RESEARCH PROPOSAL

Research Proposal



Coordinating mathematical concepts with the demands of authority: children's reasoning about conventional and second-order logical rules-Proposal

The Project

Arithmetic algorithms include two types of rules: conventional rules that can be changed by authority, and can legitimately vary from one classroom or from one country to another (for example, put sum below rather than above, number added) and logical rules that involve logic of algorithm. Changes in logical rules produce incorrect answers. Therefore these rules are not modifiable by legitimate authority. Second-order logical rules depend on particular conventions symbol system used (for example, rule for execution and value). Given symbol system used, these rules are not modifiable by legitimate authority. However, as the result of their dependence on symbol system, children may have difficulty distinguishing second-order logical rules from conventional rules. Ninety-eight children in grades 2 through 5 were interviewed about accuracy of their answers to look for alternatives to standard rules of conventional logic and second order, and legitimacy of authorities to change rules. Half children across this age range treated second-order rules of course, something like conventions, considering that the response as the result of an alternative to the second-order logical rule is correct if sanctioned by authority. With increasing age children increasingly limited jurisdiction of authority on rules of second-order logic.

Application of mathematical concepts

Logical "truths" are not under jurisdiction of authority, and if one understands that something is necessarily correct, then one will not alter this statement, because an authority insists that it is quite opposite. This is part of developing skills of children, "differentiation of definition (necessary) properties of the concept of its lack of definition (not necessary) properties" (Smith, 1993, p. 183) . While much research has investigated child development course, very little has been directly examined development of children's understanding of necessity of logical concepts in coordination with demands of authority, understanding that what is logically correct cannot be altered by authority. We rely on Piaget, 1952 and Piaget, 1963 construction studies of children with necessary knowledge in face of pressure, not authority, but of perception. For example, in studies of number portability is considered evolution of logical necessity that allows children to overcome pull of perceptual appearances that contradict logic. In this study, we investigated coordination of children in logic of mathematics with requirements of authority; we do interviews about standards used in standard addition and subtraction algorithms.

Our assumptions are as follows: Based on findings of social research (Smetana, 1983, 1981, Turiel, 1983), children in all grades is approved alternative to conventional rule, both local and remote contexts, accepting response generated by same jurisdiction and granting of authority over her (scenario 1). Based on recent research (Laupa, 2000) and pilot testing, many children try an alternative to rule of second-order logic, as if it were an agreement, accepting response generated by it and granting it authority over jurisdiction -despite fact that alternative to second order of logical rules produces the different answer to right that are obtained using standard algorithm (Scenario ...