Reasoning By Contradiction in Dynamic Geometry

This paper addresses contributions that dynamic geometry systems (DGSs) may give in reasoning by contradiction in geometry. We present analyses of three excerpts of students’ work and use the notion of pseudo object, elaborated from previous research, to show some specificities of DGS in constructing proof by contradiction. In particular, we support the claim that a DGS can offer “guidance” in the solver’s development of an indirect argument thanks to the potential it offers of both constructing certain properties robustly, and of helping the solver perceive pseudo objects

A Vowel Analysis of the Northwestern University-Children\u27s Perception of Speech Evaluation Tool

In an analysis of the speech perception evaluation tool, the Northwestern University – Children’s Perception of Speech test, the goal was to determine whether the foil words and the target word were phonemically balanced across each page of test Book A, as it corresponds to the target words presented in Test Form 1 and Test Form 2 independently. Based on vowel sounds alone, variation exists in the vowels that appear on a test page on the majority of pages. The corresponding formant frequencies, at all three resonance levels for both the average adult male speaker and the average adult female speaker, revealed that the target word could be easily distinguished from the foil words on the premise of percent differences calculated between the formants of the target vowel and the foil vowels. For the population of children with hearing impairments, especially those with limited or no access to the high frequencies, the NU-CHIPS evaluation tool may not be the best indicator of the child’s speech perception ability due to significant vowel variations

Maintaining dragging and the pivot invariant in processes of conjecture generation

In this paper, we analyze processes of conjecture generation in the context of open problems proposed in a dynamic geometry environment, when a particular dragging modality, maintaining dragging, is used. This involves dragging points while maintaining certain properties, controlling the movement of the figures. Our results suggest that the pragmatic need of physically controlling the simultaneous movements of the different parts of figures can foster the production of two chains of successive properties, hinged together by an invariant that we will call pivot invariant. Moreover, we show how the production of these chains is tied to the production of conjectures and to the processes of argumentation through which they are generated.Comment: Research report at the 40th PME Conference, Hungar

Disparity map generation based on trapezoidal camera architecture for multiview video

Visual content acquisition is a strategic functional block of any visual system. Despite its wide possibilities, the arrangement of cameras for the acquisition of good quality visual content for use in multi-view video remains a huge challenge. This paper presents the mathematical description of trapezoidal camera architecture and relationships which facilitate the determination of camera position for visual content acquisition in multi-view video, and depth map generation. The strong point of Trapezoidal Camera Architecture is that it allows for adaptive camera topology by which points within the scene, especially the occluded ones can be optically and geometrically viewed from several different viewpoints either on the edge of the trapezoid or inside it. The concept of maximum independent set, trapezoid characteristics, and the fact that the positions of cameras (with the exception of few) differ in their vertical coordinate description could very well be used to address the issue of occlusion which continues to be a major problem in computer vision with regards to the generation of depth map

Object orientation without extending Z

The good news of this paper is that without extending Z, we can elegantly specify object-oriented systems, including encapsulation, inheritance and subtype polymorphism (dynamic dispatch). The bad news is that this specification style is rather different to normal Z specifications, more abstract and axiomatic, which means that it is not so well supported by current Z tools such as animators. It also enforces behavioural subtyping, unlike most object-oriented programming languages. This paper explains the proposed style, with examples, and discusses its advantages and disadvantages

The adaptive advantage of symbolic theft over sensorimotor toil: Grounding language in perceptual categories

Using neural nets to simulate learning and the genetic algorithm to simulate evolution in a toy world of mushrooms and mushroom-foragers, we place two ways of acquiring categories into direct competition with one another: In (1) "sensorimotor toil,” new categories are acquired through real-time, feedback-corrected, trial and error experience in sorting them. In (2) "symbolic theft,” new categories are acquired by hearsay from propositions – boolean combinations of symbols describing them. In competition, symbolic theft always beats sensorimotor toil. We hypothesize that this is the basis of the adaptive advantage of language. Entry-level categories must still be learned by toil, however, to avoid an infinite regress (the “symbol grounding problem”). Changes in the internal representations of categories must take place during the course of learning by toil. These changes can be analyzed in terms of the compression of within-category similarities and the expansion of between-category differences. These allow regions of similarity space to be separated, bounded and named, and then the names can be combined and recombined to describe new categories, grounded recursively in the old ones. Such compression/expansion effects, called "categorical perception" (CP), have previously been reported with categories acquired by sensorimotor toil; we show that they can also arise from symbolic theft alone. The picture of natural language and its origins that emerges from this analysis is that of a powerful hybrid symbolic/sensorimotor capacity, infinitely superior to its purely sensorimotor precursors, but still grounded in and dependent on them. It can spare us from untold time and effort learning things the hard way, through direct experience, but it remain anchored in and translatable into the language of experience

Review article on the role of visualisation in mathematics conceptualisation and learning

Visualisation is a natural element at the root of mathematical thought, in the discovery of relations between mathematical objects, and in the transmission and communication of mathematics. This review focuses on the introductory chapter of one of the most famous books on this topic, by a widely published writer on pedagogy of mathematics, discussing at the same time implications for undergraduate teaching in the subject

Developing geometrical reasoning in the secondary school: outcomes of trialling teaching activities in classrooms, a report to the QCA

This report presents the findings of the Southampton/Hampshire Group of mathematicians and mathematics educators sponsored by the Qualifications and Curriculum Authority (QCA) to develop and trial some teaching/learning materials for use in schools that focus on the development of geometrical reasoning at the secondary school level. The project ran from October 2002 to November 2003. An interim report was presented to the QCA in March 2003. 1. The Southampton/Hampshire Group consisted of five University mathematicians and mathematics educators, a local authority inspector, and five secondary school teachers of mathematics. The remit of the group was to develop and report on teaching ideas that focus on the development of geometrical reasoning at the secondary school level. 2. In reviewing the existing geometry curriculum, the group endorsed the RS/ JMC working group conclusion (RS/ JMC geometry report, 2001) that the current mathematics curriculum for England contains sufficient scope for the development of geometrical reasoning, but that it would benefit from some clarification in respect of this aspect of geometry education. Such clarification would be especially helpful in resolving the very odd separation, in the programme of study for mathematics, of ‘geometrical reasoning’ from ‘transformations and co-ordinates’, as if transformations, for example, cannot be used in geometrical reasoning. 3. The group formulated a rationale for designing and developing suitable teaching materials that support the teaching and learning of geometrical reasoning. The group suggests the following as guiding principles: • Geometrical situations selected for use in the classroom should, as far as possible, be chosen to be useful, interesting and/or surprising to pupils; • Activities should expect pupils to explain, justify or reason and provide opportunities for pupils to be critical of their own, and their peers’, explanations; • Activities should provide opportunities for pupils to develop problem solving skills and to engage in problem posing; • The forms of reasoning expected should be examples of local deduction, where pupils can utilise any geometrical properties that they know to deduce or explain other facts or results. • To build on pupils’ prior experience, activities should involve the properties of 2D and 3D shapes, aspects of position and direction, and the use of transformation-based arguments that are about the geometrical situation being studied (rather than being about transformations per se); • The generating of data or the use of measurements, while playing important parts in mathematics, and sometimes assisting with the building of conjectures, should not be an end point to pupils’ mathematical activity. Indeed, where sensible, in order to build geometric reasoning and discourage over-reliance on empirical verification, many classroom activities might use contexts where measurements or other forms of data are not generated. 4. In designing and trialling suitable classroom material, the group found that the issue of how much structure to provide in a task is an important factor in maximising the opportunity for geometrical reasoning to take place. The group also found that the role of the teacher is vital in helping pupils to progress beyond straightforward descriptions of geometrical observations to encompass the reasoning that justifies those observations. Teacher knowledge in the area of geometry is therefore important. 5. The group found that pupils benefit from working collaboratively in groups with the kind of discussion and argumentation that has to be used to articulate their geometrical reasoning. This form of organisation creates both the need and the forum for argumentation that can lead to mathematical explanation. Such development to mathematical explanation, and the forms for collaborative working that support it, do not, however, necessarily occur spontaneously. Such things need careful planning and teaching. 6. Whilst pupils can demonstrate their reasoning ability orally, either as part of group discussion or through presentation of group work to a class, the transition to individual recording of reasoned argument causes significant problems. Several methods have been used successfully in this project to support this transition, including 'fact cards' and 'writing frames', but more research is needed into ways of helping written communication of geometrical reasoning to develop. 7. It was found possible in this study to enable pupils from all ages and attainments within the lower secondary (Key Stage 3) curriculum to participate in mathematical reasoning, given appropriate tasks, teaching and classroom culture. Given the finding of the project that many pupils know more about geometrical reasoning than they can demonstrate in writing, the emphasis in assessment on individual written response does not capture the reasoning skills which pupils are able to develop and exercise. Sufficient time is needed for pupils to engage in reasoning through a variety of activities; skills of reasoning and communication are unlikely to be absorbed quickly by many students. 8. The study suggests that it is appropriate for all teachers to aim to develop the geometrical reasoning of all pupils, but equally that this is a non-trivial task. Obstacles that need to be overcome are likely to include uncertainty about the nature of mathematical reasoning and about what is expected to be taught in this area among many teachers, lack of exemplars of good practice (although we have tried to address this by lesson descriptions in this report), especially in using transformational arguments, lack of time and freedom in the curriculum to properly develop work in this area, an assessment system which does not recognise students’ oral powers of reasoning, and a lack of appreciation of the value of geometry as a vehicle for broadening the curriculum for high attainers, as well as developing reasoning and communication skills for all students. 9. Areas for further work include future work in the area of geometrical reasoning, include the need for longitudinal studies of how geometrical reasoning develops through time given a sustained programme of activities (in this project we were conscious that the timescale on which we were working only enabled us to present 'snapshots'), studies and evaluation of published materials on geometrical reasoning, a study of 'critical experiences' which influence the development of geometrical reasoning, an analysis of the characteristics of successful and unsuccessful tasks for geometrical reasoning, a study of the transition from verbal reasoning to written reasoning, how overall perceptions of geometrical figures ('gestalt') develops as a component of geometrical reasoning (including how to create the links which facilitate this), and the use of dynamic geometry software in any (or all) of the above.10. As this group was one of six which could form a model for part of the work of regional centres set up like the IREMs in France, it seems worth recording that the constitution of the group worked very well, especially after members had got to know each other by working in smaller groups on specific topics. The balance of differing expertise was right, and we all felt that we learned a great deal from other group members during the experience. Overall, being involved in this type of research and development project was a powerful form of professional development for all those concerned. In retrospect, the group could have benefited from some longer full-day meetings to jointly develop ideas and analyse the resulting classroom material and experience rather than the pattern of after-school meetings that did not always allow sufficient time to do full justice to the complexity of many of the issues the group was tackling