Teachers usually present addition and subtraction at the same time to first and second graders. Many teachers have observed, however, that students find addition easier and more natural than subtraction. Children struggle with subtraction even when they learn "fact families" that ostensibly help them understand the relationship between addition and subtraction. Given that children continue to find subtraction difficult despite the use of time-honored practices, we suggest that teachers de-emphasize fluency in subtraction until their students become fluent in addition.

Why is subtraction difficult in first and second grade? Piaget said that subtraction is more difficult than addition because it involves a negative action that is a secondary construction of addition (Piaget 1980). He also stated that children initially think and perceive in only positive terms. For example, when young children see a red ball, they first think of it only as "red" and "a ball." A considerable time later, they are able to think of the same ball as not blue, not green, not a cup, and not a shoe.

With this theoretical insight, one of the authors (Kamii 1985) decided to ask a few subtraction questions in the first-grade classroom in which she was working. It was the middle of the school year, and these children had never been formally taught subtraction. The first question was "What's 10 take away 5?" The entire class immediately shouted, "Five!" Kamii asked, "What's 4 take away 2?" and all the students shouted, "Two!" while jumping up and down. The next question was "What's 7 take away 4?" The children began to count on their fingers.

These specific questions were selected because by October, 96 percent of the class could instantly give answers to 5 + 5 and 2 + 2, but no one knew the answer to 4 + 3. As many teachers know, "doubles" such as 5 + 5 and 2 + 2 are much easier to learn than combinations such as 4 + 3. These first graders verified Piaget's (1980) statement that subtraction is a later, secondary construction. Once children's knowledge of a sum is solid, the related subtraction is easy for them.

Recently, we conducted more systematic research to test the hypothesis that children's knowledge of differences depends on their knowledge of sums (Kamii, Lewis, and Kirkland 2001). We asked children to solve related addition and subtraction problems such as 8 + 2 and 10 - 8. If we found that fluency in subtraction was unrelated to fluency in addition, we would conclude that knowledge of differences and knowledge of sums are independent of each other. If, however, we found that fluency in subtraction was dependent on fluency in addition, we would infer that children deduce differences from their knowledge of sums.

Method

We interviewed twenty-one first graders from one class and thirty-eight fourth graders from two classes individually in a constructivist "school within a school" located in an upper-middle-class suburb. We selected the two grade levels because one of us was working in these classrooms and the...

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