Journal Entry for 4/5/2011

Traditional instruction of differential equations and conceptual learning
Discrete Mathematics and its applications
Teaching Mathematics and Its Applications
Volume 29 Issue 2

Abstract
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 with 13 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 them to 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.

First, I was immediately struck by the similarity between the concept/procedure description in this article and the concept/technique description I myself use. I feel both pleased and annoyed, and I can now imagine (in part) what Newton must have felt for Leibniz when he emerged with his own version of the calculus. Second, not even ten minutes before I found this article, I had been musing about a conversation I had had with my colleague Joe Agnich (whose DE class I had the pleasure of observing) in which I asked about how he taught the underlying concepts. He jokingly said that it was “above his paygrade” and that he focused on teaching them “how” to do these things more than “why”. (Which is not to say that he never discussed theory, just that it was not a focus of the course as written.) During my musing I was struck by the thought that “A math class without theory is like rain without a rainbow: nice enough, but ultimately unfulfilling.” Thus my response to this article was perhaps more visceral and satisfied than it might have been on another day.

This research behind this article supports the theory that deeper understanding of mathematical subject matter stems from an understanding of the concept of the material rather than the techniques and procedures involved in solving problems. the article goes on to exhibit the questions used to gather the data as well as a statistical analysis of the results. One of the implications of which was that conceptual success was a predictor of procedural success; meaning: if a student understands the concept, they is a decent chance that they will be able to apply their understanding to procedural problems.

This article was a fascinating and validating read. I have already shared it with Joe, and I look forward to hearing his take on it.