1,257 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 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
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 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
Reconsidering Arminius: Beyond the Reformed and Wesleyan Divide
No full textThe theology of Dutch theologian Jacob Arminius has been misinterpreted and caricatured in both Reformed and Wesleyan circles. By revisiting Arminius's theology, the book hopes to be a constructive voice in the discourse between so-called Calvinists and Arminians. Traditionally, Arminius has been treated as a divisive figure in evangelical theology. Indeed, one might be able to describe classic evangelical theology up into the twentieth century in relation to his work: one was either an Arminian and accepted his theology or one was a Calvinist and rejected his theology. Although various other movements within evangelicalism have provided additional contour to the movement (fundamentalism, Pentecostalism, etc.), the Calvinist-Arminian 'divide' remains a significant one. What this book seeks to correct is the misinterpretation of Arminius as one whose theology provides a stark contrast to the Reformed tradition as a whole. Indeed, this book will demonstrate instead that Arminius is far more in line with Reformed orthodoxy than popularly believed and show that what emerges as Arminianism in the theology of the Remonstrants and Wesleyan movements was in fact not the theology of Arminius but a development of and sometimes departure from it. This book also brings Arminius into conversation with modern theology. To this end, it includes essays on the relationship between Arminius's theology and open theism and Neo-Reformed theology. In this way, this book fulfills the promise of the title by showing ways in which Arminius's theology—once properly understood—can serve as a resource of evangelical Wesleyans and Calvinists doing theology together today. Abbreviations -- ix Introduction. Reconsidering Arminius: Recasting the Legacy / Mark H. Mann and Mark G. Bilby -- xi-xix Chapter 1. Consecrated through Suffering : the Office of Christ in the Theology of Jacob Arminius / Richard A. Muller -- 1-21 Chapter 2. Was Arminius an Unwitting Determinist? : Another Look at Arminius's Modal Logic / Thomas H. McCall -- 23-37 Chapter 3. Beyond Luther, beyond Calvin, beyond Arminius : the Pilgrims and the Remonstrants in Leiden, 1609-1620 / Jeremy Dupertuis Bangs -- 39-69 Chapter 4. The Loss of Arminius in Wesleyan-Arminian Theology / W. Stephen Gunter -- 71-90 Chapter 5. Jacob Arminius and Jonathan Edwards on the doctrine of creation / Oliver D. Crisp -- 91-112 Chapter 6. Convergence in the "Reformed" theologies of T.F. Torrance and Jacob Arminius / E. Jerome Van Kuiken -- 113-135 Chapter 7. Was Arminius an open theist? : meticulous providence in the theology of Jacob Arminius / John Mark Hicks -- 137-160 Conclusion. Arminius Reconsidered : Thoughts on Arminius and Contemporary Theological Discourse for the Church Today / Keith D. Stanglin -- 161-167 Contributors -- 169Published@book{stanglin2014reconsidering, title= {Reconsidering Arminius: Beyond the Reformed and Wesleyan Divide}, author= {Stanglin, Keith D and Bilby, Mark G and Mann, Mark H}, year= {2014}, publisher= {Kingswood Books}}978142679654
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 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
Critique [of Self-Evaluation of Black and White College Students by Keith D. Parker]
No full textThe article by Keith D. Parker raises interesting theoretical and methodological questions, but this review focuses on the latter. The author is correct in his critical assessment of black self-esteem research (BSER) methodology. Projective measures have been used in most cases and questions have been raised regarding the validity of such self-esteem measures and therefore about the believability of BSER findings.[1] In addition, blacks and whites tested have not been representative of the general black or white communities of the United States, yet inferences to and comparisons of the populations have been made. Finally, studies have employed non-multivariate statistical techniques which have prevented the use of controls
Can a Robot Have Free Will?
No full textUsing insights from cybernetics and an information-based understanding of biological systems, a precise, scientifically inspired, definition of free-will is offered and the essential requirements for an agent to possess it in principle are set out. These are: (a) there must be a self to self-determine; (b) there must be a non-zero probability of more than one option being enacted; (c) there must be an internal means of choosing among options (which is not merely random, since randomness is not a choice). For (a) to be fulfilled, the agent of self-determination must be organisationally closed (a “Kantian whole”). For (c) to be fulfilled: (d) options must be generated from an internal model of the self which can calculate future states contingent on possible responses; (e) choosing among these options requires their evaluation using an internally generated goal defined on an objective function representing the overall “master function” of the agent and (f) for “deep free-will”, at least two nested levels of choice and goal (d–e) must be enacted by the agent. The agent must also be able to enact its choice in physical reality. The only systems known to meet all these criteria are living organisms, not just humans, but a wide range of organisms. The main impediment to free-will in present-day artificial robots, is their lack of being a Kantian whole. Consciousness does not seem to be a requirement and the minimum complexity for a free-will system may be quite low and include relatively simple life-forms that are at least able to learn
The Petrography, Geochemistry, and Petrogenesis of Two Granite Suites, Churchill Province, Thekulthili Lake, N.W.T.
No full textTitle: The Petrography, Geochemistry, and Petrogenesis of Two Granite Suites, Churchill Province, Thekulthili Lake, N.W.T., Author: D. Keith MacDonald, Location: ThodeA petrographic and geochemical analysis has been carried out on two granite suites with contrasting morphologies and field relations. Included is a discussion of their age relationships
relative to the Nonacho Series supracrustal sequence. The respective petrographic and petrochemical features of the two granitoids is given; followed by a comparison of the two, and a discussion on their modes of petrogenetic evolution. Petrographic descriptions of selected Nonacho supplement the granitoid study. Analytical methods and a discussion of the results are included.ThesisBachelor of Science (BSc
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