Sometime before her fourth birthday, Leah was given five train engines. At home one day, she walked into the kitchen with three of them. Her father asked, "Where are the other ones?"

"I lost them," she admitted.

"How many are you missing?" he asked.

"I have one, two, three. So," she said, pointing in the air, "foooour, fiiiive ...two are missing: four and five. No! I want these to be one, three, and five," she exclaimed, pointing at the three engines. "So, two and four are missing. Still two missing, but they're numbers two and four."

In this scenario, Leah was thinking abstractly about numbers. Her abstractions allowed her to play with numbers in a similar way to the way in which she played with other things, such as blocks and toys.

Learning Through Abstract Thinking

Many people define appropriate experiences for children as "concrete and hands-on" and contrast them with abstract paper-and-pencil activities. But it's important to remember that all significant learning involves abstract thinking. We want children to be able to make generalizations from concrete experiences. For example, in order to be able to identify the color red, children have to abstract the idea from valentines, stop signs, and many other objects. The concept of something like kindness involves an even more abstract understanding.

Children abstract ideas from working with concrete objects. You can point to a chair and cup to indicate the meaning of each. While you can point to three chairs and say "three," and you can point to the numeral three and say "three," children have to abstract the idea of the number three by generalizing from many experiences.

Learning About Numbers

Let's take a look at children's development of number concepts. From birth to first grade, children develop increasingly abstract and flexible ideas about numbers and counting.

Almost from the day they are born, babies are sensitive to quantities. Generally, by 8 to 12 months of age, they can tell which of two very small collections is greater than the other. While babies are probably estimating about numbers, they are beginning the long process of learning rich and complex ideas about numbers and counting.

A significant development is made at about age 2, when children begin to engage in symbolic play. At this age, they can represent numbers with mental pictures of objects, which allows them to represent the numbers exactly. When a toddler puts out two plates, then gets two spoons and places one on each plate, he is showing the early ability to think abstractly.

Number words are important, too. At the earliest ages, children may not realize that number is an important attribute. The words help them to recognize that you can classify collections by number. They bring numbers to conscious awareness. For example, a girl is sitting in her yard with her dog when another dog wanders over. The girl says, "Two doggy!" She then asks her mother to give her two treats and gives one to each. She has made an important abstraction.

Soon, children build on these early ideas by developing their counting abilities. They string some of these words together and begin to learn to count. Never let anyone belittle this process by saying things like "that's just counting"! During the preschool years, children must learn to abstract several rules, or principles, in order to count. They include:

1. The stable-order rule. Counting words must be said only once, and in a consistent order. A child counts, "one, two, three, four, five, six, eight, seven..." each time he counts. he isn't completely correct, but he is consistent.

2. The one-to-one rule. Each counting word must be paired with one, and only one, object. Many 4-year-olds will make such mistakes as skipping an object, but will catch similar mistakes when others make them.

3. The cardinal rule. The last counting word indicates "how many" of the collection. If you ask a child who is just learning to count how many items she just counted, she may recount! But with counting practice, children learn to abstract this rule, and they find that the last number word is not an attribute of the last object counted, but an attribute of the entire collection as a whole.

4.The order-irrelevance rule. Objects can be counted in any order. As in the train-engine example, a child can label objects with different numbers and the count will remain the same.

5. The abstraction rule. Any kind of object can be collected and counted. Children can count jumps, the number of dog barks, or the missing eggs from an egg carton. As the name of the rule indicates, counting is an abstract, principled activity!

Even though young children find it hard to follow the rules, their behaviors indicate a developing understanding of them. The growing sense of number words and counting allows children to build abstract number comparisons as well. For example, after the age of 3 ½, most children can accurately compare the amounts in two collections of dissimilar objects, such as a collection of blocks and a collection of chips. They can also accurately compare collections that they don't see at the same time, such as the number of marbles in a circle with the number of drumbeats in a sequence. Between 4 and 4 ½ years of age, children can compare collections that are each made up of a mixture of different objects. This shows that they see numeration as an increasingly abstract idea that doesn't depend on the size or nature of the objects counted.

Another way children develop abstract ideas is through written symbols. Preschool children understand that written marks on paper can preserve and communicate information about quantity. For example, 3- and 4-year-olds can make informal marks on paper, such as tally marks or diagrams, to show how many are in a collection. The development of this idea is gradual, but children build on the concept as they learn more complex math skills.

Learning About Shapes

Children can learn about shapes more deeply than we often realize.

At first, children learn about shapes as "wholes"; for example, recognizing that something is a rectangle because "it looks like a door." But even this level is an abstraction. When children can separate the shape from the background, consider it, and distinguish it from other things and shapes, they have abstracted that shape.

Later, they can recognize a shape, such as a triangle, in different sizes and orientations. Indeed, they will even find that there is still more variety with certain shapes. For example, a shape can be "long and skinny" and still be a triangle. Color, thickness, and other attributes are now seen as irrelevant to the idea. They have abstracted the idea of the shape. Simultaneously, children begin another important abstraction. They come to mentally "pull out" the individual parts of shapes. For example, they begin to see a triangle not just as a shape that looks a certain way, but as one that has three sides and three corners. This gives young children a feeling of power. One girl proclaimed, "It's very pointy and very long, but I know it's a triangle. Look: it has one, two, three straight sides! "

Learning About Maps

Highly abstract and symbolic, map skills may seem developmentally inappropriate for young children. However, even 3-year-olds can build a simple but meaningful map with landscape toys, such as houses, cars, and trees. At first, they lay out a path of landmarks, and only later do they understand the relative position of, and distance between, such landmarks.

Older preschoolers can navigate and learn from maps. For example, they can learn a route through a playhouse more quickly if they see a map of it first. They have much to learn about maps, but when they can understand the layout of a real room by looking at a map or infer the location of a hidden toy in a room from a model, they are beginning to understand the sophisticated abstraction inherent in map skills.

All learning involves abstraction, not only in mathematics. Toward the end of hearing the book The Giving Tree, 4-year-old Rose said, "This book is about selfish." She had abstracted a conceptual theme from the story. Young children can and should abstract concepts from their world. When rooted in meaningful activity, such abstractions develop integrated concrete knowledge.

From the first year of life, the ability to conceptualize is necessary for understanding. As children grow, their ideas become increasingly abstract, flexible, and sophisticated.

View and download the Developmental Chart Development of Math Concepts Age by Age (PDF)