Traditional instruction of differential equations and conceptual learning

Procedural and conceptual learning are two types of learning, related to two types of knowledge, which are often referred to in mathematics education. Procedural learning involves only memorizing operations with no understanding of underlying meanings. Conceptual learning involves understanding and interpreting concepts and the relations between concepts. The relationship between these learning types has been discussed for a long time. For some researchers, procedural knowledge forms the basis for conceptual knowledge, while for others the relationship is reversed. The aims of the study reported here were first to explore the nature of students' learning in traditional differential equations (DEs) courses and second to clarify the relationship between procedural and conceptual learning. To address these aims an achievement test with13 open-ended questions, probing procedural and conceptual learning with regard to DEs, was administered to 77 candidate mathematics teachers, enrolled in a traditional DEs course. The analysis of students' responses to the test items showed that 85% of candidate teachers gave correct responses on procedural questions whilst only 30% of them gave correct responses to the conceptual questions. These findings suggest that the candidate teachers' learning was primarily procedural in the context of traditional instruction and content and that this did not lead themto develop the conceptual knowledge needed to interpret new situations properly and to produce new ideas beyond the ones they had memorized. In addition, based upon the student levels in both procedural and conceptual learning, it was concluded that conceptual learning supports and generates procedural learning but procedural learning does not support conceptual learning.