Geometric Knowledge Research Articles

Testing the effectiveness of the designing and applying process with certain circumstances in geometry teaching

According to the theory of active education and Mathematics teaching in secondary schools, there are five steps designed for geometrical knowledge and four stages for the process of applying the examined circumstances. The research objective aims to evaluate the effectiveness and feasibility of the suggestion focused on the process of constructing and manipulating situations in teaching Geometry. The research method uses the case study in the specific teaching process, which was conducted with 24 teachers who have taught Mathematics at secondary schools over three months from August 2019 to October 2019; and 30 secondary school students between May 2020 and July 2020. Finally, the result shows that all teachers agree that the design and application process is appropriate and effective for their lesson. In terms of the student's aspect, the case study reveals that the application of various steps in the process is beneficial for improving geometric deductive and spatial estimation, as well as using theoretical knowledge in the practical situation. Based on the research, this paper will clarify some solutions related to the way to design processes and utilize them in some Geometry content in secondary schools in Vietnam.

Reconnecting with the Ideal Tree: An Alternative to Heuristic Learning in Real-Time Search

In this paper, we present a conceptually simple, easy-to-implement real-time search algorithm suitable for a priori partially known environments. Instead of performing a series of searches towards the goal, like most Real-Time Heuristic Search Algorithms do, our algorithm follows the arcs of a tree T rooted in the goal state that is built initially using the heuristic h. When the agent observes that an arc in the tree cannot be traversed in the actual environment, it removes such an arc from T and our algorithm carries out a reconnection search whose objective is to find a path between the current state and any node in T. The reconnection search need not be guided by $h$, since the search objective is not to encounter the goal. Furthermore, h need not be updated. We implemented versions of our algorithm that utilize various blind search algorithms for reconnection. We show experimentally that these implementations significantly outperform state-of-the-art real-time heuristic search algorithms for the task of pathfinding in grids. In grids, our algorithms, which do not incorporate any geometrical knowledge, naturally behaves similarly to a bug algorithm, moving around obstacles, and never returning to areas that have been visited in the past. In addition, we prove theoretical properties of the algorithm.

Using Dynamic Geometry Software to Enhance Specialized Content Knowledge: Pre-Service Mathematics Teachers’ Perceptions

The study aimed to expose how pre-service teachers perceived the value of an inquiry-based geometrical task with dynamic geometry software for increasing and deepening their understanding of the task itself, and for heightening their mathematical, pedagogical, and technological knowledge in the case of reflection and transformation. Twenty-seven pre-service teachers studying to become middle-school (junior high) mathematics teachers were assigned an inquiry-based geometrical task to explore – both by conventional, low-tech means (paper and pencil) and then in a dynamic geometry software (DGS) environment – the properties of objects formed by reflections in polygons (triangles and quadrilaterals) and to offer conjectures and proofs regarding the final polygon’s shape and area concerning the original. The results show that the majority of participants indicated that DGS provided a valuable instrument for revising and expanding their knowledge, that the assignment demonstrated the value that technological tools have in teaching and learning processes, and the importance of adapting learning tasks in geometry to a dynamic geometry environment for expanding and deepening geometrical and pedagogical knowledge and reasoning skills.

Perception of geometric sequences and numerosity both predict formal geometric competence in primary school children

While most animals have a sense of number, only humans have developed symbolic systems to describe and organize mathematical knowledge. Some studies suggest that human arithmetical knowledge may be rooted in an ancient mechanism dedicated to perceiving numerosity, but it is not known if formal geometry also relies on basic, non-symbolic mechanisms. Here we show that primary-school children who spontaneously detect and predict geometrical sequences (non-symbolic geometry) perform better in school-based geometry tests indexing formal geometric knowledge. Interestingly, numerosity discrimination thresholds also predicted and explained a specific portion of variance of formal geometrical scores. The relation between these two non-symbolic systems and formal geometry was not explained by age or verbal reasoning skills. Overall, the results are in line with the hypothesis that some human-specific, symbolic systems are rooted in non-symbolic mechanisms.

POINT CLOUD EXPLOITATION FOR STRUCTURAL MODELING AND ANALYSIS: A RELIABLE WORKFLOW

Abstract. The digitization and geometric knowledge of the historical built heritage is currently based on point cloud, that rarely or only partially is used as digital twin for structural analysis. The present work deals with historical artefacts survey, with particular reference to masonry structures, aimed to their structural analysis and assessment. In detail, the study proposes a methodology capable of employing semi-directly the original data obtained from the 3D digital survey for the generation of a Finite Element Model (FEM), used for structural analysis of masonry buildings.The methodology described presents a reliable workflow with twofold purpose: the improvement of the transformation process of the point cloud in solid and subsequently obtain a high-quality and detailed model for structural analyses.Through the application of the methodology to a case study, the method consistency was assessed, regarding the smoothness of the whole procedure and the dynamic characterization of the Finite Element Model. The main improvement in respect with similar or our previous workflows is obtained by the introduction of the retopology in data processing, allowing the transformation of the raw data into a solid model with optimal balancing between Level of Detail (LOD) and computational weight. Another significant aspect of the optimized process is undoubtedly the possibility of faithfully respecting the semantics of the structure, leading to the discretization of the model into different parts depending on the materials. This work may represent an excellent reference for the study of masonry artefacts belonging to the existing historical heritage, starting from surveys and with the purpose to structural and seismic evaluations, in the general framework of knowledge-based preservation of heritage.

Learning a new geometric concept: The role of working memory and of domain-specific abilities.

It has been suggested that not only domain‐specific factors but also working memory (WM) may play a crucial role in mathematical learning included Geometry, but the issue has not been deeply explored. In the present study, we examined the role of domain‐specific factors and of verbal versus visuospatial WM on geometric learning of a new geometrical figure (trapezoid), never presented previously by the teachers participating to the study, after a lecture also involving manipulatives. Results on 105 children in their Year 4 indicated that not only some domain‐specific components (geometric declarative knowledge and calculation) but also visuospatial working memory had a significant specific impact on the ability of solving geometric problems requiring to calculate the perimeter and the area of the new figure. On the contrary, verbal WM and geometrical mental imagery did not offer a specific contribution. These findings could have important educational implications, stressing the importance of taking into account the main different aspects supporting the acquisition of geometry.

A fluid–structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU

This paper presents the development of a numerical model for violent hydrodynamics of free-surface flows interacting with flexible structures using the meshless smoothed particle hydrodynamics (SPH) method accelerated with a graphics processing unit (GPU). The present work implements a unified SPH framework to solve both the fluid and structural dynamics within the open-source SPH code DualSPHysics. Well-known deficiencies with SPH-based structural modelling (linear inconsistency, tensile instability and rank deficiency) are addressed by adopting a Total Lagrangian formulation with kernel correction and zero-energy mode suppression. The fluid–structure coupling is handled via manipulation of the existing boundary condition within DualSPHysics, which does not require any geometrical knowledge of the interface. This, together with the unified SPH framework, permits straightforward implementation within DualSPHysics and retains its parallel scalability on hardware acceleration platforms, in particular on GPUs. An extensive validation and profiling of the proposed model is carried out for a range of challenging 2-D and 3-D test cases incorporating violent free-surface flows, nonlinear structural dynamics and complex fluid–structure interactions, and its performance is characterised and quantified. The analysis shows good agreement between the proposed model and analytical/benchmark solutions from the literature. Furthermore, the GPU profiling shows speedups on a single GPU of up to 40, compared to an optimised 12-core (24 threads) central processing unit (CPU) version. These findings indicate the proposed model is a viable and efficient approach to simulating violent free-surface fluid–structure interactions.

Downsampled Imaging Geometric Modeling for Accurate CT Reconstruction via Deep Learning.

X-ray computed tomography (CT) is widely used clinically to diagnose a variety of diseases by reconstructing the tomographic images of a living subject using penetrating X-rays. For accurate CT image reconstruction, a precise imaging geometric model for the radiation attenuation process is usually required to solve the inversion problem of CT scanning, which encodes the subject into a set of intermediate representations in different angular positions. Here, we show that accurate CT image reconstruction can be subsequently achieved by downsampled imaging geometric modeling via deep-learning techniques. Specifically, we first propose a downsampled imaging geometric modeling approach for the data acquisition process and then incorporate it into a hierarchical neural network, which simultaneously combines both geometric modeling knowledge of the CT imaging system and prior knowledge gained from a data-driven training process for accurate CT image reconstruction. The proposed neural network is denoted as DSigNet, i.e., downsampled-imaging-geometry-based network for CT image reconstruction. We demonstrate the feasibility of the proposed DSigNet for accurate CT image reconstruction with clinical patient data. In addition to improving the CT image quality, the proposed DSigNet might help reduce the computational complexity and accelerate the reconstruction speed for modern CT imaging systems.

Formation of geometric knowledge and skills of primary school pupils

Formation of geometric knowledge and skills of primary school pupils

Διερεύνηση γεωμετρικών γνώσεων μαθητών με ήπιες εκπαιδευτικές ανάγκες

Geometry is a structural component of mathematics, with increased spatial and design requirements that cannot be easily met by students with mild disabilities. Systematic investigation of the difficulties encountered by students with mild disabilities in their effort to learn Geometry is a prerequisite for the implementation of effective intervention programs. However, research on this issue is relatively scarce. The aim of the present study was to assess the geometric knowledge of 54 students with mild disabilities (learning disabilities or ADHD) who attended the two last classes of elementary school. Participants were asked to recognize, describe and categorize geometric shapes and solid bodies that were presented in tactile mode and through pictorial representations. Semi-structured clinical interviews were used for gathering the data in the context of Curriculum Based Assessment and the Van Hiele’s model of geometrical thinking. Participants of both categories of mild disabilities presented difficulties in distinguishing shapes and bodies, properly using the terminology, and formatting inductive geometrical reasoning. Participants with learning disabilities had higher achievement when dealing with haptic relative to pictorial representations of geometric shapes and bodies. Sixth graders performed better than fifth graders. Results are discussed in terms of the differences between the two categories of mild disabilities as well as with regard to the implementation of intervention programs.

Geometric modeling: Background for processing the 3d objects

Processing the 3D objects is a research field of computer graphics that can be applied in 3D simulation, 3D image processing, game industry, etc. This is one of the mathematics-based research fields where geometrical knowledge is the background of existing methods. Geometric modeling is a branch of this research field based on the applied mathematics and computational geometry. It studies methods and solutions for the mathematical description of shapes or realistic objects in computer science or computer aided design. The rapid development of advanced techniques, like 3D scanners, can be used to support obtaining scanned data of any type of the real objects with different methods and solutions. This article investigates the studies based on the background of geometric modeling for processing the 3D Objects. Our contribution is focused on presentation of a completed method that is an integrated approach consisting of our previously proposed methods to reconstruct the 3D object from a 3D point cloud dataset. The method consists of four main steps, as follows: (1) Obtaining data describing the 3D objects in the real world; (2) Processing and simplifying the obtained data; (3) Reconstructing the 3D objects by filling the holes on the surface; and (4) Meshing and visualizing the 3D objects in the application. The comparison and discussion between the methods have shown the advantages of the proposed method and its application in practice.

Visuo-spatial mental imagery and geometry skills in school-aged children

The relationships between visuo-spatial abilities and geometry performances in school-aged children were examined. A battery of tests assessing non-verbal reasoning, visuo-spatial mental imagery, and academic achievement in geometry (i.e., geometric knowledge and geometric problem-solving competencies) was presented to 162 8-9.5-year-old pupils attending primary school. After controlling for age, significant associations were found between non-verbal reasoning abilities and knowledge in geometry (r = .31, p = .013) and geometric problem-solving skills (r = .35, p = .005), respectively. Similarly, using age as covariate, mental imagery abilities were significantly related to geometric knowledge (r = -.28, p < .001) and geometric problem-solving skills (r = -.24, p = .002), respectively. Furthermore, pupils with high visuo-spatial mental imagery abilities outperformed their peers with low visuo-spatial competences in the geometry tasks and further visuo-spatial abilities measure computed by their teachers. Finally, male participants showed better geometry skills than females.

VISUAL-ACTIONAL BASIS OF FORMATION OF IDEAS AND CONCEPTS ABOUT THE AREA OF A GEOMETRIC SHAPE IN STUDENTS WITH INTELLECTUAL DISABILITIES

The article analyzes the works of domestic scholars and teachers who studied the specificity of the acquisition of geometric knowledge by students with intellectual disabilities. Special attention is paid to the formation of ideas and concepts about the area of a geometric shape and the units of its measurement. The article argues that there is a connection between the level of acquisition of concepts formed at mathematics lessons and their application in practical and professional activity. The author looks for the reasons, both objective and subjective, of inadequate acquisition of geometric material, leading to the verbalism of knowledge, poor differentiation of the concepts, and difficulties in using the knowledge by students in practice. The methods of studying square measures and calculating the areas of geometric shapes in education institutions implementing adapted basic general education programs are described. The study of geometry in such organizations should be visual-actional. The methodological basis of the formation of concepts and ideas in students with intellectual disabilities involves object-based practical activity, in the process of which practical actions are described verbally, a connection is made between material and external speech forms of action, and personal experience is generalized and moves to a higher symbolic-ideal level, exerting a rehabilitation effect on the development of thinking and cognitive activity of students. The article considers the methods and techniques of preventive rehabilitation work in the process of studying this topic, such as methods of comparison, juxtaposition, finding common and distinctive features of the studied concepts, etc. The author formulates recommendations to the teachers on the implementation of interdisciplinary connections that allow using the personal experience of students at geometry lessons.

GEOMETRICAL KNOWLEDGE LEVEL ACCORDING TO VAN HIELE’S MODEL

Considering that the appropriation of geometric knowledge can help the student to understand the world around him, this work aims to show how the teacher can analyze his student's level of geometric knowledge. As a theoretical contribution, using the Van Hiele Theory which considers that the appropriation of geometric knowledge occurs at five levels, level 1 (recognition, comparison and nomenclature of geometric figures by their appearance), level 2 (analysis of figures, properties and use of them ), level 3 (precise definitions, informal logical arguments and ordering of classes of geometric figures), level 4 (demonstrations and recognition of necessary and sufficient conditions) and level 5 (formal demonstration, establishment of theorems in different systems and comparison of them) .

Šuchov’s bent networks: The impact of network curvature on Šuchov’s gridshell designs

At the turn of the twentieth century, the Russian engineer Vladimir Šuchov developed an internationally acclaimed design language for steel construction. His filigree constructions – from lattice shell to the hanging roof – are timeless in their economy, lightness and, not least, simplicity. By combining a profound geometric and mechanical knowledge, Šuchov succeeded in deducing astonishingly pragmatic designs. A decisive strategy for simplification was the deliberate, elastic and plastic deformation of Z, L and flat steel profiles into continuously curved structural elements. As a result, Šuchov was able to assemble complex doubly curved grids using only repetitive components and simple riveted connections.In this paper we conduct a geometrical analysis of these structures looking at the curvature behaviour of four exemplary structures with increasing complexity: the hyperbolic tower, the rotational hanging roof, the cylindrical lattice vault, and the parabolic, double-curved gridshell. By systematically deriving their geometry and examining the three axes of curvature of each rod within those grids, we discover previously undescribed deformations of the steel members and deduce their associated bending moments and stresses.The results reveal Šuchov’s systematic approach to utilizing elastic deformation to simplify construction, and offer conclusions about the prefabrication, the construction process, and the design approach he used for his lattice constructions. This has a direct impact not only on specific repair measures and structural analysis of his structures, but also on current research and the design and construction of similar gridshells today.

Identificação de lacunas no processo de aprendizagem dos conteúdos de geometria no ensino médio pelo método de Van Hiele

Este artigo tem por objetivo classificar o nível de conhecimento sobre geometria euclidiana em relação ao modelo de Van Hiele. Percebe-se que os estudantes, em geral, não conseguem relacionar sistemas axiomáticos, apresentando assim dificuldades em sistematizar o pensamento geométrico, bem como estabelecer conexões gráficas entre a matemática e outras áreas do conhecimento. Dessa forma, a investigação, caracteriza-se por ser uma abordagem quali-quantitativo com uma base de dados coletados em escolas da rede pública estadual, com estudantes do terceiro ano do Ensino Médio, da cidade de Bagé/RS. Como procedimento metodológico, para a realização desta pesquisa, aplicou-se questionários com questões objetivas e discursivas direcionados a um grupo composto de 280 estudantes. Com base nos resultados, conclui-se que o material desenvolvido na investigação serviu de apoio aos professores em suas práticas pedagógicas que busquem contemplar e atender características dos níveis de pensamento geométrico.

Preservice Mathematics Teachers’ Instrumental Genesis and Their Development of Geometric Knowledge in a Dynamic Geometry Environment

The emergence of dynamic geometry software influences the teaching and learning of geometry among learners of all ages. Relying on the theory of instrument, this study examined the relationship between preservice mathematics teachers’ instrumental genesis and their development of geometric knowledge when solving geometry construction problems with the Geometer’s Sketchpad (GSP). Data analysis revealed the coevolution between instrumental genesis and geometric knowledge. Two preservice secondary mathematics teachers’ explorations on two geometry problems are shared to illustrate this coevolutionary relationship. This relationship can be summarized as the following. Guided by his prior knowledge of geometry and GSP techniques, the participant first used particular GSP tools to obtain a geometric figure possibly through a non-constructible method. Through manipulating this geometric figure by various modalities of dragging and measuring as well as other instrument-mediated actions, the participant observed new geometric properties. The participant then used this newly developed knowledge to guide his GSP usage, which led him to create a dynamic figure or develop an alternative construction. This process might go through multiple iterations. It highlights the dynamic interactions among the learner, mathematical task, and technological tool, through which new geometric knowledge and meaningful ways of using technology emerge.

Online training in graphic subjects on the National open education platform

The problems of online training of students in graphic disciplines in technical higher educational institutions are discussed. The description of the «Engineering graphics» online course presented by the MSTU named after N.E. Bauman on the National open education platform is given. The course is based on a new methodological concept — an original representation of the geometric knowledge system in the form of a neural network structure, using 3D visualization of basic concepts. The goals, structure, content, complexity, and format of the course are considered. The system of evaluation of learning outcomes for the course is presented.

Warp and Learn: Novel Views Generation for Vehicles and Other Objects.

In this article we introduce a new self-supervised, semi-parametric approach for synthesizing novel views of a vehicle starting from a single monocular image. Differently from parametric (i.e., entirely learning-based) methods, we show how a-priori geometric knowledge about the object and the 3D world can be successfully integrated into a deep learning based image generation framework. As this geometric component is not learnt, we call our approach semi-parametric. In particular, we exploit man-made object symmetry and piece-wise planarity to integrate rich a-priori visual information into the novel viewpoint synthesis process. An Image Completion Network (ICN) is then trained to generate a realistic image starting from this geometric guidance. This careful blend between parametric and non-parametric components allows us to i) operate in a real-world scenario, ii) preserve high-frequency visual information such as textures, iii) handle truly arbitrary 3D roto-translations of the input, and iv) perform shape transfer to completely different 3D models. Eventually, we show that our approach can be easily complemented with synthetic data and extended to other rigid objects with completely different topology, even in presence of concave structures and holes (e.g., chairs). A comprehensive experimental analysis against state-of-the-art competitors shows the efficacy of our method both from a quantitative and a perceptive point of view. Supplementary material, animated results, code, and data are available at: https://github.com/ndrplz/semiparametric.

Shape transformation of red blood cells induced by the osmotic pressure

We present a simple red blood cell (RBC) model and solve it only based on the variational method. Such a model allows us to obtain the morphological characteristics of human mature RBC under normal conditions, and to study its shape transformation induced by osmotic pressure. This study requires only modest computational capabilities and geometric knowledge and thus is well suited to be a part of an advanced physics or biophysics course.