Pre-service teachers’ mathematical engagement in learning about the total surface areas of geometrical solids

In this article we report on pre-service teachers’ mathematical engagement regarding the total surface areas of geometric solids. Despite several attempts at improvement, the poor performance of South African learners in mathematics persists. This is attributed to instructional approaches. In the study reported on here we explored how pre-service teachers communicate conjectures, justifications, and generalisations to develop formulae for geometric solids. We employed a qualitative descriptive case study within the interpretive paradigm. Data were collected through document analysis and students’ written tasks. Four tasks were administered to 30 pre-service teachers to enable the researchers to reflect on their performance. Students’ written tasks were analysed with the aid of the model of mathematical knowledge for teaching, which served as the theoretical underpinning of the study. The findings of the study reveal that students can develop mathematical engagement and reasoning when appropriate tasks are designed to facilitate understanding of key concepts that are the cornerstone of learning about geometric solids. Certain concepts, notably, circles, radii, pi, rectangles, cones, Pythagoras’ theorem, slanting height, congruence, and prism, were crucial elements that should be explored prior to the introduction of the topic of the total surface areas of geometric solids. The study was an eye-opener to South African policy makers, mathematics teachers and lecturers in terms of identifying students’ weaknesses at pre-service level on how to develop logical methods to make sense in the learning of geometrical solids.

Os Sólidos Geométricos na Educação Brasileira: Comparativo Entre PCN e BNCC

ResumoO presente artigo faz parte de uma pesquisa de mestrado, em andamento, vinculada ao Programa de Pós-Graduação em Ensino de Ciências e Matemática (PPGECIMA), na Universidade Federal de Sergipe (UFS). A pesquisa de mestrado teve, dentre outros objetivos específicos, investigar as três dimensões do problema didático (epistemológica, econômica e ecológica) dos sólidos geométricos. Assim, o objetivo deste artigo é apresentar um primeiro estudo das dimensões econômica e ecológica do objeto matemático sólidos geométricos. Para tanto, inicialmente, realizou-se um estudo histórico do ensino de Geometria no Brasil. Em seguida, investigou-se como os sólidos geométricos estão presentes em documentos curriculares oficiais – Parâmetros Curriculares Nacionais (PCN) e Base Nacional Comum Curricular (BNCC) – implementados nas últimas duas décadas do presente século. Logo, o estudo teve como aporte teórico as investigações de autores que versam sobre a Teoria Antropológica do Didático (TAD) e as dimensões que distinguem os problemas didáticos de pesquisa, formação docente e problemáticas do ensino de Geometria. Nesse primeiro ensaio, foi possível identificar que, em relação aos sólidos geométricos, pouco se evoluiu em relação ao seu ensino na educação básica. Além disso, observa-se que a problemática quanto ao ensino de Geometria, apesar da tentativa dos documentos curriculares e da comunidade acadêmica, não foi superada. Palavras-chave: Sólidos Geométricos. Histórico do Ensino de Geometria. Modelo Epistemológico de Referência. AbstractThis article is part of a master's research, in progress, linked to the Graduate Program in Science and Mathematics Teaching (PPGECIMA), at the Federal University of Sergipe (UFS). The master's research had, among other specific objectives, to investigate the three dimensions of the didactic problem (epistemological, economic and ecological) of geometric solids. Thus, the objective of this article is to present a first study of the economic and ecological dimensions of the geometric solid mathematical object. Therefore, initially, a historical study of the teaching of Geometry in Brazil was carried out. Then, it was investigated how the "geometric solids" are present in official curricular documents - National Curriculum Parameters (PCN) and Common Base National Curriculum (BNCC) - implemented in the last two decades of the present century. Therefore, the study had as theoretical support the investigations of authors that deal with the Anthropological Theory of Didactics (TAD) and the dimensions that distinguish the didactic problems of research, teacher training and problems of the teaching of Geometry. In this first test, it was possible to prove that in relation to geometric solids little progress has been made in relation to its teaching in basic education. In addition, it is observed that the problem regarding the teaching of Geometry, despite the attempt of curricular documents and the academic community, has not been overcome. Keywords: Geometric Solid. History of Geometry Teaching. Epistemological Reference Model.

Accuracy of common stem volume formulae using terrestrial photogrammetric point clouds: a case study with savanna trees in Benin

Recent applications of digital photogrammetry in forestry have highlighted its utility as a viable mensuration technique. However, in tropical regions little research has been done on the accuracy of this approach for stem volume calculation. In this study, the performance of Structure from Motion photogrammetry for estimating individual tree stem volume in relation to traditional approaches was evaluated. We selected 30 trees from five savanna species growing at the periphery of the W National Park in northern Benin and measured their circumferences at different heights using traditional tape and clinometer. Stem volumes of sample trees were estimated from the measured circumferences using nine volumetric formulae for solids of revolution, including cylinder, cone, paraboloid, neiloid and their respective fustrums. Each tree was photographed and stem volume determined using a taper function derived from tri-dimensional stem models. This reference volume was compared with the results of formulaic estimations. Tree stem profiles were further decomposed into different portions, approximately corresponding to the stump, butt logs and logs, and the suitability of each solid of revolution was assessed for simulating the resulting shapes. Stem volumes calculated using the fustrums of paraboloid and neiloid formulae were the closest to reference volumes with a bias and root mean square error of 8.0% and 24.4%, respectively. Stems closely resembled fustrums of a paraboloid and a neiloid. Individual stem portions assumed different solids as follows: fustrums of paraboloid and neiloid were more prevalent from the stump to breast height, while a paraboloid closely matched stem shapes beyond this point. Therefore, a more accurate stem volumetric estimate was attained when stems were considered as a composite of at least three geometric solids.

SMARTPHONE USE AND THE INSPIRA DIGITAL APP IN MATH CLASSES DURING REMOTE CLASSES

This work highlights the use of the digital inspiration application in mathematics classes as a pedagogical tool, as an explanatory/demonstrative instrument of the concepts and elements of geometric solids in the 6th grade classes, at the Professora Maura de Medeiros Bezerra Municipal School – Macau/RN. The purpose of this application was to expand the possibilities of learning in the Hybrid format in the pandemic period as an inverted classroom methodology.

Assessing tree crown volume—a review

Tree crown volume is a fundamental tree characteristic. It correlates to forest biomass production and most relevant ecosystem and environmental functions, such as carbon sequestration and air pollution reduction. When researching these relationships, it is necessary to clearly define and then quantify tree crown variables in a both accurate and operational manner. In this paper, we review the reported literature on the assessment of tree crown volume. First, we compile the varying definitions of crown volume and other tree crown variables that may be used as inputs to quantify crown volume. Then, we examine the data sources for quantifying these variables, including field measurements, terrestrial photographs, aerial photographs and laser scanning. Furthermore, we compare the published approaches on translating these crown variable measurements into tree crown volume. These approaches include the approximation of simple geometric solids, approaches of computational geometry and voxelization. We also compare the reported accuracies and major challenges of these approaches. From this literature review, the reader may craft a suitable approach for the assessment of crown volume.

Pfinzing and Friends

Paulus Pfinzing von Henfenfeld embodied the gentrification of geometry in late sixeenth-century Nuremberg. A patrician merchant, Pfinzing developed and invented surveying tools, and publicized their use. He is famed for his so-called Pfinzing Atlas of manuscript maps of the Nuremberg area that he completed in 1594. His legacy in print of two now rare books from 1598–1599 on surveying and on geometric solids is less well known, as Pfinzing published them both anonymously. The former entreated the well-educated to take up surveying, while the latter presented a summary of recent scholarship on geometric solids based on Albrecht Dürer and Wenzel Jamnitzer, among other luminaries. This article investigates the ways Pfinzing used his knowledge of surveying and familiarity with Nuremberg publishing to present geometry as a gentlemanly pursuit.