A Distinction Between Conceptual Knowledge and Procedural Knowledge

Chances are, when you learned elementary mathematics, you learned to perform mathematical procedures. Known to mathematicians as algorithms, these procedures enabled you to find answers to problems according to set rules. If, for example, you think of division in terms of "divide, multiply, subtract, bring down" then you learned a division procedure (or algorithm). For another example, if you think only in terms of cross multiplying as a way of approaching problems involving proportions, chances are you learned only a procedure for solving mathematical proportions. At this point you may be wondering, "What else is there? What else would a person learn in a mathematics class?" The answer is, there is a great deal more to mathematics! These mathematical procedures are much like recipes that efficiency experts have developed to enable people to go straight to specific kinds of answers when confronted with particular kinds of well-defined problems. If we thought cooking was nothing more than following recipes developed by experts, we would be missing out on much of the joy of cooking! If there were no room in cooking for serendipitous combining of favorite ingredients, then cooking would be boring indeed.

As important as it is to be able to follow a recipe to quickly and efficiently obtain a certain kind of answer to a certain kind of problem, this is not the essence of mathematics. A true mathematician understands that an operation like division is a mental operation that may be performed by using any number of different procedures. A division problem may be solved by repeated subtraction, by repeated addition, by use of a numberline, or even by using objects and modeling the action of division. The procedure using "divide, multiply, subtract, bring down" is only one of many possible procedures. The underlying idea of division as an action is something quite different from anyone of the procedures for solving division problems. This underlying idea is referred to as the concept of division. The concept of division and the procedure of solving division problems are not the same thing. In today's mathematics classrooms we are teaching concepts first and foremost. Procedures are learned too, but not without a conceptual understanding. One of the benefits to emphasizing conceptual understanding is that a person is less likely to forget concepts than procedures. If conceptual understanding is gained, then a person can reconstruct a procedure that may have been forgotten. On the other hand, if procedural knowledge is the limit of a person's learning, there is no way to reconstruct a forgotten procedure. Conceptual understanding in mathematics, along with procedural skill, is much more powerful than procedural skill alone.