  Keller's article entitled ``Good-bye teacher...'' [J. Appl. Behav. Anal.
\textbf{1}, 79-89 (1968)] was fundamental for the development and dissemination
of Keller's Personalized System of Instruction (PSI), which was one of the
central issues in the discussions on psychology and education in the 1970s and
1980s, and nowadays has attracted attention in the context of the increasing
use of online education. Belonging to a class of approaches usually named as
mastery learning, PSI and modified PSI courses (Keller-type courses) present
several interesting results, such as final grade distributions where the
majority of students achieve the highest grades. Here, we present a simple
mathematical model underlying Keller-type individualized teaching methods,
describing, in terms of average characteristic parameters, the time evolution
of the distribution of students per unit of content, and that most students
achieve the highest grades at the end of the course. By applying this model to
a real case of an introductory electromagnetism Keller-type course, we obtained
its characteristic parameters with which we showed good agreement between the
predictions and observations. The model presented here results in a simple
formula, which is very accessible for use by a wide audience interested in
planning or investigating Keller-type or other mastering learning methods.
