Lockhart's Lament

http://www.maa.org/devlin/LockhartsLament.pdf

DISCUSS!

>>1
In summary, a lot of fart with no solid description of just what the hell happens in his dreamland classroom over the course of an hour or a year or from year-to-year.

Section-by-section commentary:

Instead of a simple statement of ideas concerning math education, the reader is confronted with a story about a musician waking from a nightmare.  The story is meant to engage the reader by describing the state of math education in this peculiar manner, by analogy with music education (which actually does not operate as described).

The author is implying that the notation and standard terminology of math is analogous to the notation and terminology of music.  The reader is meant to begin thinking that the doing of math is akin to the doing of music, and that just as music is less enjoyable if the learning involves no actual sound of music, math is less enjoyable if the learning involves no actual math which exists independent of notation and terminology supposedly as much as does the music in our ears.  Therefore, the author is claiming that learning notation and standard terminology in general is not enjoyable, and for math is akin to studying music without hearing any actual sound.  The author is claiming that because notation and terminology of math is not math, teaching notation and terminoloy does not foster desire to learn math.  The implication is that somehow the ideas of math are better learned without reference to the standard language by which we communicate ideas of math, because doing math is like playing music.  I disagree.

The author continues to imply that adherence to the standard notation of math is overly strict and unenjoyable, and is empty of actual "music" (math).  I disagree with the analogy, as I can argue that being able to read and write musical notation can be very useful.  Of course, as actual music can be played anytime too, so can doing math, but the author has yet to state what he believes "doing math" actually means.

The math-as-music idea is further described, implying that boring math skills are taught too early.  The reference to humming and silly songs is the author’s way of saying that "real" (non-standard notational and non-computational skill) math is something that children would naturally do and desire to do.  I disagree that skills are necessarliy boring.  Also, using the music analogy, learning to play an instrument involves much practice to build skill.  Perhaps the author isn’t rejecting all skills, but such specifics are not described.

The author further extends the music-math analogy, describing the pressure of standardized tests and courses that are important for college that don’t actually exist in the learning of music.  Perhaps the author means to say that we should discard standard math requirements for college.  However, if a student was to actually try to get a music degree, there typically is some kind of assessed audition prior to acceptance into a degree program.  What analogous math audition would there be?

The author writes that actual "music" (math) learning occurs late and for few students, happening only for those that specialize in math in college.  The idea of students being "bad" at music (math) is mocked as a misinterpretation of what it means to be merely bad at the standard notation and skills, which themselves have earlier been implied to not be real math.  The author wants to reader to agree that the standard notation and skills are boring and not very relevant, that facility with them is not the same thing as being good at math, and that deficiency in them is not the same thing as being bad at math.  The analogy does not make good sense when considering any aspect of music making that requires skills that require practice.  Would we all better learn math by "playing it by ear" or perhaps improvising everything that we play?

The author belabours his point that teaching music (math) in the manner described is bad.

The author tortures the reader with a whole new story, now discussing painting as analogous to math.

In the same manner that standard math education was compared to learning music without sound, the author now uses the idea of learning painting without paint.

By describing the learning of techniques that are not painting but are merely preparatory, and doing of paint-by-numbers, the author implies that students don’t get to do any actual math until rather late in their education, and even then it has no room for creative composition.  The author is trying to lead the reader to accept the idea of math as an art and that teaching it as something else is contrary to reason.

Having belaboured his point with two metaphors (music and painting), the author begins to actually write was he means more plainly.  He criticizes math education for not nurturing students’ desire to make patterns, and being practically all wrong in what is done.

The author blames bad math grades (apparently) on the math education being stupid and boring.  The author neglects other ideas that could be responsible.   Also, the author never explicitly stated that there was a bad math grades problem in the first place, but leaves it to the reader to infer what the problem is.  It could well be that the reader incorrectly infers what the author is saying wrong about the outcomes of math education; but whether or not the author intends it, the likely inference that a reader would have is that there is a problem with student math grades, or perhaps skill assessment scores, being too low.

Most peculiarly, the author now leaves behind the math-as-art metaphors, and does not write that math is like an art or has creative aspects akin to art ,or that the learning of math is analogous to the learning of art, but explicitly states that math is an art.  The statement is absurd.

ok that's as far as I got so far

People that are not mathemeticians but use some aspect of math in their employment or household duties far outnumber the people that are mathemeticians.  It is not unreasonable for schools to serve such purposes of the general public, such as understanding what percents are.  Also, mathemeticians, long before attaining the label in the vocational sense, are only permitted entry into their degree programs in the first place if they have demonstrated some level of competence by doing well enough in their prior math schooling.  If the author would write some detail about what mathemeticians really do beyond stating that they do art, his argument could be made more persuasive.  I would be curious to see if anything that they are working on is anything that I am actually capable of understanding without having studied the foundations of their topics.  In any case, the author is advocating a false dichotomy when insisting that mathematicians are innovators and not users of known methods.

The author successfully communicates that he is passionate about math but does not successfully communicate that it really is as amazing as he describes, at least not for the untrained mind.  His general and and overly grand statements do not seem appropriate when a person is faced with the arduous task of reading mathemeticians’ findings in the published journals.  A student can far more easily appreciate the output of a musician or painter.  While music and paintings can be subject to abstract analyzing, there is an immediacy involving the senses that benefits music and paintings far more so than most math.

Given how awfully complicated the work of mathematicians appears to other people, and how much explanation must be given to help people to understand the work, the author’s statement about the allure of simplicity is puzzling.  A little elaboration about how simplicity leads to the development of poorly understood financial instruments or how it is seen in multiple-page proofs, these being products of the work of mathematicians, would be informative.

The author gives a good introduction to some of the underlying philosophy of math and its consideration of objects of the mind.  A trend in math instruction has been to try to move away from abstraction and instead concentrate on solving practical problems, and to deal with what can be manipulated by the hands, leading to excesses such as overuse of things called algebra tiles.  So, the author’s describing the intangible as something that can be enjoyed is welcome.  It would be helpful, however, to avoid any implication of a false dichotomy that math must only concern itself with the world of the imagination.  Also, the typical work of mathematicians isn’t figuring out geometrical relationships that might well have been figured out by the ancient Greeks already.  I must assume that he would not have students actually attempt to deal directly with a current mathematical problem, while somehow having them do what mathematicians do on some level, to follow his desire.

Education is there to find the top 1% who will discover new technology and run the country, it's not for the remaining 99% they only need to be literate enough to read the instruction manuals and obey orders from the top 1%.

>>8
Probably true.

=(

>>8
Sadly, true. And it's a cancer to the institution of education.

>>2
>>3
>>4
>>5
>>6
>>7
Where the fuck did you retards learn how to argue? A special school? Debate.org? The Student Room?

>>11
No one's arguing, you rude son of a bitch.  Those are comments on the content linked in post 1.

http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp

continued from earlier

The author elaborates on the geometrical puzzle, clearly communicating his enthusiasm for such things.  It would be interesting to know how students are evaluated or graded on the activity if failing to solve the puzzle, or if his courses use grading at all.

Hard work does not sound appealing to me, and is surely not appealing to many students.  Wasn’t a large part of his lament the idea that there is too much time spent on fatiguing tasks?  Or are fatiguing tasks fine if they are limited to puzzling?  I imagine that they can be greatly frustrating in any case.

The author describes his solution to his triangle puzzle.  Is every student meant to figure the solution somehow?  What is done with the students who do not have a similar moment of productive insight?

His criticism is peculiar given that recent decades have seen reforms in math education that have been criticized for moving away from the very things that he claims math education has moved toward:  memorization, drills, and standard algorithms.  In my experience, it is unusual to have students memorize such formulas.  Instead, they are provided during tests.  As for introduction to such a formula in class, it is not adequate to just state the formula, as good understanding can be achieved by an explanation or demonstration related to what should be already known:  how to find the area of a rectangle or parallelogram.  The author’s puzzle method can work as a way to begin to get to that formula.  Such experience can help a student recall the formula even if memorization is not required.  As for the idea that there is nothing for the student to do when just provided with a formula for calculation purposes, as if the activity is trivial to accomplish, there are many students who do not quickly get the skill of seemingly simple substitution of values for variables and successful calculation without some practice.

>>16
I was in high-school a few years ago and I can confirm it's just a bunch of memorizing shit and plugging values into formulas.  Sadly.

>>17
Were you in the normal math class?  Did you look at Pascal's triangle and get into binomial expansions and whatnot?  There must be 30 or so formulas and things such as trigonometric identities in the last grade, and they sure as hell give you booklet of formulas instead of make you memorize them all.

>>18
and they sure as hell give you booklet of formulas instead of make you memorize them all.
What?  And no I was in the less-than-normal math classes.  Yay.

The author tempers his criticism somewhat by recognizing that formulas and facts have value, unlike what was written in the entirety of the essay so far.  He then emphasizes the value of the insight in the solving of the puzzle.

The analogy would be better expressed as that it is like showing you Michelangelo’s sculpture instead of having you imagine it and make it yourself somehow.  People do enjoy seeing ready-made sculptures, and people can appreciate a mathematical pattern without having to figure the technical parts themselves.  How much knowledge can be gained in the available time if everything is to be learned by attempting to solve puzzles?  Does the author have more than a small amount of specific mathematical knowledge that he deems worthy of knowing?  I imagine that it is limiting to rely only on the puzzling method of learning.

The author again describes how mathematics is the creative activity of making and figuring puzzles, rather than the content in an ordinary math curriculum.  His idea of what the term mathemcatics covers is narrowly defined and unkindly excludes the knowledge and procedures that mathematical works have given us.  Surely he could have promoted his methods without having to be so seemingly dismissive, excluding "facts and formulas" from the definition, notwithstanding his "not complaining" about their presence in classes.  Just as we call the products of doing art "art", we can allow people to think of formulas and their usage as part of what math is.

The author repeats again what he says math is not, by analogies to what painting is not and what astronomy is not.

The author continues to indulge in the false dichotomy of understanding versus skill.  Manipulation of symbols need not be as mindless as the author describes.  He describes graduate students who "have no mathematical talent" despite being adept at the skills that allowed the graduate students to become graduate students in the first place.  How can teachers realisticly teach elementary students to become successful graduate students in mathematics without giving them the prerequisites to being allowed in the programs?  Must elementary education point specifically to success in Master’s degree programs, when there are so many other paths to take?  Does the author imagine a world where every student can create new math at the graduate school level?

Ordinary schooling of any subject does not tend toward the writing of criticism.  Schooling in general tends toward acquisition of knowledge and skills rather than the writing of reviews.  For instance, English classes focus on writing essays involving identification of story elements and figuring implied meanings rather than personal assessments of value.  As the author likely agrees, certainly the ability to intelligently criticize the work of others is a worthwhile skill, and is good to have integrated into schooling.  What is not obvious, however, is how much background knowledge is needed in a mathematical topic to come up with more than a cursory statement of feeling about someone else's work.  It would be helpful also if the author provided guidelines for the assessment of students' criticisms.

Is the author proposing that math classes have written assignments, such as essays about triangles?  Many students find math classes appealing because they do not have to do a lot of paragraph writing, or pretend to have opinions.  Concerning coming to conclusions about Napoleon, a student has to read and remember plenty of facts and concepts related to history, geography, and civics to arrive at cogent conclusions.  Many people find that history is not so easily understood, either.  However, any ordinary high school student can read a history book and get some understanding, while reading a typical mathematical journal is beyond most adults' ability to understand.  Even mathematicians can't quickly understand all the work of other mathematicians, as areas of study are indeed numerous and have their own technical aspects.

The author purports that the society's view of math as something useful rather than as something artful is a regrettable view for society to have.  The unnecessary implication is that practical usage is not interesting, which fits with the author's adherence to a definition of math that is aligned with unfortunate false dichotomies (such as practice of skills versus figuring puzzles).

Aesthetics for the Working Mathematician
http://docserver.carma.newcastle.edu.au/150/2/01_165-Borwein.pdf

okay

The author is pretending that a discussion is taking place between an ignorant Simplicio and a wise Salviati.  The names are from a work that Galileo Galilei composed to put forth his (Salviati's) view that the planets revolve around the sun, and not around the Earth.  Apparently the Catholic pope at the time felt very insulted by his geocentric view being represented by a someone named for being a simpleton.  The author’s choice of using the Simplicio character is insulting to readers who do not agree with the author, not just for the name, but with the implication that a reader’s contrary view is akin to sticking with the idea that the sun and planets revolve around the Earth.  The start of the discussion repeats the author’s idea that math is not the same thing as the practical use of math, in keeping with his artistic aspirations for the subject.

Surely even less people are likely to be graduate students of mathematics than to encounter some utility for a math skill after finishing high school.  The author's arguments point more toward the impracticality of higher education in general, than to the needlessness of mathematical training in particular.  How many adults remember all the characters and plots of the stories they read, or the functions of all the parts of a cell, or can play all the sports as well as they did?  All knowledge and skills are more likely to be retained with practice over the long term, and no one continues to practice everything that one has ever learned, so his criticisms need not be limited to math education.  What could be discussed is the whole philosophy of education in terms of knowledge for the sake of knowledge and giving skills to people to help advance them economically.

>>26
Ironically, I was unable to understand past the start of the Gauss section, not because I had too much traditional math instruction, but too little.  I don't know what the fuck a lemniscate sine function is, and the article pretended as though, oh everyone knows what that is.  We sure as fuck aren't going to have 6th graders try to evaluate aethetics of whatever the fuck he was talking about there, without years of background study first.

>>30
We sure as fuck aren't going to have 6th graders try to evaluate aethetics of whatever the fuck he was talking about there
It's a completely different piece from Lockhart's Lament >>1.  Read the intro:

Most research mathematicians neither think deeply about nor are terribly concerned about either pedagogy or the philosophy of mathematics.  Nonetheless, as I hope to indicate, aesthetic notions have always permeated mathematics.

I shall similarly argue for aesthetics before utility.

Nothing is said about 6th graders.

>>31
I said "6th graders" genious.  His whole spiel is about all pre-college math education, not just senior high school, so it's safe to assume any subset of pre-college math education is included.

>>5
>It could be the reader misinterprets
translation: It could be I'm full of shit, and this is a meaningless ad hominem

Mathematics is a jewish pseudosciency.

Georg Cantor was born in 1845 in St. Petersburg, Russia, where his father was a merchant. The father, Georg Waldemar Cantor, had Jewish parents. -- Loren Graham, Naming Infinity, p. 25

http://en.wikipedia.org/wiki/George_Cantor#Cantor.27s_ancestry
Cantor was frequently described as Jewish in his lifetime. In 1874, Cantor married Vally Guttmann. As a Jewish (Ashkenazic) surname, Guttmann comes from the male given name Gutman.
http://www.ancestry.com.au/facts/Kantor-family-history-sct.ashx
Cantor is jewish (Ashkenazic) occupational name for a cantor, an official of a synagogue whose duty is to sing liturgical music and leads prayers

http://en.wikipedia.org/wiki/Abraham_Fraenkel
Fraenkel was a fervent Zionist and as such was a member of Jewish National Council and the Jewish Assembly of Representatives He is known for his contributions to axiomatic set theory, especially his addition to Ernst Zermelo's axioms which resulted in Zermelo–Fraenkel axioms.

http://mathforum.org/kb/message.jspa?messageID=69757&tstart=0
In 1935 Zermelo resigned from this position. There is no doubt that it
was caused by the firing and mistreatment of Jewish mathematicianss, e.g.
that of the algebraist and statistician Alfred Loewy who was removed from
his chair at Freiburg.

http://en.wikipedia.org/wiki/Aleph_number
>In set theory, Hebrew letter aleph (א)
http://en.wikipedia.org/wiki/Beth_number
>In mathematics, Hebrew letter ב (beth)



Through this instrument Klein was able to control developments in the field, since those who
worked outside his empire were never given any recognition. Eventually all the
leading journals became dependent on him, and no one could obtain even the
lowliest position without his approval. Yet even this could not satisfy Klein's
ambition for he also wanted the mathematicians of other coun-
tries to submit to his rule. Thus he drew foreign students to Gottingen and gave
them work of great interest, while young Germans, unless they happened to be
Jewish, were forced into the background. The atmosphere in Gottingen was not
only international and pacifist; it was already, in the 1890s, decidedly anti-
German. Any expression of nationalist sentiment by a young German automati-
cally jeopardized his career. In fact the Gottingen influence
was so pervasive it even created a new style among German mathematicians,
whose behavior, posture, gestures, and manners of speech were altered in imita-
tion of Jewish prototypes. Only those non-Jews who could adopt this style had
any hope of furthering their careers. Klein's dictatorship eventually led to a
barely visible but all-powerful organization across the academic landscape, and
practically every institution of higher education in Germany had its "Gottingen
Jew" on the faculty. -- David E. Rowe, Jewish Mathematics at Gottingen

It seems clear that Nazi Germany did severely persecute what it defined as “Jewish mathematics”. In his book “History of Mathematics: A Supplement” (Springer 2007) Craig Smorynski said: “… the change of mathematical direction … would reach an extreme in the 1930s with the nazi distinction between good German-Aryan anschauliche (intuitive) mathematics and the awful Jewish tendency toward abstraction and casuistry.

>>23
the ability to intelligently criticize the work of others is a worthwhile skill
That would be a waste of time. Just say "their work is crap and they should kill themselves, before you killed them". Basically, if their work don't support your ideals, their work is a useless shit. For example, if you are a white nationalist, then the writings of jewish and nigger garbage would have no use for you and your white race.

That feel when most of this thread consists of one guy deconstructing the article to oblivion.

Fuck, if I pick apart every aspect of an argument, it'll look bad no matter how well it's done.

what is it I cba to read the pdf