Pre-service Elementary Teachers\u27 Use of Spatial Diagrams: Investigations of Unbounded Shearing on Solid Figures

Mithala and Balacheff (2019) describe three difficulties with two-dimensional representations of three-dimensional geometrical objects: “it is no longer possible to confuse the representation with the object itself,” visually observed relationships can be misleading, and analysis of the representation requires the use of lower-dimensional theoretical properties. Despite these difficulties, students are routinely expected to learn about three-dimensional figures through interacting with two-dimensional inscriptions. Three-dimensional alternatives include diagrams realized through various spatial inscriptions (e.g., Dimmel & Bock, 2019; Gecu-Parmaksiz & Delialioglu, 2019; Lai, McMahan, Kitagawa & Connolly, 2016; Ng and Sinclair, 2018). Such diagrams are three-dimensional in the sense that they occupy real (e.g., 3D pen drawings) or rendered (e.g., Virtual Reality/Augmented Reality environments) spaces as opposed to being inscribed or displayed on surfaces. Digital spatial diagrams can be grasped and transformed by gestures (e.g., stretching, pinching, spinning), even though they can’t be physically touched (Dimmel & Bock, 2019). Spatial diagrams make it possible to use natural movements of one’s head or body to explore figures from new perspectives (e.g., one can step inside a diagram), as they natively share the three-dimensional space. In this study I ask: How do learners use perspective to make arguments while exploring spatial diagrams? In particular, how do participants use perspectives outside and within geometric figures to make arguments while exploring spatial diagrams? To investigate this question, I designed a large-scale spatial diagram of a pyramid whose apex and base were confined to parallel planes. The diagram was rendered in an apparently unbounded spatial canvas that was accessible via a head-mounted display. The pyramid was roughly 1 meter in height and the parallel planes appeared to extend indefinitely when viewed from within the immersive environment. I created this diagram as a mathematical context for exploring shearing, a “continuous and temporal” measure-preserving transformation of plane and solid figures (Ng & Sinclair, 2015, p.85). I report on pairs of pre-service elementary teachers’ arguments about shearing of pyramids, using Pedemonte and Balacheff’s (2016) ck¢-enriched Toulmin model of argument. Shearing is a mathematical context that is likely novel to pre-service elementary teachers and provides an opportunity to connect transformations of plane and solid figures. Participants used perspectives outside and within the diagram to make arguments about the shearing of pyramids that would not be practicable with rigid three-dimensional models or dynamic two-dimensional representations. The results of this study suggest that the dimensionality of the spatial diagrams supported participants’ arguments about three-dimensional figures without mediation through projection or lower-dimensional components. The findings of this study offer a case that challenges the constraints of two-dimensional representations of three-dimensional figures, while maintaining theoretical constraints in a spatiographically accurate representation

Knowledge-based systems and geological survey

This personal and pragmatic review of the philosophy underpinning methods of geological surveying suggests that important influences of information technology have yet to make their impact. Early approaches took existing systems as metaphors, retaining the separation of maps, map explanations and information archives, organised around map sheets of fixed boundaries, scale and content. But system design should look ahead: a computer-based knowledge system for the same purpose can be built around hierarchies of spatial objects and their relationships, with maps as one means of visualisation, and information types linked as hypermedia and integrated in mark-up languages. The system framework and ontology, derived from the general geoscience model, could support consistent representation of the underlying concepts and maintain reference information on object classes and their behaviour. Models of processes and historical configurations could clarify the reasoning at any level of object detail and introduce new concepts such as complex systems. The up-to-date interpretation might centre on spatial models, constructed with explicit geological reasoning and evaluation of uncertainties. Assuming (at a future time) full computer support, the field survey results could be collected in real time as a multimedia stream, hyperlinked to and interacting with the other parts of the system as appropriate. Throughout, the knowledge is seen as human knowledge, with interactive computer support for recording and storing the information and processing it by such means as interpolating, correlating, browsing, selecting, retrieving, manipulating, calculating, analysing, generalising, filtering, visualising and delivering the results. Responsibilities may have to be reconsidered for various aspects of the system, such as: field surveying; spatial models and interpretation; geological processes, past configurations and reasoning; standard setting, system framework and ontology maintenance; training; storage, preservation, and dissemination of digital records

Visual communication of technology: its impact on designing and innovation in industrial and engineering design education

Visual communication (VC) resources can be seen as playing an increasingly important role in delivery and learning systems in today s design and technology education. The performance of current tools and resources is the primary concern of this research, and particularly whether they take full advantage of VC when delivering technological information to industrial design (ID) and engineering design (ED) students. This thesis sought key principles behind the visual communication of technology (VCT) and its association to designing, creativity and innovation through a literature survey. The findings concluded that there were many such assertions made with little evidence concerning the associations suggested. Some guiding sources and key emerging principles (KEPs) for good VCT practices were established. A miniature-kite-designing exercise was conducted as a case study for the purpose of examining the links between VCT, designing and creativity and/or innovation. Kite-technological-information posters were used as the VCT tool for the kite-designing case. A comparative study of kite-designing was conducted in Malaysia to check the reliability of the study, and another validation study was carried out for the purpose of establishing the validity of the data gathering. Visual technological information (VTI) for kite design (or a kite-poster) was refined accordingly to the KEPs established from the literature review, and its visual impact was tested through the use of eye-tracking technology. Some selected current and historical visual tools, which have been used in design and technology communication and were recognised as having positive impacts were analysed and articulated in order to reveal a deeper understanding of the KEPs. These were further validated through eye-tracking of reading patterns of participants on those selected visuals. The perceptual responses toward those visuals were also recorded and analysed. A theoretical research framework was established to investigate VTI representation used in books by Ashby (1999) and Ashby and Johnson (2002), in new authors scholarly papers (METU, 2010), and of the author s analysis and redesign of some of those studied VTIs based on the KEPs emerging from the research. A questionnaire survey was conducted within a number of higher education institutions in 3 regions around the world in order to achieve reliable data gathering. This third case study was validated through experts discussion of the findings and related issues. Within these three case studies, a mixture of scientific (using the eye-tracker device) and conventional methods (questionnaires, interviews, discussion group and comparative studies), and also others methods such as design workshops, analysing existing resources, using own practice of design-and-redesign activities were conducted to provide quantitative and qualitative measurements to empirically validate the literature search. Evidence of links between VCT, designerly activities which involved knowledge, skills and values within the technological communication, and of facilitating creativity was obtained. Empirical evidence showed that VTIs were effective in communicating knowledge, skills and values; where the KEPs criteria had played essential roles in enriching the visual emphasis of VTIs. The redesigning exercise using the author s own practice, which articulated the KEPs through the redesign of the existing VTIs for the purpose of more effective VCT, again obtained significant evidence of visual effectiveness and easy understanding capability. Evidence from the analysis of 2 books on materials technology for ID and ED students, views from the 2 materials experts, and the literature review suggested that ID and ED students require difference types of representational models and graphical strategies of VCT in their learning. However, the empirical data from the research, which was supported by one of the materials experts, suggested that ID and ED students even with different cultural backgrounds did not require different VTIs or the use of different VCT strategies for effective communication

Making mathematics on paper : constructing representations of stories about related linear functions

This dissertation takes up the problem of applied quantitative inference as a central question for cognitive science, asking what must happen during problem solving for people to obtain a meaningful and effective representation of the problem. The core of the dissertation reports exploratory empirical studies that seek to answer the descriptive question of how quantitative inferences are generated, pursued, and evaluated by problem solvers with different mathematical backgrounds. These are framed against a controversy, described in Chapter 2, over the theoretical and empirical validity of current cognitive science accounts of problems, solutions, knowledge, and competent human activity outside of laboratory or school settings.Chapter 3 describes a written protocol study of algebra story problem solving among advanced undergraduates in computer science. A relatively open-ended interpretive framework for "problem-solving episodes" is developed and applied to their written solution attempts. The resulting description of problem-solving activities gives a surprising image of competence among an important occupational target for standard mathematics instruction.Chapter 4 follows these results into detailed verbal problem-solving interviews with algebra students and teachers. These provide a comparison across settings and levels of competence for the same set of problems. The results corroborate similar generative activities outside the standard formalism of algebra across levels of competence. Notable among these nonalgebraic problem-solving activities are "model-based reasoning tactics," in which people reason about quantitative relations in terms of the dimensional structure of functional relations described in the problem. These tactics support different activities within surrounding solution attempts and usually describe "states" in the problem's situational structure.Chapter 5 holds these activities accountable to local combinations of notation and quantity, reinterpreting results for model-based reasoning in an ecological analysis of material designs for constructing and evaluating quantitative inferences. This analysis brings forward important relations between what material designs afford problem solvers and the complexity of episodic structure observed in their solution attempts. The dissertation closes with a reappraisal of the relationship between knowledge, person, and setting and, I will argue, puts us on a more promising track for a descriptively adequate theoretical account of constructing mathematical representations that support applied quantitative inference

Mental Representation and the Construction of Conceptual Understanding in Electronics Education

Learning about abstract electronics concepts can be difficult due to the hidden nature of the phenomena of interest. Developing understanding about electronics is therefore challenging because voltage cannot be readily observed; only the outcomes of the behaviour of voltage can be observed. Consequently modelling the phenomena of interest becomes a crucial factor in supporting learners in their development of knowledge and understanding. Visualisation skills have been promoted as important when modelling knowledge in different forms, supporting learners in their development of knowledge and understanding. Current research about electronics education, however, has tended to focus on learners’ misconceptions, experimental methods and interventions focusing on theoretical aspects of knowledge. Perspectives on learners’ actual constructions of knowledge in practice are not common. The aim of this research study, therefore, was to explore the use of external visual representations in support of learning about electronics concepts, within the context of Secondary Design and Technology education. The study adopts a case study approach and uses an interpretative cross-case synthesis methodology to explore a specific case of representation use among one class of Year 10 students. The analytical framework is designed to focus on the translation of and transition between multiple representations, including computer program code, and the representation of phenomena at three levels of representation: observable, symbolic and abstract. Data collection involved the observation of learners engaged with learning activities, documents collected from these activities, individual semi-structured interviews and participant characteristics data collected from course records. The findings show that common processes of learning are accompanied by individual developments in meaning and understanding. Individual understanding was characterised with the creation of four cognitive profiles representing key learner constructs. Understanding about abstract concepts was shown to benefit from representations where concrete referents linked with practical experience. Electronics understanding was also shown to benefit from the explanatory use of program code as a supporting method with which to model and simulate circuit behaviour. The research approach involving the close observation of learners engaging with learning activities was found to provide a greater understanding of learners’ approaches to learning in practice. The outcomes are applied to the practice of teaching electronics and modifications to the research are suggested for future researchers interested in the issues of teaching, learning and concept development in electronics education

Felt_space infrastructure: Hyper vigilant spatiality to valence the visceral dimension

Felt_space infrastructure: Hypervigilant spatiality to valence the visceral dimension. This thesis evolves perception as a hypothesis to reframe architectural praxis negotiated through agent-situation interaction. The research questions the geometric principles of architectural ordination to originate the ‘felt_space infrastructure’, a relational system of measurement concerned with the role of perception in mediating sensory space and the cognised environment. The methodological model for this research fuses perception and environmental stimuli, into a consistent generative process that penetrates the inner essence of space, to reveal the visceral parameter. These concepts are applied to develop a ‘coefficient of affordance’ typology, ‘hypervigilant’ tool set, and ‘cognitive_tope’ design methodology. Thus, by extending the architectural platform to consider perception as a design parameter, the thesis interprets the ‘inference schema’ as an instructional model to coordinate the acquisition of spatial reality through tensional and counter-tensional feedback dynamics. Three site-responsive case studies are used to advance the thesis. The first case study is descriptive and develops a typology of situated cognition to extend the ‘granularity’ of perceptual sensitisation (i.e. a fine-grained means of perceiving space). The second project is relational and questions how mapping can coordinate perceptual, cognitive and associative attention, as a ‘multi-webbed vector field’ comprised of attractors and deformations within a viewer-centred gravitational space. The third case study is causal, and demonstrates how a transactional-biased schema can generate, amplify and attenuate perceptual misalignment, thus triggering a visceral niche. The significance of the research is that it progresses generative perception as an additional variable for spatial practice, and promotes transactional methodologies to gain enhanced modes of spatial acuity to extend the repertoire of architectural practice

The design and formative evaluation of computer based qualitative modelling environments for schools

This research investigated how computers might enable young learners to build models so that they can express and explore their ideas and hence they can gain understanding of the subject matter as well as developing modelling abilities. A design for a qualitative modelling environment was produced, which incorporated a simple rule-based metaphor that could be presented as a diagram. The design was founded on empirical evidence of children modelling as well as theoretical grounds. This research originated in and contributed to the Modus Project, a joint venture between King's College London and the Advisory Unit for Microtechnology in Education, Hertfordshire County Council. A prototype of the software, Expert Builder, was implemented by software engineers from the Modus team. The initial stage of evaluation, based on a questionnaire survey and widespread trialling, established that the tool could be used in a wide range of educational contexts. A detailed study of children using the qualitative modelling environment was conducted in three primary schools involving 34 pupils, aged nine to 11. They used the modelling environment within the classroom in their normal curriculum work over one school year on a variety of topics assisted by their class teacher. The modelling environment enabled cooperative groupwork and supported pupils in consolidating and extending their knowledge. A formative evaluation was used to inform the design of a revised version of the software. In addition the experiences of children using the software were analysed. A framework was developed which characterised the stages in the modelling process. Teachers in the study were observed to demonstrate the earlier stages of the modelling process and then to set tasks for the children based on the later stages of building and testing the models. The evidence suggested that the abilities to model were context dependent so that pupils as young as nine years old could undertake the whole modelling process provided that they were working on subject matter with which they were familiar. The teachers made use of computer based modelling in order to develop and reinforce pupils' understanding of various aspects of the curriculum and therefore they chose modelling tasks for the children. However in one school the children were given the opportunity to design and build models of their own choice and they demonstrated that they were able to carry out all the stages in the modelling process. A taxonomy of computer based modelling is proposed which could be used to inform decisions about the design of the modelling curriculum and could provide a basis for researchers investigating the modelling process. This would be useful for further research into the intellectual and social activities of people learning to model and for teachers seeking to develop a framework for the modelling curriculum. The National Curriculum (Department of Education and Science and the Welsh Office, 1990) specifies that early steps in computer based modelling should involve exploring models developed by others and pupils are not required to build models themselves until level 7 which is expected to be reached by more able 14 year-olds. In this thesis it is argued that a modelling curriculum should provide early opportunities for pupils to undertake the modelling process by developing simple models on familiar subject matter as well as opportunities for exploring more complex models as evidence from research reported in this thesis suggests that younger pupils are able to build models. In this way pupils will be enabled to acquire modelling capability as well as developing their understanding of a range of topics through modelling. Progression in modelling capability would involve constructing models of more complex situations and using a wider range of modelling environments

Mixed media modelling of technological concepts in electricity, methods for supporting learning styles

The overarching objective of this research is to recognize the learning styles of engineering and technology students and to propose pedagogical methods for the comprehension of technological concepts in electricity. The topic of electrical resistor-capacitor (RC) circuits has been chosen because it is fundamental to engineering and technology courses. There is substantial evidence to suggest that students find such a concept difficult to grasp. The focus of the research lies in explicating undergraduate students cognitive structures about RC circuits, and proposing a method related to students learning styles of how these cognitive structures may be enhanced. The main thesis argument claims that the transfer of knowledge from familiar RC circuit configurations to unfamiliar RC circuit configurations does not occur easily even if the problem-space is kept identical. The methodology used in this research is a mixed-method approach employing qualitative and quantitative data-gathering and analysis processes. This research concludes that the reasons for lack of transfer of knowledge stem from conceptual and perceptual constraints. Constraints involve: (a) which analogical models are employed in relation to the RC circuit, (b) how the circuit schematic diagram is drawn, and (c) relations between analogy, circuit schematic diagram, voltage-time graphs and verbal jargon used to describe circuit behaviour. The research presents a variety of novel, custom-designed learning aids which are employed within the research methodology to rectify the lack of transfer of knowledge for the RC circuits considered in the study. The design of these learning aids is based on the concept of embodied cognition and mainly makes use of visual and kinaesthetic means to appeal to students who may have different learning styles. The use of such learning aids is proposed as a complementary teaching strategy. The approach taken in this research and its outcomes are significant because they continue to inform the research and educational communities about how human development may be fostered through engineering and technology education (Barak and Hacker, 2011)

Formalizing graphical notations

The thesis describes research into graphical notations for software engineering, with a principal interest in ways of formalizing them. The research seeks to provide a theoretical basis that will help in designing both notations and the software tools that process them. The work starts from a survey of literature on notation, followed by a review of techniques for formal description and for computational handling of notations. The survey concentrates on collecting views of the benefits and the problems attending notation use in software development; the review covers picture description languages, grammars and tools such as generic editors and visual programming environments. The main problem of notation is found to be a lack of any coherent, rigorous description methods. The current approaches to this problem are analysed as lacking in consensus on syntax specification and also lacking a clear focus on a defined concept of notated expression. To address these deficiencies, the thesis embarks upon an exploration of serniotic, linguistic and logical theory; this culminates in a proposed formalization of serniosis in notations, using categorial model theory as a mathematical foundation. An argument about the structure of sign systems leads to an analysis of notation into a layered system of tractable theories, spanning the gap between expressive pictorial medium and subject domain. This notion of 'tectonic' theory aims to treat both diagrams and formulae together. The research gives details of how syntactic structure can be sketched in a mathematical sense, with examples applying to software development diagrams, offering a new solution to the problem of notation specification. Based on these methods, the thesis discusses directions for resolving the harder problems of supporting notation design, processing and computer-aided generic editing. A number of future research areas are thereby opened up. For practical trial of the ideas, the work proceeds to the development and partial implementation of a system to aid the design of notations and editors. Finally the thesis is evaluated as a contribution to theory in an area which has not attracted a standard approach

The structure and function of diagrams in environmental design : a computational inquiry

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Urban Studies and Planning, 1989.Vita.Includes bibliographical references (leaves 252-261).by Stephen McTee Ervin.Ph.D