The classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) equation for nucleation and growth transformations, which assumes random nucleation and cessation of growth where grains impinge, was originally derived by accounting for the {open_quote}{open_quote}extended volume{close_quote}{close_quote} that is obtained when overlap of the growing grains is (mathematically) permitted. Motivated by experimental and computational results that call into question the validity of this equation and by recent theoretical concerns about the appropriate content of the extended volume, a derivation of the KJMA equation is presented that does not rely on the concept of the {open_quote}{open_quote}extended volume.{close_quote}{close_quote} {copyright} {ital 1996 The American Physical Society.}