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Conceptual and procedural knowledge in mathematical cognition and generalization of acquired mathematical knowledge
Author
Lim, Zee Ying
Supervisor
Lee, Kerry
Ng, Swee Fong
Abstract
This thesis compares the effects of teaching with conceptual versus procedural emphases using generalization gradient. The distinction between conceptual and procedural knowledge has a long tradition in mathematics education (Hierbert and Lefevre, 1986). Mathematics education literature often emphasizes acquisition of conceptual knowledge instead of procedural knowledge as being an important step in learning to solve novel problems (Kieren, 1993). Yet, in the framework of memory systems in psychology, conceptual and procedural knowledge are both localized in the declarative memory system. These perspectives led to different implications for pedagogical approaches. The mathematics education literature suggests that both types of knowledge need to be emphasized, while the psychological perspective suggests that conceptual knowledge can be derived from procedural training. Hence, this study seeks to reconcile the discrepancy between these two perspectives. In the field of artificial grammar, participants were able to generalize what they have learnt to verify novel test strings through training with exemplars (Brooks and Vokey, 1991). Such generalization showed they had acquired conceptual knowledge of the abstract grammatical structure. Similarly, animal studies on operant conditioning have similarly made use of a range of novel test stimuli to obtain generalization gradients of animals’ generalization performances. In this thesis, generalization gradients of students’ performance in expansion questions were used to compare pedagogy that emphasizes conceptual versus procedural knowledge in mathematics. Results show that students taught with both conceptual and procedural emphases were able to generalize. This supported the psychological perspective that emphasized similarity between conceptual and procedural knowledge, and that students can generalize given only procedural training. Hence, differences found between conceptual and procedural knowledge in the mathematics education literature might be largely a result of how ‘qualitative’ was defined and how results from previous studies were interpreted. Results also showed that students taught with procedural emphases generally perform better given similar duration and content of teaching, though they did not generalize better. Hence, from the performance point of view, teaching with procedural emphases may be considered more efficient. This study also found that students’ acquisition of conceptual knowledge is closely tied to how teaching was delivered. For example, when handouts were introduced, with pedagogy remaining constant, the effects of pedagogy with conceptual emphases became more similar to pedagogy with procedural emphases. Using a generalization gradient yielded a more complete picture of the effects of the two different pedagogic approaches, as some differences between conceptual and procedural conditions were found only at higher levels of generalization.
Date Issued
2008
Call Number
QA14.S55 Lim
Date Submitted
2008