Conceptual Understanding

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Lapp D, Berry J and Nyman M (2010) Student Connections of Linear Algebra Concepts: A Temporal Analysis of Concept Maps
International Journal of Mathematical Education in Science & Technology, 41 No 1, 1 – 18
This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud’s theory of conceptual fields and Pirie and Kieren’s model for the growth of mathematical understanding. In addition to the existing techniques for analysing concept maps, two new techniques are developed for analysing qualitative data based on student-constructed concept maps: (1) temporal clumping of concepts and (2) the use of adjacency matrices of an undirected graph representation of the concept map. Findings suggest that students may find it more difficult to make connections between concepts like eigenvalues and eigenvectors and
concepts from other parts of the conceptual field such as basis and dimension. In fact, eigenvalues and eigenvectors seemed to be the most disconnected concepts within all of the students’ concept maps. In addition, the relationships between link types and certain clumps are suggested as well as directions for future study and curriculum design.

Headlam,C. and Graham, E. (2009), Some initial findings from a study of children’s understanding of the Order of Operations, Proceedings the British Society for the Research into Learning Mathematics Conference at Loughborough University, 37-41.

This paper presents some of the initial findings of a study into the strategies used by children to solve arithmetic and algebraic problems requiring the appropriate use of the order of arithmetic operations. The research has utilised graphics calculators which have been programmed with Key Recorder Software as a data collection tool. This has enabled the researchers to analyse the children’s approaches to some of the questions posed by observing their calculator keystrokes. Interviews with both teachers and pupils will be used to link the pupils’ strategies with the teaching methods used, and an initial analysis of observed misconceptions has been carried out. Initially this study has involved children in the UK and in Japan, where teaching methods differ substantially.

Pratt, N. & Woods, P. (2007) ‘Changing PGCE students’ mathematical understanding through a community of inquiry into problem solving’, in L. Bills, J. Hogden & H. Povey (Eds.) Research in Mathematics Education Volume 9, London, BSRLM.

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Rowlands, S., Graham, E., Berry, J. and McWilliam, P. (2007), Conceptual Change Through the Lens of Newtonian Mechanics, Science & Education, 16, 21-42.

Throughout its long history, the conceptual change literature assumed that student ‘misconceptions’ in mechanics have been formed prior to instruction. As an attempt to shed light on conceptual change, this paper examines some of the trends in the literature and argues that misconceptions may be spontaneous rather than preformed, that schema theory may be the most appropriate theory to take into account this spontaneity, that misconceptions should also be viewed through the lens of the subject as a system of well-defined concepts and that any conceptual change model may have to be prescriptive and engage the student with a metadiscourse concerning the abstract nature of the subject.

Rowlands, S., Graham, E., Berry, J. and McWilliam, P. (2005), Misconceptions of Force: Spontaneous Reasoning or Well-formed Ideas Prior to Instruction? Research in Mathematics Education, Vol. 7, 47-65.
Throughout its forty year history, the conceptual change literature assumed student misconceptions of force are formed prior to instruction. We argue that it may well be the case that misconceptions are not formed until the student considers force and motion in a scientific context for the first time. This has obvious implications for research methods. We are in the early stages of developing a research method for investigating conceptual change in mechanics. To illustrate this method, we have taken examples from one-to-one Socratic tutoring. We conclude by outlining the nest step in the research, which is to build a model that will enable the Socratic method to reveal the characteristics of misconceptions.

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