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.
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
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
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
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
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.
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.
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