Geometer’s Sketchpad Research Articles

Three Lessons on Parabolas—What, Where, Why

Using The Geometer's Sketchpad, this sequence of algebra lessons introduces quadratic functions as the product of two linear functions.

The Development of MyGSP: An Online Resource for Teaching Mathematics Based on Geometer’s Sketchpad (GSP)

MyGSP was developed following various constraints faced by teachers to integrate the Geometer’s Sketchpad (GSP) software into mathematics lessons. Based on the Waterfall Design Model, the development of MyGSP started with the analysis of user needs through literature review and simple survey. From the analysis, three main constraints were identified: (1) insufficient time to prepare for GSP-aided lessons; (2) lack of training on GSP integration in lessons; and (3) lack of references for teaching mathematics using GSP. Therefore, taking into account the computer equipment currently available at schools, the researcher developed MyGSP to overcome these constraints. In the design phase, the researcher developed teaching materials for mathematics topics at secondary schools. Teaching strategies adopting higher order of thinking skills were applied in the teaching materials. Apart from this, video tutorials on GSP usage in teaching materials were also developed. In the implementation and unit testing phase, 30 pre-service Mathematics teachers participated by providing feedback on the materials developed. In the integration phase, the modules, i.e., Teaching Material Module, Download Module, Forum Module, and Video Tutorial Module, were combined into one system known as MyGSP. The MyGSP system was also updated and enhanced from time to time in the maintenance phase. It is hoped that MyGSP will be able to fulfill teachers’ needs for integrating GSP into their classrooms.

Lesson study incorporating phase-based instruction using Geometer's Sketchpad and its effects on Thai students’ geometric thinking

Purpose – Geometry has, in many countries, proved to be one of the problematic topics in mathematics. The purpose of this paper is to report on a study that was carried to investigate the effects of lesson study incorporating phase-based instruction (LS-PBI) using Geometer's Sketchpad (GSP) on Thai students’ geometric thinking. Design/methodology/approach – The effectiveness of the LS-PBI using GSP within the targeted context was assessed in a quasi-experimental research approach involving three groups of mixed ability grade 7 students (12-year-olds) in one of the schools in Yala province, Thailand. These groups (Group 1: n=30, Group 2: n=28 and Group 3: n=29) were taught the topic of properties of 2D and 3D geometric shapes through phase-based instruction using GSP in turn by three different teachers. The teachers observed the lesson study teaching protocol. Data were collected using pre-test and post-test to assess students’ van Hiele level of geometric thinking. In addition, six students were selected to be observed in order to give supplementary data on their level of geometric thinking changed. Findings – Findings revealed that the LS-PBI was effective in improving students’ geometric thinking. Analysis of paired sample t-test showed that there was a significant difference in the pre-test and post-test scores in each group after each LS session. Analysis of covariance showed that there was a significant difference in the post-test scores among the three groups of students. Originality/value – The findings of this original study suggest that the LS-PBI using GSP was effective in enhancing students’ geometric thinking.

Pose and Solve Varignon Converse Problems

Students' conjectures relating the midpoints of the sides of a quadrilateral are inspired by using The Geometer's Sketchpad and justified by proofs.

Design of Information-educational Environment for the Establishment of New Facts in Mathematics

The aim of the paper is to present some possibilities of the GEOMETER'S SKETCHPAD (GSP) software to realize a heuristic approach in a classroom environment. It is shown how the well-known Polya's model could be used by students to discover or rediscover Plane Geometry assertions which are new for them. The approach is applied to a property of conics, that are inscribed in polygons or, which is the same, to a property of polygons, that are circumscribed round conics. In the particular case of circles the material is from the curriculum and it is designed for extra-curriculum teaching in the general case. Since the main result (theorem 1) is of mathematical interest, the details of student behavior during the discovery process are left for discussion in another publication.

An Art and Technology Approach to Actively Engage Students in the Mathematics of the Regular Polyhedra

The goal of the present work is two folded: At first the author demonstrates a step-by-step problem solving approach, to prove the (convex) Regular Polyhedra Theorem using Zome tools. Then the geometric constructions of some patterns for the decoration of solids using utilities such as the Geometer's Sketchpad will be presented.

From Visualization to Computer Animation Approaches in Mathematics Learning: the Legacy throughout History of Human Endeavours for Better Understanding

Presently, there has been growing interests in using mathematics' history in teaching mathematics [Katz, V. & Tzanakis, C. (Eds.) (2011). Recent Developments on Introducing a Historical Dimension in Mathematics Education. Washington, DC: Mathematical Association of America]. Thus, this article introduces some work of scholars from ancient East Indian culture like Bhaskara (AD 1114-1185) and Arabic culture such as Ibn Qurrah (AD 9th c) that are related to Pythagoras Theorem. In addition, some Babylonian creative works related to Pythagorean triples found in a tablet known as 'Plimpton 322', and an application of the Pythagorean Theorem found in another tablet named 'Yale Tablet' are presented. Applications of computer animation of dissection Motion Operations concept in 2D and 3D using dynamic software like Geometer's-Sketchpad and Cabri-II-and-3D. Nowadays, creative minds are attracted by the recent stampede in the advances of technological applications in visual literacy; consequently, innovative environments that would help young students, gifted or not, acquiring meaningful conceptual understanding would immerge.

Graphing Calculator and Geometers’ Sketchpad in Teaching and Learning of Mathematics

Normal 0 false false false EN-MY X-NONE X-NONE MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:Table Normal; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:Calibri,sans-serif; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Times New Roman; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:Times New Roman; mso-bidi-theme-font:minor-bidi;} This study aims to develop a module for teaching and learning of Straight Lines topics using graphing calculator (GC) technology and dynamic geometry software, Geometer’s Sketchpad (GSP) and also to determine the pedagogical usability of the module. The study is carried out in two phases. Phase 1 is the development of the module and phase 2 is a survey research using a questionnaire. A sample of 30 mathematics teachers from three different schools in district of Sabak Bernam, Hulu Selangor and Kuala Selangor are randomly selected to participate in this study. The instrument used in this study is the pedagogical usability questionnaire. The pedagogical usability criteria are the student control, student activities, objective-oriented, application, value-added, motivation, knowledge value, flexibility and response. The data are analysed using descriptive statistics such as mean, standard deviation, percentage, and frequency. The findings showed that the module developed has a high pedagogical usability . Findings from this study provide evidence of pedagogical impact in incorporating the latest trends in mathematics education, namely, integrating the GC tool and GSP software to maximize the mathematical and pedagogical benefits to students.

The Effects of Van Hiele's Phases of Learning Geometry on Students’ Degree of Acquisition of Van Hiele Levels

Geometry is a basic skill to be mastered. It is important in architecture and design, in engineering and in various aspects of construction work. However, in Malaysian education system, the process of teaching and learning geometry does not reflect its importance. The process does not emphasise the thinking skills, whereas the mathematics syllabus clearly states that thinking systematically, accurately, thoroughly, diligently and with confidence, should be infused throughout the teaching and learning process. Therefore, the aim of this study was to test the effectiveness of Van Hiele's phases of learning geometry using the Geometer's Sketchpad (GSP) on students’ degree of acquisition of Van Hiele levels. This quasi-experiment involved two groups of students; treatment and control group. The students in treatment group learned Form Two's Transformation topics through the Van Hiele's phases of learning using the GSP, while the students in control group learned the same topic conventionally. Before the study started, five students from each group were randomly selected to be interviewed to determine their initial levels of geometric thinking. The experiment took place for six weeks. At the end of the study, the same students in both groups who had been selected earlier were interviewed for the second round to analyse their final levels of geometric thinking. The results found that in their initial levels of geometric thinking, the majority of students in both groups obtained the first Van Hiele levels with complete acquisition. However, almost all students in both groups showed a low acquisition of level 2 and no acquisition of level 3. In the post interview, most of the students in control group showed an increment of geometric thinking that is from level 1 to level 2, but no one in this group achieved level 3. In contrast, all students in the treatment group showed a complete acquisition of Van Hiele level 1 and almost all of them indicated a complete acquistion of level 2. As for level 3, only one student did not achieve that level, whereas the rest showed a complete and high level of acquisition. Therefore, it can be concluded that the implementation of activities based on the Van Hiele phases of learning geometry have a positive impact on the development of higher levels of geometric thinking.

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.

Developing Pre-service Secondary Teachers' Skills of Using the Geometer's Sketchpad to Teach Mathematics through Lesson Study

The purpose of this study was to develop pre-service secondary teachers’ skills of using The Geometer’s Sketchpad (GSP) to teach mathematics through Lesson Study (LS). GSP is a dynamic geometry software program for constructing and investigating mathematical objects that adds a powerful dimension to the teaching and learning of geometry and many other areas of mathematics. Lesson Study is a Japanese model of teacher professional development in which small groups of teachers collaboratively plan, teach and revise a lesson to improve the quality of their teaching as well as to enrich students’ learning experiences. Twenty-three LS groups comprising 2 pre-service secondary teachers who attended a mathematics teaching methods course in a local public university were set up in four tutorial groups each consisting of five or six LS groups. This paper discusses how LS has helped to develop pre-service secondary teachers’ skills of using GSP to teach the topic of ‘Loci in Two Dimensions’ in one of the LS groups. Analysis of their GSP sketches in the first, second and third lessons indicates that the participants of this LS group showed positive changes in their skills of using GSP to teach the topic.

Beliefs about Teaching Geometric Transformations with Geometers' Sketchpad: A Reflexive Abstraction

A teacher's belief plays a significant role in the quality of teaching mathematics. In a fictive way, I changed my role from a researcher to a research participant in an imaginative interview. My interior other (David) interviewed me as a researcher. A single interview session was held lasting for about three hours. The interview text was used for the analysis and interpretation using the grounded theory method. I invented a substantive theory of my beliefs about teaching geometric transformations with Geometer's Sketchpad. The theory is "reflexive abstraction of my beliefs about teaching Geometric Transformations with Geometer's Sketchpad" as a personal theory characterized by some basic categories of - beliefs about the advancement of pedagogy, beliefs about pedagogical environment, beliefs about the role of students' and teacher, beliefs about self as a future teacher, beliefs about teaching learning activities, and beliefs about transitions in teaching learning. I reconstructed a synthesis of the characteristics of these beliefs. While constructing these layers of interpretive accounts, I used radical constructivist grounded theory as a theoretical base.

A Case Study of Remedial Instruction of a Sixth Grader in Solving of Proportionality Problems in a Dynamic Multiple Representation Computer Environment

We selected a sixth grade student to participate in this study. Prior to remedial instruction, the subject did not understand the concept of proportionality and used ratio formulas excessively to solve problems without any meaning on it. In addition, he could not employ multiple representations to explain his solutions. Using an electronic whiteboard and Geometer’s Sketchpad (GSP) to design the dynamic multiple representations for proportional scenarios in remedial instruction induced the subject has better understanding of proportions as well as a better grasp of the linear relationship between two variables. In the final test, the subject was able to use multiple representations to solve problems and check his calculations, therefore increasing problem solving accuracy.

STUDENTS LEARNING PATHS AS ˜'DYNAMIC ENCEPHALOGRAPHS' OF THEIR COGNITIVE DEVELOPMENT

The current study presents excerpts of my PhD thesis in which I developed a 'dynamic' hypothetical learning path (DHLP) with regard to student's cognitive development. The aim was to investigate if they would raise their van Hiele levels during and after participation in the research process. Students interacted using the interaction techniques of the Geometer's Sketchpad software and the transformations of its dynamic representations. Moreover, the role of linking visual active representations (LVARs) in the software and the role of students' instrumental decoding in the cognitive process have been investigated. Finally, examples of students learning paths, like 'dynamic' encephalographs'of their cognitive development, will be presented.

The Effects of Van Hiele's Phase-Based Instruction Using the Geometer's Sketchpad (GSP) on Students' Levels of Geometric Thinking

The purpose of this study was to test the effectiveness of Van Hiele’s phases of learning geometry using the Geometer’s Sketchpad (GSP) on students’ levels of geometric thinking. This quasi-experiment involved 94 students and two teachers. A total of 47 students were in the control group and the rest were in the treatment group. The students in the treatment group learned Form Two’s Transformation topic through the Van Hiele’s phases of learning using the GSP, while the students in the control group learned the same topic conventionally. Before the study started, students from both groups were given Van Hiele’s Geometry Test (VHGT) to identify their initial levels of geometric thinking. The experiment took place for 6 weeks. At the end of the study, the students in both groups were given the VHGT for the second round to analyse their final levels of geometric thinking. Wilcox on-t test for the design of repeated measurement was used for the data analysis. The results found that the students in both groups showed increment in their post-VHGT as compared to the pre-VHGT. However, the students in the treatment group achieved better levels of geometric thinking compared to the students in control group (t = 34.50, p<0.05). Thus, the Van Hiele’s phases of learning geometry can be applied in classrooms in order to help students achieve better level of geometric thinking.

Developing Pre-Service Teachers' Technological Pedagogical Content Knowledge for Teaching Mathematics with the Geometer's Sketchpad through Lesson Study

The purpose of this study was to develop pre-service secondary teachers' technological pedagogical contentknowledge (TPACK) for teaching mathematics with The Geometer’s Sketchpad (GSP) through Lesson Study(LS). Specifically, a single-group pretest-posttest design was employed to examine whether there was asignificant difference in the pre-service secondary teachers' TPACK for teaching mathematics with GSP afterengaging in LS which was incorporated into the mathematics teaching methods course during the first semesterof the 2011/2012 academic session in a Malaysian public university. Forty-six pre-service secondary teacherswho enrolled in the course completed both the pretest and posttest questionnaires on teachers' TPACK forteaching mathematics with GSP. The results of the paired-samples t-test indicated that there was a significantdifference in the pre-service secondary teachers' TPACK for teaching mathematics with GSP for all thesubscales after engaging in LS.

Technology Tips: Squaring the Circle with The Geometer's Sketchpad

The study of the history of mathematics can be an important approach to engaging students in learning mathematics. At the same time, mathematics history provides a context that will help students understand how and why certain types of mathematics were developed and used. The study of p has as rich a history as any mathematical topic.

Assisting Primary School Children to Progress through Their van Hiele's Levels of Geometry Thinking using Google SketchUp

This research aims to study the effectiveness of newly-developed learning modules built to assist Malaysian primary students to progress through the first three levels of widely accepted van Hiele's model of geometry thinking. The first part of the paper reviews the background of study focusing on the problems encountered by Malaysian mathematics teachers to practice this model in the classroom using Geometer's Sketchpad. It also includes brief essential information about the learning modules which was built using Google SketchUp. The second part describes methodology adopted in the research which involved quantitative approach of a pre versus post quasi-experimental design ran on a group of forty Year 6 primary school children in Malaysia. The results of comparative single-group mean analysis of the test scores suggest that the learning modules were potential in assisting students to progress through the respective van Hiele's levels of geometry thinking, with all of them demonstrated progression of at least within level.

The Activities Based on Van Hiele’s Phase-Based Learning: Experts’ and Preservice Teachers’ Views

The effective learning strategies for mathematics topics may vary according to the topic. For geometry, the activities that are suggested to be implemented are carried out using Van Hiele’s phases of learning geometry. The phases include Information, Guided Orientation, Explicitation, Free Orientation and Integration. The implementation of these phased-based activities is easier and more effective with the presence of dynamic geometry software, especially the Geometer’s Sketchpad (GSP). Therefore, the purpose of this article is to discuss the activities based on Van Hiele’s phases of learning geometry, using the GSP computer software as a tool. The developed activities were given to eight experts to get their views on the activities. Besides that, 24 pre-services teachers were also given the activities for the purpose of investigation of paedagogical usability of the developed activities. The results showed that the developed activities were well-arranged based on Van Hiele’s phases of learning geometry with the assistance of GSP. The results also showed that the developed activities met the requirements of the pedagogical usability criteria. Since these phase-based activities obtained positive views from experts and pre-services teachers, the activities can be carried out in teaching and learning geometry.

Ponderings on Pocket-sized Polyhedra

Pop-up polyhedra–three-dimensional models that can be stored for future reference–are easily constructed using The Geometer's Sketchpad and give students experience in using transformations in the plane.