Comments on ÒMathematics, The
Most Misunderstood SubjectÓ
http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp
by Prof. Robert H. Lewis,
Fordham University
These comments were
emailed to me between December 19, 2010, and January 4, 2011. I have not edited them. I have included all comments, except a
few follow-ups involving personal requests about how to learn ÒrealÓ
mathematics. I have erased all
names.
Dear
Dr. Lewis,
I just
read your excellent article, but I would like you to have included the
following point.
We live
in a world where the primary cultural educator is television. On virtually all
of the sitcoms and dramas we see displayed there, whenever a teacher is a
character the subject being taught is invariably English language or history.
This is a reflection of the right-brain orientation of those who are running
the system. Couple this with the incessant comment by personalities on the same
screen that they "hate maths, don't understand maths, etc." and we
have essentially educated our children that maths is both a non-essential and
difficult, if not impossible, subject to learn.
---------------------------------------------
Ran
across your articles and found them to be very well written. I wish my some of
my teachers would have explained the reason for math in everyday life as you
have. I would have paid more attention and would have been better prepared for
math in my later life.
---------------------------------------------
Dear Dr
Lewis,
I wrote
this email after reading your essay "What Math?, Mathematics, The Most
Misunderstood Subject". I'm not uneducated and most likely had more
mathematics then most people, but I've never really learned the real heart of
mathematics, not in the the way you described it.
I would
like to learn writing a proof (as an amateur) but I don't know where to start,
or how to recognize one, etc. Perhaps you can point me in the right direction
towards better books, online sources, etc?
------------------------------------------
I just
read your essay, linked to by slashdot.
I kept
waiting for "the meat."
What
specific changes would you suggest be made to our teaching of math? (Or
did I miss it or your point?)
I loved
your Dick and Jane analogy!
Keep up
the good work.
-------------------------------------------
Dr.
Lewis,
I just
finished your article: Mathematics, The Most Misunderstood Subject. I thought you did an excellent job
explaining why mathematics is important. I have heard teachers explain to
students that they are trying to teach them how to think and not just
math. Your examples really do a great job to help people understand your
point. I just wish I could make it required reading for every Director of
Instruction in the schools.
By the
way, your article has been slashdotted, so IÕm sure you are going to see some
additional traffic. Hopefully your ideas will spread.
Thank
you and Merry Christmas!
-------------------------------------------
Thank
you for your beautiful essay, Mathematics -- The Most Misunderstood
Subject. It was truly a joy to
read.
You'll
likely be receiving lots of attention for it over the next day or two because
it has reached Slashdot.
I
wanted to point out a little typo:
In the phrase: "no more helpful in establishing a career then, say,
philosophy" the word "then" should be "than".
Department
of Mathematics
SUNY
Potsdam
----------------------------------------------
First
thanks, the best description on the subject I have read for some time. Just a
few minor points. What's always
surprising me is the believe in simple boolean logic to describe a smart
machine.
Leaving
out: Myelin sheat (speed differential)
BAC
(Back propagating Activated Calcium,
dendrites are part of the calculation) Not all spike firings are equal
On a
low level and a lot of higher
level(aggregate) findings at Psychology level. Don't you thinks its time for a
new mathematical model to describe
a smarter machine, then just neuron firings?
BTW: I
use "Learning is the self organization of data points" as a guiding
principle.
Again
thanks for the write up,
---------------------------------------------
Dear
Dr. Lewis,
My name
is É and I am a doctoral candidate in music composition at the University of
MÉ.. I found your article "What Math?" via a Slashdot link and
fully appreciate many of your observations. I want to thank you for
taking the time to write this article and helping me appreciate the state of
math education in America. I found myself substituting the word
"Art" or "Music" for "Math" and realizing that
many of your conclusions draw a close parallel to Arts education in our country.
My only
negative comment about your article deals with your statement that "no
subject is more essential nor can contribute more to becoming a liberally
educated person than mathematics." My level of mathematics education
only extends to Calculus I so I cannot claim to have the same insights into
math as you. I would be interested to find out why you believe this is
so. I think your comments about mathematics being the language of the
universe are correct but I'm not sure math is the most ssential. I
believe that math, just like philosophy, the arts, and other sciences, are all
equally important for understanding the world around us. As an example,
if I were to listen to Western Art Music purely for the mathematical ratios of
the musical pitches then there would be an essential element of aesthetic
beauty that I would be missing. In mathematics, there is a beauty in a
perfect circle as well as a simple mathematical description. Both of
these are apt, I believe, but each tells the viewer something different about
the object as well as the viewer himself.
Again, I would very much like to thank you for
writing this article and giving me such wonderful food-for-thought.
Please keep me posted in the future of other articles you post like this.
----------------------------------------
Thank you for writing this article. As a
former mathematics teacher I think it said things that needed saying.
However, I believe that the problem starts well before high school. In
the elementary schools children are taught by people whose only real knowledge
of mathematics is that they are scared of it and dislike it. They are
very effective at communicating this.
Textbook
supported arithmetic drills assist ably in creating negative attitudes toward
mathematics. I am not saying that children should not know facts, though
the present system does not seem effective in achieving this. But
children could be learning about sets, patterns, series, logicÉ. Even
some topology. In short learning to think and wonder, not just grind away and hate.
-------------------------------------------
Dear
Robert,
I ran
across your article via Slashdot this morning while booting up my brain with
some coffee. It is a great article, and I forwarded the link to several of my
non-academic friends including a few mathematicians in linear algebra.
Many of
the points you make are also applicable to the way engineering is taught today,
at least in mechanical, aerospace, and civil engineering.
I'm a
second generation mechanical engineer. My late father, who earned his degree
before WWII, gave me his text books as a joke, expecting that the great
technical progress that had been made between WWII and the late 1970s would be
reflected in our respective generations of textbooks. He was wrong: his books
focused on the derivation of various equations used in mechanical engineering
while mine just gave the results, and in fluids, focused almost exclusively on
non-dimensional analysis so that graphs could be used to solve industrial
problems.
The
problem has only become worse during my career in academia. If I could have a
single academic wish from a genie, it would be the destruction of every book
ever written by Beer and Johnston. They have expanded into an expensive
two-volume set of books covered in an academic year material that introductory
physics courses cover in a couple of weeks on statics and rigid body dynamics,
all by applying Newton's Laws with little tricks to systems of widgets
(pulleys, levers, etc that will never be encountered in industry) and teaching
formulas. Most engineering exams are open book because professors with closed
book exams are penalized in teaching reviews. I think I could grade students
based on how much book flipping they do during the exams. During a curriculum
review that was critical of how these areas are taught, another professor and I
suggested teaching the analytical mechanics methods that are taught to
sophomore physics students and we were promptly shot down. The ultimate
solution was increased drilling on increasingly tedious problems.
The net
result is most faculty I know prefer foreign students educated in their home
countries as undergraduates to anyone educated here. When I was the chair of
graduate affairs, I had a few faculty explicitly tell me that they didn't want
any US students.
The
educational situation isn't much better at the graduate level. A friend in
physics once remarked that engineers pretend to be mathematicians because they
feel inferior to physicists. There is a lot of truth to that remark. Rigor is
often equated to notation. One of my colleagues who has won the highest
teaching awards in the UC system will mark a problem as wrong if students don't
use her notation even if the solution is otherwise correct. This is an official
policy explicitly stated in her syllabus. Students seem to like that approach.
When I taught another course where I tried to educate students on what
different fields called the same thing so they could read the literature (in
math and physics, the weak form of the momentum equation, in engineering
mechanics, the princ. of virtual work or virtual power, etc), I received
complaints.
I once
sat in on a math class for differential geometry so that I could read a
prominent researcher's papers in my field. The class was excellent -- Ted
Frankel was probably one of the best lecturers I've ever had – but what I
learned from the researcher's papers was that he wasn't using any of the tools
from differential geometry; he was just using the notation to hide conventional
work. One colleague I thought of as being very mathematical complained about
students asking why the test functions have to be zero at essential boundary
conditions -- it became clear while listening to him that he didn't know
himself.
Anyway,
enough ranting. Thanks for the excellent article and keep writing!
--------------------------------
Dear
Dr. Lewis,
It was
with great pleasure that I read your page introducing and "selling"
the virtues of mathematics (and to a much greater extent, knowledge for its own
sake) on: http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp
I
immediately forwarded it to a few friends who will most probably agree, and
also to my 15 year old brother who is stuck in the French educational system
(whose attitude towards smart students is elegantly summarized by Jean
Cocteau's famous "equality is chopping off the heads that stick
out").
Thanks
again for a great and well written web page. If I had done my undergraduate at
Fordham I would most certainly have joined your courses.
----------------------------------------
I very much agree with the
perspective expressed in you post. I particularly like ÒTeaching is not a
matter of pouring knowledge from one mind into another as one pours water from
one glass into another. It is more like one candle igniting another.Ó
------------------------------------------
Dr.
Lewis,
Thank
you for your essay
(http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp)!
It was
an absolute delight to read.
I am
very happy to read from people in academia that 'sees true'. I always want to
just go outside and shout "Yes, there are still real teachers out
there!" - when I see material of the same quality as your essay.
It
makes me cringe when I see these roving idiots that want themselves to be
called teachers but are an exact anti-thesis of their profession. Well, truth
be told, I'd rather throttle them and slap them silly. But I digress, to read
your essay was such a joy for me that I had to write and offer, meager as it
is, my support of your ideas.
Strength
to you and your ideas, and may it catch like wildfire in academia.
-------------------------------------------
Hi
Robert,
I love
your article on mathematics and education in general. Great examples!
Greetings
from a Polish teacher of web programming lecturing in Australia :)
-----------------------------------------------
Dear
Professor Lewis,
I
enjoyed reading your essay about mathematics. In particular, it looks like that
many people with power and money, even nontrivial folks like Mr. Gates
(Microsoft) do not understand the difference between education and training.
A
problem with your text similar to many others, is that people who need to read
it, like above-mentioned, will not read it online - if it were in NYT or WP...
How to correct this?
Happy
Holidays!
--------------------------------------------------
Dear
Prof. Lewis,
I hope
you're not overwhelmed with correspondence by being slashdotted. Your essay is
very interesting. Alas, I have no aptitude for math and little use for it. I
think they ruined it for me in second grade (1964-65) when they started
teaching "new math", whatever that is.
Nevertheless,
I do recognize that math has importance and value beyond the obvious, mundane
utilitarian tasks. A dear friend has a master's degree in math and speaks of it
reverently.
Your
essay opened my eyes to some things that I had barely glimpsed. You have explained what math is not and
how it should not be taught. You
have whetted my appetite. I am now curious to learn what math is and how it
should be taught. Perhaps it would take a book to explain this. Or perhaps you
could sketch something brief that would give basic information to someone like
me who is not prepared to understand anything more advanced. (By the way, the
link to the Garcia essay appears to be broken.) Your commentary is valuable. It would be especially useful
to the people who teach math K-12. I beg you to consider following up with
something more.
Thank
you and best wishes,
--------------------------------------------------
Dr.
Lewis,
I
wanted to drop a note to thank you for your essay "What Math?". I have been teaching an AP Calculus
class for the past six years. Your
comments about students memorizing procedures really strikes a chord with me. Each year I am struck by how many of my
students have "excelled" at math, yet fail to see the connections
between the pieces they have learned over the years. My challenge is to lift that veil of memorization of
formulas and to reveal the connectedness of the math to their understanding of
the world around them.
But
then, I come from a background of research (bailed from a Ph.D. in Geophysics
to the "lucrative" world of personal finance -- yes, it's been a
great decade) and not from the standard track of a high-school educator. There is an insight that comes from
using and living the mathematics that appears to have been lost in the standard
U.S. education teacher training.
This truly saddens me.
Anyway,
I will gladly pass on your insights to interested math teachers (and possibly a
few uninterested ones as well).
Since I teach a single class in the morning and then "go to
work," I don't know how much street credibility I have with the professional
teachers, but the word needs to be shared.
Again, thank
you for writing the essay. I wish
you well in your work and hope you have a Merry Christmas and Happy New Year!
----------------------------------------------------
Dr.
Lewis,
I have
read your article about mathematics being the most misunderstood subject with
great interest. The issues you describe in it are things I can relate to very
well having had a very bad mathematical education. I have ended up in the
Visual FX field where a lot of work requires problem solving and programming in
addition to traditional art training. My struggle with mathematics frustrates
me quite often and I was wondering what your suggestion is when it comes to
learning the basics of the field properly.
It
seems that most basic math books end up presenting more questions than they
answer, it also seems like the language around mathematics is rather cryptic for
people that do want to give it another try after having had bad teachers in
high school.
Any
tips or pointers in the right direction would be very appreciated.
--------------------------------------------
Professor
Lewis,
I found
your "why math" essay, and read it with great interest.
I have
a BS in Math from Va Tech, and loved studying Math. My favorite subject was
topology. I keep in touch with my undergraduate advisor, Dr George Crofts, and
we talk about the value of getting a Math degree, how it applies to industry,
etc. In the 40 years since I got the degree, I've never directly used it, but I
believe that it taught me a way to approach complex problems, and that I use
nearly daily.
I hope
I won't come across as being an ungrateful stranger, but perhaps I can offer
some friendly criticism:
Seems
to me that your essay started out great, but focused only on how most folks
have no idea what mathematicians study, do, etc. Missing was the answer, why
study it. It feels like the essay is explaining what math is not, rather than
showing
You
talk about math as a process, I agree.
Perhaps
you decided that it would a have been too long. I'm sure you know the usual
suspects (I mean, reasons) such as the study of imaginary numbers became the
foundation of all of Electrical Engineering; Turing's work becoming the
foundation of all of Computer Science.
When I
was an undergraduate, the University was just starting a Computer Science
department/degree program. I took a large number of CS courses, nearly
sufficient for a degree. But I didn't want one. I wanted the Math degree. Even
today, after being a computer developer for all of my career, I think math was
the right choice for me.
And, I
agree with you that itÕs the right choice for a far more people than know it.
The way we teach Math in high school is nearly criminal.
Have a
happy New Year.
----------------------------
Robert,
>
When people hear "mathematician", I want them to think
"poet", not
>
"accountant".
That
is a wonderful thought. My wife is an accountant, she can't believe me when I
tell stories of doing all of my math proofs in college in pen.
I
think accountants may use orthogonal parts of the brain from mathematicians.
Happy
New Year.