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Two Theories of Learning: My Thoughts September 30, 2010

Posted by contrapuntalplatypus in Music, Philosophy, Saving the World, Teaching.
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A few days ago I saw this article by Marion Brady on Twitter. Being a teacher (with a strong interest in cognitive/educational psychology) I found the article – entitled “How Ed Reformers Push the Wrong Theory of Learning” – well-written and right in line with my own theories of teaching and learning. (A shame, then, that it probably won’t get the attention it deserves, if what it says about the current educational environment is true.)

Bradley writes:

Theory T [the “conventional wisdom”] says kids come to school with heads mostly empty. As textbooks are read, information transfers from pages to empty heads. As teachers talk, information transfers from teachers’ heads to kids’ heads. When homework and term papers are assigned, kids go to the library or the Internet, find information, and transfer it from reference works or Wikipedia. Bit by bit and byte by byte, the information in their heads piles up…Measuring the success of Theory T learning is easy and precise – just a matter of waiting a few days or weeks after the transfer process has been attempted and asking the kid, “How much do you remember?” No research says how much of what’s recalled at test time remains permanently in memory, nor to what practical use, if any, that information is later put…

…Most of what we know, remember, and use, we didn’t learn by way of Theory T. We learned it on our own as we discovered real-world patterns and relationships – new knowledge that caused us to constantly rethink, reorganize, reconstruct, and replace earlier knowledge. Let’s call this relating process “Theory R.”

Theory R is why little kids learn so much so rapidly, before traditional schooling overwhelms them with Theory T. Theory R is why Socrates was famous, why project learning, internships and apprenticeships work so well, why the Progressives of a hundred years ago were so adamant about “hands on” work and “learning by doing,” why real dialogue in school is essential, why knowledge of a subject doesn’t necessarily make a teacher effective, why asking good questions is far more important than knowing right answers, why tying national standards to a 19th Century curriculum is stupid, why standardized tests are a cruel, anti-learning, Theory T joke.

Anyone who’s read my previous blog post (or for that matter spent any time ON my blog :D) will probably know that my own teaching consists almost totally of Theory R, with Theory T brought in only when absolutely necessary and then with considerable reluctance. I’m fairly sure that, in most lessons I teach, about 50% of the time is spent asking questions (“If that note is 2 beats long, how long is this note?”), about 45% making requests (“Try playing this bit after me”) and only about 5% issuing statements, like “That note is a G”.

It may, perhaps, seem self-evident that the only way to learn piano is “learning by doing”, that children cannot possibly play music they don’t understand, that asking questions in a lesson is the best way to teach children how to figure out new music at home. And yet, I’ve had, occasionally, considerable resistance thrown up to Theory R methods. More often it’s come from parents who want “quick results” – measurable, testable “progress” to, say, a given method book by a given date.

Understanding doesn’t work that way. Particularly not in music, which is partly a physical, partly mental, partly emotional and even (when playing with others) an interpersonal discipline. It’s the norm, rather than the exception, for a student to quickly master one aspect of music – say, notation – and yet struggle with, for example, the physical aspects of piano technique for years As in my own experience.

Yet I’ve encountered resistance from children as well. Most often it’s from children in the late elementary school age range, who have become so accustomed to “Theory T” methods of teaching and learning that they cannot see any alternative, and yet aren’t old enough to analyze what I am actually trying to do. Their reaction is simply bewilderment, and sometimes annoyance:

Why is my new piano teacher asking me all these questions? Is this some kind of test – will I get a mark? How am I supposed to answer to “get it right”? I don’t remember what that note is, this isn’t fair, it’s been all summer…what does she mean “you can figure it out”, sure I know the note two spaces above it, but how does that help…


(One 8-year boy I taught must win the prize, hands down, for an “extreme Theory T” approach. He played a song in his lesson that I’d assigned the week before; the notes were mostly right, but the rhythm was so erratic that it bore no resemblance to the original. I asked him if he’d read the words printed beneath each line of music, since their natural rhythm would have helped him figure it out. He gave me a look of wide-eyed, almost betrayed astonishment. “But you didn’t ASSIGN the words!” he protested.)

…And yet, after the initial shock, there’s this almost ubiquitous sense of relief – that, when they step into the piano studio, they can leave all the arbitrariness and unreasonableness of Theory T learning, and testing, behind them. This is a teacher who asks lots of questions, sure, but who will show them how to get the answers if needed. Better yet, this is a teacher who will answer any question they ask, even if it means leaving the F major scale until next week – and who will give them an answer that really makes sense. This teacher doesn’t expect them to “know” lots of stuff…only be willing to figure it out. Children have a limited vocabulary to analyze or talk about educational methods, but I’m frankly astounded by how many students have told me (in whatever words they can find) that their lessons make sense this year, like they didn’t last year with the old teacher. Something has clicked.

The smallest things can make a difference, too. One student expressed immense relief to me that I “didn’t use red pen to mark his theory questions, like the old teacher.” Of course it had never in a million years occurred to me to do that – every piano teacher keeps a pencil on hand, and when I’d come across an incorrect answer I’d merely circled it with the pencil, handed the book to him and explained why it needed fixing, whereupon he corrected it. But – thinking it over – there is something profoundly disquieting about the urge to mark in red pen (despite its convenience for large group test situations!) The color red suggests stop signs, stoplights, a rigid rule or law that has been broken, with dire consequences. And then pen is indelible. To mark something in red pen (even if you let the student fix it later!) must subconsciously suggest to the child that they got the answer wrong, and it will always be wrong, and even if they fix it a hundred times this wrong score will be forever recorded, set in stone, immutable.

Of course, I have the immense privilege of teaching students in a one-on-one situation, which allows for more flexibility. I cannot imagine how teachers in charge of large classrooms make it through the day, let alone teach (at all!) I can well imagine that it would be hard to, for example, suddenly begin using Socratic methods in a classroom of 30 first-graders. And yet there must be a better method than to furiously cram information into a child’s head, test to ensure that it (or at least a bit of it) “stuck”, and then move on to the next batch. Maybe technology holds a greater potential for interactive, “hands-on” learning? But that’s sheer speculation, and deserves a blog post in and of itself…

In any case, technology has already made certain of one thing. Theory T may never have made all that much sense, but in an time where any fact is a click of the mouse away, it is quite spectacularly pointless. Let’s try something – anything! – else, but surely even a 6-year old child could see that Theory T has become hopelessly redundant in the Internet Era?

…Oh wait. They have. It’s adults who are still catching up.

– The Contrapuntal Platypus

…and Univocalic Sonnets September 22, 2010

Posted by contrapuntalplatypus in Creative Writing, Just for Fun, Philosophy, Poetry, Through the Looking Glass.
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(Part 2…for Part 1 and lipograms click here!)

In his Ton Beau de Marot, Hofstadter also mentions a book of univocalic sonnets by the Italian writer Giuseppe Varaldo. Each sonnet uses only one vowel, while summarizing a famous work of literature (e.g. Dante’s Inferno or The Arabian Nights) in 14 lines. Though Hofstadter considers these poems “untranslatable” (i.e. one can’t retain both their content and the vowel constraint in translation) writing another univocalic sonnet on the same theme, like Dante’s Inferno, might (he can “dimly imagine”) be possible. A few years later I ran across Brian Raiter’s webpage in which he took up Hofstadter’s challenge, and produced an excellent (and very humorous) sonnet on Dante’s Inferno, containing only the vowel “i”. *

Well, as a longtime Dante AND linguistics fanatic, how could I refuse this challenge? I began writing a univocalic sonnet on the Inferno (I won’t tell you which vowel it uses, as I still hope to complete it soon). But in the process, I found myself thinking of ideas for a Purgatorio sonnet as well…and then a Paradiso one. So I worked on all three. The Purgatorio one, containing only “e”, was the first to be completed – probably as this is my favorite book of the three 🙂 Enjoy! **

He left Hell’s nether clefts, emerged – he’s freed!
Then Seeker trembles, heeds the next set test:
Steep steps meet seven levels, then the crest;
He reels, yet Helper sees; enters with speed.

Ledges where kneel men’s essences (sex, greed,
Pelf, spleen, these ever tempted; erred, yet ‘fessed)
“Be better! Perfect!” Keen, relentless zest –
Redeemed yet flesh, hence blessed end decreed.

Steeds, elders, wheels he sees creed’s secrets tell,
Scents Eden’s breeze; bereft, deserted, weeps.
Green eyes’ stern strength he meets; repents deeds, meek;
She – tender, sweet – then cedes; refreshed, he sleeps.
Wet Lethe’s creek he enters, gentle spell;
Then, reverent, ten spheres’ endless depths they seek. ***

– The Contrapuntal Platypus

* Also enjoyable are his “Short Words to Explain Relativity” and “Notes on Writing a Monovocalic Sonnet“, which gave me some ideas and inspiration. I do wish to state, however – though my respect for Brian’s creation is immense – I did not resort to Perl script-generated lists, though I admit to running some rhyming dictionary searches. 🙂

** One of the thorny parts of writing a univocalic sonnet is: does “y” count as a vowel, or a consonant? The rule tends to be to leave it out when it has a distinct vowel sound (e.g. “party”, “gypsy”) but include it when it acts as a consonant (“yes, yellow”). But what about words, such as “eyes” and “they”, when it’s entirely silent? In the end I decided to include those two – mainly because they were an intrinsic part of my two favorite images in the poem and I couldn’t bear to remove them. Take that, purists :D)

*** For the Dante keeners: Yes, I fudged a little re “ten spheres” and counted the Empyrean 😀

Just for Fun: Lipograms… September 22, 2010

Posted by contrapuntalplatypus in Creative Writing, Just for Fun, Philosophy, Poetry, Through the Looking Glass.
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(This is Part 1…don’t miss Part 2 containing a univocalic sonnet! 🙂 )

A few years back I read Hofstadter‘s book Le Ton Beau de Marot where – among various linguistic games – he explores lipograms (a challenge in which you write  without using particular letters – usually “e”). He mentions the 300-page Le Disparation by Georges Perec and Ernest Vincent Wrights Gadsby – two entire novels that do not contain a single letter “e”! (As if to make up for the omission, Perec later wrote Les Revenents, a novel which uses no vowel except “e”.)  Needless to say I thought this was all pretty cool.

Later I joined an online form and, posting in a thread on e-less lipograms, came up with the following reworking from my favorite Shakespeare play 😀


Puck’s final stanzas: “A Vision from Sixth Month’s Night”

Found our acting irritating?
Think but this, and it’s not grating:
All you did was grab a nap,
This vision just a bunch of crap.
And this sadly boring plot –
Nothing but a passing thought –
Lords, now pardon, do not sigh:
I’ll top this play by and by.
And as I am a truthful Puck,
If I had, unjustly, luck,
So to now avoid your hiss,
I’ll outdo it – I vow this,
You must know that I’m not lying.
So, good night – this hour’s flying!
Applaud us now, amigos, do!
And I’ll fulfill my oath to you.

– The Contrapuntal Platypus 😀

Time Signatures: A Socratic Experiment September 19, 2010

Posted by contrapuntalplatypus in Music, Philosophy, Teaching.
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1 comment so far

Dedicated to all those who “never totally got” time signatures in their childhood piano lessons…here is the explanation you were waiting for.


September has returned and with it a new year of piano teaching. I realized the other day that I haven’t yet posted a single entry on my teaching, even though it’s my career and (officially) one of my main blog topics! So, I decided it was high time to remedy that omission.

One of the questions I get asked a lot is: “What sort of method do you use?” I never quite know how to answer this, since there isn’t any convenient label (like “Suzuki method”) one can stick on it. I generally end up trying to explain that I have as many methods as I do students, and I use whatever technique will help the student get an in-depth understanding of the music and how it works – not simply get flashy “results” right away.

However, if anything I do comes closest to a “method”, it would be the “Socratic method”: teaching by asking questions. Though I’ve picked up many of my teaching techniques from my excellent university professors, I think the Socratic method is simply my default, intuitive way of teaching (I remember automatically doing this when I tutored children or explained concepts to my friends in school, before I’d even heard of Socrates).

The premise is simple: any student has, already, all the knowledge they need to understand any concept – though they might not be aware of it. (This tends to be true, I find, far more often than one might think.) The teacher’s job is merely to ask them the right sequence of questions – this will help the student put together the information in the right way to figure out the answer, rather than simply having it handed it to them on a silver platter, as so many teachers (music teachers in particular) tend to do

What is the advantage of this method? Wouldn’t it be more efficient just to feed the student the “facts” without all this intermediate, time consuming question-asking? I find there are two main problems with the “direct information transfer” approach. First of all, there’s no way to be sure the student really understood what you were saying without asking them at least one follow-up question to test their knowledge. But, more importantly, it’s simply boring. Nobody likes to be talked at. Socratic-style dialogue is engaging, often humorous, keeps the student involved and will stick in their memory far longer than dry, processed information. There’s a wonderful and highly entertaining illustration here of a math teacher who taught a class binary arithmetic in under 30 min using the Socratic method.

But (I can imagine the reader protesting at this point) math involves  a application of abstract laws that can be deduced through careful questioning. Music, though, is largely a human invention – surely it can’t be approached using this method of guided reasoning? Aren’t time signatures and rhythm and pitch notation merely facts to be systematically dispensed, one by one?


This summer a new family with three boys (5, 8 and 10 years old), joined my studio. They’re an attentive, enthusiastic and energetic bunch that enjoy competing (all in fun, of course) by playing all of one another’s repertoire – thereby totally confusing their teacher at times, but I certainly don’t discourage it!

One day, I was giving the oldest boy his lesson when we came to a new concept: the dreaded 6/8 time signature. Inwardly I groaned. Every time I had tried to teach this it had been a long, drawn-out process that often ended with the student looking utterly disoriented. Why were we suddenly using this strange-looking, flagged note as the “main beat?” Why was the measure now divided into six parts – or wait, was it two? Or three? What had happened to the good old quarter note which they’d understood perfectly, along with the nice familiar 4/4 time signature?

I thought back to how time signatures had been presented in my own childhood method books, starting with getting the student well-anchored in good solid 4/4 time, then 3/4 (a step that throws many of my students for a loop right away), then progressing to the thorny 6/8…and on to even more mysterious entities like 2/2 and 3/8 and 12/16, granted that the student hasn’t yet given up in frustration and dropped out. Which many do.

And suddenly a realization struck me: this was not the way to do it.

I wasn’t yet sure what the right way was, but I was absolutely certain it wasn’t presenting the student with time signatures to be learned, one by one, like capitals of the world or vocabulary words in a foreign language. All time signatures that had ever been created were produced by two simple, logical rules. Was there any simple way I could teach them the rules – in a way they understood – rather than trying to explain individual time signatures?

3. I called all three students over and we clustered around the coffee table in the middle of my teaching studio. “We’re going to talk about rhythm a bit,” I explained. (I’m wasn’t quite sure where I was going with this experiment, but had the feeling I should begin, at least, with something familiar.)

Step 1: Starting with the Familiar


“You’ve all seen this in front of most of your pieces, right?”

(Nods all around.)

“What does this “4” mean?” I point to the top number “4”.

“4 beats?” ventures the 10-year old after a bit of hesitation.

Me: “Good answer. But four beats where? Tell me more.”

8-year old: “Four beats in every bar.”

“That’s exactly right. We still don’t know very much about the beats, though. What kind of beats are they – this kind? Or this?”

(They answer “no” to both.)

“OK, can one of you draw me what the beat looks like?”

One of the boys takes the pencil and draws:

“That’s right. What kind of note is that called?”

“A quarter note.”

“Exactly! You (the two older boys) have done fractions, right? What kind of fraction has a 4 on the bottom, like this: 1/4)?”


“Can one of you draw out what a 4/4 bar will look like?”

They draw:

Step 2: Tweaking the Familiar, Part 1


“Good. Let’s try something a little different. What if I change the number of the top to a 3, like here:  How many beats are in every bar now?”


“And what kind of beat are they?”

(After a bit of hesitation: “It’s still a quarter note.”)

“Right – the number on the bottom stays the same, so it’s the same kind of beat. Can one of you draw what this bar will look like?”

They draw:

“You’ve probably seen a lot too, but maybe not .” (From one of the boys: “It’s been in a couple of my pieces…”) “Ok, good. So what happens now that I’ve got a 6 on the top? How many beats are in every bar?”


“And what kind of beat are they?”

“They’re still quarter notes.”

“So this bar would look like…”

Step 3: Tweaking the Familiar (Getting Harder)


“Okay. Let’s go back to that 4/4 time signature, with 4 beats in every bar. Now I’m going to change it a little:  Is it still 4 beats in every bar?”


“What kind of beat are they? (Pointing to the bottom 2) Are they still quarter notes, or something different now?”

“Something different.”

“What kind of note will it be?” (For the first time they look unsure, so I decide to give them some clues.) “The quarter note had a “4” on the bottom. If it has 2 on the bottom, what fraction does that look like?”

(Tentative guess:) “A half note?”

“Yes! That’s right. This bar will have 4 half notes.” I draw it:

“It looks sort of weird, doesn’t it?”

(One of the boys starts to grin and says something like: “We’ve never seen a piece like THAT!”)

“Maybe not, but there are some pieces in 4/2 time. How about another weird one: 6/2 time?”

They draw it:

Step 4: 6/8 time – explained Socratically!


“Ok, let’s change the bottom number again. What if I make it an 8, so that we have 4/8?”

“It’s an eighth note.”

“How many eighth notes in every bar?”


“What if I put a 6 on the top? How many eighth notes are in every bar now?”

“There’s six notes now:”

(At this point we’d reached the original goal of explaining 6/8 time, but given how well this was going, I decided to try a few more exotic time signatures for fun…)

Step 5: And Now For Something Completely Different…


“Let’s try a really different one now. We don’t have to put just even numbers like 2 or 4 or 6 on the top. We could put 5 if we wanted!”

(“Five?!?” they ask incredulously.)

“Yes – what would it look like?”

(I clap it for them and tell them that one composer, Bartok, wrote lots of pieces with this time signature. Then:) “We don’t just have to pick small numbers either. We can put ANY NUMBER in the world on top. What if I put 21 on top and 8 on the bottom?”

“That means 21 eighth notes in every bar!” (All three boys have started grinning by now.)

Step 6: And So It Continues…


(At this point, to my delight, the boys start asking questions.)

“Hey, can you put ONE on the top?”

“Yes! Every bar would just have one note. The piece would go by very quickly, of course!”

We draw it:

(Another question occurs to me:) “Let’s say we wanted an even shorter note. Do any of you know what it looks like?”

(One of them suggests “A sixteenth note?”)

“Right, so if we decided to write, say, 7/16?”

(Another question from the boys:) “Is there an even shorter note than that?”

“That’s a good question. The next one is called a 32nd note.  I want you to guess what it looks like, though.” (I figure that, since they’ve seen two “flagged” notes already, they should be able to pick up the pattern and add a third flag.)

(Here comes an unexpected impasse. I hand the oldest boy the pencil and he stares at the paper nervously. He obviously desperately wants to get the “right” answer and is terrified of “not guessing right”, though he seems to have an idea of what it might be.)

“Well,” I say, “somebody had to make it up first. Suppose it’s way back in prehistoric times, your friends are off hunting woolly mammoths and you’re writing music on a cave wall somewhere…” (Grins and giggles from the other two at the picture.) “You need to invent a really fast note, and it’s up to you what it looks like. There’s no right answer. You can even do a star on it, or a smiley face…” (Laughter.)

(Eventually, after taking several tentative stabs at the paper, he draws the “right” answer:)

“Well done – that’s great!” (Look of immense relief from the boy). “Though really I would have been happy if you’d drawn a smiley face too.”

One last question from the boys: “Do you just keep adding flags? What about when you run out of room?”

My answer: “Yep, you just add another flag each time…but after about 5 flags it’s humanly impossible to play anything faster! (I demonstrate at the piano). You’d need to be a robot or a computer.”

4. At this point the structured “Socratic” part of the lesson ended and we went back to individual lessons. But, as I taught the oldest boy, I could hear the 8-year old going over the more complicated examples to the 5-year old in the background, who sat eagerly listening to his older brother’s explanation.


And then, when the dad came in during the middle boy’s lesson later (he would sometimes sit in on part of the lessons) both the oldest and youngest grabbed him and started enthusiastically explaining to him that now they knew how to write ANY time signature in the world!

None of the three boys has ever had any difficulty understanding or playing any time signature since.

It’s these experiences that make teaching truly worthwhile. 🙂

– The Contrapuntal Platypus

The Contrapuntal Platypus Returns! September 11, 2010

Posted by contrapuntalplatypus in Uncategorized.
1 comment so far

First of all, a big apology to my followers for my over-a-month-long absence. I hasten to assure you that I have a bunch of very good excuses, including a very traumatic injury of my poor Rumikitty (ultimately found to be a dislocated jaw and now well on the mend), a family vacation and the start of a new year of teaching.

But, excuses or not, I’m glad to be back and eager to post some new entries. 🙂

My last few posts before leaving were all fairly serious so I’d like to do one just for fun. Here are some pictures from the trip my family and I took to Galiano Island, BC, just off the West Coast of Canada near Vancouver Island. It’s quite a picturesque island with high cliffs, sweeping views of the Gulf Islands and a huge variety of weathered rock formations on various beaches. We had three days of hiking, sightseeing and just relaxing on the beaches, and I enjoyed it all immensely. (We walked through the graveyard where local residents are buried and found that the average age is around 90…not surprising given how relaxed I felt and how well I slept each night there. It must be something in the air.)

Or maybe the light…Though I’m usually a pretty mediocre photographer, almost every picture I took on Galiano Island turned out beautifully (to my untrained artistic eye at least). There was a unique quality about the light and the radiance of the sky that transformed every otherwise ordinary picture into something really special.

For example, this shot from the top of Mount Galiano:

This one was a viewpoint called (I think) The Bluffs:

An amazing “honeycomb” rock wall down by the Bluffs:

And strange hollowed “caves” and ledges:

I found some brilliant purple starfish in the tidal pools…

…and little delicate pink-white flowers growing on the rocks (something about their shape and texture looked almost unreal…as if they were molded from sugar or Play-Doh! :D)

This shot is from another, “whale-watching” beach (unfortunately no whales were spotted this time.) If I imagine what the surface of Mars might look like, this comes close…

And, finally, I’ll end with the spooky face that I saw staring out at me from one rock formation.

…Need I say that I can’t wait to go back? 😀

– The Contrapuntal Platypus