Figure out how the child is thinking, and intervene according to what seems to be going on in his head.

Rather than focusing on the “error” the teacher should focus on the thinking patterns of the child that led him to the error.  Kamii writes, “Just as there are many ways of getting the wrong answer, there are many ways of getting the right answer” (p. 42). That means that in order to support the child’s emerging logico-mathematical understandings we need to uncover what the s current understandings are that led them to the wrong conclusion.  Correcting the error does not help the child construct knowledge. Focusing on the thinking and providing alternate ways of approaching the problem will lead to new constructions.

This past week I was observing a group of children halving and doubling numbers.  The teachers provided each child with a small dry-erase board and markers so they could illustrate the problem. (This activity was done after a week of using manipulatives to halve and double numbers).

The teacher described the problem like this:

There is a boy (Frank) and a girl (Alice) and they each have an envelope.  The teacher has 10 pictures and wants to divide the pictures evenly between the two children.  How can we figure out how many pictures each child gets?

The teacher taped 2 large envelopes on the wall and showed the children the 10 pictures.  Each of these was taped to the wall as well.  The children then tried to figure out a sensible way to divide the pictures.

One child simply wrote the number 10 on his board with some other depictions of the envelopes and then he stopped.  He wasn’t sure how to proceed. The teacher approached the child with smaller versions of the envelopes and pictures and used them as manipulatives so he could physically divide them.    In this example, the teacher considered the child’s thinking based on her observations of his attempts and provided additional support as needed.

The above drawing is called “the intuitive approach.”  The child thinks globally about the problem and eyeballs it to get the answer.  He may be right, which is a function of “chance,” but he may also be wrong as he is probably just as likely to eyeball it incorrectly as correctly.  Guessing in this way reveals that the child does not have the logico-mathematical skills to approach this problem using number sense, number rules, or one-to-one correspondence.  It is the teacher’s job to determine that the child’s thinking is not yet ready for this difficult problem and then to provide more appropriate questions for him.

In “a spatial approach” the child lines the pictures up in a one-to-one correspondence pattern in order to create 2 sets of pictures.  If these line up equally, the sets can be distributed to Alice and Frank evenly.  Once divided, the child can count the sets to determine “how many” each Frank and Alice received. In the picture below, the child lined up the numbers 1-5 and the numbers 7-10 (you can see that he missed the number ‘6’).  I am not sure but I think that this child counted each numeral and came up with ‘5’ for each set because he counted the ‘1’ and the ‘0’ as separate numerals although together, they represent one number – ’10’.

In the picture below, the child drew her own envelopes and then drew a square in each envelope (representing the pictures).  She continued doing this until she had drawn 10 squares and the 10 pictures were divided equally.  This is called “the logical approach.”  The child’s logic is well-developed as he has developed a “procedure” that will always work to solve this sort of problem.

Through observation, the teacher can determine which approach each child is using to solve the problem and then influence the child’s thinking process rather than giving the answer or solving the problem for the child.

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.