Before describing the situations that teachers can use to teach number, Kamii assumes that teachers are already optimizing children’s experiences by finding ways to put all sorts of things into relationships so that the focus of the math teaching is on the children’s mathematical thinking rather than on the quantification of objects.  She reasserts that this should be the center of the math curriculum.

Opportunities to explore and teach number are bountiful in the daily life of an early childhood classroom.  Kamii describes 6 distinct aspects of the day that can serve as opportunities to teach number organically.

1.  The Distribution of Materials

2.  The Division of Objects

3.  The Collection of Things

4.  Keeping Records

5.  Clean up

6.  Voting

Do you see how these 6 categories are connected to teaching math? How do you use the above to teach number in your program?

Over the next few weeks, I hope to look deeply at each of these topics so we can discover more and better ways to teach number.

Encourage the child to make sets with moveable objects.

Hmmmm.  Here Kamii says that focusing on one set of objects limits our ability to explore number with children.  If you present a group of objects to the child, i.e., a small basket of apples, and then ask, “How many apples are there?” There is one right answer and the child knows that.  Asking a child to count objects is not a good way to help them quantify objects.

Children are better served if we find ways for them to observe sets and make judgements about them.  For example, if you put 6 apples on the table and 3 oranges, and then ask the child to make observations about the 2 sets of objects, the child is presented with more options for thinking about number.  The child determines which has more, which has less, or if they are the same.

However, the above example is still not as useful as having the child make his own sets.  Ask the child to bring enough fruit so that every child in the group gets one piece.  The child then has to start at zero and use one-to-one correspondence to provide 1 fruit for every place setting.  The child has to decided when to stop, when he needs more, or if there is not enough.

Again, the teacher needs to understand the that there is a vast difference between a child putting one piece of fruit on every plate and the child knowing how many pieces of fruit are needed in relation to every plate.  It is the child’s mental construct of that relationship that supports the ability to quantify.

She goes on to criticize materials that are already grouped in sets that cannot be taken apart.  There are manipulatives that come in predetermined sizes representing 1, 5, 10, 100, etc.  However, the child only sees each of these objects as “1” even though the adult sees that the 5 unit is actually 5 ones put together.  The 10 unit is 10 ones put together.  However, if the child cannot manipulate those units on his own, they can only represent a unit of 1, regardless of their lengths.

Children can take these units rods and put them in order from shortest (unit of 1) to longest (unit of 10).  However, according to Piaget (as cited by Kamii, p. 39) children are using their spatial knowledge of creating a “stair step shape” to put them in order rather than their understandings of number.  This is observable.  They line them up and move them around until they form the shape you see above.  The logic-mathematical construct that the 2 Unit is made up of 2 individual 1 Units and the 5 Unit is made up of 5 individual 1 Units, etc., is an internal, and therefore non observable mental activity.

The way to know if this is true is to present the young child with a set like the one above and have him put 10 unit rods in order.  Then present the child with a box if individual Unifix cubes (all of which are 1 Units) and see if he can make 10 rods with lengths that differ by one cube each.  Not as easy as it looks.

Encourage children to quantify objects logically and to compare sets (rather than encouraging him to count).

When encouraging children to quantify rather than count, Kamii describes a typical interaction that takes place every day in every child care center around the country.  The teacher asks a child to place cups on the table for snack time.  She asks, “Will you bring 8 cups?” Kamii says it is better to say, “Will you bring just enough cups for everybody?” so that they child herself, can determine how to best accomplish the request.  Rather than telling her exactly “how many” cups are needed, the teacher can provide a more substantial opportunity for the quantification of objects by encouraging the child to figure how many are needed, and then to bring just enough so that everyone has one.

Each child will approach this dilemma with their own ideas about how to solve it.  For instance, some children will bring far more cups than they need so they are sure to have enough.  This group will learn that they then have “leftovers” and those must be returned to the cabinet.  Other children may make trips, one by one, back and forth to the cabinet, only carrying one cup at a time.  This, they will find, is not very efficient, especially if there are many children in the group.  Both attempts have their problems which create additional opportunities for the children to quantify, compare, and estimate.  Compare that with the first example when the teacher tell the child how many are needed.  This allows the child to count out the correct number of cups, but nothing else is explored.

Watch how this child tries to count out the correct number of cups for snack time.  He counts them one-by-one but then puts them underneath the pile so there is no beginning or end to the set.  He finally just takes some and begins to set the table.

Kamii goes on to explain that counting is important and is of of the foundational skills required for addition and subtraction.

Take a look at a short video that describes how we understand number sense and why children need to develop strong early skills so they stay strong in math later on.

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.

Constance Kamii was born in Switzerland and studied under Jean Piaget himself for many years.  She is currently a professor at the University of Alabama where she continues her work as an early childhood researcher and professor.  She continues to focus her energies on children, math and Piaget.

Over the next several Tuesdays, I am going to introduce this book to you hopefully to spark some discussion about this very specific approach to teaching children “number”.