An Approach that Works
We now have ample evidence that traditional, lecture-based forms of mathematics instruction do not work for most students. Passively listening to someone else explain mathematics is a poor way to learn it. To help students learn, they must be actively engaged in making sense of the mathematical ideas.
What does this mean in practice?
Effective active learning is more than keeping students busy in the classroom. Real learning happens when students are involved in making meaning of the new mathematical ideas they encounter, when they are challenged to discover relationships among and uses for the mathematical ideas they know, to prove their own theorems.
IBL is a pedagogical approach with a long pedigree, rooted in the work of Dewey and Piaget. It is a structured approach to learning mathematics, leading students through a succession of ever more challenging problems and proofs that force them to build a coherent understanding of mathematical ideas and concepts. Its goal is nothing less than equipping them to determine when and how particular techniques or approaches might be used in solving an unfamiliar problem. It takes the focus off the instructor and onto the students who spend class time wrestling with problems and proofs they have never seen before, but have been carefully chosen so that they can find their way through to a solution.
A ninety-second introduction to IBL by Michael Starbird, University of Texas Distinguished Teaching Professor in the Department of Mathematics at the University of Texas at Austin: