The third kind of knowledge is logico-mathematical knowledge – this is knowledge that is constructed within the mind of the learner.  It is based on the foundation of physical knowledge.  If you have a blue ball and a red ball (the color of the balls is observable and is therefore an example of physical knowledge) but that there is a difference in the color of the balls is logico-mathematical.  It is the relationship between the objects that needs to be constructed.  Understanding and knowing that both balls bounce is physical knowledge but comparing the heights of the bounces is logico-mathematical.

Piaget argues that knowing number is not an inherent trait but something that is constructed within the minds of human beings because number is a construct of relationships.

In Chapter 1 – The Nature of Number, Kamii explores how children learn number through expansive descriptions of Piagetian Conservation Tasks.  Young children cannot conserve number and quantity until they are nearly through the early childhood years.  It is Kamii’s contention that we don’t “teach” conservation because children develop conservation through their own constructive of logico-mathematical knowledge.

Take a look at this video below to see a typical child performing a conservation task.  See how quantity and the relationship between the objects needs to be internalized.

Next week we will look at Chapter 2 to see how Kamii sees the teacher’s role in teaching number.