What Mathematical Concepts Do Infants and Toddlers Learn?
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.
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