Van Hiele Theory Research Articles

The van Hiele Levels at Czech Secondary Schools

This paper focuses on the results of a pilot study based on the van Hiele theory of geometric thinking at Czech secondary schools. The van Hiele theory describes how young people learn geometry. It postulates five levels of geometric thinking. Usiskin later tested the ability of the van Hiele theory to describe and predict the performance of students in secondary school geometry. Our pilot study aimed to verify the test structure of the Usiskińs test in the Czech secondary schools, and to compare the outcomes with American results.

Development of Arithmetical Thinking: Evaluation of Subject Matter Knowledge of Pre-Service Teachers in Order to Design the Appropriate Course

One of the key courses in the mathematics teacher education program in Israel is arithmetic, which engages in contents which these pre-service mathematics teachers (PMTs) will later teach at school. Teaching arithmetic involves knowledge about the essence of the concept of “number” and the development thereof, calculation methods and strategies. properties of operations on different sets of numbers, as well as the properties of the numbers themselves. Hence, the question arises: how to educate PMTs in order to supplement their mathematical knowledge with the required components? The present study explored the development of arithmetic thinking among pre-service teachers intending to teach mathematics at elementary school. This was done by matching the van Hiele theory of the development of geometric thinking to arithmetic. Analysis of findings obtained both in the present study and in many studies of geometry teaching indicates that this approach to considering the learners’ level of thinking development might lead to meaningful learning in arithmetic course for PMTs.

An Assessment of the Readiness of Grade 10 Learners for Geometry in the Context of Curriculum and Assessment Policy Statement (CAPS) Expectation

In 2012 January, as part of the ongoing process of curriculum revision which began in 1994, the South African Depart of Education (DoE) introduced the Curriculum and Assessment Policy Statement (CAPS). The CAPS brought Euclidean geometry with its formal proof back into the compulsory mathematics curriculum. It also implied that all learners needed to perform at level 4 (Deduction: with formal proof) of the van Hiele levels in Euclidean geometry in all grades in the Further Education and Training (FET) phase-Grades 10-12. The main aspect of the van Hiele theory is that of children's understanding of geometric concepts can be characterised as being at certain specific levels within a hierarchy of levels from 0 to 4 with level 0 being the lowest. This paper reports on the assessment based on the van Hiele model of geometrical thinking levels of a sample of South African Grade 10 learners in Euclidean Geometry. It is a follow up of an earlier published study with 191 grade 10 learners by the same authors. Data for this study were generated from 359 Grade 10 learners in five senior secondary schools in one Education District. These schools were selected through purposive sampling. The study made use of both quantitative and qualitative research techniques for data collection. In the former, multiple choice questions and in the latter, structured interviews were made use of. The quantitative and qualitative data were anlysed using Microsoft Excel 2007 and thematic analysis, respectively. The results from the study revealed that the majority of the learners were at level 0 despite the CAPS expectation that learners are to perform at level 3 in order to be ready for level 4 thinking in Grade 10. The findings from the study assisted the authors to suggest recommendations that could be made use of by curriculum developers and implementers to improve the instructional strategies of geometry learning and teaching.

Global van Hiele (GVH) Questionnaire as a Tool for Mapping Knowledge and Understanding of Plane and Solid Geometry

This paper presents the Global van Hiele (GVH) questionnaire as a tool for mapping knowledge and understanding of plane and solid geometry. The questionnaire facilitates identification of the respondents' mastery of the first three levels of thinking according to van Hiele theory with regard to key geometrical topics. Teacher-educators can apply this questionnaire for checking preliminary knowledge of mathematics teaching candidates or pre-service teachers. Moreover, it can be used when planning a course or granting exemption from studying in basic geometry courses. The questionnaire can also serve high school mathematics teachers who are interested in exposing their students to multiple-choice questions in geometry.

The Van Hiele Geometry Thinking Levels: Gender and School Type Differences

This research shows that it would be very appropriate to divide the teaching of geometry into different levels according to van Hiele theory. Secondary schools, with regard to their specialization, should determine what levels they want to achieve, and adapt teaching geometry to that goal. Our research also shows that Van Hiele levels are equally suitable for both genders.

Competencies in using Sketchpad in Geometry Teaching and Learning: Experiences of preservice teachers

The subject of teacher competencies has been a key issue in mathematics education reform as the quality of an education system is fundamentally defined by the quality of its teachers. The study reported in this article attempted to identify and analyse displayed preservice teacher competencies and challenges encountered in early experiences with a Sketchpad-mediated learning of geometry. The instrumental approach to technology integration in the classroom as articulated by Trouche, Artigue and others was used as an analytical framework together with the van Hiele theory of geometrical thought development. A qualitative research approach was used to investigate preservice teachers' emerging competencies. Data were gathered through worksheet productions, lesson observations, open-ended questionnaires and an exit focus group interview. Twenty third-year mathematics major preservice teachers participated in workshop and microteaching sessions involving the use of the Geometer's Sketchpad dynamic geometry software in the teaching and learning of the geometry of quadrilaterals. The competencies displayed by the participants were described in terms of the instrumental theory for technology use and the van Hiele theory for geometric thought development. The competency levels showed that the preservice teachers had difficulties with computer hardware and software usage and the associated classroom organisation and management initially but gradually improved and gained confidence. The participants' own evaluations of their competencies affirmed that early experiences were unpleasant for many and barriers could be overcome by early exposure to computer environments. More computer resources need to be availed for a day-to-day integration to be sustainable.

Teoria de van Hiele: contribuições para a forma/ação de professores de Matemática

As crenças dos professores, suas concepções, sua linguagem, sua postura, qualidades e defeitos, tornam-se, de certa forma, seus instrumentos de trabalho, exercendo influências tanto positivas quanto negativas no processo de ensino e aprendizagem. Nessa perspectiva, para o ensino da Matemática, surgem questionamentos sobre a arte de ensinar Geometria. Procurando respostas para este questionamento, teceram-se reflexões sobre a Teoria de van Hiele, visando proporcionar mais um aporte para sustentar a prática pedagógica dos professores em formação. Para tanto, foi feito um levantamento bibliográfico sobre a Teoria de van Hiele, e os dados encontrados apresentaram nuances favoráveis à forma/ação dos professores de Matemática, pois a Teoria de van Hiele possui uma base estruturalista, que estabelece estratégias metodológicas que favorecem a resolução de problema e a interdisciplinaridade numa visão não linear, capaz de conduzir o aluno a superar as lacunas da aprendizagem em Geometria.

Improving the Levels of Geometric Thinking of Secondary School Students Using Geometry Learning Video based on Van Hiele Theory

An educational video called Video Pembelajaran Geometri (Geometry Learning Video = VPG) is developed in this study. The video is aimed to help year 9 students in improving their level of Van Hiele geometrical thinking. VPG is designed as an alternative aids to overcome the limitation of information and communication technology (ICT) usage in Parepare, South Celebes, Indonesia. Based on Van Hiele learning model, all topics in geometry for Indonesian Secondary School (Sekolah Menengah Pertama = SMP) are included in VPG in order to support the students who are going to continue their study further. The effectiveness of VPG was tested on 180 students categorized as 90 students from level 0 (L0), 60 students from level 1 (L1), and 30 students from level 2 (L2). Comparative Analyses on the results of adapted Van Hiele Geometry Test (VHGT) pre and post VPG usage indicated that there is a significant difference between the mean scores. It was also discovered that a significant improvement in geometric thinking level is occurred in the majority of the students. DOI: http://dx.doi.org/10.11591/ijere.v2i1.1935

A Survey of South African Grade 10 Learners’ Geometric Thinking Levels in Terms of the Van Hiele Theory

The main thesis of the van Hiele theory is that childrens' understanding of geometric concepts can be classified into a sequence of five hierarchical thinking levels, with levels 1 and 5 being the l owest and the highest.However, an additional lower level, level 0, was added in by other researchers. This paper reports on a part of a larger study which focused on the van Hiele levels of geometric thinking amongst a group of Grade 10 learners. The sample consisted of 191 Grade 10 learners from five senior secondary schools in one Education District in the Eastern Cape Province of South Africa. The respective mathematics teachers in each of these schools assisted in the selection process. The schools were selected through purposive sampling. The necessary ethical r equirements were met. Participants completed a test on van Hiele levels of geometric thought by responding to questions on basic geometric concepts including the classification and properties of triangles and quadrilaterals which constituted the basis for space and shape component of the South African Grade 10 Mathematics curriculum. The data were analyzed by manual counts and by using Microsoft Excel. The study found that the majority of learners were at level 0, which is a cause for concern. The paper recommends that educators who facilitate geometry l earning in grade 10 need to familiarize themselves with the van Hiele levels in order to achieve effectiveness in the teaching/ learning interface of geometrical concepts.

Comparing the old and new 6th - 8th grade mathematics curricula in terms of Van Hiele understanding levels for geometry

This study was conducted with the aim of comparing the behaviors and attainments related to the plane geometry in the old and new 6th - 8th grade mathematics curricula (MC) in terms of Van Hiele geometry understanding levels. With this purpose document analysis method was used. In the study, the levels of behaviors and attainments in the old and new 6th – 7th grades MCs were determined according to Van Hiele theory. As a result of the study, it was found that although the number of attainments in the old MP related to plane geometry decreased in the new one, the percentage of levels of some of the topics was increased. Moreover it was found that both the old and new MCs involve higher order behaviors and attainments for 6th grade such as deductive reasoning.

Reform-Based Curriculum and Motivation in Geometry

The aim of this study was to compare motivation of sixth-grade students engaged in instruction using reform-based curriculum with sixth-grade students engaged in instruction using a traditional curriculum. There were 273 sixth- grade mathematics students, 123 in the control group and 150 in the treatment group, involved in the study. This study took place in North Florida. The researchers used a questionnaire, the Course Interest Survey (CIS), administered to the students before and after a five-week of instruction. The paired-samples t-test, the independent-samples t-test, and ANCOVA with α = 0.05 were used to analyze the quantitative data. The study showed that there was a statistically significant difference in motivation between the groups favoring the treatment group. In other words, the reform-based curricula designed on the basis of van Hiele theory, compared to a traditional one, had more positive effects on students’ overall motivation in learning geometry at the sixth grade level.

LISTEN TO THE SILENCE: THE LEFT-BEHIND PHENOMENON AS SEEN THROUGH CLASSROOM VIDEOS AND TEACHERS’ REFLECTIONS

This study uses classroom videotapes to examine the phenomenon of students who are left behind during whole-class teaching. Zooming in via these videos enabled us to analyze these classroom situations by means of a compact, multi-perspective set of theories – van Hiele theory, conceptualization, and visual perception. The analysis provided a picture of the classroom interaction in which the sound of silence of those students who might not share understanding during the class discourse is ignored. Based on a sample of Taiwanese junior high school geometry classes, the study demonstrated how investigating situations of silence uncovered the possibility of teachers’ unawareness of student difficulties or their unsuccessful efforts to cope with them. Teachers’ post-lesson interviews and questionnaires shed additional light on the perspective of left-behind students. Implications of this method for pre- and in-service teachers’ programs are briefly discussed, and some recommendations for change are proposed.

From geometrical figures to definitional Rigour: Teachers’ analysis of teaching units mediated through van Hiele's theory

Abstract: The present study reports on curriculum design implemented in eight parallel classes of high school, with students aged 14–15, by means of work‐sheets dealing with properties of the principal plane geometric figures. Van Hiele's theory was used to facilitate teachers’ discussion of student performances on work‐sheets in order to enhance their didactical awareness. We hypothesize that van Hiele's theory is a proper means to provide encouragement and support for teachers in rethinking their teaching goals and practices. This environment enables teachers to manage their students’ transition from the perception of a figure to its definition. In this paper, we will recount a teachers’ discussion as it unfolded in a school‐university work group. The analysis of students’ work on work‐sheets interweaves with an analysis of the teaching actions that foster the process of understanding definitions.

Developing geometry thinking through multimedia learning activities

In this study, a multimedia learning software program named GeoCAL is described. It is based on van Hiele's geometric thinking level theory, which consists of four levels: recognition, visual association, description/analysis, and abstraction/relation. In addition to presenting the software design, this study also explores the learning effects of GeoCAL on each of the geometric thinking levels and on overall geometric thinking. The subjects of this study are second-grade elementary school students of an average age of eight who have not previously had formal lessons in geometry. The experimental results indicate that, with the exception of recognition ability, GeoCAL produces significant learning effects on visual association, description/analysis and abstraction/relation as well as overall geometric thinking.

Connecting arguments to actions— Dynamic geometry as means for the attainment of higher van Hiele levels

New technology requires as well as supports the necessity to raise the level of geometric thinking. Freudenthals view of van Hiele's theory corroborates a dynamic multi-level curriculum that offers material support for higher levels. For levels higher than 2, the dynamic locus capability of Dynamic Geometry software is crucial, e.g. in the study of loci of orthocentres and incentres. Discrepancies between their algebraic and geometric descriptions can motivate a deeper involvement with basic curve theory on the side of the teacher, who thereby can predict in which cases the students may succeed in restructuring the construction to overcome the discordance.

Geometry content knowledge: Is pre-service training making a difference?

Huge financial investments are made annually in teacher training in order to prepare prospective teachers for a career in teaching. The impact of pre-service training on prospective mathematics teachers' (n=124) content knowledge (using the Van Hiele Theory) is investigated. Results seem to indicate that prospective teachers exit their school career with higher geometric understanding than after three years of mathematics content and methodology training or after four years of methodology training. The deterioration in prospective teachers' geometry content knowledge indicates that pre-service education does not assist in maintaining or improving levels of geometric understanding. Possible contributing factors are identified such as the fragmented nature of pre-service teacher education programme and the absence of a long-term strategy connecting teacher education to teaching practice in a meaningful way. The results have serious implications for pre-service mathematics education programmes and some suggestions on the features of an improved programme are made.

Using dynamic geometry to expand mathematics teachers’ understanding of proof1

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers’ understanding of other functions of proof than just the traditional function of ‘verification’. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called ‘reconstructive’ approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry.

Grade 7 teachers' and prospective teachers' content knowledge of geometry

The geometry content knowledge of Grade 7 teachers (n =18) and prospective teachers (n =100) was investigated, using the Van Hiele theory and acquisition scales of Gutierrez, Jaime and Fortuny. Results indicated that both teacher and prospective teacher populations failed to reach the level of geometric thinking and degree of acquisition expected from successful teachers. The impact of teaching experience and different pre-service time frames (3 years vs 4 years) on the level of geometrical thought was also investigated. The conclusion was that teachers and prospective teachers do not have adequate control of the Grade 7 geometry subject-matter they have to teach. This holds implications both for pre-service and in-service teacher education as well as classroom practice. South African Journal of Education Vol.23(3) 2003: 199-205

Tactual shape perception in relation to the understanding of geometrical concepts by blind students

Tactual shape perception is a synthesis of many parameters which lead blind students to make sense of external stimuli. It is vital to investigate this field of perception because it reveals many differences between passive and active touch. The van Hiele model provides a tool for investigating and classifying students' thinking in geometry. In a study of 19 blind and visually impaired students in a special school in Greece, we sought to understand their own way of approaching, developing and understanding concepts of shape. The findings support on one hand the appropriateness of the van Hiele theory and on the other the complexity of tactual shape perception and its uniqueness with respect to every student's thinking. Teachers and specialists can enrich their understanding and be more effective when teaching geometrical concepts that require taking into account variables such as: touch, movement, posture, type of shape, language, prior knowledge and the conditions of the task.

Increased Knowledge in Geometry and Instructional Practice

This study examined the effects on instruction of an intervention program designed to enhance teachers' knowledge of geometry and their knowledge of research on student cognition in geometry. Forty-nine middle-grade (4-8) teachers participated in a 4-week program consisting of a content course in geometry and a research seminar on van Hiele theory. The pretest and posttest results showed significant gains in content knowledge and in van Hiele level. The analysis of a lesson-plan task revealed a significant shift in goals and expectations to the next higher van Hiele level. Follow-up observations of 8 teachers found marked changes in what was taught, how it was taught, and the characteristics teachers displayed. Teachers attributed these changes to increased geometrical content knowledge and research-based knowledge of student cognition.