“Oh, did you drop your _______?”

What fun it is for the child to drop the same item again and again, exploring the depths of her caregiver’s patience.  Exploring the concepts of “where?” and “how far?” are a very young child’s entrance into the world of spatial sense.

Children develop spatial thinking the skills associated with imagining objects in different positions as well as their movements – over a long period of time and are  necessary building blocks in constructing logico-mathematical knowledge.

How do we support these emerging skills?  As with all other mathematical concepts, the introduction of vocabulary is one sure way to begin and reinforce the ideas.  When you present a puzzle for a young child to solve, be sure to support his attempts with spatial language such as; “try it upside-down,” “turn it over,” “move the piece up or down,” etc. You can also support these emerging concepts by playing games, presenting challenges, using math manipulatives, and in dramatic play.

Try hiding an important toy in the classroom and then give spatial clues so the children can search for it.  Hide an object in the sand table and create a simple map that leads the children to the treasure.  Play “hotter and colder” as children try to locate an item.  Later, tangrams and more sophisticated puzzles will challenge the children’s thinking and support their growing spatial sense.

Some of the dollhouses seemed far too delicate for play. Some were very complicated. They all had intricate details that revealed a slice of life from a certain place and time so carefully designed that you can almost see teeny families living inside of them.

How does “housekeeping” or “dollhouse” play connect to math?  As young children explore family life, they consider the relationships that exist between people and between the roles they play.  Imagine the young child putting the baby in his crib.  She is thinking about the babies she knows in her life.  They are small and they are young.  Their growing understanding is that babies are smaller than some other people and younger than some other people.  It is the relationship between the baby’s age and size as well as his role relative to others in his family.  Once the young child begins making these observations and putting this information together, she is constructing logico-mathematical knowledge.

Arranging the furniture and sorting it into the appropriate rooms is another way the young child develops mathematical competence while playing with dollhouses.  The places and spaces she encounters will challenge her social constructs about how these things compare to her own home and family.  She will have to develop new schemas for new information as her current schemas are challenged.  Even if your classroom dollhouse doesn’t look exactly like the homes your children live in, they provide a framework for play that encourages creative thinking.

“In” and “out” are almost as simple and can be used every day when you go outside or when it is time to come back inside.

Even though the sign for “outside” is different from “out” keeping your signs simple will help encourage even the very youngest children to use them.

Directionality is an early math concept that will later incorporate all other geometric concepts.  Remember that these ideas are found within the context of a “relationship” – Up is in relation to Down and In is in relation to Out.  This means that  these understandings are logico-mathematical.  It may seem simple, but children build the foundations of understandings from the simple to the profound.

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.

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.