“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.

When I was little, my big sister set up the basement so we had to follow an obstacle course that kept us moving over and around the furniture.  The big motivator to keep us on course was that she told us there were “crocodiles” in the carpet and if we touched it we would get eaten.  Big sister fear is a REAL thing.

I like the idea of setting up each part of the course with open-ended options for the children.  Since development is bumpy, some children may want to jump on two feet, while others are working on hopping on one.  You can place paper plates on the rug with numbers on them and have the children move from 1 to 2 to 3, etc.  After that, they must move to a table and crawl under it (use an arrow to indicate where to go) and then over to a block balance beam.  Once they’ve completed the balance beam obstacle, they can go over to a table where they have to put 4 Legos together before they move on.

These are simple ideas that may encourage children to try new things, follow simple directions and then make personal choices about how to complete the tasks.  Try and develop a few obstacles that encourage mathematical thinking and spatial awareness.

This is another way to keep everyone moving during these very cold winter days.

Each time they tossed the dice, the teach tried diligently to get the child to recognize and name the numbers.  She used her best teacher voice saying things like, “Ooooh, what number is that?  Can you count to 4?  Are the number the same?  Are they different?”  Not once did the child respond to her prompts.  He didn’t even look up or acknowledge her attempts at engaging him in the numbers.

I watched this interaction for a little while longer and came to the following conclusion:

The teacher was wasting her time and energy focusing on what she thought was important when using dice as a manipulative rather than observing the child and following his lead to uncover what he thought was important about the dice. 

If the teacher had been paying closer attention to the child’s cues, she would have noticed that he wasn’t rolling the dice to see which numbers came up.  He was rolling the dice to see how close he could get them to travel without falling off the edge of the table.  His game was about space and distance, about near and far, about staying on and not falling off.  His engagement with the dice was intense and exciting.  His game had rules and was fun.  His game didn’t include a relentless string of questions with right and wrong answers.

If the teacher had been paying closer attention she would have seen an opportunity for an authentic interaction between the two of them based on the genuine interests of the child.  She would have found many opportunities to encourage mathematical thinking, spatial recognition and math vocabulary.

Math opportunities exist in all areas of the early childhood classroom and in all sorts of adult-child interactions.  Playing with dice may seem like a super obvious example of an “opportunity” to explore number with a young child.  However, the obvious is not always the best choice.  Follow the child’s lead.  It may take you somewhere far more interesting.

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.