rsl12 I wonder why your teacher doesn't want you to use a metronome. If anything needs a strict beat, it's ragtime.
Although one hand is supposed to maintain a strict beat, I thought the syncopation in the other hand really isn't supposed to be steady for ragtime. (I've always been able to do the syncopation for The Entertainer, but it's only in the past few years that I started getting the barest feel for the swing+syncopation that I think ragtime calls for.) I thought it's that particular need for (usually) the right hand to range semi-freely within a measure which calls for no metronome, since that's harder to pick up for most people than following a steady beat.
At least, that's my hypothesis; if anyone has a different one, I'm happy to entertain other ideas. (I've always known that there was something too rigid about the way I played ragtime, so I rarely tried. But a few years ago I was trying a simple piece that my son was working on, and all of a sudden my right hand sort of unhinged itself for a few measures, and it finally felt very, very right. 😃 )