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This paper discusses the theories of conceptual change and how they explain the difficulties in mathematics learning, especially in enlarging the number concept. Two studies are presented, one of which examines the research questions: What is the role of prior knowledge in students' answers to questions about the density of rational and real numbers? and Do theories of conceptual change explain the difficulties students have in learning these concepts? Ten university professors of mathematics were interviewed about their personal learning histories as professional mathematicians, and were asked what they remembered about the initial learning of the real number concept and how they currently conceptualize the numbers system. The main result from the interviews was that most of the professors remembered that learning to understand the notion of real numbers was something unique, and it required them to move into a new kind of abstract thinking, free from the constraints of prior mathematical intuition. In the second study a number concept test was administered to 640 calculus students (aged 17-18 years) from 24 randomly selected Finnish upper secondary schools in the usual classroom conditions. The test included identification, classification, and construction problems in the domain of rational and real numbers. The role of the students' prior knowledge was then analyzed using their answers to four critical questions. Results from the second study indicate that the process of knowledge acquisition, especially in mathematics, involves restructuring as well as enriching one's prior knowledge structures, which can take place at many different levels. (Contains 30 references.) (ASK)

This meta-analysis integrates 296 effect sizes reported in eye-tracking research on expertise differences in the comprehension of visualizations. Three theories were evaluated: Ericsson and Kintsch's ("Psychol Rev" 102:211-245, 1995) theory of long-term working memory, Haider and Frensch's ("J Exp Psychol Learn Mem Cognit" 25:172-190, 1999) information-reduction hypothesis, and the holistic model of image perception of Kundel et al. ("Radiology" 242:396-402, 2007). Eye movement and performance data were cumulated from 819 experts, 187 intermediates, and 893 novices. In support of the evaluated theories, experts, when compared with non-experts, had shorter fixation durations, more fixations on task-relevant areas, and fewer fixations on task-redundant areas; experts also had longer saccades and shorter times to first fixate relevant information, owing to superiority in parafoveal processing and selective attention allocation. Eye movements, reaction time, and performance accuracy were moderated by characteristics of visualization (dynamics, realism, dimensionality, modality, and text annotation), task (complexity, time-on-task, and task control), and domain (sports, medicine, transportation, other). These findings are discussed in terms of their implications for theories of visual expertise in professional domains and their significance for the design of learning environments. (Contains 9 tables and 2 footnotes.)