Student evaluation of mathematical statements using dynamic geometry software

The study aims to determine the views of final-year pre-service mathematics teachers towards their experience of the use of dynamic geometry software in teaching, following the implementation processes that they carried out when using this software in a real classroom environment. The study was designed as a case study, which is one of the qualitative research methods. The study was conducted with four pre-service teachers studying in the mathematics teaching department at the education faculty of a university in Turkey during the 2016 to 2017 academic year. As one of the results of the study, it was observed that the pre-service teachers, following the theoretical lessons that they had in the education faculty prior to implementation, gave more limited opinions, whereas during the school experience application of dynamic geometry software, following its application in a classroom environment, they mentioned more features of this software for use in teaching. The pre-service teachers, in relation to their real classroom experiences, expressed their views on the features “concretisation and visualisation”, “possibility of presenting many examples”, “making students feel that mathematics is valuable”, “permanence and facilitation of learning”, and “preventing erroneous mental representations” with regard to the role of dynamic geometry software (DGS) use in teaching and learning. Yet, prior to using DGS in the classroom, these pre-service teachers had given importance only to the “concretisation and visualisation” and “permanence and facilitation of learning” characteristics. Key words: Dynamic geometry, pre-service mathematics teacher, geometric shapes, views.

This paper reports on an aspect of a larger research study conceptualised within a teacher development project in Mthatha, Eastern Cape Province. The project was initiated with the objective to develop appropriate skills to use dynamic geometry software (DGS) effectively and strategically as a teaching and learning tool for mathematics. The study reported in this paper aims specifically to ascertain how selected mathematics teachers integrated co-developed technologically aided visualisation tools in the observed lessons. The case study involved two teachers from different schools. The data sources were the classroom observations followed by stimulated reflective interviews with the teachers. The data were analysed to study the use of DGS visualisation tools in relation to Kilpatrick’s framework of teaching proficiency. The lessons evidenced a displayed alignment with the elements of teaching proficiency in the context of teaching geometry. The dynamic visualisation opportunities offered by DGS proved to supplement the teaching repertoire for the participating teachers. Pedagogical practices influence the use of DGS, as evident from the lessons when the participating teachers incorporated collaboratively developed GeoGebra applets into their classrooms. We argue that the collaborative engagement between teachers appears to be a positive way forward in closing the gap between having access to technology and adapting it for effective use in mathematics classrooms.

In the present study, the effects of using dynamic geometry software in active learning framework on students’ problem posing skills and their views about problem posing were examined. The participants consisted of 16 eighth-grade students. Data were collected by problem posing tests, open-ended questions, student diaries, and dynamic geometry software supported tasks. The study, designed with the embedded mixed method, lasted 13 weeks. The dependent t-test was used in the analysis of quantitative data, and descriptive analysis was performed in qualitative data. Problem posing skills of students examined according to use of mathematical language, grammar and expression, suitability to acquisitions, quantity and quality of data, solvability, originality, solution of the problem criteria. In the present study, it was determined that the use of dynamic geometry software in active learning framework developed students’ problem posing skills. The problems that students posed in the dynamic geometry software during the implementation process indicated improvement in terms of problem posing skills as the weeks passed. It was found that students had positive views about the problem posing process, while they experienced some difficulties in this process.

This chapter discusses how the use of Dynamic Geometry Software (DGS) can be used to support students’ engagement with the Standards for Mathematical Practice as outlined in Common Core State Standards for Mathematics (CCSS-M). In particular, the aim of this chapter is to (1) describe what students’ strategic use of appropriate tools might entail in a DGS environment, and (2) argue that for students to engage in these practices in a DGS environment, they must construct meaning for and with these tools in the process of instrumental genesis. Illustrative examples are provided from three secondary mathematics classrooms, and the chapter concludes with recommendations for future research and teacher education in this area.

This research firstly aims to reveal the pedagogical potentials and limitations of dynamic geometry software (DGS). For this purpose, research on Cabri (2D/3D/Plus), the Geometer’s Sketchpad, GoogleSketchUp, GeoGebra and Logo software was reviewed and compiled. As a result of the research, it appears that DGS offers an effective pedagogical environment because of its ability to interact with digital materials and allow for geometric objects to be seen from all sides. However, we found that teachers have some problems in combining DGS with class pedagogy. In this study, the effects of gender, age, education level, skill in using technology and the influence of professional experience were also examined in relation to primary school mathematics teachers’ preferences in using DGS. As a result of this research, it was found that GeoGebra and Cabri were the most-used DGS packages by primary school mathematics teachers, while other geometry software was less preferred by teachers.

The use of digital tools in mathematics lessons has recently gained in significance, especially because of ongoing technical developments. Particularly in the context of mathematical modelling, digital tools have become more and more important. They have been deployed for many years and are currently being intensively discussed from a didactical point of view. This paper discusses to what extent modelling processes using digital tools can be described theoretically, and surveys significant empirical findings in this field. Based on a quantitative control study with 709 students, we especially investigated the competence of mathematising. We compared the competence development of a test-group that worked with digital tools, to a control-group that worked with paper and pencil on the same tasks during a four-lesson intervention on geometric modelling tasks. We find a comparable improvement of mathematising in both groups. This competence development was also investigated in relation to the influence of attitudes towards the software used and program-related self-efficacy. We find program-related self-efficacy, but not attitudes towards the used software, to be a significant predictor of the gain in competency. These results are discussed with respect to different performance studies examining the use of dynamic geometry software.

Este trabalho é um recorte de uma pesquisa realizada no Programa de Pós-Graduação em Educação em Ciências e Matemática da Universidade Federal de Goiás (UFG). Foram realizados o mapeamento e a análise das pesquisas publicadas entre 1987 e 2017, que abordam o ensino de geometria plana no ensino médio com o uso de softwares de geometria dinâmica como recurso didático. Para tanto, os procedimentos metodológicos do Estado do Conhecimento nos guiaram no planejamento, na organização, análise e apresentação dos dados encontrados. O campo empírico baseou-se nos bancos de dados das bibliotecas de instituições que possuem mestrado e doutorado relacionados à Educação Matemática e no Banco de Teses e Dissertações (BDTD) da CAPES. Definiu-se como marco inicial o ano de 1987, por ser o ano de criação do 1º software de geometria dinâmica e 2017 como marco final devido ao de que o levantamento foi realizado em 2018. Os resultados indicam o Geogebra como o software mais utilizado nas experiências com o ambiente dinâmico. Outro aspecto que emerge dos dados coletados são influências da teoria dos registros de representação semiótica e da didática francesa. As perspectivas são permeadas por uma discussão que valoriza o “aprender a aprender” e a formação para o mercado de trabalho, focando nos aspectos técnicos em detrimento da construção de conceitos geométricos.

Interactive geometry software brings new opportunities and challenges for how to make the teaching and learning process more effective and simultaneously more interesting. However, the majority of computer-aided activities for geometry courses are oriented especially on basic problems of pre-college mathematics curricula. This article concerns the use of dynamic geometry software in teaching high-level geometry courses for students in mechanical, civil, and other engineering programs at the tertiary level.

The mathematical and technological nature of tasks containing the use of dynamic geometry software in middle and secondary school mathematics textbooks

The purpose of this mixed-methods study was to investigate the relationship between teachers’ conceptions of mathematics and their use of intelligent tutoring systems for mathematics instruction. Intelligent tutoring systems are adaptive computer programs which administer mathematics instruction to students based on their cognitive state. A conception is a mixture of beliefs and knowledge. The participants in this study were 93 junior high school mathematics teachers from three school districts in the Midwest. Data were gathered using a two-part online survey. The first part of the survey contained questions about their use of intelligent tutoring systems, graphing calculators, Desmos and dynamic geometry software. The second part of the survey contained Likert questions from the teachers’ version of the Conceptions of Mathematics Inventory. Desmos is a website providing interactive classroom activities and a user-friendly graphing calculator. Dynamic geometry software is a class of interactive geometry programs. The quantitative analysis revealed no statistically significant interactions between teachers’ conception scores and intelligent tutoring system use, or between teachers’ conception scores and how intelligent tutoring systems were used. There were statistically significant interactions between teachers’ conception scores and their use of graphing calculators, Desmos, and dynamic geometry software. The qualitative analysis revealed that teachers used intelligent tutoring systems for differentiation. Teachers used graphing calculators, Desmos, and dynamic geometry software for visual, computational, and exploratory purposes. Teachers exclusively using intelligent tutoring systems to incorporate technology should also incorporate technology which promotes student exploration.

<p style="text-align: justify;">Preservice mathematics teachers' beliefs about actions related to the use of the technological tools in teaching mathematics may affect how they are going to use them in their classroom activities. However, there is a limited evidence of what beliefs they hold on their intended actions of using technological tools in teaching mathematics. This study presents two preservice high school mathematics teachers' actional beliefs related to their intended actions in teaching geometric transformations (GTs) using Geometer's Sketchpad (GSP). The study comprised of a series of five task-based qualitative interviews with each of two senior undergraduate preservice teachers at a medium-sized public university in the Rocky Mountain Region of the United States. This study used a radical constructivist grounded theory (RCGT) with five assumptions—symbiosis, voice, cognition, adaptation, and praxis as a theoretical framework to guide the study process. The thematic findings of the study included four in vivo categories of their beliefs associated with actions of teaching GTs with GSP – assessment of student learning, engaging students in a group activity in exploring GTs with GSP, engaging students in individual activity in exploring GTs with GSP, and exploring GTs with GSP as 'suck it up and do it.' Pedagogical implications of these categories have been discussed.</p>

Contents: Preface. Part I: Why Teach Geometry? E.P. Goldenberg, A.A. Cuoco, J. Mark, A Role for Geometry in General Education. K. Gravemeijer, From a Different Perspective: Building on Students' Informal Knowledge. D. Chazan, M. Yerushalmy, Charting a Course for Secondary Geometry. R.L. Devaney, Chaos in the Classroom. Part II: Studies of Conceptual Development. J. Pegg, G. Davey, Interpreting Student Understanding in Geometry: A Synthesis of Two Models. R. Lehrer, M. Jenkins, H. Osana, Longitudinal Study of Children's Reasoning About Space and Geometry. R. Lehrer, C. Jacobson, G. Thoyre, V. Kemeny, D. Danneker, J. Horvath, S. Gance, M. Koehler, Developing Understanding of Space and Geometry in the Primary Grades. D.H. Clements, M.T. Battista, J. Sarama, Development of Geometric and Measurement Ideas. M.T. Battista, D.H. Clements, Students' Understanding of Three-Dimensional Cube Arrays: Findings From a Research and Curriculum Development Project. J.A. Middleton, R. Corbett, Sixth Grade Students' Conceptions of Stability in Engineering Contexts. K. Raghavan, M.L. Sartoris, R. Glaser, Interconnecting Science and Mathematics Concepts: Area and Volume. D. Dennis, J. Confrey, Geometric Curve-Drawing Devices as an Alternative Approach to Analytic Geometry: An Analysis of the Methods, Voice, and Epistemology of a High School Senior. K.R. Koedinger, Conjecturing and Argumentation in High School Geometry Students. Part III: Defining a New Semantics of Space: Computers, Software, and the Electronic World. E.P. Goldenberg, A.A. Cuoco, What Is Dynamic Geometry? M. de Villiers, An Alternative Approach to Proof in Dynamic Geometry. J. Olive, Opportunities to Explore and Integrate Mathematics With the Geometer's Sketchpad. D.L. Watt, Mapping the Classroom Using CAD Program: Geometry as Applied Mathematics. L. Zech, N.J. Vye, J.D. Bransford, S.R. Goldman, B.J. Barron, D.L. Schwartz, R. Kisst-Hackett, C. Mayfield-Stewart, The Cognition and Technology Group at Vanderbilt, An Introduction to Geometry Through Anchored Instruction. K.A. Renninger, S.A. Weimar, E.A. Klotz, Teachers and Students Investigating and Communicating About Geometry: The Math Forum.

The ability of a student to obtain, analyze, measure and compare the many instances of a mathematical proposition in a dynamic geometry environment gives him/her opportunities to make conjectures and test a proposition. These roles for dynamic geometry software are widely acknowledged as having the potential to enrich the teaching of geometry. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by ‘‘what if’’ and ‘‘what if not’’ questions, has the potential to promote links between empirical and deductive reasoning. A classic set of problems in Euclidean plane geometry consists of finding the path of a point that is subject to given constraints. Its systematic study goes back to a lost work (in two books) of Apollonius of Perga, Plane loci (Botana and Valcarce 2003). Except for the simplest loci, such as lines, circles and perhaps the conics, this subject is usually avoided in most geometry texts. This is due to the common difficulties faced when mentally visualizing various objects with different movements. With the emergence of dynamic geometry software, the locus problems have attracted new interest from researchers. Using the locus generation features of these software one can easily obtain the locus of a point under some constraints. Although there are some minor differences, different dynamic geometry software, behave in a similar way when obtaining loci. In general, two objects must be selected: the first one, usually a point called the driving point or mover, is bound to a path, whereas the other, the locus point, must depend somehow on the first one. Since the element

I have always used concrete marupulatives, such as marshmallows and toothpicks, to create models for my geometry and discrete-mathematics courses. These models have come in handy when discussing volume, introducing the 4-cube, or illustrating isomorphic or bipartite graphs. However, after discovering what a dynamic geometry–software package could do for geometry teaching, which has been well documented by research (e.g., Battista and Clements [1995]), I realized that this type of technology also had much to offer for teaching graph theory in my discrete-mathematics course. Although this article discusses The Geometer's Sketchpad 3 (Jackiw 1995), any software that can draw, label, and drag figures can be substituted for Sketchpad.

The power of The Geometer's Sketchpad® (GSP) for teaching and learning lies in its potential for enabling students to explore geometric concepts and recognize relationships among these concepts as illustrated through the use of multiple representations. This article explores two problems and examines how the use of tables along with GSP sketches may assist students in understanding these concepts. The first activity investigates the relationship of the length of a trapezoid's median (or midsegment) to that of its bases. The second involves showing how the area of a regular polygon approaches that of a circle when the number of sides increases. In both activities, we will start with the geometric object, look at the measurements, tabulate these, and make a graph to illuminate the situation.