Some of children’s first thinking happens when they begin to distinguish between things that they have seen before and new stimuli (Wiebe et al., 2006). A child becomes habituated to objects in their environment. When this happens, they tend to pay attention to it less and less. To recognize something as familiar means that the child knows that they have previously experienced it. When a new or novel stimulus is introduced, the child will spend more time examining it than the familiar object (E. M. Brannon, Abbott, & Lutz, 2004; E. M. Brannon et al., 2006). The implication of this is that infant’s minds are actively engaged in observing new objects and stimuli in their environment. Every aspect of a new object fascinates them. Once it becomes familiar they spend less time looking at it because their brains are more accustomed to it. Infants are acting on objects not only with their hands, mouths, and eyes, but with their brains.

Research shows us that even very young babies have an abstract understanding of quantity and a more concrete, object-related sense of number. Infants as young as 6 months can tell the difference between large sets of objects based only on number, provided the ratio of difference is great enough (they can discriminate 8 vs. 16 but not 8 vs. 10) (Barth, Mont, Lipton, & Spelke, 2005; Feigenson et al., 2002; Jordan & Brannon, 2006; Xu et al., 2005).

Research has also shown that children as young as 7 months of age are developing an understanding of mathematics (E. M. Brannon, 2002, 2003; Feigenson et al., 2002; McCrink & Wynn, 2004; Wynn, 2000). Consider the following observation in an infant-toddler program.

The teacher shakes a rattle in front of a child and says, “See the rattle?” The infant, sitting in a high chair, reaches for the rattle and grabs hold. She then drops the rattle. When the teacher picks up the rattle and gives it back to the child, she drops it again on the right side of the chair with eyes steadfastly fixed to the floor. The teacher picks up the rattle, saying “Oops! The rattle fell on the floor!” as she returns it to the child. The child then takes the rattle and drops it on the left side of the chair, again watching it fall and staring at it on the floor.

The infant originally drops the rattle to test the precepts of gravity. When the teacher picks up the object and gives it back to the child and she drops it again, the child is testing the constant nature of gravity. When the teacher gives the child the object again and the child drops it again, but this time on the other side of the chair, the child is testing whether gravity works on both sides of the chair. The child is testing theories about her environment. She is learning, and the teacher responds, “Did the rattle fall? Can you do it again? Can you make it fall over here?”

Children are actively and systematically constructing the world around them. They are acting as scientists to construct physical knowledge of object. This is the knowledge of an object that can be observed. Physical properties like shape, texture, and color are examples of physical knowledge. Social knowledge is knowledge acquired through interactions with others. Mathematics incorporates many types of social knowledge, such as “saying the numbers to 10” or the “names of the shapes,” but should not be the primary focus for infants and toddlers.

Most mathematical learning at this age is what Piaget called logico-mathematical knowledge. This is the knowledge of relationships between objects. Piaget felt this knowledge was the basis for logical and mathematical thinking. These mathematical ideas are not a tangible physical property of an object or group of objects or something that needs to be taught by someone else. They are constructed by making relationships between the objects (Piaget, 1952; Kamii, 1985). For example, a rattle and a teddy bear can be put into many different kinds of relationships. One relationship might be that they are “different” because the rattle makes noise and the teddy bear does not. Another relationship might be that they are the “same” because both are toys. “Difference” or “sameness” is not a physical property of either item and one does not have to be taught this. Instead, the objects are put into a comparative relationship in the mind.

Another logico-mathematical relationship is “two.” The “twoness” is not a physical property of either the rattle or the teddy bear, but exists in the mind in the form of a relationship. Mathematics is based on our ability to put things into mathematical relationships (Kamii & Chemeketa, 1982).

Children as young as age one try to find answers systematically, as can be seen in the previous example. While, this may seem like a haphazard process, they are mentally organizing their thoughts and their worlds mentally with an often remarkable ingenuity. Children will repeat actions or introduce other methods in order to test the regularity of the observed phenomena. Answers raise new problems and questions in the children’s mind, which they immediately set about resolving.

Bobby, a 2-year-old, stacks a set of blocks on the floor. He builds a tower four blocks high and then pushes it over with his hand. He seems pleased by his actions. He then builds the tower again and knocks it over by throwing a ball at it. Again he seems pleased. Bobby then builds the tower five blocks high and pushes it over with his hand, then rebuilds the five-block tower and throws the ball at it.

This example shows how a child acts like a scientist by testing many different variables, including the height and number of blocks, as well as experimenting with the method of knocking over the tower. Infants use logic and scientific processes to reconstruct, mentally, the world around them. (Forman, 1982; Sinclair & Kamii, 1995). Progressive organizing behaviors, such as the following, exist at a very young age:

- making comparisons between and among objects based on similarity
- putting one object in one hole (one-to-one correspondence)
- putting objects into a series from smallest to biggest as preparation for more complex number concepts

As teachers of children of this age, we have to be mindful of the complex constructions going on in their minds. Every time they touch, see, smell, taste, or move something they put it into their minds in a certain way (Sinclair & Kamii, 1995). The brain instinctively wants to make sense of this new information and puts it into a framework that Piaget called a schema. For infants and toddlers, these schema are very concrete and are directly linked to their senses and motor activity. From 24–36 months, children are developing some representational thought, that is, they begin representing their knowledge using language, drawings, and objects. As the child grows, representational thought is going to be very important in the construction of mathematics. A large part of mathematics is based on representational thought. Representation is the process of making one thing stand for another.

For example, we use the numeral “4” to stand for | | | |. We use all sorts of symbols and signs to represent more abstract ideas in mathematics such as addition (+), subtraction (−), and multiplication (×). In the infant and toddler years, the simple act of supporting a child’s burgeoning representational thought process is supporting his future mathematical ability.

Teachers of children from 2–3 years are on the cutting edge of fostering this vital intelligence. Children this age will begin to incorporate what they know into their imaginative play and this becomes a fertile area for mathematics. When a child pretends to use a block as a telephone, he is using representational thought. He has made a block stand for the phone just like “4” stands for | | | |. Many times, children develop a better understanding of these concepts if they are presented in a play context.

For example, children playing in the housekeeping area could be asked questions like “how many apples or spoons?” or “how much soup will you need?” They could be asked to put toy cans in their places in the pantry, which supports one-to-one correspondence. The children are also developing rudimentary counting and number skills and can be asked to count, even if they are not always accurate. “Let’s see how many spoons are here. Let’s count together: 1, 2, 3, 4. . .” Even before infants can count, however, they begin to discern similarities and differences in their environment that form the basis for forming mathematical relationships (E. M. Brannon et al., 2004; Kuhlmeier, Bloom, & Wynn, 2004; McCrink & Wynn, 2004), as in the following observation:

Twelve-and-a-half-month-old Xu is given a set of cups that fit inside one another and a group of sticks of different lengths. He takes the second-to-longest stick, looks at it, and keeps it in his left hand. With the other hand, he takes the second biggest cup. With the stick he firmly touches the cup. Then using the rod he firmly touches the largest cup, then the next smallest cup, and then the cup he is holding.

As children try to apply order to their environment, through mentally and physically acting on objects in their environment, they are thinking logically and even mathematically. When they do very simple actions such as using a stick to touch three cups in sequence they are systematically applying order to their environment using number, and logic (Sinclairs 1989). This is the child making sense of the world. As teachers in programs for young children, we need to recognize the rich mathematical learning that is occurring, and that the environment is vital to the infant’s construction of mathematics. Setting up a stimulating environment is the most important thing a teacher can do for infants and toddlers. Make objects available that children can observe, sort, and act on mentally, as we saw in the vignette with Xu. Interactions with objects will aid the child in developing the basic concepts needed for higher-level mathematics, such as one of the first basic concepts they will learn: the concept of “more.”

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