High School Geometry Research Articles

Developing Secondary Students’ Understanding of Proof through Constructing, Critiquing, and Revising Arguments

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.

Promoting Geometric Reasoning through Artistic Constructions

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.

Elementary Derivations of the Euclidean Hurwitz Algebras Adapted from Gadi Moran’s last paper

“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.

High school geometry textbooks’ proving opportunities of Common Core theorems

AbstractThe 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.

Conceptual Misunderstanding: On the Interplay of Understanding and Misunderstanding in Geometry

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.

A triangular gift box and problem solving

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.

An old problem with a new twist

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.

A geometric approach to Fermat principle as a demonstration of high school geometry and physical optics

Fermat’s principle states that a light ray refracted across different media will traverse the fastest path as the physics for Snell’s law. A geometric proof of Fermat’s principle will be demonstrated as an intuitive approach to learn high school geometry and physical optics. It will be proved explicitly by showing that all alternative paths need more traversal time to complete their journey.

THE ERROR METHOD IN THE ORGANIZATION OF STUDYING THE TOPIC «THE CROSS SECTION OF THE POLYHEDRONS»

The main task of the school today is not just to equip students with a certain set of knowledge, skills, and abilities, but also to form their ability to learn, work in a group. Based on the above, it is advisable to use interactive teaching methods in math lessons. One of the most popular strategies for interactive learning is collaborative learning. The article discusses interactive teaching techniques and their features in teaching geometry in high school. Given examples of the application of the «error method» in the study of the topic «The cross-section of the polyhedrons» are given. It is shown that, when discussing erroneous decisions in collaborative learning, students acquire the skills to solve such problems. The article fills in one of the gaps that exist in the methodological literature on the description of interactive methods of teaching geometry in high school.

Pre-service mathematics teachers’ semiotic transformation of similar triangles: Euclidean geometry

Some pre-service mathematics teachers in South Africa are nervous about the content of Euclidean geometry because they did not study Euclidean geometry in high school but will be expected to teach same when they start their teaching career. Because of this, graduating pre-service mathematics teachers were enrolled for a six-week intervention programme that metamorphosed to this research. This research reported a mixed study design that explored how pre-service mathematics teachers’ respond to tasks on similar triangles with respect to treatment and conversion transformation of semiotic representations. Sixty-five pre-service mathematics teachers response to a written task was analysed using descriptive statistics while the extract from semi-structured interview were analysed thematically. The main finding of this research revealed participants’ determined attempt to improve their geometry content knowledge for teaching through visual representation. The research also revealed that many participants were able to answer those items requiring treatment of registers but struggled with those requiring conversion of the register. The interview confirmed some of these struggles experienced by the participants in solving the given task. This research recommended careful design of programmes for upgrading pre-service mathematics teachers’ content knowledge for teaching school topics.

Students' Interest in Understanding Geometry in South African High Schools

The major contributors of challenges with teaching and learning of Euclidean Geometry relate to teaching pedagogy, teaching methodologies, students' lack of interest in Geometry, lack of understanding of many geometric concepts and lack of use of technology. The performance of students in high school Geometry, Grade 10-12, is poor. Students with high self-efficacy that is linked to interest, are more likely to make efforts to complete a task and persist longer in those efforts. The purpose of this paper was to investigate students' interest in understanding Geometry in high school Further Education and Training phase. Participants were 390 Grade 10-12 Mathematics students from Mpumalanga and North West in South Africa. A mixed-methods approach was utilized following the convergent parallel sequential design. Qualitative data were collected through the survey questionnaire, with open-ended and closed-ended questions to determine students' interest in Geometry. Atlas ti was used to analyse qualitative data. The quantitative aspects were analysed with SPSS. It was found that students indicated that the teachers continue to teach even if they do not understand. The results also showed that more than half (57.1%) of the students felt calm and relaxed when they were attempting Geometry problems. About 60% of the students were confident enough to take Mathematics at high school level, which included Geometry. It is recommended that teachers take into consideration the students' interests when planning and preparing Geometry lessons.

Mathematical Design Thinking in the Classroom through Graphic Art

This classroom study immersed high school geometry students in the creative and intellectually challenging design task of developing unique logos using mathematics and technology. The students applied and deepened their knowledge of transformations while using dynamic geometry software. One of the main aims of the task was to elevate student creativity andautonomy within the mathematics classroom while they engaged in mathematical design thinking to create their logos. The discussion provides insight into considering student work from a design perspective, which can offer students new ways to engage with mathematical concepts and make their thinking more explicit

AS PESQUISAS SOBRE O USO DE SOFTWARES DE GEOMETRIA DINÂMICA NO BRASIL

Este trabalho é um recorte de uma pesquisa realizada no Programa de Pós-Graduação em Educação em Ciências e Matemática da Universidade Federal de Goiás (UFG). Foram realizados o mapeamento e a análise das pesquisas publicadas entre 1987 e 2017, que abordam o ensino de geometria plana no ensino médio com o uso de softwares de geometria dinâmica como recurso didático. Para tanto, os procedimentos metodológicos do Estado do Conhecimento nos guiaram no planejamento, na organização, análise e apresentação dos dados encontrados. O campo empírico baseou-se nos bancos de dados das bibliotecas de instituições que possuem mestrado e doutorado relacionados à Educação Matemática e no Banco de Teses e Dissertações (BDTD) da CAPES. Definiu-se como marco inicial o ano de 1987, por ser o ano de criação do 1º software de geometria dinâmica e 2017 como marco final devido ao de que o levantamento foi realizado em 2018. Os resultados indicam o Geogebra como o software mais utilizado nas experiências com o ambiente dinâmico. Outro aspecto que emerge dos dados coletados são influências da teoria dos registros de representação semiótica e da didática francesa. As perspectivas são permeadas por uma discussão que valoriza o “aprender a aprender” e a formação para o mercado de trabalho, focando nos aspectos técnicos em detrimento da construção de conceitos geométricos.

An Investigation of the Relationship Between Two Norms of the Instructional Situation of Geometric Calculation with Algebra in U.S. High School Geometry

An Investigation of the Relationship Between Two Norms of the Instructional Situation of Geometric Calculation with Algebra in U.S. High School Geometry

METODOLOGIAS DE ENSINO: ARTICULAÇÃO DOCENTE PARA O ENSINO DE GEOMETRIA ESPACIAL À ESTUDANTES, EM TEMPOS DA COVID-19

A pandemia da COVID-19 impactou significativamente a sociedade, seja por questões econômicas, educacionais ou de saúde. A partir dela, exigências governamentais foram postas, e uma delas envolveu a adoção de medidas de distanciamento para controle da disseminação do SARS-CoV-2 (vírus da COVID-19), inclusive no contexto escolar, em que o ensino presencial precisou ser suspenso. Nesse contexto, como alternativa para continuidade de atividades educacionais, muitas instituições de ensino brasileiras passaram a trabalhar o ensino remoto, tendo como suporte Tecnologias Digitais da Informação e da Comunicação (TDICs). As TDICs podem ser um instrumento importante no desenvolvimento das práticas docentes, inclusive em atividades de ensino matemático, como por exemplo, geometria espacial. Neste sentido, objetivamos nessa pesquisa analisar, quando houvesse, a utilização de recursos metodológicos do docente para o desenvolvimento do ensino de geometria espacial à estudantes do Ensino Médio, em tempos de COVID-19. Metodologicamente, a pesquisa foi classificada numa abordagem qualitativa do tipo descritiva, e teve como técnica de coleta de dados o uso de um formulário eletrônico, com questões abertas. Como participantes tivemos três professores de matemática de uma escola da rede estadual de Camocim de São Félix-PE, que estavam ensinando ou iriam ensinar geometria espacial no ensino médio na modalidade de ensino remoto. Utilizamos a análise de conteúdo para analisar os dados obtidos. Das inferências, contatamos que os docentes utilizam vários recursos tecnológicos para o desenvolvimento de suas práticas para o ensino de geometria espacial, para facilitar a compreensão e visualização do conteúdo. Esses recursos remetem dos professores investigados certa articulação para aplicação de novas metodologias de ensino, nas aulas de matemática. Verificamos ainda, alguns desafios apresentados pelos professores, como a falta de recursos dos discentes e a formação docente. Assim sendo, concluímos que o uso das TDICs tem sido um grande aliado na articulação e apresentação do conteúdo de geometria espacial, mas para que sua potencialidade seja aproveitada no processo de apresentação do conteúdo de geometria espacial é necessário que os professores estejam instrumentalizados quanto ao uso desses instrumentos. Ademais, verificamos a necessidade de reflexão acerca dos desafios advindos do modelo de ensino remoto.

USE OF AUGMENTED REALITY MEANS IN THE PROCESS OF STUDYING PLANIMETRY IN INSTITUTIONS OF SECONDARY EDUCATION

Visualization of educational material in secondary education institutions is one of the successful components of qualitative learning of this material and its further application in the learning process. In a special need of visualization are the school subjects that are difficult to understand and, essentially, do not have a wide practical application at the beginning of the study, that is, subjects of the mathematical cycle. This is why the aim of this study is to analyze the mobile tools that can be used to visualize educational material when studying a geometry course in high school. The study used such research methods as analysis and justification of the choice of mobile augmented reality tools for studying mathematics. As a result of this analysis, two augmented reality tools, Arloon Geometry and Geometry AR, were identified as ones that can be used to visualize educational material and create a problematic situation in order to gain new knowledge in high school geometry lessons. The analysis of these tools made it possible to say that the use of augmented reality tools in geometry lessons creates the conditions for positive emotional interaction between the student and the teacher. Firstly, students understand that a mobile device can be used to organize the learning process, to intensify it and to build a personal learning trajectory. Secondly, the emotional component of learning creates conditions for better memorization of educational material, promotes their mathematical interest, actualizes their creative potential, introduces conditions for finding different ways to solve geometric problems. Thirdly, the ability to find solutions of simple geometric problems with the help of augmented reality instills into the student a positive attitude towards success, and the solution of more complex problems becomes faster and more intense.

Materialist epistemology lends design wings: educational design as an embodied process

Materialist design is presented as an embodied perspective on educational design that can be applied to redesign of classroom-based learning environments. Materialist design is informed by a framework of materialist epistemology, which positions material innovation on equal placement with symbol-based formal theory. Historical examples of Einstein’s conceptual reliance on trains for his Theory of Relativity and the Wright brothers’ use of wind tunnels in aeronautics illustrate how materialist design drives progress on complex design problems. A key aspect is the application of scale-down methodology, where complex systems are reconceptualized as interactions among nearly decomposable subsystems that can be redesigned and integrated back into the entire system. The application of materialist design is illustrated with the redesign of an embodied video game that uses real-time motion capture technology to promote high school geometry reasoning and proof, following its use in an ethnically and linguistically diverse classroom. Our embodied perspective offers particular insights for understanding and implementing designs of complex learning environments, and assessing their influences on educational practices and student outcomes.

A variation on an old geometric theme

ABSTRACT The purpose of this note is to introduce two challenging problems in geometry that were inspired by some well-known problems in high school geometry. These problems can be explored in the dynamic geometry environments of computer construction programmes.

Subject Matter Knowledge of Geometry Needed in Tasks of Teaching: Relationship to Prior Geometry Teaching Experience

This study proposestask of teachingas an organizer of dimensionality in teachers’ subject matter knowledge for teaching (SMK) and investigates it in the context of measuring SMK for teaching high school geometry (SMK-G). We hypothesize that teachers use different SMK-G in different aspects of their teaching work and that such differences can be scaled and associated with key elements of instruction. Analyses of 602 high school teachers’ responses to two sets of items designed to measure the SMK-G used in two particular tasks of teaching—understanding students’ work (USW) and choosing givens for a problem (CGP)—suggested the two scales of SMK-G to be distinguishable and differently related to experience in teaching high school geometry.

Factors Affecting Mathematics Achievement In Central Asian Specialized Universities

This study examines variables explaining student’s academic performances in mathematics from specialized engineering institutions. A survey consisting of 42 items was conducted from 127 students and statistical multiple regression was carried out to analyze the data set. Based on Fennema-Sherman Mathematics Attitude Scales followed by the result of stepwise linear regression, found a significant impact of high school geometry grades in mathematics performance. Au-thors suggest that mathematics instructors in higher education should pay attention to improve their student’s confidence, which in turn would decrease the anxiety level towards mathematics. The high school teachers should not advise their students to go to technical sciences in higher education unless the student’s confidence and high school math grade are sufficiently high