Cognitive units, concept images, and cognitive collages : an examination of the processes of knowledge construction

The fragmentation of strategies that distinguishes the more successful elementary grade students from those least successful has been documented previously. This study investigated whether this phenomenon of divergence and fragmentation of strategies would occur among undergraduate students enrolled in a remedial algebra course. Twenty-six undergraduate students enrolled in a remedial algebra course used a reform curriculum, with the concept of function as an organizing lens and graphing calculators during the 1997 fall semester. These students could be characterized as "victims of the proceptual divide," constrained by inflexible strategies and by prior procedural learning and/or teaching. In addition to investigating whether divergence and fragmentation of strategies would occur among a population assumed to be relatively homogeneous, the other major focus of this study was to investigate whether students who are more successful construct, organize, and restructure knowledge in ways that are qualitatively different from the processes utilized by those who are least successful. It was assumed that, though these cognitive structures are not directly knowable, it would be possible to document the ways in which students construct knowledge and reorganize their existing cognitive structures. Data reported in this study were interpreted within a multi-dimensional framework based on cognitive, sociocultural, and biological theories of conceptual development, using selected insights representative of the overall results of the broad data collection. In an effort to minimize the extent of researcher inferences concerning cognitive processes and to support the validity of the findings, several types of triangulation were used, including data, method, and theoretical triangulation. Profiles of the students characterized as most successful and least successful were developed.Analyses of the triangulated data revealed a divergence in performance and qualitatively different strategies used by students who were most successful compared with students who were least successful. The most successful students demonstrated significant improvement and growth in their ability to think flexibly to interpret ambiguous notation, switch their train of thought from a direct process to the reverse process, and to translate among various representations. They also curtailed their reasoning in a relatively short Period of time. Students who were least successful showed little, if any, improvement during the semester. They demonstrated less flexible strategies, few changes in attitudes, and almost no difference in their choice of tools. Despite many opportunities for additional practice, the least successful were unable to reconstruct previously learned inappropriate schemas. Students' concept maps and schematic diagrams of those maps revealed that most successful students organized the bits and pieces of new knowledge into a basic cognitive structure that remained relatively stable over time. New knowledge was assimilated into or added onto this basic structure, which gradually increased in complexity and richness. Students who are least successful constructed cognitive structures which were subsequently replaced by new, differently organized structures which lacked complexity and essential linkages to other related concepts and procedures. The bits and pieces of knowledge previously assembled were generally discarded and replaced with new bits and pieces in a new, differently organized structure

A Rush of Connections and Insights, a Glorious Moment of Clarity

For several years we have been interested in pre-service teachers\u27 memory for mathematical episodes. Partly this is because memory is such a vital aspect of mathematical problem solving. Long-term declarative memory is the sort of memory involved when a person talks, writes, draws, or otherwise consciously represents their recollections. Warner, Coppolo & Davis (2002) identify long-term declarative memory as a key ingredient in flexible mathematical thinking - the ability to apply mathematical solution processes in different settings and across different representations - and Davis, Hill & Smith (2000) emphasize long-term declarative memory as a key feature in effective teaching of mathematics. Memory, broadly speaking, has three aspects: formation, storage and retrieval (Squire,1994). An aspect of memory that is of particular importance for this study is that of the emotional color of a memory. Le Doux (1998, 2002) has argued in recent years for the importance of emotional color in all aspects of memory, and Thurston (1997) alludes to colorizing his memory for written mathematics in order to try to understand what it\u27s really getting at rather than just what it says

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail

Flexible thinking and met-befores : impact on learning mathematics

In this paper we study the difficulties resulting from changes in meaning of the minus sign, from an operation on numbers, to a sign designating a negative number, to the additive inverse of an algebraic symbol on students in two-year colleges and universities. Analysis of the development of algebra reveals that these successive meanings that the student has met before often become problematic, leading to a fragile knowledge structure that lacks flexibility and leads to confusion and long-term disaffection. The problematic aspects that arise from changes in meaning of the minus sign are identified and the iconic function machine is utilized as a supportive strategy, along with formative assessment to encourage teachers and learners to seek more flexible and effective ways of making sense of increasingly sophisticated mathematics

FUNCTION MACHINES & FLEXIBLE ALGEBRAIC THOUGHT

We explore how college students understand ideas of functions, and which representations are productive for them in promoting their ability to work flexibly across representations. We use pre- and post-test scores, and triangulations via student self evaluations, to generate a hypothesis related to flexible thinking and success in algebra. We use confidence intervals to provide evidence for a highly significant change in student flexibility in algebraic thinking, and to assist in generating a plausible model of how the use of function machines in a developmental algebra course is instrumental in stimulating that flexibility