Check out Engineering Explorers, our newest online resource!

Number is Not Directly Teachable

by Early Math Counts

so says Kamii in the first paragraph of the 3rd chapter of Number in Preschool & Kindergarten.  She argues that the development of logico-mathematical knowledge is learned indirectly – and therefore, taught indirectly.

Kamii lays out her 6 Principles for Teaching Number broken down into three categories. The first principle of teaching number is about creating all kinds of relationships (I discussed this a coupe of weeks ago).

Principle 1 – The creation of all kinds of relationships.

Encourage the child to be alert and to put all kinds of objects, events, and actions into all kinds of relationships.

The teacher’s role is to create the “social and material environment that encourages autonomy and thinking.”  If we agree that children construct their own understandings of the world around them, then they need ample opportunity to do so with ample materials to do so with.  We want children to think for themselves and not simply do what they are told so adults must provide an environment that indirectly encourages this.  As children problem-solve, play, pretend, work, and engage with their peers, they are developing and examining all sorts of relationships in a wholly organic way.

Kamii even explains that conflict creates opportunity for children to put things into relationships.  Notions about fairness and equality are rooted in perceived hierarchical relationships.  As children develop logical thinking as well as morality these relationships adjust accordingly.  Negotiating conflict resolution requires that children consider fluid alternatives to problem-solving.  The less a child is required to simply “obey” adult authority, the more they are able to negotiate human relationships, choices and outcomes.

Kamii describes this through the following vignette:


When two children fight over a toy, for example, the teacher can intervene in ways that promote or hinder children’s thinking.  If she says, “I’ll have to take it away from both of you because you are fighting.”  Alternatively, the teacher can say, “I have an idea.  What if I put it up on the shelf until you decide what to do? When you decide, you tell me, and I’ll take it down for you.”  Children who are thus encouraged to make decisions are encouraged to think.  They may decide that neither should get the toy, in which case the solution would be the same as the one imposed by the teacher.  However, it makes an enormous difference from the standpoint of children’s development of autonomy if they are encouraged to make decisions for themselves.  ….An alternative solution might be for one child to have the toy first and for the other child to have it afterward.  Traditional ‘math concepts’ such as first, second, before-after, and one-to-one correspondence are part of the relationships children create in everyday living, when they are encouraged to think. (p.30)

More often than not I observe teachers responding to the described conflict the way the first teacher did.  The teacher takes it upon herself to solve the conflict which may be the easier of the two choices, and definitely the quicker of the two.

What I tell my students and what I am telling you now, is that every time you solve a problem for a child, you rob him of the opportunity of solving it for himself.  When you think of it in terms of “robbery” it becomes much easier to make the more difficult and time-consuming decision.

Next week we will look at another Principle of Teaching Number.

Leave a Reply

Your email address will not be published. Required fields are marked *