Norma C. Presmeg

Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics

The growing recognition that imagination, far from being a peripheral adjunct plays a central role in reasoning has important implications for the teaching and learning of mathematics. This paper has two main parts. In the first part, examples from high school case studies are used to illustrate prototypical mathematical images and the use of imagery in metaphoric and metonymic ways in mathematics. In the second part, pattern imagery and other types of imagery are discussed as central components in a model of mathematical reasoning. It is suggested that different forms of imagery may be used in ways that make abstraction and generalization possible in mathematics. The more abstract forms are idiosyncratically constructed by individuals. However, it is suggested that the activity of the imagination in the model presented here is central to meaning-making, and that the process makes shared meanings possible. Some social consequences for mathematics classrooms are explored briefly.

Transitions between contexts of mathematical practices

Acknowledgements. List of Contributors. Editors Prelude: Researching mathematics learning: the need for a new approach. 1. Mathematics learners in transition G. de Abreu, et al. 2. Immigrant children learning mathematics in mainstream schools N. Gorgorio, et al. 3. The transition experience of immigrant secondary school students: dilemmas and decisions A. Bishop. 4. Thinking about mathematical learning with Cabo Verde Ardinas M. Santos, J.F. Matos. 5. Exploring ways parents participate in their childrens school mathematical learning: cases studies in multiethnic primary schools G. de Abreu, et al. 6. Transitions between Home and School Mathematics: Rays of Hope Amidst the Passing Clouds M. Civil, R. Andrade. Editors Interlude: Theoretical orientations to transitions. 7. Towards a cultural psychology perspective on transitions Between contexts of mathematical practices G. de Abreu. 8. Mathematical acculturation, cultural conflicts, and transition A. Bishop. 9. Shifts in meaning during transitions N. Presmeg. Editors Postlude: The sociocultural mediation of transition. Author Index. Subject Index.

Visualisation and mathematical giftedness

Recent research findings are described which uncover the fact that visualisers are seriously under-represented amongst high mathematical achievers at senior high school level. Possible reasons for this phenomenon are discussed, and kinds of imagery which overcome some of the limitations of visual processing in mathematics are described.

Uncontrollable Mental Imagery: Graphical Connections Between A Function And Its Derivative

The view that imagery might be a disadvantage on certain tasks might surprise some mathematics educators who contend that a learners conceptual understanding is increased whenever visual imagery is used. One of the limitations of imagery found in the literature comes to bear on a unique aspect of mathematics teaching and learning. This is the notion of an uncontrollable image, which may persist, thereby preventing the opening up of more fruitful avenues of thought, a difficulty which is particularly acute if the image is vivid. Although one calculus students images supported high levels of mathematical functioning, occasionally his vivid images became uncontrollable, and the power of these images did more to obscure than to explain. This type of imagery can be a major hindrance in constructing meaning for mathematical concepts — contrary to the ‘panacea’ view of imagery which is sometimes expressed.

Metaphoric and metonymic signification in mathematics

Abstract This paper explores the roles of metaphor and metonymy in making sense of the ambiguities inherent in representation of mathematical constructs. Using the metaphors of “chains of signification” and “descent into meaning” for metonymies and metaphors respectively, these literary figures are discussed with regard to their use in mathematics. Synonymy, homonymy and polysemy are viewed as explanatory constructs in an analysis of ways in which metaphor and metonymy aid learners and mathematicians alike in making sense of mathematical ideas and resolving ambiguities.

Visualization and Affect in Nonroutine Problem Solving

The purpose of this research was to investigate the presence, role, extent, and constraints of visual thinking in conjunction with affective states and with metaphors in the problem-solving processes of graduate students as they solved nonroutine problems. This article gives details of the solving of 3 word problems by 4 students. Visual imagery was evidenced by 3 descriptors, namely drawing, verbal report, and gesture. Visual imagery was used by each of the 4 students for each of the 3 problems. Visual imagery was reported even in instances where no diagram was drawn and the solution appeared to be purely algebraic. The roles of visualization were investigated in 4 main moments of the solution processes, which we have called preparation, solution, conclusion, and hindsight. The types of imagery and their roles in these moments cause us to differentiate between use of imagery to make sense and to solve, as 2 distinct aims of visualization. Base knowledge, spatial reasoning, metaphors that may enable or constrain, and particularly affective issues all played a role in the graduate students use of visualization.

Semiotics in mathematics education

1. Introduction.- 2. Semiotics in theory and practice in mathematics education.- 3. A summary of results.- References.

School Mathematics in Culture-Conflict Situations.

In times of cultural change, education plays an especially important role. The writer suggests that even mathematics curricula, which have traditionally been considered culture-free, have a role to play in fostering mutual understanding amongst members of different cultures, after a period of cultural upheaval. Anthropological and educational sources are used to suggest points of relevance when a mathematical curriculum is designed for multi-cultural classrooms.

Balancing Complex Human Worlds: Mathematics Education as an Emergent Discipline in Its Own Right

I taught a course on informal geometry — a content course — to students at The Florida State University who are prospective high school mathematics teachers. In the first week I asked them to bring or wear to the next class, something which had geometry in it, and to come to class prepared to tell why they had chosen that particular item and to talk about its geometry.