Levels Of Geometric Thinking Research Articles

Geometric Thinking of Prospective Mathematics Teachers: Assessing the Foundation Built by University Undergraduate Education in Ghana

This study investigates the geometric thinking levels of final year prospective mathematics teachers in Ghana, utilizing the van Hiele model to evaluate their proficiency. The main purpose was to assess whether university undergraduate mathematics education provides a sufficiently strong foundation for teaching senior high school geometry. A descriptive survey design was employed, involving 1,255 prospective mathematics teachers from three universities: University of Education Winneba (UEW), University of Cape Coast (UCC), and Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development (AAMUSTED). The van Hiele Geometry Test (VHGT) was administered to measure participants’ levels of geometric thinking. The results revealed that 8.8% of participants attained van Hiele Level 1 (visualization), 30.0% reached Level 2 (analysis), and 32.4% achieved Level 3 (abstraction). However, only 15.9% and 12.9% of prospective teachers reached Levels 4 (deduction) and 5 (rigor), respectively. These findings indicate a significant gap between the current geometric thinking skills of prospective teachers and the expectations of the Ghanaian mathematics curriculum, which anticipates higher-order thinking skills. The study concludes that the current undergraduate mathematics education programs in Ghanaian universities may not be adequately preparing future teachers to teach senior high school geometry effectively. It is recommended that these programs be revised to include more focus on developing higher-order geometric thinking skills, with an emphasis on deductive reasoning, formal proof-based learning and rigor in geometry thinking. Enhancing the curriculum and teaching methods could narrow this gap and improve the overall quality of geometry education in Ghana.

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.

Examination of Seventh Grade Students' Van Hiele Geometric Thinking Levels and Their Mistakes on 'Quadrilaterals'

In this study, it was aimed to examine seventh grade students' Van Hiele geometric thinking levels and their mistakes on 'Quadrilaterals'. The research was conducted with ten seventh grade students and the case study model, one of the qualitative research methods, was used. The Van Hiele geometric thinking levels test developed by Usiskin and the diagnostic test developed by Başısık were rearranged and administered to the students. According to the research findings, seventh grade students have misconceptions about diagonal, height concepts and special quadrilaterals. One of the most important of these is that the square positioned on one corner cannot be recognized by students at the analysis level, but is recognized by students at the visualization level. Students at the analysis and visualization levels have different misconceptions regarding the trapezoid. While there is a misconception that all side lengths must be different from each other as a condition for being a trapezoid, there is also a misconception that having four sides is sufficient. Another one is related to the concept of diagonal. There is a misconception that shapes with equal side lengths also have equal diagonal lengths. While no errors or misconceptions were encountered in students at the informal level, misconceptions were encountered in students at the analysis level, and more errors were encountered in students who were visualization and could not be assigned to any level.

The Trend of Geometric Thinking Studies: A Systematic Review

This study focuses on reviewing information from several articles published in educational and social sciences journals worldwide to document the trend of geometric thinking studies in the Indonesian context. The current study shows that the number of publications on geometric thinking has increased in the last four years. The studies conducted in Indonesia and for Indonesian students used a systematic review guided by PRISMA to examine a hundred articles indexed by Scopus, Google Scholar, and Science and Technology Index (SINTA) between 2011 and 2021. Van Hiele’s Geometric Thinking Test, developed by Usiskin, is extensively used when examining geometric thinking levels in the intervention. In addition, descriptive analysis and a t-test are also employed. The result shows that the most common type of geometric thinking experimental research is quasi-experimental design, with the qualitative approach found to be the most used. Therefore, educators should increase the quantity of qualitative and quantitative studies from various nations about geometric thinking. Furthermore, it is also essential to examine the correlation between student performance in geometry and the teacher’s level of geometric thinking. Finally, combining the van Hiele Geometry test with interviews is strongly advised to provide comprehensive and in-depth data for examining geometric thinking.

Pre-Service Elementary Mathematics Teachers’ Evaluations of Meetings with Expert and Peer Participation Based on van Hiele Geometric Thinking Levels

In this study, it was aimed to present the pre-service elementary mathematics teachers’ (PEMT) evaluations of meetings with expert and peer participation based on van Hiele geometric thinking levels (vHGTL). The case study method was used and the study group of the research consists of three PEMT who are studying in the third year of a state university. Data were collected with meeting records, reflection reports of the PEMTs, researcher’s field notes and interviews and were analyzed simultaneously by two researchers to ensure coding reliability with content analysis method. As a result of the research, it was seen that there was a positive change in PEMTs’ criteria for determining vHGTL. It was determined that they had difficulties in clearly distinguishing the differences between the levels at the beginning, and that in the last meetings, they insisted on the criteria related to the levels they determined and made more accurate determinations about the levels. As a matter of fact, the teacher candidates also stated that meetings enabled them to determine the achievements’, solved and unsolved problems’ vHGTL better. In addition, they stated that allowing them to freely express their ideas in meetings contributed to their professional development in geometry teaching knowledge. Suggestions regarding experts and peers participatory meetings were presented.

The Relationships Between Linguistic Literacy and Geometric Thinking in 2nd-Graders

ABSTRACT The correlation between linguistic literacy and geometric thinking was investigated in this study, which was conducted among 99 native Hebrew-speaking 2nd-graders. Current results suggest a positive correlation between the study measures. Higher linguistic literacy achievement was linked to higher geometric thinking achievement. Significant differences were found in most literacy measures between children with low and medium-high geometric thinking, in favor of the latter group. The findings suggest that students with a lower level of geometric thinking are also at literacy risk, and that the two processes share cognitive aspects. These relationships appear in 2nd grade, a relatively young age, which can be critical for children’s future development in both domains. Identifying children at risk for developing literacy and geometry difficulties can be essential in planning an integrated curriculum that refers to literacy in all its aspects, as a tool to prevent future failure both in geometry and language arts.

Some Results and Evaluation of Training for the Development of Students’ Spatial Visualization

This article discusses the results and evaluation of a training course focused on developing spatial visualization using interactive dynamic tools. The course aims to effectively integrate technology with the Descriptive Geometry course by combining content, pedagogy, and technology while considering the course's characteristics. The training course consists of three phases: seeing, imagining, and drawing. Each phase has different tasks that involve manipulating 3D objects in a virtual environment, corresponding to the level of geometric thinking and cognitive steps of mental rotation. The results of the organized training showed a statistically significant increase in student's spatial skills, and the evaluation of students using the SURE (Structure-Oriented Evaluation) model was deemed sufficient.

Examination of 9th Graders' Levels of Geometric Thinking

Robust understanding of geometry is important for students’ future studies in mathematical sciences, many careers, and for understanding the world around them. Therefore, the aim of this study was to understand the geometric thinking levels of 9th graders in Ghana before they enter the Senior High School. This study used the van Hiele theory (Van Hiele-Geldof, 1957) to understand geometric thinking levels of 400 9th graders in western Ghana. These students were given the van Hiele Geometry Test (VHGT) developed by Usiskin (1982). The results showed that 56.2% of 9th graders demonstrated thinking at the pre-visualization level, 30.75%of the graders attained level 1(visualization), 12.5% reached level 2 and only 0.5% demonstrated level 3 thinking. These findings indicate that students are struggling to meet the Ministry of Education’s curriculum standards. Additional analysis points to reasoning about properties of geometric figures using definitions and informal deductive reasoning, instead of relying on visual characteristics of diagrams as a specific obstacle for students. Recommendations are made for improving the teaching and learning geometry based on our findings.

Errors and Misconceptions in Euclidean Geometry Problem Solving Questions: The Case of Grade 12 Learners

Euclidean geometry provides an opportunity for learners to learn argumentation and develop inductive and deductive reasoning. Despite the significance of Euclidean geometry for developing these skills, learner performance in mathematics, particularly geometry, remains a concern in many countries. Thus, the current study examined the nature of learners’ errors in Euclidean geometry problem-solving, particularly regarding the theorem for angle at the centre and its applications. Van Heile’s theory of geometric thinking and teacher knowledge of error analysis were used as conceptual frameworks to make sense of the nature of learners’ errors and misconceptions. Using a participatory action research approach, the study was operationalised by five mathematics teachers from four secondary schools in Motheo district in the Free State Province of South Africa and three academics from two local universities. The study analysed 50 sampled midyear examination scripts of Grade 12 learners from four schools. The findings of this study revealed that most learner errors resulted from concepts on Van Heile’s operating Levels 0 and 1, while the questions mainly required Level 3 thinking. The study recommends that teachers determine their learners’ level of geometric thinking and integrate this knowledge in their lesson preparations and material development.

Relationship between Mathematics Teachers’ van Hiele Levels and Students’ Achievement in Geometry

Using non-experimental quantitative correlational design, the van Hiele levels of thirty grade 9 mathematics teachers, as well as the achievement of 1489 students in geometry were investigated in this study. Results showed a significant difference with a substantial effect size of .64 between the achievement of students whose teachers are operating at level 5 in the van Hiele Levels of Geometric Thinking and those of students whose teachers are operating at level 2. This study underscored the importance to leverage on teachers’ van Hiele level of geometric thinking as it is highly correlated to student achievement. Developing teachers’ geometric thinking ability at the expected levels in order to improve students’ achievement may require trainings, further studies, or curriculum improvement for pre-service teachers. Moreover, a deeper study focusing on other content areas and skills in mathematics may be done to further assess their interrelationships with teachers’ van Hiele levels and student achievement. In addition, it would be of interest to investigate the grade level where student mathematics achievement starts to decline.

Analisis Peningkatan Level Berpikir Geometri Mahasiswa Berdasarkan Teori Van Hiele dengan Pendekatan Konstruktivisme

The provision of problem-solving, analytical, systematic, critical, and creative skills is important to improve human resources that can be obtained from learning mathematics. Geometry is a field of mathematics that aims to develop logical thinking skills, develop spatial intuition, and interpret mathematical arguments. Several studies show that students' understanding of geometry material is still at a low level, so it is necessary to develop or improve the geometry learning process. The purpose of this study is to describe the increase in students' geometric thinking levels based on Van Hiele's theory with a constructivism approach. The method used in this research is descriptive qualitative with data analysis techniques using the Miles & Huberman model in the form of data reduction, data presentation, and drawing conclusions. The results showed that before the learning process using a constructivist approach, students' geometric thinking skills were at levels 1 to 3 or the analysis stage to a formal deduction with the average being at the informal deduction and pre-formal deduction stages. Meanwhile, after the learning process using a constructivist approach was carried out, geometric thinking skills increased at levels 2 to 4 or the informal deduction stage to the rigor with an average increase in the formal deduction stage.

Analysis of Changes in Robot Activity Levels in Mathematics Classes Based on Van Hiele Theory

This study designed robot programming classes toward mathematical concepts and curriculum to introduce robotics, a representative technology of the Fourth Industrial Revolution era, into the mathematics classroom, and analyzed the changes in students’ van Hiele levels. This study presented the robot activity levels (RLs) per the van Hiele levels of geometric thinking (VLs), and applied each RL to the class’s design, including tasks. The students’ learning process and class results were analyzed to ascertain whether their VL increased. The class was conducted with gifted middle school students, and utilized LEGO Mindstorms EV3, an extensively used educational robot. While performing RL based tasks, students sequentially completed tasks RL1~RL4. This study focused on the task performance process of VL3 students to investigate how they transition to a RL beyond their VL. As students performed the five tasks designed to help them transition from VL3 to VL4, reflecting the rise in the van Hiele levels of geometric thinking, they gradually progressed through the active inquiry and finally reached VL4 by completing RL4 tasks.

Mathematics Teachers’ Geometric Thinking: A Case Study of In-service Teachers’ Constructing, Conjecturing, and Exploring with Dynamic Geometry Software

<p style="text-align:justify">Many research studies have been conducted on students’ or pre-service teachers’ geometric thinking, but there is a lack of studies investigating in-service teachers’ geometric thinking. This paper presents a case study of two high school teachers who attended the dynamic geometry (DG) professional development project for three years. The project focused on the effective use of dynamic geometry software to improve students’ geometry learning. The two teachers were interviewed using a task-based interview protocol about the relationship between two triangles. The interviews, including the teachers' work, were videotaped, transcribed, and analyzed based on the three levels of geometric thinking: recognition, analysis, and deduction. We found that the participating teachers manifested their geometric skills and thinking in constructing, exploring, and conjecturing in the DG environment. The study suggests that the DG environment provides an effective platform for examining teachers' geometric skills, and levels of geometric thinking and encourages inductive explorations and deductive skill development.</p>

Prospective Mathematics Teachers’van Hiele’s Geometry Thinking and Habits of Mind: A Description of Hard Skill and Soft Skill by Gender

Geometric thinking level and habits of mind were important hard and soft skills required of prospective mathematics teachers. The purpose of this study is to describe the van Hiele geometric thinking level and habits of mind of prospective mathematics teachers. The descriptive quantitative research method was used. There were 31 female and 17 male prospective mathematics teachers in the sample. The research instrument for measuring geometric level was derived from Usikin's (1982) van Hiele geometric thinking test and a habits of mind questionnaire. According to the findings of this study, prospective mathematics teachers' ability to think geometrically has progressed to the stage of analysis and informal deduction. Furthermore, male prospective mathematics teachers achieved the highest level of geometric thinking, known as rigor. Furthermore, both as a whole and in terms of indicators, prospective mathematics teachers' habits of mind fall into the strong category. Male prospective mathematics teachers have more positive habits of mind than female prospective mathematics teachers. The study's findings can help lecturers, particularly those in the mathematics education study program, determine students' hard and soft skills at the end of the year. Prospective mathematics teachers can also be assigned by institutions based on their geometric thinking and habits of mind.

Rethinking strategy on developing students’ levels of geometric thinking in Sokoto state, Nigeria

<span lang="EN-US">Geometric thinking skills remained a topical issue in mathematics education. The purpose of this research is to investigate the van Hiele levels of geometric thinking skills of the students in Sokoto state to provide a clear picture of the students’ levels for the appropriate development of learning activities, and better understanding. The study involves three mathematics teachers and 200 students (100 students each of basic and senior secondary school students). The samples of the teachers were purposely selected and students were randomly selected. There were two instruments used in the study; interviews for the mathematics teachers while a van Hiele test for geometric thinking was adopted to collect data for the students’ van Hiele levels of thinking. Thematic analysis (for teachers’ interview), descriptive, and Mann-Whitney U test for the analysis of students’ van Hiele levels of thinking was used. The result shows that all the teachers indicate that the traditional approach is the dominant method used in teaching and learning and that students in the state lack basic skills in school geometry. Also, the result indicated that the majority of students sample among Basic and Senior secondary schools in Sokoto state were operating at level 0 respectively. Furthermore, a significant difference between the two independent groups was found in favor of senior secondary school students. It is hoped that in the future, educational institutions could use the present research as a guide for the development and design of modules, learning activities based on the van Hiele levels to bridge the gap in the state.</span>

Effect of Two-Dimensional Geometry Learning on Geometric Thinking of Undergraduate Students During the COVID-19 Pandemic

This study aimed to analyze the effect of two-dimensional geometry learning on the geometric thinking of undergraduate students during the COVID-19 pandemic. This qualitative research involved students in the sixth semester. Data were collected using documentation, test descriptions, and interviews. The results showed there are some reasons why the students faced difficulty in mastering a higher level of geometric thinking: the lack of understanding of the concepts in geometry, the lack of knowledge of the definitions of terms and statements, and how to use them to prove. This study concludes that the variation in the geometric thinking level of undergraduate students shows that there are various students' abilities in two-dimensional geometry, starting from mastering concepts, definitions, and theorems to their use in proof. The suggestion is that several strategies are needed to serve these variations, starting from learning assistance and using diverse learning media. Stakeholders, teachers, and prospective teachers can use the results of this study to improve their understanding of geometry and how to teach it in schools.

Pengembangan Media Pembelajaran Matematika berbasis QR Code untuk Kemampuan Berpikir Geometris Siswa Kelas IV SD

The purpose of this study was to design a mathematics learning media based on qr code to find out the level of geometric thinking of grade IV elementary school students. This research belongs to Research and Development (R&D with ADDIE models, namely Analysis, Design, Development, Implementation, Implementation, and Evaluation). Learning media that have been developed will be assessed and validated by experts who are competent in their fields The instrument used in this study was a questionnaire in the form of teacher responses and student responses to the qr code-based mathematics learning media Results: 1) The author has produced a qr code-based mathematics learning media on the circumferential material and building area flat for grade IV SD / MI students, 2) Expert assessment in material validation, media validation, and language validation which gets a percentage of 96% at highly valid intervals, 3) responses of grade IV SD / MI students on the practicality of the media based on mathematics based learning qr code obtained an evaluation percentage of 87%. This value is at a very, very practical interval.

GEOMETRIC THINKING LEVELS OF MIDDLE SCHOOL STUDENTS IN THE SULTANATE OF OMAN

The study aimed to identifying geometric thinking levels of 8th grade students of Middle School in the Sultanate of Oman, and the significance of statistical differences in geometric thinking according to the gender variable. The sample of the study consisted of (136) students from the schools of the General Directorate of Education in Al Dhahirah Governorate. To achieve the objectives of the study, the researcher applied the geometric thinking test to the study sample. The results showed a low level of geometric thinking of the study sample, as (17.65%) of the sample did not exceed the visual level, while (55.88%) of the sample was classified on the first visual level, (17.65%) of the sample was on the analytical level, and (2.94%) of the sample was on the semi-inductive level. The results also showed that there was a statistically significant difference in the geometric thinking test between males and females in favor of the female sample. This may be due to the difference in teaching practices and employment Teaching aids in teaching geometric subjects, and the researcher recommends the importance of training mathematics teachers to design educational activities based on Van Hill levels of geometric thinking.

Analisis Kecerdasan Spasial Siswa dalam Menyelesaikan Soal-Soal Geometri Berdasarkan Tingkat Berpikir Van Hiele

The purpose of this research is to describe, identify and analyze students' spatial intelligence in working on geometry problems according to Van Hiele's theory. The method used in this research is descriptive research with a qualitative approach. This research was conducted at SMP Negeri 2 Tarakan. The research instrument used was the Van Hiele geometric thinking level test and spatial ability test, interview guides and documentation. Prior to use, the test, an interview guide, was first validated. The data analysis technique will be used in 4 stages, namely collecting, reducing, presenting, and concluding all the data obtained. The results of this study are that students with high spatial intelligence (KST) are at the level of informal deductive thinking (level 3) where at this ability students are able to know the relationship between shapes. While students with moderate spatial intelligence (SCC) are at the level of analytical thinking (level 2), namely the ability to know and mention the properties of flat shapes. Students with low spatial intelligence (KSR) are at the level of visualization thinking (level 1). In conclusion, it is important for teachers to know students' abilities in learning geometry by paying attention to the level of geometric thinking, being able to use the Van Hiele Geometry learning model, increasing spatial intelligence and van Hiele levels can be done through training involving geometric media or software (geogebra).
 
 Keywords: spatial intelligence, geometry, level of thinking, Van Hiele

Effects of Dynamic Geometry Software on Students’ Geometric Thinking Regarding Probability of Giftedness in Mathematics

Gifted students have potential to improve countries and this potential can be revealed and developed in schools where they spend most of their times with other regular students. However, these classrooms have some limitations for them; hence, they need some differentiated activities. Usage of dynamic geometry in mathematics lessons could be an opportunity to provide differentiated activities. Therefore, the aim of this study is to explore effects of mathematics lessons integrated with dynamic geometry activities on students' van Hiele geometric thinking levels controlling their probability for mathematical giftedness. Participants of the study were fifty-three fifth graders from a private school in Marmaris, Turkey. These students were grouped in terms of their probability of mathematical giftedness. Seven dynamic geometry activities about properties of line segments, triangles and quadrilaterals were developed and implemented in classrooms with tablets. Van Hiele geometric thinking level test was administered to the participants as pre-test and post-test. Results showed that dynamic geometry activities help students to move from geometric thinking level about recognizing shapes with visual clues to higher level about geometrical properties of shapes, namely relationship among shapes and their properties. Moreover, interaction between their probability for mathematical giftedness and improvements in geometric thinking levels were found. This study may contribute both to the gifted education and mathematics education fields by exploring the improvements in geometric thinking level and differentiated opportunities for gifted students. As a suggestion, a more comprehensive experimental study with larger samples so as to obtain generalize the findings could be conducted as a further study.