1,721,813 research outputs found
Theoretical frameworks for the learning of geometrical reasoning
No full textWith the growth in interest in geometrical ideas it is important to be clear about the nature of geometrical reasoning and how it develops. This paper provides an overview of three theoretical frameworks for the learning of geometrical reasoning: the van Hiele model of thinking in geometry, Fischbein’s theory of figural concepts, and Duval’s cognitive model of geometrical reasoning. Each of these frameworks provides theoretical resources to support research into the development of geometrical reasoning in students and related aspects of visualisation and construction. This overview concludes that much research about the deep process of the development and the learning of visualisation and reasoning is still needed
The mediation of mathematical learning through the use of pedagogical tools: a sociocultural analysis
No full textA sociocultural analysis suggests that pedagogical artifacts employed in the teaching and learning of mathematics both enable and constrain learning. This paper summarises three classroom studies of mathematics learning that have utilised a sociocultural approach. Each of the studies indicates how insight can be gained into the ways in which students attempt to make sense of the mathematics they encounter
Opportunities for the development of geometrical reasoning in current textbooks in the UK and Japan
No full textDeveloping a good model of the school geometry curriculum continues to be one of the most important tasks in curricular design in mathematics. This paper reports on an initial analysis of current best-selling textbooks used in lower secondary schools in Japan and the UK (specifically England and Scotland). The analysis indicates that, following the specification of the mathematics curriculum in these countries, Japanese textbooks set out to develop students’ deductive reasoning skills through the explicit teaching of proof in geometry, whereas comparative UK textbooks tend, at this level, to concentrate on finding angles, measurement, drawing, and so on, coupled with a modicum of opportunities for conjecturing and inductive reasoning. The available research suggests that each approach has its own strengths and weaknesses. Finding ways of capitalising on the strengths and mitigating the weaknesses could prove helpful in formulating new curricular models and designing new student textbooks
The shaping of student knowledge: learning with dynamic geometry software
No full textThe focus of this paper is a software genre usually referred to as ‘dynamic geometry’ because of the ability of the user to dynamically manipulate geometrical figures created with the software tool. Using data from a longitudinal study of 12-13 students’ use of dynamic geometry software, the focus of the analysis is on the interpretations the students make of geometrical objects and relationships when using this form of software. The analysis suggests that the students’ mathematical reasoning is shaped by their interactions with the software in that their ability to explain geometrical facts and relationships evolves from imprecise, ‘everyday’ expressions, through reasoning that is overtly mediated by the software environment, to mathematical explanations of the geometric situation that transcend the particular tool being used. Such findings suggest that curriculum initiatives that encourage the use of dynamic geometry software are appropriate but that the incorporation of such software into classroom practices is unlikely to be straightforward
Visualisation, imagery, and the development of geometrical reasoning
No full textThis report focuses on some aspects of the nature and role of visualisation and imagery in the teaching and learning of mathematics, particularly as a component in the development of geometrical reasoning. Issues briefly addressed include the relationship between imagery and perception, imagery and memory, the nature of dynamic images, and the interaction between imagery and concept development. The report concludes with a series of questions that may provide a suitable programme for research and lays the foundation for further work of the BSRLM geometry working group
Exploratory talk within collaborative small groups in mathematics
No full textThis report describes one aspect of a wider research study on exploratory talk within collaborative small groups in secondary mathematics lessons. It outlines students’ views of using collaborative activity to learn mathematics. The fuller research study explores the extent to which exploratory talk occurs in collaborative peer groups in secondary mathematics classrooms
The process of re-designing the geometry curriculum: the case of the Mathematical Association in England in the early twentieth century
No full textThis paper examines a key period of change in geometry teaching in England. Our focus is the character and nature of the recommendations of the 1902 geometry report of the UK Mathematical Association. We analyse historical documents of the Mathematical Association using a theoretical framework informed by work in the sociology of education. Our analysis shows that the character and recommendations of the Mathematical Association report were influenced by various factors including: that Mathematical Association members at the time still respected the traditional Euclidean approach to geometry as a basis for school geometry; that the academic and ‘power’ resources available to the Mathematical Association at the time were not sufficient to enable a complete change from the traditional approach; that a lack of consensus between the various members of the Mathematical Association prevented a more radical proposal; and that the general climate in schools at that time was not prepared for far-reaching changes to the teaching of geometry. These findings accord with other research on educational reform which indicates that curriculum change processes are invariably complex and often subject to much politicking
Researching geometrical intuition
No full textThis paper describes an exploratory study of the nature and role of geometrical intuition in the solving of geometrical problems. The findings suggest that geometrical intuition plays an important part in the critical decisions that problem solvers make when tackling geometrical problems
Using imagery to solve spatial problems
No full textThis report focuses on the use of imagery to solve a range of spatial problems. The research projects reviewed in this report offer some insight into the range of strategies used by solvers of spatial problems and point to relationships between spatial and verbal skills
The process of re-designing the geometry curriculum: the case of the Mathematical Association in England in the early 20th Century
No full textThis paper examines a key period of change in geometry teaching in England. Our focus is the character and nature of the recommendations of the geometry report of the UK Mathematical Association in 1902. We analyse historical documents of the Mathematical Association using a theoretical framework developed from Cooper’s model. Our analysis shows that the character and recommendations of the Mathematical Association report was influenced by various factors including: that the Mathematical Association members still respected the traditional Euclidean approach to geometry as a basis for school geometry; that the academic and power resources available to the Mathematical Association at the time were not sufficient for a complete change from the traditional approach; that conflicts between the various members of the Mathematical Association prevented a complete consensus; and that the climate outside the teaching committee of the Mathematical Association was not ready for radical reform at that time
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