Four Types Of Mathmatical Knowledge......

     Magdalene Lampert points out in: "Knowing, Doing, and Teaching Multiplication," Cognition and Instruction ;
the following:

    There are four types of mathematical knowledge: intuitive, concrete, computational, and principled conceptual knowledge. Intuitive knowledge represents an understanding that is derived from specific contexts and relates only to those contexts. Computational knowledge enables one to perform activities with numerical symbols according to previously determined and generalizable rules. This is the most common form of mathematical knowledge presented in schools. Concrete knowledge involves knowing how to manipulate concrete objects or representations of them to solve a problem. Finally, principled conceptual knowledge represents the understanding of abstract principles and concepts that govern and define mathematical thinking and procedures. Some cognitive scientists suggest that real mathematical understanding is a combination of these different ways of knowing mathematics.