Herbert P. Ginsburg on misconceptions about mastering early math

Sharon Griffin and her colleagues have done the field a great service in addressing key issues of mathematics education for young children. First, she clarifies what is meant by learning to understand number. Early “numeracy,” as some people refer to it, does not involve only rote memory. Memorizing the first part of the “number string” (numbers up to about 12 in English) is required, but even more crucial is the linking of spoken numbers with ideas of quantity. Another way of saying this is that from the outset the young child needs to learn abstract ideas about number and to become familiar with the meaning and uses of different kinds of number representations (like the number line on a thermometer). As Griffin has described elsewhere, the child is engaged in learning “central conceptual structures”—deep-seated cognitive principles—about number. Lesson 1 then is that from the outset, learning mathematics is an abstract activity even for young children. No doubt this is also true in other areas of mathematics like spatial relations, shape, and pattern, topics that Griffin has not investigated intensively. Mathematics for little children is not baby mathematics.

As is widely known, low-income children do relatively poorly in school and need extra help to succeed there. Griffin argues that these children may not be sufficiently exposed at home to the kinds of activities that can promote the adequate development of the conceptual structures required to serve as a foundation for school learning. In particular, low-income children may have insufficient experience with adult generated language that can help them organize mathematical experiences. In any event, the effective solution is not remedial education after children fail; it is prevention in the early grades.

Many have described the problem; few have done anything about it. Griffin’s work represents a major exception. Her Number Worlds program is a comprehensive, organized attempt to help low-income children, from preschool onwards, to develop the kind of conceptual understanding of number that is essential to education. The Number Worlds curriculum assumes that the low-income children are capable of the work and does not involve them in intellectually impoverished activities.

In implementing Number Worlds, Griffin has learned a great deal about difficulties teachers experience in teaching early mathematics. First, she notes that pre-service students often have a misconception of what it entails: many think that early number is all about manipulating symbols, not understanding them. Further, she observes many inadequate teaching practices in the classroom. For example:

• Teachers introduce written symbols before insuring that children’s number words are linked to ideas of quantity.
• They sometimes ask children use manipulatives to solve symbolic problems before children understand what the symbols refer to.
• Under the pressure of high-stakes testing, teachers sometimes spend more time on drill than on using language to make the material meaningful.

These examples suggest that the one of the key tasks of early mathematics education is to address teachers’ conceptions of what mathematics education requires and to improve their practice. A fine curriculum like Number Worlds cannot succeed without professional development of this type.

Herbert P. Ginsburg is Jacob H. Schiff Foundations Professor of Psychology and Education at Teachers College, Columbia University.