We saw in the previous chapter that people follow rules when they carry out actions. A challenge that has likely confronted teachers from the very beginning of teaching is how to convey the meaning of the rules. Do students really understand what they are doing, or have they rotely memorized a set of procedures?
Although production rules model procedural knowledge, they were not designed for modeling declarative knowledge specifying how we organize concepts. It is not even clear how the production rules, themselves, are organized. Are there simply huge lists in long-term memory consisting of thousands and thousands of rules?
One view of understanding is that it consists of conceptual (declarative) knowledge that underlies the meaningful use of procedures. Hiebert and Lefevre ( 1986) argue that the distinction between conceptual and procedural knowledge is useful for thinking about mathematics learning in order to help us better understand students' failures and successes. Conceptual knowledge, in their formulation, is knowledge that is rich in relationships of many different kinds. In contrast, procedural knowledge consists of rules or procedures that are used to complete tasks. The primary relation in procedural knowledge is "after," which is used to order the procedures. For instance, we perform actions in a specified order when doing multicolumn subtraction. When a column requires subtracting a larger number from a smaller number, it is necessary to borrow from the column to the left as the next action.
The challenge, then, is to find a way to model declarative knowledge that could organize the rules and provide conceptual understanding of what the