I have been using the standard B flat fingering found in most books since I stated playing the flute. Now my teacher has told me to use only the left thumb B flat lever on the bottom of the flute to play a B flat because he it eliminates a finger placement. Ok that makes sense and I think I can adapt pretty quickly but all the method books show the standard fingering that involves the right index finger. So which is correct? I'm

There will be occasions (for instance with B natural B flat sequences) when it is preferable to use the right index finger rather than roll the thumb. Some teachers consider use of the Bb thumb lever as lazy but beginners often play more securely using it because they don't have to remember to play Bb. In this instance, the fingering alternatives are really a matter of preference rather than changing the ease of playing or slight pitch variations.

Both are correct fingerings (in fact, there are 3 "correct" fingerings for Bb). Acoustically, the thumb Bb, or the fingering employing the Bb shake may be slightly better, but almost never does using the first finger RH make a noticeable difference, so this factoid really has no bearing on your playing. As ekdavies pointed out, in certain situations (for example, playing a B Major scale, or jumping from Bb2-F#3), the thumb Bb becomes awkward because you need to roll your thumb to change from Bb to B. In this case, using the 1+1 fingering (with the right index finger on the F key), or fingering B, but adding the Bb shake (that curvy little key that extends over the F# key and kind of looks like a third trill) is the best option. However, other circumstances call for the thumb Bb for ease of technique. For example, it may be easier to play a Bb Major scale using the thumb Bb. You need to become familiar with all three options, and figure out when each should be used. Personally, my default choice is the thumb Bb, unless the situation calls for something else, but you may find you're more comfortable with one of the others.

There's definitely something to be gained from using 1+1, but the same holds true for the Thumb Bb, or the Bb shake....They all have different uses and are necessary for different situations. I choose to use the thumb Bb as my default fingering for Bb, but by no means does that mean it's the only fingering I use. I was brought up using the 1+1 fingering, but once you're able to do so, why not simplify your life by using the thumb Bb when appropriate? Really, as long as you can get the results that you want, it doesn't matter what you're doing. As I said above, become comfortable using all three "normal" Bb fingerings, and then decide what is most appropriate for a given situation.

Well my teacher wants me to use the B flat key so I guess I'll use it but I have gotton used to the 1+1.

bottom line, learn all those fingerings...I use the thumb Bb whenever I can but learn all of them...

good point.

My teacher told me that thumb Bb is fine but only for 1st and second octaves. On the third octave, tuning may be affected for the highest notes.

I found one problem using it, when the piece has B's and Bb's. When a B comes, it can be hard to take the thumb off quickly enough, so it must be taken into account in advance.

All that really matters, though, is whether you can play the note. Keep your eye out for a teacher the next time you need a better one that is a bit more of a pragmatist and less "by the numbers". Usually these tend to also be good jazz or improv players, which is where you want to end up eventually, IMO, because they teach you things that you can't get out of classical methods.