Conceptual and Procedural Knowledge : The Case of Mathematics

TLDR: The evidence presented in this paper suggests that learning-by-being-told is an inaccurate model of the kind of arithmetic learning that actually occurs in classrooms and that arithmetic is learned by induction: the generalization and integration of examples.

Abstract: : According to a common folk model, students learn arithmetic by understanding the teacher's explanation of it. This folk model suggests that other, more complicated procedural skills are also acquired by being told. The evidence presented herein suggests that learning-by-being-told is an inaccurate model of the kind of arithmetic learning that actually occurs in classrooms. Rather, arithmetic is learned by induction: the generalization and integration of examples. Contents: 1) Schematic vs. teleological knowledge; 2) Three ways that arithmetic could be learned; 3) The conservative evaluation of the induction hypothesis; 4) A liberal evaluation of the induction hypothesis; 5) Learning by analogy; 6) Learning by being told; 7) Summary; 8) Concluding remarks; 9) Appendix.