by Constance Kamii
October 2000
Teachers need as much scientific knowledge about how children learn mathematics as physicians have about the causes of illnesses. Because of this need, teacherpreparation programs must change. Specific examples from classrooms illustrate this need.
I once wondered why some first graders were getting such answers as 3 + 4 = 4. By watching them, I found out that they were putting three counters out for the first addend and then four for the second addend, including the three that were already out.
Errors of this kind result from prematurely teaching a rule to follow. According to this rule, one must put counters out for the first addend, more counters for the second addend, and count all of them to get the answer. This rule works for children who already know that addition is the joining of two sets that are disjoint. However, the rule is superfluous for those who have constructed this logic, and it causes errors for those who have not constructed it.
Another example of imposing a rule that is either superfluous or premature is teaching countingon to children who are countingall. Countingall refers to solving 3 + 4 by counting out three counters, then four other counters, and counting all of them again ("onetwothreefourfivesixseven"). In countingon, by contrast, children say "fourfivesixseven."
With scientific research replicated worldwide, Piaget showed that all children construct, or create, logic and number concepts from within rather than learn them by internalization from the environment (Piaget 1971; Piaget and Szeminska 1965; Inhelder and Piaget 1964; and Kamii 2000). Studying the research leads teachers to understand that addition involves partwhole relationships, which are very hard for children to make and which cannot be taught through practice and memorization. To add two numbers, children must put two wholes together ("three" and "four," for example) to make a higherorder whole ("seven") in which the previous wholes become two parts. When young children cannot think simultaneously about a whole and two parts, they countall by changing both the "three" and the "four" into ones. Making them counton is harmful when they cannot mentally make the partwhole relationship necessary to counton.
When teachers study Piaget's theory and replicate the aforementioned research, they can understand why some first graders cannot counton. When children have constructed their logic sufficiently to make the partwhole relationship of countingon, they give up countingall, just as babies give up crawling when they can walk. I hope that the day will come when teachers entering the classroom and those already in the classroom have as much scientific knowledge about how children learn mathematics as physicians have about the causes of illnesses. To reach this vision, the teacherpreparation programs must change.
References
Inhelder, Barbel, and Jean Piaget. The Early Growth of Logic in the Child. New York: Harper & Row, 1964.
Kamii, Constance. Young Children Reinvent Arithmetic. 2nd ed. New York: Teachers College Press, 2000.
Piaget, Jean. Biology and Knowledge. Chicago: University of Chicago Press, 1971.
Piaget, Jean, and Alina Szeminska. The Child's Conception of Number. New York: W. W. Norton & Co., 1965.
Constance Kamii, ckamii@uab.edu, is a professor of early childhood education at the University of Alabama at Birmingham. She studied under Piaget for parts of fifteen years to become able to use his theory in early childhood education. 
