Applicability of conceptual knowledge and operational levels of knowledge representation

The "same perimeter problem" is a task that asks whether the area of a transformed parallelogram in which only the angles have been changed is equal to the area of the original parallelogram. This problem can be solved easily by using an area formula. However, because the 2 perimeters remain the same, even university students sometimes judge incorrectly that the 2 areas are equal. Published research has suggested that the origin of this misjudgment may be not only in problems with inert knowledge, but may be also due to difficulties in the operation of knowledge representation. In the present study, 106 university freshmen and sophomores were presented with 2 versions of the same perimeter problem. One version had a hint about using the area formula, and the other, no hint. The students completed a questionnaire that measured their operational level of knowledge of the formula. The results were as follows : (1) the high level of knowledge no-hint group could answer correctly at a high rate ; (2) the hint had little effect on the performance of the low level group ; and (3) the students who answered incorrectly after being given the hint considered the formula to be only a calculation procedure. It was suggested that learners' operational level of knowledge representation is an important factor in predicting whether they can apply their conceptual knowledge.