Kamii describes the difference between the “construction of number” and the “quantification of objects” as these two activities if you will, are vastly different and should be seen as so by educators. The main difference and perhaps the most important is that the quantification of number is partially observable through a close examination of a child’s actions and behaviors.  The construction of number, however, is not observable.  It is an activity that takes place within the child’s mind.

The example provided is that we can observe a young child setting the table in the classroom.  As she attempts to place enough cups around the table for each of her classmates she is quantifying the objects.  The thinking that is going on in her mind is not observable and it can’t be known, but it is there.  That thinking is the construction of number.  Kamii asserts that the child needs to have the “mental structure” of number in order to quantify (count out) the cups,  so it stands to reason, that the inverse must also be true, that the practice of setting out the cups will solidify the mental structure.  Therefore, the quantification of number in the classroom, at home, and in a young child’s life is part of the foundation for the construction of number. Opportunities to do this must be made available to young children regularly.

Kamii goes on to explain that the teachers’ focus should not be that the child quantifies objects correctly but on the thinking process itself.  As the young child constructs mental structures around and about number she will attempt and fail repeatedly, which is necessary to build the solid structure of number.  That structure will underscore the rest of the child’s logico-mathematical abilities.  Therefore the first principle of teaching number should be to encourage the child to put all sorts of “objects, events, and actions into all kinds of relationships.”

This part of the chapter got me thinking about the ways I did this with my own children when they were young.  One thing that really stands out is both of my boy’s engagement in activities where they were putting smaller objects into larger objects.  We had nesting dolls which were a family favorite.  Louie would line them up from smallest to biggest and then put them inside one another.  Inevitably, one would get left out and in order to get it into the right spot, he had to disassemble the entire thing and reline them up.  The relationships he was focusing on were the size of the dolls, as well as the tops as opposed to the bottoms.

How do you encourage logico-mathematical thinking with your children?  Does it happen naturally or is it designed by the adults?  How can you increase  opportunities for children to engage in meaningful play around logico-mathematical knowledge?

There is one more small part of Chapter 2 that I am saving for next week.