There are two ways of approaching music theory - as a music theorist (who may or may not actually play) or as a music practitioner (that is: a musician). The knowledge requirements are obviously different and in the context of this question I will focus chiefly on the practitioner's perspective.
Let's think about what (some) knowledge of music theory is worth to a musician. I can see at least three reasons to learn at least some theory:
Communication with other musicians,
Let's take a look at each of them in turn.
Let's give it a name, so we can have conversations about it.
(the Mighty Mighty Bosstones on the origin of the term skacore)
As musicians we need to be able to talk about the finer points of music with other musicians and everyday language is not always up to the task. Sure, you can say "play this very quietly", as opposed to pianissimo, but most times the only alternative to standard musical terms that signify what you want ("play this bit with a 6/8 feel", for example) is to actually demonstrate what you're trying to say. Clearly, being able to express a potentially complex concept with a simple term is preferable.
Since we're talking about teaching, the two musicians who need to communicate are the teacher and student. It is, of course, possible to teach someone to perform a musical piece without addressing theory at any point (simply by demonstrating all the motions the student needs to go through in the context of the performance), but it will go a lot quicker if you are able to communicate on a conceptual level. Lest we forget, such basics as note names and musical notation are also parts of music theory.
We can look at a piece of music as a long series of notes and rests that get us from the beginning to the end - and even learn to play by memorizing it as such, or we can look at it as construct made up of smaller building blocks. Crucially, these building blocks will appear in other music as well.
What kind of building blocks? Scales, chords, arpeggios, sequences, rhythmic patterns, standard forms, cadences etc. Everytime we are able to spot something we already know in an otherwise unfamiliar piece of music, our work becomes that much easier. Rather than examining and learning each note in turn, we can instead apply higher-level concepts that we are already familiar and proficient with.
In order to become familiar and proficient with such "building blocks", however, we must first become aware that they exist. One of the functions of music theory is identifying such building blocks, naming them and examining how they fit together.
Sooner or later, we may find ourselves needing to actually create some music, as opposed to learning what someone else has written. This may be as involved as writing an entire piece from scratch, or as simple as "could you come up with a part over this bit?"
Again, it is possible to do the work by trial and error. However, it will be quicker and easier if we have an idea of what is likely to work and what isn't - if we are familiar with the options available to us and what they sound like.
A knowledge of music theory allows us to both become familiar with such options and to learn a system of classification that makes it easy for us to keep track of them.
A practical approach to teaching music theory
With the above in mind, two things become apparent: that it will be highly useful to the student to learn at least a bit of music theory; and that learning the theory should be subservient to the learning of musical practice.
In other words, the student should, first and foremost, be learning to perform music. However, as they learn to play, they should also be gradually learning what it is that they're playing and how it fits in with other things that they have already learned.
Let's try to compile a list of the most basic music theoretical concepts that every student should learn:
- A solid knowledge of rhythm (including rhythmic notation) is an absolute must for everyone. I always say that learning to count is the most important skill to master as a musician.
The great thing about rhythm is that it is a skill that can be practiced from day one, even when the mechanics of producing sound from an instrument are still a deep, dark mystery. In my teaching practice, I have always started out my beginning students on clapping exercises, letting them get a feel for rhythm without having to deal with the fiddly motions of playing an instrument.
- The basic names of the sounds one is learning to produce should be taught at the earliest opportunity, though probably no sooner than when the student is first called upon to play them.
Students should be learning the names of the notes they play (trivial, I know, but nevertheless important), chords (especially with pattern-intensive instruments, such as the guitar), articulations and so on (specifics will depend on the instrument).
The reason for this should be clear: the student will be called upon to play these things over and over again in the course of their study. The teacher needs a concise way to communicate what the student needs to play and it is much easier for the student to be thinking along the lines of "here I'm playing the C major chord", as opposed to "I need to put this finger here and this finger here and that finger there and..."
It is also likely that the student will at some point in the future find themselves playing with others outside the classroom and here the ability to have meaningful conversations about what is to be played will be invaluable.
Teaching some appropriate form of musical notation is to be encouraged with regards to this particular issue. It need not be standard musical notation from the beginning, if a simpler commonly used system exists (I am thinking of tablature especially, in this case, because it offers an unambiguous notation system for an instrument where the same note can potentially be played in six different places, or more). The point is for the student to have some way of taking notes of what they need to play, for when they don't have a teacher to show them, and to be able to explore written music on their own.
- As the student learns to play, they should also be developing a mental framework that allows them to understand what they're playing and how it fits in with what they already know.
Faced with an instrument we are presented with a sea of musical possibilities. We need a way to break them down into bite-sized chunks.
Each instrument has its own set of "tricks" to make it easier (for example, the piano keyboard lets one "see" the note names; the guitar, on the other hand, makes it easy to transpose music to different keys, by shifting fingering patterns to different position), so specifics will depend on what is being taught.
On the other end of the spectrum, there are things universal to all (or most) instruments: things such as musical form, tonality, rhythmic templates.
In the context of such a framework, it is possible to isolate concepts common to a large number of specific pieces and to practice these in isolation. If we know how to play a C major arpeggio, say, we can fall back on this knowledge whenever we encounter it in a hitherto unknown piece of music. We can practice it in different keys, building up mechanical proficiency (that is: proficiency with the movements needed to execute a musical passage) and preparing ourselves for a time when we're required to play it in a musical context.
In all cases, however, we must first become aware that such methods for breaking music down exist, before we can turn them to our advantage.
As previously stated, for a musical practitioner, playing should come first. Theoretical concepts need only be brought into the picture as they appear in the music the student is learning to play.
All things rhythm will need to be introduced pretty much from the beginning, even if we begin with something as simple as playing (or clapping) quarter notes while counting "one-two-three-four".
Similarly, as the student is learning to play individual notes or chords, we should be introducing the names we will henceforth be referring to them by. The ability to say "now play a B note", as opposed to "play with that finger there", is so important it is scarce worth mentioning. If the student is playing chords, they should know what those chords are called - with several notes being played simultaneously, the ability to "dump" a large amount of information into a single name is that much more important.
Once the student has a grasp of the basics, and learns to play some simple pieces, we can gradually introduce higher-level concepts. Musical form is something that is often forgotten, when we think of music theory, but it is both crucial to mastering longer and more complicated pieces and a fairly simple concept even for a beginner. By demonstrating how a seemingly monolithic piece of music can be broken down into "chunks" that are repeated and intertwined we are showing the student how to make their work of learning a new piece easier.
When the student begins to expand into different keys, we should be gradually explaining how the concept of keys works and how the various parts of the tonal system (scales, chords, cadences) can be isolated, practiced and applied. The goal is for the student to look at a piece of music and think less in terms of a long series of notes, but rather in terms of musical blocks that they can spot, identify and know how to play.
Practice makes permanent, so it is important to keep reinforcing these concepts in the course of our teaching. The student should learn and practice these elements of musical theory until they become ingrained. The easiest way to do this is to guide the student through analyses of the music they play - especially new pieces, once the student has a fairly good grasp on what they've been taught so far. Remember, the goal is to make it easier for the student to apply things they already know and can play in the context of learning new music.
From a teacher's perspective, this means we should be approaching our lessons analytically as we prepare them. We should have a solid idea of the theoretical concepts present in the music we will be teaching the student and look for ways that we can exploit them to the student's advantage.
Does a piece contain some concept the student is familiar with? Make a note of it and point it out.
Does it share a concept with other music the student already knows, that would make learning easier and faster? It may be time to introduce the concept and show the student how it is applied.
Have you already equipped the student with an analytical toolkit of their own? Great! Begin the learning of new music by looking over the score with your student and discussing all the places where the student can apply their existing skills.
If we want our students to take away a knowledge of musical theory that will help them with their musical practice, it should be clear that we need to examine our own. I firmly believe all teaching is also learning (Schoenberg seems to agree). Can we go beyond teaching our students how to play a specific piece of music, towards building skills and knowledge that will allow them to play different things with the minimum amount of extra work? If not, perhaps it is time to brush up on our music theory.
Edit: an illustrative example
After I posted this answer, I got to thinking that it could do with an illustration of the final goal we might hope to achieve in teaching theory to practitioners and it so happens I have a good personal anecdote.
At my sister's wedding I was asked to sit in with the band as a lead guitarist. They were playing a bunch of appropriate standards that I was familiar with, but had never played before. They asked me briefly before going on stage, so any kind of rehearsal was out of the question.
Luckily, the keyboard player had some brief notes (basically, chords and lyrics), so we'd agreed I'd just be looking over his shoulder. Then he tells me:
"There's just one thing: I'm transposing my keyboard to make my life easier, so the actual keys I'll be playing in will be different."
What I'd ended up doing is having him tell me the actual key before he started a song. Then I would mentally apply a quick Roman numeral analysis to the chords he had written out to get an idea of where I am, harmonically, and that would give me an idea where I had to go melodically. Worked out rather well.
The moral of this story is that in order to pull it off, I had to go beyond what information I was presented with (which was both incomplete and not quite accurate). The only way to do so was to first isolate the general concept and then come up with an appropriate application, in terms what I actually played. Without a solid grounding in basic music theory (or a quick ear and perfect pitch) it would have been impossible.