An advanced multiple scattering algorithm for the Monte Carlo simulation of electron transport problems is developed. Unlike established multiple scattering algorithms, this new method, called transport condensed history (TCH), is a true transport process - it simulates a transport equation that approximates the exact Boltzmann transport process. In addition to having a larger mean free path and a more isotropic scattering operator than the Boltzmann equation, the approximate transport equation also preserves the zeroth- and first-order angular moments of the exact equation. These features enable TCH to accurately predict electron position as a function of energy (path length) and to move particles across material boundaries and interfaces with acceptable accuracy and efficiency. Numerical results and dose calculations are shown to reveal the advantages of TCH over conventional condensed history schemes.