Identification of numerical principles prerequisite to a functional understanding of place value

Abstract

The purpose of this study was to find some remedy to frustrations engendered when children fall to grasp the essential principle of place value after several attempts at reteaching. It was hypothesized that these children must have failed to acquire understanding of some numerical principle(s) prerequisite to understanding the place value aspect of the numeration system. Four plausible prerequisite principles were identified (1) synthesis of ordinal and cardinal properties of the numeration system, (2) both the addition and subtraction operations, (3) understanding of counting by groups, and (4) understanding of exchange equivalences such as one ten for ten ones, etc. It was hypothesized that understanding of analog clock reading was also dependent upon understanding of the same four prerequisite principles.

By conducting four pilot studies, six interview protocol instruments were developed to measure levels of understanding for the four prerequisite principles and the place value and clock reading criterion principles. Three levels of understanding: no understanding, transitional understanding, and competence were designated to correspond with Plagetian stages in the development of a new operation. Forty-eight children, twenty with second grade completed and twenty-eight with third grade completed, were tested on all six instruments.

Hypotheses tested were: (1) if the four identified prerequisite principles are necessary to understanding of place value, then subjects will demonstrate a level of understanding on the place value measure no higher than their lowest level of understanding achieved on the four prerequisite measures; and (2) if the four identified prerequisite . principles are necessary to understanding of clock reading, then subjects will demonstrate a level of understanding on the clock reading measure no higher than their lowest level of understanding achieved on . the four prerequisite measures.

The findings were that both hypotheses were supported at the .01 probability level. Analysis of the research design and examiner observations suggested possible explanations for anomalous aspects of the obtained data. Limitations, directions for further research, and implications for teachers were also discussed.