[sci.math] Subtle Math Questions

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.edu

ghot@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 +=+

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. 

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

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

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)

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.EDU

brs@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?

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)

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