When you hear or see the word “math,” what do you think of? Your high school algebra class? Balancing your checkbook? A geeky engineer with pocket protectors? When you add “early childhood” to “math,” what do you think of then? A little one learning to say, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10”? A bright poster with a circle, triangle and rectangle neatly labeled? All of these are common ideas about what math is and how math starts, but none of them are what I mean when I say “foundational math.” Before I tell you what I do mean, I want you to try something.

Look at this image:
Consider this question:

Which of the figures are the same?

Often when I ask this, a person says, “They are all different from each other.” Another says, “They are all the same; they are all shapes.” Both of these answers make sense, but I often ask people to keep looking to see if anyone can come up with another answer. Usually, people then generate these six answers:

• top two shapes are both orange
• bottom two shapes are both green
• left two shapes are both striped
• right two shapes are both solid
• top left and bottom right are both circles
• top right and bottom left are both triangles

In fact, although none of the two shapes are identical to each other, any two of them are “the same” in some way. Figuring this out involves logical thinking about the attributes of the shapes.

This shape activity demonstrates one definition of mathematics – a logical way of thinking that allows for increasing precision. We can use math to make sense of the world. We can use math to solve problems. To use math in these ways, though, we cannot just memorize facts. We must build our own understanding, so that we can think flexibly in different situations. Without a strong foundation, a tall building would not stand for long. Likewise, without a strong foundation in mathematical concepts, children can struggle to understand the more complex mathematical thinking they need later in life.

At the Early Math Collaborative, we have developed a set of 26 “Big Ideas” – key mathematical concepts that lay the foundation for life-long mathematical learning and thinking. While these concepts can be explored at any early age, they are powerful enough that children can and should engage with them for years to come. As you engaged in the shape activity earlier, you were using two of the Big Ideas:

• Attributes can be used to sort collections into sets.
• The same collection can be sorted in different ways.

Most likely, you were not thinking about these ideas consciously; rather, you were looking at the shapes and thinking about them. You were using math to make sense of the puzzle I posed and to come up with a solution. This type of math may not match your prior notion of math as quickly-recalled facts and properly executed procedures. You may need to set aside some of those notions in order to develop a deep understanding of foundational math that will help you have fun doing math with children.

The bucket balance asks children to figure out which side is heavier, which side is lighter, and to consider notions of “more and less.”  But perhaps more interesting to the egocentric child, is the way the bucket balance can be used to find the “same” weight.

Why is making the balance even more interesting than the exploration of more and less?  Because the young child has an innate interest in issues of fairness and equality.  They want to know how power differentials come to be and why some people seem to fit while others don’t.  They have a vested interest in why their older siblings get to stay up later than they do and why some kids seem to always push to the front of the line.  These are issues of social parity and push them to explore the concept that “fair doesn’t always mean equal.” Making things exactly the same is the ultimate test and one that is difficult  to accomplish even for adults.

As children put rocks in one of the buckets in order to get it to balance, they see how difficult it is to make it exactly even.  This is especially true when using nonstandard units such as rocks.  They work very hard testing and retesting the sides, lining up their faces with the table looking for the smallest discrepancies in the weight distribution.

Often they have to trade one heavy rock for two smaller rocks.  This challenge continues until they are satisfied that it is even and balanced.  Try this in your own classroom with rocks or seashells or twigs and see what happens.

Even very young children will recognize that there are 3 animals in each story – same. They will also notice that both tales have a wolf – same.   From there, the similarities get murkier while the differences become more obvious.  The 3 Pigs are brothers but the 3 Bears are a Mommy, Daddy, and Child bear – different.  There is a little girl in the bear story named Goldilocks but there are no people in the pig story – different. The Wolf is a bad guy in both stories – same, but he experiences very different fates (depending on the version you are telling!)

Even within each of these stories there are opportunities to compare and contrast.  As Goldilocks encounters different areas of the Bear’s house she notices that all of the items are the same (beds, chairs, porridge) but they are also different (firmness, size, temperature).  The Pigs all build houses but the houses are very different.

All of these similarities and differences can be described with and by the children especially if you have visual representations of the characters.  A felt board with all of the characters would work great.  Create a line down the middle of the board so the children can manipulate the characters depending on the questions you are asking.  Once you have explored both stories with the children, leave the felt board and associated pieces out for the children to explore on their own.  It is through access to the materials that they can practice telling and retelling the stories to their hearts’ content.