What do I need this course for? The eternal student question!!!
(also, why is learning math so hard?)
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
PeskyCat@aol.com

Someone sent me this quote today. It seems in consonance with what you've said.
"In the world of the future, the new illiterate will be the person who has not learned to learn," Alvin Toffler

Shelley
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Spencer Kagan <spencer@KaganOnline.com>

I say life has many turns. Today I run a publishing firm and am glad I took print shop and Journalism in high school although at the time I had no idea I would ever apply those skills. When I decided to go for a Doctorate in psychology, I had to learn statistics, and applied algebra at the time I learned it I thought was useless. When I built a ranch in Mexico, I applied Geometry. I had enjoyed geometry very much in high school, but did not foresee it would help me out when I was in rural Mexico, trying to calculate the height of supporting beams on a slope, where no one could physically measure. With the change rate so fast, we only dimly imagine the kinds of jobs we will have, and some of our "academic" learning can actually be quite practical.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Tedd Kelly <tkelly@CERR.COM>

Ted, when I taught eighth grade English, I tried many answers to that or similar questions. I found that kids would not accept answers as logical yours, so I responded "OK, so what are you going to study the rest of this hour?" A shrug of shoulders and back to Macbeth.

It works with a lot of questions. It worked with my three year grandson today. "Why do I have to eat peas?" "Well, if you don't eat the peas what are you going to eat?" After a long staring session with the peas that he could stare a way, he ate them. His mother was going into the kitchen, heard the question but not the answer. She returned at about the time he was finishing his peas and asked me "How did you do that?" "Just a little
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
   David Lustick <lustick1@MSU.EDU>

It beats flipping burgers!      Works everytime
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Paul Ellis <ellis@NKU.EDU>

Dr. Keith Devlin, NPR's "Math Guy" and author of The Math Gene: How Mathematical
Thinking Evolved and Why Numbers Are Like Gossip, recently gave an entertaining and
informative talk on the Northern Kentucky University campus about math and the skills necessary
to do math, which is also the subject of his book.

Devlin's explanation for why learning math is so hard is illustrated by a comparison: Math is like a
TV soap opera. The relationships in the soap opera world are very complex -- wife / daughter  /
husband / lover / ex-lover # 1 / ex-lover #2 / ex-lover #1 who is also step-father #2 / etc.
Understanding a soap opera may seem easy because we are very familiar with the terms,
concepts, and relationships that are needed to understand what goes on in the soap opera world.
The relationships in mathematics are no more complex than the relationships in a soap opera, but
most of us are not so familiar with the terms, concepts, and relationships in the math world. If you
didn't know the term lover, the term daughter, the concept of incest, then the equation lover +
daughter = incest would be very perplexing. We can focus on the drama because the terms,
concepts, relationships are relatively well-known. In math, most of us have not been and are not
immersed in the terms, concepts, and relationships. We need to find a way to teach math so
students can focus on the drama of math.

Why do students need to learn mathematics? Well, Dr. Thomas L. Magliozzi (aka Click, or is it
Clack?, on NPR's "Car Talk") once ranted rather seriously that math courses are designed only to
prepare students for the next math course, which most students never take. He also debunked the
idea that "math is good for the mind" -- is there an instructor of a college course that will admit that
his or her course is NOT good for the mind? Perhaps we ought to require more physical
education, since there is at least some hard evidence that physical activity improves mental ability.

Why do students need to learn math? Because educated people should be numerate as well as
literate. But to my mind, being numerate does not mean Algebra I to prepare students for Algebra
II to prepare students for College Algebra to prepare students for Calculus I or whatever next
level math course most students have no intention of taking.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Jerry Taylor <JerryTaylr@AOL.COM>

Here's one possible response. Tell him/her:
I'll make a deal with you. You tell me exactly what career you're going to be
doing for the REST OF YOUR LIFE, and I'll teach you ONLY what you need to
know for that specific career... nothing more, nothing less. (We could
probably compress your time in school by several years, too.) But remember,
if you ever decide to CHANGE careers, tough! You won't have ANY
skills/abilities for another job. And you won't know how to LEARN them,
either!

(It's estimated that the average worker changes jobs 10 times and careers
three times in a working lifetime.)
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Nicola Simmons <simmonsn@ADMIN.HUMBERC.ON.CA>

Ahhh - where possible, a metaphor to demonstrate the need.

I recently worked with a faculty member who was trying to teach
pharmacy assistant students the importance of learning math. She
had tried bags of candy to represent percentages, and all manner
of examples, but could not get them to understand in their
calculations why it mattered where they put the decimal point - "I
got the numbers right, so I should only get half a mark taken off.."
etc.

The solution (pun intended) we brainstormed was this: The
following class she went in and set up two sets of medicine cups of
liquid (it was juice) at the front of the room. She told the students -
I want you to come and select your medication - the set on the
right was prepared by a pharmacy assistant who calculated
correctly, and the second set are correct except that the assistnat
got the decimal point in the wrong place. They are ten times as
strong as they should be, and are toxic. Please choose your
medicine. Surprise, no-one wanted to choose from that set.

She could have explained all that, but the graphic illustration sunk
in, and while her students didn't become overnight math geniuses,
they never again raised the question of decimal placement.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Walter Whiteley <whiteley@MATHSTAT.YORKU.CA>

Just to follow up that 'use' of math. When a couple of Math Educators (Celia Hoyes and Richard Noss)
followed around nurses and observed how they actually computed proportions for medicine doses, they found that
the nurses used methods which were NOT those taught in school.  They were correct, but involved more steps during
which the person could keep the 'proportions' clearly in mind. I would prefer medicine done by those nurses to ones done by a student tricking to mimic a calculation learned in school but not digested into some 'big picture'.
+++++++++++++++++++++++++++++++++++++++++++++++++++++
Annette Gourgey <statsense@RCN.COM>

I have an advantage in teaching statistics because the question of 'when am
I ever going to use this' is more easily answered than it may be for
subjects like algebra or calculus that are taught with fewer applications.
But maybe this approach could be used with other subjects as well.
I bring in as many newspaper articles as I can find, to show how the subject
is used in the real world. There is lots of financial information using
math of various sorts as well as statistics. I give a couple of leading
questions and have students discuss the articles in a CL group, or in class,
or write something about them at home. I try to drive home the point that
once they know some practical math, doors are opened to them. They don't
have to take someone's word for it, but they can judge for themselves.

For example, is Bush's plan to privatize social security in the stock market
a good idea? I give them an article that discusses it. The statistical
concepts in the article are average return and variation (volatility). Once
they understand how these affect investments, they can make their own
evaluation of the president's proposal. I also give them graphs of market
trends, poll results, etc. After 9/11 someone even wrote an article about
probability and figuring how high your odds are of becoming a victim of
terrorism (lower than you think).

Every day there is something interesting in the paper that uses math; I just
file them by topic as I come across them and go back to the files when I
need something for a particular topic.

I believe that, with some thought, real-world applications can be generated
for other topics in math as well. For example, I once taught algebra at a
Manhattan school. I became aware that our street addresses, with our
numbered streets, followed a linear function. So I taught that function in
a lesson on how the slope and intercept work. Another time I tutored an
algebra student; just telling her that algebra prepares people for calculus,
which can be used in engineering and building, was enough to help her get
over her resentment so she could become receptive to the tutoring.
I think what makes math so hard is that it is so abstract. If it is made
concrete, students have a much easier time making sense of it and connecting
with it.

My students really go for this. They say the articles are the best part of
the course. Since beginning this I've never heard the complaint of 'when am
I ever going to use this' again. The only people that have given me
resistance over it are some math professors who think the only math worth
teaching is 'pure' math.
+     +     +     +     +     +     +     +     +     +     +     +     +
I have comments in response to three of the posts to this thread, which I'll
put together below.

On why learning math is so hard:  I see students more caught up with the
precision of math than they are in other subjects.  The answer has to be
exact and it has to be what the teacher wants.  They don't seem to have as
much of a sense of what it means in terms of their own internal sense-making
as they do in other things I've taught (psychology, reading).  It seems to
be arbitrary and external to what is meaningful to them.  My personal
opinion is that this has more to do with how math is taught than math
itself, but I suppose it is complex and can be debated.

On applications for algebra and other math topics:  The Consortium for
Mathematics and Its Applications (www.comap.com) has wonderful teaching
materials on this.  They have lesson modules indexed by math topic (basic
math, elementary algebra, precalculus, trig, calculus, etc.) that give
examples of real-world applications.  The materials are classified by high
school and college; the high school materials are excellent for
remedial/developmental math (e.g. I got some very good stuff on teaching
percentages from them).  Some of it I never had occasion to use in the
classroom because I didn't teach those topics, but I read just for my own
enjoyment and expansion.  One of my favorites was compound interest,
following the investments of two hypothetical people to see who ended up
with more money and why.

I, too, have sometimes told people that even if they never like my subject,
they will get pride from doing something they never thought they'd be able
to do.  Some students have related to this, especially after seeing their
grades improve as a result of their efforts.  I've also asked them why they
came to a liberal arts college rather than a community college or vocational
school, and what they think the purpose of that kind of education is.

I have to admit, though, that the answers to their question that are based
on any form of "because it's good for you" are really answers for us rather
than for students, who are usually not mature enough to appreciate that yet.
That's why I like the applications--they're fun and they hit students on a
level that has more immediate reality to them.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Janeane Weprin <weprinj@AII.EDU>
Hi Annette,

We have two developmental and two college level math courses.  The problem
we had is how do you take art and design students who may only know
arithmetic and prepare them for college level mathematics in two eleven-week
quarters?!  There was no way we were ever going to cover all the content in
a traditional Elementary and Intermediate Algebra sequence, (although we
tried at first.)  We had to ask ourselves what is important.  What do we
want our students to get from our mathematics course?  Of course we wanted
life skills, and to prepare students for their other courses for which math
was a prerequisite, but we wanted more.  We want them to be able to think
and apply what they learn.  We have found that most  students are able to
think conceptually even when their nuts and bolts math skills (arithmetic,
algebraic) are weak.

So first, I worked with the general education science and business
instructors and the directors of the major departments to find out what
mathematics the students truly needed.  I discovered they would need only
about 25% of the content we were covering in our developmental courses, and
we were not covering some of what was needed.  So we bit the bullet and
started eliminating content, choosing depth over breadth.  It wasn't easy!
As so much of math is interconnected, we had to be very careful about what
we decided to include, so that students had what they needed to move through
the curriculum. Next we looked for "reform" text books which presented math
in context.  We chose the texts by the Consortium for Foundation
Mathematics.  We focused on the function as the central theme, and
incorporated the TI-83.

Our college level course is modeled after the Illinois Articulation General
Education math course (IAI M1 904) and the General Education Statistics
course (IAI M1 902).  (See www. Itransfer.org for more information on the
IAI.)  Our focus in these courses is more conceptual and less technical.
They are far from perfect and we are continually modifying all of our
courses, looking for better ways to teach them and integrate them with other
disciplines.  (We include visual/creative projects with mathematical themes
such as the golden ratio and spiral, tessellations,  symmetry, and
fractals.)

I would be happy to discuss this further with anyone interested.  As to the
a.k.a., I have known some who think my ideas are radical and anti-math.
Maybe it is my UC Berkeley education!
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Annette Gourgey <statsense@RCN.COM>

That is an excellent point about the unpredictability of life.  I have a
story like that too--never expected to end up teaching statistics, but a
tutoring job opened up, one thing led to another and then that was what I
wanted to do.  My original plan, to teach psychology, didn't pan out, first
because of a lack of jobs and then because I developed other interests that
grew out of the statistics tutoring (math learning, math anxiety).
Sometimes I've told students this story and they're always surprised.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"John M. Flanigan" <johnf@HAWAII.EDU>

I have given three answers to the second question:

Philosophical: Since you are in college, you are presumed to be seeking a
college degree. People who have college degrees are expected by society
to be "Educated Persons." An educated person is one who has a broad
understanding of many things not directly directed at earning a paycheck.
(I often add that we are not just in the business of turning students into
employees; we are in the business of helping students become Educated
Persons.)

Recreational: If you have a good, broad, knowledge of how various systems
operate, it makes thinking more fun. Understanding how things work -- from
economic plans and the behavior of people to ocean tides, seasons,
weather, and rainbows -- is an enormous and inexhaustible source of
pleasure.

Practical: Other things being equal, advancement in any job, and access to
improved situations will normally fall more readily to those who show
evidence of being Educated Persons -- who can hold their own in written
and oral communication with others of various areas and degrees of
knowledge, and who can make decisions based upon broad knowledge of
politics, society, psychology, science, art, ... -- whatever is appropriate to the occasion.

This answer to the first question is from Stephen Pinker's _How the Mind
Works_: "Mathematics is ruthlessly cumulative." In no other subject is it

required to learn and remember *everything* from each semester in order to
do well in the next. Students just aren't used to that stringent
requirement.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
MaryLiz Pierce <MATH4FOBIX@AOL.COM>

My response to the second question has two parts.
(1) You will always function several levels below the highest level you have taken in math, as well as other
subjects. Raising the level you have studied will raise the level you work at every day. You will see new and
easier ways to solve problems.

(2) You will be using the same problem solving skills in all areas of your life. When starting the idea of
problem solving skills, I first plan a trip I want to take with them - to show the relationship between the ski
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"Alix E. Peshette" <apeshet@PACBELL.NET>

When I was teaching world history I responded to the "why do we have to
learn this" question this way. History and time is sort of in a spiral.
Events and things do get repeated but not exactly in the same way. It's
very helpful to know how another time and place dealt with an event and how
that solution worked or didn't work.

My best example to give the kids was the analogy of the advent of the
printing press and the advent of the Internet. We can learn a lot about the
accuracy of the information that is offered by studying how the printing
press flooded the world with varying degrees of good information.
Kids seem to really understand the comparison to something that they use
here and now.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Cynthia Breeding <cbreed@IO.COM>

For the second question I tried to think back to the 17 years I taught math
from 4th grade to college age. That question was often used on me, but I
never really had a pat answer. It is something a math teacher shows more
than explains.

I always partnered up with a teacher in another discipline to work on
problem solving, project planning, job shadowing, anything that would get
students involved in using some solid math in complex, but meaningful
situations, regardless of the level.

Once they get hooked, they will do a lot of symbol manipulation, practice,
etc., just to get to the next authentic thing.

It is true that people can live a very satisfying life without algebra or
any higher level mathematics. But they never know what a fantastic life they
can have if they know how to do some math!
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"Susan Nasser" <Susan.Nasser@cdha.nshealth.ca>

I forwarded your posting on this subject to my son, who is in his third year of math studies at the University of Waterloo in Ontario, mentioning that you are a math professor.

While he quickly followed his reply (which I am reproducing below) with a second message saying he realized that he had not actually answered the question, I found his comments interesting and thought you might as well.

I am somewhat in awe of the mathematical bent of both of my children (my daughter is still in high school) and enjoy hearing their thoughts on what it is like to understand math.

Anyway, here is what he said:

"I believe the most accurate answer to this isn't much of an answer at all. It's really just something you're born with, it seems. Certain people understand certain things which others don't. For example, even within the math faculty, I find that there are 2 types of people: those that understand algebra, and those that understand calculus. Less than  1% of math students at UW understand both. So an algebra person might ask a calculus person why calculus is so hard, and the calculus person won't have a clue what the algebra person is talking about.

That doesn't really address what to tell them though... I know most people wouldn't be satisfied with "it's just a different way of thinking". With luck, it might give you some ideas though.There, now you can post to the list! You might even mention that your son is in combinatorics and optimization. Since he's a math teacher, there's a chance he'll understand that."
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Barb Deardorff <barb_deardorff@YAHOO.COM>

My standard response (Chemistry & Physics) is that you will use these thinking skills for the rest of your life.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Rolland Gaudet <rgaudet@USTBONIFACE.MB.CA>

Content vs ... It sounds better in French: "contenu vs contenant".

If you're the typical math major, or taking less university math than that, you have as much chance of hitting the lottery as "needing" calculus in later life. You're "only" developing your mind's ability to function when you take math. This is not unlike the runner who exercises not because he's training for the olympics but because "it's good for him". With this training under your belt you can do anything. I tell every one of my students that if they can get a B+ in my classes they can do anything. And they believe me. A specific case of "contenu vs contenant": the difference between a bachelor's degree in administration and an MBA. You cover somewhat similar material, but not at the same intensity or tempo, since in an MBA you're supposed to already know how to work and think. This is somewhat related to the futility (or waste of time) in doing a second bachelor's degree, for most thinking people.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"Arthur L Crawley" <acrawle@lsu.edu>

Ted, the rub you speak of is true for all disciplines. However, I do believe it is in the language of mathematics, the use of signs and symbols to make sense of reality which is stressed, e.g. e=mc2 that makes mathematics "so hard" . "A solution" is to connect your disciplinary understandings to real world problems, thereby making the theoretical visible and accessible to our learners. Because some of our students (those that are often more like our faculty-- it's called replication) get the abstractions quickly and may not prefer the collaboration of peers or practice to deeply understand the concepts and the thinking involved. I
believe many of our undergraduate learners need concrete examples and hands on experiences before the theoretical abstractions can take hold.

Good teachers build a bridge to those learners and in doing so help both types of learners make sense of what they observe, know, understand, apply, and may I add, value. And it is simply more fun and motivating to the majority of our students and to the professors to do so.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"Theall, Michael" <Theall.Michael@uis.edu>

A quick response: I try to find relevant, real-world situations where the
content can mesh with some necessary or meaningful task or set of conditions.

Here's my example.
My son took chemistry last year (as I did in high school and then, in
college). We both were taught in the same way, learning the terminology and
doing things like balancing chem. equations. Since 1961, I have never had
any occasion to balance another equation. My son will be a music major and
the same will likely apply to him. What both of us would have profited
from, would have been the connection of chem. to something more relevant
than a lab experiment and the periodic table. An ecological approach would
have helped both of us to understand the how and why of the subject, and to
see how its workings affect us on a daily basis. This seems important
particularly in high school courses and/or for those who will not be
directly involved with chemistry as a professional field. I'm not proposing
that we drop the theory/science, just that we treat it in a different way
for those who do not intend to go into the field. I know some will say that
this idea somehow violates the sanctity of the subject or is pandering to
less able students. But the issue you raise is not how to make great
chemists or mathematicians, it's how to get the larger population of
students (probably with weaker background or interest levels) to be willing
to invest the time and effort necessary to acquire the necessary knowledge
of both the content and its importance. One way is to make the content more
meaningful and thus, more understandable. Another is to be willing to teach
those not gifted or initially interested in the field. Courses that have
the purpose of weeding out 'those who can't hack it' serve only elitist
purposes. As Lee Chronbach said, education should be a talent development
effort, not a competition.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"Ann J. Pace" <paceaj@UMKC.EDU>

I thought that Edward Thorndike found nearly 80 years ago that there was no support for this "mental discipline" perspective,
i.e., that the learning of certain subjects such as mathematics enhances thinking generally. Also, hasn't much more recent research on thinking and problem-solving pretty much discredited this general transfer view? (An exception would be the kind of general heuristic represented by IDEAL, which has wide applicability as an approach-to-task, if used explicitly.)

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Catherine Scott <c.scott@UWS.EDU.AU>

Why is maths hard to learn? Because people do not want to. Why don't they
want to? Because people work on a need to know basis for deciding what to
learn and they do not think they need to know most of what maths teachs them.
And unlike other irrelevant but verbally based material, there is
insufficient redundancy or 'commonsense' in the material to allow them to
'surface learn' sufficient to get through the final exams. To even scrape
by they have to work quite hard.

'Another bad effect of commerce is that the minds of men are contracted,
and rendered incapable of education. Education is despised, or at least
neglected........' Adam Smith
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Thomas Bruggeman <tombrugg@goodnews.net>

This is actually a story of trying to pick up on the end of your
response: "... very few real world problems modeled through math that most people
would encounter." And, yeah, my response has not received ovations.
STORY: Schools that have a 'core curriculum' (especially if it is BIG)
Can get into very interesting discussions of 'how much this', 'how much
that' in the core. In the late eighties, in preparation for a North Central
visit, some of us proposed a multidisciplinary core course discussion of
human development (maybe count it as a social science?). Question #2
was to be addressed, if not explicitly, at least as part of a tradition. We
almost succeeded in gaining consensus, except that the course was NOT a
part of the tradition... go figure. Some of us suggested that what was
being learned about learning and about the use of tech in addressing
needs of those with disabilities as well as others SHOULD become part of a new
articulation of the tradition. Didn't happen, but the question was
raised.

Math models of learning and human development would have been included -
computer simulations and some artificial intelligence topics as well.
The implications about learning/teaching styles and predominance of
'lecture' style, in my opinion, became a basis for 'wait and see' and allowing
more time for further research, discussion, etc.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Brad Helland <helland_b@FC.SD36.BC.CA>

The earliest Math we learn is in "number concepts". If you don't know how
to add, subtract, divide and multiply, then how are you going to carry out
life? How do you assume what your bill should be when you have been
sitting in a restaurant or bar for hours? How do you work out your
"savings" when something is 25% off? How do you know your Telco company
isn't ripping you off?

Larger scale. Let's assume AT&T, because most listers here are probably
American, charges an extra one cent on everyone's monthly bills for 2002.
That's twelve cents per house, and let's work with 30% of your population.
Now you're talking about approximately 75 million people? Am I right?
250 million in the USA? Seventy-five million people. That's a profit of
$900,000 for doing NOTHING! There's your economy of scale.
How about physics? A penny dropped 6 inches will barely be felt. A penny
dropped from the top of the Empire State Building would kill you if it hit
you on the street.

Math is integrated in everything we do. Introduce them to Fibonacci.
He's got explanations for most architectural concepts. Show them a
cross-section of a pine-cone. Fibonacci has an explanation for the spiral.
As for it being abstract? I would say it is ENTIRELY the opposite to
begin with, and only becomes abstract in upper division multiplication at
the university level. LOGIC is NOT abstract. It is sequential.

What grade do you teach? Your response may not be well-received because
of the grade level you're explaining it to. Perhaps they don't know what
you're talking about. I'm really lost when you say that "there are very
few real world problems modeled through math that most people would
encounter". I challenge you to come up with any sort of problem having to
do with construction, destruction or technology that does NOT involve math.
And learning math is only hard if you're a right-brained person. Talk
about left brained and right brained people because the right brain
controls your ability to be "creative" whereas your left brain controls
the "logic" that we need in our daily lives. Learning math is EASY for
many people and difficult for many others. Maybe the onus is on us to
make it easier for those who have difficulty. Perhaps we need to explain
it in other ways instead of "our way" which we have a tendency to fall into.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Derrel Fincher <dfincher@GOL.COM>

Two answers that frequently fit:
Q. Why is learning math so hard?
A. It isn't. We just make it that way as we have no idea how to connect it
to anything remotely useful as we really don't know how to do so ourselves.
But we like to tell you how important it is.

Q. What the heck do I need this for?
A. You don't need it for squat because we obviously have no idea why you
would need it or it would have been obvious in this class. But it makes us
feel important.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"michaels" <michaels@always-online.com>

What you did is the way I handle it on the first day of class.  My question to the students
is "Who feels they are going into a field of work where they will not need math?"  I have
had very little trouble showing were math was necessary in almost any job and then I show
them that the more education they seek, the more math they will need (Statistics, Operations Annalysis, etc.).

So the more we know about the practical uses of our subject, the better we can explain
it to our students.  Yes it helps you think better (and can be just plain fun) but that is not
what our students want to hear.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++
  Donna Kotsopoulos <Donna_Kotsopoulos@EDU.YORKU.CA>

Answering this questions has been my focus for almost two decades. In
grade 10, some eighteen years ago, my math teacher told me that I couldn't
count.  I took this at face value and accepted it as truth.  What other
options do adolescence have?

His proclamation haunted me for most of my adult life.  At at the age of
30, I returned to university to try to discover why learning math was so
hard and if I was truly incapable of doing so.  My curiosity has led me to
consider the discipline from multiple perspectives. It is my firm belief
(and my thesis topic!) that as educators we fail to make the critical
connection to the reality that mathematics is a language and as such
should be taught using the theories and principles of second language
acquisition.   The abstractions need to be taught in a language students
can understand - clearly with the intention of ultimately developing the
formal language.  It is ultimately the educators responsibility to
facilitate this code transfer.

I would suggest that this might make learning mathematics easier.
++++++++++++++++++++++++++++++++++++++++++++++++++
Margo Husby-Scheelar <husbyscheelm@SHAW.CA>

What a wonderful thought! When taking Linguistics as an undergraduate, I
discovered that solving linguistic problems was akin to algebra, only with
words. Looking at math from a linguistic perspective would make it so much
more accessible to people like myself who barely made it through high school
math and undergraduate statistics. If I could find a math teacher who used
this approach, I would cheerfully retake high school math and may even dare
some university level courses simply to prove to myself that I'm not
mathematically stupid.
+++++++++++++++++++++++++++++++++++++++++++++++++++
Walter Whiteley <whiteley@MATHSTAT.YORKU.CA>

Let  me jump in with a slightly contrary view.
[For other listers, I should indicate that I have worked with
Donna, so some of this will not be a comlete surprise to her.]

In the human brain, it appears that much of mathematics
is NOT done in the language parts of the brain,
(though some parts, such as memorizing multiplication tables
do seem to be).  More of it is done in the visual / kinesthetic
parts of the brain - the parts that do eye-hand coordination,
and seem to include an analogue number line, active at age 3 days.
[See books by Butterworth: the Mathematical Brain, Macmillan, 1999;
Dehaeme: The Number Sense Oxford University Press, 2000 ... ]

Of course, many texts and some teachers filter the communication of
math through language, either surpressing the true visual nature
or mangling the visual forms by showing them without teaching them.
When using visuals, teachers and texts tend to ingore the fundamental
observation that different people to NOT see the same thing when
staring at the same image (something that is even more true with
moving and dynamic images).  As the Cognative Scientists teach is:
we create what we see.  We see, in part, with learned attention
to some features and learned ignoring of other things.
This learning, including learning to shift attention, is critical
to how we extract information from visuals in math, how we
do math in visual forms, etc.  It is learned, but iti sseldom taught.

Algebra is a particular example of what, to me, is a highly visual
area of mathematics.  I say to my students that algebra is about
cosmetics, not about surgery.  Many ways to change the appearance
of something, changes that shift my attention and suggest further
steps and transformations, without changing the underlying content
(i.e. what the 'solutions' will be).   I am working on a power point
presentation which illustrates some of this, with the working title
'On Picture Writing' - the title of a famous little article of George
Polya, who wrote 'How to Solve It' and 'Plausible Reasoning'.

I note that failure to at least show visual mathematics as one
possible way to do math causes serious damage to the diversity of
people practicing mathematics.  It excludes people with certain packages
of abilities and disabilities (e.g. certain forms of autism and
dyslexia), it gives a dishonest  public face to what mathematicians
like me actually do in the quiet of our offices or group convesations,
etc.  In the process, it drives away, in the transition through high school,
many of the students who, by final year undergraduate studies could be the
most efective and creative.

All of that does not say that the analogy to language is going to
be irrelevant.  One is learning a complex cognitive skill.  Even
language uses multiple parts of the brain (text and context).
It is, however, only an analogy.  Visual Literacy is also only
and analogy, but it gets some necessary first steps taken.

   Some people even talk as if one can only think with words.
Aristotle spoke otherwise. He said (in translation):
  A soul never thinks without a mental image.
We do think with images, at least as much in mathematics as in
any other field.  [Images are not scattered into pieces on
leaving the eyes - those externalized parts of the brain.
We create true maps of images in the visual system, V1, as
well as many decomopositions in other parts of the visual system.
Moreover, in the mind's eye we use words and associations to
cause such images to appear, to rotate etc.  See Kosslyn, Image and Brain,
MIT Press 1996.]  Whatever happens in the brain, as we do math,
must be reusing parts of the brain evolved in humans prior to
the evolution of math.  (See Lakoff and Nunez: Where Mathematics Comes From,
Perseus Books, 2001.)
For a dramatic autobiographical description of this, see the book:
Temple Grandin: Thinking in Pictures: my life as an autistic,
Vintage Books, New York 1996.  (Grandin is a university faculty member,
and I have had autistic graduate students in mathematics.)

I have some larger powerpoint presentations under the title:
'to see like a mathematician' and recently had a short article
commissioned and accepted by Time Educational Supplement on this
topic.  Sometime later this spring, I hope to put these (and others)
on my web site, with annotations, for downloading.  I have been
teaching a first year seminar on 'information in visual form'
and it is truly fascinating how we can change the way we see.
This too has a profound effect on how we do mathematics and science.
++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Julie Sessions <JUSESSIONS@AOL.COM>

I was thinking along the lines of what Jerry said.  Ask your student at what age he was able to start
seeing his future and if he could read yours.  This usually gets a laugh but also gets a point across.  I
never thought I would go into a field that would require math and science because I had such a hard
time in middle and high school with them.  I picked social work as my beginning major but after 2
years switched schools and got a nursing degree.  Needless to say my math and science teachers
from high school were dumbfounded because they saw how hard I struggled with both subjects.
Nursing isn't for me though and had to get out of it and work only occasionally just to keep my license
up.  Too much lifting for my back.  So, when kids say, "why do I need this?  I will never use it."  I am a
prime example of why they sho! uld never say  never.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Jean Federico <federico@STRATO.NET>

I teach ESE kids...my answer to "Why do we need to know this?"  is simple -

If you want a CD that Walmart has, Walmart tells you what the cost is.  You
either pay the cost or do without the CD.  The State of Florida has the high
school diploma you want.  They say the price is passing this math class.
You either pay the price or do without the diploma.
+++++++++++++++++++++++++++++++++++++++++++++++
  Brenda Tiefenbruck <tiefe002@TC.UMN.EDU>

I like your website problems but the last one should include the word
"Explain" in order to be considered 'reformed'.

The second question is one that plagues us all. You response is what we as
teachers want but our students are only looking to answer questions or more
often, work problems. Have you read Lipping Ma's book on Teaching and
Learning Elementary Math?

I teach Introductory Statistics and like Annette can find much in everyday
life to assist my students. When Algebra is my teaching assignment the
questions are more difficult to answer. Thanks, as always, for making us
think.
+++++++++++++++++++++++++++++++++++++++++++++++++
 Shevawn Eaton <CD0SBE1@WPO.CSO.NIU.EDU>

I have  hear students ask this question about much more than math....why
do they need to take gen eds, why does the university think they need
all those stupid "core requirements", why does the university allow this
old/insane/incompetent/incomprehensible/whatever person teach that
course that everyone has to take?

 My response, which always gets a laugh and gets students out of their
victim mentality is simply this....

Because you need to be in courses that will build your character.

Hard courses, seemingly stupid courses and courses that are badly
taught all contribute to one's character in a very positive way.  They
test your perseverence, your tenacity, your assertiveness and your will
to survive!  It is a phenomenal accomplishment to succeed under such
circumstances and you have to develop a number of skills in the process
that are as much a testament to your fiber as they are to your capacity
to learn XXX.  And those life skills are the important ones.

The next time you encounter something difficult, regardless of the
reason, you know in the fabric of your being that you can do this
because you got through that awful (fill in the blank) experience in
college.  If you got through that, you can get through anything!
+++++++++++++++++++++++++++++++++++++++++++++++++
 Janeane Weprin <weprinj@AII.EDU>

Hi Listers,

The question "Why is learning math so hard?" is closely related to the
question "What do I need this for?"  I would like to address the latter
question.

I was a math teacher for years.  (I am now the Director of General Education
at our college.)  I have had to answer this question for my students and for
myself.  I have attended numerous mathematics conferences in which many (too
many?) sessions are devoted to convincing students that what we are teaching
is important for them to learn.  Why do we have to try so hard to justify
what we teach?  Why does the subject not justify itself, if it is so
important?  Something is wrong and something needs to change.  Is there
perhaps some truth to the idea that students do not need as much math or the
type of math that high schools and colleges require?  Are we using math as a
filter?  Perhaps it should be an elective?  Imagine a math classroom filled
with students who really want to be there.

It often seems to me that we assume everything we teach is important and
then find the justification, rather than determining what is important and
then teaching it, essentially putting the cart before the horse.  Much of
education is a self-perpetuating system - things are the way they are
because this is how they have been for years.  I know that there are
educational systems in other countries in which students are awarded
baccalaureate degrees and which require no mathematics beyond arithmetic.  I
am not arrogant enough to say their degrees are less valid than the ones
granted in the US.  I think we may need to step back and re-evaluate what is
truly important.  I believe we are still too content focused, with content
often separated from context.  I believe the academic disciplines remain
artificially separated.  Is it surprising our students are not able to
integrate and synthesize information when they have been taught each subject
separately most or all of their academic careers?

Trying not to be hypocritical, at our school, we have created a mathematics
curriculum which is focused, integrated, and applicable to our students
majors and lives.  Our courses are not easy, to which our students will
attest.  We emphasize math in context and as a tool for problem solving and
understanding the world.  We rarely get the question "What do I need to
learn this for?" any more.
+++++++++++++++++++++++++++++++++++++++++++++++++
  Helmut Lang <langhr@UREGINA.CA>

Math teaches students how to think.  Really?
Ann Pace raises a valid observation regarding the theory of formal
discipline.  However, even if the product of learning to think occurred,
that which is transferred may well be low level.  Too often, school math is
little more than memorization of steps.  If one uses the chesnut taxonomy
of Ben Bloom (other taxonomies could be used), memorizing steps is at the
RECALL level.  Students then follow the steps to solve a problem--the
APPLICATION level--bypassing COMPREHENSION. Too few math teachers have
students ever operate at the ANALYSIS, SYNTHESIS and EVALUATION levels.
Teaching for thinking?  Hardly.  Teaching for transfer?  Hardly.
++++++++++++++++++++++++++++++++++++++++++++++++
 Michael Agostini <michael@HEZEL.COM>

Ted,
Often I think students ask why they should study a subject when they can't
understand how that knowledge is used outside of the classroom.  Perhaps
each student will not use all the concepts studied but at least knowing how
other people use the concepts may help demonstrate the value. One way to
help them make that connection is to explain to them the kinds of
professionals who use each of the math concepts being studied and how they
use these formulas/theories. Inviting an engineer, physicist or economist to
your class to share his/her work and give some specific examples of the
problems they encounter would probably be enlightening to your students.

Having students solve "real" problems by using their math may be more
effective in demonstrating the value of mathematics than trying to explain
some of the intangible benefits.  At the K-12 level, an organization called
National Math Trail (www.nationalmathtrail.org) is showing teachers how to
help their students apply basic math concepts to the world around them and
incorporate technology into this learning process.
+++++++++++++++++++++++++++++++++++++++++++++
  "Navarro, Virginia L." <virginia.navarro@UMSL.EDU>

I wanted to share an example I use with my students to illustrate Ann Pace's
and Helmut Long's points. As a reasonably good student and school
game-player, I successfully completed high school geometry at a private
college prep school. Theorems were like puzzles to me; all the pieces needed
to fit together in a certain way.  I had absolutely NO CLUE about what
geometry was for, about, etc.  As a result of this failure to move to a
"scientific" from a "spontaneous concept understanding (to borrow from
Vygotsky's notion), I never built on my geometry knowledge despite my
academic meaningless A. It became dead knowledge in the worst way in that it
was unavailable conceptually to move my cognitive functions forward in new
situations. Transfer, I believe,  is vastly over-rated when dynamic
conceptual understandings are never achieved.
+++++++++++++++++++++++++++++++++++++++++++++++++
  John Mason <j.h.mason@OPEN.AC.UK>

I suggest (and i tested this out numerous times) that when someone
askes "why am I doing this?", "what are we doing this for?", and
other variants, it signals lack of confidence, loss of facility, loss
of self-esteem.  I find I never as these questions when things are
going well.  I suspect that others are similar.  So when a student
asks it, I choose to treat it as a statement: "I can't do these
questions", "I am lost".

How did I investigate this?  I trained myself to answer, without
batting an eyelid and remaining very still having given my answer,
that "It is absolutely vital  in the steel industry".  That of course
dates me, for it was tat a time when the UK actually had a steel
industry.  Students facers went absolutely blank.  This was clearly
not what they were asking.  Try out a version for yourself and I am
confident you will see that they were not asking for applications,
nor for some philosophical justification in terms of mental powers etc..
++++++++++++++++++++++++++++++++++++++++++++++++++
  Kathleen Verner <kverner@TEMPLE.EDU>

 A good reason to learn math and math concepts is that math is the primary
universal language or so I believe.
++++++++++++++++++++++++++++++++++++++++++++++
Suskie, Linda [mailto:lsuskie@TOWSON.EDU]

Ted, your second question is REALLY relevant these days, as we're
discovering that many students, in particular many students from
non-European backgrounds, learn better (i.e., achieve deep, lasting
learning) and more easily when they can relate a topic to their own lives
and what they have already learned.  I think that's why some students have
so much trouble with math.  I wish every teacher began every year and every
unit by explaining why we're learning this and what impact it will have on
us.

I'm probably going to draw flames from the math teachers for even bringing
this up, but I'm starting to question more and more why we have the high
school math curricula that we have.  For the thousands of high school
students who will never need or take calculus, are binomials, geometric
proofs, trigonometry, and their kindred really the best ways to achieve the
ends Ted describes?  Could we achieve many of the same ends through an
algebra-statistics sequence, which would have the advantage of being far
more relevant to most students?  (And better prepare those interested in the
social sciences and health sciences for college?)
++++++++++++++++++++++++++++++++++++++++++++++++++++
James Allen <allenj@MAIL.STROSE.EDU>

I find Linda's suggestion regarding the traditional math curriculum very
interesting. As a former secondary math teacher (5 years) and a former math
learning skills counselor at the university level (6 years) working with
students in both math/science and social science majors, I also began to
think along the same lines as Linda. Based on my teaching experience (and
not being aware of the relevant research literature on this topic), it seems
that much of the secondary math curricula does not serve a majority of
college-bound students well (and even less so for those who do not attend
college). The traditional curricular sequence does seem to be a benefit to
those who choose a major in which Calculus is a foundation (Math,
Engineering, Economics), but not for most liberal art and social science
majors. However, the most likely alternative (early "tracking" of students
into non-Calculus sequence) is not one I favor either. I've come to believe
that it might be more beneficial to focus on very fundamental structural
components of all math. For example, an extensive and deep understanding of
the nature of "patterns" that is found in any branch of mathematics. I think
I am arguing for a "less is more" (depth vs. breadth) approach to the
teaching of math, where the "less" that is taught is focused on the
fundamental structural elements of mathematics. From my limited knowledge of
upper elementary math curricula, it seems that this approach is attempted,
but is dropped (perhaps too quickly before it is deeply understood) once one
begins a secondary sequence in math. From a cognitive development
perspective, I would hypothesize that for many students the focus on
"patterns," etc. might be more appropriate at an early secondary level
(8th-10th grade).

I'd certainly be interested in hearing from math curricula experts or anyone
on these lists that is knowledgeable about research in this area.
+++++++++++++++++++++++++++++++++++++++++++++++
Scriven@AOL.COM

It may be time to say that this is a bad question. It's not hard for some, is
hard for others, and there's always something that's hard for everyone. I got
conscripted into mathematics because I needed it to become a flight officer
in the air force, but managed a couple of degrees in it when the war was
stopped with an oversupply of pilots, leaving me with nothing to do but go to
college. But I'm hopeless at music and Latin. And some parts of mathematics
are very hard for me, others easy.

The only good question is how best to teach it. Even there it seems clear
that what works for some does not work for others. And I take it that at a
general level the answer is much as people have been discovering throughout
education; there has to be a very large amount of highly interactive process
with frequent evaluative feedback, its content highly individualized to the
interests and ability level of the student. This isn't hard to do unless you
insist on highly heterogeneous classrooms (by ability/achievement level) and
won't use both games and the programmed text approach on the computers that
are being underutilized.

Oh, well, nothing new about this...
+++++++++++++++++++++++++++++++++++++++++++++++++++
"Frazier, Sundee" <Sundee.Frazier@TUI.EDU>

I just heard about a math remediation site on the web that sounds good,
in case you're interested in passing on to students struggling with
math.  They can use the program on-line to increase their math
proficiency, prepare for high school exit exams or CLEP testing, or just
to learn to love math more!  The site is:   www.etap.org
++++++++++++++++++++++++++++++++++++++++++++++++
David Warlick <davidwarlick@MAC.COM>

Ted,
I taught math my first year as a teacher.  The year went so badly that
by Christmas I registered to take the civil service exam so that I could
become a rural postal carrier.  I would like to address your second
question, because it is one that I think about a lot.  The simple answer
is that we teach what we do and students learn what they're taught,
because someone in the state capital put it on a set of standards, and
they probably put it there because they were taught it in approximately
the same grade.

I suspect that there is much that we are teaching that will not be of
use to our students, and trying to rationalize it only degrades our
profession in front of them.  "How many long division problems have you
performed on paper in the past month?"  I believe also that your
justification, although true, is a little lame as well.  If we are
teaching students to think, then we should teach them to think within a
context that is relevant, if not to their future, then to their current
lives.

My answer, although I haven't had the opportunity to express it
compellingly yet, is "context".  Students should learn science in order
to understand the context of their being, where they are, and where they
are going -- why the world behaves the way that it does.  We teacher
social studies, which is in no way less important, for the very same
reason, to create a context for our being, who we are with, and with
whom we are going -- why people behave the way that they do.  It is all
in order to sustain and enrich our culture to create a higher quality of
life.

The question we should be asking is not, "why are we teaching what we
do?", but, "what should we be teaching?"

Exactly two cents worth!
+++++++++++++++++++++++++++++++++++++++++++++++++
 Michael Simkins <msimkins@SANTACRUZ.K12.CA.US>

I have not followed this discussion since David's contribution below, so
please forgive me if I repeat something others have said subsequently.

David wrote, "If we are teaching students to think, then we should teach
them to think within a context that is relevant, if not to their future,
then to their current lives."  I agree.  My 3rd cent is: it's most important
that we find context in their current lives.  It's there to be found, I'm
sure.  There may be different relevance for later in their lives, but if
there is any relevance for their lives today, we should find it and use it.
++++++++++++++++++++++++++++++++++++++++++++++++++
Veronica <yatesrv@BELLSOUTH.NET>

Negative attitudes......Having the fruit without the tree.

The study of mathematics is not only a practical art it is a theoretical
art. It is has been used as a tool to improve the quality of life. Many
professions such as engineers, doctors, and even businessmen have uses
some form of mathematics. Mathematics is not limited to numbers; it
expands to logical, reasoning, and/or critical thinking. Although I have
not used long division in a number of years, this algorithm has
contributed to the development of my cognitive skills.

Furthermore, speaking of relevancy, I can not recall the last time I
used the word cantankerous, yet it was a requirement for succeeding in
my language arts course. It seems that it is "okay" to question the
level of mathematics that is learned, but not the level of language that
is learned. It seems that it is "okay" to have poor skills in
mathematics, but it is a tragedy to be a poor reader. Double standards?
In my experience, I have observed that poor mathematics student were
generally poor readers and have had bad attitudes towards mathematics.
Could this be a result of the negative attitudes conveyed throughout our
modern society?

I believe we must prepare students for the future as well as their
current lives. After all, a mathematician initiated the concept of the
computer
+++++++++++++++++++++++++++++++++++++++++++++++++++
Veronica <yatesrv@BELLSOUTH.NET>

Well said.
I'll make a deal with you. You tell me exactly what career you're going to be
doing for the REST OF YOUR LIFE, and I'll teach you ONLY what you need to
know for that specific career... nothing more, nothing less. (We could
probably compress your time in school by several years, too.) But remember,
if you ever decide to CHANGE careers, tough! You won't have ANY
skills/abilities for another job. And you won't know how to LEARN them,
either!
++++++++++++++++++++++++++++++++++++++++++++++++++
 Bonnie Bracey <BBracey@AOL.COM>

In a message dated 3/18/2002 5:35:44 AM Eastern Standard Time, davidwarlick@MAC.COM writes:

   The question we should be asking is not, "why are we teaching what we
   do?", but, "what should we be teaching?"

I agree with David. The point of some math textbooks is more of the same until you memorize it. I
guess to simplify things in many school systems, there is but one way to do special kinds of math.

I am proud to say that my school children learned math in many different ways, understanding that in
the system there was a preferred way of doing things, but with the freedom to get the answers in a
variety of ways. Two of them are currently in math and technology at Berkeley.

Here is what happened to me. I was teaching base 2, 5, and some other, as the book prescribed and
this child   sat and made a database taking this learning through 25 different numbers. It was then that
I decided I had to let math children have wings. So I started doing other kinds of math instruction as a
part of the
work I do. Interdisciplinary problem solving, content related.

I also did a day which was not drill and kill math. I also used as many math game programs, and
simulation projects as I could. We spent a month once working any time we had a free period or time
in learning to beat the computer using probability, and with some lessons on geometry where you had
to apply the skill, and the simple kinds of problem solving that used to be only for the gifted and
talented,

I found web sites with special resources but there was at the time also the National Geographic
Kidsnetwork , which was also a different kind of science, math and geography.

There is a researcher who has written to this point.
Equity in Math Cooperative Groups
http://www.terc.edu/wge/coopgroups.html
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
  Cindy Firak <higuys@OHIO.NET>

I tell students who balk at math that Math is a language, it is also an artform.
If god was an artist/creator he must have also been a mathmetician because there is so much math in nature.
Math has much magic to it. This explaination seems to stimulate more interest.
Take it from here starting with magic squares, or the secrets to some math magic or the geometry of a flower.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Connie Hudgeons <connie@HANDYWERKS.COM>

As a non-math teacher,  I would like to respond to the second part of Linda's response.

> I'm probably going to draw flames from the math teachers for even bringing this up, but I'm starting to question more and more why we have the high school math curricula that we have.  For the thousands of high school students who will never need or take calculus, are binomials, geometric proofs, trigonometry, and their kindred really the best ways to achieve the ends Ted describes?  Could we achieve many of the same ends through an algebra-statistics sequence, which would have the advantage of being far more relevant to most students?  (And better prepare those interested in the social sciences and health sciences for college?)
>
My former high school pondered this question.  We were looking at making all curriculum meaningful and relevant.  The math department decided to split the traditional math track after geometry.  They offer the traditional calculus/stats class and as a companion class offer and "advanced applied math/stats class."  The kids who are interested in engineering/technical stuff do the traditional trace, while the applied math is geared to the medical/science applications.  The hs has 5 PhD's who left the "professional" world to become math teachers (they laugh when they say that!!) There is one biochemist, one industrial chemist, one "techie" (holds multiple patents for computer stuff!), one theoretical math, and one in philosophy -- they worked together to redesign the course structure and as a result have increased the numbers of kids in upper division math classes by almost 60% in 2 years.  Lots of kids who didn't want the calc approach are taking the applied/stats approach and loving.  The math profile scores from the SAT have gone up by almost 25 points in the same 2 years -- so we have more kids taking the test and still there is an increase.  These same math teachers convinced the lower division teachers to include math vocabulary and math research into the curriculum, so kids are learning the vocabulary & skills to actually do "word problems"  -- so across the board these kids problem solving skills are going up by working on the concepts that literacy requires application.

Interesting thing about the high school, is that 3 of the 5 teachers who did the revamping have left the school and gone on to other teaching positions. Yet the success of the "experiment" is paying off with more kids feeling good about math.  As a math phobic, I really, really appreicate that!!!

So, Linda, I hope that you and I both don't get burned!!
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
  <phendric@EGREEN.WEDNET.EDU>        Peter Hendrickson, Ph.D.

Hello List--
     As another non-math teacher, I would add my voice to Connie's
provocative program description.  It strikes me that there even more
possibilities for students in the discrete math path through
algebra/statistics.  The opportunities for ties to the sciences, social
sciences, and health/fitness arenas are tantalizing.  For many students
statistics, especially applied statistics, gives them a tool to use in
other classes while they are yet in high school.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Bonnie Bracey <BBracey@AOL.COM>

I thought about this title again . I am at SITE. I have had the priviledge of
attending some workshops on thinking, teaching, learning , and retention.

What do we see as math? How do we define it? FOr some of us it is the way we
were taught , so theoretical math, problem solving, thinking math may be
areas of difficulty for us. Often the way we were taught frames the way we
understand math.

What do students see as math? Do they realize how in their daily activities
math is used. In the TIMMS studies it is said that we teach math a mile wide
and an inch deep.

As a teacher I might also be bold enough to say that we teach it on the
calendar and the same kinds of subjects are left until the end of the school
year a lot if we do not preassess and always follow the book.

Theoretical math can be taught in a lot of different ways. There is
www.enc.org to help. There is math forum , there is learning to problem
solve, there is thinking about the use of math in real life, real time and
involving the students in thinking math activities as well as hands on
demonstrations.

I am proud to say that I was often ridiculed for doing hands on math past the
third grade. I am proud to say that I used games and software and problem
solving nearly every day in some way.

I even used cuisenaire rods when they were no longer the fashion, and so I
had them when they came back in.

Parental use of the software, projects, and handson also was a help.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Bob Zenhausern <drz@PANIX.COM>

Math is hard because we approach it as if we lived in the 19th century
not the 21st.  In the 19th arithmetic was the most basic and essential
tool for higher math.  One major accomplishment of the 20th century
was to eliminate the need to labor over complex arithmetic and use
calculators and computers to increase accuracy and allow theretofore
computations.  The 21st should build on the 20th, not regress to the
19th.

Consider the typical school curricula where children take up to 4
years to learn basic arithmetic computation.  It is a grueling process
that turns off many children and is never mastered by others.

I have two questions of my own?

1) How much of what we call math disability is simply arithmetic
disability?

2) If arithmetic were invisible, what math would you teach a first
grade child?

I think math should be taught with a two pronged approach that
reflects the world in which we live.

Think about how you deal with the bill in a restaurant.  You do not
add to the nearest penny, but estimate.  On the other hand, if you are
doing accounting and need accuracy to the nearest penny, you use a
calculator or computer.  This is the way children should be taught.

Rather than memorizing tables and learning to "carry", "borrow", etc.
teach a child to estimate answers, first with single digits then
multiple.  Grade a child not on how many answers are perfectly correct
but on how close the child is to the correct answer.  If absolute
accuracy is required teach the child to use the tools of the 21st
century.  If the child is allowed to practice estimation with
immediate feedback, they will learn to be totally accurate not only
with the usual tables but to a certain extent two digit numbers.  In
addition, they will also be within 10% accuracy in the estimations.

My preferrence is to teach math with a spreadsheet which makes
arithmetic invisible and allows a child to concentrate on mathematics.
I have also developed a prototype spreadsheet to teach estimation.
If you are interested, take a look at
http://www.enabling.org/estimate/estimate.xls

I have not tried to use this on a computer that does not have the
Excel spreadsheet program and would be interested in feedback on this.
+++++++++++++++++++++++++++++++++++++++++++++++++++++
 Anne Pemberton <apembert@EROLS.COM>

Ted,
The main reason I've seen to have kids study higher math, above that which
is daily practical, is to find out who can go into the discovery science
part of math. Only the strong survive, but society as a whole needs those
strong to survive, and we need to find potentials to that end and help
potentials develop.

But, sadly, I'm not very good at teaching math, tho I can do a great job
with technology, reading, and a decent job with social studies, and the
simpler sciences .... I was a good math student in and directly after high
school, but "lost" the ability during adulthood, and was never any good at
explaining what I could learn ... There was a time when I could figure
interest compounded daily using logarithms .... now the computer does it
for me when I happen to need it.

In your joke, the example of "new Math" from the seventies(?) was
priceless!  I was a parent during the "New Math" phase ... and remembered
the remediation in late middle school so the kids could learn computation
so they wouldn't bomb in high school ...

Perhaps the answer is "you learn math because society reflected in the
experts and politicians wants as many people to learn as much math as
possible, so that society will always have mathematicians." ...

Perhaps for the same reason that we study Philosophy .... so that the study
continues through our society ...
+++++++++++++++++++++++++++++++++++++++++++++++++++
Janet Robbins <Janet.Robbins@ANU.EDU.AU>

Ted, ideally, I would always try to set up a scenario in which the student spontaneously recognises the need.  I n my children's "free" primary school (sort of John Holt style)the teacher and kids would establish a "village" (Greek, mediaeval, whatever) and building (geometry) shops, trade, money and maths would inevitably naturally result. It was brilliant! Higher maths happened more formally, but often in the context of play (what other purpose is there for pure maths?).
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Richard VanAmerongen <rvanamer@pcc.edu>

"What the heck do I need this for??????!!!!!"  (As a Serious Question.)
I trust you will receive many other replies, but here is my two cents.
Two possible ideas come to mind.

     A sincere question of concern from your student.
     A question to simply escape the reality of undergraduate education.

I use a simple eye-to-eye check to decide the status of the question, especially in pre-college level
(Algebra I & II, Intermediate Algebra).  I hear this question in all other non-Calculus theme
courses as well.  So, you are in good company.
For your question I think I would reply with a short comeback such as

      " ... this is an excellent  opportunity to add to your extra credit work.  I think a research of
     the uses for Mathematics in the academic area you are electing to pursue is a choice for
     you."

(Partly tongue-in-cheek, but effective in placing the ball back in their court!)

The recent years I have been offering Bonus options ("free" to choose, only positive points).  This
question, when a real "concern", fits the criteria for extra work and I would offer it for "Bonus"
point(s).  A real career analysis project!  It says to me I an no longer the "authority" guy, and
allows "free" research project assignment to bring to class a answer.  (Being fair, this could be
offered to all students individually, or possibly as a teams, dividing the class into pairs or three
person teams.

Recently, I even held a class vote (ala Florida results) on the Bonus status (yea or nay).

I have used this approach to answer well-meaning ideas (reinforcing "critical thinking") with some
success.  Weak students still avoid these chances to bring their other skills into use.  When asked
by a student not contributing to the class, I have to carefully weigh the odds that a question is the
"sidetrack" attempt rather than their sincere concern.  (Happiness is when a "responsible"
learner accepts accountability for issues they present to the class and follows through with
a reasoned solution.)
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
  CoolMath2@aol.com       Karen

Luckily, the last time a kid asked me this (in front of an entire class that looked at me like, "YEAH!"), I
boldly asked, "What's your major?"

She replied, "Music."   Whew!  I had an earfull for her!  Everything from the musical scale to how they
use logarithms to fade out the songs on a CD.

No one else had the nerve to ask me that again... that semester.  8-)

Yeah, the pat response of "it trains you to think" usually falls flat.  Why?  I don't think most of them
care about thinking.
Scary.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
CKMont@aol.com Cal Montgomery

I've generally either made a joke ("you should study logic" -- my area -- "so
that when you're hanging out with people you want to impress, you can make
what look like complex statements about things you know little about, that
are actually tautologies, and you'll never get caught in a mistake and people
will think you know more than you do") or gone for some kind of application
that looks like something the students might use, for at least some examples
(we'll leave out the Venn Diagram Dating Service, in which the men were
classified according to the standards Tall, Dark, and Handsome <grin>).

Two real-world examples:

1.  When working in a store in a mall, I had to make up weekly schedules for
employees' shifts.  Each week the home office gave a total amount of money I
was allowed to spend on payroll;  I was given a list of employees who were
salaried and therefore entitled to a certain number of shifts of a certain
length each week.  Other than that, I had to assign workers to shifts that
they could work, according to store need and according to their pay rates.
If P is the amount I'm allowed to spend on payroll, and S1 and S2 are the
weekly salaries of salaried employees, then I had P-(S1+S2) dollars to spend
on part-time workers.  If some workers are getting paid at H1 dollars an
hour, others at H2, and still others at H3, then I have x+y+z hours to hand
out, where x*H1+y*H2+z*H3=P-(S1+S2).  (Obviously, this permits a range of
values for x, y, and z.)

Note that if better workers are paid more (which was not necessarily the
case, and I need a certain number of hours just to cover the store (to ensure
that there is always someone on the sales floor to discourage shoplifting),
and P is low for the week, I have a strong incentive to give the most hours
to the worst workers.  (There are shifts when all you want is a body;  there
are other shifts when you really don't want the worst workers you have
running things.)

Note also that, knowing how many hours I need coverage on the floor, I can
make a case for going over budget in a given week before I start preparing
the schedule -- and, looking ahead (and knowing the formula which gives me P,
based on sales targets for each week), I could also give predictions about
when I could get labour dollars (year to date) back under the allowed amount.

Aside from this, I had to report, each week, how much our goal was (the
amount of money brought in during the corresponding week the previous year),
and how much our goal to date was;  and also how much we had indeed brought
in for the week and for the year to date.  I then needed to compute the
percentage "up" (over goal) or "down" (under goal) we were for the week, and
for the year to date.  A good grounding in basic arithmetic allowed me to
"eyeball" the figures and catch bad errors.  My boss, on the other hand, who
had never encountered the word "protractor" before (we sold some educational
toys and a customer asked her in front of me), couldn't do that.  And when
you report that you're "up 45%", when your goal is $2,165 and you brought in
$2,243 (this is just a random example), you're going to hear from your
district manager, and she's not going to be happy.

Note that this was basically a minimum wage job.  But being able to do these
things meant that (a) I got more of the shifts where documents for the home
office were prepared than I otherwise would have (bigger paycheck) because I
was good at preparing them, and (b) I had a lot more control over my work
schedule than I otherwise would have because I often made schedules.  (And,
if I'd stayed with the company, it might well have helped me move up into
positions of more authority and better pay.)

My boss, on the other hand, tended to make schedules and then calculate the
total labour costs -- and then go back and redo her work until she was in
better shape.  I didn't have the same hit-or-miss approach, and the work was
much easier for me.  (I could go very slow and easy and still look to the
uninitiated as if I were accomplishing huge amounts in tiny periods.)

I had particular luck with this stuff when tutoring a co-worker in algebra.
It was clear to her that having mastered the material she was working on made
a difference in the workplace.  And since I was simultaneously training her
on the paperwork for the home office (hoping to get her more hours), we
looked at the formulae that we were supposed to use and tried to get to the
point where, even if she couldn't remember them, she could reconstruct them.

2. When you go to college (I'm not sure how this usually works in primary and
secondary education), you will discover that a lot of your professors will
grade "on a curve."  Most students (or at least the ones I've taught) seem to
believe that a curve means you get some number of points handed to you "for
free" and added to your point score, which you can then convert to a
letter-grade according to some easy algorithm.  (This was a shock to me, to
discover that they didn't have any idea how we were assigning grades even on
"objective" tests where points were given or taken away for very specific
things.)

In fact, many professors will agree, up front, to assign a certain number of
A's, a certain number of B's, and so on.  So that in a 50 person class, 5
people are going to get an A- or better on the midterm, 10 additional people
are going to get a B- or better, 20 more people are going to get a C- or
better, and so on (or, depending on what the grades are supposed to mean, the
pattern may be different).  This helps professors adjust for the fact that
what they think is easy isn't always going to be easy for the students, and
that it's hard to write an exam that'll give you the score distribution you
want.  A score of 53 out of 100 doesn't mean anything at all, then, until you
see the distribution of scores.  I grew up with 64 and under being an F, but
I've also handed out Bs to students with grades in the 50s, if the test was
hard enough.

The problem is, if you're in a class with a lot of very talented people,
they're going to affect the curve.  If there are 5 people in your 50-person
class who are always going to be better than you on this material no matter
how much you study and the professor gives no more and no fewer than 10% A's,
you're looking at a B.

Practically, this is something to consider if you have a handful of talented
people in your programme.  (It also raises questions about how well,
especially in small classes with students who want to be there, the curve
system works.)

Wholly aside from this, most professors will give you an English-language
statement about how they calculate grades on the syllabus.  This isn't
entirely algorithmic;  if your quiz grades get steadily better over the
semester, you're likely to do better than if your quiz grades get steadily
worse, even if you end up with the same total, and there's room for judgment
calls (I once successfully argued to a professor that a student who was very
involved, and very good, in a discussion section but whose exam scores were
lower than I'd've expected, that "participation" ought to be weighed
especially heavily -- that it was poor test-taking skills rather than poor
mastery of the subject that was holding her back).  But it's useful to be
able to work out the formula.

Suppose your quiz grades are Q1, Q2, Q3, and Q4, your homework grades are H1,
H2, H3, H4, H5, H6, H7, and H8, your midterm grade is M, your final grade is
F, and your course grade is G (all on a 0-100 points measure), then knowing
that "The homework will count for 15% of your grade;  the quizzes and the
midterm will count for 25% each;  and the final will count for 35%," you know
that the formula is roughly

G = ( (15*((H1+H2+H3+H4+H5+H6+H7+H8)/8)) + (25*((Q1+Q2+Q3+Q4)/4)) + (25*M) +
(35*F) ) / 100

And you can calculate, going into the final, what score you need to get the
grade you want -- and you can also go lobby your professor to weigh some
things more heavily than others ("I was really struggling at the start, and
my first quiz and my first three homeworks show it -- but if you look, you'll
see that I had mastered that material by the midterm, and the second half of
the course I did really well") if you look like you're near a cutoff.

I have known very few students who aren't interested in the precise details
of how their grades are assigned.

Only fairly rarely have I been successful with "this material will teach you
to think in a certain way," because people who don't have the skills to think
in that particular way aren't usually aware of precisely what they're
missing, and they seem to think they've been doing just fine without it.
(I'm having this problem right now, actually, with someone.)  My most
spectacular success was with a third grader who, having been told the year
before that you couldn't divide odd numbers by two, reacted to his teacher's
announcement that she was going to explain how to divide odd numbers by two
(in other words, that she was going to introduce fractions) by becoming very
distraught (this impressed me significantly more than it impressed his
teacher or principal).  The explanation that the ground rules in mathematics
changed on a regular basis (the introduction of imaginary numbers so that you
could find the square root of -1, for example) seemed to be useful to him.
But he wasn't exactly typical.

On the other hand, demonstrating that thinking in the way I'm talking about
can give me some insight into the kind of problems that students can imagine
themselves encountering does appear to have an impact.

(Standardized tests can sometimes offer an answer to "Why should I learn
this?" but they don't motivate everybody.)

Another problem I've faced (and this is primarily in general philosophy
classes) is that some students who have always been successful in certain
academic areas suddenly fail -- previous teachers may have expected them to
master a body of knowledge and give it back on tests, but in philosophy
(which, like math, is primarily a skills class until a certain level) we
expect critical assessment about the information they've acquired.  I've done
reasonably well starting out with, "It's really clear from this paper that
you're someone who is used to much better grades than you're getting.  Let's
talk about what I'm looking for, that your previous teachers probably didn't
ask of you."  If I focus on the ways they have failed, they don't come back.
If I focus on the academic strengths that are apparent (since I've only
taught in the post-secondary environment, I'm used to students who have had
at least some academic success in the past), I do better at nudging them
toward the ways of thinking that I'm supposed to be imparting.

If you can track it down (I haven't had a copy for years), there's an Isaac
Asimov story in the book "Nine Tomorrows" about a young person in a society
where skills are essentially downloaded into brains who's shut out of that
system.  It's a story that's had a powerful impact on many of the people I
know who think seriously about how they should be educated.  Another
interesting book with a chunk about what we should study and how and why (the
meeting with the academic advisor in particular) is Robert Heinlein's "Space
Cadet".  (This had a great impact on me at a time when I'd given up on the
idea that classes were going to be accessible to me -- I wasn't in the public
system and thus not covered under IDEA -- and that if I wanted an education I
needed to figure out how to get it for myself, so perhaps I'm overstating.
On the other hand, I passed my tests okay even without having a clue what
happened in the lectures, and I give at least some of that credit to
Heinlein.)  I grant that you're unlikely to be assigning science fiction in
the middle of a math class, but these are reasonably accessible items that
I've found useful for places to start.

And, since most of my experience is with students who have chosen to be in a
university, but are taking the class I'm working in to meet a requirement,
I've often started the semester with a "Why I think this is worth learning"
discussion.
++++++++++++++++++++++++++++++++++++++++++++++++++++++
Michael Chejlava <chejlavm@mail.lafayette.edu>

Here is an excerpt from a message that I sent to the Chemed list about this
problem.

"There have been several comments about math and how it affects other
subject areas.  Back in the dark ages when we still used slide rules and
quill pens the math that was taught was mostly the "pure" math with few
applications.  Pure math is a self-contained system starting with a few
basic postulates and is then built up with theorems and proofs etc.  This
pure math can stand alone, but without applications it is just and
interesting mental game.  I have gotten the sense that the math community
is changing and starting to focus more upon applications.

The only caution I have to give you if you are a bibliophile like me, is
that even with the low prices, you have to use some restraint."

Numeracy is what students really need.  They need to understand numbers and
what they can use them for.  A basic understanding of statistics would help
people to understand risks.  A basic understanding of compound interest
would keep many people from charging their credit cards to the limits.  The
ability to use proportional reasoning would point out that when a person
throws away "just one" aluminum can rather that recycling it they could
then calculate how much aluminum would be wasted if every person in the
U.S. threw away "just one" aluminum can.  If you cannot do this
calculation, then you are not really an educated person.

Unfortunately in most schools, the only place where numeracy is taught is
in the chemistry and physics classes.

The current common method of having students memorize addition,
subtraction, multiplication and division tables at an early age with little
understanding of what they are doing, or even worse, having them punch
numbers into calculators leads students to follow rote mechanical
procedures with little understanding of what they are doing.  In this
method they are also often taught that there is only one "right" way to add
columns of numbers etc.

Somewhat better ways that I have read about are having students develop
their own addition tables by counting groups of items and then discovering
that it is faster to put together a table and memorize the results of their
counting so that they can save time.  Similar methods can be used for the
other operations.

Another big cause for innumaracy is the fixed time, variable learning
method used in schools, where all students are passed along at the same
rate.  Eventually many students get behind enough that they simply memorize
operations without understanding.

Schools should also add frequent diagnostic testing of each students
strengths and weaknesses, rather than just looking at total scores.  This
practice is what keeps the Sylvan and other commercial learning centers in
business.  Often a single misunderstanding by a student will show up later
a failing grades.

I believe that ALL evaluation of student learning should be continuous,
diagnostic, AND comprehensive.
+++++++++++++++++++++++++++++++++++++++++
Michael Chejlava <chejlavm@mail.lafayette.edu>

At 10:49 AM 3/18/02 -0600,  Michael Theall wrote:
>Ted,
>
>A quick response:    I try to find relevant, real-world situations where the
>content can mesh with some necessary or meaningful task or set of
>conditions.  Here's my example.

I agree that real world examples are important, but students needs the
basic skills before they can do anything useful with any knowledge.

>My son took chemistry last year (as I did in high school and then, in
>college).  We both were taught in the same way, learning the terminology and
>doing things like balancing chem. equations.   Since 1961, I have never had
>any occasion to balance another equation.

Chemical equations are the sentence of chemistry.  Without an understanding
of chemical equations, stoichiometry, the mole, and the periodic table, you
are not doing or learning chemistry, but rather a bunch of facts.

I have not diagrammed a sentence since I was in sixth grade.  I do not
remember all of the terms used in grammar, but I can write with correct
grammar, in most cases, and I can recognize when what I am reading is
grammatical.  Since I am not an English teacher, should I have not bothered
with grammar?
+++++++++++++++++++++++++++++++++++++++++++++++++++++++
"Theall, Michael" <Theall.Michael@uis.edu>

Michael,

I'll excerpt & respond, but I think the bottom line is that we basically
agree about what's really important.

"I agree that real world examples are important, but students needs the
basic skills before they can do anything useful with any knowledge."

I never said not to teach basic skills, only to teach them in ways that were
more successful for more students.

"Chemical equations are the sentence of chemistry.  Without an understanding
of chemical equations, stoichiometry, the mole, and the periodic table, you
are not doing or learning chemistry, but rather a bunch of facts.

I have not diagrammed a sentence since I was in sixth grade.  I do not
remember all of the terms used in grammar, but I can write with correct
grammar, in most cases, and I can recognize when what I am reading is
grammatical.  Since I am not an English teacher, should I have not bothered
with grammar?"

Again, I never said to ignore equations, but to put them in a more
understandable context.  English (or whatever language) is everyone's
everyday mode for communication with everyone else.  Thus everybody needs to
be able to use it well.  Only chemists talk to each other in equations.

"When I learned to play a trumpet, I was taught proper fingering and
embouchure. I played scales, did countless exercises and learned to read
music.  Since I never planned to be a professional musician, should I have
just been given the instrument and be left to play with it and make any
sound that I could get out of it?"

Again, I never proposed this kind of extreme approach or the abandonment of
solid basics.

"Unfortunately environmental chemistry is tremendously complex.  It requires
an understanding of stoichiometry, kinetics, physical chemistry,
photochemistry, equilibrium calculations and many other topics far outside
the realms of chemistry, including geology, meteorology, biology, economics
etc.  To show a students a few  massively simplified examples and passing
it off as chemistry does them little good."

Sounds as if I hit the sanctity of content button.  I didn't say to
massively simplify anything and I used the term 'ecological' to mean a more
comprehensive, systematic presentation that incorporated applications and
integration.  The same applies to statistics, for example, where many
courses treat the material only as mathematics, and students never get a
solid understanding of how and why the content works.  For most, it's not
the derivation of the formula so much as the application.  I know,
understanding the derivation helps to understand how the application works.
My objection is to skipping the integration and application altogether.

"Then why should all students be required to read Shakespeare, or Milton, or
any of the "great masters"  if they are not going to teach literature and
all they need to do is to read a newspaper or job application?

Why should anyone ever need to know about the War of 1812?  The war is over
and we know the final score.  I doubt that the conditions that brought
about this war or the methods of fighting it will  ever be of much
practical use."

Social science and the humanities deal with human interactions that are in
one way or another, a part of everyday life.  People act, react, and behave
in ways that can be illuminated by history and literature.  On a day to day
basis, the student who never gets anything except equations and the periodic
table won't be able to see the connections of chemistry to daily life.  Even
then, our daily existence is characterized by multiple human interactions
that involve us more than the chemical content of our toothpaste.  Note that
I said "involve".  This doesn't deny the importance of understanding how we
are affected by science, but it does mean that the vast majority of people
do not have to interact in the same way with math and science as they do
with other people. To be expert in the methods of fighting in 1812 is not
pertinent to many, I agree.  And in the same sense, to be expert in
recalling the formula for acetone isn't very pertinent either.

"It does not take a genius level IQ to balance chemical equations.  In the
act of learning to balance chemical equations, the student should learn
that in chemical reactions that matter is not created or destroyed, but
that the way the atoms are bonded is changed."

Never said it did & I agree with the underlying, important learning you
note.

"I totally agree with the above two comments. However, teaching just a few
"stories" about chemistry without some basic understanding is worse than
useless, since the students would then think that they knew chemistry."

I never argued for just telling "a few stories", nor did I claim that they
would suffice as learning.
++++++++++++++++++++++++++++++++++++++++++++++++++
Doug Karpa-Wilson <dkarpawi@indiana.edu>

> "When I learned to play a trumpet, I was taught proper fingering and
> embouchure. I played scales, did countless exercises and learned to read
> music.  Since I never planned to be a professional musician, should I have
> just been given the instrument and be left to play with it and make any
> sound that I could get out of it?"

This did catch my eye as, since the approach making music relevant to the
student is part of the foundation of the Suzuki method (If I understand it
correctly) Learning an instrument is far more likely to happen if the
student engages and music has real life relevance.  In this case, the skill
is learned by imitating parents and peers, weaving the skill (scales, etc)
into a social fabric which gives it relevance for the child's life.

So, perhaps, yes, learning music organically might not be so bad.  As a
child's appreciation grows, I suspect the child would have have greater motivation
to do scales.I'm curious, do you still play?  If not, perhaps 'playing around'
might have not been so bad. Ofcourse, having more time would probably help too, I'm sure!
++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Gary Zak <gzak@pacificcoast.net>

The best answer I have ever seen...quote
----------------------------------------------------------------------------------------
Why study math?
Teachers of mathematics are bombarded with questions like this every day, so I think one has to
know how to respond to it without the sort of made up "pain is good for you/it will teach you how
to think/it's going to be on the test" crap.

Generally, I tell them the truth:  except for arithmetic (including fractions), some stuff involving per
cent, and some idea of ratio, you probably will not use any of this for anything in your daily life.
And you can probably find some sort of work-around even for that.  So very little of it is of any
use at all for the average person.

Here are some of the responses I give when my students ask "What's this good for/Why do I need
to know this?"  None of them are absolutely convincing:

1. It's fun.  Not the systematic, follow-the-rules-and-apply-the-formula sort of stuff, of course.
That's not much fun at all (there's pride in a job well done, yes, but not much joy).  Unfortunately,
the fun stuff usually requires you to know the systematic stuff "cold".  If you don't, you usually
don't even understand that  the question is interesting (if you even understand it!), let alone how to
get an answer.  Just wait until you jump out of your seat and knock over your friend's soda
because you just realized there was a line-and-a-half proof of why there are 63 games played in
the NCAA basketball tournament.  You'll see.

2. It does teach you how to think.  Not "logically", really, but in three important ways:
a) It teaches you to go back and forth between the "big picture" and the "details" (the forest and
the trees).  Solving any mathematical problem that has more than one or two steps usually requires
you to do that, and the more you do it, the better you get at it.
b) It teaches you to be careful.  If you're not careful in mathematics, you're the bug, not the
windshield.
c) It teaches you that if you can figure out the rules, and follow the rules, you will get the answer,
just like magic.  Even better, you will look "really smart", when all you really did was follow the
rules.  (I have personally been coasting on this for the better part of three decades.)
Each of these three ways of thinking is extremely useful in everyday life, whether you are a car
mechanic (like my father), a surgeon, or an Assistant Customer Service Representative Associate.
Each is also absolutely unrelated to any particular problem or concept in mathematics: it's a way of
thinking, not a piece of knowledge.

3. It depends on what you plan to do with your life.  Tell me what you plan to do for a living, and I
will try to tell you where you might use something.  If you aspire to be Chief Burrito Folder at
Taco Hut, you probably won't need any of this (well, maybe just a little dynamical systems). But if
you want to do anything at all technical (business administration, engineering, computer anything)
you will need lots of math. Will you need to know the point-slope form for the equation of a line?
Maybe, maybe not.  Tell me who you are going to take a course from, or get a job with, five years
from now. I'll call her up and ask her, OK?

4. You don't actually need math for anything at all.  Well, that's assuming you are a God, or if not,
that you have a really excellent graphing calculator (with MIndRead? interface to push the buttons
for you). Mathematics was invented because we aren't smart enough to intuit the answer to every
question we might be interested in. And the dumber you are, the more math you need.  (And you
wondered how come I know so much....)
Mark Snyder,    Dept. of Mathematics,   Fitchburg State College
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 Craig Andres <candres@KETTERING.EDU>

One answer that I give now is "because that is what our accreditation board
requires".  We are primarily an engineering school (which has more obvious
reasons for requiring math), but I still get the question "Why do I have to
learn Calculus, I have talked to every engineer on this planet and not one of
them ever uses Calculus in their job, so why do I have to take it" (that might
be a bit exaggerated).

I have used many methods to try to convince students that it is an important
part of their development.

A) I tell them that any natural science degree is really an applied mathematics
degree, because that kind of degree requires a higher understanding of math.

B) I tell them that Calculus is the base language of engineering and science.  I
then invite them to open an advanced engineering text or two and see what kind
of symbols and notations they see.

C) I try to explain the difference between an Engineer/Scientist and a
Technician.  What are the expectations of the two positions?  How do they differ
in education and training (this is where the math comes it)?  How do they differ
in pay and mobility?  I acknowledge that Engineers are sometimes really
Technicians, and sometimes Technicians become Engineers, but the degree is more
likely to affect pay and mobility.

D) my short response is "to get your degree", which does not satisfy students
but is sometimes appropriate.
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Guillermo Uribe <omsaret@EMAIL.ARIZONA.EDU>

My take on question #1
Math is hard because we, the teachers, make it harder than it really is.
For instance, students will say that math is about memorizing a bunch of
formulas. No matter how much we argue against this perception, the fact is
that most Math textbooks place a  lot of emphasis on the formula. Students
get the idea that regurgitating the formula is all they need.

In my experience teaching algebra, I only emphasized the axioms of the real
numbers. I made sure the students understood that any other formula was
just a shortcut to which they could get time and time again using the
axioms only. Using the formula was not required, solving the problem
was.  When they could not remember formulas, I simply asked to solve the
problem any way they could. More often than not, students can think a valid
solution if they are given the chance; many times the student will use the
formula correctly without "remembering it" that is, without regurgitating
the formula first and then plugging the data. If Math is about thinking,
our teaching must reflect that goal.

My take on question #2 is this:
How do students know they are NOT going to need this stuff?

A reverse example: I was the only Math major NOT to take the computer
science emphasis while I was in College. I did, however take the three
computer science courses that were required as part of my course work.  My
graduate work did use computers but did not specialized in computers. Then
Life happened and I am making a living of the knowledge I acquired about
something I did not consider "worthy" when I was in College. Yet, when it
happened, I was ready to jump in the field, because I HAD to take those
courses. How many of us can relate to this story?

We could initiate a discussion in which the students define "useful" or
"practical."  We could explore specifically what is useful under that
definition. We could "move back in time" and use that definition to
"discard" stuff that was deemed "useless" at the time (but that, of course,
they can not live without these days). Examples abound. The moral is to get
them to understand that the more prepared they are, the more doors may open
in a future that they can not determine now.
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  "Sedillos, Marlene" <SedillosM@BARTON.CC.KS.US>

That was great to hear.  Willard Daggett's, a featured speaker at the NADE
conference, questioned those same things.  Other countries my be passing us
up in teaching what is needed in the "real world" while the US is stuck in
the "old academic world" which hasn't changed for generations. "Because I
said so" doesn't work.
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Craig Andres <candres@KETTERING.EDU>

Though you have to admit that what got our civilization to where it is currently
is the education system that was in place from the turn of the century through
the 70's and maybe into the 80's.  Change for the sake of change is not
acceptable, and I believe that has happened over the two decades.  It seems to
me that it changed with the goal that more people pass math with higher grades,
not that they became better at math.  Consider that with a greater dependance on
technology instead of learning how to think, we may be slowing our future
progress, it may be difficult to recover from that trend.

Maybe when some of these changes were made the right questions were not asked.
Instead of just asking how it would affect all students succeed in math (or at
least think they succeeded), it should have been asked "How will this affect our
future scientists, mathematicians, and engineers?  Will they have the necessary
background to do their jobs?  Will they be able to take us to the next level?
Are we preparing students well enough for math in college where many of these
programs are still very traditional in their approach?  I believe those
questions were not considered, and every year our students seem less prepared
and less willing to compensate for their weak math background.
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