Comments on  “Mathematics, The Most Misunderstood Subject

 

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 “real” 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.

 

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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.

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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?

 

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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.

 

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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 I’m sure you are going to see some additional traffic.  Hopefully your ideas will spread.

 

Thank you and Merry Christmas!

 

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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

 

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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,

 

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Dear Dr. Lewis,

 

My name is … and I am a doctoral candidate in music composition at the University of M…..  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.

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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, logic….  Even some topology. In short learning to think and wonder, not just grind away and hate.

 

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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!

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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.

 

 

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 I very much agree with the perspective expressed in you post. I particularly like “Teaching 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.”

 

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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.

 

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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 :)

 

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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!

 

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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,

 

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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!

 

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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.

 

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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 it’s 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.

 

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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.