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.

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.

If a child is presented with a configuration of 8 coins placed in a random pattern on the table and asked to put the same number of coins out as the model, the child will respond in one of the following 4 ways:

The four levels are:

Level 0: Inability to even understand the adult’s request.

Level I: Rough, visual estimation or copy of the spatial configuration.

Level II: Methodical one-to-one correspondence

Level III: Counting

“Counting does not become a perfectly dependable tool for young children until the age of 6,” reports Kamii.  Children need to be able to 1) say the words in the correct sequence, 2) count objects – make one-to-one correspondence between the words and the objects, and 3) final, choose counting as a dependable and desirable tool to come up with the correct answer.  Before these skills are secure, the child may feel more confident in choosing a different method for coming up with the answer.

If the 8 coins look like this;

The child may place coins so they take up the same amount of space as the ones above. The child is not concerned with “number” here, he is concerned that it looks the same.

This makes perfect sense for a child who is tricked by appearances.  It “looks” the same to the child, so it is the same.

The child may also line up her coins so they match the model exactly.  By doing this, she is using one-to-one correspondence rather than counting as a preferred method of answering the question.

                      Here, you can see that the child made sure that for each coin, he placed a corresponding coin.  This method is quite appropriate and will result in the correct answer.  It isn’t the most efficient nor does it use “counting” as a tool to arrive at the answer.

Kamii uses these examples to encourage teachers to allow children to choose their preferred method for arriving at the answers.  The goal is that all children eventually choose counting as the most efficient, practical and mathematically appropriate avenue to accomplish a counting task.

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.

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.