# Mathematics Development in the Elementary Years

### From Foundations For Numeracy

What are the important aspects of cognitive development that influence how children acquire mathematical facts, knowledge, and procedures in the elementary school years? Children’s cognitive development from ages six to twelve shows considerable growth. Six-year-olds are moving beyond intuitive and externalized interactions with the world. They start to form more complex mental representations of their own knowledge and that of others. This ability to consider another’s perspective (referred to as having a “theory of mind”) ties into increased working memory skills, both in terms of the amount of information children can retain, and in the availability of processing resources. The ways in which children encode and store information are still not exactly like those of adults, but they become more logical and more organized in how they make connections between different pieces of knowledge. This increase in memory capacity and memory organization develops gradually in the age range of six to twelve, but can also show periods of rapid change. Every child is an individual and so progresses on a unique trajectory – for teachers, this can make student learning in their classroom frustratingly variable. Nevertheless, as a group, children’s developing ability to process new information, retain it, and thus to solve increasingly complex problems takes elementary school children from the rudiments of symbolic number system knowledge to basic algebra. Furthermore, as children progress through the stage described by Piaget as *concrete operational* (i.e., a period when children have a better understanding of mental operations and begin to think logically about concrete events; ages 7-12), to the beginnings of *formal operational* (i.e., a period when skills such as logical thought, deductive reasoning, and systematic planning emerge and the ability to think about abstract concepts develops; ages 12-adulthood), they become ready to question their existing knowledge and challenge their teachers and peers.

In the elementary years, children’s mathematical development is closely tied to the experiences that they have in school. Initially, teachers and students must expend considerable effort to master the symbolic number system. Children must learn number vocabulary, knowledge of place value, rules for generating larger numbers, the conventions related to number labels and number symbols, and they must connect this symbolic knowledge to underlying representations of quantity. In Kindergarten, for example, a child who can label the symbol “37” with the verbal label “thirty-seven” is showing a reasonable level of knowledge for his or her age. By the end of Grade 1, children are expected to be producing labels for numbers in the hundreds and to understand how those symbols represent quantities. Symbolic knowledge allows children to compare and contrast large quantities in a precise way. Knowing that 134 is a smaller quantity than 341, for example, requires children to link quantities to symbols through conventional rules. Ideally, children start to connect quantities of numbers in the hundreds and thousands to other representations (such as a physical number line); by Grade 2 we expect that many children can accurately place a number such as 678 appropriately when shown a line with the endpoints labeled 0 and 1000.

Exactly when children acquire certain knowledge is of concern in relation to curriculum guidelines. It is important to realize that the goal is for all children to continue to learn and develop their knowledge. At times, children may be ahead of the curriculum and at other times they may be behind. Overall, the acquisition of knowledge will vary according to experiences, disposition, and even brain maturation.

Children are also expected to build up a repertoire of mathematical procedures in the first few years of school, starting with basic addition and subtraction in Grade 1 and moving to multiplication and division in Grades 3 and 4. Learning these operations involves more than memorizing “facts;” a conceptual understanding of additive composition helps children to understand subtraction and addition as complementary operations. Curriculum standards have helped to expand the scope of what children are exposed to in school. The National Council of Teachers of Mathematics (NCTM) has provided “focal points” to help teachers identify the central aspects of knowledge acquisition that help children to build a reasonably complete package of mathematical knowledge (www.nctm.org/standards/). In combination with an understanding of cognitive development in the elementary years, this information can help teachers to build up their own understanding of what constitutes successful mathematical learning in this age range.

Along with exciting increases in children’s knowledge organization and capacity for learning come some other developments that can be frustrating for teachers. Because children in elementary school still view the world in concrete terms, they may over-apply rules that they have learned. This lack of flexibility, which may be generalized or may be specific to certain areas, might help children to acquire knowledge that is rule based, but may prevent them from seeing beyond the rule to the general principle. Teachers need to be aware that even when children exhibit excellent procedural skills in mathematics, they may nevertheless fail to understand *why* those procedures work, or may be unaware of when to apply certain rules. For example, some 9- and 10-year-olds may insist on solving a problem like 14 + 7 – 7 by adding 14 + 7 and then subtracting 7. Even if they know that 7 – 7 = 0, they might not be confident in using the knowledge in the context of another problem. This tendency to apply superficial and rigid rules may, in part, account for another very typical mistake that children make. When presented with problems like 5 + 6 + 3 = 4 + ?, they may simply add up all the numbers. By applying a typical procedure that usually works – that is, “add numbers to get the answer” – they are showing their lack of understanding of the meaning of the equal sign. Luckily, these kinds of misconceptions, once they are revealed, are relatively easy for teachers to anticipate and avoid. Understanding that children at this age tend to learn rules without necessarily extracting the bigger picture is useful in understanding mathematical development.

Although many children between six and twelve show both increasing capacity and a tendency to focus on rules and procedures, teachers also need to be aware that cognitive development and skill learning is variable and does not always seem to move in one direction. This means that a learner may revert to a less sophisticated strategy, even after showing evidence of using something more advanced. A researcher by the name of Robert Siegler has described development as a series of overlapping waves, where some skills are moving forward, some appear to be moving backwards, and in other cases, little learning appears to occur. As a group and over a reasonably long period of time, children’s knowledge moves forward, but on a day-to-day basis, changes may not always be improvements! Experienced teachers will learn to recognize these apparent periods of decline in children’s learning in particular areas and recognize that individual children may show different patterns of learning for different skills. This tendency for knowledge acquisition to act like waves in the sea is one reason why a variety of teaching approaches and learning activities will be beneficial for both teachers and students. Sometimes children just need more practice to make progress in learning a skill, but other times it will be more important to present the procedure or concept differently so that it gives children the opportunity to reconfigure their knowledge.

Knowing about general principles of learning can also be helpful for teachers. Many of the characteristics that describe children’s learning apply more generally. Adults may appear to learn faster because they already have more knowledge that they can link to new information. Nevertheless, anyone learning a new skill shares similar challenges. Consider the situation of a North American who moves to the UK and wants to be knowledgeable about the game of cricket. She may have knowledge about similar games (such as baseball) that can form the basis of her mental representation for cricket. She will need to learn new vocabulary, new rules, and cope with inconsistencies where knowledge of baseball or other sports actually interferes with learning how the game of cricket works. If she perseveres, acquires the knowledge, goes and watches the games, or even participates, her understanding will increase. She may experience “ah-ha” moments where a particularly obscure referee’s call suddenly makes sense. But her knowledge will accrue gradually, undergo reorganizations, sometimes desert her at critical moments, and most importantly, will require effort to acquire. Compare such a learning process with children’s developing capacities of storage and processing efficiency, and it is clear that mathematical development from six to twelve will be variable across skill domains, across individual children, and as a function of the instructional milieu.

In summary, children’s cognitive development from six through twelve allows them to acquire new knowledge and procedures, as well as the conceptual structures that support mathematical learning. Development will not always be smooth and uni-directional. Teachers can encourage student success in mathematics by becoming knowledgeable about children’s cognitive development.

More detailed information about the various stages of development in relation to mathematical learning can be found in the following book:

- Siegler, R., & Alibali, M. W. (2004).
*Children’s thinking*. Upper Saddle River, NJ: Prentice Hall.

To learn more about children’s development of a “theory of mind,” refer to:

- Astington, J. W. (1993).
*The child’s discovery of the mind*. In J. Bruner, M. Cole, & A. Karmiloff-Smith (Series Eds.).*The developing child*. Cambridge, MA: Harvard University Press.

For a link to learning in schools, consult:

- Hattie, J. (2009).
*Visible learning*. New York: Routledge.