Kamii’s lists several types of games that support early math concepts. These are:

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

My favorite game – completely made up by the fabulous teachers at my former preschool- was called “Steal the Jewels.”

The premise is simple.  Take a whole bunch of shiny strings of beads (the New Orleans Mardi Gras kind work really well) and make a pile of them in one area of your indoor gross motor area, or outside, if it is warm.

The children decide the rules.  They may choose to make it a chasing game, where some children are the Stealers and some children are the Rescuers.  The Stealers run to the jewels and steal one string (or two, if the children choose that) and try to get their jewels to the Stealers’ place.  The Rescuers try to tag the Stealers, and if successful, the Stealers give up their jewels and they are returned to the original spot.

Encourage the children to make rules that prohibit grabbing the jewels from each other, or tagging too hard.  It works best if the Rescuers assign one child to be the Protector of the jewels, and only the Protector can hover around the pile.

At the end of the designated time (5 minutes, 10 minutes), play is stopped and the children  either count the jewels or weigh them.  There are now two piles of jewels: the original jewels and the rescued jewels.  Compare the weights or the number of jewels, and play again. This game is fun, exciting, engaging, and the math possibilities are strong.

What kinds of gross motor games do you play?

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.

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.

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.

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.

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.

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 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.

Kamii describes the difference between the “construction of number” and the “quantification of objects” as these two activities if you will, are vastly different and should be seen as so by educators. The main difference and perhaps the most important is that the quantification of number is partially observable through a close examination of a child’s actions and behaviors.  The construction of number, however, is not observable.  It is an activity that takes place within the child’s mind.

The example provided is that we can observe a young child setting the table in the classroom.  As she attempts to place enough cups around the table for each of her classmates she is quantifying the objects.  The thinking that is going on in her mind is not observable and it can’t be known, but it is there.  That thinking is the construction of number.  Kamii asserts that the child needs to have the “mental structure” of number in order to quantify (count out) the cups,  so it stands to reason, that the inverse must also be true, that the practice of setting out the cups will solidify the mental structure.  Therefore, the quantification of number in the classroom, at home, and in a young child’s life is part of the foundation for the construction of number. Opportunities to do this must be made available to young children regularly.

Kamii goes on to explain that the teachers’ focus should not be that the child quantifies objects correctly but on the thinking process itself.  As the young child constructs mental structures around and about number she will attempt and fail repeatedly, which is necessary to build the solid structure of number.  That structure will underscore the rest of the child’s logico-mathematical abilities.  Therefore the first principle of teaching number should be to encourage the child to put all sorts of “objects, events, and actions into all kinds of relationships.”

This part of the chapter got me thinking about the ways I did this with my own children when they were young.  One thing that really stands out is both of my boy’s engagement in activities where they were putting smaller objects into larger objects.  We had nesting dolls which were a family favorite.  Louie would line them up from smallest to biggest and then put them inside one another.  Inevitably, one would get left out and in order to get it into the right spot, he had to disassemble the entire thing and reline them up.  The relationships he was focusing on were the size of the dolls, as well as the tops as opposed to the bottoms.

How do you encourage logico-mathematical thinking with your children?  Does it happen naturally or is it designed by the adults?  How can you increase  opportunities for children to engage in meaningful play around logico-mathematical knowledge?

There is one more small part of Chapter 2 that I am saving for next week.