Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

Developing Property-Based Geometric Reasoning

Recognizing the complex nature of students’ geometric reasoning, we present guidelines and suggestions for implementing a Guess My Shape minilesson that focuses students’ attention on properties and attributes of geometric shapes.

OVERVIEW OF GEOMETRIC REASONING ABILITY OF SIXTH-GRADE STUDENTS IN SOLVING FLAT PLANE GEOMETRIC PROBLEMS AT THE INTEGRATED ISLAMIC ELEMENTARY SCHOOL AL-MAWADDAH SEMARANG IN INDONESIA

The purpose of this study is to determine the reasoning ability of sixth-grade students in solving basic geometry problems on a flat plane. The subjects of this study were 6 students from 24 students of the Integrated Islamic Elementary School "Al-Mawaddah", representing the leading public and private elementary schools in the city of Semarang. For research on three intellectual abilities, namely intelligent, moderately intelligent, and less intelligent, two students were obtained for each on the recommendation of the homeroom teacher. The research method used is a mixed method, which is a type of research in which a researcher combines elements of a qualitative and quantitative research approach. The data collection techniques were observation, written test, and interview test. The results showed that the research value exceeded the completeness value (= 70). The validity of the data was carried out by triangulation of different times, and valid data were analyzed to draw conclusions. The following is a profile of students' basic geometric reasoning abilities in solving problems as a form of mathematical ability. The results showed that subjects with high, medium, and low abilities met the indicators of ability and basic geometric reasoning skills, including visual, verbal, drawing, logic, and applied skills.

Categories of intuitive reasoning and GeoGebra 3D: an experience with Brazilian students

This work presents the result of the application of a didactic sequence designed to understand the concept of the Cavalieri’s Principle, supported by the GeoGebra application in its version for mobile phones - 3D Calculator. For this study, the Theory of Categories of Intuitive Reasoning, by Efraim Fischbein, was used as a conceptual basis. The objective of this work was to elaborate and develop a didactic sequence aiming to subsidize the learning of the Cavalieri’s Principle from GeoGebra, as a way to help the student in the construction of geometric reasoning, through visualization, perception and intuition. The methodology of this work is qualitative research, exploratory type, being carried out from a didactic sequence developed in two meetings remotely, due to the scenario of the COVID-19 pandemic. The target audience of this research is a group of students aged 15-17 years from a public school in Fortaleza - CE, Brazil. In summary, it is pointed out that the intuitive reasoning categories mobilized from the use of GeoGebra have great potential to stimulate the evolution of the student's geometric thinking, through the development of perception, intuition and geometric visualization.

‘Experiences Are Not Successful Accompaniments to Knowledge of the Truth’

During the last decades of the seventeenth century, the Roman Inquisition investigated a group of Neapolitan lawyers and legal scholars who, passionate about scientific and geometric reasoning, tried to reformulate the very concepts of science and truth. Leaving aside theoretical questions about the heuristic methods of this group, this chapter focuses on the response of the most reactionary wing of local society to this challenge, both in Naples and in Rome. Worried by the emancipation from the principle of authority taking shape in both lay and ecclesiastic culture, the Inquisition sought to maintain control over the different forms of religious and intellectual dissent unfolding in Naples.