Using Dynamic Geometry Software to Gain Insight into a Proof

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 study was designed to investigate how 13-year-old students reason about 2-dimensional shapes while working with Dynamic Geometry Software (DGS).

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.

This chapter develops around two fundamental ideas, namely, that (1) the perception of the affordances of a certain digital tool is essential to solving mathematical problems with that particular technology and that (2) the activity thus undertaken stimulates different mathematising processes which, in turn, result in different conceptual models. Looking thoroughly, from an interpretative perspective, at four solutions to a particular geometry problem from participants who decided to use dynamic geometry software at some point of their solving activity, our main purpose is to illustrate the ways in which the same tool affords different approaches to the problem in terms of the conceptual models developed for studying and justifying the invariance of the area of a triangle. Their different ways of dealing with the tool and with mathematical knowledge are interpreted as instances of students-with-media engaged in a “solving-with-dynamic-geometry-software” activity, enclosing a range of procedures brought forth by the symbioses between the affordances of the dynamic geometry software and the youngsters’ aptitudes. The analysis shows that different people solving the same problem with the same digital media and recognising a relatively similar set of affordances of the tool produce different digital solutions, but they also generate qualitatively different conceptual models, in this case, for the invariance of the area.

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.

Students’ conceptions through the lens of a dynamic online geometry assessment platform

The article describes the development and analysis of a quantitative study investigating to what extent the potential of dynamic geometry environments (DGEs), in relation to mathematical reasoning competency, are utilized in lower secondary schools in Denmark (grades 7–9). The study entails a questionnaire, which was developed on the basis of an extensive review that uncovered four potentials of DGEs in relation to reasoning competency: feedback; dragging; measuring; tracing. 220 Danish lower-secondary mathematics teachers completed the questionnaire. Analysis indicates that the potentials of measuring and dragging are utilized to some degree, feedback to a lesser degree, while tracing is almost non-existent. Furthermore, there are signs that DGEs are used as a substitute for the paper-and-pencil environment to solve tasks that were originally designed for paper and pencil. Possible improvements of praxis are discussed and the integration of the results into praxis in further research is elaborated upon.

This paper describes the problem solving behavior of two preservice teachers as they worked individually on three nonroutine geometry problems. A dynamic tool software, namely the Geometer’s Sketchpad, was used as a tool to facilitate inquiry in order to uncover and investigate the patterns of metacognitive processes. Schoenfeld’s (1981) model of episodes and executive decisions in mathematics problem solving was used to identify patterns of metacognitive processes in a dynamic geometry environment. During the reading, understanding, and analysis episodes, the participants engaged in monitoring behaviors such as sense making, drawing a diagram, and allocating potential resources and approaches that helped make productive decisions. During the exploring, planning, implementation, and verification episodes, the participants made decisions to access and consider knowledge and strategies, make and test conjectures, monitor the progress, and assess the productivity of activities and strategies and the correctness of an answer. Cognitive problem-solving actions not accompanied by appropriate metacognitive monitoring actions appeared to lead to unproductive efforts. Redirection and reorganizing of thinking in productive directions occurred when metacognitive actions guided the thinking and when affective behaviors were controlled.

The purpose of this study was twofold. Firstly, to extend the Technology Acceptance Model (TAM) to assess secondary mathematics school teachers’ intention to use Dynamic Geometry Software (DGS) in geometry teaching and, second, to examine the relations between the parameters of TAM and the role of external variables. We extended TAM by integrating in the model the notion of ‘perceived pedagogical-learning fit’, which refers to evaluating the pedagogical learning appropriateness of teaching geometry with DGS based on a cognitive-learning model. The results of the study showed that perceived pedagogical-learning fit and attitude are the strongest predictive factors of intention to use a DGS in geometry teaching. The perceived usefulness did not significantly affect their intention to use it. Personal innovation, facilitating condition and computer anxiety had weak to moderate effects on perceived pedagogical-learning fit and perceived ease of use, and indirect effects on intention to use.

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.

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 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.

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.

What makes Dynamic Geometry Environment (DGE) a powerful mathematical knowledge acquisition microworld is its ability to visually make explicit the implicit dynamism of thinking about mathematical geometrical concepts. One of DGE’s powers is to equip us with the ability to retain the background of a geometrical configuration while we can selectively bring to the fore dynamically those parts of the whole configuration that interest us. That is, we can visually study the variation of an aspect of a DGE figure while keeping other aspects constant, hence anticipating the emergence of invariant patterns. The aim of this paper is to expound the epistemic value of variation of the Dragging tool in DGE in mathematical discovery. Functions of variation (contrast, separation, generalization, fusion) proposed in Marton’s theory of learning and awareness will be used as a framework to develop a discernment structure which can act as a lens to organize and interpret dragging explorations in DGE. Such a lens focuses very strongly on mathematical aspects of dragging in DGE and is used to re-interpret known dragging modalities (e.g., Arzarello et al.) in a potentially more mathematically-relevant way. The exposition will centre about a specific geometrical problem in which two dragging trajectories are mapped out, consequently resulting in a DGE theorem and a visual theorem. In doing so, a new spectral dragging strategy will be introduced that literally allows one to see the drag mode in action. A model for the lens of variation in the form of a discernment nest structure is proposed as a meta-tool to interpret dragging experiences or as a meta-language to relate different dragging analyses which consequently might give rise to pedagogical and epistemological implications.

Spoken and written language change when students (age 14–15) work with digital tools in the mathematics classroom. The digital tool not only offers new experiences and actions that directly influence language, but also provides lexical expressions such as buttons or technical expressions that students use. This article studies language used in the mathematics classroom empirically when working with dynamic geometry software (DGS), especially in the context of geometrical constructions. The results show the variety of different lexical phenomena in the students’ use of language. The use of such lexical expressions also mirrors the use of the digital tool itself. Qualitative analysis of the videotaped paired task-based clinical interviews shows both obstacles and potentials of the lexical expressions that students use when working with dynamic geometry software.