Introduction

ideas through problem solving, communication, the active construction of mathematicalrepresentations, and metacognition. According to Hiebert and Carpenter (1992),conceptual knowledge is knowledge that is understood. Procedural knowledge isknowledge of the rules and symbols of mathematics (e.g., the algorithm used tosolve a computation question). Conceptual understanding and procedural knowledgeare complementary. Conceptual understanding helps students with long-term understandings;procedural knowledge helps students to connect conceptual understandingwith symbolic language.Students who are assigned activities that emphasize only the rote acquisition ofprocedures (procedural knowledge) without promoting an active understanding ofthe concepts underlying such procedures are at a disadvantage, especially in thelater grades, when they encounter more abstract concepts. If mathematics becomesnothing but procedures, students attain only a superficial understanding – one that,over time, may disappear completely.The surface knowledge that results from the rote learning of procedures is exemplifiedby the student who is asked to solve the following problem: “Some children aretravelling on a school trip. If there are 98 children and each bus holds 30, how manybuses are required?” The student uses the division algorithm to solve the problem butthen is unable to ascertain what the remainder of 8 represents (8 buses? 8 children?)or what the implications of the answer are (does it mean 8 /30 of a bus?). Interestingly,younger students who do not yet know the algorithm can solve the question quiteeasily, using blocks and make-believe buses, and they have little difficulty declaringthat the 8 leftover children will need to get a small bus or have someone drive them.Students remember learning that makes sense to them. If they understand a concept,they can solve problems with or without memory of the related procedure. For instance,students who forget the multiplication fact 6 x 8 but understand the concepts behindmultiplication can easily make a connection with a known fact such as 5 x 8 and thenadd on the additional 8. Students who do not have the conceptual understanding mayflounder with a vaguely remembered and incorrect answer such as 68 and not havethe understanding to recognize that their answer is incorrect.Sometimes the solving of a problem promotes conceptual understanding. Throughthe process of interacting with a problem that involves the combining of two-digitnumbers, for example, students can develop a deeper understanding of place valueand its effect on addition.Principles Underlying Effective Mathematics Instruction 25