@mightymquinn,
Here's a stab in the dark but there was a 1953 movie called
The 5,000 Fingers of Dr. T. This really fits your description...about a music teacher and kids practicing piano.
http://www.imdb.com/title/tt0045464/combined
Peter Lind Hayes (future TV star) was the music instructor. Also in this movie was actor Hans Conried (Uncle Tanoose from Danny Thomas' 'Make Room for Daddy').
The kids wore beanies...etc...and, as I recall, pullover sweaters, I recall. The boy, actor Tommy Rettig, who went out to play Lassie's master, was the young boy in this movie.
"Of course this is not really a movie for children. The structure of the movie is similar to "Wizard of Oz", beginning with a child which seems to be frustrated with the real life and then dreaming into another world, full of magicians, witches, fabulous adventures ... "
"A small boy plots to upset the grand performance by THE 5,000 FINGERS OF DR. T. to be held in the sinister Terwilliker Institute.
The whimsical world of Dr. Seuss first saw expression in a Hollywood feature film in this fast-paced fantasy which examines a child's musical nightmare. Although it was a financial & critical disappointment when initially released, it has established itself comfortably as a nostalgic favorite for Baby Boomers who first discovered it decades ago.
"Dr. Seuss (Theodor Seuss Geisel, 1904-1991) wrote the original story, co-authored the script and penned the lyrics in his own inimitable style. The action plays itself out over vast, curvaceous sets which will immediately seem familiar to readers of his books, while the brightly colored costumes make the players look like characters from the good Doctor's stories come to life."