Posts Tagged ‘Education’

The purpose of a college education

March 5, 2010

Facebook thread:

“One oddity of the St. John’s curriculum is its refusal to engage the
modern sciences. One could argue that graduating “Johnnies’’ are among the best-educated young men and women in the country. However, it is doubtful that any could score a C-minus on a freshman biology test at a community college.”


I graduated St. Johns in the class of ’71. I have never regretted those four years. The nature of the program was ideal for my temperament, strengths and weaknesses.

Obviously there are only so many areas or topics that can be covered. I am sure that a biology major could not pass an elementary examination in accounting.

We must ask ourselves what the real goal of a college education is. No matter how well we memorize and master some body of facts or techniques we will gradually forget them unless we regularly exercise ourselves by referring to them and using them.

The head of the computer science department at the University of New Haven had a PhD in chemistry. The first companies to have a great need for computers and the funds to afford them were chemical companies which took some chemists and trained them as programmers. He told me that he could no longer remember the chemistry he had mastered years ago.

I had a long talk once with a fellow who had earned a masters degree in biology from Yale. During the course of our discussion I pointed out to him how the Sickle trait provides a survival advantage over people with normal hemoglobin in regions where malaria is endemic.

He seemed rather embarrassed that he had totally forgotten that fact although it was in his field of specialty. It is often the case that people who graduate from Ivy League schools with advanced degrees have a knack for absorbing an entire textbook over a weekend and acing the exam and forgetting much of the material after a year has passes.

In these days Internet search engines it is not WHAT we remember that is important so much as knowing what to LOOK for, how to FIND it and having found it HOW to put it to good use.


I find the words REFUSAL and MODERN amusing in this sentence: “One oddity of the St. John’s curriculum is its refusal to engage the
modern sciences.

There has hardly been any age in antiquity where someone did not refer to themselves as MODERNS in contrast to the ANCIENTS. Darwin was certainly MODERN and cutting edge in his day as was Galen, Ptolemy, Aristotle, the pre-Socratics, etc.

As a “modern nation” we are divided over the notion of evolution vs. creationism vs. intelligent design. I have read statements to the effect that evolution is no longer a “theory” but has been scientifically demonstrated as a fact.

For hundreds of years “atomism” was one among many theories. I remember reading The Alembic Club reprints recounting the long epic struggle from Dalton to Cannizaro to promote atomism from the status of theory to an indisputable fact.

I have read that the last 30 years have been devoted to research on string theory and Euler’s famous formula without one single shred of experimental evidence. I read that several years ago so perhaps something has now been substantiated. I remember when “black holes” were simply a theory but now actual black hole singularities have been discovered.

I am under the impression that there are two main goals behind education and training. In a voting democracy of a government “of the people and by the people” it is essential that all adults achieve a certain kind of literacy and foundation so that they may make sound decisions regarding voting issues and party platforms. The other goal of specialized education and training is to produce an expert who can be successful in difficult, intricate tasks such as neurosurgery, “rocket science”, entrepreneurial executive leadership, etc.

A military general in the 20th century once observed that we are technological giants and ethical infants. We have unraveled the mysteries of genetics and can alter genes to perform new tasks and yet we have no trepidation about playing god and perhaps unleashing some demon from Pandora’s box which can never be put back.

I see the average American as woefully undereducated and misinformed when it comes to voting the likes of someone like Palin into a position of authority.

It is barbaric to mix religion and politics. Locke felt that any citizen who did not fear some higher power would not be able to raise their hand and swear an oath to tell “the truth and nothing but the truth, so help me God.” Suppose someone has lost their right arm during military service? Does that mean they should raise their LEFT HAND? And suppose they have lost both arms in combat? Does that mean they cannot give testimony in a court of law. And if they place their hands on some edition of the Bible do they know with certainty that it is a perfect translation? Do they even OPEN that Bible to assure themselves that it is not filled with blank pages. Have they every read through that Bible once in their lives? Can they intelligently discourse about the meaning and purpose and historical evolution of Biblical analysis and scholarship. Pray tell WHOSE god are we swearing to? And please explain to me who it is that has access to “the TRUTH.” If you place me under oath and ask me “how much is two plus two” I shall most certainly testify “four” but does that constitute “THE TRUTH” or am I merely prattling hearsay from the rote memorization of my early grade school education? Am I of the caliber of a Gauss or Bertrand Russell or Alfred North Whitehead that I can PROVE that 2 plus 2 equals four? What utter nonsense. We see governors and presidents caught in lies and perjury frequently. They all pay lip service to some pathetic Protestant doctrine so they can get elected. Their so called religious convictions do not make them exempt from sexual misconduct or graft or corruption or abuse of their office to pursue personal agendas.



The phrase “two plus two equals five” (“2 + 2 = 5”) is a slogan used in George Orwell’s Nineteen Eighty-Four as an example of an obviously false dogma one must believe, similar to other obviously false slogans by the Party in Nineteen Eighty-Four. It is contrasted with the phrase “two plus two makes four”, the obvious – but politically inexpedient – truth. Orwell’s protagonist, Winston Smith, uses the phrase to wonder if the State might declare “two plus two equals five” as a fact; he ponders whether, if everybody believes in it, does that make it true? Smith writes, “Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.” Later in the novel, Smith attempts to use doublethink to teach himself that the statement “2 + 2 = 5” is true, or at least as true as any other answer one could come up with.

Orwell had used the concept before publishing Nineteen Eighty-Four. During his years of employment at the BBC, he became familiar with the methods of Nazi propaganda. In his essay “Looking Back on the Spanish War”, published in 1945 (four years before the publication of Nineteen Eighty-Four) Orwell wrote: Nazi theory indeed specifically denies that such a thing as “the truth” exists. […] The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future but the past. If the Leader says of such and such an event, “It never happened”—well, it never happened. If he says that two and two are five—well, two and two are five. This prospect frightens me much more than bombs […]

In Fyodor Dostoevsky’s Notes from Underground, the protagonist implicitly supports the idea of two plus two making five, spending several paragraphs considering the implications of rejecting the statement “two times two makes four.”

His purpose is not ideological, however. Instead, he proposes that it is the free will to choose or reject the logical as well as the illogical that makes mankind human. He adds: “I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too.”

According to Roderick T. Long, Victor Hugo had used the phrase back in 1852. He objected to the way in which the vast majority of French voters had backed Napoleon III, endorsing the way liberal values had been ignored in Napoleon III’s coup.

Victor Hugo said “Now, get seven million five hundred thousand votes to declare that two and two make five, that the straight line is the longest road, that the whole is less than its part; get it declared by eight millions, by ten millions, by a hundred millions of votes, you will not have advanced a step.”

Victor Hugo here is echoing earlier French thought – Sieyes, in his “What is the Third Estate?” uses the phrase, “Consequently if it be claimed that under the French constitution, 200,000 individuals out of 26 million citizens constitute two-thirds of the common will, only one comment is possible: it is a claim that two and two make five.”

Ivan Turgenev wrote in prayer, one of his Poems in Prose “Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: Great God, grant that twice two be not four.” Also similar sentiments are said to be among Leo Tolstoy’s last words when urged to convert back to the Russian Orthodox Church: “Even in the valley of the shadow of death, two and two do not make six.” Even turn-of-the-century Russian newspaper columnists used the phrase to suggest the moral confusion of the age (e.g. Novoe vremia (New Times), 31 October 1900.

In case you had any doubts, here is a proof that 2+2=4.

Of course, if by “2” we mean “apple” and by “4” we mean “orange”, then the statement is false. It should be clear that “2+2=4” has a specific meaning, and if we change any of its meaning, we’ve changed the statement. Natural numbers, such as 2 or 4, have specific meanings. They are things which obey the Peano axioms. If they don’t obey the Peano axioms, they are not really natural numbers, and we might as well be talking about apples and oranges.

The Peano axioms thoroughly logical and simple to state. But I’m not going to cover it in detail, since you can just peruse the Wikipedia article for more.

For every natural number n, the Peano axioms define the “successor of n”, or S(n). Every natural number, except zero, is the successor of another natural number. All natural numbers can be expressed this way:


We have names for each of these numbers: 0, 1, 2, 3, 4, …

And so, by “2+2=4”, we really mean this:

S(S(0)) + S(S(0)) = S(S(S(S(0))))

Not only do natural numbers have a specific meaning, but the symbol “+” has a specific meaning. It is defined with the following two axioms:

n + 0 = n
n + S(m) = S(n + m)

So here’s the rest of the proof:

S(S(0)) + S(S(0)) = S( S(S(0)) + S(0) )
= S( S( S(S(0)) + 0 ) )
= S(S(S(S(0))))

Fairly simple, eh? But, hey, maybe if you find a way to tap into the power of the other 90% of your brain, you will prove the impossible. Either that or your dreams will be crushed and the resulting cynicism will negatively affect the rest of your life.

A harder problem would be to prove that n + m = m + n. I think you might even have to use the axiom of induction for that one.

We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about ‘and.’
—Sir Arthur Eddington

The complete proof of a theorem all the way back to axioms can be thought of as a tree of subtheorems, with the steps in each proof branching back to earlier subtheorems, until axioms are ultimately reached at the tips of the branches. An interesting exercise is, starting from a theorem, to try to find the longest path back to an axiom. Trivia Question: What is the longest path for the theorem 2 + 2 = 4? Orange julia butterfly with 2+2=4 dots. Credit: N. Megill 2004. Public domain.

Trivia Answer: A longest path back to an axiom from 2 + 2 = 4 is 150 layers deep! By following it you will encounter a broad range of interesting and important set theory results along the way. You can follow them by drilling down this path. Or you can start at the bottom and work your way up, watching mathematics unfold from its axioms.

2p2e4 (2+2=4) <- 2cn <- 2re <- 1re <- ine0 <- mul01i <- subidi <- negsubi <- negidi <- negid <- pncan3 <- subadd <- subaddi <- negeui <- addcan <- addcani <- cnegexi <- cnegex <- cnegextlem3 <- cnegextlem1 <- axrnegex <- negexsr <- pn0sr <- distrsr <- dmaddsr <- addclsr <- addsrpr <- enrer <- addcanpr <- ltapr <- ltaprlem <- ltexpri <- ltexprlem7 <- ltaddpr <- addclpr <- addclprlem2 <- addclprlem1 <- ltrpq <- recclpq <- recidpq <- recmulpq <- mulcompq <- dmmulpq <- mulclpq <- mulpipq <- enqer <- mulasspi <- nnmass <- omass <- odi <- om00el <- om00 <- omword1 <- omwordi <- omword <- omord2 <- omordi <- oaword1 <- oaword <- oacan <- oaord <- oaordi <- oalim <- rdglim2a <- rdglim2 <- rdglim <- rdglimi <- rdgfnon <- tfr1 <- tfrlem13 <- tfrlem12 <- tfrlem11 <- tfrlem9 <- tfrlem7 <- tfrlem5 <- tfrlem2 <- tfrlem1 <- tfis2 <- tfis2f <- tfis <- tfi <- onss <- ordsson <- ordeleqon <- onprc <- ordon <- ordtri3or <- ordsseleq <- ordelssne <- tz7.7 <- tz7.5 <- wefrc <- epfrc <- epel <- epelc <- brab <- brabg <- opelopabg <- copsex2g <- copsexg <- eqvinop <- opth <- opth1 <- opi1 <- snex <- snprc <- n0 <- ne0 <- ne0f <- noel <- dfnul2 <- eldif <- elab2g <- elabg <- elabgf <- vtoclgf <- hbeleq <- hbel <- hbeq <- hblem <- eleq1 <- eqeq2 <- eqeq1 <- bibi1d <- bibi2d <- imbi1d <- imbi2d <- pm5.74d <- pm5.74 <- anim12d <- prth <- imp4b <- imp4a <- impexp <- imbi1i <- impbi <- bi3 <- expi <- expt <- pm3.2im <- con2d <- con2 <- notnot2 <- pm2.18 <- pm2.43i <- pm2.43 <- pm2.27 <- id <- mpd <- a2i <- ax-2

The list above was produced by typing the commands "read" then "show trace_back 2p2e4 /essential /count_steps" in the Metamath program. By the way, the complete proof of 2 + 2 = 4 involves 2,452 subtheorems including the 150 above. (The command "show trace_back 2p2e4 /essential" will list them.) These have a total of 25,933 steps — this is how many steps you would have to examine if you wanted to verify the proof by hand in complete detail all the way back to the axioms.

We selected the theorem 2 + 2 = 4 for this example rather than 1 + 1 = 2 because the latter is essentially the definition we chose for 2 and thus has a trivial proof, 1p1e2 (shown primarily to support a lame attempt at humor, with apologies to frequent contributor Mel O'Cat). In Principia Mathematica, 1 and 2 are cardinal numbers, so its proof of 1 + 1 = 2 is different: see theorems pm54.43 and pm110.643 for a translation of its theorems into modern notation.

One of the reasons that the proof of 2 + 2 = 4 is so long is that 2 and 4 are complex numbers—i.e. we are really proving (2+0i) + (2+0i) = (4+0i)—and these have a complicated construction (see the Axioms for Complex Numbers) but provide the most flexibility for the arithmetic in our database. In terms of textbook pages, the construction formalizes perhaps 70 pages from Takeuti and Zaring's detailed set theory book (and its first-order logic prerequisite) to obtain ordinal arithmetic, plus essentially all of Landau's 136-page Foundations of Analysis.


The value of brute memorization in education

February 23, 2010

I grew up in a world where rote memorization meant everything. From 7th grade onward my life was non stop memorization and then spewing it back on exams and quizzes. Now we have the Internet and search engines like Google. Just this morning I was thinking about my 8th grade history assignment to find information about the Concordat of Worms and also the False Decretals. I search our substandard libraries high and low, and could not find the information. My point is simply that now I do not have to remember the 10 commandments or the 12 apostles or the beatitudes of the sermon on the mount or the Gettysburg address, or Hamlet’s soliloquy for the simple reason that I am always seated in front of a computer which has Internet access and I may quickly access all such information. So, brute force memorization is an artificial test for whether someone has done their homework but plays no role in knowing HOW to think, or what to think, or developing a genuine love for learning and information and creative writing.


Well, quizzes on memorized material are attempts to quantify, yet whatever we memorize will soon be forgotten if not constantly used. Hence, memorization is not only a waste, but is perhaps harmful if one’s goal is to instill in a student a lifetime love of learning and writing for its own sake.

These FB threads are not our doctoral dissertations. I take all my pills and metamucil, drink my coffee, and try to use what is left of my brain by browsing various threads and trying to respond in some stream-of-consciousness association fashion as in : you say po-tay-to I say po-TAH-to. Sorry if my comment seemed a bit off topic, but surely it is not a total non sequitur,

Actually today I saw several other threads posted by life-time career teachers complaining how their pay scale is tied to a classes performance. So, at times, when I respond to one thread, I am actually reacting to several threads.

With each passing year I feel more and more strongly that physical classrooms are inefficient, antiquated and not cost effective when all textbooks may be on line and in public domain while students participated in chat rooms and message board threads and watch videos of lectures. And I think Wi-Fi should be available to all for free so that children may have equal access to Internet resources. Obviously there are issues with counter-productive activities of game playing and porn watching but surely our society and technology is clever enough to work that out. Computer illiteracy is a serious impediment to a government “of the people and by the people.”

Social Networks, Google and Continuing Education

December 11, 2009

With the Internet, we can tap into each others stream of consciousness and with search engines we can explore anything in considerable depth in as little as an hour (because we are reading peoples recollections of digests of summaries, which may sound superficial but can really be quite deep.) This is why, when I add 20 friends a day and several of them ask if they know me, I think to myself “they really don’t GET it… you WANT to add people you DON’T know, to expand your mind.. what will you gain if your status about eating an apple or having irregularity is read by your grade school buddy across town and your cousin that you visit on Thanksgiving?

Terrified of Westell Modem & Verizon DSL

November 15, 2009

Posted at

A member wrote:
You plugged a XP machine into the modem/router and nothing happened. So I was curious why you would think a Linux based machine would make a difference. Sorry if it came off as snotty. And I have always said, the only stupid question is the one un-asked.

The other thing you could look into is

Sun VirtualBox. I have Ubuntu and Win7 RC installed inside it. But you can install all kinds of OS’s, mess with them all you like. That way you aren’t limited to just that version Linux, you could install them all, as long as you have the hard drive space.

My reply:
Aha, you misunderstood. I have NEVER until now had the guts to plug anything into the Westell router, because we have little money, and I cant afford the headache and expense of messing up my home internet. But now, from what you tell me and what I have read in google (and I wrote a long support question to Startech, who will probably get back to me next week about their Ethernet card which is only $10, and claims to have Linux drivers).

Also, here is a funny true story. Last week, I thought my stepdaughter was telling me that she has an old tower she doesnt want, and I could wipe out the drive with a pure Ubuntu install. She just called to say her husband was bringing it over with a monitor (which I can keep)…. but when I told her my plans, she was horrified, and said that she wants me to CLEAN UP THE VIRUSES and give it back to her, because right now it will only boot in F2 Safe mode. Ha ha! I am glad I asked, and she should have been less vague and cavalier. But now I have little fear about plugging the cable from the Westell jack into the ethernet jack. Besides, it stands to reason what when any software is about to modify anything, it would prompt.

Although, here are two things I didnt understand about Verizon DSL. Once, I changed the password on the Verizon email account, and that DISABLES the modem that is connected to that account, and a service technician must do a number of things over the phone to restore the DSL service. That was a painful lesson. Another time, DSL was down, and I though I should take a pen and hit the reset button on the Westell. I didnt realize that doing that WIPES OUT EVERYTHING in the Westell, and you have to get on the phone for an hour with a technician to reset the Westell. So perhaps those two horrible experiences contribute to my fear and hesitancy to mess with the Westell.

Old saying: once burnt, twice shy. Plus, if it ain’t broke dont fix it. Years ago we had our first home DSL with AOL. My stepson who was only 13, decided one day to reinstall AOL from a new diskette that came in the mail. That WIPED OUT THE DSL, and AOL was NEVER able to restore it, and their tech dept was not too worried about restoring it.

Thanks for your feedback, it did help! I wasn’t hesitant about hooking up an Ubuntu machine. I am just scared to death of Verizon and Westell. I did take the courage to go into the Westell setep to enable Wifi with WEP key protection. 3 years ago in my 27 story high rise, one could pick up all sorts of unprotected wi-fi. NOW, all the channels are protected and one is even named “GET AWAY!!!”. People began to realize that other people could steal bandwidth.

I would like to close by suggestion something that I think Thomas Jefferson would suggest were he alive today: namely FREE PUBLIC WIFI in every city. Jefferson asked that only two accomplishments be mentioned on his tombstone: that he signed the Declaration of Independence, and that he founded the University of Virginia. Jefferson suggested free education for all, which in his day seemed like a strange idea. After all, education was only for nobility and the wealthy. The peasant farmers and laborers didnt need to know how to read. But Jefferson realized that unless ALL were educated, even laborers and farm workers, then there could NEVER be a government “of the people and by the people.”

What does this have to do with free wi-fi, you ask? Everything. Today, computer illiteracy is the new danger. In 1960, my mother had to fight to get me a $2000 World Book encyclopedia so I could do my homework in 6th grade, because there were no libraries close by. But today, who buys an encylopedia, when there are search engines and things like Wikipedia?
Every child NEED computers and internet access. True there are dangers and abuses in chat rooms etc, but then there are dangers in the postal mail system. That does not mean we should do away with the post office. It means that we should find safe guards and establish punishments for crimes committed using the U.S. mail or the internet. We must make it a crime to shout FIRE as a joke in a public auditorium, but we must preserve freedom of speech. And if there is ANYTHING at all in this world which will unify all peoples and nations and creeds it is the communication and understanding provided by the Internet. I know because I have a friend of 10 years now in Tehran Iran, who loves Plato and Tolstoy and Dostoevsky and everything that is western and free.

In a very poor neighborhood in one large city in India, where there are many high tech computer firms, as an experiment, they installed many rugged computer terminals in the alley ways where homeless street children dwelt. Those children quickly learned how to be very

Homework Pros and Cons

August 31, 2009

This is a PLURK.COM thread posted to by various high school teachers

Pros and Cons: Should Homework be Abolished? Alfie Kohn’s RETHINKING HOMEWORK.

Comments by various posters:

Homework should be logical not worksheets! Creating and developing multimedia projects not memorizing facts.

I believe some homework is important to practice skills, sharing learning with family, and develop responsibility. Guidelines of 10 min per
grade level is good. Being read to or reading independently should be in additioin to this. When homework becomes a battle between child & parent, something is wrong with the system and should be explored and rectified.

Also, homework should be checked but not used as a grade since it is impossible to tell if the student did it.

We don’t include homework in our gradebook, but are told by the high schools that we don’t give enough. They say we’re not preparing our students well enough for high school by not giving them plenty of homework now. They average 3 hours a night at HS

*sigh* we assign roughly 1 – 1.5 hours of homework per night. Now this is flexible, since we assign it over the entire quarter, but it works

out to about that per night. And that’s focused work. I think it’s the right thing to do. I’m tired of this debate. Ours is not trad work.

Wow! 3 hrs/night? How about extra curricular activities? Aren’t we looking for well rounded citizens?

If my kid got 3hrs/night I’d be talking to teachers about why and what is the value of giving such work!

I can’t wait to see the responses. For the record, I am all for getting rid of homework. As a matter of fact,

I very rarely give any homework to my students. If they don’t finish classwork they may have to finish at home, but I don’t give homework.

I had TONS of homework from 7th grade through 12th grade in a public school sys in CT I am age 60 and I still have occasional nightmares of school anxiety during sleep As a Junior in Highschool, I would arrive home at 3:30pm exhausted, eat dinner, go to bed, get up midnight, and study til time to leaveInAM

I think homework should only be assigned when necessary to provide practice for skill acquisition – not necessary in all content areas

This is an interesting topic. I posted the plurk page link on Facebook, but I think because it is a PRIVATE plurk, others cannot read

I agree we need to rethink homework but not whether we should or shouldn’t but more at what we do with it. To often homework is used to fill

class time or repeats learning that has already been accomplish or to introduce a lesson only to create fear for the student.

abolished? no. carefully constructed, necessary, and meaningful? YES.

For me the new illiteracy issue is computer internet illiteracy. Free Wi-Fi and lowcost computers should be available to all.

In 1960 my Mom had to buy a $2000 World Book Encyclopedia, because we lived far from the small seedy public library with limited hrs.

I carried SO MANY books to school each day that my arm and posture were affected.

Now, with Internet, search engines, CD’s and Memory sticks, expensive bulky textbooks are unnecessary!

The problem with homework is that if the child knows how to do the work, it’s unnecessary torment, and if they don’t know how to do it, they don’t have a readily available teacher to assist. As a parent, I don’t like having that responsibility passed to me for two reasons. One, I’m a lousy teacher. There’s a reason I didn’t join the profession. Two, I don’t know every subject. Never took Spanish, for example. I think the school day should be extended until 5:00, with the last couple of hours being ‘independent work’ but with teachers available for extra assistance.

I’m not a fan of homework as busywork or as mere extension of the homework. Homework should enrich and expand exisitng knowledge, and then it should stop.