There are at least three kinds of mathematical knowledge:

1. Conceptual knowledge
2. Procedural knowledge
3. Principled knowledge

Conceptual knowledge pertains to knowledge that is rooted in a firm understanding of principles. In mathematics, this refers to the “why” aspect.

Procedural knowledge pertains to knowledge that is driven by the use of procedures and steps. In mathematics, this refers to the “how” aspect.

Principled knowledge, a term borrowed from Lampert (1986), pertains to knowledge that is both conceptual and procedural. In mathematics, this refers to being able to perform the operations and to explain the validity of the operations.

With students with learning disabilities in math, special education teachers must provide both principle-based and procedural information, that is, they should explain why a method works in principle and how it is done.

Procedure + Principle strategy works better than Procedures-only strategy

Example 1: In using algebra tiles:

Good thing to do:
We’re going to add this two polynomials, x2 and 3x2. We could do this because both are similar terms (use manipulatives to show a piece of a red square and three pieces of another red square). So in all I have 4 red squares and this means 4x2.

Not so good:
We’re supposed to add x2 and 3x2. Copy x2 and just add 1 and 3. You get 4x2.

Example 2: Suppose students with learning disabilities in math have been asked to determine the area of a square whose side is 5 cm.

Good thing to do:
In finding the area of this square, we know that a square is a rectangle with equal sides. So if you remember the formula for the area of a rectangle, A = length x width, then the area of a square is A = side x side. Hence, the area of this square must be A = 5 x 5 which equals 25 square cm. Don’t forget to include the correct unit of measure. Since we’re dealing with an area of a figure that has both length and width, the measure has to be in square units.

Not so good:
We’re supposed to get the area of this square. So A = s2. Just plug in the numbers and you’re all set. So A = 52 which equals 25.