School Geometry Research Articles

We re-examine a Malawian teacher’s lessons which, using the framework of mathematic problem-solving developed by Polya, the typology of levels of task demand from Stein and colleagues and Malawi’s description of learner-centred education (LCE), were described as teacher-centred and evaluated as ‘not good’. We studied seven video-recorded circle geometry lessons taught by the teacher and analysed these first using an LCE framework and then the Mathematics Discourse in Instruction (MDI) framework, adapted to suit the analysis of geometry lessons. The LCE analysis revealed that while the lessons were undoubtedly teacher-centred, they were not at the extreme end of an LCE continuum. Analysis using an adapted MDI framework showed that the teacher’s use of mediational means opened opportunities to learn mathematics. We argue that LCE frameworks are useful in mathematics education research as they do not dichotomise teaching practices, but they are insufficient. They can obscure opportunities made available to learn mathematics. Frameworks that illuminate such opportunities are needed to fully describe mathematics teaching practices. Furthermore, we identified links between the elements of LCE exhibited and the mathematical mediational means in use. These suggest that supporting teachers to strengthen their mathematical mediational means in use could enable movement towards more learner-centered teaching.

The aim of this paper is to explore a framework for articulating the relationship between teachers' practices and students' self-efficacy in teaching and learning of Euclidean Geometry (RTP&SS-EG) in high school. The effective learning of Mathematics, especially Euclidean Geometry by students, is directly influenced by teachers' confidence and competence. The RTP&SS-EG framework was developed based on the gap in literature revealed through international studies, indicating a direct relationship between teachers with a higher level of self-efficacy, and students' achievement. However, the researchers did not come across national studies promoting the relationship between the teachers' practices and students' self-efficacy on the learning of Euclidean Geometry in high school. The question asked was: To what extent is the relationship between the teachers' practices and the students' self-efficacy in teaching and learning of Euclidean Geometry in High school? Geometry Students Self-Efficacy (SGS-EQ) and Geometry Teacher Self-Efficacy questionnaire (GTS-EQ) were utilized. The participants were 81 Mathematics teachers, and 390 Grade 10-12 students from the Mpumalanga and North West provinces in South Africa. Reliability for the entire scale of both instruments computed in terms of the internal consistency was found to be .83. To establish the content validity of the scores from both instruments, the exploratory factor analysis was computed using varimax rotation. The four-factor solution accounting for 43.03% of the total variance was obtained from 36 items of the GTS-EQ. A five-factor solution for 45.37% of the total explained variance was obtained from the 46 items of the SGS-EQ. In RTP&SS-EG, three education theories, constructivism, social constructivism and social cognitivism, policies, as well as teacher and student self-efficacy principles enjoin the practice enacted from the relation between the teachers' practices and the students' self-efficacy in the learning of Euclidean Geometry. The findings revealed that the learning value associated with the relationship between teachers' practices and students' self-efficacy may be integrated into the everyday preparations of teaching-learning plans in Euclidean Geometry. It is critical that policymakers, curriculum developers, teachers and students make use of this model to ensure that self-efficacy of both teachers and students is taken into consideration in Euclidean Geometry.

In this article, in geometry lessons of primary school geometry, the theoretical and methodological aspects of the development of issues and increasing their interests have been stated in the science. The article also examines scientific and practical theories of great mathematical scientists and scientists on mathematical knowledge and the importance of today's education.

This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of $\mathfrak{osp}(1\mid 4)$ and choose $\mathfrak{osp}(1\mid 2) \times \mathfrak{sp} (2)$ as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [10].

The problem of studying mathematical models in the course of school Geometry is discussed. An example of realizing the main requirements of the Federal standards of the main general education and secondary general education in the framework of mathematical modeling process is cited. Attention is paid to working with pupils from non-profession oriented mathematical groups and groups oriented to the professions connected with constructing geometric models. It is noted that mathematical modeling decreases the tension when studying plane geometry and solid geometry at school, providing a possibility to have deeper insight in the features of geometric figures, construct them in the projects in other subjects and fields. It is concluded that study of mathematical models makes teaching Geometry more efficient and provides quality basis to form professional competences, connected with creating and using geometric objects, in future.

Popularly used in marketing and business, gamification has been gaining interest in educational contexts for its potential to invigorate otherwise mundane or difficult processes. A gamified environment transfers motivational elements of games to learning activities thereby engaging learners in the learning task thus transforming dull classroom environments to smart ones. This paper presents the design process of a gamification intervention in geometry at elementary level, based upon Huang and Soman (Gamification of education. Research report series: behavioural economics in action, 29. Rothman School of Management, University of Toronto, Toronto, 2013) model. We describe how insights from various sources helped us to refine an intervention previously used in one school. The design focuses on gamifying the tangram, an unplugged resource, through incorporating game-based elements of leader boards, points/stars and challenge levels to motivate young learners individually and in teams. Cognitive and motivational scaffolding undergird five challenge levels to bring affordances to self and social elements for learner participation in increasingly complex geometry tasks. There are limited theoretical models to guide educational researchers, especially ones that do not require digital resources. This paper presents our insights and recommendations to support scaffolded learning in student-centred gamified learning environments.

ABSTRACT We examined high school geometry students’ written work on four proof tasks where they posed a conjecture, drafted an argument, provided written critiques, then revised their argument based on peer feedback. Students’ written work across the tasks was analyzed to determine whether the instructional sequence supported them in improving their arguments and attending to key aspects of proof (justifications, generality, clarity, structure). Claim-level analysis for each of the key aspects revealed minor changes between students’ draft and revised arguments with results varying by task. That said, students attended to the key aspects of proof through the critiques they provided each other with most critiques, if appropriately addressed, having the potential to help improve the draft argument. Students’ reflections also showed this process helped them think about the clarity and level of detail in their arguments. Implications for this study include the benefits of providing proof tasks that offer fewer supports for students, alongside multi-faceted analysis of their written arguments, in terms of providing insights into students’ current understanding of proof.

With the transition to the digital age, changes have emerged in the skills expected from the individuals of the 21st century, and accordingly, the preparation of curricula to develop these skills has become the main goal of all countries in the world. In our country, studies have been carried out to develop curricula in this direction, and with this research, it is aimed to examine the secondary education mathematics (2010, 2011, 2013 and 2017) and geometry (2011) teaching programs in the context of 21st century skills. The research is a survey study aimed at examining the secondary school mathematics and geometry course curriculum in terms of 21st century skills. As the data source of the research, secondary school mathematics course and secondary school geometry course curricula shared on the official website of the Ministry of National Education were taken. Document analysis method was used in the collection and analysis of data in the research in which these teaching programs were accepted as documents. Curriculums specified within the scope of document analysis were analyzed with descriptive analysis method based on 21st century skills within the scope of Partnership for 21st Century Learning [P21]. The skills included in the curricula were supported by direct quotations from the curricula. According to the results of this research, it has been determined that the curriculum is not qualified to cover all 21st century skills. The fact that media literacy, leadership and responsibility skills are not included in the curriculum, and that the evaluation elements of the programs are insufficient in the context of 21st century skills are among the remarkable results. The findings obtained at the end of the research were discussed with the support of the literature and suggestions were made for future research.

In order to promote geometric understanding, teachers frequently use hands-on activities. Such activities can be used to expound upon the declarative statements and theorems of geometry. Using a compass, straightedge, and protractor, students are able to actively build conceptions involving bisectors, midpoints, and perpendicular lines. Additionally, activities that require students to problem-solve and formulate problems, using their construction knowledge and skills, can reinforce and strengthen that which they have learned. This article describes STEAM instruction with high school geometry students designed to productively integrate geometric constructions, digital technology, elements of art, and principles of design to enhance students’ geometric reasoning.

We present the result of an eight-year didactic experiment in two primary school classes involving comparative geometry activities: a comparison between Euclidean plane geometry and spherical geometry that took place over five years. Following the didactic experiment, three years on from the end of the experiment, final questionnaires were administered and codified in order to evaluate the project’s effect on the pupils’ school performance and attitude, especially with regard to mathematics.

“Real Normed Algebras Revisited,” the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions, and the octonions, as well as proofs of their properties, using only what was known to 19th-century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above-mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. His reconstructions underline an interesting connection between Euclidean geometry and these algebras, and avoid the advanced machinery used in previous derivations of these results. The goal of this article is to present Gadi’s derivations in a way that is accessible to a wide audience of readers.

This study aims to develop a learning media in the form of multimedia geometry flipbook that are suitable for use by elementary school students. This research uses Research and Development Method, with The phases in this research consist of; (1) product analysis, (2) design products, (3) Development, validation and revision. This research used is a mixed approach. The results of the research at the analysis stage, the researcher found problems through a geometry ability test, analyzed the curriculum, and the students needs. The results of the design stage, namely designing the story board, preparing the content/material to be included in each menu, and compiling the learning media assessment instrument. The results of the development stage, namely product manufacture and validation and revision. The results of expert validation by material experts are included in the very worthy criteria (96.95%). The results of the validation by media experts showed very worthy criteria (98.18%). The results of the teacher's response as learning practitioners showed very worthy criteria (89.04%). Based on these results, the multimedia flipbook learning media for elementary school students in the Mattiro Bulu sub-district developed is suitable for learning geometry in elementary schools.

Abstract The Common Core State Standards for Mathematics (CCSSM) explicitly states many specific theorems for students to prove across multiple domains (i.e., congruence, similarity, circles, and coordinates) in high school geometry. This study examined five high school geometry textbooks for how they approached proof of 17 theorems stated in the congruence domain focused on lines and angles, triangles, and parallelograms. Results showed that although textbooks provided 75 student opportunities to prove these theorems, no textbook provided student opportunities to prove all 17. Textbooks rarely had students write proofs from general conditional statements, and instead typically provided consistent hard scaffolding including the given, what to prove, and a diagram for all proof opportunities. Some of the textbooks used novel scaffolding such as partially completed proofs, flowchart proofs, second proofs, and hints in the back of the book. Textbooks need to continue shifting more responsibility to students proving the CCSSM theorems by incorporating more diverse scaffolding along with a process for removing the scaffolding as student learning progresses.

Analyzing the communication and work of high school geometry students through error-prone topics such as transformations, constructions and employing various thinking processes such as factual and procedural knowledge could reveal insight to students understanding of Geometry. Such a body of work [1, 4, 5], could reveal suitable or unsuitable considerations of a topic combined with suitable or unsuitable uses of a thinking process thereby producing suitable or unsuitable conclusions. Directed by the objective of the study, three high school geometry students' interpretations were analyzed in order to explore: (1) connections among topics and thinking processes; (2) relationships among suitable and unsuitable uses of the topic and thinking processes; and (3) whether unsuitable uses of topics and unsuitable thinking processes can yet lead to suitable understanding. The results revealed no definitive association among some topics and some thinking processes, whether suitable or unsuitable in either realm. Unsuitable applications of factual, procedural, and conceptual knowledge were not associated simply with the lack of these thinking processes, at times. However unsuitable thinking processes were present, deliberate, purposive, and not absent. By implication, there is a need for more studies regarding student use and misuse of thinking processes regarding other mathematical subjects.

Modern trends in the development of technology are based mainly on successful engineering activities. The combination of the most modern computer technologies and the traditions of using a pencil, drawing, drafts, and, of course, a living fruitful thought is a guarantee of the education of a genuine Engineer. On the one hand, educational technologies in engineering universities at the initial stage should be based, in our deep conviction, on school geometry. On the other hand, descriptive geometry and engineering graphics should contain examples for immersion in the chosen specialty. By order of the Ministry of Education and Science of the Russian Federation, engineering classes, in which lecturers of technical universities were involved, were organized 10 years ago in Moscow schools. The creation of engineering classes is caused by the need to instill in students an interest in the engineering profession. To successfully solve this problem, it is necessary to increase the level of students» knowledge in geometry and drawing. On the basis of these subjects, one should develop design skills and skills of using computer modeling tools. The result of this cooperation should be the involvement of schoolchildren to participate in competitions and olympiads. The Department of Engineering Graphics, Bauman Moscow State Technical University, has accumulated 30 years of experience in cooperation with specialized schools in order to develop students» interest in the engineering profession. The article provides an example of the introduction of a common cutting tool, i. e. a drill, which demonstrates the depth of engineering thought formed over a thousand-year history, into the educational process when studying geometry at school and descriptive geometry at a technical university. The drill as an object comprehensible to first-year students is used to provide the basic knowledge of descriptive geometry. In addition, the problem related to the improvement of the technology of end-to-end training throughout the course of obtaining an engineering education is touched upon.

The purpose of this study was to describe the steps of implementing the Realistic Mathematics Education (RME) approach in teaching geometry in elementary schools. This research uses collaborative action research between lecturers, students, and teachers in 2019 at one of the primary schools in Central Java. Data collection techniques through observation, interviews, and tests. The result of the research is that the implementation steps of the RME approach include: (a) Understanding contextual problems. (b) Describing contextual problems, (c) Solving contextual problems, (d) Comparing and discussing answers, and (e) Concluding. The implementation of the RME approach is proven to improve mathematics learning outcomes about geometry.

In this note we discuss how a modular triangular origami gift box can be made with six rectangular sheets of printing paper. We also derive a formula for the volume of the box based on high school geometry and algebra.

Teaching gerrymandering in our high school geometry classrooms provides students with a unique opportunity to use mathematics to describe, analyze and make sense of the world around them. Our purpose is to provide our students the opportunity to apply learned definitions and formulas of area and perimeter to a sociopolitical context. We present a unit that we designed for teaching high school students about gerrymandering and describe how teachers may implement this unit in their classrooms. In this unit, students discovered the mechanics of gerrymandering. They made calculations and observations about area, perimeter, and the ratio of area to perimeter. They considered compactness as a potential tool to indicate gerrymandering. They used proportional reasoning to measure the fairness of the partisan split for a given map. We provide supporting examples of student work and discourse and make recommendations for future iterations of this unit.

The purpose of this note is to introduce a challenging problem in geometry that was inspired by a well-known problem in high school geometry. In this note we will explore geometry problems that involve six-petal rosettes inside a regular hexagon.

Areas are actively used in the solution of many geometrical problems. In this work,smart and laconic solutions are found for various problems by means of the area method. The well-known trigonometric addition formula is also proved. In the area method, the given formulas are divided into the parts, whose areas are then calculated using problem data.