Placing the thumb on its side across two strings and using it as a note-playing finger (known as thumb position) is one of the defining skills of advanced cello playing. I typically start teaching thumb position when my students are comfortable shifting in first through eighth positions (Serial Shifting, The Shifting Book for Cello, Part Two, and Finger Exercises for the Cello, Book Three). They need to have learned Tenor Clef and should know their three-octave cello scales (Three-Octave Scales for the Cello, Book One).
Depending on the student, I will start with either Thumb Position for the Cello, Book One, or Thumb Position School for the Cello. I usually begin with Thumb Position for the Cello, Book One, which focuses on moving in and out of thumb position and teaches the notes one at a time, providing a simple introduction to the idea of thumb position. Thumb Position School for the Cello, on the other hand, is an in-depth study of the notes and finger spaces in thumb position, with exercises and short pieces in different keys.
Teaching cello thumb position involves a fair amount of specific guidance regarding thumb placement (I like the thumb on its side, across the A and D strings, with the knuckle on the A string and the side of the thumb next to the nail on the D string.) The wrist should remain straight so that the arm can help support the hand. Care should be taken to avoid strain and tension from a high left shoulder or from a bent wrist. Around 5 minutes of thumb position each day is sufficient time to build beginning thumb technique without overdoing it.
After students have studied some thumb position in the method books, I like to have them start studying pieces that use thumb position: Duport's Sonata in G major, Breval's Concerto No. 2, Vandini's Sonata in G, Senaille's Allegro Spiritoso, and Popper's Gavotte.
After some proficiency has been gained, books such as Thumb Position for the Cello, Book Two, and Thumb Position Studies for the Cello, Book One can be helpful in building further thumb technique.
Cassia Harvey is a cellist, a cello teacher, and writes technique for strings.