Geometric Thinking Research Articles

Didactic Design of the Concept of Surface Area of Flat-Sided Prism Based on van Hiele’s Theory in Online Learning

This research aimed to develop a didactic design of the concept of the surface area of a flat-sided prism by considering the stages of van Hiele’s theory as a learning trajectory. The didactic design of the concept of the surface area of a flat-sided prism based on van Hiele’s theory has been adapted to the pandemic situation and implemented to online learning. The research method employed was a qualitative method with data collected through observation, interviews, and documentation. The initial step in this research was to test the concept of the surface area of a flat-sided prism on 53 9th-grade students for the 2019/2020 school year to identify learning obstacles. Following the identification of the learning obstacles, an initial didactic design was then drawn up by applying the phases in van Hiele’s model of geometric thinking. The didactic design prepared was subsequently implemented online to 8th-grade junior high school students. The results of the implementation of the didactic design were analyzed as the final product. The conclusion from this research is that by using a didactic design that considers the stages of van Hiele geometry in understanding the concept of surface area of a flat-sided prism, it can help students understand the concept of a flat-sided prism correctly. It was found that students' understanding of the concept of the surface area of a prism improved from visual level to informal deduction.

CLUSTERS OF PREVALENT PATTERNS OF GEOMETRIC THINKING LEVELS AMONG MATHEMATICS STUDENTS

Geometric thinking skills are the perceived abilities of an individual to think and reason in geometric contexts. These skills acquired by students in geometry remain poor and unsettling because of the misconceptions that hinder the students in learning the components of geometry. The study described the common unplaceable patterns in geometric thinking of 153 mathematics education students in a state university in Eastern Visayas, Philippines. Frequency Analysis was employed in the study to determine the number of occurrences of the patterns stressing the cause for students placed under level 0 or unplaceable. Van Hiele Achievement Test was used to gather the students’ performance in geometry at all levels, namely: visualization, analysis, informal deduction, deduction, and rigor. The findings attested that only 13.1% of the students managed the third level of the Van Hiele Levels while 43.1% of them were unplaceable. Common patterns were drawn and describe to understand the consequences in geometric thinking ability at level 0. These observable patterns were grouped into core-remedial, topical-corrective, and close-corrective groups. The clusters will enable educational institutions to address the individual gaps in geometry.

Experts and designated users evaluations on visual tools screencast SketchUp Make (ViToS-SUM)

The problem and the aim of the study. Students at secondary school are facing problems in learning Mathematics for topic Geometry. The purpose of this study is to examine the validity of a learning strategy for 3-dimensional Geometry, using Visual Tools Screencast SketchUp Make, called ViToS-SUM. Research methods. ViToS-SUM consists of four components: level of van Hiele geometrical thinking, visual spatial skills, visual tools and video tutorial screencast SketchUp Make. A topic in form 3 mathematics, Plans and Elevations was chosen for this study. The whole process of design and development of ViToS-SUM adopted the five cyclic stages of ADDIE instructional design model. This article addresses the details of the final two stages specifically, implementation and evaluation prior to pilot test. Twelve students from a secondary school and three experts involved in this study. Quantitative approaches were used to collect data as well as to analyse the experts’ and students’ views on the appropriateness of ViToS-SUM. Results. The experts agreed that both visual spatial skills (mean = 5.00) and level of van Hiele geometrical thinking (mean = 4.61) should be embedded in ViToS-SUM. Moreover, the experts also agreed that the content of ViToS-SUM is suitable (mean = 4.51) with the mathematics content for topic Plans and Elevations. The pre and post test showed that there were significant differences in mean scores of visual spatial skills, before and after learning via ViToS-SUM (t=12.21; p<0.05). Furthermore, the pre and posttest also revealed that there is a significant difference in students’ level of van Hiele geometrical thinking before and after intervention (Z =-3.18; p < 0.05). Thus, ViToS-SUM had supported most of students in constructing concepts of Geometry. Meanwhile, the findings revealed that all experts agreed that ViToS-SUM served as pedagogical learning strategy for Geometry. Conclusion. This learning strategy should be integrated in the mathematics curricular for secondary schools to increase students’ performance in Geometry. Training is needed for teachers in order to deliver the concepts of Geometry effectively using this mode of teaching. More computer facilities should be equipped to schools in order to encourage teachers and students to utilize technology in teaching and learning.

Effect of the VAN Hiele Instructional Model on Students’ Achievement in Geometry

The purpose of this study was to determine the effect of the Van Hiele instructional model on students’ achievement in Circle Geometry at Daffiama Senior High School in the Daffiama-Bussie-Issa District of the Upper West Region in Ghana. The purposive and simple random sampling techniques were employed to select a sample of 75 participants for the study. The sample involved two groups: the experimental group and the control group. While teaching based on the Van Hiele model was carried out in the experimental group, teaching with the traditional method was carried out in the control group. The study employed a quasi-experimental research design. The instruments used for data collection were tests, interviews, and classroom observation. Findings from the data analysis suggested that participants were at the prerecognition level before the intervention, improved from the prerecognition level to level 2 after the intervention as the model facilitated learning. It was recommended that teachers determine the geometric thinking levels of students before instruction; the Van Hiele learning and instructional model is adopted in curriculum design and applied in the teaching of geometry and other areas of mathematics.

Descriptive content analysis of theses related to geometric thinking levels

Geometrik düşünme, geometri öğretimi alanında önemli bir yere sahiptir. Geometrik düşünmenin düzeyleri Van Hiele teorisi ile belirlenmiş ve birçok araştırmaya konu olmuştur. Alanyazındaki çalışmaların nasıl bir çerçeve oluşturduğunun ve güncel eğilimlerin belirlenmesi amacıyla bu çalışmada, Türkiye’deki geometrik düşünme düzeyleri ile ilgili lisansüstü tezlerin betimsel içerik analizinin yapılması amaçlanmıştır. Bu amaçla Yükseköğretim Kurulu Ulusal Tez Merkezi, “geometrik düşünme düzeyleri”, “geometri anlama” ve “Van Hiele” anahtar kelimeleri kullanılarak taranmıştır. Bu bağlamda, 2005-2019 yılları arasında yayınlanan ve erişime açık olan 71 teze ulaşılmıştır. Tezler doküman inceleme tekniği ile incelenmiştir. Ardından verilerin içerik analizi yapılmıştır. Bulgular 11 kategori altında düzenlenerek frekans ve yüzdeler halinde tablolarda sunulmuştur. Çalışmanın bulgularından hareketle konuya ilişkin yüksek lisans tez sayısının doktora tezlerine kıyasla fazla olduğu ve tezlerin belirli üniversitelerle sınırlı olduğu görülmüştür. Tezler yaygın olarak ilköğretim ana bilim dalına bağlı programlarda yürütülmüştür. Çalışmalarda genellikle 100 kişiye kadar olan örneklem büyüklüğü tercih edilmiştir. Örneklem grubu genelde ilköğretim öğrencilerinden seçilmiştir. Spesifik bir geometri konusundan ziyade geometri bütün olarak çalışılmış ve öğrencilerin geometrik düşünme becerilerinin yanında başarı, tutum gibi bilişsel ve duyuşsal özellikler de incelenmiştir. Yüksek lisans tezlerinde nicel, doktora tezlerinde ise nitel desen daha fazla tercih edilmiştir. En çok tercih edilen ölçme aracı ise testlerdir. Veriler analiz edilirken çoğunlukla kestirimsel tekniklerin kullanıldığı sonucuna ulaşılmıştır.

Geometric Thinking of Future Teachers for Primary Education—An Exploratory Study in Slovakia

Various studies show that the level of knowledge achieved by pupils is influenced by the level of knowledge of their teachers. In this article, we focus on geometric thinking and the solutions for geometric tasks through a study of future teachers of primary education. The research sample consisted of 59 master’s students from the Teacher Training for Primary Education (TTPE) program. To determine the level of geometric thinking of TTPE students, the van Hiele geometric test was used. Two geometric multi-item tasks were proposed and the students’ solutions to these tasks were quantitatively and qualitatively evaluated. The main goal was to analyze students’ misconceptions while solving tasks and to compare and reveal the connections between their solutions and their achieved level of geometric thinking. A statistical implicative analysis was used for a deeper analysis, namely the statistical software C.H.I.C. The research findings show that more than 40% of TTPE students in the research sample did not reach the required level of geometric thinking. The achieved level of the geometric thinking of students is also influenced by the type of high school education. We observed problems with understanding the concept of the triangle and square in TTPE students. The connections between the solutions of two geometric tasks and the achieved level of geometric thinking were also revealed.

Prospective teachers constructing dynamic geometry activities for gifted pupils: Connections between the frameworks of Krutetskii and van Hiele

The Swedish educational system has, so far, accorded little attention to the development of gifted pupils. Moreover, up to date, no Swedish studies have investigated teacher education from the perspective of mathematically gifted pupils. Our study is based on an instructional intervention, aimed to introduce the notion of giftedness in mathematics and to prepare prospective teachers (PTs) for the needs of the gifted. The data consists of 10 dynamic geometry software activities, constructed by 24 PTs. We investigated the constructed activities for their qualitative aspects, according to two frameworks: Krutetskii’s framework for mathematical giftedness and van Hiele’s model of geometrical thinking. The results indicate that nine of the 10 activities have the potential to address pivotal abilities of mathematically gifted pupils. In another aspect, the analysis suggests that Krutetskii’s holistic description of mathematical giftedness does not strictly correspond with the discrete levels of geometrical thinking proposed by van Hiele.

An Axiomatic System for a Physical or Digital but Continuous 3-Dimensional Euclidean Geometry, Without Infinite Many Points

This paper is concerned with finding an axiomatic system, so as to define the 3-dimensional Euclidean space, without utilizing the infinite ,that can imply all the known geometry for practical applied sciences and engineering applications through computers , and for more natural and perfect education of young people innbsp the Euclidean geometric thinking. In other words by utilizing only finite many visible and invisible points and only finite sets, and only real numbers with finite many digits, in the decimal representation. The inspiration comes from the physical matter , rigid, liquid and gaseous, which consists of only finite manynbsp particles in the physical reality. Or from the way that continuity is produced in a computer screen from only finite many invisible pixels . We present such a system of axioms and explain why it is chosen in such a way. The result is obviously not equivalent, in all the details, with the classical Euclidean geometry.nbsp Our main concern is consistency and adequacy but not independence of the axioms between them. It is obvious that within the space of a single paper, we do not attempt to produce all the main theorems of the Euclidean geometry, but present only the axioms.

Assessing van Hiele’s geometric thinking levels among elementary pre-service mathematics teachers

Teachers’ geometric thinking is crucial to teaching efficacy in geometry since teacher knowledge or thinking serves as a basis for the quality of instruction provided for students’ learning. Teachers’ thinking about geometry has attracted much attention among mathematics education researchers. This study therefore aimed at assessing elementary pre-service teachers’ geometric thinking within the first three levels of van Hiele’s model. The study was guided by three objectives. The objectives were to (1) assess the distribution of van Hiele’s geometric thinking among the study participants, (2) determine if the participating pre-service teachers’ geometric thinking is significant for teaching geometry, and (3) find out if any difference in geometric thinking of the pre-service teachers existed with regard to gender. The study used the descriptive survey design. The study participants were prospective mathematics teachers drawn from four Colleges of Education in the Bono Region and Ashanti Region of Ghana. The Colleges were randomly selected for the study. The study participants comprised 217 pre-service teachers. The van Hiele’s test instrument was adapted and pilot tested to assess the internal consistency of the items in the various levels. The calculated reliability coefficient of the instrument ranged from 0.71 to 0.74. The instrument was administered to the study participants on the scheduled date. Data generated from the participants were analysed based on the study objectives. Findings from the analyses show that pre-service teachers have limited geometric thinking within the first three levels. However, their geometric thinking of the levels assessed was found to be significant which could have some impact on teaching geometry. Findings also reveal gender differences in pre-service mathematics geometric thinking. It is recommended that conscious effort must be made by mathematics teacher educators in the Colleges of Education to deepen the pre-service mathematics teachers’ geometric thinking.

BERPIKIR GEOMETRI LEVEL VISUALISASI SISWA SEKOLAH MENENGAH PERTAMA MELALUI TOPIK SEGIEMPAT MENURUT TEORI VAN HIELE

This research aims to describe the level of geometric thinking and geometric thinking processes of Junior High School students according to van Hiele's level of thinking on the topic of quadrilaterals. The qualitative approach is the research method used in this study through a case study method by testing the Van Hiele Geometry Test (VHGT) which was adapted from Usiskin's CDASSG and conducting interviews about the thinking process in the form of identifying, defining, and classifying which was adapted from the interview guide of Burger and Shaughnessy (1986). The subjects of this study were 297 grade VII and VIII students from two schools located in the Lembang sub-district. The results of the VHGT test showed that there were 81 students counting level 0 (visualization). The results showed that the students of class VII and VIII level 0 were as follows: 1) students were able to recognize the types of quadrangle but still affected by the prototype, 2) students were not able to classify quadrilaterals, and 3) overall description of the geometric thinking process level 0 in the form of identifying, defining, and classifying aspects according to van Hiele's thinking characteristics in general.

Using Tangram as a Manipulative Tool for Transition between 2D and 3D Perception in Geometry

Creating a mental image of our spatial environment is a key process for further abstract geometric thinking. Building a mental representation can be understood as a part of the process of visualisation. From the wide concept of visualisation, in this article, we will focus on the part where the mental representation of spatial relations, mental objects and mental constructions are created, and their manifestations as a 3D physical object and its plane representations arise. Our main goal is to follow the transition between 2D and 3D representations of physical objects and also to observe how and when such a transition happens in students’ thinking. For that purpose, we also use Tangram, because manipulation with the Tangram pieces in space and filling out planar figures by them indicates the transition between 3D and 2D. Our research, using an action research methodology, was conducted on the students of three 5th grade primary school classes as a part of a larger long-term project. We pointed out a relationship between spatial abilities and the perception of 2D–3D relationships in students’ mind.

Incorporating a Metacognitive Learning Model to Improve Geometric Thinking in High-School Students

Thinking development processes among high-school students is an important and significant issue that has been widely investigated (Leviathan, 2012; Ball, 1996; De Risi, 2015). A few studies discuss the development of mathematical thinking as this field contains additional difficulties to the traditional factors, teachers, students, and parents, and is one of the most important areas taught in school, according to De Risi (2015). Due to the importance of this subject, the challenge facing researchers, mathematicians, and educators is how to improve students’ abilities and achievements in mathematics. In recent years, researchers have found that in order to improve students’ achievements and abilities in mathematics, one can use self-direction. Self-direction is a strategy by which the learner acquires the ability to cope with learning from several aspects and contributes to inking development. In this study, we showed that self-directed learning with an emphasis on metacognition would improve students’ understanding of the subject in question. Using the metacognitive guidance model, the students acquire and develop learning skills that contribute to developing their geometric thinking. In this study, there is the added value of using a learning model based on metacognitive guidance and its significant contribution to combining multiple subjects into one problem.

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

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

Developing reasoning within a geometric learning progression: Implications for curriculum development and classroom practices

Promoting reasoning is the goal of mathematics education. While reasoning behaviours can be observed, how to characterise them and nurture their growth remains ambiguous. In this article, we report our effort in drafting a learning progression and geometric thinking model and using them to investigate Australian students’ geometric reasoning abilities. The data were taken from a large-scale study into the development of mathematical reasoning. Rasch analysis resulted in eight thinking zones being charted. Using a mixed method, we analysed 446 Year 7 to 10 students’ responses on a task that requires them to enlarge a logo, state its coordinates and calculate the enlarged area. In-depth, fine-grained analysis of students’ explanations revealed the range of skills and techniques students used to reason about the situation. The findings suggest that higher level reasoning was characterised by evidence of increased visualisation skills and proficient use of mixed mediums to communicate intent. The implications of the findings for curriculum and classroom practice are discussed.

An Investigation of the Geometric Thinking Levels of Middle School Mathematics Preservice Teachers According to SOLO Taxonomy: "Social Distance Problems"

The aim of this study is to examine the geometric thinking levels of middle school mathematics preservice teachers regarding the problem situations regarding the concept of social distance according to the SOLO taxonomy. The research was conducted using the special case method, one of the qualitative research methods. The working group, studying at a state university in Turkey in middle school mathematics teaching department consists of 80 preservice teachers. While determining the participants, purposeful sampling method was used because the preservice teachers who took "Basics of Mathematics II" and "Special Teaching Methods II" courses were selected. In the study, semi-structured interviews were conducted with 15 preservice teachers since it was aimed to examine remarkable situations from the answers given to open-ended questions. Descriptive analysis technique was used while analyzing the data. Most of the answers given in the study were found to be below the relational geometric thinking level. As a result, it was determined that most of the answers given reflected quantitative learning. In line with this result obtained in the study, it is suggested that open-ended questions about life should be included frequently in teaching in order to reach the relational and extended abstract level answers that reflect qualitative learning.

STUDENTS’ GEOMETRIC THINKING ON TRIANGLES: MUCH IMPROVEMENT IS NEEDED

A look into students’ misconceptions help explain the very low geometric thinking and may assist teachers in correcting errors to aid students in reaching a higher van Hiele geometric thinking level. In this study, students’ geometric thinking was described using the van Hiele levels and misconceptions on triangles. Participants (N=30) were Grade 9 students in the Philippines. More than half of the participants were in the van Hiele’s visualization level. Most students had imprecise use of terminologies. A few had misconceptions on class inclusion, especially when considering isosceles right triangles and obtuse triangles. Very few students correctly recognized the famous Pythagorean Theorem. Implications for more effective geometry teaching are considered.

Rooting the concept and skills of engineering sense in Egyptian and Arabic research and studies

Geometric sense is one of the types of mathematical sense in its general concept and there is a difference between researchers in Egypt and the Arab countries about the concept of geometric sense and its main and sub-skills and there is also a confusion among researchers between geometric sense measurement tests and achievement tests in geometry. The aim of the research is to accurately define the concept of geometric sense that eliminates differences between researchers, to refute the geometric skills identified by researchers within Egypt and the Arab countries for geometric sense, which confuses geometric sense, geometric thinking, geometric proof and achievement in geometry, to identify key and sub-skills for geometric sense according to the writings of the American Council of Mathematicians NTCM so that researchers can commit to them in future research.

Level Berpikir Geometri Peserta Didik Berdasarkan Teori Van Hiele pada Materi Bangun Ruang Sisi Datar

According to Van Hiele, in understanding geometry, students need to understand five stages, namely stage 1 (introduction stage), stage 2 (analysis stage), stage 3 (sequencing stage), stage 4 (deduction stage), and stage 5 (accuracy stage). Each student has different geometric thinking abilities. This study aims to examine the geometric abilities of junior high school students in the material of flat-sided geometry according to Van Hiele's Theory. This research belongs to the type of qualitative research with a case study design. It was conducted at three schools in Padang, West Sumatra, Indonesia. The subjects were students of SMP Negeri 1 Padang, SMP Negeri 7 Padang, and SMP Negeri 25 Padang. Data collection was carried out using tests, interviews, and documentation. Data analysis was carried out with the stages of data reduction, data presentation, and data verification or drawing conclusions. The data originality technique was carried out with the triangulation technique, where the researcher compared the test data with the results of interviews with students. Of the 94 students (male = 38, female 56) tested based on the geometric thinking level of Van Hiele Theory, 92.55% of the students reached the Van Hiele thinking stage at the visualization level. As many as 45.74% of students reached the level of geometric analysis, and 6.38% of students reached the level of abstraction. For the level of deduction and accuracy, no one has been able to solve the problem. The results obtained in this study are that most students are still at stages 1 and 2, unlike van Hiele's opinion, which revealed that the level of geometric thinking for junior high school students is at level 3. According to the study results, it is essential to carry out further checks on the level of geometric thinking of junior high school students in other areas.

Analisis Berpikir Siswa dalam Memecahkan Masalah Dimensi Tiga Berdasarkan Van Hiele

In learning mathematics, innovation is still needed to make mathematics more easily understood by students, especially in the field of geometry. This becomes interesting when examined and is associated with van Hiele's level of geometrical thinking which is modified by Polya's problem solving steps. This study aims to determine the thought process of vocational students majoring in Modeling Design Engineering and Building Information in solving three dimensional problems. This type of research is a qualitative descriptive study. The research subjects are students of pre-visualization, visualization, analysis, and informal deduction level. The research subjects worked on the three dimensional problem test and were interviewed. The results obtained are pre-visualization level students fulfilling several indicators at the visualization level, students at the visualization and analysis level meet all indicators at their respective levels, while informal deduction level students are unable to meet one indicator when implementing a plan that is having another way of solving problems. This can be seen from the results of problem solving given and interviews conducted on research subjects.
 Kata kunci: Berpikir, Pemecahan Masalah, Dimensi Tiga, Van Hiele

Profile of Hard skills and Soft skills of Mathematics Education Students

This study analyzes some hard skills (problem-solving abilities, level of logical thinking, and geometric thinking) and soft skills (students' self-concept and mathematical habits of mind) mathematics education students explicitly in the first year. The research method used is descriptive quantitative. From the population of all mathematics education students in the first year, one group of students was selected as a random sample using analysis techniques, data presentation, and conclusion drawing. Based on the research results, mathematics education students' overall ability in the first year is already good. Students were mastering five reasoning on the test of logical thinking (TOLT). It means they have solved problems with reasoning associated with proportional or ratio, control variables, probability, correlation and combinatorics. Most students have reached the level of thinking geometry at the analysis stage; that is, students have already understood the properties of concepts or geometry based on informal analysis of parts and component attributes. However, students do not have good soft skills. Even though they have a strong habit of mind, students' self-concept is quite sufficient