Traditional teaching approaches are generally teacher-directed, where students are taught in a manner that is conducive to sitting and listening. It is often argued that these approaches may not provide students with valuable learning skills, and rather that non-traditional approaches to teaching and learning might better endow students with such skills. The teaching of mathematics that is usually referred to as "non-traditional" uses constructivist philosophies as its bases; by implicating strategies in which the individual develops, constructs or rediscovers knowledge in an attempt to make sense of his or her universe, or by employing social constructivist ideals which suggest that group work, language and discourse are vital for learning. Also, it is often argued that non-traditional teaching is done using a problem solving approach, where the learner is the problem solver. Non-traditional teaching and learning (NTTL) in mathematics and engineering needs to be well understood so that appropriate and meaningful comparisons with traditional techniques can be made. The computer based teaching technology known as "e-learning" is now often used in tertiary level mathematics and engineering teaching. The e-learning methodology is considered to be more in-line with the non-traditional than the traditional teaching approaches. This paper critically reviews the literature that makes comparisons of the traditional and non-traditional teaching approaches in mathematics and engineering education, and specifically examines the advantages and disadvantages of these approaches along with the manner in which they influence the academic performance of students in mathematics and engineering courses.