Conference rusure::math

    Try to get the students to apply their intuitions into math !

    dont teach them to mechanically how to solve a certain equation,
    or solve some abstract mathematical problem, but first explain what the 
    equation or the math means physically or geometrically..

    and one more thing, always outline where what you are explaining fits
    in the overall scheme of things, before jumping into detailed, dont
    let them lose sight of the forest because they are just looking at the
    tress , always step back, and explain WHY you are doing what you are
    doing, and WHY you are not using another method instead and always try
    to give physical or geometrical correlations and examples of what
    you are working on means...

    i guess that inspiring students to think for themselves is more important 
    than teaching the mechanical steps needed to solve something...

    /nasser
    ps. one thing of what i mean, is this:
    
    looking at integrating a single valued function over a certain range, it 
    might help to let them think of integration as a BALL that starts 
    rolling from top of snow mountain down to the bottom, as it rolls it
    collect snow and it grows larger, when it get to the bottom, the weight
    of the BALL is the value of the integral !

    the BALL is the function ! since it accumulates snow as it rolls, the
    integration is really an accumulator ! it is summation , at every step, 
    it picks up weight i.e. it increases in value, different functions means 
    different shapes of objects rolling down, or sphers that have different
    surface proprties that allow them to accumlate more snow than other
    when rolled over the same path, a discontinuity means a hole 
    in the path of the sphere, and hence we cant integrate (i.e roll )over it !
    since the sphere will be lost there (ie. undefined at that point) 
    etc.. etc..

    with more imaginatins one might be able to explain other things along 
    these lines such as limits, etc..etc..

    i dont mean this is a good example, but something i just thought of to 
    tell you what i mean, and this is supposed NOT to replace real math
    analysis by any mean,s it just an example of how to help students
    get a FEELING to what things mean, and then you go and teach them analysis,
    but then they have a FEELING for things, and they'll be able to 
    processed with the math easier...

    just my 2 cents offcourse..
    
    

    I'd second Nasser's ideas.  Show the big picture before you get into
    the details.  Help them build mental images of the problems they need
    to solve.
    
    From my experience, it's important to let students see you make a
    mistake and regroup and continue solving the problem.  Too often
    lectures are so well prepared that the student never learns what to do
    when s/he has made a mistake.
    
    Build their self esteem by letting them fail, try again, fail, try
    again, and finally succeed.  The "get it right the first time" is fine
    theory for manufacturing, but not for learning.  They need to believe
    they can recover and that no failure is permanent.
    
    Remember, there are only three kinds of successful teachers,
                                                             
    	1) Teachers who are enthusiastic about their subject and can
    pass that enthusiasm to their students.
    
    	2) Teachers who have a passion for learning and can transmit 
    that to their students.
    
    	3) Teachers who care for their students and move mountains to help
    them learn.
    
    /rab
    

	My college calculus teachers were all turkeys.  One was an aging
	bigot; one was an aging oh he wasn't too bad.  I don't remember
	one at all.  The first one, calculus in electronics, was OK.  That
	was a practical class.  The trig teacher in high
	school was a turkey; she catered to her favorite students.  
	A geometry teacher in junior high school
	was pretty good.

	Each person needs to learn their own way.  Be prepared to teach 
	the same problem two or three different ways.  As you learn about the
	students in a particular class, you will better choose which 
	approaches to cover, but always use a few different ones.
	Read How to Solve It and teach it to the class as part of the
	class.  Does this problem look like a problem I know how to solve?
	Can I break up this problem into tractable pieces?  Can I look it up
	in 1000 Solved Problems in Calculus? ;-)  can I write the problem in
	different way? Can I discard some of the information?

	It is interesting to wonder whether to take Hardy's approach and teach
	math with no reference to the physical world, or to use the physical
	world.  Beats me.  I like the physical world, but usually come to
	understand some math completely independently of it, and then
	apply it.
	Tom

    I taught lowerclass math at my University for several grad (not "grand")
    years.  A professional grad student passed this idea along to me, and I
    tried it one year with tremendous success:
    
    Say we're teaching Rolle's theorem (IF f is differentiable on (a,b),
    continuous on [a,b], f(a) = f(b) = 0 THEN there exists a pt. x on (a,b)
    satisfying f'(x) = 0).  (Or whatever is the topic du jour).
    
    So you go thru the lecture and prove it, and maybe work one or two
    examples.  Then turn the tables on the class.  For the last 10-15
    minutes or so of the hour, hand them a problem that requires them
    to actually use what they've just learned.  Something really simple,
    but which makes use of the lesson.  E.g., "apply Rolle's theorem to
    f(x) = x� - 7x + 12".  Maybe even prod them along with subdivisions
    of the problem, e.g., "(a) Find two points a and b where f(a)=f(b)=0",
    "(b) Therefore point x is _____".
    
    Have them hand it in at the end of the class.  Grade it and count it
    like a problem on an exam.
    
    It is a real effort on the teacher's part to come up with good problems
    that can be applied every day, in fact you can't do it every day, but
    twice a week is readily done.
    
    Once they got the hang of it the students really appreciated the chance
    to apply learning while it was fresh in their mind.  And, of course, it
    pretty much forced them to pay attention in class.
    
      John

re .-1

You can even give only problems where x ends up equal to (a + b)/2,
and then ask the students to prove that is always true or find a
counterexample.  O:-)

Dan

    Hi, Jim!
    
    The characteristictic that distinguished my distinguished math teachers
    is that they were ... distinguished!  Which isn't to say that they were
    aloof or haughty, or lacked humor.  Just think of Cary Grant.
    
    Be alert.  And give attention to each of the students; they're all
    people and have value.  But why am I telling *you* this?  I'm sure
    you'll be more than fine.  Good luck with this career change.
    
    					- Gilbert
    
    P.S.  Say hello to Brenda.  I've probably misplaced your address (again).
    What is it?  Please *don't* answer (a+b)/2 !

    
    I've always thought kid's would benefit from becoming engaged in the
    philosophical foundations of mathematics. Too often mathematics is
    presented as a 'done deal'. I wish I'd been exposed to the case for
    Truth and Beauty at an earlier age. 
    

To me, the most important thought that's been expressed so far is: different
people learn different ways.  Know how your students learn, and be prepared
to teach it in a way that they'll understand it.

Some people prefer to learn details before the overview.  Others prefer to
learn in exactly the opposite way.  The best way is what is good for the 
student.

Relevance is always a problem.  Students need to see that the what they're
learning is relevant; they need to see it used for "real."

I also agree that the problem solving aspect of math needs to be stressed.
The skills used in solving a math problem are identical to those used to
solve many other types of problems: trial and error, decomposition, divide
and conquer, etc.  Using a math class to teach about those skills independent
of straight math might be fun.

Beware of roadblocks.  About 15 years ago, a study was done of people who
said they were bad at math.  The researchers found that many of them had
stopped learning after reaching a "roadblock."  These came in the form of
some concept or algorithm, usually a simple one, that they just never under-
stood.  Until the roadblock was removed, the people were unable to learn more
math.  I don't know if the study has stood the test of time...

Also:

READ, READ, READ.  Find every book you can on recreational mathematics and
look for ways to tie those puzzles in to what you're doing in class.  Martin
Gardner books, Scientific American, two books by Ivars Peterson: The
Mathematical Tourist and...  (the name of the other escapes me).  You'll
find lots of material that you can use in class to help make math real.

Subscribe to the Mathematics Association of America journal.  There's one
for high school level. Find one in a library and read through it -- it's great!

Anecdotes:

DEBUG YOUR TEACHING: I once substitute taught four sessions of a calc. class
at a local tech school.  Subject was integration using trigonometric
substitutions.  I prepared for hours.  I wrote down each step in 
each derivation.  I did every problem explicitly, in case anyone had any
questions.  I taught, then assigned homework.
    Next class, I asked people to come up to the board and put up the answers
to their homework.  The results were abysmal.  I went over it again.  Assigned
more homework.
    Next class, back to the board.  They just didn't seem to get it.  What's
more, they couldn't quite figure out what it was that they didn't get.  So I 
said, "OK.  I'm going to show you the solution to one problem step by step.
When I do something you don't understand, yell."  I went through the most
complicated steps with nary a whimper from the class.  Suddenly, I
simplified an equation by substituting cos^2(x) for 1-sin^2(x) and the class
yelled.
    So, I spent ten minutes deriving cos^2 + sin^2 = 1, cot^2 + 1 = csc^2, etc.
    If your students aren't doing well, it's so important to find out why.

I've just started a Masters in Computer Science degree, after being out of
college for 15 years.  Last semester, I took a course on computer language
theory (grammars, automata, etc.).  The teacher recommended a book called
"Godel, Escher, Bach: The Eternal Braid," a fascinating 750 page book that
deals with many topics related to computer language theory.
    One day in class, we got an assignment to prove a theorem about the
relationship between recursive and recursively enumerable languages.  I
completed the proof with no trouble at all, even though it made little sense
to me.  Later that week, I just happened to read (in GEB) about that very
theorem, presented in an intuitive way (along with a picture) and I was amazed
(and thrilled) to finally understand the theorem.  A new approach,
a picture, made all the difference.
    The point: pictures and intuition make difficult concepts seem so simple.

Try to encourage students to read about math: mathematicians, puzzle books,
recreational math.  Not all of them will, but for one or two students a year
it might be a turn-on.

I wish you the best of luck...

Dave

re: .7

>>    I wish I'd been exposed to the case for
>>    Truth and Beauty at an earlier age. 

Amen.

I was fortunate to have two teachers in high school who successfully made
me aware that there is an inherent beauty in Mathematics.  That Math for
its own sake was an honorable pastime.  That the end goal is not arriving
at a numerical solution, but rather an understanding of the total problem,
which can be stated in a definitive, nonambiguous manner.

re: roadblocks

My sister ran into a roadblock with Algebra.  She has not passed it by,
even 25 years later.

My roadblock was differential equations (too much memorization, that's
what I blame now.)  I still don't understand them, but in college
I made and end-run around the roadblock and finished my Math requirements
with number theory, without any significant problems.  I believe that
if I had the need, I could go back and try to grasp the missing concepts
of differential equations.
    
JPB

Interest the unmotivated.
Challenge the motivated.
Inspire the gifted.
Good luck!

    re base note
    
    First of all, good luck on your new career.
    
    I would recommend that any mathematics you teach should be motivated,
    either by a historical understanding of the problems that gave rise to
    the subject your are doing, or showing something insightful about the
    mathematician who got the result. What was his angle?, what motivated
    him?
    
    As you know geometry came into being when the Greeks abstracted the
    surveying techniques of the Egyptians.  Show the *relevence* of the
    abstract geometrical theorem (when possible) by tying it to a *real*
    problme in space measurement or physics.
    
    Likewise for calculus. Calculus was invented to handle velocity and
    acceleration. When function theory is being taught tie it in to the
    physics which motivated it (if possible).
    
    Also show mathematics as a *human* activity, not an excercise in
    marblized ideas and icon worshipping.
    
    Hey man, good luck!!!
    

re .8
    
>"Godel, Escher, Bach: The Eternal Braid," a fascinating 750 page book that
>deals with many topics related to computer language theory.
    
    Godel, Escher, Bach: The Eternal Golden Braid
    -      -       -         -       -      -
    
    Agreed, a very stimulating book, recommended to anyone who hasn't
    already read it (and even those who have). Author is (from memory)
    Douglas Hofstadter (sp?).
    
re .0
    
    I've always thought that the saying about "those who can, do, ..." is a
    bit unfair and short-sighted because one generation of bad teachers
    will lead to the next generation of bad scientists, engineers,
    mathematicians, etc. I think that enthusiasm for the subject is the most
    important thing a teacher can impart to a student.

    I can't keep away from this subject.
    
    Roadblocks!
    
    	My wife's roadblock, and that of many students I have known, was
    FRACTIONS, as in 4-6th grade fractions.  Beware of problems in basic
    arithmetic, they can be like rocks just under the surface of a stream. 
    People who were taught arithmetic as memorization are likely to have
    little insight into generalizations we take for granted.  For example,
    commutativity and associativity of addition and multiplication.  The
    distributive laws are frequently mismash in a student's mind.  They
    don't need to know the words, but they MUST feel the rightness of the
    rules.
    
    	FEAR of failure is the biggest roadblock of all.  I have had
    students come up to me on the first day of class saying "I am going to
    flunk this course!"  The only response I could come up with was "If you
    believe that, you will make it come true!"  I had one student who
    clutched so badly on exams that she submitted to post-hypnotic
    suggestion to help herself keep her cool.  Very distressing.
    
    	They need to be able to think about the problems on the exam and
    not on what will happen after the exam is over.  That's hard when you
    are afraid.  
    
    	If you've never felt the fear of math, find someone you know well
    who has.  Ask them to tell you what its like.  It can be very
    debilitating.
    
    fwiw,
    
    /rab

	Thanks everyone -- I appreciate the input.  Lots of great ideas
	and important references.

	I'm leaving via the EEP (Engineers into Education Program) and
	have pointed other participants over as well.

	-Jim

	p.s. -- I don't buy into the "those who can't, teach" bit either.
	It's sort of like one of those racist epithets only people of
	that particular race can get away with.

There have been *many* good points made in the previous replies... to 
highlight/paraphrase a few: 
	
	o inspiring students to think for themselves
	o acknowledging that students are people and that each one 
	  is different 
	o providing and maintaining context (both mathematical and 
	  historical) around the topic of the day
	o exhibiting your passion for the subject and for learning 
	  in general
	o applying physical realities to theoretical concepts 
	o reading recreational math books and finding ways to integrate 
	  their content into your classes (I believe you will find that 
	  once you have done the reading, the integration will happen with 
 	  little effort on your part.)

In addition to all of these suggestions (although implied by most of them), 
I believe it is critical that you are an *excellent* listener. 

One of the most effective ways a math class can be conducted is through 
the voices of the students. Listen... they will amaze you with how much 
mathematics they can construct on their own. 

Obviously frameworks and contexts need to be established and an appropriate 
atmosphere needs to be maintained, but once these needs are addressed, try 
to minimize the lecture and maximize the input from the other people in the 
room. 

				Jack 

P.S. Good luck!

    I have to add this anecdote, vis-a-vis listening.
    
    Back in the dark ages, when I was in high school, we had an algebra
    teacher named Nellie Loss.  (You can guess what her nickname was:-))
    
    Well, "Total" couldn't understand students' questions, much of the
    time, because she didn't know what kinds of confusions "lurk in the
    minds of men".  I frequently found myself saying things like "What Bill
    wants to know is ..." because I could empathize with my buddy (knew
    where he was coming from, for you kids out there).
    
    Don't just listen to your students in class, listen in the hallways, at
    sporting events, at dances, on the campus, wherever.  You may not want
    to "use" their dialect, but you're damned if you can't interpret it.
    
    /rab

    My friend has a replacement for "...those who can't, teach."  He
    prefers:  "Teach, those who can, do!"
    author:  Jay Bonstingl
    
    David Gertler

I overheard some people talking (you can't really call it eavesdropping --
I was in a ten passenger limo in the seat directly in back of these people,
and they were speaking very loudly).

Anyway, one of them remarked that he worked with some engineers who were 
so advanced, and who were working on projects so complex, that it was 
impossible for them (the engineers) to explain to "mere mortals" just what
it was that they were doing.

The other person remarked that the fact that the engineers' work could not
be explained is not a testament to their incredible knowledge.  What it 
probably meant was that these people didn't really understand what they 
were working on.

The point is: in my opinion, it takes a great deal of expertise to teach
effectively.  You have to know your material inside and out.  When students
ask, "what's this used for?" nothing beats personal experience.

You've probably left the company by now, but if you see this.. Good luck again.

Dave

>The other person remarked that the fact that the engineers' work could not
>be explained is not a testament to their incredible knowledge.  What it 
>probably meant was that these people didn't really understand what they 
>were working on.

Richard Feynman had this point of view... he even attempted to get across
a reasonably accurate view of quantum electrodynamics.

For my own part, I never really feel I understand something until I
can explain it to myself in simple terms.  If it's something I need
to use (like some mathematical abstraction...) it is always more
reliable for me to play with the idea till it can be seen with the
minimum of advanced machinery.

Some things are a bit difficult to tell the truth about from first
principles though.  For example, why the cube can't be "duplicated"
or why exp(pi*sqrt(167)) is very nearly an integer...

Most engineering stuff I understand is actually simpler than these
mathematical facts.  For example, wideband FM, how wire antennas work,
etc.  But these can be well obfuscated in engineering textbooks :-)

- Jim