Rather than spending your time reinforcing correct answers and correcting wrong answers, time is better spent allowing  children to exchange ideas with their peers so they can discover and uncover number concepts on their own.  If children become accustomed to adults as their only source of feedback, they don’t learn to trust their own instincts and argue or defend their positions.

Think about a time when a child looked up at you after completing a task (let’s say, distributing a classroom newsletter into each of the cubbies) to find out if she is right.  Imagine now, that you don’t tell her that she is or she isn’t, but encourage her to ask her friends to help her.  That means that rather than “fixing and responding” you “wait and see” how the children figure out how to solve the problem.  The friends might come over and examine each of the cubbies to look for the newsletter.  They might find an empty cubbie or a cubbie with multiple letters.  Those children will have to figure out how to explain why the task wasn’t completed accurately and then help correct it.  This interaction requires social negotiation by both parties as well as a pooling or skills to fix it.

It is the conflict between the children that creates the space for negotiation.  It is the negotiation that requires a deep and meaningful examination of each child’s own number concepts in contrast to their peer’s.  This internal chaos demands the child to examine her own belief’s or understandings and then make the necessary adjustments to construct new understandings.

Kamii explains that children who only look to adults to reinforce their ideas only find approval and disapproval.  Rather than encouraging autonomy in children, this sustains heteronomy and children continue to mistrust  their own abilities to solve mathematical problems.

This portion of Chapter 3 concludes with a discussion of group games as a wonderful vehicle for an exchange of ideas amongst peers.  Games provide a format for children to check each other’s math – “You moved 4 squares, not 3”-and children are then required to go back and investigate the mathematical question.

When autonomy becomes the aim of education, educators will attempt to increase the areas of overlap between the area of overlap between the two circles.

In Chapter 2 of Number in Preschool & Kindergarten, Kamii  lays out one of the most basic principles of Piagetian theory.  According to Piaget, the goal of education must be to develop the child’s autonomy, the ability to self-govern.  Because schools rely heavily on the use of rewards and punishments (grades, stickers, teacher approval, time-outs, etc.) children spend the vast majority of their time in these structured learning environments governed by others – heteronomy.

She goes on to elaborate that it should not be the goal of education to simply create obedient children, as obedience is about a fear of consequences rather than an internal gauge of right and wrong.  When we apply this to learning, children who learn in order to please others or because they are afraid of the negative consequences of not doing what they are told, most often, cannot think for themselves.

What do you think of that?

Next week, I will finish exploring Chapter 2 “Objectives for Teaching Number” where Kamii discusses the difference between the “construction of number” and the “quantification of number”.  I hope you are looking forward to it as much as I am.