Mathematical Operations

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 develop number sense. 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 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 × 2 = 18, but so does 3 × 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 that it can also, and 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.

Perhaps the most significant cognitive advancement toward understanding mathematical concepts is the ability to conserve number. If children begin with two equivalent quantities (i.e., numbers) and can comprehend that the quantity of 5 does not change when one group is spread out or bunched together, then they have begun to understand the invariance of numbers in the environment. With such knowledge, school-age children can better predict relationships between numbers and quantities.

Learning more demanding mathematical concepts also requires some memorization. Even though math educators argue that understanding is more important than rote memorization, students still need to memorize rules that can be applied to a variety of problems. Usually, school-age children either intentionally or unintentionally memorize their multiplication tables, as well as other mathematical “facts” or “rules.” Over time and with practice, children also tend to invent their own strategies to facilitate the learning of mathematical concepts and operations (Ernst, 2000). Once certain operations become automatized, attentional capacity becomes available to apply to the remainder of the problem. Learning how to identify the greatest common factor in the numerator and denominator of a fraction, for example, would be much easier if you could focus your attention on what those common factors are, rather than struggle to determine which factors go into either the numerator or the denominator.

Finally, mathematical concepts are learned best if they can be linked to something that is familiar or known to the child. In other words, if the concepts can be linked to previously learned math concepts or embedded in practical problems that make sense to children, they will be able to assimilate the information more easily. Using other terms, one’s knowledge base can and will facilitate the decoding and storage of this new information.

While the developmental progressions described in this chapter are relatively universal, there are individual differences in learning rates and styles among school-aged children. During the past decade in particular, researchers have determined that if the task is made more familiar, children will typically exhibit cognitive abilities, such as decentration, multiple-perspective taking, strategy generation, and even mathematical skills, at a much earlier age than Piaget predicted (Siegler, 1996). Despite the appearance of these skills in highly supportive environments, however, there is a tendency for children to approach the world in the patterns outlined earlier.

The patterns of skill development portrayed in this chapter describe “average” learners who share a common cultural background with their teacher, class, school, and curriculum material. Much of how children learn and what they learn can also be traced to their parents’ educational values, the pedagogical practices of their teachers, their classroom environments, and the support they receive from their schools and communities.