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How to Find a Good Teacher? (A Lesson from Aikido…) September 25, 2011

Posted by contrapuntalplatypus in About Me, Music, Teaching.
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About a month ago, my Twitter friend @mwforhr and I were talking about piano teaching. Since I’m a piano teacher and she has a  daughter in music lessons, she asked me what questions she should ask a new piano teacher to find out if they were ideal.

I’m rather embarrassed to realize I never answered…mainly because any question I could think of didn’t seem to get to the heart of the matter. And people lacking competency in a given field nonetheless often have an amazing ability to BS their way through such questions (as pretty well any university student knows! :D) Level of education, performing experience, “method”, number of years teaching…none of it seemed to pin down what makes a “good teacher”.

But yesterday ago I had an interesting experience along these lines. Not as a teacher, but as a student – of Aikido. I’m a total beginner, never having studied any martial art or even gymnastics. I’m not even naturally “good” at tasks involving kinesthetic awareness and coordination…which is precisely why I’m taking Aikido. (Well, that, and it’s a great workout – and very satisfying when I “get it”!)

Yesterday was my first class. The local aikido club works on more or less a drop-in basis, and that day I was the only beginner. The others were running through a rather complex series of techniques they needed for a test of some sort. I could see, as I watched the sensei, that each move consisted of a number of steps, and I was sure if I could rehearse each step in isolation I could learn the entire technique. But even at “slow motion” speed the demonstration blurred together too quickly for me to grasp any part of it securely. Should I step forward with my left foot first, or my right, grasp or deflect the attacker’s arm, push or pull them to the mat…it was all very confusing.* The sensei was very patient – but, obviously, he couldn’t be everywhere at once and he had other students to correct.

For one exercise I found myself paired off with another woman – I’ll call her Alex. Alex had been there for several months, and obviously had a rather intuitive grasp of aikido. As I tried both the attacking and defending roles, she delivered various corrections in an increasingly exasperated tone. Finally, she burst out with: “Stop thinking about it! Just let your body feel it and do what’s natural!”

I stopped and looked at Alex. “I’m an analytical person,” I told her, trying not to lose my own cool. This is how I learn. If I can break it down, I can get it. I’m trying to break it down.”

Alex seemed taken aback. “Oh,” she said and blinked. Her expression implied she had not even considered this as a means of learning, that what she was doing had come to her naturally – so intuitively she’d never had to analyze it, even to herself. Over the next few minutes, she tried for my sake…but obviously it was a foreign way of thinking to her.

For a later exercise, I was paired with a different student who we’ll call Terri – friendly, warm and above all, patient. As soon as she saw I was having trouble grasping the technique in question, she started reducing it to individual motions (I didn’t even need to ask). After a few minutes of drill I managed to run the basic version successfully – and it felt wonderful. 🙂

As I walked home I realized: this is what makes a good teacher. The ability to break something complex down into steps that anyone – even a rather klutzy beginner like me – can grasp.

Want to find a good music teacher? First of all, ask for a sample lesson (most teachers will do this at no charge – if they charge, it’s probably a bad sign already.) Don’t take them your most polished piece. Instead, bring in a piece that you have trouble with.

Better yet, make it a piece that reflects your weakest point. Have trouble grasping syncopated rhythms? Bring in a ragtime arrangement. Fast scales? Take a Mozart sonatina marked Presto. If this makes you feel embarrassed, remember, you’re not auditioning for them – you’re auditioning them as a teacher.

Then go to them and ask one very simple question:

“Tell me why this isn’t working.”

They’ll have you play the piece in question and, if they’re a good teacher, right away they’ll start getting you to play some simpler form of it. If it’s fast scales, they might make you play it at half the speed, focusing on your hand position. If rhythms, they’ll make you slowly analyze how a single measure works, then expand that to a line, then (maybe) a page…and when the lesson is over, the problem may not be “solved” in its entirety, but you’ll have made significant progress.

Now, if they’re a bad teacher, one of two things will happen:

(1) They’ll offer some vague, handwaving answer. (My own favourite was a teacher who told me, “You just have to move your hands faster!”) Whatever it is, it’ll make you frustrated and confused without helping you get any better. Bad sign – get out while you can!


(2) They’ll go, “Well, just play it like this…” and sit down and demonstrate. This means they teach mainly by demonstration, and either they don’t really understand what they’re doing, or they understand but can’t transmit that knowledge to you in any other way than by doing it. Which isn’t very helpful either.

The thing is, a good teacher will stand back and let you pick up things intuitively when that works for you. If, say, you find arpeggio technique easy, they won’t give you months of arpeggio etudes. But when the day comes that you run into a brick wall – and even the most “gifted” students WILL have these moments – they’ll be there to patiently help you through it. One brick at a time. 😀

– The Contrapuntal Platypus

* It didn’t help that the sensei insisted that we continually switch sides when running each drill — apparently this is standard aikido practice. It’s meant to ensure that you learn how to do both mirror-image versions of the technique, but it had only the effect of scrambling my brain just when I’d _almost_ gotten it…


The Miniatures Project (#1: Grieg!) July 4, 2011

Posted by contrapuntalplatypus in Music, Recordings, Teaching, The Miniatures Project.
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Short version: Some Grieg for you to enjoy, recorded by Yours Truly aka The Contrapuntal Platypus! 😀

Dedicated to my wonderful friends Ana and Keith, and Joanne and Josh, recently engaged/married…

…And to my high-school best friend Yun-Yun, as she prepares for her wedding this August…

(…And to all the happy couples in New York state who, as of July 24, 2011 will legally be able to marry!)

I wish you all many happy years of joy, companionship and love together. 🙂


Long version:

I had my year-end private studio recital last week (just in time for the holiday long weekend, which added to the excitement and the celebratory atmosphere!) I am amazed, proud, and utterly blown away by my students’ playing. They were all superbly prepared and played very well (no devastating memory slips, tears, breakdowns or other disasters… :S) On a big Steinway piano, in a large hall, some of their performances sounded downright magical. Seriously, I was repeatedly on the verge of tears (good ones!) 🙂

At the end, I played (and recorded) a bit of Grieg I’d wanted an excuse to learn for years: Wedding Day at Troldhaugen.

Some friends of mine had been clamoring for me to post recordings here, so I thought I’d oblige. For a while I’d been thinking about a new project. See, in my B. Mus and M. Mus years, I was all into BIG stuff. Beethoven sonatas, Bach fugues, and my senior recital pièce de résistance, the Liszt Dante sonata. 😀 Nary a Grieg or Chopin or Mendelssohn piece to lighten the mood. No, I scorned miniatures – “salon” music, I thought, light, trite and not worth my time.

Problem is, after you graduate and are asked to play for weddings, churches and so on, people want to hear Chopin and Debussy and so on. (They certainly don’t want the Liszt Dante Sonata or the Bach-Busoni Chaconne, I can tell you.) And – with the reduction in practice time that comes with 40 students – even I began thinking it might be time to tackle something less…enormous.

So, here is my challenge to myself: over the next 6 weeks, I aim to learn, polish and record at least 10 miniatures – pieces under 5 minutes in length. (The Grieg is an exception as it’s mainly repeats. :D) Hopefully I’ll end up with a solid repertoire of wedding-ready music…and a variety of short recordings for my blog as well!

– The Contrapuntal Platypus

Positive Piano Teaching May 12, 2011

Posted by contrapuntalplatypus in Childhood, Music, Teaching.
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“How can you stay so positive?

Last year I wrote a post entitled “Nix the Negativity, Please” and I thought the above would make an excellent springboard for a discussion of the opposite – being positive 🙂

Last Wednesday, I was talking with the mother of two of my students. Let’s call her Katy, and her two boys Joel and Travis. Joel is the oldest of the two boys; though highly talented at math and other logic-based subjects, he finds kinesthetic activities – like tapping or moving to a beat – far more difficult. Travis, on the other hand, is the epitome of the “right-brained” child: dance, gymnastics, art all come to him with great ease.

We’d just had the lesson, in which both boys had made good progress and passed several of their songs. Katy and I were chatting in the car on the way to the bus afterwards, and all at once she burst out with, “I don’t understand it! How can you stay so positive?”

“Well, I really love teaching…” I ventured tentatively.

“No,” she elaborated on her theme. “It’s more than that. If it were me teaching them, there’s no way I could truthfully say ‘That’s great, you’ve made so much progress on this piece.’ Of course when I help the boys practice I try not to be critical, it’s my job as a parent to be encouraging. But listening to them from the other room, I could hear so many things wrong with their playing – and all I could think was “That really sucked, how could you mess up there again, why can’t you get it?””

Part of what fueled it, of course, had been a week of frustrating practice with Joel. I’d assigned him a simple metronome exercise, and asked Katy to help: tap along with him, one beat per metronome tick, then fade out and let him take over. Then two beats per metronome tick, alternating right and left hand (just like playing the bongos).

It had driven her crazy, or nearly so. “I just don’t understand how he can’t get it. He tries to tap along, but he’s *waiting* for the tick and taps after it, too late. I don’t think he’s getting any better. Travis finds it easy, of course.”

“Actually, Joel was much better at it this week – at least with me,” I assured her. But she remained skeptical. “It’s great that you can be so positive about it. But I really wonder if he’ll ever learn it.”


A week later, I was back at Katy’s house for the boys’ next piano lesson (in which they both did quite well, passing most of their pieces and showing a lot of improvement on the others.) In the car after the lesson the subject of Joel came up. “I have to say,” ventured Katy, “he does seem to be getting better at the rhythm thing. He found the metronome exercise a lot easier this week.” (Her expression told me she’d found it much easier as well. ;))

“That’s great!” I responded. “And then,” she went on, “we were sitting in the car, driving, and there was pop music playing with a heavy bass beat. And all of a sudden I saw him moving to the beat – and tapping along! ‘My teacher said I needed to practice this,’ he told me.” (I was astounded to hear this – even I hadn’t expected him to practice in his “free time”, and in such a creative way!)

And I realized I had my answer…this is why I stay positive. Not because I’m self-deceptively optimistic or naive or walking through life with stubbornly rose-colored glasses, but simply because I’ve discovered two general truths about learning:

1. It is pretty much possible for anyone to learn any skill, no matter how “bad” they seem to be at it at first.

2. Intelligent, persistent practice generally pays off much faster than anyone imagines it will.

Our society, as I’ve mentioned in previous posts, puts far too much emphasis on the myth of “talent.” Note that I’m not denying there’s such a thing as – let’s call it – “aptitude”. Obviously everyone finds some things easy and other things not so easy; that’s universal. But the idea of “talent” – that you’re endowed with a particular genetic heritage which makes you good at some things and bad at others, and will determine everything from your hobbies and interests to your career path – is absolutely a myth.

It’s amazing how much resistance to this idea I’ve gotten from fellow musicians in particular. (There was a piano forum in which I stated, very seriously, that anyone could achieve concert-level performance ability with enough persistence and a good teacher. I was thoroughly laughed at, but I still stand by that comment.) Perhaps it’s because we have a bit too much invested in this idea of talent? That we, as musicians, were showered from On High with an ineffable, special, divine talent which mere drudgery alone will never match? That if (horrible thought!) anyone could match our achievements under the right circumstances, maybe we’re not such amazing, “gifted” people after all.

Well, of course, we are…just because we’re human. 😀 But not because of our so-called talents. Because we are all incredibly adaptive, creative beings who can learn to do pretty much anything we’re interested in, and who can’t be pinned down by labels like “klutz” or “tone-deaf” or “unmusical.” (Or, for that matter, “dyslexic” or “hyperactive” or “slow” or “autistic” or “unimaginative”…or any more of the Negativity-labelled pigeonholes adults will often try to stick children into.)

…And really, what more reason does one need to be positive? 🙂

– The Contrapuntal Platypus

Asian vs. Western Education: A Third Way? (My response to Amy Chua, Part 1) January 8, 2011

Posted by contrapuntalplatypus in Music, Saving the World, Teaching.
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(This is Part 1 of my response to Amy Chua, focusing on her article though a music teaching/educational lens. UPDATE: Part 2, consisting of some more general thoughts on parenting and how to “motivate” children to learn, is now up here. Comments, questions, and debates are welcome!)

A friend of mine tweeted an article today titled “Why Chinese Mothers Are Superior”, which he suggested would more accurately be called “How to Rob Your Child’s Life and Live it For Them.” It’s written by an Asian mother, Amy Chua, who quite genuinely believes that the Asian method of education – rote memorization and practice, insistence on high achievement, and strict discipline – produces the most successful, and in the end happiest, children. When I tweeted back my initial reaction to the article, my friend responded, “She forgets that in order to become a musician, you need talent first and then you can practice like hell.”

I found this assertion extremely interesting, but limiting – just as I found the author’s stance more compelling than one might think at first glance, but also ultimately flawed. The Asian model assumes that rote practice and strict discipline can ensure “success.” In contrast, the Western model argues that inherent “talent” – either you have it, or you don’t – is first and foremost essential. Frankly, neither of those educational models seems ideal. Isn’t there any other alternative?

Though there are aspects of Ms. Chua’s article I quite strongly disagreed with (never allowing one’s children to have a playdate or be in a school play? Requiring them to be the top student in every academic subject?), I felt she hit some things right on the mark. She states:

What Chinese parents understand is that nothing is fun until you’re good at it. To get good at anything you have to work… Once a child starts to excel at something—whether it’s math, piano, pitching or ballet—he or she gets praise, admiration and satisfaction. This builds confidence and makes the once not-fun activity fun…One of the worst things you can do for your child’s self-esteem is to let them give up. On the flip side, there’s nothing better for building confidence than learning you can do something you thought you couldn’t.

Agreed 100%. This is where the Asian theory, for me, comes out far ahead of the Western theory of “talent”. The Western model is horrendously disempowering, not to mention disheartening; children tend to assume, if they can’t get something right away, that they “aren’t good at it” and should just “give up.” (Of course, if I’d done this there’s no way I’d be a pianist today.) Say what you like about the Asian system – at least it provides some scope for improvement.

It’s only after this point that, for me, the Asian theory goes totally off the rails…

Chinese parents demand perfect grades because they believe that their child can get them. If their child doesn’t get them, the Chinese parent assumes it’s because the child didn’t work hard enough. That’s why the solution to substandard performance is always to excoriate, punish and shame the child. The Chinese parent believes that their child will be strong enough to take the shaming and to improve from it.

Ouch! The problem with the Asian model is that it assumes any failure to do well (in academics, music, etc) is due to lack of motivation, practice, or work ethic. As a teacher I know that far more often it’s due to a fundamental lack of understanding.  The child’s current level of comprehension is on one side, the concept is on another, there’s a gap in between – and no amount of screaming, punishment or shaming will give the child the ability to jump over the gap, any more than yelling at me would make me able to leap a fifty foot wide crevasse. Excoriating the child for “not getting it” won’t make the child any more likely to get it, only make them feel guilty for something that isn’t their fault. Rather, it’s the teacher’s responsibility to give them – one by one – the tools they need to bridge the gap on their own, and guide them through the process.

As an example of what can go so terribly, horribly wrong with the Asian model, I’ll hand it over to Ms. Chua once more.

Here’s a story in favor of coercion, Chinese-style. Lulu was about 7…working on a piano piece called “The Little White Donkey”…incredibly difficult for young players because the two hands have to keep schizophrenically different rhythms. Lulu couldn’t do it. We worked on it nonstop for a week, drilling each of her hands separately, over and over. But whenever we tried putting the hands together, one always morphed into the other, and everything fell apart. Finally, the day before her lesson, Lulu announced in exasperation that she was giving up and stomped off. “Get back to the piano now,” I ordered.

“You can’t make me.”

“Oh yes, I can.”

Back at the piano, Lulu made me pay. She punched, thrashed and kicked. She grabbed the music score and tore it to shreds. I taped the score back together and encased it in a plastic shield so that it could never be destroyed again. Then I hauled Lulu’s dollhouse to the car and told her I’d donate it to the Salvation Army piece by piece if she didn’t have “The Little White Donkey” perfect by the next day…I threatened her with no lunch, no dinner, no Christmas or Hanukkah presents, no birthday parties for two, three, four years. When she still kept playing it wrong, I told her she was purposely working herself into a frenzy because she was secretly afraid she couldn’t do it. I told her to stop being lazy, cowardly, self-indulgent and pathetic….I used every weapon and tactic I could think of. We worked right through dinner into the night, and I wouldn’t let Lulu get up, not for water, not even to go to the bathroom. The house became a war zone, and I lost my voice yelling, but still there seemed to be only negative progress, and even I began to have doubts.

Then, out of the blue, Lulu did it. Her hands suddenly came together—her right and left hands each doing their own imperturbable thing—just like that.

Lulu realized it the same time I did. I held my breath. She tried it tentatively again. Then she played it more confidently and faster, and still the rhythm held. A moment later, she was beaming.

“Mommy, look—it’s easy!” After that, she wanted to play the piece over and over and wouldn’t leave the piano…When she performed “The Little White Donkey” at a recital a few weeks later, parents came up to me and said, “What a perfect piece for Lulu—it’s so spunky and so her.”

A success story for coercion? Not quite. Not being a child psychologist, I won’t speculate on what sort of damage can be done to a 7-year old by repeated insults, bullying and threats (not to mention withholding water and bathroom breaks – surely this would count as physical abuse?) As a piano teacher, however, I can pretty firmly state that no real progress was made here.

Here’s the first warning sign: “We worked on it nonstop for a week, drilling each of her hands separately, over and over. But whenever we tried putting the hands together, one always morphed into the other, and everything fell apart.”

This is pretty much a universal problem among pianists – playing left and right hands separately, and putting them together, are two very different mental and physical tasks! It was certainly not due to the 7-year old being “lazy”, “self-indulgent”, “cowardly,” or “pathetic.” Even less was it due to a fundamental “lack of talent” (as per the Western approach). Rather, it is simply a technical problem the child hasn’t learned to deal with. Tackling a problem like this is not the child’s responsibility nor, in fact, the parent’s (unless that parent has extensive training in music.) It is the teacher’s, and Ms. Chua should have taken the issue to him or her in the next lesson. Trying to “shame” a 7-year old into playing two-hand cross-rhythms, without help from an experienced teacher, is like trying to “shame” her into solving algebra problems when she’s never been introduced to the concept before. Lack of understanding is not something one should feel guilty about.

However, it’s pretty clear Ms. Chua did not herself have the musical experience to analyze the problem and break it down into a form that Lulu could solve. Rather she tried to solve the issue by brute force – and for hours on end this was unproductive. But then, the light dawned…or did it?

Then, out of the blue, Lulu did it. Her hands suddenly came together—her right and left hands each doing their own imperturbable thing—just like that.

Just like that! Magic! And all it took was hour upon hour of bullying, threats, verbal and physical intimidation, insults and screaming, right?

Not so fast. Unfortunately, by Ms. Chua’s account, it sounds as though her daughter stumbled across the solution by exhaustive (literally!) trial and error more than anything else. All piano students have moments where something that didn’t work before, magically starts to fit together. Unfortunately “magic” can only go so far; the student is left with little or no understanding of what they did to make it work, or how to fix it if – almost inevitably – it stops working again at some point in the future (usually the night before a major performance!)

Threats and insults aside, the problem with rote drill as a learning method is that it tends to entirely bypass the important long-term goals (musical understanding) in favour of more peripheral short-term ones (learning “The Little White Donkey” for the next day.) As a result, the child gets “good” at playing a lot of individual pieces but often has little or no general musical comprehension, artistic expression, or even music reading ability. (More on the weaknesses of the rote system, in my own and others’ experience, here.)

An ideal educational model neither relies on inherent “talent” nor “discipline” as primary strategies for teaching. Certainly, it can be useful to exploit a child’s strengths as a teaching tool (case in point: one of my students had an amazing memory for pattern and harmony, though he had difficulties reading music. I assigned him various technical exercises in the circle of fifths pattern – usually something only more advanced students would tackle. He learned them easily.) Likewise, rote memorization and repetition can be a useful tool for learning. However, neither works as a good fundamental teaching model.

Above all, a good teacher needs the ability to be both creative and analytical (pop-culture notions of right- and left-brained people aside, the two are not mutually exclusive!) Creativity is needed in order to formulate a problem in new ways, using analogy, stories, visual images, metaphor or whatever else is needed to get an idea across. Analysis is essential for breaking down a problem into steps and then handing the student the tools they need to process those steps, one at a time – so that the previously unleapable gap becomes a series of shorter hops, or if necessary easy steps.

This dual strategy of creativity and analysis produces short-term results as well as long-term understanding. But, more important than either, it gives the student the tools they will need to solve, not only musical problems, but any problem requiring creative and/or analytical ability. Ms. Chua’s assertions aside, bullying and insulting a child won’t give them self-esteem. However, neither will telling them that “everyone has different talents” and “someday you’ll figure it out” (what kid can’t see through that sort of rhetoric to hear the underlying “you’re not good at this” and “I’m not really sure why you can’t get it now”?) What will give a child self-esteem, however, is showing them the strategies they will need to solve any problem they encounter. And, unlike a late-night, tearfully forced mastering of “My Little White Donkey”, these are positive, empowering lessons that will last a lifetime.

– The Contrapuntal Platypus

* I taught an Asian student last year who transferred into my studio. After several years of study under a “rote” method, her technique and imitative ability were quite good but she was unable to read even a simple 2-line piece in C major. Furthermore she had no comprehension of how the music was constructed, what the composer’s markings meant or what the piece was trying to portray.

I told her mother that we would need to spend some time working through the lower RCM levels to improve her reading ability, after which we could return to more advanced music. We did so and she made excellent progress, moving through about 5 RCM grades in as many months. However, about 6 months after she began lessons, her mother called to tell me that she was leaving my studio. The reason (as I found out when I asked to speak with the daughter; her mom wouldn’t tell me) was that, apparently, she “wasn’t progressing fast enough.”

I asked the mother how quickly she expected her daughter to progress – and how passing six grades in half a year could possibly be insufficient. From her answer I gathered that she wanted her daughter to be playing the sort of “flashy” music that she’d been working on before and that goals such as long-term understanding, music reading and expressivity were, for her, secondary to surface brilliance and showiness.

Similarly, when my sister taught English in Japan for a year she found that the educational system was almost entirely structured around drill. Students would memorize, word-for-word, a small set of formulaic conversations with little understanding of their meaning or structure. As a result they would be able to say, for example, “I’d like a cup of coffee, please” but have limited ability to modify the sentence to ask for two cups of coffee, or a cup of tea instead. Many of her students who had studied English in school for years still were entirely unable to have a basic conversation with a native speaker.

I don’t mean to suggest with these anecdotes that all Asian teachers or students fall into this stereotype. However, it does seem to be a major weakness of the rote system that it promotes quick results over true understanding.

Beethoven’s Phone Number November 5, 2010

Posted by contrapuntalplatypus in Flights of Fancy, Just for Fun, Music, Teaching, Through the Looking Glass.
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Another teaching anecdote…this one’s not deep or profound, but it IS hilarious. 😀

Today one of my younger students began working on “Shepherds’ Song” from Piano Adventures Book 1. It’s an arrangement of the theme from the last movement of Beethoven’s Pastoral Symphony (which is one of my favorite pieces. If you haven’t ever heard it…please do take a moment and remedy this omission before going on :D)

I played the piece for him and explained it was taken from one of the symphonies of Beethoven, a very famous composer. He nodded sagely as he pointed to the words “Ludwig van Beethoven” written on the right-hand side of the page above the music.

Then he indicated Beethoven’s birth and death dates (1770-1827), printed below the name. “And here’s Beethoven’s phone number,” he proudly announced.

Okay, admit it…could you have stopped yourself from nearly falling over laughing? Didn’t think so. Me neither.

When I got a handle on myself again I explained to him that, regretfully, Beethoven had been dead for quite some time and, that even when he was alive, he was almost entirely deaf…so if telephones had been around in his time he wouldn’t have been able to use them at all.

Still, I had to admit I liked the idea – a telephone directory of Famous Composers of Western Music. Now the question is who would you call first? I’m torn between Bach (my all-time favorite) and some of the Renaissance/Baroque composers, like Josquin du Pres, whose lives we know so little about. If only…

– The Contrapuntal Platypus 😀

Never Be Afraid To Improvise (Teaching Lesson of the Day) November 4, 2010

Posted by contrapuntalplatypus in Flights of Fancy, Music, Serendipity, Teaching.
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Today was one of those days that make any piano teacher rejoice…a day when things “click” for students, not just once but in almost every lesson. One student who, week after week, had done essentially zero work on her pieces (I’d begun to dread her lessons) came in having prepared them so well that I had to stare in amazement. :O Another who announced she’d “forgotten” (??) to work on a song nonetheless sat down and played it perfectly. We went on to try some theory and ear training – new concepts for her, but she caught on quickly and enjoyed them.

But the best sort of lessons are the ones where I learn something too…and one of those happened today as well.

I hate to admit it, but I’ve always been terrified of teaching young beginners. It’s a horrifying sense of responsibility: I’m providing the foundation for the rest of their musical lives. I continually worry that I’m not getting new concepts across in a way they understand, as they’re often only five or six years old. It’s also difficult to hold a really young child’s attention for half an hour – especially because pieces for beginners use just a few notes and are often not that “interesting.” I often have this nagging feeling that if I were a TV program, they would have pressed the “change channel” button on the remote long ago. 😉

My beginner student today – let’s call her Jasmine – had just started learning how to read music a week or two ago. Today we were tackling a new piece from the Primer level of Piano Adventures. Now, this is by far the best series of “beginner” books out there on the market: the pieces are fun, unique, and impressive at an early stage. They don’t contain huge distracting or “babyish” illustrations like so many primer-level books. The upper-level pieces (books 2b and on) are, really, masterpieces of “miniature” composition; I’m continually in awe at how the writers (Nancy and Randall Faber) managed to create so much from a few five-finger patterns or chords.

However, even the Fabers had, apparently, some trouble making the first 5 notes of C major interesting. And as I played today’s piece (“Fourteen Little Frogs”) for Jasmine, she appeared distinctly underwhelmed.

“Fourteen little frogs…sat upon a log…”

No, this wasn’t going anywhere. And honestly, what self-respecting five-year old would want to learn a song about Fourteen Little Frogs when all the amazing diversity of rock and pop and jazz and country and, yes, classical music was an iPod button push or a YouTube click away? In fact, I had to respect her simply for coming and sitting in front of a piano for half an hour when the rest of her friends were all too audibly laughing and talking in the next room.

As my mind was busy grappling with this sad truth, my fingers were approaching the end of the song:

“One by one they jumped into the little waterfall…”

Jasmine’s eyes were glazing over, and at that moment I knew I had to do…well, SOMETHING quick. Something like:


…I suddenly heard myself shout, while my hands hit a big loud “tone cluster” – also known as “random group of black and white piano keys all next to one another”. 😉

Jasmine jumped, startled. (Believe me, I was as surprised as she was.) Then her face lit up. Her eyes brightened. “SPLOOSH!” she echoed, banging a similar handful of random notes.

All of a sudden she wanted to learn Fourteen Little Frogs. I told her, of course, that she had to play all the notes in the song before she could do the surprise ending. That was fine by her, as long as she was allowed to do the “sploosh” at the end. I’ve never seen someone so eager to learn a beginner song.

It’s amazing what a simple tone cluster can do, correctly placed. 🙂

The moral of the story: never be afraid to improvise. Your students will thank you for it…and they’ll pay a lot more attention if you have some surprises up your sleeve. Who doesn’t like pounding out a sudden tone cluster or sweeping a glissando down the piano? And they make a much better reward for diligent practicing than chocolates or even stickers… 😉

Even more importantly, though, it keeps the lessons fun for the teacher as well.

– The Contrapuntal Platypus

Nix the Negativity, Parents…A Personal Plea October 17, 2010

Posted by contrapuntalplatypus in Music, Philosophy, Saving the World, Teaching.
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This past week I came across Dan’s extremely powerful and moving blog, Single Dad Laughing. I was struck by one entry in particular – You Just Broke Your Child. Congratulations – in which he describes how he saw a dad emotionally and physically abusing his young boy in a grocery store. I began writing a reply on the comments thread which turned, you guessed it, into a rather detailed blog post. It talked about the power that adults in general – including music teachers – wield over a child’s confidence and self-image and how easily it can be abused (and the drastic consequences). I spent several hours writing it and was about to publish it here (don’t worry, I will still post it in a day or two…)

Then something happened which really brought home to me what Dan was talking about…and how often this happens.

A student who I’ll call Naomi (not her real name) came to make up some lesson time she’d missed. When Naomi started lessons with me over a year ago she was tentative, unhappy, convinced that she “couldn’t do it” and was “bad at piano.” Neither of her parents had much musical training and her support base was minimal. She’d also been though a year of mediocre teaching and some fundamentals had been missed, making reading music a challenge for her. A year later she’s made astounding progress. More importantly she’s a happy, radiant girl who grins every time she comes to piano and is eager to play me the week’s pieces. She’s got an amazing sensitivity for harmony and we’ve been doing some experimenting with simple counterpoint and traditional rounds like “Frere Jacques”. And she enjoys every moment of it.

The lesson today went extremely well. She had fixed everything I’d highlighted in one of her pieces (I was amazed at how thorough her practice had been!), we played a round together and the time flew by. Soon enough her dad came to pick her up. When he walked in the door she was trying out a bit of her new piece – a rather difficult one with some unfamiliar notes and rhythms that I gave her some help on. She walked towards the door smiling.

And then it happened. As Naomi was getting her things together, her dad, frowning slightly, commented that she still seemed rather hesitant about learning new music. In fact – he warmed to his theme – she often had some difficulty in playing through pieces smoothly, without stopping to look for notes.


I looked over at Naomi and her face had fallen.

Literally fallen. Where a moment before I’d seen a laughing, enthusiastic child, she was staring sadly at the floor, the smile gone from her face. Her dad was disappointed in her. Again.

I looked back at her dad in disbelief. Why would he say something like that with his child right there? Can’t he see the effect it has on her?

Raging inside, I firmly told Naomi and her dad that (a) she’d been making amazing progress and that I was especially proud of how well she’d done with her piece that week and (b) that reading new music and playing without stopping were the hardest things in piano. And I’d found them hard too.

Naomi – looking a little reassured at that – and her dad walked out the door to my studio and I went into the next room, actually on the edge of crying.

Now, I am not trying to imply that Naomi’s dad is remotely equivalent to the cruel, abusive father that Dan observed in Costco. I know for a fact Naomi’s father cares about her very much and I’m sure he was speaking out of good motivations…certainly not any deliberate horrible wish to belittle or upset his daughter. Who doesn’t worry from time to time about their child’s progress? Who doesn’t want their daughter to excel in everything she tries?

But it is really, really saddening to watch the unintentional effect his words had on a sensitive child who I’ve only recently begun to convince that she is capable, she is musical and she can learn piano.

Please, parents, adults everywhere…let’s Nix the Negativity.

Some thoughts:

(1) I have absolutely nothing against a parent sharing their concerns with me – in fact I encourage it. E-mail, telephone, call to set up a meeting. Any time. My door is always open. But not in front of your child. That goes for anything, not just music. If they’re having trouble with Subject X, openly talking about their Subject X issues in front of them is just going to make them more unsure and hesitant and certain that they’ll never be good at it, because apparently their own parents don’t even believe in them.

(2) One of the best things about my parents, growing up, was how they always emphasized What We Did Well. A favorite quote of my mom’s was “Soar with Your Strengths”…in other words focus on what a child is good at, not what they have trouble with. Of course improvement’s always possible, but it won’t help to obsess over the subject a child is struggling with and let them know that they’re “behind” or “not progressing fast enough”.

…No, my childhood wasn’t perfect and my parents and I had the usual (healthy) dose of fights and arguments (:D). My mom in particular had high standards and I would often complain that she was being “picky” or “too critical” about details. But they would never have “cut me down” in that way. Discussing areas of concern with a teacher…sure. But in front of me they emphasized my strengths…always. And it had a tremendous impact on how I see myself today.

(3) More than anything I wish I could have said to the father: “Yes, Naomi seems hesitant at times. Terrified, even – terrified of making mistakes and doing “something wrong.” Why do you think that is? Maybe it’s because you’ve made her hesitant and unsure by being so negative? Maybe if you talked about how happy you were that she learned her song really well this week, she might feel a little less unsure and worried about her playing?

Also, stepping back a bit…what is the ultimate goal here? Is Naomi really taking piano lessons in order to learn to play pieces up to speed without stopping? Or is it to develop an appreciation and love for music that will last all her life? Is Negativity really going to help her accomplish that?”

And finally there’s:

(4) Progress takes time, effort and lots and lots of patience. I am fiercely anti-elitist in my teaching because I was not a “talented student” and if I learned to play at a “concert pianist” level, anyone can. I tell every student that they are talented, they are musical and they are capable because I believe with all my heart that every human being is talented, musical and capable of everything they put their mind to. I give them the tools they need in order to realize that potential I see in them and then, little by little, give them greater independence in their practicing and learning until they can literally be their own teachers.

What will kill this process is Negativity.

Negativity implies categorization. You’re not a talented student. You’re not making progress “fast enough”. You haven’t learned a piece “up to speed”. You’ve fallen short in whatever way. It draws a line in the sand and invariably sticks the child on the wrong side. The worst bit is that an adult with a highly critical perspective will always focus on a child’s shortcomings…and almost every music student finds something difficult, whether it’s rhythm or technique or memory or note-reading. Therefore to Negativity’s judgmental gaze all children except a lucky few (the “elite”) are weighed and found wanting.

If Negativity tells me I have no potential, what’s the good in practicing? If I’m not “musical”, why should I bother trying to express myself in this piece? If I’ve put my best effort into lessons this year, and my progress isn’t “good enough”, then what’s the point of working hard next year?

It’s not just parents. Teachers can be negative too. I can be (and have been) very negative. With a few students it’s helpful, even necessary…the confident, happy ones who believe in themselves and know they can play well but didn’t really feel like practicing the past few weeks. 😀 Fine. But even then it has to be done in a spirit of encouragement and good humour. When a child doubts their own ability, Negativity will only make it worse.

If in doubt…Nix the Negativity. Please.

– The Contrapuntal Platypus

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

Time Signatures: A Socratic Experiment September 19, 2010

Posted by contrapuntalplatypus in Music, Philosophy, Teaching.
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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