## Development of Conceptual Structures

Conceptual structures about numbers develop early and allow children to experiment with calculations in their preschool years. They quickly master many addition and subtraction strategies, carefully selecting those that are best suited to a particular problem. As they apply their algorithms, they mentally determine how much time it took them to make the calculation and the likelihood that the result is correct. Siegler (1989) studied children using these strategies, and he concluded that they compile detailed statistics on their success rate with each algorithm. Gradually, they revise their collection of strategies and retain those that are most appropriate for each numerical problem. Here is a simple example. Ask a young boy to solve 9 − 3. You may hear him say, “nine . . . eight is one . . . seven is two . . . six is three . . . six!” In this instance, he counts backward starting from the larger number. Now ask him to calculate 9 − 6. Chances are that rather than counting backward as he did in the first problem, he will find a more efficient solution. He counts the number of steps it takes to go from the smaller number to the larger: “six . . . seven is one . . . eight is two . . . nine is three . . . three!” But how did the child know this? With practice, the child recognizes that if the number to be subtracted is not very close in value to the starting number, then it is more efficient to count backward from the larger number. Conversely, if the number to be subtracted is close in value to the starting number, then it is faster to count up from the smaller number. By spontaneously discovering and applying this strategy, the child realizes that it takes him the same number of steps, namely three, to calculate 9 − 3 and 9 − 6.

Exposure at home to activities involving arithmetic no doubt plays an important role in this process by offering children new algorithms and by providing them with a variety of rules for choosing the best strategy. In any case, the dynamic process of creating, refining, and selecting algorithms for basic arithmetic is established in most children before they reach kindergarten.

Exactly how number structures develop in young children is not completely understood. However, in recent years, research in cognitive neuroscience has yielded sufficient clues about brain development to the point that researchers have devised a timeline of how number structures evolve in the brain in the early years. Sharon Griffin (2002) and her colleagues reviewed the research and developed tests that assessed large groups of children between the ages of 3 and 11 in their knowledge of numbers, units of time, and money denominations. As a result of the students’ performance on these tests, they made some generalizations about the development of conceptual structures related to numbers in children within this age range. Their work is centered on several core assumptions about how the development of conceptual structures progresses. Three assumptions of particular relevance are as follows:

- Major reorganization in children’s thinking occurs around the age of five when cognitive structures that were created in earlier years are integrated into a hierarchy.
- Important changes in cognitive structures occur about every two years during the development period. The ages of 4, 6, 8, and 10 are used in this model because they represent the midpoint of the development phases (ages 3 to 5, 5 to 7, 7 to 9, and 9 to 11).
- This developmental progression is typical for about 60 percent of children in a modern, developed culture. Thus, about 20 percent of children will develop at a faster rate while about 20 percent will progress at a slower rate.

## Structures in Four-Year-Olds

The innate capabilities of young children to subitize and do some simple finger counting enables them by the age of four to create two conceptual structures, one for global quantity differences and one for the initial counting of objects (Figure 2.1). Looking for global quantity, they can tell which of two stacks of chips is more or less, which of two time units is shorter or longer, and which of two monetary units is worth more or less. On a balance scale, they can tell which side is heavier and/or lighter and which side of the beam will go down. Children at this age are still relying more on subitizing than counting, but they do know that a set of objects will get bigger if one or more objects are added or smaller if one or more objects are removed.

Counting skills are also developing. They know that each number word occurs in a fixed sequence and that each number word can be assigned to only one object in a collection. They also know that the last number word said indicates the size of the collection. Most can count to five, and some can count to 10. Yet, despite these counting capabilities, these children still rely more on subitizing to make quantity determinations. This may be because the global quantity structure is stored in a different part of the brain from the counting structure and because these two regions have not yet made strong neural connections with each other.

## Structures in Six-Year-Olds

Children around six years of age have integrated their global quantity and initial counting models into a larger structure representing the mental number line. Because this advancement gives children a major tool for making sense of quantities in the real world, it is referred to as the central conceptual structure for whole numbers. Using this higher-order structure, children recognize that numbers higher up in the counting sequence indicate quantities that are larger than numbers lower down. Moreover, they realize that numbers themselves have magnitude, that is, that 7 is bigger than 5. The number line also allows them to do simple addition and subtraction without an actual set of objects just by counting forward or backward along the line. This developmental stage is a major turning point because children come to understand that mathematics is not just something that occurs out in the environment but can also occur inside their own heads.

Now children begin using their counting skills in a broad range of new contexts. They realize that counting numbers can help them read the hour hand on a clock, determine which identical-sized money bill is worth the most, and know that a dime is worth more than a nickel even though it is smaller in size. Unlike four-year-olds, they rely more now on counting than global quantity in determining the number of objects, such as chips in a stack and weights on a balance.

## Structures in Eight-Year-Olds

Children at the age of eight have differentiated their complex conceptual structure into a double mental counting line schema that allows them to represent two quantitative variables in a loosely coordinated fashion. Now they understand place value and can mentally solve double-digit addition problems and know which of two double-digit numbers is smaller or larger. The double number line structure also permits them to read the hours and minutes on a clock, to solve money problems that involve two monetary dimensions such as dollars and cents, and to solve balance-beam problems in which distance from the fulcrum as well as number of weights must be computed.

## Structures in Ten-Year-Olds

By the age of 10, children have expanded the double number line structure to handle two quantities in a well-coordinated fashion or to include a third quantitative variable. They now acquire a deeper understanding of the whole number system. Thus, they can perform mental computations with double-digit numbers that involve borrowing and carrying, and can solve problems involving triple-digit numbers. In effect, they can make compensations along one quantitative variable to allow for changes along the other variable. This new structure also allows them to translate from hours to minutes and determine which of two times, say three hours or 150 minutes, is longer. They find it easy to translate from one monetary dimension to another, such as from quarters to nickels and dimes, to determine who has more money, and also to solve balance-beam problems where the distance from the fulcrum and number of weights both vary.

## References

Griffin, S. (2002). The development of math competence in the preschool and early school years: Cognitive foundations and instructional strategies. In J. M. Rover (Ed.). Mathematical cognition: A volume in current perspectives on cognition, learning, and instruction (pp. 1–32). Greenwich, CT: Information Age Publishing.

Siegler, R. S., & Jenkins, E. A. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum.