# Laying the foundation for learning math

Foundational mathematics is mathematical thinking that develops prior to primary school. It anticipates arithmetic and does not use written numerals, but is clearly mathematical, and can be taught.

At the Early Mathematics Education Project, we like the term “foundational mathematics,” because it:

- suggests its important role in preparing children for elementary school mathematics;
- distinguishes it from elementary school mathematics, making it clear that curriculum for older children must not be “pushed down;” and
- legitimizes this mathematics as content that should be the focus of pre- and in-service early childhood teacher training in mathematics.

## Introducing abstractions

Too often, early math is viewed as basic math. As adults who long ago mastered the fundamental concepts, we underestimate just how abstract early math really is. Take the idea of “three” for example.

Three is a counting word that stands for a quantity. It’s the quality that three boats, three bears, and three bananas all have in common. In order to see “three” you have to see past the objects themselves, and see only how many of them there are. It’s a bit of a mental trick, or abstraction.

The abstractions abound, because numbers are used in a lot of very different ways, which means there is a lot to learn about how numbers work. For example, it can be the “3^{rd} of February” or “3 o’clock.” Do small children understand that the reason they are three is that they have been alive for three years but not yet for four? Most often they don’t.

By preschool, most children know something about three, but they need many opportunities to experience “threeness” before they will develop an adultlike understanding. This basic idea of naming quantity is one of the very early mathematical concepts that undergird children’s later use of symbols and algorithms in elementary math. Only a solid yet flexible conception of what three is—a real abstraction—makes the more complicated idea of “three plus one” meaningful.

## A teacher’s task

An early childhood teacher needs to have deep, interconnected knowledge of mathematical ideas such as cardinal numbers, repeating and growing patterns, ordinality, classification, size and shape, and location and direction. She needs to know how to present them in engaging lessons. At the same time, she must understand how a child’s developing mind encounters, integrates, and sometimes misunderstands these big ideas

Only such a teacher will be ready to do things like:

- explain why we always say the number names in the same order when we count;
- use a child’s (mistaken) idea that a triangle is “upside down” to explore the language of shape and relative position; or
- ignore a “3” written backward in order to maintain group focus on the cardinal meaning of the number.

Our professional development prepares teachers for these challenges.