Abstract In this paper we document the advantages of utilising technology to enhance teachers’ instructional activities. In particular we showcase the potential and impact that the use of Geometer’s Sketchpad may have on the teaching and learning of geometry at school. A series of five, two-hour teacher development workshops in which Geometer’s Sketchpad was used were attended by 12 Grade 11 and 12 teachers. The findings revealed that teachers had a better understanding of the same geometry that they initially disliked. This finding was supported by a quantitative analysis which showed a positive change in the understanding of and beliefs about geometry from when the teachers started to the end of the workshops.

In 2010 a course called MathsLab was designed and implemented in a Johannesburg secondary school, aimed at Grade 9 learners, with the objective of using technology to explore and develop mathematical concepts. One module of the course used Geometer’s Sketchpad to explore concepts in Euclidean geometry. This research report investigates whether technology can result in progression in the zone of proximal development as described by Vygotsky. Progression was measured through the use of a pre- and post-test designed to allocate Van Hiele Levels of geometric thought to individual learners. Changes in the Van Hiele Levels could then verify movement through the zone of proximal development. The results of the pre- and post-tests showed a definite change in learners’ Van Hiele Levels, specifically from Van Hiele Level 1 (visualisation) to Van Hiele Level 2 (analysis). This observation is in line with research that places learners of this age predominantly at these levels. Some learners showed progression to Van Hiele Level 3 (ordering) but this was not the norm. The value of using technology in an appropriate and effective manner in mathematics education is clear and is worthy of further research.

This study investigated how Grade 10 learners conceptualise an introductory activity to the sine function with The Geometers' Sketchpad. In a study by Blackett and Tall (1991), the initial stages of learning the ideas of trigonometry, are described as fraught with difficulty, requiring the learner to relate pictures of triangles to numerical relationships, to cope with ratios such as sinA = opposite/hypotenuse. A computer approach might have the potential to change this by allowing the learner to manipulate the diagram and relate its dynamically changing state to the corresponding numerical concepts. The learner is thus free to focus on specific relationships, called the principle of selective construction, as stated by Blackett and Tall (1991). The use of this educational principle was put to test to analyse the understanding of Grade 10 learners' introduction to the sine function. Data was collected from a high school situated in a middle-class area of Reservoir Hills (KZN) by means of task-based interviews and questionnaires. Given a self-exploration opportunity within The Geometers' Sketchpad, the study investigated learners' understanding of the sine function only within the first quadrant: A) as a ratio of sides of a right-angled triangle B) as an increasing function C) as a function that increases from zero to one as the angle increases from 0° to 90°. D) as a relation between input and output values E) the similarity of triangles with the same angle as the basis for the constancy of trigonometric ratios. The use of Sketch pad as a tool in answering these questions, from A) to E), proved to be a successful and meaningful activity for the learners. From current research, it is well-known that learners do not easily accommodate or assimilate new ideas, and for meaningful learning to take place, learners ought to construct or reconstruct concepts for themselves. From a constructivist perspective the teacher cannot transmit knowledge ready-made and intact to the pupil. In the design of curriculum or learning materials it is fundamentally important to ascertain not only what intuitions learners bring to a learning context, but also how their interaction with specific learning experiences (for example, working with a computer), shapes or changes their conceptualisation. The new ideas that the learners' were exposed to on the computer regarding the sine function, also revealed some errors and misconceptions in their mathematics. Errors and misconceptions are seen as the natural result of children's efforts to construct their own knowledge, and according to Olivier (1989), these misconceptions are intelligent constructions based on correct or incomplete (but not wrong) previous knowledge. Olivier (1989), also argues that teachers should be able to predict what errors pupils will typically make; explain how and why children make these errors and help pupils to resolve such misconceptions. In the analysis of the learners' understanding, correct intuitions as well as misconceptions in their mathematics were exposed. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2004.

This thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometer’s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some known, classical centers. Particularly, we show that the Parry reflection point is the common point of two triads of circles, one associated with the tangential triangle, and another with the excentral triangle. More interestingly, we show that a certain rectangular hyperbola through the vertices of T appears as the locus of the perspector of a family of triangles perspective with T, and in a different context as the locus of the orthology center of T with another family of triangles. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

This study investigated how Grade 11 learners from a school in KwaNdengezi, near Pinetown, in Durban, understood the cosine function with software known as The Geometer’s Sketchpad. This was done on the basis of what they had learnt in Grade 10. The timing was just before they had covered the topic again in their current grade. The researcher hoped, by using The Geometer’s Sketchpad, to contribute in some small way to teaching and learning methods that are applicable to the subject. This may also, hopefully, assist and motivate both teachers and learners to attempt to recreate similar learning experiences in their schools with the same or similar content and concepts appropriate to them. In this research project, data came from learners through task-based interviews and questionnaires. The school was chosen because of the uniqueness of activities in most African schools and because it was easily accessible. Most learners do not have access to computers both in school and at home. This somehow alienates them from modern learning trends. They also, in many occasions, find it difficult to grasp the knowledge they receive in class since the medium of instruction is English, a second language to them. Another reason is the nature of the teaching and learning process that prevails in such schools. The Primary Education Upgrading Programme, according to Taylor and Vinjevold (1999), found out that African learners would mostly listen to their teacher through-out the lesson. Predominantly, the classroom interaction pattern consists of oral input by teachers where learners occasionally chant in response. This shows that questions are asked to check on their attentiveness and that tasks are oriented towards information acquisition rather than higher cognitive skills. They tend to resort to memorisation. Despite the fact that trigonometry is one of the topics learners find most challenging, it is nonetheless very important as it has a lot of applications. The technique of triangulation, which is used in astronomy to measure the distance to nearby stars, is one of the most important ones. In geography, distances between landmarks are measured using trigonometry. It is also used in satellite navigation systems. Trigonometry has proved to be valuable to global positioning systems. Besides astronomy, financial markets analysis, electronics, probability theory, and medical imaging (CAT scans and ultrasound), are other fields which make use of trigonometry. A study by Blackett and Tall (1991), states that when trigonometry is introduced, most learners find it difficult to make head or tail out of it. Typically, in trigonometry, pictures of triangles are aligned to numerical relationships. Learners are expected to understand ratios such as Cos A= adjacent/hypotenuse. A dynamic approach might have the potential to change this as it allows the learner to manipulate the diagram and see how its changing state is related to the corresponding numerical concepts. The learner is thus free to focus on relationships that are of prime importance, called the principle of selective construction (Blackett & Tall, 1991). It was along this thought pattern that the study was carried-out. Given a self-exploration opportunity within The Geometers' Sketchpad, the study investigated learners' understanding of the cosine function from their Grade 10 work in all four quadrants to check on: * What understanding did learners develop of the Cosine function as a function of an angle in Grade 10? * What intuitions and misconceptions did learners acquire in Grade 10? * Do learners display a greater understanding of the Cosine function when using Sketchpad? In particular, * As a ratio of sides of a right-angled triangle? * As a functional relationship between input and output values and as depicted in graphs? The use of Sketchpad was not only a successful and useful activity for learners but also proved to be an appropriate tool for answering the above questions. It also served as a learning tool besides being time-saving in time-consuming activities like sketching graphs. At the end, there was great improvement in terms of marks in the final test as compared to the initial one which was the control yard stick. However, most importantly, the use of a computer in this research revealed some errors and misconceptions in learners’ mathematics. The learners had anticipated the ratios of sides to change when the radius of the unit circle did but they discovered otherwise. In any case, errors and misconceptions are can be understood as a spontaneous result of learner's efforts to come up with their own knowledge. According to Olivier (1989), these misconceptions are intelligent constructions based on correct or incomplete (but not wrong) previous knowledge. Olivier (1989) also argues that teachers should be able to predict the errors learners would typically make. They should explain how and why learners make these errors and help learners to correct such misconceptions. In the analysis of the learners' understanding, correct understandings, as well as misconceptions in their mathematics were exposed. There also arose some cognitive conflicts that helped learners to reconstruct their conceptions. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2010.

Digitala verktyg har numera en betydande roll i matematikundervisningen på gymnasiet, där dynamiska matematikprogram är ett tänkbart verktyg. I denna litteraturstudie sammanställs kvantitativ forskning för att ta reda på om undervisning med dynamiska matematikprogram leder till bättre elevprestationer i matematik, och i så fall inom vilka matematikområden detta gäller. Vetenskapliga artiklar insamlades genom litteratursökning i ERIC och UniSearch för att kvalitetsvärderas och analyseras innan de sammanställdes till ett resultat. Resultatet bygger på 14 kvasiexperimentella studier och visar att undervisning med dynamiska matematikprogram leder till bättre elevprestationer i matematik. Ytterligare faktorer kopplat till elevprestationer framkom också i resultatet, exempelvis att undervisning med dynamiska matematikprogram leder till att eleverna minns stoffet bättre och att de lättare kan förstå nya matematiska koncept. Dynamiska matematikprogram användes framgångsrikt i matematikområdena: Aritmetik, algebra och funktioner; Logik och geometri; Trigonometri; Problemlösning, verktyg och tillämpningar. Resultatet implicerar att matematiklärare på gymnasiet har vetenskapligt stöd för att använda dynamiska matematikprogram i sin undervisning inom flera matematikområden. Detta gäller särskilt GeoGebra som var programmet som användes i 13 av 14 artiklar. / Today, digital tools play a significant role in mathematics education in upper secondary school, where dynamic mathematics programs are one possible tool. In this literature review, quantitative research is compiled to find out if education with dynamic mathematics programs leads to improved student performance in mathematics, and if so, which areas of mathematics this applies to. Articles from academic journals were gathered through literature search in ERIC and UniSearch to be valued in quality and analyzed before they were compiled to the result. The result is based on 14 quasi-experimental studies and shows that education with dynamic mathematics programs leads to improved student performance in mathematics. Additional factors related to student performance also appeared in the result, for example that education with dynamic mathematics programs leads to students remembering the teaching content better, and that they can more easily understand new mathematical concepts. Dynamic mathematics programs were used successfully within the following areas of mathematics: Arithmetic, Algebra and Functions; Logic and Geometry; Trigonometry; Problem Solving, Tools and Applications. The result implies that mathematics teachers in upper secondary school have scientific support to use dynamic mathematics programs in their teaching within several areas of mathematics. This is especially true for GeoGebra, which was the program that was used in 13 of 14 articles.

本文主要的目的是藉由The Geometer's Sketchpad (GSP)軟體的設計，幫助我們得到一組連續型模糊樣本。另外對於模糊數的無母數檢定我們提供了一個較為一般的方法，可以針對梯型、三角型，區間型的模糊樣本同時進行處理。 藉由利用GSP. 軟體所設計的模糊問卷，可以較清楚地紀錄受訪者的感覺，此外我們所提供之對於模糊數的無母數檢定方法比其他方法較為有效力。 在未來的研究裡，我們仍有一些問題需要解決，呈述如下：當所施測的樣本數很大時，如何有效率的在網路上紀錄受測者所建構的隸屬度函數？ / The purpose of this paper is to develop a methodology for getting a continuous fuzzy data by using the software The Geometer's Sketchpad (GSP). And we propose a general method for nonparametric tests with fuzzy data that can deal with trapezoid, triangular, and interval-valued data simultaneously. Using the fuzzy questionnaire designed by GSP. can help respondents to record their thoughts more precisely. Additionally our method for nonparametric tests with fuzzy data is more powerful than others. Additional research issues for further investigation are expressed by question such as follows: how to record the membership function on line, especially when the sample size is large?

It was the purpose of this design study to explore the Geometer’s Sketchpad dynamic mathematics software as a tool to model the derivative in introductory calculus in a manner that would foster a deeper conceptual understanding of the concept – developing domain specific understanding. Sketchpad’s transformation capabilities have been proved useful in the exploration of mathematical concepts by younger learners, college students and professors. The prospect of an open-ended exploration of mathematical concepts motivated the author to pursue the possibility of representing the concept of derivative in dynamic forms. Contemporary CAS studies have predominantly dwelt on static algebraic, graphical and numeric representations and the connections that students are expected to make between them. The dynamic features of Sketchpad and such like software, have not been elaborately examined in so far as they have the potential to bridge the gap between actions, processes and concepts on the one hand and between representations on the other. In this study Sketchpad model-eliciting activities were designed, piloted and revised before a final implementation phase with undergraduate non-math major science students enrolled for an introductory calculus course. Although most of these students had some pre-calculus and calculus background, their performance in the introductory course remained dismal and their grasp of the derivative slippery. The dual meaning of the derivative as the instantaneous rate of change and as the rate of change function was modeled in Sketchpad’s multiple representational capabilities. Six forms of representation were identified: static symbolic, static graphic, static numeric, dynamic graphic, dynamic numeric and occasionally dynamic symbolic. The activities enabled students to establish conceptual links between these representations. Students were able to switch systematically from one form of (foreground or background) representation to another leading to a unique qualitative understanding of the derivative as the invariant concept across the representations. Experimental students scored significantly higher in the posttest than in the pretest. However, in comparison with control group students the experimental students performed significantly better than control students in non-routine problems. A cyclical model of developing a deeper concept image of the derivative is therefore proposed in this study. / Educational Studies / D. Ed. (Education)

The subject of teacher competencies or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competencies necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major 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 triangles and quadrilaterals. Five intersecting categories of teacher competencies were identified: mathematical/geometrical competencies. pedagogical competencies. computer and software competences, language and assessment competencies. / Mathematical Sciences / M. Ed. (Mathematical Education)

The purpose of this study was to examine the effects of the use of technology on students’ mathematics achievement, particularly the Florida Comprehensive Assessment Test (FCAT) mathematics results. Eleven schools within the Miami-Dade County Public School System participated in a pilot program on the use of Geometers Sketchpad (GSP). Three of these schools were randomly selected for this study. Each school sent a teacher to a summer in-service training program on how to use GSP to teach geometry. In each school, the GSP class and a traditional geometry class taught by the same teacher were the study participants. Students’ mathematics FCAT results were examined to determine if the GSP produced any effects. Students’ scores were compared based on assignment to the control or experimental group as well as gender and SES. SES measurements were based on whether students qualified for free lunch. The findings of the study revealed a significant difference in the FCAT mathematics scores of students who were taught geometry using GSP compared to those who used the traditional method. No significant differences existed between the FCAT mathematics scores of the students based on SES. Similarly, no significant differences existed between the FCAT scores based on gender. In conclusion, the use of technology (particularly GSP) is likely to boost students’ FCAT mathematics test scores. The findings also show that the use of GSP may be able to close known gender and SES related achievement gaps. The results of this study promote policy changes in the way geometry is taught to 10th grade students in Florida’s public schools.