Circle Theorems Research Articles

The use of Van Hiele’s geometric thinking model to interpret Grade 12 learners’ learning difficulties in Euclidean Geometry

The 21st-century mathematics classrooms should equip learners with well-grounded knowledge and thinking skills pertaining to geometry. However, Euclidean geometry remains one of the challenging, if not the most difficult topic for many learners. As a result, the purpose of this article is to interpret Grade 12 learners’ learning difficulties in Euclidean geometry. We use Van Hiele’s geometric thinking model and Hoffer’s skills to argue an interpretation of learning difficulties in Euclidean geometry as a focal point towards creating effective teaching and learning of this important topic. This explanatory sequential mixed-methods approach involved 60 Grade 12 learners who wrote a geometry test and completed a questionnaire based on Van Hiele’s geometric thinking levels. In addition, semistructured interviews were conducted with a sample of 12 learners and four educators to investigate their views about geometry learning difficulties. The findings of the study revealed that learners had poor conceptualisation of properties of shapes, visualisation skills, circle theorems and geometry terminology, resulting in them experiencing learning difficulties. The recommendations are that, during instruction learners should be given the opportunity to manipulate real geometric objects to enhance their visualisation and visual thinking skills. In addition, we recommend that educators should teach level-specific geometry vocabulary to enable learners to understand concepts at different Van Hiele’s levels. Furthermore, we recommend that educators should use constructivist teaching approaches that encourage learners’ conceptual understanding instead of traditional methods that promote rote memorisation of geometric facts. Educators should develop learners’ broad knowledge of geometry to overcome geometry-related errors and misconceptions.

Engaging Neural Plasticity in Senior High School Students: The Impact of Guided Discovery Teaching Method on Achievement in Circle Theorems

Purpose: The study's objective was to ascertain how circle theorem achievement in SHS students was affected by the guided discovery teaching method, a student-centered approach associated with improved neural plasticity.
 Materials and Methods: Two Form 2 classes from various Wa Municipal schools were chosen for the study using convenient, purposeful, and straightforward random sampling techniques. The study adopted a non-equivalent quasi-experimental design to compare students who are taught with the guided discovery method and the traditional method of teaching circle theorems concepts. The sample size was composed of 164 students. Using a separate t-test and descriptive statistics, the Geometry Achievement test was investigated. A pre-test was given before the experiment (post-test) began. The students who engaged in guided discovery instruction outperformed than those who did not when teaching and learning Circle theorems.
 Findings: The findings suggested that student-centered methods like guided discovery can greatly improve students’ achievement in the study of circle theorems. One of the implications derived from the study indicated that guided discovery teaching approach offers students the chance to put a method of learning into practice after they have used it. This was done by using illustrations of diagrams on cardboard. This must be considered in the planning of educators and subject-matter experts. For pre-tertiary education, the government must make it mandatory to use cardboard, mathematical instruments, and instructional sheets as teaching aids. Since visual representations of Euclidean Geometry diagrams bring reality to teaching and learning through pictures and diagrams, the importance of visualization and experimentation in learning circle theorems should not be underestimated by teachers or students. This will improve their conceptual understanding.
 Implications to Theory, Practice and Policy: One can recommend that in order to enhance the performance of SHS students, it will be most advantageous to introduce guided discovery teaching methods to pre- and in-service teachers, through promotion by Ghana Education Service and/or other stakeholders in the education sector. This introduction may be distributed through workshops and seminars for mathematics teachers' instructional techniques and skills will improve as a result. This study has added a lot to our understanding of the world. The guided discovery teaching method approach to teaching circle theorems has been strengthened and expanded as a result of this research, first and foremost. This thesis thus makes a substantial enhancement of the body of knowledge. The research also explains and backs up the notion that guided discovery methods aid students' academic endeavors. This shows that the teaching process engages students' attention and improves their capacity for memory and recall.

Learner Centred Teaching Strategies That Can Be Incorporated with Guided Discovery Method on Senior High School Students’ Achievement in Circle Theorems

Learner Centred Teaching Strategies That Can Be Incorporated with Guided Discovery Method on Senior High School Students’ Achievement in Circle Theorems

EFFECTIVENESS OF USING GEOGEBRA IN TEACHING AND LEARNING CIRCLE THEOREMS ON STUDENT-TEACHERS’ PERFORMANCE

This paper aims to explore the effect of using GeoGebra in teaching and learning circle theorems on student-teachers’ performance. This paper is based on the results of the four-week experiment of GeoGebra software in teaching and learning the concepts of circle theorems of the geometry of student-teachers at the College of Education. The effectiveness of using GeoGebra software was explored by measuring the achievement level with the help of achievement tests and perceptions of using GeoGebra software through the questionnaire. The data were analyzed and interpreted by using a frequency distribution table, percentage, mean, standard deviation, and t-test. The findings showed that there was a higher mean score in achievement in circle theorems of the experimental group, that is, those taught with GeoGebra software than that of the control group, that is, those taught without GeoGebra software, and the difference was significant at 0.05 level. Similarly, the student-teachers of the experimental group had a positive perception to support the use of GeoGebra software in teaching and learning circle theorems in geometry and mathematics in general. Thus, the use of GeoGebra software is an effective tool to increase achievement; promote curiosity and creativity; make clear sense of concepts and encourage overall learning of students in mathematics. Finally, this study suggests that mathematics teachers need to use ICT tools including GeoGebra for effective teaching and learning.<p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0073/a.php" alt="Hit counter" /></p>

Generalized Necessary and Sufficient Robust Boundedness Results for Feedback Systems

Classical sufficient conditions for ensuring the robust stability of a dynamical system in feedback with a nonlinearity include passivity, small gain, circle, and conicity theorems. We present a generalized version of these results for arbitrary semi-inner product spaces. Our result is purely algebraic, and holds even when the conventional discrete or continuous-time causal dynamical systems are replaced by general nonlinear relations, where there need not exist a notion of time. Our result clarifies when the sufficient conditions for robust stability are also necessary, and explains why stronger assumptions such as linearity and time-invariance are typically needed to prove necessity in the conventional dynamical systems setting.

Fixed ellipse theorems in metric spaces and an application of discontinuous activation function in neural networks

In this article, we establish new fixed ellipse theorems that will provide a geometrical perspective to the classical metric fixed point theory. Also, we provide novel fixed ellipse consequences as the expanding of some existing and known fixed circle theorems in literature to ellipse case, and give non‐trivial illuminating examples, which show the impressiveness of our results. On the other hand, using the given fixed ellipse, we give some solutions for the open question which is posed by Rhoades with respect to the presence of contractive mappings that permissive discontinuity at the fixed point. Furthermore, we give an application to usage respecting activation functions handled at neural networks via the discontinuity and the obtained results of fixed ellipse.

Using activity method to address students’ problem-solving difficulties in circle geometry

This action research sought to address the problem-solving difficulties of students in circle geometry problems using activity method of teaching and learning. The study in investigated students’ difficulties in solving circle geometry problems. It also found out how activity method of teaching and learning helped address the students’ difficulties of solving circle geometry problems. Finally, it established the extent to which the activity method of teaching and learning affected the performance of students in circle geometry. The accessible population was all form two students at Zion College in the Volta Region of Ghana. It numbered 499. The sample was general science 2B students. The sample size was 27 students made up of 10 girls and 17 boys. The instruments for data collection were pre- and post-test. The pre-test was conducted to establish the existence of the difficulties of students in solving circle geometry problems. After establishing the problems, activity method of teaching and learning was used as an intervention to remedy the situation. A post-test was administered to measure the improvement. The data was presented using tables and figures. Inferential statistics was also used to analyze the data. From the analysis, activity method help the students solve complex questions, which demand multiple circle theorems to solve. It was recommended that mathematics teachers should always use activity method of teaching and learning in teaching circle geometry.

Some common fixed-point and fixed-figure results with a function family on $ S_{b} $-metric spaces

<abstract><p>In this paper, we prove a common fixed-point theorem for four self-mappings with a function family on $ S_{b} $-metric spaces. In addition, we investigate some geometric properties of the fixed-point set of a given self-mapping. In this context, we obtain a fixed-disc (resp. fixed-circle), fixed-ellipse, fixed-hyperbola, fixed-Cassini curve and fixed-Apollonious circle theorems on $ S_{b} $-metric spaces.</p></abstract>

Acceptance of the GeoGebra Application in Learning Circle Theorems

The learning area of circle theorems is one of the most difficult topics in geometry, resulting in low student performance. GeoGebra has been shown in studies to enhance learners' proficiency in circle theorems. However, pre-service teachers' use of GeoGebra is not at the expected level in Eswathini. The adoption of an information system is reliant on its acceptance by individuals. However, little is known regarding pre-service teachers' use of GeoGebra to understand circle theorems. The goal of this study was to investigate pre-service teachers' perceptions of GeoGebra's suitability for learning circle theorems. A cross-sectional survey design was used in this investigation, with a total of 187 pre-service instructors as participants. The model explained 74.9% of the variance in the acceptability of GeoGebra for learning circle theorems by Eswatini pre-service teachers. According to the findings, task-technology fit, system quality, system compatibility, perceived ease of use, perceived usefulness, perceived attitude toward, and user satisfaction account for 74.9% of the variance in actual use. The study's findings revealed that rural Eswatini pre-service teachers' reported attitude toward using the mathematics software application GeoGebra for learning circle theorems was the strongest direct predictor of actual use. This research shows that pre-service teachers' views toward technology integration in education should be positive for educational learning applications to be successfully adopted in Eswatini teacher training institutes.

Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

AbstractIn A Treatise on Electricity and Magnetism, Maxwell determines the angles of intersection for which one may use Kelvin's inversion method to obtain the perturbed electric potential upon placing intersecting spherical conductors into a region with a known potential. There are numerous modern applications utilizing this geometric construction in potential theory and hydrodynamics, and generalized circle and sphere theorems play a foundational role in this area of mathematical physics. In his work, Maxwell gives an intuitive argument for obtaining the perturbed potential based on intersecting planar conductors and a spherical inversion, and in this paper we extend his ideas to a full proof using rotational transformations and reflections. In the process, we disprove results in [Proc Lond Math Soc., 1966:3(16)] and [Stud Appl Math., 2001:106(4); Z. Angew. Math. Mech., 2001:81(8)] on boundary value problems in hydrodynamics involving intersecting circles and spheres, and we detail the angles of intersection for which these theorems are viable. Moreover, our proof recovers a special case overlooked by Maxwell for which Kelvin's inversion method may be utilized to obtain full solutions.

Conceptualising TPACK Within Mathematics Education: Teachers’ Strategies for Capitalising on Transitions Within and Beyond Dynamic Geometry Software

This article investigates the knowledge arising in mathematics teachers’ planning of how to manage transitions within and beyond dynamic geometry environments in the topic of circle theorems. The notion of situated abstraction is used to elaborate the central TPACK construct within mathematics education and address previous criticisms of the framework, specifically to clarify the distinction between the central construct and the dyadic constructs. Four case-study teachers each participated in a semi-structured interview based upon a pre-configured GeoGebra file. The teachers were asked to demonstrate how they would use the GeoGebra file to introduce students to the circle theorem that the angle at the centre of the circle, subtended by an arc, is double the angle at the circumference subtended by the same arc. The visual and audio aspects of the GeoGebra interviews were recorded and the TPACK framework used to analyse teachers’ knowledge arising in the four interviews. The central TPACK construct is illustrated with examples of teachers’ strategies for capitalising on transitions within and beyond dynamic geometry environments for the purposes of teaching circle theorems and contrasted with the dyadic construct of TCK. The utility of the theoretical elaboration of the TPACK construct within mathematics education is demonstrated and implications discussed.

Effect of Inquiry-based Teaching Approach on Students Achievement in Circle Theorems

This study investigated the effect of inquiry-based teaching approach on students’ achievement in Circle theorems in Senior High Schools. The study used sequential exploratory mixed method research design to collect quantitative and qualitative data to answer the various research questions. A sample of 105 students and 6 mathematics teachers from the two schools were randomly and conveniently selected respectively for the study. Circle Theorems Achievement Tests (CTAT) was administered to both intact classes (control and experimental) as pre-test and after the intervention a similar CTAT was administered as post-test. During treatment, the experimental group were taken through inquiry- based teaching approach instruction while the traditional instruction was applied to the control group. Results from paired sample t-test showed that participants in the experimental group had increment in their post-test as compared to the pre-test. However, independent samples t-test results revealed that students in the experimental group achieved better in the post-test as compared to those in the control group. Interview data showed students negative attitudes and teachers’ teaching methods (use of traditional teaching method) were the main cause of students’ poor performance in circle theorems. The observation data also revealed that time factor was challenging since inquiry class activities needed more time to complete and also forming the small groups was a challenge in the class due to large class size and classroom not spacious. In conclusion, inquiry-based teaching approach was found to increased students’ achievement in circle theorem than the traditional instruction and hence recommended for teachers to implement it in their teaching.

Teaching and learning circle theorems with Dynamic Autograph technology: Should we employ the social-interaction or the self-exploration strategy?

The use of technology as a pedagogical practice is growing at a phenomenal rate due to the availability of numerous computer mathematics software. As a result, conscientious efforts need to be made by teachers to integrate technology effectively to enhance the teaching and learning processes and consequently learning outcomes of students. This non-equivalent quasi-experimental study sought to explore students’ achievements and their attitudes after using a dynamic Autograph software technology to learn circle theorems. One hundred and fourteen Senior High School students purposively selected from three different schools participated in the study. While students from two of the schools were taught using the Autograph technology in a social-interaction and self-exploration environment respectively, the group of students from the last school were taught conventionally without the use of the Autograph technology. The findings of this study showed that the three teaching strategies had a significant positive effect on students’ achievements. However, the social-interaction Autograph-assisted group significantly outperformed, followed by the self-exploration Autograph-assisted group, and then the conventional group. The study also showed that though the self-exploration group showed higher positive attitudes than the social-interaction, the Autograph-assisted groups in general reported overall positive attitudes towards using the technology indicating further that the Autograph-support made lessons practical, stimulating, and provided imagery simulations and multiple representations that enhanced visualisation and understanding. Implications of the Autograph technology use in teaching, while engaging students in small groups are discussed.

EFFECTS OF PROBLEM-BASED INSTRUCTIONAL STRATEGY ON SENIOR SCHOOL STUDENTS’ PERFORMANCE IN CIRCLE THEOREMS

This study was conducted with the aim of determining the effects of Problem-Based Instructional Strategy (PBIS) on students’ performance in circle theorems in Mathematics. The study is a quasi-experimental type of non-randomized, nonequivalent pre-test and post-test control group involving a 2 x 2 x 3 factorial design, indicating instructional strategies at two levels (Problem-Based Instructional Strategy and conventional method), gender at two levels (male and female) and scoring level at three levels (high, moderate and low level). The target population was all Senior Secondary School students in Ogbomoso, Nigeria. A total of 244 students participated in the study. The research instrument used is Circle Theorems Performance Test (CIRTPT). t-test statistics and ANCOVA were employed to analyze the data. The findings of the study revealed that: there was a significant difference in the performance of students taught circle theorems using PBIS and those taught with conventional method (t(242)=2.87; p<0.05); there was no significant difference in the performance of male and female students taught circle theorems using PBIS (t(110)=0.52; p>0.05); and there was no significant difference in the performance of students on the basis of score levels when taught circle theorems using PBIS (F(2,108)=2.31; p> .05).

The impact of concept-based instruction on senior high school students’ achievement in circle theorems

The influence of two teaching methods namely, concept-based instruction and traditional method on senior high students’ achievement in circle geometry was investigated using the non-equivalent quasi experimental design. Purposive sampling technique was used to select two schools which were designated as experimental and control groups. Pre and post-tests were conducted to assess participants’ achievement before and after the intervention. Paired sample t-test and an independent sample t-test technique were used to analyse the data. The findings of the study revealed a significant difference in the mathematics achievement of students in the experimental group taught using concept-based instruction compared to those taught with the traditional method (control). The implications of the findings are discussed.

Digitalizing the Teaching and Learning of Mathematics at the Senior High Schools in Ghana; the Case of Flipped Classroom Approach

Introduction: The 21st century comes with its own challenges so far as the teaching and learning of mathematics is concern. This is because most of the senior students have access to digital devices that allow them to navigate the social media channels like the WhatsApp, Twiter, instagram, SnapChat, Facebook, TikTok and the likes. Unfortunately these handles are loaded with various forms of videos and other stuffs that these innocent young ones turned to love to follow more than their studies, thereby affecting their academic achievements.
 Purpose: The motive of this study was to enhance students’ conceptual understanding in circle theorems using the flipped classroom approach.
 Methodology: A sample of Fifty (50) Remedial students from the Bekwai Municipal Assembly in the Ashanti Region of Ghana was selected by means of purposive sampling method for the study. With the use of quasi- experimental design, a pretest post test method was use to gather the data on the students to ascertain the impact of the interventional design. In addition to the tests, a questionnaire was also used to find out how students feel about the flipped classroom pedagogy in teaching and learning.
 Findings: The analysis of the questionnaire came out that students enjoyed the lessons and their performance appreciated as a result. A statistical significant improvement in the students’ performance made the researcher reject the null hypothesis and concluded that flipped classroom approach impacted positively on students’ performance.
 Recommendations: It was therefore recommended that the flipped classroom should be adopted as one of the pedagogical strategies in teaching mathematics at the Senior High school.

Fixed point to fixed circle and activation function in partial metric space

Abstract We familiarize a notion of a fixed circle in a partial metric space, which is not necessarily the same as a circle in a Euclidean space. Next, we establish novel fixed circle theorems and verify these by illustrative examples with geometric interpretation to demonstrate the authenticity of the postulates. Also, we study the geometric properties of the set of non-unique fixed points of a discontinuous self-map in reference to fixed circle problems and responded to an open problem regarding the existence of a maximum number of points for which there exist circles. This paper is concluded by giving an application to activation function to exhibit the feasibility of results, thereby providing a better insight into the analogous explorations.

Effect of Geogebra on Pre-Service Teachers' Achievement and Perception of Circle Theorems at E. P. College of Education, Bimbilla-Ghana

The study determined the effects of Geogebra on pre-service teacher's achievement and perception in circle theorems. The study was carried out in Evangelical Presbyterian (E.P.) College of Education, Bimbilla. Three research questions were formulated to guide the study. A sample size of 88 pre-service teachers (PSTs) was used. The control and experimental groups had 45 and 43 pre-service teachers respectively. The study used a quasi-experimental design. Data were collected from the subjects and analyzed quantitatively using SPSS version 18 on the close-ended questionnaire and the achievement pre-test and post-test. The pretest and post-test were analyzed using an independent sample t-test while the perception questionnaire used percentages, mean, and standard deviation in the analysis. Results from the study revealed that students taught circles theorem with Geogebra teaching strategy achieved a higher mean score than those taught through the conventional method with large effect size. The experimental group showed a positive interest in the Geogebra teaching strategy of teaching circles. The paired sample t-test also confirmed that the experimental group gained a mean difference of 9.83 in post-test with a large effect size too. The study recommends the integration of Geogebra into the teaching of circle concepts and theorems at the colleges of education in Ghana.

Impacts of implementing geogebra on eleventh grade student’s learning of Euclidean Geometry

The aim of the study was to evaluate the efficiency of applying GeoGebra software as a teaching tool to enhance high poverty, rural grade 11 students’ understanding of Euclidean geometry circle theorems. The experimental group received instruction on how to use, make simple constructions, and measure elements in GeoGebra. The control group was taught in a traditional lecture method. Using a quantitative research design, the findings indicated that the use of GeoGebra had a statistically significant effect on the students’ ability to correctly complete problems regarding some circle geometry theorems. Additionally, using a Likert scored questionnaire, students demonstrated that they valued the use of GeoGebra and appreciated this learning innovation. Altogether, students who interacted with GeoGebra were highly engaged in the learning processes and actively collaborated with other students rather than remaining passive learners. The students who participated in the study evaluated GeoGebra as an appropriate tool to assist them in the learning of mathematics.

Singularity Induced Interior Stokes Flows

Three complex variable circle theorems for studying the two-dimensional Stokes flows interior to a circular cylinder are presented. These theorems are formulated in terms of the complex velocities of the fundamental singularities in an unbounded incompressible viscous fluid. Illustrative examples are given to demonstrate their usefulness.