Using the Socratic Method to Teach Notation

Would it surprise you to learn that you can use the Socratic method to help students "discover" music notation AND explore why it exists as it does?

Maybe it's a stretch, but I had in mind to use a Socratic-/Inquiry-based model to teach notation to a General music classroom somewhere between 6th-12th grades. Allow me to demonstrate with a hypothetical dialog... and let me know what you think I could improve on. Thanks!

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[Play a note on a piano or any musical instrument.]

Class, think about trying to "write" this note for someone else to play. What could you use?

[Confused looks from students...]

Well, what do we use to write words?

"Letters!"

Good! So, someone come to the board and show me how you would write "circle."

[student comes up to the board and writes "CIRCLE"]

Okay, someone else show us another way to write circle.

[Eventually, a student will come up to the board and draw a circle]

So, are words the only thing we use to communicate?

"No. We can use shapes."

Great! So, we can use words or shapes to communicate. What are the simplest shapes we can use?

"Circle/Square/Triangle"

Good. Which one of these can you draw the fastest without changing the shape?

[Confused looks... maybe some guesses]

Let's play a game. I need three volunteers. [Hands go up, pick three students to come to the board]

Let's time who can draw the most triangles, circles, or squares the fastest without changing the shape in ten seconds. Countdown from 10 to 1, class. Ready? Set. Go!

[Circles should be the fastest because there are no angles in a circle]

So, if we wanted to use the fastest symbol we can write accurately, which one would we use?

"Circles!"

Great. So, let's use a circle to represent this pitch. How do we know what pitch it is?

[Confused looks]

Is this higher or lower than this note? [play a note higher or lower than the original note]

"Higher/Lower"

So, how could we show the difference between these two pitches? Someone come up to the board and show me.

[student will more than likely draw one circle at a lower part of the board and another note at a higher part of the board.]

Great, so this pitch is this note, and this pitch is that note. [Play both notes as they relate to "higher" and "lower" pitches.] Let's have another note. [Play a different note]

Someone draw that note on the board compared to the other two notes. Here, I'll play them again.

[student comes up to the board and draws a third note]

Okay, so we have three notes. (Lets assume kids already practice "scales" on an instrument or with their voice)

[Play a major scale, probably the most familiar scale to students]

How many notes did I play?

"Seven/Eight"

Which is it, seven or eight?

[Make students explain why it's seven or eight.]

"It's eight, the top note sounds the same as the bottom note, but it's higher." (Almost correct)

"It's seven, the top note restarts the scale." (Actually correct)

If we use symbols to locate pitches, what can we use to "label" them?

"Letters."

Okay, so what letter should we start with?

[students may call out any letter of the alphabet]

Think about the alphabet. What letter did you learn first?

"A."

Would it be simpler to begin with a random letter or the first letter of the alphabet?

"The first."

I agree. So, let's start with A. What's next?

"B." Next? "C, D, E, F, G..."

How many letters is that?

"Seven."

Do we have enough letters to locate each pitch of the scale?

"Yes."

Great!

If we can draw circles to represent pitches, and we can label those pitches with letters, is there a way for us to show which circle represents which pitch without using letters?

[Confused looks...]

Can we use lines and spaces?

"Maybe."

Let's check. [Draw two or three lines and some circles/ovals, making sure to include the use of at least one line and one space] Is this note in this space higher or lower than the note on this space?

"Higher/Lower."

Good, so we can use lines AND spaces, right??

"Yes."

Dialog to be continued...

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And yes, I'm assuming quite a bit here in terms of what students will shout out and what they'll struggle with, but it's piecemeal enough to explore the foundation of notation and give students the opportunity to "think" about the symbols they use. If they're instrumentalists, they might fall back on music they've seen and jump ahead. Keep them on pace with the dialog and follow-up with questions if they're not arriving at the basics they need for understanding notation. This dialog can go on forever, but I absolutely think a basic understanding of notation from this method will lead to inevitable discoveries about "intervals," "scales," "enharmonic spellings," and so on if continued over several class sessions, broken into concepts. It may enhance the experience of practicing as well, as students begin exploring the fundamentals while practicing their music will awaken more cognitive processes... leading to a musical meta-cognition of music theory and performance.

But yeah, thought I'd just perform the exercise and see what I could come up with and more or less "simulate" here. What do you think?

:hmmm: socratic? :hmmm: method? :eyebrow:

To be honest, I hate this method of teaching with passion.

If you want to teach someone something specific, tell them. Doing it like this however makes the assumption that your specific method (in this case western music notation) is the "logical and natural" way of doing things and should thus naturally come to your students - when in fact, it's simply a reached consensus that has become somewhat fixed over the centuries.

The problem is that teachers imagine such dialogs and imagine the answers of the students at the same time. That's a big no-no for me. Never assume that the students are going to think a particular thing, because usually that means you won't actually be listening to them, but just be grasping for things they say which you can interpret as what "suits your plan".

What if the students answer to your questions in a different way and in the process come up with a notation form that is actually much more intuitive and a lot different? Do you then tell them, "well yeah, but you're wrong", or "my attempt failed, so I'll just tell you the real way"? Or do you keep up the game and continue asking questions in the hope to dissuade them from their method and still find your solution? That would, IMHO, defeat the whole point of this method and make yourself no longer appear very credible.

The major problem is that many people take certain facts for granted. Most of us take the connection between "high pitches" and a "high position" on a staff line for granted - but by doing that we're ignoring that we're simply equating them because they both contain the word "high", even though it has an entirely different meaning in both cases. "High" in regards to frequencies has extremely little to do with the spatial "high", except that for certain (of course explainable) reasons, the two once were associated with each other and people began calling certain pitches "high" and others "low" (even though they might as well have called some pitches "hot" and the others "cold", for instance). By letting the students make the connection that "high sounds" should "logically" be placed "high up" on the staff, you're giving them the impression that "what sounds alike fits together", which might lead them to justify quite some weird reasoning in other cases…

In this particular case, it would also be misleading for the students even if the dialog went exactly as planned and they arrived at your music notation method. It would make them think that all of notation is the logical result of such reasoning, which may confuse them a lot when they are later confronted with conflicting methods and extensions of that system. It would be a lot better to make it immediately clear to them that methods of communication (languages, music notation, whatever) aren't held to because they do things in a particularly logical or intuitive fashion, but because holding to a historical consensus facilitates communication in many cases.

This form of teaching only works if you're honestly ready to accept any answer from the students and are able to react to them to come to any conclusion, which may differ from your preconceptions. For these kinds of subjects, the method is fine, but for trying to lead students in a particular, preconceived direction, it simply comes off as presumptious. I've had teachers that did that and I hated them for it.

So, are letters the only way to write words?

"No, we use shapes."

LOL

Gardner ftw as always. I also don't like this method one single bit.

This mildly works in mathematics when a)not dealing with notation b)not dealing with something extremely complex. In mathematics, there is only one right conclusion for each process. If you have the hours needed you can guide someone by the Socratic method to the correct answer, but if you want to be strict about it, you'll have to be willing to explore their ideas completely until they are wrong.

I think about a problem like this:

http://www.cut-the-knot.org/Generalization/Menelaus.shtml#proof

I can guide someone through the first proof easily, but I cannot motivate the 2nd proof by questions as easily, and that's the one with the greatest insight needed. Sometimes, there's a need to lay out the information and motivation behind something in order to not waste time and get the kid where he needs to go.

While I don't believe teaching how to solve problems, when to use equations or how to think about problems is helpful (thus I don't teach that), I do think it's important to give the building blocks and then write questions that allow the student use those building blocks and extend them. So if you were ever step into one of my classes, it would essentially be a 10 minute lecture filled with theorems, scattered important mentions of things to keep in mind and lastly a set of questions that I expect kids to answer by next week.

I do this because as a student, I was never good at staying awake in class and as a professor I became worse at it. Thus I encourage everyone to see me during one of my 2-4 hours office hours so we can spend some one on one time and actually get to the root of what they don't understand, instead of guessing what the class doesn't understand as a whole.

During those 1-1 times I mostly guide by asking a kid why he's doing what he's doing, because I need to understand how he's thinking. That's basically the closet I get to this method of teaching.

Neat work, Shaun. The Socratic method seems like it's for people who either talk to themselves or have no friends :P

I hate the Socratic Method in general. But as I read this, I laughed heartily.

So, are letters the only way to write words?

"No, we use shapes."

or

How do we show which notes are higher and which notes are lower??

"Lines!"

Okay, what do we have between two lines?

"Space."

I mean really!

Well, since most people are bashing this I'll interject as one of the few people here who took a non-intro graduate level course.

At lower level studies, most people are absolute correct that this method isn't exactly the most effective method in the world. However, from my one month at Law School (don't ask) and my time paying my dues in high level mathematics, I can say this method is extremely effective where they are gray areas.

If you basically require the students to read what they need to read before class, present a problem and then beat them down by asking questions, you essentially make the student really think about what he read on a level not readily done using another teaching method. So yes it may suck when you are trying to teach something that is pretty solid, like notation, but when it comes to:

What defense is best for this case?

or

What is the best way to find the regularity of solutions of Euler equations? (Which has not been solved as of it)

then performing this method is actually quite useful. In fact, when talking two people who have deep knowledge of a certain field, you'll often find that they general speak in question form to each other. It's interesting, but it makes a lot of sense when you are talking about things that are not absolute yet.

Of course, we were merely discussing the application for this particular topic. My "I hate this method of teaching with passion." was poorly phrased in this sense of course, as I certainly agree with you that it can be very useful in certain other situations (which I also mentioned in my post). If a teacher does an analysis of a Stravinsky piece with a (moderately advanced) group of students, it is generally much more useful to approach it by asking questions, since in this case the teacher (hopefully) also doesn't have a preconceived, fixed "solution" in mind, but is honestly ready to accept the students' input. For me, that kind of honesty is the crucial part. I don't like questions that are merely rhetorical. I like questions where the teacher is honestly interested in advancing a thought process through whatever answers the students may give.

Well, just keep in mind that I threw this whole thing together in about, maybe, an hour. It was just something to experiment. There are some particularly funny moments in it, too. I don't happen to think the Socratic Method is the end-all be-all of education, but I do think it's a LOT better than simply telling students information without having them work at it themselves. This was my attempt to build notation from the ground up, at least the granular aspects of it. Getting kids/young musicians to think about even something as simple as why we use ovals instead of other shapes in Western music.

We're talking about a lesson like this introducing notation in some general music class occurring between 6-12 grades. It's obviously not perfect, but it's not entirely devoid of potential either, at least I don't think it is, yet, anyway. Besides, and this is really my beef with Tokke's and SSC's responses so far, if you don't -like- the Socratic Method then why are you even posting here? You aren't offering anything constructive, so just get out of my thread if you're not going to offer suggestions.

@ Gardener and BD: Personally, I like the Socratic Method. It'd be neat to structure a dialog that uses it to engage students in thinking for themselves earlier in their education. At the secondary school level, too many teachers seem more interested in making it easy, spoon-feeding their classrooms with bits and pieces of information... and kids writing notes in notebooks furiously. And I don't want to be that kind of teacher because that form of education is worthless to me. I'd rather create a conversational environment that's more engaging and gets kids to participate in class.

As far as a preconceived, fixed solution in mind, yeah, at the level I'm considering using this, it's not that much of a biggie. I believe Western Music notation follows a logical pattern, and as long as the bits and pieces follow that pattern, the Socratic Method makes sense to use. Just the very idea that traditional notation uses ovals instead of any other shapes, which are easier to draw quickly and consistently than other shapes, is something that can be connected to talking about shape notation further down the road. At least then there's a context for explaining it because students have been forced to think about how music has been written in Western Music and why it's been written that way (strictly from a notation standpoint).

I dunno, I wanted to experiment. So, sue me if my first attempt sucks. :P

Neat work, Shaun. The Socratic method seems like it's for people who either talk to themselves or have no friends :P

Ouch, John. That one stung. :)

Again, i have nothing i can say about success or richness of content of this post, but why something that involves asking asking questions and getting answers is by default Socratic Method? Maybe i'm too sensitive, but i feel there's hardly anything Socratic about it. But maybe i'm wrong, so, please, show me from which Plato's dialogue you get the idea that Socrates was a knowing teacher asking some group of people to form rather empirically obvious statements without irony, without reductions of their initial beliefs and logics, without the famous underlying Socratic position "i know, that i don't know"?

As for now i see nothing Socratic in there.

You could name this as well "Method of a caring, but rather overtly assuming, parent of all ages".

Sincerely, the title is dead and an insult to Socrates. Even though you might not be the first one to come with such a pornographic name having only one parameter adequate, that of a dialogue (but what do i know, here's 21 Century, maybe dialogue is not natural in teaching).

Rant over.

On a side note, when i was bored from studying Plato, i would draw lines, circles, squares and use various thickness to represent music at lectures. So, maybe you're right, there's Socrates somewhere there in his dialogues.

Oh, and just an anecdote, Plato, who wrote all Socrates' dialogues, had very strange relation to arts, and in his "Republic" he suggested we better exile all poets, and censor all music and painting, since music, badly used, may cause harm on one's soul.

...and this is really my beef with Tokke's and SSC's responses so far, if you don't -like- the Socratic Method then why are you even posting here? You aren't offering anything constructive, so just get out of my thread if you're not going to offer suggestions.

On the contrary, I believe the socratic method has its place and can be extremely useful in such situations. In philosphy for example, or for subjects, as noted by Drake, where there are shades of gray. But for something as concrete as basic notation, which has been standardized over several hundred years, there's no real debate about it. Sure, I would ask them questions about certain parts of it, but I would do it more like this:

Teacher: Here's a staff and some dots on it. These dots are called notes. You'll notice that they're ovals. Why do you think they're ovals?

Students: [Myriad responses.]

Somethign like why they're ovals isn't set in stone. Does anyone actually know? Because I don't think speed of writing was the only reason. Indeed, the first notes were other shapes too, squares and triangles and what not. But the simple fact that they are ovals in standard notation (disregarding shape-note notation which is a whole other ball game), is not in dispute. For all the kids know, the notes could have been squares and it would have been perfectly fine.

Just something to think about.

Thanks for your response, Tokke. These are certainly good things to think about.

I guess my "model" for the way the method is used (and how I encountered it when it was used in my experience as a student) flows along the lines of this dialog:

http://www.garlikov.com/Soc_Meth.html

Note, he's making a pretty clear case for teaching by asking instead of by telling. That's more or less where I found my inspiration to do this... just to see if it could be done. I'd really like to have this kind of "inquiry" structure for an introduction to notation... because it just involves students more than simply "telling" them how music is written. They actually have to think about how they'd represent sound on paper. Maybe because music is so abstract from the "concrete" world (meaning you can only demonstrate music by how it sounds, not how it looks), I've been forced to assume some things and just resolved to myself that I would have to make adjustments during the actual lesson.

As far as note shapes, I think it makes sense why we still use ovals today. Consider what it was like being a copyist. The simplest shapes to draw are lines and circles/ovals. Try it and see if that's not the case. If you can draw consistent squares faster than you can draw consistent circles, then maybe there's another explanation. It just makes sense to me why notes stopped being written as neumes in chant and are now written today as ovals. Must have caught on for some reason... but there's no basis in the literature I've read to suggest that was the reason this changed.

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Pliorius, I have no intention of getting into a pissing match over Socratic method, Plato, and Socrates. I just call it like I've understood and experienced it. If you think it deserves a better description, by all means, please suggest one. I'll happily consider it.

well, i'm not entering pissing match, actually you would easily lose it. i piss alot.

but if you use something and call it "Socratic Method" i assume you have an idea what was Socrates Method, so i'm only asking which dialogue you modelled yours on?

It's like i would call something "Music in Bach Style" and proceed with drone metal, would it be in any sense meaningful (except for the gags)? Sure, they both would have sounds in it. But you must clearly see it's not enough.

As i said in many other posts, the clearing of your concepts, when you present them, is your problem and should be a big part of your thinking proccess, since, after all, at least minimal rsponsibilty is required, wouldn't you agree?

What you presented here i do not think bears any relation to Socrates whatsover, so why include (the name) it?

There were tons of books written in dialogue style after Plato. You have a wide pool to select from.

There's "Socratic Method" in the name only here, not in content or form. That's all i have to say.

What, then, should we call this method of inquiry in a classroom environment?

I mean, this is a "method" of instruction in education that we "call" Socratic method, even if it's only loosely associated with Socrates. Why split hairs over this when it's obvious this isn't a "philosophical" discussion? I see your point, but I guess I'm a little baffled at what I should do about what you're saying. I mean, if it's that significant, why not spearhead a movement to change "Socratic" method to something else...?

I guess I really just don't see the point in over-complicating the discussion with a deep, philosophical analysis of why "Socratic" method is or is not "Socratic."

And I don't want to be that kind of teacher because that form of education is worthless to me. I'd rather create a conversational environment that's more engaging and gets kids to participate in class.

Sure, that makes sense. But if the method is the actual goal here, why not choose a topic to which it can be applied a bit more broadly? There's nothing wrong with also tackling some broader issues with kids - maybe issues that won't give them any factual knowledge right away, but that get them involved in active discussions and thinking. Here, it seems a bit to me like you want everything at once: You want them to gain information that is very specific, but somewhat arbitrary in content, and at the same time you want them to discover stuff on their own by an open dialog. Those things clash.

You can still do many of the things you said (letting them try out writing different shapes quickly etc.) if you want, but at the same time make it clear that there's a specific method you want to teach them and that those "games" on the sideline are merely a way of helping them understand it. That seems like a more sincere method than trying to feed them knowledge while making them think they arrived at it on their own (mostly because the latter rarely really works out like that).

So, you're more against the game in the course of the instruction? Or are you more against the inquiry model for teaching notation at all?

I mean, we're talking about kids being introduced to music in a format other than just listening to it... and there's plenty to discuss that's more broad and subjective, sure. But why avoid the inquiry model for discovering some of the pretty sensible reasons why notation is written the way it is? What is so "un-programmable" about notation in teaching it this way? I guess I just don't see the Socratic/Whatever Method being strictly used in subjective areas. Math is pretty objective and logical in concept, and the method was used in that context as well. What's so different about Western music notation?

I don't mind games. I mind asking questions without being truly interested in what they may answer.

Math has, as mentioned, generally objective and logical solutions. As long as you make no mistakes in answering the questions of the teacher, you will logically eventually arrive at a correct solution. There usually are no alternative solutions that are equally sound, yet "not considered valid".

That's totally different for something like music notation: Here, there are -tons- of entirely logical solutions one could come up with and the only thing that makes our established form of notation the "right one" is, well, that it's established.

Sure, ovals are easy to draw. But they are by far not the -only- easy thing to draw. It might as well be small diagonal dashes, half circles, or whatever. And depending on what kind of writing instrument you use, different shapes are written easier. I mean, after all there's a reason why we didn't -always- notate in ovals. Or, one might argue that it's extremely tedious to even write down individual symbols for every single note and not just draw whole "note configurations" or contours in one single shape or line. (Such as we did with Neumes - which, contrary to what you said above, are of course a lot quicker to write than tedious individual notes.) Or we might not write down the pitch at all, but instead what to do in order to produce a pitch (i.e. tabulature). Or maybe the students won't even see the importance of notating that specific pitch and would rather notate their piece as the sentence "Play something happy that ends with huge chords".

Giving the impression that our form of notation is the direct result that logically comes from any practical consideration is -definitely- misleading.

P.S. Oh, and good luck teaching rhythms that way. Rhythmic notation is clearly -totally- unintuitive the way it's established in western music notation.

That's totally different for something like music notation: Here, there are -tons- of entirely logical solutions one could come up with and the only thing that makes our established form of notation the "right one" is, well, that it's established.

If this was the case, it seems reasonable that we'd have several different forms of "logically" derived systems of notation that all developed in different ways throughout the past 400 years. But we don't. Why not? Just because it was -a system- of notation doesn't mean its existence is purely due to "established norms". Sure, there is an "established" method of teaching music that involves introduction to music using the traditional tonal system... that's true. But within that context, traditional notation -is- a logical derivative of that system.

Sure, ovals are easy to draw. But they are by far not the -only- easy thing to draw. It might as well be small diagonal dashes, half circles, or whatever. And depending on what kind of writing instrument you use, different shapes are written easier. I mean, after all there's a reason why we didn't -always- notate in ovals. Or, one might argue that it's extremely tedious to even write down individual symbols for every single note and not just draw whole "note configurations" or contours in one single shape or line. (Such as we did with Neumes.) Giving the impression that our form of notation is the direct result that logically comes from any practical consideration is -definitely- misleading.

Sure, but when we take ovals, fill them in, place stems and beams on them, and begin discovering how Western notation deals with the temporal aspects (rhythm, duration) of music, again these things logically follow out of the context of a traditional tonal syntax. So does harmony. It's not misleading to give the impression that notation is the direct result that logically comes from the tonal system. Of course it's not the only system, and no one is saying it is. Unless you're suggesting I should not introduce tonal music to general music students at all, I'm still struggling with this notion that the symbols of the notation system that have been used for centuries are entirely based on "established practice." It's only established practice in the broader context of world music, but that's not where General music generally starts.

There's a difference between teaching kids that tonal music is the only "real" music out there (outright false) and using the tonal system (and its emerging properties, like notation) as a window into the discovery of "writing sound" within a context that is familiar to students. If, by and large, students listened to and were familiar with music that didn't use the traditional notation system at all - or if they were playing that music with any regularity - it would make perfect sense to me to introduce music using a different system of notation. But within the context of traditional music and that notation system that emerges with its use, I disagree that there's any ambiguity in how notation developed, that it's this "establishment" within the established tonal system of various centuries. Please, if you have evidence of other systems emerging and being used regularly within the centuries of tonality feel free to present them as a demonstration of your claims. Show me what else logically follows from it if you know of any.

I don't see too many students confusing this as being "the only natural way" to write music, and if it's taught well, students will form a greater appreciation for the notation systems that exist with the understanding that they are intelligently designed from the music that inspires/necessitates their use. In fact, there could even be an assignment near the end of the course where students write music in a "style" of music that's never existed and create their own notation system to go with it. That's actually something I might implement in my curriculum, I think. It sounds like fun.

P.S. Oh, and good luck teaching rhythms that way. Rhythmic notation is clearly -totally- unintuitive the way it's established in western music notation.

WHAT?! Is this a joke?

A whole note divides into two half notes, which divide into four quarter notes, which divide into eight 8th notes, which divide into sixteen 16th notes, and so on. The space of one measure of music can be accounted for at infinite levels of division. What is "unintuitive" about that?

Are you referring to the shapes used (stems, beams, flags, etc.)? If so, I'm pretty sure I can come up with a way to bridge into rhythm by introducing new shapes and manipulating existing shapes. How far that would go to retain student interest is another matter entirely, but there's probably a way.

Yes, I'm talking about the graphical representation.

Now, I won't even start about measures/times etc., but let's simply look at how different note lengths are expressed in the notes:

Let's say we start out with a whole note. Empty oval. Fine. Now we want to designate a duration that is half as long. There are many ways we could do this, but sure, filling out the circle is easily doable, so let's just do that. Still fine. We now have established a practice of halfing note duration by filling out ovals.

Oh, wait, we even need more durations. Now we have a problem! The circle is already filled! So let's just invent a totally different graphical element to add to the mix! Let's draw a line from the oval in some direction.

Now we already have established two entirely unrelated graphical elements to signify halfing of note duration.

Next, we need even shorter notes! Our circle is filled, so we can't do that again. But we could add more lines sticking out of the note head! Hmm, for some reason we don't do that. Instead we add totally different looking, smaller lines that stick out of the previous line. Woo! Another completely new graphical element!

Well, finally, at that point, they seemed to have realized that it might be best not to invent new stuff again and again, and simply add more of those small lines to the big line, to make the note durations even shorter. Clever!

Hmm, the problem is, now all our note durations always have the duration of a power of two of the smallest one. We need something for those inbetween. So, we again need to add something completely different to the mix. Let's add a dot!

Hmm, but while that gives us a duration that is 1.5 times as long as our previous one, what if we want one that is only a third as long? I know! Let's add triplets! Another element!

After that we only need to still throw in confusing meters, barlines, tempo indications, fermatas, rests, ties etc. to complete our funny cocktail. We surely need dozens of different ways of displaying note duration, all used at the same time!

Some kid might argue that this is awfully complicated and have the alternative idea that since the horizontal axis of our sheet represents the passing of time, it would be logical to forego this wild mix of symbols and work with different horizontal distances, the same way as we work with vertical distances for pitch. But alas, the kid would be sorely mistaken! That's not how it's supposed to be done! Horizonal spacing should have nothing to do with rhythm, no sir!

The point is: Notation has developed not so much by first establishing parameters to be notated and then developing a means to easily and clearly write them down. Rather, notation was always oriented at what was currently considered important to write down. And when new things began to matter in music, people drew from existing notation systems and expanded them in all kinds of directions. Eventually some of these developments would be forgotten as people agreed with each other on one specific such development that was maybe not quite as confusing as the others. But the ultimate problem is the same as the ultimate problem of any language and the ultimate problem of any computer operating system and many more things: They started out with relatively simple, experimental things and over time people simply threw more stuff on top and bended them in all kinds of directions, because that was easier to adapt to than to invent a new, logical system into which -all- the important parameters would fit equally well.

We can still be happy though, of course. I wouldn't have wanted to live in Bach's time, where tons of totally conflicting forms of notation were flowing around. Luckily it became somewhat fixed on -one- method after some time, which may be flawed, but it's at least manageable.

The zany little anecdote here is quite amusing.

Thing is, at each level of subdivision, you're not "just adding" more things...

In fact, the only thing that changes between the symbol of a whole note and a half note is the addition of a stem. So, without changing what a half note comes from, the open oval symbol of the whole note, the half note quite intuitively follows from the whole note. The only "difference" between the symbols is the existence of the stem. Then, the next breakdown to the quarter note, everything but the open oval stays the same. The stem of the half note remains. The eighth note follows quite intuitively from the quarter note - the only change is the addition of a flag or beam.

And so on. These symbols didn't just appear out of thin air. The symbols undergo the most "gradual" change necessary to indicate they are different without becoming disconnected from the longer durations they come from. And just as you add to the symbols as you divide and subdivide them, you remove those changes in reverse. Every symbol is connected to the other according to the manner in which they are divided. SO, getting back to the beginning of your dialog/anecdote...

There is nothing connecting the shape of the whole note to the shape of a filled oval with no stem, the example you're using. These are only similar in shape, but not directly connected to each other - you can't see the open oval shape of the whole note in this hypothetical half note of yours. Yet, looking at an actual half note, you can see the whole note shape it comes from... the visual link exists. Just like with a quarter note, the filled-in oval is different, but the connection to the shape of the half note is the stem. The connection between the eighth note and the quarter is the filled oval and stem, you just add the beam or flag to indicate the next division. This progresses quite intuitively if you observe what remains from the previous symbol as you progress through the divisions.

So, for an inquiry-based dialog for this, you might begin with a riddle/challenge. Starting with the whole note, demonstrate the division by 1) adding/changing something as long as 2) the symbol/change connects to the original, undivided symbol, 3) you can identify what the symbol was divided from in the new division, and 4) the oval "note-head" shape remains constant. So the dialog might look something like this...

Okay class, we've explored pitch frequency (assume this has been explained) and how we can represent it symbolically with these things we call "notes." Let's explore how we can divide these notes in time. (Simple explanation of measures could follow here...)

Here is the context we'll use for dividing notes: 1) The new note must have exactly one added element, 2) The new symbol should look exactly like the symbol it divides without the added element, 3) the added element must connect to both the old symbol and the new one, and 4) we cannot change an oval to any other shape to represent division in this exercise. Okay, let's begin.

We have an "open" oval. What can we do to this to divide it into two notes that would be equal to it?

"We can fill it in."

Okay, let's do that. [Fill in the oval]

Does this have exactly one added element?

"Yes."

Does it look exactly like the symbol it divides without the element?

"Sure!"

Really? Show me the open oval.

"There is no open oval. That's what we changed."

Yes, but remember we have to keep an element of the symbol that's being divided. If we color this in, that's a filled oval. Is there a way we can indicate the division without losing the element, the open oval, that represents where this new division comes from?

"I don't know..."

Could we do this?? [Draw a stem on the whole note] Does this add a new element?

"Yes."

Does this indicate what we're dividing?

"Yes."

Does this connect to the pre-existing symbol?

"Yes."

Since we're calling the open oval without the stem a "whole" note, what do you think we should call the note with a stem?

"A divided note?"

We divided the note from one whole note into two notes, so can we be more descriptive with what this note is since we divided it into two notes?

[Possible follow up here if you get, "I don't know how to describe it." If we have a whole note, and it take two of these other notes to equal one whole note, then "how much" is one of these notes? - "One Half?" Yes, so what should we call this note? "A half note?" Sounds good to me... let's continue.]

Okay, what did we add to make the division?

"A line."

Yes, a line. In music, we call this a "stem." Now that we have a stem indicating what this divides, do you think we can fill in the oval?

"Yes/No"

Which is it? Yes or no?

"No, because the open oval tells us what note we are dividing." (Wrong)

"Yes, because the open oval doesn't tell us what we're dividing, the stem tells us what we're dividing." (Correct)

Yes, the stem is telling us what we're dividing. Remember, the open oval tells us what that stem divides. The stem tells us what the filled-in oval divides, right?

"Right."

And the dialog would possibly continue in that way... so, yeah. The parameters involved in the symbols we use in traditional notation to indicate rhythm are quite concrete and intuitive if you examine the connectivity between notes and form a context. It's my job as the instructor to understand what that context is and be sure it's followed during the inquiry. Otherwise, class will progress exactly as you described it, ultimately leading to mass confusion in the end.

If we explore notation this way, we're not saying "this is the only way we do it," we're simply exploring "why notation follows the patterns that it does." It makes more sense when you see a half note on music to recognize that it divides from a whole note and that it divides into a quarter note, to see the use of the stem as a connecting element in the division of note durations. I might not even approach all of notation this way, honestly, but I think it -could- be done effectively.

Yes, I mixed up some stuff there with my imaginary half note. Seems I forgot how those are written! Must be hard to remember after all!

Seriously though: Your reasoning is rather silly. First of all: My point was that there's no convincing reason to use so many entirely different symbols, when it all could be represented by much fewer, in a clearer way.

Furthermore, your arguments to justify that first adding a stem, then filling out the oval is intuitive and logical, yet doing it the other way round isn't is totally unconvincing, as is your unquestioned acceptance of stems and beams as "intuitive" means of note division, the acceptance that we should even consider filling the oval at some point (instead of adding beams right away) - or even just the acceptance of the fact that note shape should represent rhythm. You're just desperately trying to find means to justify whatever our current form of musical notation currently happens to consist of.

You can ALWAYS find apparently logical reasons why things happened to be standardized in a certain way. And there usually ARE reasons. But it's naive to deduce from this that what we have now is the most logical way of doing it.

I'm done arguing here, because honestly, reading those "teaching scripts" makes me shudder a bit.

Seriously though: Your reasoning is rather silly. First of all: My point was that there's no convincing reason to use so many entirely different symbols, when it all could be represented by much fewer, in a clearer way.

Sure, just go ahead and demonstrate this by writing a simpler form of notation using a Beethoven or Mozart Sonata accounting for the specific, granular elements of music like rhythm, pitch, etc.

Let's see it. That's all I'll say for the moment.

*Draws from his personal experience as a Professor*

Never ever underestimate a kids ability to baffle you, ever. Here's a real life example from a time long ago when I was motivated to teach.

Keep in mind, this is an advance course, so perhaps the kids here were a bit brighter than most, but nevertheless it's amazing.

M - Me

K - Kid

G - Girl next to Kid

B - Boy next to Girl

-------

M - so it's simple, every complex number z has what in polar?

K - a polar representation

M - good, and what's the restrict?

K - R > = to 0.

M - good, we remember Calculus. So what can we say about Hermitian endomorphism A in relation to polar?

K-No clue

M - How about you? *points at G*

G - No clue

M - *points at B*

B - Every Automorphism A should have two unique multiplicative deomcpositions

M - Awesome, what are those?

*No Answer*

M_ What's unitary and what's Hermitian?

G - R is unitary and U, V are Hermitian

M - Cool, now put it together

*No answer*

M - What's positive definite?

class together - The Hermitian

M-Awesome, now put it together

*No answer*

M You *points at B* how did you know every automorphism A has two unique multiplicative decompositions

B - Each Automorphism A is positive-Definite Hermitian endomorphism and I think that means it has spectral decomposition.

M - Awesome, what's the form?

B - I don't know.

M - Anyone?

*No answer*

M - ok, that's fine, let's think about this, how can we define these two matrices.

*now this is where it gets interesting.*

G - As a linear mapping

M - yes we could

B - Can't we formulate this into a differentiable function?

M - Yeah definitely we could under what condition?

B - with respect to several variables

M - Yeah

K - isn't it easier to just write it as summation?

M - Could be, why?

G - but we can handle automorphism under the umbrella of analysis easier than in Algebra.

M - we can?

G - yeah because we can transpose it to a Riemann Sphere if it's differentiable?

M - wait we can?

G - yeah and from there I think just reduce this into something from complex analysis, I don't remember what

M - *mind is boggled*

K - yeah because isn't that homomorphic?

it goes on like that

But essentially they reasoned what I always thought to be a simple 1 2 3 problem into a 20 page proof that was pretty much correct minus some loose ends. I couldn't say they were wrong, I couldn't say they were right either. Eventually, I had to go back and show them what the simple way was, but the point still stands. If you assume kids will go along with your idea, you will be sorely mistaken. If you assume a kid will end up at your conclusion you're also wrong. What ends up happening is this a)kid plays along and gets to where you want or b)you shut out that one creative kid who's ideas are not illogical or wrong, but just not standard. Personally, I think I'll do anything to avoid shutting out the one creative kid, because once those people feel like they have no input their mind is shut out, and I think by simply following your script you do block out those people. You have to be absolutely 100% willing to follow their ideas out completely and then be ready to backtrack and explain why your method is better or simply take their way.

You have to ask yourself, what happens when the leader of the pack in a class is completely off the wall out of the box thinker? Do you just keep saying, ok that's fine but what about this? It becomes obvious really quickly that you have a game plan and that they aren't really involved in the process, they're just getting to where you want to go, especially with something like music notation which I wager most kids have seen before.