was "by paying attention to the X, because that's the answer." The Japanese system, generally bad-mouthed as authoritarian and mindless, seems to me heavy on playfulness, thoroughness and deep understanding. In international comparisons of children's mathematical abilities, the Hungarians come out close to the top, up there with the Japanese. Anybody got anything to report on Hungary. (And am I correct in noting the oddity that the Hungarian and Japanese languages are related, rare members of the Somethingorother Altaic family?) -dlj. ---
david.lloyd-jones@rose.uucp (DAVID LLOYD-JONES) (05/04/91)
> >Nevertheless, the good teacher of mathematics will have a deep appreciation >of the way mathematics is actually used in the world at large and not just >a good understanding of a traditional list of arithmetical and algebraic >algorithms and formulas. The student who sees his mathematics teacher as >inadequate, not only in the internal mechanics of the subject, but in >success in making the subject relevant to the world at large, will correctly >reason (YES, students are capable of reasoning) that this person has nothing >of importance to tell him. I think this is very good, and is much more than the bland truism it looks like. Have you thought about how you would identify people with such a deep appreciation? What would you consider sound tests or screening processes? What would you think sound recruiting processes for such people? * * * Just as the coach of the football team is >normally expected to be a good athlete well beyond the capabilities of >the average high-school athlete, so should a high-school or even elementary >school mathematics teacher be a source of inspiration. Here, by contrast, I disagree with you; on the first half, not the second. The coach of a football team can operate from a wheelchair, or from crutches in the stands. What would be important, it seems to me, would be knowledge of the game and the intention to teach the kids the visualisation of learning strategies toward that knowledge. I agree with you, though, about inspiration. Again, as above, how do you intend to identify it? Or what would you consider a reasonable and testable proxy? * * * > >Currently, of course, we cannot attract people with the requisite combination >of people and technical skills into the school system, particularly at the >lower levels. There's an answer to this: it's called money. How much? When the candidates are half male, half female, you know you're hiring from the labour pool, not the cheap-labour pool. That's your first cut. Then you run your screening tests. If that doesn't give you enough, then you crank up the money to get more candidates. Some other benchmarks: if average income across the economy is $45,000 per family, then this should probably be your entry-level salary for teachers. (Or do you think teachers should be from the below average sector?) If the normal income of an anaesthesiologist is around $245,000 then you've got the range for a person with supervisory responsibility in currciculum. If the President of a major corporation usually gets $300 to 700,000, then you know what you ought to be paying State and Provincial top educational administrators. * * * > >Of course, the mathematics community itself is not immune to criticism in this >regard. Take the college level, at which I have some experience. The >"sexy" subject, regarded as the principal goal of a good engineering and >science student is Calculus, which, in my experience, is one of the most >bizarre and arcane subjects students ever encounter, being obsessed with >complex derivative and integral calculations of dubious value. > >The dreary subject, reserved for "slow" student and non-specialists, is >"Finite Mathematics". In my opinion, the topics in this course are far >more relevant to the ordinary experiences of people than first year calculus. Calculus was the fun paradigm for the age when change became general. I agree with you bout finite math: appropriate paradigm for the digital age. -dlj. ---
me@csri.toronto.edu (Daniel R. Simon) (05/05/91)
In article <1991May2.133856.8338@psych.toronto.edu> grant@psych.toronto.edu (Stuart Grant) writes: >I agree that watered down courses in which students are not expected to learn >are not much use to anyone. However, I don't think that this is >the biggest problem with the math instruction in primary and secondary >schools. _Any_ math course taught at a college or university will be at >least as sophisticated as what teachers will be teaching in primary and >secondary schools. Not knowing how to do differential equations is not >the greatest problem math teachers have. > >Calling the education faculty math courses wimpy, and making math teachers >take "regular" math courses is not the answer. The quality of math >instruction will improve if teachers are given more training in the >teaching of math. Teaching math is difficult, Motivating students and >getting across abstract concepts that the students have not used before >is, I believe, the greatest difficulty. Would it be acceptable for, say, a foreign language teacher to have a grounding in that language that barely goes beyond the high school level, or for anyone to argue that "*any* history course taught at a college or university will be at least as sophisticated as what teachers will be teaching in primary or secondary schools"? Is it sufficient for high-school art or music teachers to have "any" college- or university-level course before they teach? "There *is* confusion worse than death" Daniel R. Simon -Tennyson (me@theory.toronto.edu)
me@csri.toronto.edu (Daniel R. Simon) (05/05/91)
In article <1991May3.124454.12758@watdragon.waterloo.edu> mcramer@watdragon.waterloo.edu (Mert Cramer) writes: >The notion that, within the present framework, any change in maths instruction >will make a difference is naive. An informative discussion of the development >(or not) of maths skills in pre-school children is in a BBC documentary called >"Four plus four equals the wings of a bird". Among the points it makes: >1. For most people math is something you do at a desk and has no relivance to > life problems. >2. The formal method of teaching math makes the subject which the student > encounters which is NOT concrete (numbers apply to anything) hard to > visualize. Both of these points, as far as I can tell, apply equally well to absolutely any discipline that involves the least bit of abstraction. In most other disciplines, of course, developing the skills of abstraction is understood by everyone to be one of the key goals of teaching the subject, whereas mathematics teachers instead face constant skepticism from others about the value of learning the abstractions they teach. >3. The teaching of math concepts by exploration rather than by lecture is > a more effective technique. [......] >One the major points in the film is that the curiosity about math and numbers >is largly destroyed by the usual techniques of promary teaching. You might say >that anyone who has an interest in math by the time they get to university >has survived in spite of all formal education has tried to do to them. I, for one, would like to challenge, based on my personal experience, the common perception that the "usual techniques" for teaching mathematics, or any other subjects, are anything like what most people seem to think they are. When I look back casually on my elementary school years, I remember (oddly enough, given that I was raised in the sixties and seventies) exactly the same kind of interest-destroying tedium described above, not just in math class, but in every single one of the subjects I studied. On the other hand, if I reminisce with a little more care, I can vaguely remember that a huge fraction of my in-school time was subject to a virtually uninterrupted stream of attempts to make my education more "creative" and entertaining, most of which left me with feelings of enjoyment and enthusiasm, and virtually nothing else. My clearest memories by far are of learning "the old-fashioned way" through conventional, structured lessons, which imprinted their material in my brain much more effectively than the myriad "projects" and "activities" which (I seem to remember) delighted me at the time, but whose purpose now escapes me entirely. Am I really unique in having had such a diverting and unproductive time in public school? "There *is* confusion worse than death" Daniel R. Simon -Tennyson (me@theory.toronto.edu)
csuyx@cu.warwick.ac.uk (Wally..) (05/05/91)
In article <1991May2.192705.17581@news.larc.nasa.gov> kludge@grissom.larc.nasa.gov ( Scott Dorsey) writes:
[loadsa math-related ideas deleted..]
Now, either mathematicians shouldn't be allowed within two yards of a
computer, or mathematicians should realise that uw.general is not, I repeat
*not* a maths newsgroup.
I can see how this might confuse some mathematicians, but uw.general is a
University of Warwick general newsgroup. Not a maths forum in disguise.
Now, please remember this when pressing 'f' or 'F' for follow-up and read the
newsgroups line closely. If it includes 'uw.general', then remove it from the
list please. Cross posting is a pointless, dull and rude exercise.
Thanx, I feel better now.
Regards,
Wally..
--
O O 'They're coming to take me away, hi-hi, ha-ha, ho-ho. O O
o Away to the funny farm..' o
\_/ .siggy fault(core dumped) Regards, Wally.. \_/merigh@cpac.washington.edu (Mohamed Merigh) (05/05/91)
> I can see how this might confuse some mathematicians, but uw.general is a > University of Warwick general newsgroup. Not a maths forum in disguise. Funny, I used to think that it is a University of Washington newsgroup until I saw lots of postings from University of Waterloo... Are they all called UWs (You Dubb)? Mohamed.
ndallen@contact.uucp (Nigel Allen) (05/05/91)
David L. Elliott mentioned the idea of giving a talk to a school's math or science club. Teachers who would like help starting a math club may want to get in touch with Mu Alpha Theta, which is *not* a fraternity. It's a national high school and junior college mathematics club, and is sponosred by the Mathematical Association of America and the National Council of Teachers of Mathematics. It publishes a newsletter, and sponsors an annual (August) convention. For more information, write to: Mu Alpha Theta 601 Elm, Room 423 Norman, Oklahoma 73019 or phone (405) 325-4489 voice. (I have nothing to do with the group, but my father used to be editor of the newsletter.) Nigel Allen ndallen@contact.uucp
hrubin@pop.stat.purdue.edu (Herman Rubin) (05/05/91)
In article <1991May3.200312.10109@cci632.cci.com>, brs@cci632.cci.com (Brian Scherer) writes: ..................... > I wish to disagree on the point that math teachers do not learn much. > I went to a college that was a teachers college and a science college. > The only difference between the major in Math and the major in Math > education was areas outside of the math department. Like english etc. > The math ed people had to take courses in lesson preperation, > and the sych(sp) courses. The math courses were geared to the science > arena and you had to have a B or better in them. I have posted elsewhere on this topic. The math courses now taught to undergraduates rarely do anything appreciable towards understanding the concepts. > I would like to make a comment about the so-called teachers that > teach at a college. Do you know that most never have to take any > educational courses to be able to teach? Many do not know how > to write lesson plans, have good examples (worked out ahead of time), > and really know how to present the material to the students. It is a very good thing that they have not had that (insert your own derogatory term). Most of them have plenty of examples; too many of them are not very good, I will agree. The text usually has plenty of examples, if the students can read :-). Presenting the details of manipulations, which is what is now stressed too often, is utterly deadly, and of little use. How can concepts be taught? By being carefully presented, and then by making the students USE them in unusual situations until the light dawns. The concept of a proof cn be taught, but how to prove cannot be taught at all. The idea of using symbols can be taught, but how to formulate cannot; incorrect formulation can be pointed out, and examples of correct formulation given, but giving rules and buzz-words are more likely to cause harm. These topics are important, not how to compute sums and products, to solve formulated problems, to differentiate and anti-differentiate. > As an ex-secondary math teacher, who still teaches on the side, not only > in the math arena, but in the computers (micro) and also for the bouy scouts > I would guess that the whole area of education needs to be looked > at and re-done. I agree with the last sentence. But redone from the point of learning, not memorizing and passing examinations. If you have faced a group of undergraduates who have had the full two years of calculus, and cannot use their calculus on a take-home exam in problems not that much different from homework thoroughly discussed in class, I believe that you would agree that their calculus was worthless. We teach what can be forgotten after the final, and examine accordingly. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 Phone: (317)494-6054 hrubin@l.cc.purdue.edu (Internet, bitnet) {purdue,pur-ee}!l.cc!hrubin(UUCP)
dtate@unix.cis.pitt.edu (David M Tate) (05/06/91)
In article <11891@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes: > >How can concepts be taught? By being carefully presented, and then by making >the students USE them in unusual situations until the light dawns. The concept >of a proof cn be taught, but how to prove cannot be taught at all. The idea of >using symbols can be taught, but how to formulate cannot; What makes you say these things? I teach how to prove; I teach formulation. Yes, it's difficult as hell, but that doesn't mean you shouldn't even bother to try. >incorrect formulation >can be pointed out, and examples of correct formulation given, but giving rules >and buzz-words are more likely to cause harm. These topics are important[...] You equate "teaching" with "giving rules and buzz-words"!? No wonder you think it's impossible. Today is the first day of summer classes here at Pitt. This afternoon, I'll be teaching a required sophomore course which, if it had a name, would be called something like "Problem formulation and modelling". The whole point of the course is to teach the Art of Problem Formulation and "Solution" (with the scare quotes emphasized). It's about all those things you say can't be taught. I've taught this course once before, with striking improvement in the students' skills. Of course, I'd rather get at them in high school, but it's the best we can do... Of course, from reading your previous postings I know that you do not really think of teaching in the cynical terms you have used here. So I'm curious: why can't we teach the Art of Modelling the same way we teach other Arts? I took violin lessons for years; they weren't just "rules and buzz-words". (Of course, this requires that the instructor be an artists of sorts himself, and not a mere technician...) -- David M. Tate | "Your telegram has been sent, sir. You should be dtate@unix.cis.pitt.edu | receiving it in about an hour. We've sent your Motto: | bags ahead to your hotel. Where will you be Gramen artificiosum odi | staying?" --Firesign Theater.
bdb@becker.UUCP (Bruce D. Becker) (05/07/91)
In article <CSUYX.91May4191028@lily.warwick.ac.uk> csuyx@cu.warwick.ac.uk (Wally..) writes: | |Now, either mathematicians shouldn't be allowed within two yards of a |computer, or mathematicians should realise that uw.general is not, I repeat |*not* a maths newsgroup. | |I can see how this might confuse some mathematicians, but uw.general is a |University of Warwick general newsgroup. Not a maths forum in disguise. | |Now, please remember this when pressing 'f' or 'F' for follow-up and read the |newsgroups line closely. If it includes 'uw.general', then remove it from the |list please. Cross posting is a pointless, dull and rude exercise. "uw.general" is a newsgroup at the University of Waterloo in Waterloo, Ontario, Canada. It is also likely in existence at the University of Washington in Washington State, USA. "york.general" is York University, near Toronto, Ontario, Canada; "ut.general", aside from also being at the University of Texas, is also at the University of Toronto. I think you'll find an ongoing discussion here about these subjects. The fact that the University of Warwick is included is most likely an artifact of uunet's willingness to carry every possible newsgroup... -- ,u, Bruce Becker Toronto, Ontario a /i/ Internet: bdb@becker.UUCP, bruce@gpu.utcs.toronto.edu `\o\-e UUCP: ...!utai!mnetor!becker!bdb _< /_ "The really important problems require greater earnestness" - J. Cage
harkcom@spinach.pa.yokogawa.co.jp (Alton Harkcom) (05/14/91)
In article <a44e6214dec62822e944@rose.uucp> david.lloyd-jones@rose.uucp (DAVID LLOYD-JONES) writes: =}My daughters, now 14 and 8, are in the regular Japanese school system, I don't have any children in school yet, but one of my hobbies is education (not teaching though I have done that on occasion, but examining texts and schools to see how things are done). In particular I am interested in Mathematics as that is the subject which gets the worst treatment in US schools. =}The Japanese system, generally bad-mouthed as authoritarian and mindless, =}seems to me heavy on playfulness, thoroughness and deep understanding. Most of the criticism of Japanese schools teaching by rote may be true, but the math education in Japan is excellent. The children are drilled in the basics to the point that even long after finishing studying, they remember the processes involved. Math is taught at a much faster pace than in the US. I believe the foundations of algebra are layed early in elementary school and continued through to high school. The reason for this is thatEvery student is EXPECTED to learn it. A case in point: operators If given 5 + 8 x 2 x 0, most high school graduates can give the correct value that it represents (even if the don't like math)... They won't know that the value could change depending upon the notation or why the value would be what it is, though. It might be interesting if people emailed me what they feel the value of X would be and which notation they are using... Al
harkcom@spinach.pa.yokogawa.co.jp (Alton Harkcom) (05/15/91)
In article <HARKCOM.91May14084703@spinach.pa.yokogawa.co.jp> harkcom@spinach.pa.yokogawa.co.jp (Alton Harkcom) writes: =} A case in point: operators =}If given 5 + 8 x 2 x 0, most high school graduates can give the correct =}value that it represents (even if the don't like math)... They won't =}know that the value could change depending upon the notation or why =}the value would be what it is, though. I would like to thank all of the people who took the time to reply to this. I have already had a large enough response to satisfy my curiosity, but I still welcome any more comments. Everyone got the correct answer: '5' in standard notation as the multiplication operator has higher priority. I doubt the response would have been as good if I polled a wide selection of HS graduates in the US though. I would guess that the majority of high school graduates in Japan would have gotten it right. Al
duncan@ctt.bellcore.com (Scott Duncan) (05/15/91)
In article <HARKCOM.91May15113004@spinach.pa.yokogawa.co.jp> harkcom@spinach.pa.yokogawa.co.jp (Alton Harkcom) writes: > > Everyone got the correct answer: '5' in standard notation as the >multiplication operator has higher priority. I doubt the response >would have been as good if I polled a wide selection of HS graduates >in the US though. I would guess that the majority of high school >graduates in Japan would have gotten it right. I'm concerned that we care whether some other country's kids can get the right answer or not. If education in this country cannot produce students who can get the correct answer (even better, explain their reasoning, even if the answer was wrong), then who cares? Would we feel better about education in this country if it was as disappointing in others? (Perhaps yes since it seems fear over competition from the Russians in the 50's and now the Japanese in the 80's-90's is the major concern of many people.) Speaking only for myself, of course, I am... Scott P. Duncan (duncan@ctt.bellcore.com OR ...!bellcore!ctt!duncan) (Bellcore, 444 Hoes Lane RRC 1H-210, Piscataway, NJ 08854) (908-699-3910 (w) 609-737-2945 (h))
g_harrison@vger.nsu.edu (George C. Harrison, Norfolk State University) (05/16/91)
In article <1991May15.121134.29777@bellcore.bellcore.com>, duncan@ctt.bellcore.com (Scott Duncan) writes: > In article <HARKCOM.91May15113004@spinach.pa.yokogawa.co.jp> harkcom@spinach.pa.yokogawa.co.jp (Alton Harkcom) writes: >> >> Everyone got the correct answer: '5' in standard notation as the >>multiplication operator has higher priority. I doubt the response >>would have been as good if I polled a wide selection of HS graduates >>in the US though. I would guess that the majority of high school >>graduates in Japan would have gotten it right. > > I'm concerned that we care whether some other country's kids can get the right > answer or not. If education in this country cannot produce students who can > get the correct answer (even better, explain their reasoning, even if the > answer was wrong), then who cares? Would we feel better about education in > this country if it was as disappointing in others? (Perhaps yes since it seems > fear over competition from the Russians in the 50's and now the Japanese in the > 80's-90's is the major concern of many people.) > > Speaking only for myself, of course, I am... > Scott P. Duncan (duncan@ctt.bellcore.com OR ...!bellcore!ctt!duncan) > (Bellcore, 444 Hoes Lane RRC 1H-210, Piscataway, NJ 08854) > (908-699-3910 (w) 609-737-2945 (h)) Let me ask a very simple question: How many people in society (Japaneese or American [USA]) get asked such a question. If a person who has graduated from high school knows the solution to a question like 5 + 6 x 7 x 8 x 0, I would say "nice!" It would mean that they have remembered the rules. Would it mean that they would necessarily be productive members of society??? If a person does not KNOW the correct answer, should we assume that they are not worthy of being Chariman of the Board of a major company? I have a Ph.D. (Piled higher and Deeper) degree in [PURE!!] mathematics. I see the great growth in the 1990's being based in inventiveness and not in who KNOWS WHAT. For those who hire.... I have a question: "Would you HIRE a 4.0 student who has done nothing but do extremely well in college grades, or would you HIRE a 2.859 student who was President of the local academic club, raised 2 children, and has worked 40 hours a week as a manager at Hardies (MacDonalds, Burger King, Pizza Hut, etc)? George... George C. Harrison, Professor of Computer Science Norfolk State University, 2401 Corprew Avenue, Norfolk VA 23504Internet: g_harrison@vger.nsu.edu Phone: 804-683-8654
duncan@ctt.bellcore.com (Scott Duncan) (05/16/91)
In article <980.28319d22@vger.nsu.edu> g_harrison@vger.nsu.edu (George C. Harrison, Norfolk State University) writes: > >For those who hire.... I have a question: "Would you HIRE a 4.0 student who >has done nothing but do extremely well in college grades, or would you HIRE >a 2.859 student who was President of the local academic club, raised 2 >children, and has worked 40 hours a week as a manager at Hardies (MacDonalds, >Burger King, Pizza Hut, etc)? Well, I don't hire, but since George posted this in response to things I said, I'll dive in. Realistically, depending on what kind of company one works for (and its size), you may never even see the resume of the person with a 2.859 compared to a 4.0! I have been rejected by the Personnel folks at 3 of my last 4 jobs only to get an interview with an actual technical manager and end up with a job (or an of- fer of one which I turned down for a better offer at a place that had no formal Personnel screening process). I, by the way, was far from college by this time, but in at least one of the cases, I found out that the degree from 1969 was a problem for a job in 1983 despite 11+ years of industry experience! Some places have preestablished "standards" for degree expectations as well as grade point averages and have little or no orientation toward industry (or any other kind) of experience. They prefer people they can mold their own way. On the other hand, if I did get both resumes, none of what George mentions would be sufficient for me to give a job to either person. As I implied in my posting, I'd be interested in seeing how the applicants express themselves and their reasoning ability. I'd also be interested in what sorts of things they value in a job educationally, technically, etc. and why. My assumption (perhaps erroneous) based on what George says is that the 4.0 student probably went straight through to get the degree while the 2.859 stu- dent may have had to take more than 4 years to get the degree given their family and work situation. Thus their life experiences might be very different and that would probably affect how they present themselves and what they have come to value. While I do not make hiring decisions, I have over the years at several jobs done much interviewing and made recommendations to folks who do hire. I have not tended to end up favoring the "4.0 with nothing else to recommend them" kind of people versus the "2.859 who has thought about work and life" sort of person. However, I'm also a person who turned down going for a PhD because I liked to teach undergraduates (who often needed the help) rather than the graduate students (who at least acted a lot like they didn't need anything) and was clearly told that serious faculty didn't do the former. >George C. Harrison, Professor of Computer Science >Norfolk State University, 2401 Corprew Avenue, Norfolk VA 23504Internet: >g_harrison@vger.nsu.edu Phone: 804-683-8654 Speaking only for myself, of course, I am... Scott P. Duncan (duncan@ctt.bellcore.com OR ...!bellcore!ctt!duncan) (Bellcore, 444 Hoes Lane RRC 1H-210, Piscataway, NJ 08854) (908-699-3910 (w) 609-737-2945 (h))