Kamii describes several great examples of group games appropriate for even the youngest children.

• Aiming Games
• Hiding Games
• Races and Chasing Games
• Guessing Games
• Board Games
• Card Games

In the coming weeks, I will explore each of these categories further as I finally wrap up the discussion of this great book.

We usually see programs/teachers/classrooms accomplishing this through “table setting” when children are asked to distribute the cups and napkins, etc.  This is a good example of how the distribution of these items asks that children consider “how many” children there are, “how many” of each item they need, if they have “too much,” or “too little,” and what they might need to do to correct the situation.

There are other opportunities you can seize throughout the day that will give more children a chance to distribute items.  Items that need to “go home” are often put into backpacks or cubbies which is a perfect way children can “distribute.”  Children can give out “classroom mail” and books for independent reading.  Other children can collect the books and distribute them back onto the book shelf.  Ask children to “hand out” items at group time and then mix it up by asking, “Can you bring over enough instruments so that each child gets one?” or “Make sure that everyone gets 2 books so they can choose which one they want to look at.”  This way you allow the child to use her internal mathematical thinking strategies to problem solve.

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 think about number and quantities of objects when these are meaningful to him.

Kamii says that rather than setting time aside each day as “Quantification of Objects Time” teachers should seize opportunities to work with quantities, groups of objects, and comparisons as they arise in the classroom and as they are directly connected to what the children are actively interested in.

We often see young children sitting in circle time discussing the calendar or the number of days in the week, etc. with their teacher.   Nearly every teacher who I visit does this.  I am not directly opposed to this activity as I don’t think it is harmful or dangerous, yet I don’t think it is engaging in a meaningful way for children. This calendar activity was originally designed as a part of a  kindergarten  math curriculum, so when it is done with children who are younger, it is either watered down or it is not really developmentally appropriate.

There may be some 3 and 4 year-olds who care about how many days they have been in school or whether or not this month has 30 or 31 days, but for the most part, this is an activity that adults think children are supposed to care about.  So what do preschool children care about?  They care about themselves.  They are egocentric and primarily concerned about their lives, their worlds, their ideas, and their feelings.  So, we need to find ways to quantify objects that children care about.  They care about how many objects they have (i.e., large blocks, dolls, cars).  They care about their own families (sisters, brothers, pets, grandparents).  They care about equality and justice for themselves (not necessarily their classmates).  Therefore, teachers must find ways to encourage children to quantify objects when they are involved with the above objects or discussing the above concepts.

Above, when I said this principle should be applied to all teaching with young children, I was referring to meaningfulness. As educators of young children, we need to find ways to connect to what children care about rather than what we think they should care about.  That can come later.  For the time being, meeting children where they are at, rather than expecting them to come around to our us, will serve them in much more appropriate ways.

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.

Kamii writes that once the child has the concept of the number seven, s/he can then learn that the number seven can be represented by either symbols or a sign.  Symbols are figurative representations of a number such as tally marks and the sign for seven is the numeral 7.  Recognizing the signs that represent number concepts (numerals) may come before the child is able to represent those same number concepts with symbols. That means you may think that the child has mentally constructed a concept for the number 7 but what she has done instead is memorized the sign for the number 7.  These are two very different things.

Kamii goes on to say that early childhood teacher focus too strongly on the signs for numbers rather than the symbols.  Counting numbers is important and recognizing numbers is good but teachers should spend more time focusing on the mental construction of number concepts through symbolic representations rather than through memorization of word series or numerals.

I have posted this picture before but I am re-posting it because I think it speaks directly to the above argument.  The teacher is not focusing solely on numerals and counting.  She is actively representing number concepts through symbols.  See how she uses pictures to represent each boy present in class and another to represent each girl in class.  When the children make the connection that each symbol represents each child and all of the symbols together represent a quantity, they are developing number concepts that are deeply rooted in logico-mathematical knowledge rather than the social construct of numeral recognition.

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.

We know that children find this effect fascinating, but what mathematical concepts are they learning when they participate in this activity?  Read the article to find out and in the meantime, check out this video of 4000 dominoes.  Amazing.

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.

Check it out.