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Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course

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Abstract

The purpose of this study was to investigate pre-service teachers’ views about the history of mathematics course in which GeoGebra was used. The qualitative research design was used in this study. The participants of the study consisted of 23 pre-service mathematics teachers studying at a state university in Turkey. An open-ended questionnaire was used as a data collection tool. Qualitative data obtained from the pre-service teachers were analyzed by means of content analysis. As a result, it was determined that GeoGebra software was an effective tool in the learning and teaching of the history of mathematics.

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... • use of CAS (Burke & Burroughs, 2009;Kidron et al., 2001) or DGS (Baki & Guven, 2009;Caglayan, 2016;Isoda, 2004;Meadows & Caniglia, 2021;Zengin, 2018); • mathematical topics such as algebra (Baki & Guven, 2009;Jankvist et al., 2019a), calculus (Aguilar & Zavaleta, 2015;Chorlay, 2016), geometry (Arzarello et al., 2007;, or probability (Bakker & Gravmeijer, 2006; Chorlay 2015), or ...
... The study by Kidron (2004), presented as one of the three examples, is also of this kind, invoking "theory without going beyond the mere invocation at such" (Niss, 2007(Niss, , p. 1309. Similarly, in Zengin (2018), theoretical perspectives were mentioned along the way, yet it is unclear what role they played in the analyses of the empirical studies reported. In Zengin (2018), several of the mathematics education references made are to HPM homegrown constructs, such as Jankvist's (2009) distinction between history as a tool and history as a goal. ...
... Similarly, in Zengin (2018), theoretical perspectives were mentioned along the way, yet it is unclear what role they played in the analyses of the empirical studies reported. In Zengin (2018), several of the mathematics education references made are to HPM homegrown constructs, such as Jankvist's (2009) distinction between history as a tool and history as a goal. However, more often than not, it seems to be the case that references to HPM literature serve the purpose of placing a given study in the (HPM) research landscape, or mirror some of the arguments for resorting to history in the first place. ...
Article
This article is a review of the literature on the use of history of mathematics in combination with use of digital technologies in the teaching and learning of mathematics. The review identifies 33 peer-reviewed publications (book chapters, journal articles, and papers in conference proceedings) that address an actual interplay between use of history and digital technologies. Of these, 24 concern the use of primary historical source material, i.e., mathematical texts produced by past (historical) mathematicians. Besides asking the questions of which publications actually incorporate this interplay and what purposes the use of history and digital technology serve, we also address the question of the role of mathematics education theoretical perspectives (or the lack thereof) in the identified publications. A reading of the identified publications shows that the ones involving use of primary historical source material often have a clearer connection to theoretical constructs from mathematics education research (outside the area of History and Pedagogy of Mathematics, or HPM). Still, only a small number of the identified publications make use of mathematics education theoretical constructs specifically addressing the use of digital technology.
... • use of CAS (Burke & Burroughs, 2009;Kidron et al., 2001) or DGS (Baki & Guven, 2009;Caglayan, 2016;Isoda, 2004;Meadows & Caniglia, 2021;Zengin, 2018); • mathematical topics such as algebra (Baki & Guven, 2009;Jankvist et al., 2019a), calculus (Aguilar & Zavaleta, 2015;Chorlay, 2016), geometry (Arzarello et al., 2007;, or probability (Bakker & Gravmeijer, 2006; Chorlay 2015), or ...
... The study by Kidron (2004), presented as one of the three examples, is also of this kind, invoking "theory without going beyond the mere invocation at such" (Niss, 2007(Niss, , p. 1309. Similarly, in Zengin (2018), theoretical perspectives were mentioned along the way, yet it is unclear what role they played in the analyses of the empirical studies reported. In Zengin (2018), several of the mathematics education references made are to HPM homegrown constructs, such as Jankvist's (2009) distinction between history as a tool and history as a goal. ...
... Similarly, in Zengin (2018), theoretical perspectives were mentioned along the way, yet it is unclear what role they played in the analyses of the empirical studies reported. In Zengin (2018), several of the mathematics education references made are to HPM homegrown constructs, such as Jankvist's (2009) distinction between history as a tool and history as a goal. However, more often than not, it seems to be the case that references to HPM literature serve the purpose of placing a given study in the (HPM) research landscape, or mirror some of the arguments for resorting to history in the first place. ...
Article
This article is a review of the literature on the use of history of mathematics in combination with use of digital technologies in the teaching and learning of mathematics. The review identifies 33 peer-reviewed publications (book chapters, journal articles, and papers in conference proceedings) that address an actual interplay between use of history and digital technologies. Of these, 24 concern the use of primary historical source material, i.e., mathematical texts produced by past (historical) mathematicians. Besides asking the questions of which publications actually incorporate this interplay and what purposes the use of history and digital technology serve, we also address the question of the role of mathematics education theoretical perspectives (or the lack thereof) in the identified publications. A reading of the identified publications shows that the ones involving use of primary historical source material often have a clearer connection to theoretical constructs from mathematics education research (outside the area of History and Pedagogy of Mathematics, or HPM). Still, only a small number of the identified publications make use of mathematics education theoretical constructs specifically addressing the use of digital technology.
... Tidak hanya itu, upaya lain yang dilakukan untuk menjawab permasalahan ini adalah pengembangan lembar kerja siswa [17]. Solusi lain yang diterapkan pada pembelajaran tatap muka pada materi sudut adalah dengan pemanfaatan program Geogebra [18] karena efektif digunakan pada kelas matematika [19]. ...
... Geogebra merupakan sebuah inovasi dalam pembelajaran matematika [21] yang membantu memvisualisasikan materi-materi geometri [22], [23]. Pemanfaatan Geogebra terbukti dapat mewujudkan pembelajaran yang efektif [19], [24]. Selain itu, manfaat lain penerapan Geogebra adalah meningkatkan motivasi [25] dan kemandirian belajar siswa [26]. ...
Article
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Penelitian ini bertujuan untuk menghasilkan modul pembelajaran dengan Aplikasi GeoGebra untuk meningkatkan pemahaman konsep matematika materi Sudut dalam Dimensi Tiga pada siswa kelas XII SMA. Langkah pengembangan modul mengadopsi langkah-langkah desain pengembangan 4D yang dikembangkan oleh Thiagarajan. Agar menghasilkan modul dengan kualitas yang baik dilakukan Uji coba produk yang melibatkan ahli isi, ahli desain pembelajaran, siswa pada uji perorangan, uji kelompok kecil dan uji lapangan, serta guru matematika pada uji lapangan. Data kelayakan produk dikumpulkan dengan angket. Sedangkan, untuk menguji efektivitas modul dalam meningkatkan pemahaman konsep siswa dilakukan pretest-posttest yang dianalisa dengan uji t berpasangan. Modul Geogebra yang dikembangkan memiliki kelayakan 70% dari ahli isi dan 88% dari ahli desain pembelajaran, 90% dari uji perorangan, 81% dari uji kelompok kecil dan 87% dari uji lapangan. Penggunaan modul pada pembelajaran terbukti dapat meningkatkan pemahaman konsep matematika berdasarkan uji-t yang mana nilai signifikansinya 0,000. Hasil ini didukung dengan adanya peningkatan nilai rata-rata dari pretest ke posttest sebesar 4,97. Dari hasil angket, siswa menyatakan bahwa penerapan modul Geogebra dalam pembelajaran juga membantu meningkatkan motivasi belajar siswa. Dengan demikian penerapan modul Geogebra dapat mendukung pembelajaran matematika secara online maupun pembelajaran tatap muka dengan pemanfaatan teknologi.
... The studies that are relevant to HoM in mathematics teacher education generally focus on improving mathematical knowledge that prospective teachers are going to teach (Arcavi et al., 1987;Clark, 2012;Jankvist et al., 2020;Mersin & Durmuş, 2021), developing positive attitudes, motivation and beliefs towards mathematics (Furinghetti, 2007;Phillippou & Christou, 1998;Zengin, 2018), and providing instructional materials, pedagogical tools, and techniques for teaching mathematics (Jankvist, 2009). A meta-analysis of the HoM literature reveals that the empirical research in mathematics teacher education programs mainly investigates the reflections of PMTs in terms of using HoM as a pedagogical tool to improve their future mathematics teaching practices (Baki & Bütüner, 2018). ...
... These studies rarely attend to the instructional tools to teach mathematics history. One exception is the use of GeoGebra in a HoM course, where prospective teachers report their conceptual learning of mathematics (Zengin, 2018). In addition to this, incorporating popular films into the HoM courses is considered as one of the ways of learning the historical development of mathematical knowledge together with mathematicians" experiences while pursuing their mathematical works. ...
Article
Bu makale, matematik tarihi dersinde popüler filmlerin kullanımına ilişkin matematik öğretmen adaylarının görüşlerini incelemektedir. Çalışma kapsamında yer alan matematik tarihi dersinde farklı matematikçilerin farklı zaman ve bağlamlardaki deneyimlerini ve mücadelelerini tasvir eden Agora, Sonsuzluk Teorisi ve Akıl Oyunları olmak üzere üç film kullanılmıştır. Dersin tamamlanmasını takiben, on iki ortaokul matematik öğretmeni adayı ile yüz yüze görüşmeler yapılmıştır. Matematik öğretmeni adayları, popüler filmlerin ilham verici olabileceğini ifade etmişlerdir. Matematik öğretmen adayları için popüler filmler, matematikçilerin yaşam tarzlarının ve deneyimlerinin daha görünür hale geldiği bir bağlama oturtmaya yardımcı olmaktadır. Diğer bir bulgu, popüler filmlerin öğretim materyali olarak kullanımına ilişkin olup, öğretmen adayları filmlerin ilgi çekici olması gibi avantajlara ve matematikçilerin hayatlarını fazla dramatize etmesi gibi dezavantajlara sahip olduğunu ifade etmektedirler. Ayrıca, matematik öğretmeni adayları popüler filmlerin kâr odaklı bir sektörden geldiğinin farkındadır ve ticari amaçların matematikçilerin hayatındaki meseleleri anlama konusunda sınırlı olabileceğini düşünmektedirler. Bu çalışmanın sonucunda, matematikçilerin hayatını anlatan popüler filmler, sınırlılıkları dikkate alındığı takdirde, matematik öğretmen eğitimi derslerinde kullanılabilecek etkili öğretim materyalleri olarak önerilebilir.
... Although research has documented the use of Dynamic Geometric Software (DGS) in teaching the history of mathematics (HoM) (Zengin, 2018), the focus of this research specifically targets the development of proof for pre-service teachers by utilizing DGS to revisit historical proofs with a modern lens. In particular, this article examines whether and to what extent, geometrical constructions using Dynamic Geometry Software (DGS) along with incorporating the history of mathematics (HoM) encourage the understanding of pre-service teacher"s understanding of proofs. ...
... The findings in this study concur with Fujita et.al. (2010), Zengin (2018), and Conners (2007) them with an illustration that may be used in proofs. PSTs not only used DGS, but also performed these procedures on paper with a marked straightedge and compass to experience the actual methods used in antiquity. ...
Article
Full-text available
Pre-service mathematics teacher (PST) education often addresses within Geometry Classes how to utilize Dynamic Geometric Software (DGS). Other classes may also incorporate teaching pre-service teachers about the history of mathematics. Although research has documented the use of Dynamic Geometric Software (DGS) in teaching the history of mathematics (HoM) (Zengin, 2018), the focus of this research specifically targets the development of proof for pre-service teachers by utilizing DGS to revisit historical proofs with a modern lens. The findings concur with Fujita et.al. (2010), Zengin (2018), and Conners (2007) work on proof. The novelty of this article was the combination of incorporating the history of mathematics (HoM), dynamic geometry software (DGS), and Toulmin’s model of argumentation. A pedagogical approach appeared to emerge: DGS’s dynamic nature allowed PSTs to see several examples of a method to provide them with an illustration that may be used in proofs.
... This approach has proved fruitful also in a context of higher education (Barnett, Lodder, Pengelley, Pivkina, & Ranjan, 2011). A noteworthy effect achieved by the use of original sources is the so-called dépaysement épistémologique (Barbin, 1997) described as "the astonishment of the learner facing a posture, a framework, a process or a particular argument, far from those of today" (Guillemette, 2015); even in combination with the use of ICT (Massa Esteve, 2012) or incorporating software like GeoGebra (Chorlay, 2015;Zengin, 2018). However, Jankvist (2014b) stated that "not much emphasis has been put on the use of original sources in teacher education" (p. ...
... We have also seen in task 3 that some participants turn to GeoGebra to support their discourse or even to better understand it. In fact, facilitating understanding and visualization of concepts are benefits of using GeoGebra in combination with history of mathematics (Zengin, 2018). ...
Article
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The use of original sources is a useful resource not only to be used with secondary school students but also with prospective mathematics teachers. In this work, we designed a series of tasks based on a fragment excerpted from Clairaut’s Éléments de Géométrie to be carried out with 24 participants enrolled on a Masters’ Degree in Secondary School Mathematics Teaching. This fragment was chosen both due to its content and to its narrative structure and our main goal was to determine which elements of professional knowledge were used by prospective secondary mathematics teachers when reading this fragment. In order to do so, we used the MKT model as an analytical tool and we also assessed some aspects related to literacy skills. The prospective teachers were able to recognize mathematical and pedagogical components within the source that relate to their future practice. In addition, the participant’s literacy skills seem to play a role in the richness of their reading.
... Taking into account that the literature evidences that the incorporation of ICTs into the classroom and the use of Dynamic Mathematics Software for mathematics education is beneficial for learning [10,12,13,27], there's a limited application of TUIs in more abstract domains such as trigonometry [28]. These findings led to consider an intervention in the learning process of undergraduate students, where the influence of a TUI could be beneficial to facilitate learning of trigonometry. ...
... One of the key differences offered by the interface-as opposed to traditional teaching styles-is the freedom to err and to navigate in a non-linear way. The "learning by doing" approach [12,27] encouraged experimentation and acceptance of "failure" within the participants. In addition, the continuous validation of acquired knowledge during the session enabled students to feel empowered to incorporate more complex concepts progressively. ...
Article
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In mathematics education, studies reveal difficulties in the teaching-learning of trigonometry in secondary and higher education, due to the fact that students are not encouraged to achieve a deep conceptual understanding of abstract concepts. Several studies demonstrate that incorporating digital technologies has a positive impact on students’ learning. However, most of the existing technologies do not consider the use of the body and multiple senses. Tangible User Interfaces (TUIs) in contrast, can host bodily interactions that have the potential of enhancing learning. Nonetheless, there is a lack of applications of TUIs for trigonometry education. This study consisted in designing and validating a tangible interface for the teaching-learning of basic concepts of trigonometry. The interface hosts a pedagogical experience that privileges exploration through physical manipulation and fosters intuitive and collaborative learning. A Pre-Test was applied to 121 students to determine previous knowledge, yielding a 29.1% performance. After two sessions using the interface, the results of a Post-Test reveal an increase of 37.1%, confirming the educational effectiveness of the interface and the pedagogical experience to facilitate learning of basic concepts of trigonometry.
... GeoGebra is one of the technological tools used in mathematics education. GeoGebra software makes contributions to students' learning of mathematical concepts in a dynamic environment by constructing and testing them (Zengin, 2018b). In addition, GeoGebra allows the construction of multiple representations of mathematical concepts (Preiner, 2008) and provides opportunities for students to build connections between different representations (Dikovic, 2009). ...
Article
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The purpose of this study is to examine the effect of a technology-enhanced collaborative learning environment on secondary school students’ mathematical reasoning in the concept of triangle. The participants of the study are 30 secondary school students. This study was carried out with the embedded design, one of the mixed methods designs. The quantitative aspect of the study was carried out with the quasi-experimental design including comparison group design. While the experimental group received training in a technology-enhanced collaborative learning environment, the control group students continued their education in a traditional informal-collaborative learning environment. The qualitative aspect of the study included data belonging to a group of four students chosen among the experimental group. The data of the research comprised students’ audio and video recordings, screenshots, dynamic mathematics software GeoGebra files, and written products. The quantitative data were collected via open-ended questions including ten items with the intention of revealing mathematical reasoning and the qualitative data were gathered with the designed mathematical tasks. While independent t-test was used to analyse the quantitative data, the qualitative data were analysed with Toulmin’s model and dialogical approach. As a result of data analysis, it was found that technology-enhanced collaborative learning environment has a positive effect on students’ mathematical reasoning in the concept of triangle.
... While studies encourage the integration of the history of mathematics and instructional technology such as dynamic mathematics software (Zengin, 2018) and e-book (Yevdokimov, 2007) to support mathematics instruction and there is little study on investigating the views of students about integrating the history of mathematics into mathematics lessons through the use of technology, especially for gifted students. There is a need to conduct some research studies on the effectiveness of the history of mathematics as a supportive factor as well (Goktepe & Ozdemir, 2013). ...
... En palabras de los autores: "estas tecnologías ofrecen valiosas herramientas para (…) aplicar el razonamiento covariacional y analizar e interpretar funciones asociadas con situaciones dinámicas" (Carlson et al., 2002, p. 374). GeoGebra es un ejemplo de un sistema de geometría dinámica que, aparte de dar movimiento a las construcciones realizadas, genera un entorno de aprendizaje interactivo y visual que permite explorar cómo se relacionan las variables con la solución geométrica (Zengin, 2018). ...
Article
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The study aims to analyze the change in the mental actions of a student associated with the use schemes that he mobilizes when solving an approximation task to the área of a region with GeoGebra. Theoretical elements related to the covariational reasoning approach and the instrumental approach were considered. The student, after generating, mobilizing and refining his use schemes, made visible behaviors that are associated with mental actions, such as drawing a polygon to approximate an area and recognizing that, the more points on the graph the distance between them is smaller and a better approximation is obtained. That is, observable behaviors can be associated with levels of reasoning or with changes in associated mental actions, which indicate a transition from one level of reasoning to another.
... The third way begins as the first way but concentrates on the integration of dynamic geometry software. A similar approach, i.e. the incorporation of the dynamic mathematics software into History of Mathematics, has been attempted by other researchers with positive results (see Zengin, 2018; Meadows & Caniglia, 2021). ...
Article
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The well-known from the History of Mathematics "hundred fowls problem" is the topic of this paper. Our main aim is to analyze the historical background of the problem and to present some ways of solving it, which have different characteristics, and each one represents a particular strategy. The interaction of Eastern and Western mathematics and the combination of different fields like Algebra and Technology in order to solve the problem has been highlighted. The study of the specific topic revealed some teaching perspectives, which we note at the end together with some concluding remarks. This paper might have implications in future research in the field of indeterminate analysis but also in Mathematics Education.
... If we give an example to these studies from literature, it is seen that Jacinto and Carreira (2017) created and used a framework for analyzing the processes of mathematical problem solving with technology (GeoGebra). The study of Zengin (2018) that he incorporated GeoGebra into a history mathematics course refers to the use of the software in teaching and learning of different mathematics subjects. On the other hand, Kovacs et al. (2020) explored a way to accommodate GeoGebra Automated Reasoning Tools (GGb-ART) in a computer-supported geometric reasoning ecosystem. ...
Article
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The purpose of this study was to investigate the trends of GeoGebra related studies. We focused on the articles published between 2009 and 2021 and existed in Web of Science database. The bibliometric analysis of the GeoGebra related articles exposed four-clustered structure according to the degree of relationships among terms that reflect the articles. The clusters were identified as learner anatomy (the use of GeoGebra to improve students' mathematics learning), teaching analytics (the role of GeoGebra in mathematics teaching), technological focus (the technological and procedural issues), and conceptual extension (the attempts considering the features of GeoGebra, mathematics topics and skills). This study offers evidence to show trends and practices in GeoGebra related studies, as well as to clarify their relationships with other concepts. The investigation into GeoGebra's dynamics would help researchers, teachers, students, and policymakers gain insight into areas that need further research, as well as plan for future studies.
... The third way begins as the first way but concentrates on the integration of dynamic geometry software. A similar approach, i.e. the incorporation of the dynamic mathematics software into History of Mathematics, has been attempted by other researchers with positive results (see Zengin, 2018; Meadows & Caniglia, 2021). ...
Article
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This paper proposes a new goodness-of-fit for the two-parameter distribution. It is based on a function of squared distances between empirical and theoretical quantiles of a set of observations being hypothesized to have come from the gamma distribution. The critical values of the proposed statistic are evaluated through extensive simulations of the unit-scaled gamma distributions and computations. The empirical powers of the statistic are obtained and compared with some well-known tests for the gamma distribution, and the results show that the proposed statistic can be recommended as a test for the gamma distribution.
... Todos los elementos del recurso educativo tienen permiso de descarga, dada la concepción de ser un recurso abierto a la comunidad académica. En cada componente del estudio de cónicas se utilizaron los registros semióticos de lenguaje común y algebraico, orientados a través de preguntas que promueven el diálogo y la argumentación del profesor en formación, en el esquema gráfico como registro semiótico se utilizaron dos representaciones semióticas como son el seccionar el cono y las representaciones en el plano bidimensional, en estos últimos casos fue utilizado el programa computacional gratuito para el estudio de la geometría dinámica, el cual tiene gran aceptación en la comunidad de profesores (Zengin, 2018), como lo es GeoGebra. ...
Article
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The objective of the present study was to identify, analyze, and characterize semiotic registers of conic sections by teachers in training. The methodology used a mixed approach based on didactic engineering. The quasi-experiment was developed in three phases: 1) preliminary analysis of conceptions and forms of teaching, 2) a priori design and implementation of educational digital resources, and 3) a posteriori analysis. The results showed learning and teaching difficulties, especially in the cognitive conversion activity. A technological didactic intervention was designed and implemented based on an open access digital educational resource supported by Duval's semiotic representation theory of registers. It is concluded that the resource is a mediator for the study of conics and the acquisition of new knowledge that allows the mobilization of mathematical knowledge from perspectives other than graphic approach or common language.
... However, many previous studies focusing on learning mathematics only employed GeoGebra without clear learning strategies, such as [33]- [36]. These studies only used GeoGebra as a tool in learning mathematics without applying effective learning strategies that might increase students' motivation. ...
Article
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Motivation is an aspect heavily emphasized in a mathematics class to produce a quality human capital. However, students’ motivation towards mathematics is still low while students’ performance in mathematics is declining. The students' lack of motivation toward mathematics could be attributed to ineffective learning strategies in mathematics classrooms. In addition, many previous studies have only focused on an inductive reasoning strategy and the separate use of GeoGebra. Therefore, this study aims to identify the effects of learning through an inductive reasoning strategy assisted by GeoGebra on students’ motivation for the Functional Graph II topic. The research design was quasi-experimental which involved 94 Form 4 students from a secondary school in Johor. The research sample was divided into three groups: (1) Study Group 1 (an inductive reasoning strategy assisted by GeoGebra); (2) Study Group 2 (an inductive reasoning strategy without GeoGebra); and (3) Control Group (a conventional strategy). The research instruments consisted of a motivational questionnaire set and an inductive reasoning strategy assisted by GeoGebra and without GeoGebra. The inductive reasoning strategy was guided by the Marzano’s Inductive Reasoning Model consisting of an observation process on specific examples and patterns that determined if a generalization was true. The MANOVA test results show that the overall motivation level for Study Group 1 is high in terms of attention and relevance. With regard to confidence, the results indicate that Control Group and Study Group 1 show the same motivation level. As for satisfaction, the motivation level for Control Group is the highest compared to other groups. In conclusion, learning through an inductive reasoning strategy assisted by GeoGebra can increase the students’ motivation in mathematics specifically for the Functional Graph II topic.
... DGS, CAS, spreadsheets reflective thinking by providing multiple representations. For example, in relation to a use for many years has been that students have lost the opportunity to experience classical geometrical intuitions, which are not replaced by a haze of algebraic symbols; DGS begins to offer a chance to re-experience some ageargumentation and illustration are to be found in the HPM proceedings (Isoda, 2000b;, the ESU proceedings (Aguilar & Zavaleta, 2015;Bruneau, 2011;Chorlay, 2015;Hong & Wang, 2015;Jankvist, Misfeldt & Aguilar, 2019), and in miscellaneous other channels (Baki & Guven, 2009;Burke & Burroughs, 2009;Caglayan, 2016;Erbas, 2009;Kidron, 2004;Olsen & Thomsen, 2019;Papadopoulos, 2014;Zengin, 2018). Yet, only about a handful of these may be considered as actual empirical research studies. ...
Conference Paper
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In this paper, we argue for the use of digital technologies in making original sources more accessible to students. W digital technologies 005) framework of instrumental genesis, and a. The combination of original sources and use of digital technologies appears to be a somewhat overlooked area in the HPM research, not least in terms of empirical investigations. Yet, in this paper we lay down the theoretical bricks for such further investigations.
... We were guided by the recommendations of the researchers [73][74][75][76][77][78] as well as our personal experience in choosing software for student learning. GeoGebra software has proven to be a great package for learning mathematical content because of its dynamical nature and ease of use [79]. ...
Article
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The presented study analyzes the usage of the didactic efficiency of multiple representations in a computer environment in learning absolute value functions and equations. It is known that the axis of symmetry of the graph of the absolute value function is the y-axis. The research was applied at the University of Novi Sad, Serbia. The data were collected by testing a group of 226 students: major chemistry and physics students at the beginning of their common calculus course. The students worked individually in two groups: the experimental and control group. The experimental group of students practiced using GeoGebra software, and the control group of students practiced using paper and pencil. At the end of the experiment, which lasted for two weeks (six school classes), both groups were tested with a post-test of knowledge without using a computer. It can be concluded that GeoGebra software had a positive influence on the students’ achievements in solving absolute value equations.
... Integrating Geogebra will possible encourage students to reflect how to find the parts of a geometry [15]. Geogebra is also able to help students investigate and represent geometry [16]. This study focuses on identifying the impact of the use of Geogebra Classic application towards geometry learning outcome of fifth graders. ...
Article
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This study aimed to identify the impact of Geogebra Classic application towards geometry learning outcome of fifth graders. The method employed was experimental research using Non-equivalent Control Group Design. Serving as the subjects of this study were students of fifth grade in an elementary school in Malang, Indonesia. The data were required through statistical tests and analyses through SPSS 21.0 for Windows. The results of the study showed that the average difference of the score of fifth graders after having treatment of using the application were 66.4 for the pre-test and 81.2 for the post-test. In the meantime, the t-test on the study revealed that the value was 0.002 < 0.05 meaning that the hypothesis was accepted. This indicated that there is an impact of the use of Geogebra Classic application on geometry learning outcome of the fifth graders. Therefore, it can be concluded that Geogebra Classic application was able to help students understand geometry-related learning materials.
... Gómez-Chacón (2011) identified that GeoGebra software supported students in making mathematical connections; thus, a mathematical concept helped them to understand the connection between a graph and its equation. Zengin (2018b) also determined that GeoGebra supported some pre-service teachers in developing their mathematical thinking and mathematical connection skills by constructing a bridge between mathematical concepts and the history of mathematics. Therefore, this finding is compatible with the results of these studies (e.g, Gómez-Chacón 2011; Zengin 2018b). ...
Article
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The purpose of this study was to examine the effect of GeoGebra software on pre-service teachers’ mathematical connection skills. The participants of the study comprised 22 pre-service mathematics teachers. A mathematical connection self-efficacy scale and an open-ended questionnaire regarding mathematical connection skills were used as data collection tools. The implementations undertaken with the participants lasted 13 weeks in the dynamic learning environment. After GeoGebra implementations, the quantitative data were analyzed using a dependent t-test and the qualitative data obtained with the open-ended questionnaire were analyzed using descriptive analysis. Based on the results, it was determined that GeoGebra software could be used as an important tool for the development of mathematical connection skills.
... • HTML5 and Javascript, drawing inspiration from the success of both GeoGebra, a dynamic mathematics software [11], [12] and PhET, a platform for interactive simulations for Science and mathematics [13], [14] • MATLAB TM a popular numerical mathematics software widely used in engineering [15] and • Mathematica TM [16] with the powerful computer algebra system and rich visualization capabilities ...
Article
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This study aims to analyze changes in mental actions associated with the covariational reasoning of one pair of students when working on tasks to approach the concept of definite integral through Riemann sums. Mental actions are gathered from student interactions in GeoGebra, from their written reports, and from a semi-structured interview applied to one pair of students from an engineering school in Mexico. Data are analyzed with the theoretical construct of covariational reasoning. The results show an evolution in the way students coordinate simultaneous changes between the variables involved in tasks, from the no coordination level to the chunky continuous covariation level. It is concluded that questions to propose conjectures on the use of rectangles in approximation processes to the area of a region, including their justification, and the process of inferring behaviors from hypothetical situations, reveal behaviors associated with higher levels of covariational reasoning.
Article
This study examined gifted students’ views on integrating the history of mathematics embedded videos into mathematics classrooms. The research was conducted with 30 fifth-grade students who were identified as gifted with the WISC-R Intelligent Test Score and aptitude test. Data were collected through students’ video reflection papers after watching the videos on biographies of mathematicians. Content analysis was conducted to analyze data collected with reflection papers. The findings were grouped as history as a tool and history as a goal. Under the title of history as a tool, the students’ reflections were categorized as history as a cognitive tool and history as a motivational tool. As a cognitive tool, students stated that the history of mathematics embedded videos expanded their knowledge of mathematicians’ early work and their occupations. As a motivational tool, the history of mathematics embedded videos helped raise students’ curiosity about mathematicians and their works, and mathematical concepts, increased their motivation for invention, and developed a positive attitude toward mathematics and learning. On the other hand, under the title of history as a goal, the students stated that the history of mathematics embedded videos broaden their knowledge of mathematicians and their contributions, mathematical concepts, and mathematical evolution. Thus, the videos can be used as enrichment activities for gifted elementary students in mathematics classrooms.
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Pre-service education and in-service teacher professional development (collectively termed teacher professional development or TPD here) can play a pivotal role in raising teaching quality and, therefore, learning outcomes for children and young people in low- and middle-income countries (LMICs). However, TPD opportunities in LMICs are limited, unsustained, and often not informed by recent research evidence, and outcomes are mixed. Educational technologies offer potential to enhance formally provided programmes and informal peer-learning forms of TPD. We present the first systematic review of the literature pertaining to technology-mediated TPD for educators of school-aged learners in LMICs, aiming to characterise appropriate and effective uses of technology along with specific constraints operating in those contexts. An in-depth synthesis of 170 studies was undertaken, considering macro-, meso- and micro-level factors during TPD design and implementation in the 40 LMICs represented. Volume of publications increased dramatically over the review period (2008–2020), indicating that the field is rapidly developing. Results largely showed benefits for teachers, but evidence for sustainability, cost-effectiveness or tangible impacts on classroom practice and student outcomes was thin. Promising, locally-contextualised forms of technology-mediated TPD included virtual coaching, social messaging, blended learning, video-stimulated reflection, and use of subject-specific software/applications. We report on the variable effectiveness of programmes and limited attention to marginalised groups. To maximise effectiveness of technology-enhanced TPD, the role of facilitators or expert peers is paramount – yet often glossed over – and the interpersonal dimension of teacher learning must be maintained. Recommendations are made for researchers, policymakers, teachers and teacher educators.
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In this article, we provide an empirical example of how digital technology; in this case, GeoGebra may assist students in uncovering—or whiteboxing—the content of a mathematical proof, in this case that of Proposition 41 from Euclid’s Elements. In the discussion of the example, we look into the impact of GeoGebra’s “dragging” functionality on students’ interactions and the possession and development of students’ proof schemes. The study and accompanying analysis illustrate that, despite the positive whiteboxing effects in relation to the mathematical content of the proposition, whiteboxing through dragging calls for caution in relation to students’ work with proof and proving—in particular, in relation to students seeing the necessity for formal proof. Moreover, caution must be paid, e.g., by teachers, so that students do not jump to conclusions and in the process develop inexpedient mathematical proof schemes upon which they may stumble in their future mathematical work.
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GeoGebra is a mathematical software that supports learning mathematics to make it easier to visualize geometric shapes. Geography software can create geometric images that are 2 dimensional or 3 dimensional so that GeoGebra can help students to represent real geometric shapes. This research aims to view the effectiveness of GeoGebra software in improving students’ visual representation ability. This study used a one-group pre-test/post-test design pre-experiment. The sample in this ability were students of eight grades SMP Negeri Sukoharjo who studied the material of flat-sided building. The instruments used in this study were the pre-test with and post-test of mathematical representation abilities. The results showed that there was an effectiveness of using GeoGebra in improving students’ visual representation skills with the difference in the mean score of students after being given geoGebra treatment was 61.3 for the pre-test and 70.5 for the post-test.
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The objective of the study was to determine the level of understanding and characterize the type of ideal geometric thinking that teachers demonstrate in solving problems with the use of the GeoGebra software. The research was of mixed character, with sequential exploratory design. It was structured in two phases. The population was made up of 120 teachers who attended the course entitled: Innovate and transform the teaching of mathematics with GeoGebra. The virtual course was developed by the National University of Education during the period 2019–2020. In the first phase of the quantitative phase, a diagnostic questionnaire of descriptive analysis was applied. The second qualitative phase applied an analytical questionnaire on the polygons issued by the participants. The results show that the participants understood the activities in a dynamic way. To learn geometry, the level of understanding was very good when using the GeoGebra. It was possible to characterize that 15% of the participants developed a basic geometric representation without connection, demonstrating a submissive and linear style in the development of activities. 35% of the participants worked on contrasting geometric polygons, representing a constructivist design of visual variables. Finally, 50% of the participants designed axial geometric polygons, representing a higher level of complexity and totalitarian figuration. In conclusion, the software generated a great interest and motivation in the pedagogical activities. With the use of GeoGebra a level of effective understanding is acquired in the development of geometry.
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The inclusion of History of Mathematics (HoM) in the official mathematics textbook published by Indonesian Ministry of Education and Culture became an essential aspect in supporting the national curriculum of mathematics education. This study aimed to examine Indonesian prospective mathematics teachers' pedagogical attitudes and beliefs in using HoM to teach mathematics. 305 prospective teachers from nine universities in Indonesia were involved as the respondents. Data were collected using Questionnaires of Attitude and Belief in using HoM to teach mathematics. The results showed that prospective teachers had positive attitudes and beliefs in using HoM to teach mathematics. However, those who had taken the mathematics history course tent to have a higher score compared to those who had not taken the course yet during their study times. Hence, this study recommends that involving the HoM course is vital in supporting the prospective teachers' learning process as a part of their future professional developments. However, both groups showed low scores in the questionnaire item of self-efficacy beliefs. This implies that teacher education programs should facilitate the improvement of prospective teachers' self-efficacy beliefs and skills, especially of using HoM in teaching mathematics.
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Bu araştırmanın amacı, yedinci sınıf “Çember ve Daire” konusunun öğretimine yönelik geliştirilen GeoGebra destekli öğretim materyaline ve öğrenme ortamına ilişkin öğrenci görüşlerini yansıtmaktır. Araştırma özel durum çalışması ile yürütülmüştür. Araştırma, Isparta ilinde yedinci sınıfta öğrenim gören 30 öğrenci ile gerçekleştirilmiştir. Veriler, 5’li likert tipi 25 madde ve 4 açık uçlu sorudan oluşan anket ve sınıf içi gözlemler yoluyla toplanmıştır. Nicel veriler SPSS 17.0 paket programı ile analiz edilmiştir. Nitel veriler betimsel ve içerik analizi tekniği ile çözümlenmiştir. Araştırmada GeoGebra destekli öğretim materyalinin öğrenmeyi kolaylaştırdığı, görsel olarak zihinde canlandırmayı güçlendirdiği, günlük hayatla ilişkilendirilebilmeyi desteklediği, daha kolay çizimler yapma ve keşfetme fırsatı verdiği belirlenmiştir.
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Latest interactive and dynamic geometry software introduces new practices in Mathematics education, adding virtual 2D or 3D animated graphs to traditional artifacts and compass-and-straightedge constructions. This work presents the results of a teaching experiment in primary school, realized using Montessori method and setting math education through perceptual-sensory inputs. Cheap material was exploited to introduce pupils to one of the most important deductive process: the geometric proof. An all brand-new artifact was realized in order to explain Pythagorean Theorem using a hydro-mechanical system. The proof was performed starting from practice, direct observation by real and virtual tools. Pupils used GeoGebra, a dynamic geometry freeware: they built shapes, determined areas and verified the equivalence. Pupils repeated the proof using crayons, producing pictures full of creativity. Finally, we show a comparison between new technological tools and traditional real tools. Teacher's role was to involve students in an educational project with real and virtual tools.
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There is concern about mathematics success and its related pedagogy. Society has seen rapid changes in the economy and technology with a call for these changes to be reflected within the classroom. New methods of teaching mathematics are being sought with the purpose to improve teaching and learning while making mathematics relatable to the new generation of learners. The incorporation of technology within the classroom has been seen an option to make this change. The purpose of this research was to determine how effective the use of the GeoGebra app is in allowing learners to successfully discover the properties of straight line graphs. Furthermore, the research looked at learner responses to using the app. A qualitative research design was used with data generated through a task-based investigation, as well as individual interviews. Results of the research showed that the use of GeoGebra aided learners successfully in discovering the properties of straight-line graphs with the majority of learners understanding both concepts. The results also showed that learners had a positive outlook to the use of the app and enjoyed the experience. Keywords: GeoGebra, iPad technology, mathematics teaching, linear functions, software manipulation.
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This study examines pre-service teachers’ points of view about learning history of mathematics during their undergraduate education. An open-ended questionnaire was administered to one hundred and twenty pre-service teachers, during the fall semester of the 2013-14 academic year. The participants indicated that learning history of mathematics could increase their content knowledge as they understand how formulas, theories and relations were developed over time. In addition, it could develop them intellectually as they learn life stories of mathematicians. Also, it could help them to hold the attention of students, and answer some of the why questions. Particularly, they reported using history of mathematics knowledge while teaching Geometry and Numbers.
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The rapid growth of technology for learning includes the introduction of educational software. However, rare reports were found that provides evidence on the effectiveness of these software. This study investigates the effectiveness of using GeoGebra software on Mathematics learning among 62 students in Malaysia. Results show that students have positive perception towards learning (m =4.26) and have better learning achievement using GeoGebra (p < 0.05). Available free online, GeoGebra can benefit students Mathematics learning and diversifying learning in classrooms.The overflown of resources triggered students’ interest to learn Mathematics however, the selection of software has to be properly planned.
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The use of information technology has been used to improve capability and success in teaching and learning environments. This case study has been carried out with a view to replying the question of how we can make the subject of integers, which students in the 6th grades find too difficult to learn, easier and more permanent. The subject of Integers is one of the most difficult subjects that students have difficulty in grasping because they have only learnt natural numbers in Math. In general, when students come across a new number system, they have difficulty in learning it. Also, as they haven’t conceptualized the subject, they do not like the subject and cannot correlate the mathematical links. However, as teachers and mathematics educators, our duty is to make our students enjoy the subject of integers and ensure its retention. For this purpose, so as to eliminate the problems faced during the learning process, we have tried to conceptualize the subject by employing new software, Geogebra, by which we aim to provide a permanent learning for our students. The subject of integers has been taught with different methods to two homogeneous classes. One of the classes was taught with traditional teaching techniques, whereas the other was taught by Geogebra. In order to evaluate how much they understood the subject, a test has been applied immediately after the lesson. Two weeks later, a new test was applied to see how much of the subject they remembered. Thereafter, by comparing the results of both exams, the contributions and benefits of Geogebra in teaching Math have been analyzed.
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This paper discusses ways of enhancing the teaching of mathematics through enabling learners to gain stronger links between geometry and algebra. The vehicle for this is consideration of the affordances of GeoGebra, a form of freely-available open-source software that provides a versatile tool for visualising mathematical ideas from elementary through to university level. Following exemplification of teaching ideas using GeoGebra for secondary school mathematics, the paper considers current emphases on geometry and algebra in the school curriculum and the current (and potential) impact of technology (such as GeoGebra). The paper concludes by raising the implications of technological developments such as GeoGebra for the pre-service education and inservice professional development of teachers of mathematics.
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This paper describes a study aimed to identify most common impediments related to the introduction of an open-source mathematical software package GeoGebra. We report on the analysis of data collected during a three-week professional development programme organised for middle and high school teachers in Florida. The study identified challenges participants face during workshops and evaluated the difficulty levels of GeoGebra tools. Findings of the study, complexity criteria of software tools and commonly occurring difficulties, provided the basis for the development of several new materials assisting workshop activities and contributed to the improvement of introductory GeoGebra workshops. (Contains 1 table, 2 figures, 3 online resources and 1 note.)
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In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers’ belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the ‘laboratory of mathematics education’ carried out in a 2year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.
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Despite the wide interest in combining mathematics education and the history of mathematics, there are grave and fundamental problems in this effort. The main difficulty is that while one wants to see historical topics in the classroom or an historical approach in teaching, the commitment to teach the modern mathematics and modern mathematical techniques necessary in thepure and applied sciences forces one either to trivialize history or to distortit. In particular, this commitment forces one to adopt a Whiggish approach to the history of mathematics. Two possible resolutions of the difficulty are (1) radical separation – putting the history of mathematics on a separate track from the ordinary course of instruction, and (2) radical accommodation – turning the study of mathematics into the study of mathematical texts.
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This is a theoretical article proposing a way of organizing and structuring the discussion of why and how to use the history of mathematics in the teaching and learning of mathematics, as well as the interrelations between the arguments for using history and the approaches to doing so. The way of going about this is to propose two sets of categories in which to place the arguments for using history (the “whys”) and the different approaches to doing this (the “hows”). The arguments for using history are divided into two categories; history as a tool and history as a goal. The ways of using history are placed into three categories of approaches: the illumination, the modules, and the history-based approaches. This categorization, along with a discussion of the motivation for using history being one concerned with either the inner issues (in-issues) or the metaperspective issues (meta-issues) of mathematics, provides a means of ordering the discussion of “whys” and “hows.”
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This study considers the results of a diagnostic test of student difficulty and contrasts the difference in performance between the lower attaining quartile and the higher quartile. It illustrates a difference in qualitative thinking between those who succeed and those who fail in mathematics, illustrating a theory that those who fail are performing a more difficult type of mathematics (coordinating procedures) than those who succeed (manipulating concepts). Students who have to coordinate or reverse processes in time will encounter far greater difficulty than those who can manipulate symbols in a flexible way. The consequences of such a dichotomy and implications for remediation are then considered.
Article
The merits of incorporating history into mathematics education have received considerable attention and have been discussed for decades. Still, before taking as dogma that history must be incorporated in mathematics, an obvious question is, Why should the history of mathematics have a place in school mathematics? Answering this question is difficult, since the answer is subject to one's personal definition of teaching and is also bound up with one's view of mathematics. Fauvel's (1991) list of fifteen reasons for including the history of mathematics in the mathematics curriculum includes cognitive, affective, and sociocultural aspects. My purpose in this article is not to provide complete and satisfactory answers but rather, on the basis of theoretical arguments and empirical evidence, to attempt to pinpoint worthwhile considerations to help high school teachers think about what history really can do for the curriculum and for their teaching.
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Information and Communication Technologies (ICT) in education provide a new learning environment where the student builds his own knowledge, allowing his visualization and experimentation. This study evaluated the Geogebra software in the learning process of Calculus. It was observed that the proposed activities helped in the graphical interpretation of the covered content.
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Upper division mathematics courses often leave students with the sense that mathematics is essentially an abstract subject whose development is founded on formalized reasoning within formalized contexts. But mathematics is more than this. A dynamic geometry environment like GeoGebra offers professors in prooforiented geometry courses an alternative to immediate formalization. In this chapter, we describe two classroom situations, one online and one face-to-face, in which students developed and experimented with dynamic geometry models, leading to the formal construction, refinement, and validation of mathematical concepts and theories at the heart of the formal content goals of the courses.
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These classroom notes offer methods of solving the quadrature of lunes, that is, the area of croissant-shaped plane figures bounded by two intersecting non-congruent circular arcs, using Hippocrates of Chios’ area conservation and similarity arguments. I also offer a method of using history in the classroom with students via dynamic geometry snapshots presented in a manner that complements the analytic and the visual approaches.
Article
Günhan Caglayan discusses methods of solving the quadrature of parabola, the area of the region bounded by the parabola and a chord, using Archimedes' ideas of infinite sums and limits. He demonstrates an exploration of this problem based on snapshots used in GeoGebra technology, a dynamic geometry software (DGS) that intertwines algebra, geometry, and spreadsheets environments. The rationale for using a DGS in the exploration of the quadrature problem is founded in the view of experimental mathematics in which the role of technology manifests in gaining insight and intuition, discovering new patterns and relationships, and graphing to expose math principles. The snapshots used have been a useful asset in the discovery of new patterns and in the setting of the stage for formal proofs.
Article
The use of the history of mathematics in teaching has long been considered a tool for enriching students’ mathematical learning. However, in the USA few, if any, research efforts have investigated how the study of history of mathematics contributes to a person's mathematical knowledge for teaching. In this article, I present the results of research conducted over four semesters in which I sought to characterize what prospective mathematics teachers (PMTs) understand about the topics that they will be called upon to teach in the future and how that teaching might include an historical component. In particular, I focus on how the study and application of the history of solving quadratic equations illuminates what PMTs know (or do not know) about this essential secondary school algebraic topic. Additionally, I discuss how the results signal important considerations for mathematics teacher preparation programs with regard to connecting PMTs' mathematical and pedagogical knowledge, and their ability to engage in historical perspectives to improve their own and their future students' understanding of solving quadratic equations.
Article
Today's Internet provides a large number of freely available interactive materials for mathematics learning and teaching. Most of these mathlets are ready to use but not ready to be modified or extended. In this article we present a collaborative environment of open source tools around the dynamic mathematics software GeoGebra that gives educators the freedom to create new and modify existing materials in an online community. We describe how to do this based on design principles for multimedia learning and experiences from our work with teachers. See http://www.joma.org for an interactive version of this article. Technologies Used in This Article This article uses Java (1.4.2 or later) and JavaScript for several interactive figures created with GeoGebra. Please install Java from www.java.comand activate JavaScript in your browser if necessary.
Article
Florida State University owns a collection of twenty-five cuneiform tablets, acquired from Edgar J Banks in the 1920s. We describe their rediscovery, present an edition of one of them (a twenty-first century BC labour account from the Sumerian city of Umma), and discuss their potential for use in undergraduate mathematics education.1 1We are very grateful to Steve Garfinkle, Denise Giannino, John Larson, Lucia Patrick, Plato L Smith II, and Giesele Towels for their help in the research and writing of this article. View all notes
Article
In Clark and Robson (20081. Clark , K M . 2008 . “ ‘Heeding the call: history of mathematics and the preparation of secondary mathematics teachers’ ” . In Inquiry into mathematics teacher education , Edited by: Arbaugh , F and Tayler , P M . 85 – 95 . Association of mathematics teacher educators . View all references), we described the re-discovery, translation, and edition of a collection of 25 cuneiform tablets that reside in the Robert Manning Strozier Library at Florida State University (FSU). In addition, we discussed the construction of a classroom instructional unit, intended for use with Grade 4 and 5 students (aged 9 to 11) in Eustis, Florida—the retirement home of Edgar J Banks, who sold the tablet collection to FSU in 1922. Here I describe piloting the instructional unit with students, their teachers, and district supervisors in Fall 2008. The unit was created from the historical text found on FSU 22 and FSU 23 and the grade level mathematics standards from Florida's Next Generation Sunshine State Standards (NGSSS) for mathematics (2007).
Article
The integration of history into educational practice can lead to the development of activities through the use of genetic ‘moments’ in the history of mathematics. In the present paper, we utilize Oresme’s genetic ideas – developed during the fourteenth century, including ideas on the velocity–time graphical representation as well as geometric transformations and reconfigurations – to develop mathematical models that can be employed for the solution of problems relating to linear motion. The representation of distance covered as the area of the figure between the graph of velocity and the time axis employed in these activities, leads on naturally to the study of problems on motion by means of functions, as well as allowing for the use of tools (concepts and propositions) from Euclidean geometry of relevance to such problems. By employing simple geometric transformations, equivalent real life problems are obtained which lead, in turn, to a simple classification of all linear motion-related problems. When applied to a wider range of motion problems, this approach prepares the way for the introduction of basic Calculus concepts (such as integral, derivative and their interrelation); in fact, we would argue that it could be beneficial to teach the basic concepts and results of Calculus from an early grade by employing natural extensions of the teaching methods considered in this paper.
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An analytical survey of how history of mathematics has been and can be integrated into the mathematics classroom provides a range of models for teachers and mathematics educators to use or adapt.
Article
Everybody knows that mathematics is an abstract subject. It follows that most people who have studied the problem of learning such an abstract subject, would agree that some passage from the concrete to the abstract must be mapped. By concrete, we mean usually, our immediate contact with the real world. We come into contact with objects and events and we re-act to them. This is the concrete level at which all organisms behave until they are able to organize their re-actions to events into re-actions to sets of events. This is the first stage towards abstraction — when the organism begins to classify. It is of interest to try and investigate the details of the abstraction process, not only from concrete experiences to classification, but to the learning of extremely complex abstract systems such as exist in mathematics and which more and more people, including children, are called upon to learn. The difficulties, in the way of such studies, are great, one of the main difficulties being that mathematicians on the whole are not interested in learning, and psychologists on the whole do not know enough mathematics to be able to formulate the problem in a way in which a possible solution might be sought. At the Sherbrooke Psychomathematics Center, for the past few years, we have been studying the abstraction process as it proceeds from the concrete to the final stage of wielding a mathematical formal system. We have rather few laboratory results of this aspect of our work as yet but we have enough classroom evidence that we can postulate certain regularities that seem to occur and certain pre-requisites that appear to have to be satisfied before certain stages of learning can successfully be undertaken.
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Incl. bibliographical references, app., index
Article
This paper discusses an investigative approach to problems concerned with approximations. Two problems are analysed in detail and both are inspired by their historical origins. The first examines an ancient Chinese formula for calculating the area of a segment. The second looks at Dürer's construction for the trisection of an angle. It is argued that historical problems can provide a rich source for such an investigative approach. Discussion of the second problem also raises fundamental issues about the nature of ‘conceptual tools’ and ‘conceptually exact solutions’ to geometric construction problems, particularly in the light of using dynamic geometry environments such as Sketchpad.
Article
The study reported in this article deals with the observed actions of Turkish pre-service mathematics teachers in dynamic geometry environment (DGE) as they were learning Khayyam's method for solving cubic equations formed as x3 + ax = b. Having learned the method, modelled it in DGE and verified the correctness of the solution, students generated their own methods for solving different types of cubic equations such as x3 + ax2 = b and x3 + a = bx in the light of Khayyam's method. With the presented teaching experiment, students realized that Khayyam's mathematics is different from theirs. We consider that this gave them an opportunity to have an insight about the cultural and social aspects of mathematics. In addition, the teaching experiment showed that dynamic geometry software is an excellent tool for doing mathematics because of their dynamic nature and accurate constructions. And, it can be easily concluded that the history of mathematics is useful resource for enriching mathematics learning environment.
Research in education: evidence-based inquiry
  • J H Mcmillan
  • S Schumacher
Motivating students in learning mathematics with GeoGebra
  • K S Choi
Kuramdan uygulamaya matematik eğitimi
  • A Baki
Qualitative research practice: a guide for social science students and researchers
  • D Snape
  • L Spencer
Omar Khayyam and dynamic geometry
  • S Burns
Semiotic mediation: from history to mathematics classroom
  • M G Bartolini-Bussi
  • M A Mariotti
Sosyal bilimlerde nitel araştırma yöntemleri [Qualitative research methods in social sciences
  • A Yıldırım
  • H Simşek