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 (onetoone 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 onetoone 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:
Twelveandahalfmonthold Xu is given a set of cups that fit inside one another and a group of sticks of different lengths. He takes the secondtolongest 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 higherlevel mathematics, such as one of the first basic concepts they will learn: the concept of “more.”
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