Mathematical Operations in Middle Childhood
In the preschool years, children begin counting by reciting the number words in sequence (e.g., one, two, three, four, five). This rote counting helps them to develop a sense of number. They learn that numbers come before or after each other and that numbers are smaller or larger than one another. They learn that one counting word corresponds with each object they count (i.e., one-to-one correspondence), and they also develop a sense of sequence and pattern (Franke, 2003). Counting, sequencing, and detecting patterns are often considered the basis of the elementary school curriculum. Comprehension of these concepts is necessary to support students in developing the skills and understanding necessary to operate on numbers and to solve a variety of mathematical problems that are introduced in the early elementary school grades, such as addition and subtraction.
Multiplication, division, fractions, decimals, ratios, and geometry are some of the mathematical concepts and operations typically introduced during middle childhood. A number of developmental changes described in the previous sections may help explain why children in middle childhood might be cognitively “ready” to address these mathematical skills and concepts. For example, Piaget researched extensively how children develop the concept of number. A developmental pattern he observed was that as children exhibit decentration they can begin to generate multiple ways to solve a mathematical problem. For example, 9 x 2 = 12, but so does 3 x 6; and you can solve the addition problem 9 + 4 by adding 4 units to 9 or knowing that 10 + 4 is 14 and taking away 1.
Children who showed an understanding of seriation demonstrated a more sophisticated understanding of mathematical concepts. They were able to assimilate not only that one number can be larger than another but also that it can—at the same time—be smaller than another number. Seriation allows for a comprehension of number lines as well as greater- and/or less-than problems. Understanding reversibility allows a child to appreciate the relationship between addition and subtraction. Taking two apples out of a basket after adding two reverses or “undoes” the first operation. The same understanding of reversibility can be applied to multiplication and division.
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