Sorry if this reads a bit dis-jointed-- I kept thinking of more random stuff, and I need to run so no time to edit...
As for quarter notes, I have my beginning students walk around the room, and tell them, "beat's in the feet." The feet keep the tempo, like a marching band walking to the beat 1-2-3-4.
I tell them that the feet are the quarter notes. Then I have them march in place, and clap and say,"quarter, quarter, quarter, quarter" on each step.
The next lesson is half note, where they begin a clap on the left side of their body, and hold hands as they cross their mid line, to release the clap and open their hands on the right side. Say "Half" on the left side, and "note" on the right side.
Repeat. (so, you do two foot steps for one half note)
A whole note (4 beats) is "whole note hold it" and you begin a clap on the left side, and hold it L to R and L to R before releasing it.
A dotted half note is called, by me, a half note dot, with a count of 1-2-3. This one is harder to clap out, because you start L R L, then R L R, repeating...starting a new clap and hold each new note.
By this time, the student is learning how to mark measures and has done some theory in recognizing line vs. space notes, and repeated notes, and steps vs. skips, and can even do some "music math" such as, one quarter note plus one half note equals how many beats? 3.
They will have drawn notes and done workbook pages on all of this, and I would check and review at next lesson.
For beginners, it takes a primer book and a level one book before you even get to eighth notes, let alone rests! But there is SO much to cover!
for the same clapping method, where before you clapped and said "quarter" in one beat/clap, you would now say "two-eights" or "two eighths" as one clap/beat per side. (2 8th notes equal 1 beat, or 1 quarter note)
Music truly is another language, and some of my parents seem to think that a 30 minute lesson with 10 minutes of practice the day of will be ok. Well, I take what I can get, but it takes a lot of work to learn an instrument.
Good for you for asking questions. The only wrong question is the one never asked. The teacher should not "look at you as if you have three heads."
As for positions on the piano, perhaps you mean transposing into different keys? Many books have you learn in C position, or middle C position, and some may then have you "transpose" to G position. Once you learn a bit about sharps and flats, and scale building, you can transpose to other keys, but it takes lots of practice and a good ear and eye. Not all instruments are even in the same key. A trumpet is a B flat instrument, so if I tried to play the identical music along with a trumpet player, we would sound off. One of us would have to adjust.
That is why a conductor's score has a line for each instrument, because the notes really are different, or may utilize different clefs, as another poster explained.
Also, pod to the idea that a performer does not always make the best teacher. Some things that they just know naturally, may be very hard for them tho explain. But a good teacher should have several ways to explain the same concept.
Not that I am so wonderful, but above how I wrote for quarter notes and such, may take a few lessons for a six year old, but one lesson for an 8 or 9 ear old, and while one student may take a year or more to get through one book, another may take one semester. And for older students, I draw them a "measuring cup of music" and show how a "whole" is broken into "4 equal quarters" (just like a game, or money) and it must measure up.
I adjust my language and lessons as the students' light bulb moments increase or decrease.
I also have a three ring binder of extra lessons I have found from free music teacher websites. Some students just "get" concept X, but are stuck on concept Y. It is MY job to help them through, and not belittle them.
I also utilize an assignment notebook, where we have ongoing questions and conversations. I sometimes make up a quick theory assignment (mark these measures, draw 10 whatevers, define this vocab word, etc...) and I record each week what they are to do for practice. Some of my better students, somehow, are the ones that ask more questions, and read over the notes...hmmm...
Also, understanding practice vs. playing helps.
I have had parents tell me,"Oh Susie practiced all week!" But what she did was play her favorite easy song 17 times in a row, and never touched her homework.
This is like reading a library book for fun, but never reading the school book assigned for lit class.
(Not that the OP is doing that-- I am doing a random teacher post, and hit my own nerve -ouch!)
Since you are an adult, you have to do the "listen to yourself" as you practice. Sort of a step outside your body, and watch your posture, hand position, tension, etc.. You do not have a parent to help you determine if passage A sounded identical in rhythm to passage B, as it should, just with different notes.
You could record yourself, though!
Having the student write out their own definitions of music vocab helps me to see if I explained it well and if they understand the concepts. if we all used the exact same wordage all the time, in exactly the same way, we would be very boring. Music is NOT boring!!
There are music forums for teachers and students, and I bet you can find some good inspiration there, also.
oh-one last thought-
there are several methodologies in teaching. If he just want you to memorize, he might have more of a Suzuki method in mind, which is sound and shape more than theory, which will come later. Suzuki is much better for young children, who do not have to unlearn "bad" habits like fly-away fingers, or stiff and tense hands, or not using all 10 fingers, etc...and true Suzuki will have you spend weeks on one song, one drill, and build slowly from there.
So, again, my advice is that you and he need to have an away meeting from your lesson, and find common ground and understanding.