Conference rusure::math

    re .0 (Naser)
    
         I agree with you on the point that learning to play math tricks on
    big number multiplication is a waste of time since it is just some
    silly algorithm that has very little to do with the fundamentals.
    
         However, I must say that there is a need to learn to add, subtract
    and memorize the multiplication table.  Math education is like
    constructing a building and those things are part of the foundation
    building process.  What you suggest sound a lot like the "New Math"
    stuff.
    
         Other than that most countries that are successful in math
    education (such as Japan) begin their math classes with arithmetics. 
    
    Eugene
                 

        Besides, when you are at the store you need to be able to
        tell whether the big peanut butter or the little peanut
        butter costs more per pound.
        
        Dan

    re .2
    
    At my local supermarket the price on the shelf is given in guilders per
    kilogram (as well as the actual price) so we don't even have to be able
    to do the division.

    re .3
    
    Yes, but it would be good if they knew whether 5.9 $/unit is
    more or les than 5.10 $/unit
    
    -Joe
    

.3>    At my local supermarket the price on the shelf is given in guilders per
.3>    kilogram (as well as the actual price) ...
    
    Now let's see...Dutch guilders are small relative to dollars in about the
    same proportion that kilograms are large relative to pounds.  Therefore
    the price ELIS::GARSON sees on the shelf is (a) more than what you'd
    see in the U.S., (b) less, (c) about the same or (d) none of the above
    
    Not seriously folks,
    
      John

    Ok guys, thats not what i really meant: a kid not knowing if a cockie 
    that cost 10 cents is more expensive than one that cost 5 cents because 
    10 > 4 .
    
    my point deals with more important stuff (i hoped), how to initiate young
    kids to math in a 'natural' way, it is obvious that dumping
    multiplication tables on them is not working, (for that majority,
    for some it worked , like with the fine math participants of this note file)
    
    but if US school kids are the lowest in math skills among the
    industrlized countires , some thing is wrong in the way we are
    introducing math to them.
    
    I conjucture that it is the emphasis on 'numbers' other that concepts
    is what is causing it.
    
    but again, iam not a teacher, so i could be wrong, the cause could
    be the video arcades or MTV or why bother with math when you can make
    a bundle more money running with a bal in your hand, or etc....
    
    /naser

    ref .-1 (me)
    > that cost 10 cents is more expensive than one that cost 5 cents
    > because 10 > 4 .
                  ^^^
                   5
    
    you see, just numbers dont work with me either, i must have missed
    the day they introduced them :-)
    
    /naser
    
    

    re .3
    
    In my supermarket in Acton, Ma., the unit prices provided are
    frequently wrong, occasionally grossly so, despite being printed on neat
    computer generated labels.  So I still have to do division. :-)

re .6
    
>    but if US school kids are the lowest in math skills among the
>    industrlized countires , some thing is wrong in the way we are
>    introducing math to them.

I've wondered how it is that the U.S. government can convincingly argue
that we are mathematically disadvantaged w.r.t. other countries when
they are simultaneously claiming that we don't even have a uniform
mechanism for comparing American schoolchildren against one another on
a common basis (the apparent purpose of the proposed national testing
programs).

What is the objective basis for such a claim ?
Will it really stand up under a careful analysis ?

(* caution: politically sensitive statements dead ahead *)

Is this yet another example of a Reagan-Bush smoke and mirrors operation
to make it appear that there is a 'problem' that they will solve by talking
up a storm and then doing nothing ?

This seems to be a favored technique of recent adminstrations and their
conveniently pliable congresses:
(1) Make a big public fuss over some problem that is going away by itself
(example: claim that a police state will 'fix' a drug problem, when what's
'fixing' it is actually shifting demographics.)
(2) Wait a few years while the problem naturally remits.
(3) Claim victory and congratulate your 'policy' for its 'success', and
use it to bolster further totally unrelated political agenda
(e.g. 'See how well it worked ? We need a more tightly controlled police
state')

(* end caution zone *)

I saw what I consider a good (and peripherally related to this topic) example
of this in a local paper recently.

Apparently to bolster his standing to local constituents, one of our
congresspeple has been pushing the idea that more school hours are the
important variable that will 'fix' the 'school performance problem'.

After the editorialising, a small table showing the number of school hours
and relative rank of about 10 countries was displayed.

A brief examination of this table cast serious doubt on the value of
aggregate school contact hours as a predictor of school performance.
For example, Sweden (among the lowest number of school hours comparable
, if not smaller than, the U.S.) ranked 3rd for overall performance right up
there with Japan and Germany, and several other countries with greater hours
(comparable to Japan and Germany) were near the bottom.

My conclusion is that there is more evidence of a math performance crisis
among congressional aides (products, no doubt, of the late 50's through
60' high school generations) than with the general school age population :-)

All this being said, I do not deny that I do see lots of evidence of
'innumeracy', but I have no strong evidence to suggest that this has not
always been so, or that things are really getting worse (at least among
the people I encounter).

.6>    I conjucture that it is the emphasis on 'numbers' other that concepts
.6>    is what is causing it.

    I conjecture that teaching methods have little to do with it; that the
    problem is that the kids don't study and their parent(s) don't take
    responsibility for their children's education.  Because their parents
    have been lulled into the left-wing notion that somehow the government
    should take care of things "for them".

    At least that is an explanation that is consistent with lack of
    correlation between money spent and/or school hours and/or test scores.
    
      John

I'll agree with .9 that we don't have solid evidence that a problem exists, how
bad it is or whether it is getting worse.

But for the sake of the discussion, I'll assume the following (which corresponds
to my gut feel):

	there is a problem
	it is bad enough to decrease our standard of living in the next century
	it is getting worse

.0>they instead should teach students things like what a function means, and 
>what the difference between a rational and irrational number, and where 
>does Pi come from. and what does an equation mean and etc..
>I mean some basic classical mathematics concepts. and numbers will follow
>by themselfs.

As .1 says, this is New Math, introduced in the 1960s in the US.  If anything,
it seems to have made the problem worse.  I don't think this is the direction
to move in.

.0>f you were this country's schools chief, what will change and do
>differently to really teach school kids real Mathematics ?

I'd begin by testing my gut feel: a lot of things (see .6) contribute,
but the big problem is the bizarre modern idea that learning is never hard work
or painful, if you are doing it right.

If testing failed to confirm my gut feel, I'd either resign or rethink the whole
problem.

If testing did confirm it, I would go public with it, telling everybody, 
parents, students, teachers and educationists that we need to learn math, and
that part of learning it was going to be real hard work.  Then I'd put a bunch
of folks to work analyzing math skills and concepts:

	what is easy to learn and what is hard and what is fun?
	what does every modern worker need to know?
	what do many modern workers need to know?
	what does every responsible citizen need to know?

Then I'd put a bunch of folks to work designing a curriculum, writing textbooks,
creating teacher training tools, creating PC student tools, designing tests to 
measure acquisition of skills and concepts, and testing all the above.  I'd aim 
for a program which included a lot of hard work, with fun stuff at regular 
intervals, as a reward and to show the point of the hard work.  I'd can 
what failed and distribute what worked.

    Designing curriculum, new textbooks, use of computers, and all the
    other trappings of educational science, as it were, will do
    little good in my opinion.  My understanding is that generation
    after generation of educators has tried this and we can see the
    results...

    What is really required is good teachers, and a society that values
    learning.  Without that, there's little that can be done about it.

    Also, I'm unconvinced about how "bad" students are these days,
    the fact that kids at a checkout counter can't make change
    notwithstanding.

    - Jim

.12>    Designing curriculum, new textbooks, use of computers, and all the
>    other trappings of educational science, as it were, will do
>    little good in my opinion.  My understanding is that generation
>    after generation of educators has tried this and we can see the
>    results...

If this was meant as a reply to .11, please note that all the ed sci stuff came
after we had identified and announced a new direction.  The ed sci stuff does 
not mean much by itself, but it is how we implement the new direction.

And "generation after generation" does not reflect my (admittedly limited) 
knowledge of the history of math education.  Back in the old days, elementary
math was taught mainly by memorizing tables and rules, with lots of practice 
on large numbers.  Then word problems were emphasized; I think that was back 
in the 1920s.  The New Math, introduced around 1960, emphasized understanding 
concepts and deemphasized memory.  Since then there has been some tinkering,
toning down the New Math ideas and cutting back on the hard work.  There have 
been minor forays towards programmed learning, computers and mental arithmetic,
but they have little impact on the mainstream.  All the changes to the 
mainstream in the last 40 years have been in the same direction: away from
memorization, basic skills and hard work, towards concepts, advanced ideas and 
fun.  Reversing that direction may not be a good idea, but the failures of the 
recent past are not evidence against it.

>    What is really required is good teachers, and a society that values
>    learning.  Without that, there's little that can be done about it.

You could probably scrape together a few dozen elementary school teachers good
enough to teach New Math or the ideas in .0.  Most of the teachers currently 
teaching math are probably good enough to teach the ideas in .11.  How good 
people have to be depends on what you are asking them to do.

Also, remember a Deming rule of thumb: 20% of the quality problems in a system 
can be traced to the people, 80% to the process they are asked to carry out.

I agree about valuing learning.  I think we value learning more now than we did
ten years ago, and we will value it a lot more ten years from now.  This is an
opportunity that the schools need to take advantage of.

The big problem I could see with the ideas of .11 and this note are that they
could be presented and implemented on a very superficial level.  "Hard work
is good for you, so we will make you do the hardest work we can find."  This
would be rejected by educationists and students alike.  That's why I emphasized 
the evaluation of skills and concepts in .11.  Once we know what students need
to learn, we can say "This will be hard work, but you need to learn it."  I
think students, although perhaps not educationists, would accept this.

    
    There is a problem in this whole mess that bothers me.  There is
    extreme confusion of mathematical skill with arithmetical skill.
    Most teachers I've met don't know these are not related.  I was told at
    the end of eighth grade that I had no mathematical ability but I had a
    correctly low opinion of the teacher's understanding of anything more
    complicated than hopscotch and I rectified the problem of his
    recommending against my being in the better math classes as I went
    along.  As arithmetic became less important in school, I got better
    grades.  As a graduate student, I found life truly fine in that my
    habit of adding 5 and 9 and getting say 12 no longer lost points.
    
    Now my daughter says math is a problem for her but I tell her
    arithmetic is the problem - she has yet to see math.
    
    This is not an uncommon problem.  My father in law is a moderately well
    known expert in PDE's and operator theory but is incapable of
    determining whether or not he was short changed.
    
    I wish we could learn to distinguish between math and arithmetic - not
    everyone who should discovers the lack of relationship.
    
    Chuck

    Re:    <<< Note 1453.14 by HIBOB::SIMMONS "Tristram Shandy as an equestrian" >>>
    
.14>    There is a problem in this whole mess that bothers me.  There is
>    extreme confusion of mathematical skill with arithmetical skill.
>    Most teachers I've met don't know these are not related. 
    
    I'll agree with you that math and arithmetic skills are not the same,
    but there is a relationship which is relevant to the way that most
    people think about and solve practical problems.
    
    When I am solving the problems I get paid to solve, people expect me to
    produce an actual number, often preceded by a dollar sign.  They expect
    the number to be correct, and would not be happy if I tell them that
    the method is correct, but the number may be off by a decimal point or
    so.  Since my arithmetic skills are not too solid, I try to use a
    calculator or a computer to do that part of it.  But even so, I have to
    do a hand calculation or two to check the calculator or computer.  If I
    had no arithmetic skill at all, I would never know whether any number I
    produce is correct.
    
    I have a similar situation when I am using my math skills as a citizen.
    
>    grades.  As a graduate student, I found life truly fine in that my
>    habit of adding 5 and 9 and getting say 12 no longer lost points.
    
    True for the small minority of elementary students who are going on to
    graduate school in math.  As a graduate student in chemistry, I lost
    points for incorrect arithmetic.
    
>    known expert in PDE's and operator theory but is incapable of
>    determining whether or not he was short changed.
    
    There are many more people whose jobs depend on making change than
    people whose jobs depend on doing operator theory.
    
    One of the problems for New Math is that it assumed that elementary
    school math classes should teach math, defined as what mathematicians
    do.  Among many other problems, this ignored the fact that only a very
    small minority of workers and citizens need to do what mathematicians
    do.  The skills actually needed by the large majority got lost in the
    shuffle. 
    
    If you did not mean to suggest a connection between your note and the
    topic, then I apologize for this irrelevant comment.

    I also think there is a lot of math in arithmetic.  For one thing, it
    teaches the idea of rigorous algorithms.  Also the structure of group,
    ring, and field are all there.  By the time of abstract algebra, the
    students will already have some intuitive idea and some examples of
    those thing.
    
    Eugene

    re .15 and .16
    
    Both interesting but of course we all know New Math was a hopeless
    failure - only token vestiges remain in text books (thank heaven).
    Anyway, rings and fields and stuff caught my fancy early but didn't
    help my arithmetic skill nor anyone elses to my knowledge.  Also a
    mathematical education didn't help.
    
    What I was driving at is don't tell someone he can't do math just
    because he can't do arithmetic or the other way 'round either.
    
    The tests for math skills usually also fail to differentiate math and
    arithmetic so the error in thinking is widespread, rampant if you like,
    in educators at the elementary and high school level.
    
    Anyway, people with poor arithmetic skills gravitate to jobs requiring
    little in that line, e.g. I've been an engineer for many years.  That I
    don't get a job requiring math is another story.
    
    Anyway, I think our educational methods in arithmetic and mathematics
    are not in any way superior to those 100 years ago simply because the
    distinction, which should be taken advantage of, is simply ignored.
    
    Chuck

    re .17,
    
    But Chuck, I will bet doing a lot of arithemtics (even with a lot of
    1+1=3) helps in understanding abstract algebra.  It maybe subconcious,
    but somehow I can't imagine someone learning group or ring theory
    without doing a lot of add and multiply as a kid.
    
    Other than that I agree with you.  By the way, I also make a lot of
    mistakes when doing arithmetics.  I mean really stupid ones.  That is
    why I studied mathematics.  Sometimes, I have a feeling that I am
    dyslexic.
    
    Eugene

    By the way, I am working late, and right now there are a few guys peddling
    quick arithmetics on TV.  Really pointless, if you ask me.  For the
    mathematically matured, these things are for amusement at most.  For
    little kids, they are quick fixes for easy A's.  Sort of fitting right
    in with the general trend of the socity--Don't wanna work on the time
    consuming fundamentals; just ask for quick fixes.  Wait, a college
    student just came up...  Well, she is doing great with her
    arithmetics...  Business major...  Got the number just right.  Guess
    very useful for calculating weekly payroll...  Now the fat guy in
    orange is talking about self-esteem...  Well, time to turn off the TV.
    
    Eugene
          

    ref .17 (Eugen)
    >  why I studied mathematics.  Sometimes, I have a feeling that I am
    >  dyslexic.
     
    You'r in good company, Did not Albert Einstein Also had some dyslexic ?
    
    ref .18 (Eugen)
    I agree with you on this thing about expanding least amount of effort
    just for the sake of getting good grade even thought the student might
    not have REALLY learned any solid material.
     
    for example, there is a paid commercial for 'if there is a well there
    is an A' or something like that, I have nothing against getting an A
    offcourse (I always try to get one myself), and I beleive that 
    organisation in the way you study and think will help, but the 
    feeling I get from watching that TV deal is the the students they show 
    are so happy that they are only have to study 1/2 the time now, and 
    get better grade. learning is like something thet they HAVE to do, and 
    if they can do the least of it, but still get good grade, hick then 
    why learn anything more, lets spend the extra time we SAVED watching MTV !
                               
    Ok, I think I said enough, I still have more, but Iam gonna control
    myself.
    
    /Nasser
                                                  
    

    I haven't been here for over a year, but I can't resist this topic.
    
    Thesis:  There is educational value in arithmetic for mathematicians
    and in mathematics for arithmeticians.
    
    To start out with, the natural, rational, and real number systems are
    _systems_, which can be shown to obey some rules that can be stated
    systematically.  It is easy to forget that many people who appear to be
    mathematically ignorant, demonstrate it by their failure to understand
    this abstract, but powerful concept of regular, systematic, dependable
    behavior.   I can't recall the source, but I do remember a mathephob
    writing about his/her worries that the answers to algebra problems were
    purely arbitrary.  This demonstrates that this person never grasped the
    rule based nature of mathematics.
    
    Arithmetic, aside from its practical applications, is intended to
    provide the intuitive grounding for the abstract concept of regular,
    rule based behavior.  If teachers fail to identify or point out
    this value, the students are getting short changed.  
    
    I'm sure we all remember things like casting out nine's and
    applications of commutability that we use as tricks to reduce the grunt
    work of mental arithmetic.  That illustrates mathematics applied to
    arithmetic.  Students must also learn that effort spent on tools can
    pay off in increasing mental productivity.  
    
    I rest my case.
    
    Dick