Reactive navigation in partially familiar planar environments using semantic perceptual feedback

This article solves the planar navigation problem by recourse to an online reactive scheme that exploits recent advances in simultaneous localization and mapping (SLAM) and visual object recognition to recast prior geometric knowledge in terms of an offline catalog of familiar objects. The resulting vector field planner guarantees convergence to an arbitrarily specified goal, avoiding collisions along the way with fixed but arbitrarily placed instances from the catalog as well as completely unknown fixed obstacles so long as they are strongly convex and well separated. We illustrate the generic robustness properties of such deterministic reactive planners as well as the relatively modest computational cost of this algorithm by supplementing an extensive numerical study with physical implementation on both a wheeled and legged platform in different settings.

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.

Visual Foundations of Euclidean Geometry

Geometry defines objects that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we examined the possibility that representations of small, 2-dimensional visual forms provide cognitive foundations for geometric knowledge, and in particular Euclidean knowledge. We asked two questions: First, are humans sensitive to form variations that are relevant to Euclidean geometry (e.g. changes in angle)? Second, can observers easily disregard variations that are irrelevant to Euclidean geometry (e.g. changes in position and orientation)? One hundred and twelve participants from the U.S. (age 3-34 years), and 25 participants from the Amazon (age 5-67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of metric proportions (hereafter, ‘shape’) or global size, but only U.S. adults drew distinctions between mirror images (i.e. forms differing in sense). Similarly, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size interfered with the detection of size or shape deviants, respectively. Children and adults from both cultures thus analyzed visual forms according to their Euclidean properties, even if they had not received formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded.

Conhecimento dos professores sobre geometria nos anos iniciais do ensino fundamental: uma visão do estado da arteTeachers' knowledge of geometry in the early years of elementary school: a state of the art

ResumoNeste artigo, a partir de um estudo do estado da arte, analisa-se o tema conhecimento geométrico de professores dos anos iniciais em pesquisas em educação matemática realizadas no Brasil num intervalo de 19 anos, entre 2000 e 2019. A leitura de 31 estudos em nível de mestrado e doutorado apontam que entre os aportes teóricos que embasam o conhecimento geométrico do professor, destacam-se os de Shulman (1986, 1987) e Tardif (2002). Ao longo dos anos, esses modelos teóricos tornaram-se as principais referências para análise do conhecimento/saberes de professores. Em relação aos objetivos, a maioria dos estudos buscava analisar ou identificar como a formação em serviço ou formação continuada pode influenciar na mobilização de conhecimentos/saberes pelos professores. Embora o objeto de análise fosse praticamente o mesmo e os estudos utilizassem uma abordagem qualitativa, os procedimentos metodológicos eram diversificados, incluindo estudo de caso, pesquisa-ação e análise documental. Como instrumentos de coleta de dados destacaram-se diagnósticos; registros produzidos pelas participantes; diário de campo da pesquisadora; gravações em áudio e/ou vídeo e produção de sequências didáticas, entre outras. Observou-se uma tendência em coletar informações por meio de encontros formativos, oficinas e laboratórios de matemática, possivelmente para que os pesquisadores interviessem no desenvolvimento do conhecimento geométrico dos professores. Os resultados das pesquisas analisadas apontam para fragilidades no conhecimento conceitual e prático dos professores em relação à geometria. Também indicam que os processos formativos possibilitam mudanças no conhecimento conceitual e na prática educativa a partir da reflexão dessa prática e da construção de aprendizagens.Palavras-chave: Geometria, Conhecimento de professores, Educação Matemática.AbstractIn this article, based on a state of the art study, we analyse the theme of geometric knowledge of teachers of the early years in mathematics education in research carried out in Brazil between 2000 and 2019. The reading of 31 studies at the master’s and doctoral level points out that among the theoretical contributions that support the teacher’s geometric knowledge, Shulman’s (1986, 1987) and Tardif’s (2002) stand out. Over the years, these theoretical models have become the main references for the analysis of teachers’ knowledge/know-how. Regarding the objectives, most studies sought to analyse or identify how in-service education or continuing education can influence the teachers’ mobilisation of knowledge/know-how. Although the object of analysis was practically the same and the studies used a qualitative approach, the methodological procedures were diverse, including case study, action research, and documentary analysis. As instruments of data collection, we highlight the diagnoses; registers produced by the participants; researcher’s field diary; audio and/or video recordings and production of didactic sequences, among others. There was a tendency to collect information through formative meetings, workshops, and mathematics laboratories, possibly for researchers to intervene in the development of teachers’ geometric knowledge. The results of the studies analysed point to weaknesses in the teachers’ conceptual and practical knowledge of geometry. They also indicate that the education processes enable changes in conceptual knowledge and educational practice based on the reflection of this practice and the construction of learning.Keywords: Geometry, Teachers’ knowledge, Mathematics education.ResumenEn este artículo, basado en un estudio del estado del arte, se analiza el tema conocimiento geométrico de los docentes de los años iniciales en investigaciones en educación matemática realizadas en Brasil entre 2000 y 2019. La lectura de 31 estudios a nivel de maestría y doctorado señala que entre los aportes teóricos que sustentan el conocimiento geométrico del docente, se destacan los de Shulman (1986, 1987) y Tardif (2002). A lo largo de los años, estos modelos teóricos se han convertido en los principales referentes para el análisis del conocimiento docente. En cuanto a los objetivos, la mayoría de los estudios buscaban analizar o identificar cómo la formación en servicio o continua puede influir en la movilización de conocimientos por parte de los docentes. Aunque el objeto de análisis fue prácticamente el mismo y los estudios utilizaron un enfoque cualitativo, los procedimientos metodológicos fueron diversos, incluyendo el estudio de casos, la investigación-acción y el análisis de documentos. Como instrumentos de recolección de datos, se destacaron los diagnósticos; registros producidos por los participantes; diario de campo del investigador; grabaciones de audio y/o video y producción de secuencias didácticas, entre otros. Hubo una tendencia a recolectar información a través de reuniones formativas, talleres y laboratorios de matemáticas, posiblemente para que los investigadores intervinieran en el desarrollo del conocimiento geométrico de los docentes. Los resultados de las investigaciones analizadas apuntan a debilidades en los conocimientos conceptuales y prácticos de los docentes en relación con la geometría. También indican que los procesos de formación posibilitan cambios en el conocimiento conceptual y la práctica educativa a partir de la reflexión de esta práctica y la construcción de aprendizajes.Palabras clave: Geometría, Conocimiento de los profesores, Educación matemática.

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.

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.

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) .

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.

Los problemas espaciales: una propuesta alternativa para enseñar geometría en la Educación Secundaria ObligatoriaSpatial problems: an alternative proposal to teach geometry in Compulsory Secondary Education

ResumenPresentamos un esbozo del problema de investigación que forma parte de un trabajo de tesis. Tras analizar el currículo español de matemáticas y algunos manuales escolares, hemos encontrado una ausencia de las cuestiones a las que responden los conocimientos geométricos propuestos para la Educación Secundaria Obligatoria (alumnos de 12 a 16 años). Después de haber revisado diferentes estudios relacionados con la enseñanza de la geometría, postulamos que los problemas espaciales pueden ayudar a encontrar una posible razón de ser de esos conocimientos geométricos. Así, pretendemos identificar y abordar algunos problemas espaciales a fin de confirmar dicha hipótesis.Palabras-clave: Teoría Antropológica de lo Didáctico, Geometría elemental, Conocimientos geométricos, Problemas espaciales, Modelización Espacio-Geométrica.Abstract.We present an outline of the research problem that is part of a doctoral thesis. After analysing the Spanish mathematics curriculum and some textbooks, we have found an absence of the questions to which the geometric knowledge proposed for high school (students aged 12 to 16 years) answer. After reviewing different studies related to the teaching of geometry, we postulate that spatial problems can help to find a possible justification for such geometric knowledge. Thus, we intend to identify and address some spatial problems in order to confirm this hypothesis.Keywords: Anthropological Theory of Didactics, Elementary Geometry, Geometric Knowledge, Spatial Problems, Space-Geometric Modeling.