Intermediate Cognitive Organizations in the Process of Learning a Mathematical Concept: The Order of Positive Decimal Numbers

This investigation found that children use three implicit and incorrect rules when comparing decimal numbers and that these rules are intermediate levels of response in the process of learning decimal numbers. The rules are intermediate in that: (a) the frequency of their use by children decreases from the fourth grade to the seventh grade; and (b) when a child is able to give a correct answer to the comparison of two decimal numbers, it is possible to induce use of the rules by giving the child more than two numbers to compare. These findings are interpreted as evidence that intermediate cognitive tools, which respond successfully to most of the tasks encountered early in learning, continue to coexist with more advanced tools.