The Route: Progression of Understanding

Research has uncovered an awareness of number in infants shortly after birth. The ability to represent number and the development of informal strategies to solve number problems develop in children over time. Many studies have explored how preschoolers and children in the early elementary grades understand basic number concepts and begin operating with number informally well before formal instruction begins (Carey, 2001; Gelman, 1990; Gelman and Gallistel, 1978).

Children’s understanding progresses from a global notion of a little or a lot to the ability to perform mental calculations with specific quantities (Griffin and Case, 1997; Gelman, 1967). Initially the quantities children can work with are small, and their methods are intuitive and concrete. In the early elementary grades, they proceed to methods that are more general (less problem dependent) and more abstract. Children display this progression from concrete to abstract in operations first with single-digit numbers, then with multidigit numbers. Importantly, the extent and the pace of development depend on experiences that support and extend the emerging abilities.

Researchers have identified two issues in early mathematics learning that pose considerable challenges for instruction:

  1. Differences in children’s experiences result in some children—primarily those from disadvantaged backgrounds—entering kindergarten as much as two years behind their peers in the development of number concepts (Griffin and Case, 1997).

  2. Children’s informal mathematical reasoning and emergent strategy development can serve as a powerful foundation for mathematics instruction. However, instruction that does not explore, build on, or connect with children’s informal reasoning processes and approaches can have undesirable consequences. Children can learn to use more formal algorithms, but may apply them rigidly and sometimes inappropriately (see Boxes 3.1 and 3.2). Mathematical proficiency is lost because procedural fluency is divorced from the mastery of concepts and mathematical reasoning that give the procedures power.

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