Learning Mathematics From Examples and by Doing

This article demonstrates the feasibility and effectiveness of teaching several mathematical skills by presenting students with carefully chosen sequences of worked-out examples and problems - without lectures or other direct instruction. Thinking-aloud protocols of 20 students learning factorization by this method are analyzed to determine the kinds and depth of understanding students attained. Students did not simply memorize procedures but were able to recognize when the procedures were applicable and to apply them. Most students were also able to use their understanding of the concept of factorization to help learn the procedures and to check their results. The method of learning from examples has now been tested successfully with a class covering the entire 3-year curriculum in algebra and geometry in a Chinese middle school.