Education: We could be all totally wrong. But then, again ...

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We could be right.

One thing young students complain about is that teachers don’t tell them everything.

Teachers don’t tell students “everything” for various reasons, not all of them good ones.  Sometime the teacher doesn’t know everything.  Sometimes the teacher doesn’t have time to provide complete and detailed information about everything; it depends on what the teacher sets out to do.  Sometimes the teacher just leaves something for the student to discover on his or her own.

It is the easiest thing in the world for a teacher to cut down the volume of topics that he or she sets out to supply the student with, and give complete and detailed information about those few topics.  Honestly, the pressure on a teacher to do exactly that is enormous, and many feeble-minded teachers who are more anxious to come away looking good than to provide students with material do indeed go that route.  You can just imagine a teacher saying: well, if I set out to teach multiplication all the way up to “16 times,”  a lot of the students are not going to be able to do it.  In fact, the larger the multiplication table, the less ‘success’ my students will have!  So I think I will stop at “5 times,” and do it really well.

Obviously, these are two extremes, and there is a lot to be said for stopping at, say “10 times,” which really gets the students ready for long multiplication some day in the future.  But the example was just an analogy; the less a teacher undertakes to teach, the better the teacher can perform the task of teaching.

All this is, in principle, the problem of curriculum.  How much to teach, and what choices to make, are ultimately best left to the professionals, who understand the pros and cons of each choice; it is best if parents stay out of those decisions.  There’s nothing to prevent a parent from supplementing what a child learns at school with more information at home; this is a wonderful thing you can do, if it is done judiciously.  I remember my mother teaching me the idea of algebra, and solving an equation when I was about 10, and my teachers were, justly, furious, when I used that method to solve an “inheritance” problem, which began something like: “A rich man died, and left a large fortune.  One half was given to ... ,” and the problem ends with “... and the parakeet got $20.  What was size of the original fortune?”

In my opinion, there is a great benefit in (1) teaching considerably more than you can teach perfectly, and (2) leaving large gaps for the student to fill.  As my colleague puts it, the volume of information out there is truly vast, and getting more vast every second; our students must learn to be “lifelong learners.”  What we are teaching them most importantly is how to teach themselves.  The older we get, the less someone is going to sit down and show you exactly what is going on.  That explains (2).  But what about (1)?

Some of the ideas that young people have to be presented with in these modern times are very complex, and very subtle.  Understanding, if it comes at all, comes in layers; certain ideas must sink in, before certain other ideas can begin to make sense, and usually for some of these ideas and concepts, the time is longer than one semester.  Unfortunately, the response of certain members of the teaching profession to this problem has been rather defensive.

“If you can’t teach the entire thing all at once,” some would say, “wait until the student is ready someday in the future.  I like to make sure that if I start an idea, I can completely finish it.  It’s very dissatisfying to the student to be taught a mere glimmering of an idea, and then be left waiting for the rest of it.”  But wait: is the whole point to make things satisfying to the student?  I tend to think that student satisfaction is only a small part of the equation.

“Another way to do it,” more thoughtful teachers would say, “is to introduce some new halfway idea, and explain that partial idea completely, so that the rest of the thing can be built up from that partial idea.”  But what if that partial idea is unsatisfying to the student?  Well, to these teachers, teaching a synthetic partial idea is satisfying to teach, even if it is unsatisfying to the student, for whom the completion of the chain of reasoning doesn’t come until much later.  So the curriculum is often cluttered up with these halfway-houses invented by creative teachers, who plan to complete them someday into the more sophisticated concept that they cannot deliver right away.  But this scaffolding that they have dreamed up might not be recognized by the people who are in charge of the student’s education when the time is right.  This modularization of the curriculum, as I call it, has its pros and its cons, and this is one of the cons.  Teachers invent vast masses of half-concepts that they teach students, so that some day the concepts can be completed.

At any rate, the business of touching upon ideas that cannot have all their loose ends tied up right away is an unavoidable thing; some teachers are just better at giving the impression that the loose ends are tied up, but they’re not; they’re simply closed off with color-coded masking tape for a future teacher to deal with.  This is widely considered today as a good thing, since untidiness of any sort is frowned upon.  Honestly, if you can’t have sophistication, tidiness is the next best thing.  We must simply not mistake tidiness with sophistication.  As you can readily see, this leads to a great deal of what can only be described as deferred gratification for the student.  That is part of the price you have to pay to learn science, which is more of an edifice than the Humanities or the Arts.

I touch on a huge spectrum of ideas that there is simply no time to deal with in complete detail, which frustrates the students in my class who simply want to learn enough to past the test, and escape from my clutches.  They tend to grill me closely about ‘exactly what is the test going to be like?’  The unspoken question is, what are the actual questions going to be, and what are the answers?  I often give what are called review sheets, which are intended to be practice questions.  But students would rather be told that they’re getting a sample test.  I put at the top of the review sheet: There is no relationship whatsoever between the form and the content of this review sheet, and those of the test you are preparing for.  What is the point of this detailed cross-examination of the teacher?  I have taught them 42 ideas---at the very least---in 42 class meetings.  Why not study all 42?  Oh, no; I must tell them the five most important things I have taught them, because they have no time for the rest.  Modern kids are busy executives.  After all, if busy executives only have time for the five most important things anyone has to tell them, why not kids?  After all, their minds are smaller, right?  And they have to get ready to be busy executives, after all.

Why do I deliver such a large volume of ideas without doing a proper job of it?  A lot of it has to do with connections.  Facts anyone can understand; the harder thing is to be aware of the connections.  I would be only too delighted if my students were such avid readers of magazines and all sorts of stuff they come across as I am.  For instance, I tell my trigonometry students that sines and cosines are used in the compressed music files they listen to on their MP3 players or Ipods, and in the pictures they browse on the Internet.  They look at me blankly.  How, they wonder, can you possibly use sines and cosines for pictures?  I try to explain the structure of a musical sound, and what distinguishes the sound of, say, a flute from that of an oboe or a clarinet.  I go on to describe how the sound, for a fraction of a second, can be represented by a formula that consists of an infinite number of sines and cosines added together, with other numbers weighting them.  We can save a list of a few thousands of these weights, and then go on to the next moment in time, and do it for that note, and so on.  This sampling is done 44,000 times a second, roughly, and that’s essentially what is stored in the MP3.  (The miraculous thing is actually how these numbers are used to reconstruct the sound; that was the genius of Fourier, who pioneered the idea.)  A similar thing is done for image files, and that is how Jpeg files are made.  You can look it up: it’s called the Discrete Cosine Transform.  If a kid knows how to Google something, he or she has got started on the most basic method of self-education.  But that’s just the first step; you must go on to learn how to filter out the useless information quickly, before you lose interest completely in your question.


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Vocalise, by Sergei Rachmaninov

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It is only recently that I became familiar with this piece by Rachmaninov (Rachmaninoff): it is a song without words for soprano, I believe originally written for Soprano and Piano, the last of a set of 14 Romantic Songs (or romances), but subsequently transcribed for voice and orchestra.  Here it is, so that you can know what I'm talking about:



If you would like to hear the piece in a form closer to how it was originally intended to sound, let me find you a link to a clip on YouTube ... here is Kiri Te Kanawa singing it.  The voice can be a little distracting, so here is an orchestral performance, conducted by the Korean Chung Myung-Whun (brother of the well-known violinist).  (Incidentally, this is a beautiful performance that does the piece justice, without a lot of self-conscious self-promotion.)

It is amazing at many levels, but on the face of it, it is not very hard to appreciate.  The structure is easy to describe: it has two broad sections, and they are usually described by symbols such as A and B, which are simply names.  Section A is repeated, and then B is played twice, and then there is a really lovely ending, which we can call C.  This structure is called binary form, generally speaking.  [The whole piece is then AABBC.]

Part A, which itself is in two parts, is the most characteristic section, and most likely to be remembered at a first hearing.  It comes to rest in the middle, then continues with a little more agitation, to end with a sort of question, or at least a complaint.  (I know that sounds a little peculiar for something to say about a musical piece.)

Part B takes off, elaborating on the complaint, expressing increasingly great agitation and frustration, and ends with the principal voice coming to a halt, as if it has reached the end of its energy.  (This brings us to the whole issue of musical rhetoric, which is an idea that derives from the word rhetoric as applied to public speaking or polemic writing.  In that context, the word means the techniques of presenting ideas with a view to persuasion.  In music it has a slightly different meaning, which is a little vague; it seems to me that musical rhetoric has to do with repeated phrases, to express a sort of insistence.  The connection is that of course in speeches, repetition is used to emphasize points, and is merely a particular device in a much wider spectrum of devices available in written and spoken language, as opposed to music.)

Part C, the ending (called a coda, which simply means tail) continues almost seamlessly with the ending thought of Part B, with another, contrasting voice reminding us of the opening melody, while the main voice bestirs itself to provide a countermelody, and the two tunes come to a close, with which the piece ends.

(In one of the descriptions of the piece, someone has suggested that the whole piece is nostalgia for the homeland of Rachmaninov, namely Russia.  This is possible, but it might also be pure conjecture.  Music written in the latter half of the 19th, and the early 20th centuries often had these inspirations, which were sometimes stated by the composer, but most often not.  If it is true, it certainly provides a plausible explanation for the development of the piece: sweet thoughts of the homeland come to the exile, and gradually turn to frustration and bitterness, but then pleasant memories return to moderate the bitterness, and provide comfort.)

An interesting feature of the piece is that it is a highly individual melody over a chordal accompaniment; this sort of music is called homophonic, a main melody, and harmonic accompaniment.  But the accompaniment often has counter-melodies; in fact the counter-melodies almost characterize the piece.  You can almost hear the piano accompaniment of the original version in the repeated chords, with a very rhythmically interesting bass line that is highly syncopated, that is, it often comes in off the beat.  The pianist would normally play the bass note, and then use the left hand to play all sorts of other stuff, until another bass notes is played, followed by more chords, chords, chords.  If you look for other versions of Vocalise on YouTube, you hear numerous recordings of amateur instrumentalists, performing it with a piano.  It's one of those tunes that everyone would love to learn and play.

The harmony, too, is fascinating.  There are several instances of what is called the Pathetic Cadence, a melody and underlying chords that are very recognizable:



The key (pitch level) might not be the same, but the chords should sound familiar.  (I should have used the same key as the piece, I suppose ...)

Furthermore, Rachmaninov has his parts creep about in semitones, so that the chords they make aren't traditional chords, but are the distant descendants of the chromatic harmonies of Wagner, which in turn are distant descendants of chromatic harmonies and melodies of Mozart.  The general principle of chromatic harmony is that when a note is altered by making it a sharp (or even a double-sharp), as we find here, it wants to keep moving upwards to the next higher note, and notes altered by lowering them, that is, making them Flats, want to keep going downwards.  But here we have notes that are flattened, but keep coming back up, and notes sharpened, that keep going down.  The meandering chromatic notes feel both restless and slightly threatening, but never so threatening that we feel frightened, but only unsettled.

[Added later:

With classical harmony, it is possible for most musicians to get a sense of the piece and where it's going, at each point, enough to cobble together the tune and the accompaniment by ear, as it is called.  This was not the case with Vocalise when I entered it into the editor; I simply entered it in, observing bits of the harmonic detail in passing, without it making much sense to me.  This morning, however, I was able to play bits of it at the piano, so I have a very slight idea of what it is trying to do.

It struck me that the composer has deliberately written a piece that struggles to go from one harmonic center to another, but is thwarted from doing so.  It is very clever, compositionally, and this sense of pleading in the melody is very clear, as if the voice is pleading with captors to be permitted to do something; it isn't clear exactly what: take care of a sick child, be allowed to visit a grave; who knows?  This is the essence of romance: to portray a feeling that seems to be clear, but whose details are open to some degree of interpretation.  It is entirely possible that Rachmaninov was representing emotions that were indeed inspired by his exile (I don't know enough of his life story to be able to speak about what sort of exile that might have been), but that he was portraying a range of feelings, a lot broader than those brought on by exile.  Who knows?  Perhaps exile is the worst feeling a Russian can experience.]

Some of the chromatic countermelodies were considered so important by Rachmaninov that he had them doubled in octaves; in the orchestral score, for instance, you'll hear violins and cellos going parallel.  Now Italian opera composers often had violins and cellos going furiously in octaves, to underline the highly romantic melodies that they wrote, or plain old melodies that they wanted romanticized further, to suggest high drama.  Here, too, Rachmaninov wants to underscore the drama of the mood, but at a slightly more intimate level.  The melody, for all its expressiveness, is not a big bold melody, but rather an introspective one.  Kiri Te Kanawa makes it sound big and bold, but one gets the feeling that the melody resents being presented in such an extrovert romantic manner.

Finally, this piece, which is on the face of it a homophonic composition (melody + chords) has so much inner melody in it, that I thought it might be illuminating to transcribe it as a chamber piece.  So the performance is for solo violin, viola, cello, and two flutes, and chorus double-basses.  I would have used a single double-bass, but they have rather rough tone, and the double-bass section sounds smoother.  The counterpoint does sound a lot more transparent in this presentation.

Enjoy!

Arch

Hooked Up! A Survey of Consumer Technology

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I’ve been  into computers for a long time, well before they became accessible to everyone back in the eighties.  Recently, young people have got into technology in a big way, and have drawn their parents and elders into it as well!  This happened with me, too.

My daughter decided to join a beginning computer class I taught one summer, but didn’t really have the motivation to get into it.  But, shortly afterwards, she joined a group in her high school that programmed Lego robots for competitions, and had a wonderful time.
 When she graduated from college, she had considerable computer knowledge, but decided on an Apple Mac for her personal computer, which was a surprise.  But it was even more a surprise when she decided on a Droid smart phone.

All the technology that’s around us is unavoidable for adults who need to use it professionally, but a lot more of it is just for fun.  Most households have someone who is hooked in, as they say, but sometimes a kid can be strictly interested in stuff the adults in his family aren’t inspired by, such as video-games, and so on.  Our family is fortunate to have Junior feed us information about technology that is both useful and interesting, possibly to a wider audience.  I’ll first describe the hardware, and then go over the software, though in principle it is really hard to divide them completely.

Phones

Gradually, people are getting themselves smart phones.  The two most popular types are Iphones, from Apple, and Droids, essentially from Google.

The Iphones are fully functional phones that can take calls as usual, but they can also connect up to the Internet.  Droid phones are essentially similar, using an interface developed by Google, called Android.  The Ios interface for Iphones and the Android interface of  Droids are really operating systems, which are programs that run constantly, managing the services of the piece of hardware.

Older phones had a display screen, and a keypad.  Increasingly, phones have a touch-sensitive screen that throws up a keyboard when needed, so that a separate keyboard is not required.  Some people like a keyboard, even a tiny one, and a few phones have a sliding keyboard that is recessed when not being used.  But the keyboard is a point of structural vulnerability; a solid phone without moving parts is less fragile, in principle.  (The touch-sensitive screens are made of a very sturdy, thin glass-like material developed by Corning Glass, called Gorilla Glass.  But my students have managed to crack even that material, but the phone still somehow continues to function. Note: if you crack the glass, replacements are available.  It is possible to replace it yourself, I'm told.)

Depending on your Plan, your phone calls, both incoming and outgoing, are metered, and you get billed.  In addition, when you use the Internet, you get billed for the time you’re connected to the Internet, as well as the volume of data that goes out to and comes in from the Internet.  Every time you connect up, there’s some data going out, and some data coming in, every request is a little data, and then there’s everything that comes in: websites, pictures, text, information, music, etc.

Cellphones function by connecting the phone to Cell Towers that are located everywhere, on actual towers in the country, and on top of tall buildings, in the cities, and the calls are relayed from tower to tower.  In places where there are no towers of your phone provider, they may have a reciprocal agreement with another provider (called Roaming), or you may not be able to make or receive the call (a “dead spot”).

Most services allow your phone to use a Wireless Internet connection, in your home, for instance, in which case they won’t count any Internet use while you’re on the home network.  You have to provide the password for your home wireless network one time, usually, and the phone remembers it for future connection whenever you’re at home.  Similarly at work, you need to provide a password one time.  At bookstores and Starbucks restaurants and airports, etc, there are wireless hot spots (provided for laptop computers that passengers might have with them), which the phone can use for data access.

Voice and Text

In addition to phone calls, most phones today which have text capability (a keyboard, or even just the usual keypad), with which they can send text messages.  (This uses a channel that was intended to send information about phone calls, but which was under-used, and which the phone companies decided to use to send something that was essentially like e-mail.)  The text message arrives at its destination and makes a small alert sound, and then waits.  The message can be picked up whenever the recipient desires.  It can then be deleted, or kept forever, depending on how much space your phone has.  Not everyone’s phone had text capability, of course.  But phones with Internet capability began offering the ability to actually send e-mail.  Today’s smart phones can do all three: voice, text message, or e-mail.
For various reasons, cell phones needed to have a record of the exact time at the location of the phone, so of course they display this time.  In addition, the phone can triangulate its location, and send exact geographical information to the phone company, which can be used for GPS-like services, and driving directions, etc.  The only additional piece of information they need are road maps, which the phone can carry on board, or retrieve from the Internet for a small additional fee.  Recently, this service is provided at very little or no cost.

Tablet Computers

Tablet computers have been around for years, but attracted little attention before the Ipad was introduced by Apple.  The entire face is touch-sensitive, and it usually has no moving parts at all: no fan, no conventional disk drive. These little computers do have large capacity flash memory (the sort that is found in thumb drives), and most of them have back-lit displays.  In other words, they may as well be enormous cell-phones!  Note: at time of writing (May 2013) there are two types of Tablet computers: Wireless Only (which link to the internet), and 4G which also link to your phone service.  The former kind, since it only uses wireless Internet, which you have already paid for separately, is less expensive to buy, and less expensive to pay for as you go along.  The latter kind has to be paid for as a phone, and costs the usual $50 or so per month that a phone costs these days.

When any Tablet first powers up, the touch screen shows an array of about twenty icons that you can touch with your finger, like clicking with a mouse.  Each one triggers a program, such as the Internet browser, or your e-mail, or the music player, or a word-processor, or your photo album, or the settings manager (a little like Control Panel, in Windows), or almost anything you want to load on.  On most tablets, you can go to a second “page” of icons by swiping from right to left, like turning a page.  All your programs and most important files are represented by icons.

Once you ‘click’ on an icon, that program will usually fill up the whole screen, and you get to navigate through that program with a line of icons, either at the bottom, or across the top, just like the menu or toolbars of a conventional program.  If text input is required, as you would in a word-processing program, a keyboard pops up at the bottom.  There are ‘shift’ buttons, which switch the keyboard from lower-case to upper-case, or to symbols and numbers.  There is usually a button that removes (“dismisses”) the keyboard, and you’re back to using your finger like a mouse.

Around the edge of the pad there are sockets for various things: the power cord for charging it, a socket for earphones, a socket for a flash-card, possibly an HDMI connector for output to a TV, and so on.  The fact that there are no moving parts means that the devices can run with very little power.  On the other hand, they can’t supply power to a peripheral device, like a CD burner, or anything that draws power as much as a USB connector.  (In the future, peripherals might draw a lot less power than they do now, in which case Tablets could connect to that new generation of peripherals.)

A well-known drawback of the Ipad family of tablets is that they don’t handle Flash Video very well, or hardly at all.  Certain sorts of video can be played on a tablet PC, but for some reason flash video does not, on some tablets.  (Flash video is the sort you get on YouTube.)  The Ipad interface is very similar to that of the Iphone, and the operating systems of the two are essentially the same.  It is possible, as mentioned earlier, to get an Ipad with phone capability, so that you can connect to the Internet even when you’re nowhere near a wireless hot-spot, using cell service.  Basically it is as though you’re connected through a ‘modem’ via the phone service (though the connection is direct, and nothing at all like a traditional modem.  (Modems actually used sound, long ago, but it has been a long time since computers connecting through a phone line actually used sound, though Fax machines still continue to use an audio signal.  It won’t be long before Fax machines are replaced with something a little more sane.)

Blue Tooth
Both Tablets and Phones connect to each other and to printers and keyboards and mice and so on using a certain short-range variety of wireless called BlueTooth.  BlueTooth is a special kind of wireless that needs two devices to be “paired together”; each device must identify the other positively, after which they merrily continue to be linked through a little private wireless connection provided they’re close to each other, within a few feet.  This is an easy way for a tablet or a phone to connect to a printer, for instance, and wires are not needed.
The pairing process is sort of interesting.  Each device has to be set to ‘find’ the other.  Any Bluetooth device in the vicinity could be paired with any other, so there has to be some method for one of them to establish the identity of the desired partner.  If a Bluetooth mouse is to be paired with a computer, for instance, the computer insists that the mouse should click on a particular spot.  If that is done successfully, the computer singles out the signal from that mouse, and ignores all other Bluetooth-type signals it may be getting.  In a crowded office with many Bluetooth mice and computers, this is obviously quite a trick.  But wireless communications are pretty sophisticated now; consider that a single cell tower is linked to hundreds of cellphones, but manages to get the right message to a given phone most of the time.  This is all part of the wireless encoding technology.

E-readers such as Nook and Kindle and Kobo (and earlier, Sony E-reader) are special-purpose tablets, with a tablet display especially designed for ease of reading text.  Some of them have buttons for turning pages and selecting the book, others use a touch-sensitive screen.  There is flash memory built in that holds the book displaying program, as well as the books, usually a few Gigabytes.  A single book is usually about half a megabyte, so a single Gigabyte can hold 2000 books of that size.  (Books with illustrations will be much bigger.)  Some e-readers are black-and-white, others are full color; some need light to be read from, others have internal illumination, such as the Nook Glow.

Ipods and MP3 players are little devices on which you can store music files.  They have a socket for headphones, and a little screen with a menu, and a couple of buttons with which you scroll around to find the tune you want.  You get to play a single song, a playlist of songs, a whole album, or go through your entire collection in any desired order, or randomly.  You can either connect them to your computer with a USB link, or using Bluetooth, or by inserting a flash card into it.  If you connect it to your computer, software on the computer can be used to organize the music files in folders, make a database of them, locate information on them from the Internet, and even buy new music from various provided, e.g. Amazon, Google, or Itunes.  My entire collection can probably be saved in 64Gb of storage on a MP3 player.  (Each piece is around 1 Megabyte, or 3 Mb for large-scale classical works.)

Software

Because of the close similarity between the Ios used on Ipads and Iphones on one hand, and Android used on Droids and Android tablets such as Xoom and Kindle and Nook e-readers on the other, using a Droid feels very much like using a Xoom tablet.  (I really haven’t used one, but this is what I’m told: they’re quite similar.)  It is no surprise that the programs that run on phones and tablets are similar.  In fact, when my wife got her Iphone, she was surprised to find that all her purchases of music and games and software from Itunes (which is an online store for all sorts of things that run on Iphones and even on PC computers) would be automatically loaded onto her Iphone as soon as she logged in, without further intervention.  Apple products are all designed that way: the gadget starts working without the user having to wait to find out how to use it properly.  In contrast, for Android, there is a brief but significant wait time until you figure it out.

Phone, Text

The phone and text messaging is very intuitive.  There are icons on both kinds of systems that are fairly self-explanatory.  There is a stack of people whom you called most recently (or who called you); you get to select one of these, or one from your “favorites”, or your “contacts”, or a new number directly from the keypad.  Then you start talking when your party picks up, if your calling, or you get to type in your message, if you’re texting.
Browsing, and E-mail

On Ios, the preferred, pre-packaged browser is usually Safari, but you can download any browser you want.  The pages you see are scaled-down versions of the page you would get on a computer, unless it is a page that has a special “App”, which is essentially a web-page written specifically to look good on a phone or a tablet, using an interface that has a family resemblance to all the other Iphone Apps, or Droid Apps.  The Facebook page, for instance, is missing the side columns with the advertisements; instead, the advertisements are at the foot of the page, or interspersed right between the posts of your friends.

A lot of us have personal e-mail accounts on a website.  On the phones and the tablets, the e-mail is a separate app, designed specifically for a phone or a tablet, with all the other paraphernalia removed, or put in a hierarchy of menus that only expand on demand, because the usual nonsense takes up too much space, and space is at a premium.

The browser always remembers where you were when you opened it up last, and that’s what it opens up, rather than a home screen.

On Android, the Search engine is Google, by default, and is a separate App (application, meaning program) entirely, tailored for phone / tablet use.  In fact, you can talk at the screen, and it figures what you’re searching for by voice interpretation.

Common Websites

Popular Web destinations, such as Imdb, YouTube, Facebook, Amazon, NFL Today, CNN, and so on, all have dedicated icons, and you go directly to them without going through the Browser.

Other Apps

There are some interesting application that Junior discovered:

Shazam: this one identifies tunes that are playing near you.  You activate the program, and hold it up to the source of the music, and after a few seconds it displays the title and the performer, and offers to sell it to you.

Nook: this is an e-book reader, and if you have a Nook account with Barnes and Nobles, it has your library ready to go.  You can either read them off the cloud (in this case, the server at Barnes and Noble), or download the books to the filespace on your phone or tablet.

Kindle: similar to the above

Sudoku:  There are several alternative implementations of this game, that provides random puzzles for you to solve.

Camera:  Most phones come with a camera, with both forward-facing and backward-facing lenses.  There is a button icon which you press, and you get a photo, or even a video.  You can choose to keep or discard.

Note Everything:  A multi-purpose recording program that records video, sound, doodles, or typed-in text.

Amazon MP3:  Plays MP3’s you’ve bought from Amazon, or browses for new purchases.

Created by the Fractal app

Contacts:  Manages your contact list that you maintain on Gmail.  This is a centralized location for e-mail, snail-mail addresses, and phone numbers, which you can update from either your desktop or your phone, and synchronize on demand.

Calendar:  A calendar program.  You get to set alarms to remind you of events you program in advance.

Calculator:  Most phones have a basic calculator.  Optionally you can download a free App that emulates a graphing calculator, that will draw curves and surfaces.

Gallery:  A gallery of photos you’ve taken, or downloaded.

Fractals:  A number of programs that will generate fractals for you, most commonly of the Mandelbrot Set, a furiously complicated set of points on the XY plane, discovered by the late Benoit Mandelbrot.

Quick Office:  Basic implementations of the most common Microsoft Office Suite programs, e.g. PowerPoint, Word, Excel.  Make a slide presentation from your photos, text, and material from the Internet, in fact, practically anything within reach of your phone.

Skype:  The popular Video chat program, now implemented for hand-held phones and Tablets.  You can make space-age video calls to your friends, provided they have Skype installed.

A Little Fun Geometry for Y'all

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I’m surprised to discover that people enjoyed Geometry a lot more than I thought they did.  Even more surprisingly, it appears that math people hate Geometry, while everybody else either likes it, or has a mild dislike for it, nothing like the deep dislike that some of my mathematics students seem to have!

Because of the great variety of attitudes towards the subject, people in different places seem to have very different degrees of geometry experience, and what I’m posting today might be old hat to some, and quite refreshingly new to others.

Triangles are very interesting things, mostly because they provide a framework for lots of other structures.  One very easy exercise is to make a circle that goes through all three points (vertices, or corners) of the triangle.

Constructing a Circle through the Three Vertices of a Triangle

The principle is simple, if you take two points at a time.  Suppose our triangle is ABC.  Considering just A and B to start with, there is a line of points an equal distance from A and B, as shown at right.  Each of these points might be a different distance from A, but they’re the same distance from B.

We can do the same with the points B and C; again we get a whole new line of points the same distance from B and C.  Now, it is clear that these two lines of points have one point in common!  Let’s call it O.  This point O is the same distance from A and B, and the same distance from B and C.  (As we say in mathematics, by transitivity, it is the same distance from C as from A, but we won’t fuss about that.)  It follows that we can draw a circle centered at this point O which will pass through all three of A, B, and C.

How does one make the first line, which is all the points equal distances from A and B?  Easy: join AB, find the midpoint.  So this is one of the points.  Now make a perpendicular; that gives us the whole line.

When anyone sees the triangle with the circle around it, it merely looks as if someone had taken a circle, and picked three random points on it, and joined them to make a triangle.  It is a little more convincing to have the three points  move on circles of their own, and observe the circle around the triangle follow along.  Even here, the eye is deceived into believing that the circles are the fundamental thing, but a little thought would make you see that this is unlikely, since it is difficult to make a circle dance.  Here is a video showing the dancing triangle, with its circumcircle.  The music is by J. S. Bach, and it is the (organ) Fugue in A minor, BWV 543, played using a Marimba sample.



[Without appearing to obsess over whether and why people have a strong aversion to mathematics, I’d like to pursue what the root causes might be, perhaps in a later post.

My suspicion is that it is partly because people give up too quickly on mathematics, which leads to the second problem, namely that Mathematics is a vast area of knowledge, and a lot of it is at the intermediate level, about the level of high-school study, so that most laymen have already abandoned relating to the subject by that time.  Of course, an enormous amount of mathematics is at the advanced level, and this part of mathematics is very diverse, because it has all sorts of made-up mathematics, created almost exclusively for the purpose of getting someone a doctorate or a publication, and of no use whatsoever, but the rest of it is quite successful attempts, I have to concede, to organize and illuminate what is already known, by underscoring the commonality of it.

But this middle level has been around for ages.  It just so happens that only a small part of it has to actually do with numbers and arithmetic; the rest is all about logical relationships.  This is not exactly your everyday logic, but more complex reasoning.  Here is an example: for every pair of fractions you can give me, I can give you a number that lies between them that is not a fraction.  That sentence is quantified, which means that the phrases “for every” and “I can find you” are used, which are definite, but not the normal logical equipment people use.]

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An Ideal World for Conservatives

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I have been thinking about the wild and crazy rants of people such as Rish Lumbaugh, on the one hand, and your typical Tea-Partyers on the other hand, and, on the third hand, my conservative buddies, with whom I have to be careful not to get into arguments.

They love to complain about everything: the deficit, the Federal Government, Taxes, Regulations, … the list is never-ending.  I would like to explore whether their complaints make sense.

The Federal Government

Well, this is the major complaint all conservatives have, be they fiscal conservatives, or social conservatives:  The Federal Government is too big and cumbersome, too expensive, and too intrusive.
OK.  Suppose we cut down the Federal Government to its bare minimum.
We want a strong defense, so we would keep the Armed Forces pretty much the way it is now, except that we would probably have to negotiate just how much money we want to spend on Veterans.  Conservatives are divided on this issue: some of them agree with the liberals that Veterans deserve the best possible care, though they would prefer that it should be funded by private donations and charity, but, failing that …  Others would throw them to the wolves, and insist that the government should have no responsibility for them.  They should get jobs like everybody else.
I think most conservatives would be all in favor of reducing Congressional salaries and benefits to their absolute minimum, since there isn’t a fair way of paying conservative congressmen a higher wage than liberal congressmen.  Ideally, congressional salaries should be paid by their constituents, without being a national burden.
We could get rid of the Department of Education, giving more control to local school districts.  (Low-income conservatives will be all for this, not realizing the implications of giving complete control to school boards.)  Pretty soon, private schools will be the only game in town, with church-funded school available in those districts where the churches are well-supported.  One would expect that the curriculum would favor faith over science, but churches can surprise us.
Here is where conservatives usually miscalculate:  Federal Controls on Goods and Services.  The Department of Agriculture requires that food and drug items be labeled with certain mandatory consumer information.  Suppose * cars were not labeled with their gas mileage, gasoline was not labeled with the octane, refrigerators were not labeled with the annual cost to operate them.  Die-hard conservatives claim that some businesses will label their products, others will not.  Those that are consumer friendly, they claim, will prosper, others will not.  And, they say, consumers have recourse to litigation if labeling is inaccurate.  Enormous tons of money saved by eliminating some of the most annoying offices of the Federal Government.
Good luck.  Consumer information not required to be accurate by law?  The consumer information will be just as useful as the warnings on cigarettes.  (In retrospect, perhaps they have a point.)

State and Local Regulations

Let’s look at a single instance: mufflers and catalytic converters on cars, and safety standards.  Again, my friend is fond of saying that manufacturers will voluntarily make their cars safe.  But considering Ford Mustangs, we know (because the information was leaked) that Ford Mustangs have mufflers that are barely within the noise parameters.  There are no regulations against motorcycle noise.  So, once regulations are no longer enforced, the din and the pollution will be intolerable.  The wealthy can move into quieter neighborhoods.  They will destroy virgin wilderness to find quiet, and build highways to their quiet homes, and pretty soon, of course, noisy cars will roar up past their quiet retreats.
What about zoning?  Is zoning a restriction that conservatives will hang onto, or will they do away with it, giving everyone greater freedom?  Interesting point.  Again, I suppose, voluntary zoning will miraculously happen.

Our own local government is presently controlled by Republicans.  But there is a noticeable lack of loosening of regulations.  Most conservatives who stand for election claim to be “moderates”, and invoke the ire of Tea-Party-ers and the like towards liberals only when annoying liberals need to be kept in their place.  In other words, the conservatives in power only pay lip-service to anti-Government rhetoric; they definitely believe in big government.

If not for the eight years of Bill Clinton, the Government of The United States of America would be in much greater debt.  The surpluses of the Clinton Era enabled George W. Bush to indulge in the spend and spend and no tax behavior that we saw.  And present-day fiscal conservatives will ultimately allow President Obama to raise revenue, while they wash their hands of all responsibility, thus enabling their own man the liberty to lower taxes and spend, if they get their person elected in 2016.  If there are sixteen years of uninterrupted Republican rule, they will either adopt broadly liberal values, or they will lose the vast majority of their less affluent voters.  Either way, the country will look very different from its present moderately reasonable state.  And the infrastructure flavor the government will put in place will make it impossible to get back to this heaven, even if both liberals and conservatives want it, and the rich will emigrate to flee the hell they create.

The disaster scene that we can anticipate if extreme conservatives and libertarians have their way gets funnier and funnier.  Colleges and Universities will have to raise tuition, which brings the dream of the upper-class to reality: no competition from poor, smart kids.  College will be the privilege of the wealthy few, as they believe it well should be.

Social Security, Medicare, and similar services will be privatized.  Aging middle-class Baby Boom-ers will hurt, but those of that generation who advocate privatization will have it made; it is the less affluent who will suffer.  If dividing the country --even conceptually-- into rich and poor, and channeling each part towards different resources of different quality is class warfare, this is class warfare.


What about roads, bridges, airports, regulation of land, sea and air traffic?  Well, only certain important places will be developed, based on local resources.  So certain wealthy suburbs will boast small, private airports, with their flight paths over the poorer residential areas, while large, public-access air terminals will have to fend for themselves.  There will probably be a lot more private, long-distance flights in small planes, to accommodate the very wealthy, and the middle-class will probably have to use land transport, just as many of them do now.

Gasoline will probably be cheaply available, though the higher grades will only be available at selected locations in wealthy suburbs.  Without government regulation of any sort, and without very much taxing, oil companies could set prices as high as they like.  On the assumption --by the general population-- that oil reserves are infinite, oil companies will sell the gasoline at a price that will maximize profits for the short term, since they know that it is a short boom, before the bust that is sure to come.  There will be a few private stockpiles, out of which gas will be sold at exorbitant prices once the flow of new oil is exhausted.

Most of all, food will be suspect.  Without regulation, cautious wealthy Americans will have to get their food from trusted, small, expensive providers, while the rest of us will have to take our chances with unregulated supermarket food.  Some supermarkets will certainly enforce internal standards, but they're more likely to simply take out insurance against customers who take ill and sue them, rather than really watch the reliability of their suppliers.  Or, we can grow our own.

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