Geogebra

While most of the studies done on GeoGebra has been focused on the dynamic nature of GeoGebra (e.g. Aydin & Monaghan, 2011; Karaarslan et al., 2013) and the effects of GeoGebra on students’ learning (e.g. Bulut & Bulut, 2011; Edwards & Jones, 2006; Icel, 2011), this study explored teachers’ perspectives by describing their goals regarding the mathematics software GeoGebra and their beliefs about the utilisation of this software. Furthermore, instructional strategies and methods while integrating GeoGebra used by those teachers were investigated; and teachers’ favourite topics to integrate GeoGebra are further examined. Thus, the research questions are:

1. What are the middle school mathematics teachers' goals and beliefs about the use of GeoGebra in the seventh-grade middle school mathematics classes in Turkey?
1.1. For what purposes do they use GeoGebra?
1.2. What do they believe about the effectiveness of using GeoGebra?

2. How do middle school mathematics teachers use GeoGebra in the seventh grade middle school mathematics classes in Turkey?
2.1. In which topics do they use GeoGebra?
2.2. Which instructional strategies and methods do they utilise to integrate it?

Este artigo é um relato de experiência interdisciplinar com os componentes curriculares Matemática e Física numa turma da Educação de Jovens e Adultos – EJA, o Ciclo VI, em 2018, numa escola estadual da Paraíba. Neste projeto, com a finalidade da aprendizagem de gráficos de funções, foram realizados experimentos práticos nos quais os alunos mediram, observaram, preencheram tabelas, responderam a algumas questões, traçaram gráficos e debateram sobre a aplicação de tais conteúdos no cotidiano. Além de aulas expositivas, discussões, exercícios de fixação da aprendizagem, fizemos uso de planilhas eletrônicas e do software GeoGebra com a finalidade de conjecturar fórmulas, traçar gráficos e criar arquivos para cálculo de áreas, distâncias, alturas e velocidades à luz da Teoria da Orquestração Instrumental.

Cores dinâmicas em regiões retangulares podem ser obtidas de conceitos básicos da geometria euclidiana. No GeoGebra, interessantes imagens são reveladas por meio de exibição de rastros adicionados num sistema aditivo RGB. Com o objetivo de relacionar Retas de Euler numa composição de mapas de cores, neste artigo é apresentado um esquema pontual e dependente da posição cartesiana dos vértices de um dado quadrilátero. Os resultados das construções geométricas e as ilustrações dinâmicas obtidas reforçam um caráter lúdico e atrativo do GeoGebra. Assim, suportado por apresentação rigorosa dos conceitos geométricos, foi possível revelar figuras com padrões intrigantes e que não eram visíveis numa observação direta de fórmulas matemáticas.

Resumen: La investigación presenta el estudio de una situación didáctica desarrollada por estudiantes universitarios, mediada por el GeoGebra, para introducir la noción de suma de Riemann a partir de la suma de áreas de rectángulos. La comparación entre los análisis a priori y a posteriori evidenció que se redujeron las dificultades para plantear la suma de áreas de rectángulos, cuando se introdujo notaciones simbólicas.

In this chapter of the Technology in Mathematics Education:Contemporary Issues, we explore the nature and the importance of dynamic visualizations for teaching and learning probability. The chapter begins with a discussion of importance of probability as one of the key elements of risk literacy. We identify, in a literature review, the features of dynamic visualizations that make them more suitable for learning mathematics and what differentiates them from static representations. Through the case study of learning conditional probability in the classroom, we describe how students use and transform representations from inert static to kinesthetic/aesthetic representations. We also discuss limitations of static representations and illustrate how those limitations could be resolved using dynamic representations.

This monograph oriented implementation and achievement of learning activity cycle in science 3 Elementary School, describes some of the expertise in the use of educational media by a professor of the group's schools school Les Houches in Haute Savoie. These can be considered to a repository of techno pedagogical expertise; in particular, the organization and procedural methods covering of cognitive activity of pupils in the field of identification calculations with GeoGebra tool on the touch pad. This study is carried out on both active and complementary methods: self-confrontation interviews during which the school teacher, faced with the observation of his publicized activities, reveals the intentions that guide its practice, and analyzes teaching instrumented that can account for its professional schemes.

Los recursos tecnológicos digitales han mostrado la posibilidad de profundizar en la naturaleza del conocimiento matemático y establecer nexos entre diferentes conceptos. La investigación didáctica ha mostrado que es posible usarlos para apoyar el aprendizaje de la demostración en geometría. La investigación en este campo se ha enfocado ampliamente a la geometría plana, dejando poco explorado el contexto de la geometría espacial. En este taller presentamos un conjunto de problemas de construcción de geometría espacial resueltos con GeoGebra 3d, dirigido a estudiantes con alta capacidad matemática de últimos cursos de Primaria y primeros cursos de Secundaria, con los que se pretende ayudarles en el aprendizaje de la demostración.

Neste artigo, pretendemos apresentar um recorte parcial de uma pesquisa de mestrado, no qual se descreve especificamente, uma Situação Didática Olímpica (SDO) que é estruturada com o suporte do software Geogebra. Uma intervenção realizada de forma adequada pela exploração de softwares, permite aos estudantes ultrapassar entraves que eventualmente possam surgir na construção e aplicação de conceitos geométricos. De um modo particular, a SDO descrita propõe ao aluno a possibilidade de vivenciar o processo de aprendizagem sob o ponto de vista metodológico da Teoria das Situações Didáticas (TSD). A teoria das situações analisa e decompõe esse processo em quatro fases distintas e interligadas – ação, formulação, validação e institucionalização – nas quais o saber tem funções diferentes e o aprendiz não tem a mesma relação com o saber. Ademais, a análise dos elementos matemáticos e a identificação das suas propriedades, através da modelização da situação didática, permite os estudantes solucionar o problema proposto e validar o teorema de Pitot. Desse modo, a SDO representa uma alternativa para aulas direcionadas às olimpíadas de matemática, sua indicação e estruturação, descreve elementos atinentes à mediação didática durante o processo de ensino e aprendizagem, os quais enfatizam pormenores que detêm a possibilidade de controlar/prever as ações dos estudantes, assim como, proporcionar um sentido mais significativo para o estudo da geometria no contexto olímpico. Abstract: In this article, we intend to present a partial cut of a master's research, in which we describe specifically, an Olympic Educational Situation (SDO) that is structured with the support of Geogebra software. An intervention carried out adequately by the exploitation of software, allows students to overcome obstacles that may arise in the construction and application of geometric concepts. In a particular way, the described SDO proposes to the student the possibility of experiencing the process of learning from the methodological point of view of Theory of Educational Situations (TSD). The theory of situations analyzes and decomposes this process into four distinct and interconnected

Las nuevas tecnologías de la información y comunicación ofrecen un medio para que los estudiantes exploren, conjeturen, redescubran, construyan nuevos conocimientos y desarrollen habilidades tanto matemáticas como digitales. Los programas de geometría dinámica se han convertido en uno de los recursos informáticos que mejor permiten la interactividad del alumno con las ideas matemáticas. Geogebra es un software de geometría dinámica que es de fácil uso y de carácter abierto, es por eso que es una herramienta poderosa en el proceso de enseñanza – aprendizaje de la geometría. La experiencia de mosaicos con geogebra fue realizada con las alumnas del segundo grado de educación secundaria del colegio Sagrado Corazón – Sophianum. Se realizó en tres fases, la primera de instrumentalización ya que las alumnas no sabían utilizar el software; la segunda, la de descubrimiento de las propiedades de las isometrías (simetría axial y puntual, rotación y traslación) con el uso de Geogebra; y la tercera, de aplicación de las propiedades de las isometrías por medio de la construcción de mosaicos. Los objetivos de la experiencia son: utilizar el software GeoGebra para la generación de conjeturas referidas a los conceptos de isometrías, que las alumnas descubran que dentro de un mosaico existen diferentes isometrías e introducir en el aula de matemática un software de geometría dinámica. La aplicación de la experiencia resultó valiosa porque se logró que las alumnas reconozcan las propiedades y elementos de cada una de las isometrías y utilizaron su pensamiento geométrico de manera creativa en la construcción de los mosaicos con la ayuda del Geogebra.

This chapter focuses on environments that support the fostering of SDML. This research drew on constructionism as theoretical lens and its implementation by means of microworlds. The aim of this chapter is to determine what features of microworlds enable them to be supportive multimodal learning environments for SDL. The design features of selected microworlds are investigated, focusing on the requirements of SDL, problem-solving, opportunities for collaboration and the specific resources available to
students and facilitators. The qualitative study that this chapter is based on was completed from within a pragmatist paradigm and involved document analyses and multimodal content analyses to do an exploratory evaluation of the selected microworlds. The exploratory evaluation focused on the creation of a rubric for evaluation of the features of microworlds. Specific aspects in the selected microworlds were identified that can potentially support SDML.

Mathematics Logarithmic functions are an important part of the school curriculum. Also, they are important tools for solving problems in mathematics, physics, chemistry, biology, economics etc. The teaching of these functions is done in different ways in school curricula. This article focuses on the discovery of logarithmic functions using alternative methods for efficient didactic transposition. Our work focuses on: A bit of history concerning the invention of the logarithm function; Demonstration of the existence of a function which transforms the product in sum; Discover the logarithm function using the GeoGebra software; Explore the parameters of the function f (x) = a logb (x + h) + k using the GeoGebra software; Application of the GeoGebra software to solve problems containing the logarithm function. We concluded this work by clarifying the usefulness of the Geogebra tutorial in the didactic transposition for the teaching of logarithmic functions for the pupil as well as for the teacher. Also, this study assesses students' motivation in learning mathematical concepts. In addition, we propose didactic perspectives of the use of Geogebra in improving the teaching of mathematics. Résumé Les fonctions logarithmes, occupent une part importante dans les programmes scolaires, elles sont des outils importants pour résoudre des problèmes pour les mathématiques, la physique, la chimie, la biologie, ou

Учеб.-метод. пособие / С.Р. Мугаллимова; Департамент образования имолодёж. политики ХМАО-Югры, Бюджет. учреждение высш. образования ХМАО-Югры “Сургут. гос. пед. ун-т”. – Сургут : РИО БУ “Сургутский государственный педагогический университет”, 2019 – 55 с.

We investigate which practices are performed by teachers, when they scaffold students' learning to solve mathematical problems together, with technologies that facilitate group learning in a whole-classroom context. We describe teacher practices in order to evaluate their contribution to the Whole-Class Scaffolding, in the context of a course that was meant to facilitate learning to solve mathematical problems in small groups. Sixteen and twenty four junior high students took part in two iterations of a design research, correspondingly. We identified six teacher practices. These practices included: (1) Presenting strategy-oriented problems, in order to familiarize students with mathematical problem solving strategies and heuristics; (2) Decomposing a problem into stages; (3) Modeling the use of the dedicated technologies; (4) Preliminary Learning to Learn Together (L2L2) talk; (5) Routine L2L2 talk; and (6) Summative L2L2 talk. We conclude that scaffolding full-year learning to solve mathematical problem together in the context of a whole class is feasible, with suitable technologies. This study sheds also theoretical insights on Whole-class scaffolding and on the ways group learning could enhance learning to solve mathematical problems by an individual.

Circle theorem is one of the dreaded topics in mathematics many teachers find difficult teaching to the understanding of their students. When the researcher was doing his internship at St. Francis’ college of education, Hohoe, he identified that trainees find it difficult solving problems on circle theorems. This prompted the researcher to find out why the problem existed and how to help solve the problem. The researcher then gathered data on the problem using questionnaire and test technique. He administered pre-test to confirm the problem identified. He took the trainees through an intervention process using Geogebra and then administered post-test to evaluate the effectiveness of the intervention. The outcome of the post-test showed a great improvement in trainees’ performance in circle theorem indicating that, the Geogebra used by the researcher during the intervention process has helped to enhance trainees’ understanding of circle theorems. It is therefore recommended that teachers should use concrete and semi-concrete materials in their teaching to enhance the understanding of learners. Teachers are also advised to integrate ICT into the teaching process to facilitate the teaching and learning process.

The integration of mathematical modeling with technology and the advantages of technology to modeling process have become more important in today's fast-growing society. The studies about how the technology affects the mathematical modeling process and how to use the technology more effectively are of importance. The purpose of this study is to illustrate how to use GeoGebra in the process of mathematical modeling. In this study, GeoGebra was used in solution process of a problem designed in accordance with
mathematical modeling and the intended uses of GeoGebra were described in the mathematical modeling process. The solution of the Height-Foot Length Problem designed by the researchers was carried out taking into account the seven step modeling process. With this study, it was exemplified how the mathematics teachers will be able to use the mathematical modeling and the GeoGebra in their lessons. It is thought that GeoGebra will contribute to the uncovering and the development of modeling skills and will be provided
more conceptual and mathematical thinking by preventing losing in procedures.

The 2013 official national curriculum published by the Turkish Ministry of
Education formally required high school mathematics teachers to actively incorporate computer software in their teaching. The primary purpose of this study was to demonstrate the development of an inquiry-based teaching unit especially geared for high school mathematics students and teachers for the general concept of integral calculus. The main theme chosen as a case for this proposed inquiry unit was
on definite integral and volumes of solids of revolution of real life daily objects. As a result, the primary purpose was to provide the process of developing a practical example of using pedagogically driven dynamic mathematics software (GeoGebra), a 3D digital model coupled with hands-on real life examples, all embedded in a constructivist learning environment. Also, within this study, the perceived effectiveness of the developed teaching unit by in-service high school mathematics teachers based on their experiences was reported.
Key words: Mathematics education, inquiry-based learning, integral calculus, definite integral, constructivist learning, modeling, dynamic mathematics software, GeoGebra

Resumen El GeoGebra es un software diseñado principalmente para la enseñanza de la Matemática en cualquier nivel educativo, es de código abierto y es gratuito; pero que además puede ser utilizado en otras áreas del conocimiento como la Física y por supuesto para la enseñanza de la misma, en este caso particular para construcción de diagramas de cuerpo libre y como un complemento para la edición de imágenes, gracias a que con GeoGebra cuenta con elementos como vectores, circunferencias, rectas paralelas y demás objetos geométricos, que como es sabido también son indispensables en la Física, tanto para explicación de conceptos, resolución de problemas, así como para la edición de material didáctico. Abstract GeoGebra is a software designed primarily for teaching of Mathematics at any educational level, it is open source and free. Can also be used in other areas of knowledge such as Physics for the construction of free body diagrams and as a complement for image editing. GeoGebra has elements such as vectors, circumferences, parallel lines and other geometric objects, essentials in Physics for explaining concepts, solving problems, and editing teaching materials.

Penelitian ini dilatarbelakangi oleh siswa SLTA yang masih merasa kesulitan dalam mempelajari geometri, dan lebih khususnya lagi materi dimensi tiga. Siswa mengalami kesulitan dalam melakukan proses abstraksi, padahal abstraksi merupakan hal yang penting dalam pembelajaran matematika. Dalam mempelajari geometri dibutuhkan suatu alat peraga, seiring dengan perkembangan teknologi dan komunikasi dalam pendidikan yang berdampak pula pada alat peraga untuk geometri. Salah satu alat peraga tersebut berbentuk virtual manipulatives atau alat peraga yang dapat dimanipulasi, yaitu program GeoGebra. Salah satu fungsi GeoGebra adalah program untuk membantu siswa mempelajari geometri. Penelitian ini bertujuan untuk melihat deskripsi lengkap tentang proses abstraksi siswa yang belajar dimensi tiga dengan bantuan program GeoGebra serta melihat deskripsi lengkap tentang proses abstraksi siswa yang belajar dengan metode konvensional. Penelitian ini merupakan penelitian kualitatif dengan subjek penelitian kelas sepuluh MAN Cilegon. Teknik pengumpulan data dalam penelitian ini menggunakan observasi, wawancara, dan tes. Hasil penelitian di analisis dengan teknik analytic induction dan constant comparison. Penelitian ini menghasilkan kesimpulan bahwa proses abstraksi siswa yang mempelajari materi dimensi tiga dengan bantuan GeoGebra adalah proses abstraksi empiris yang terjadi selama proses pembentukan konsep. Sedangkan proses abstraksi siswa yang mempelajari materi dimensi tiga dengan metode konvensional adalah proses abstraksi teoritis yang berlangsung pada proses asimilasi, akomodasi dan ekualibrasi dalam satu rangkaian pemrosesan informasi.

Kata Kunci : Abstraksi, Virtual Manipulatives, GeoGebra.

Bu çalışma, bir ilköğretim matematik öğretmen adayının, öğretmenlik uygulaması dersi sırasında, geometride üç boyutlu cisimlerin farklı yönlerden görünümleri konusunu anlatırken Teknolojik Pedagojik Alan Bilgisi'ndeki değişimi incelemek için yapılmıştır. Örneklem bir öğretmen adayı olarak Ankara'daki büyük üniversitelerden birinden seçilmiş, katılımcı anlaşmalı özel okulda stajını yaparken gözlemlenmiştir. Çalışmada kullanılan veri toplama araçları görüşmeler, gözlemler, alan notları, GeoGebra dosyaları ve ders planlarıdır. Araştırmanın sonucu düşünüldüğünde, ilköğretim matematik öğretmen adayının Teknolojik Pedagojik Alan Bilgisi'nde (TPAB) öğretmenlik uygulaması dersi boyunca bir gelişim olduğu görülmektedir. Müfredatı teknoloji ile birleştirme ve öğrencilerin teknolojiye erişimini sağlama konusundaki gelişim küçük olsa da, teknoloji aracılığıyla öğretme ve öğrencilerin öğrenmelerini sağlama konusunda öğretmen adayının TPAB seviyesinde önemli bir artış gözlemlenmiştir. Sonuçlar göz önünde bulundurulduğunda, öğretmen eğitimi programlarında öğretmenlik uygulaması dersi sırasında öğretmen adaylarını destekleyici teknoloji derslerinin gerekliliği ortaya çıkmaktadır çünkü matematik sınıflarında etkili bir öğrenme ortamı sağlanabilmesi için öğretmen adaylarının gerekli temel becerilere sahip olması beklenmektedir.

Özet
Uzamsal yeteneğin insan yaşamı için önemli ve gerekli bir özellik olduğu düşünülmektedir. Uzamsal yetenek gerektiren durumlardan biri de kişinin farklı açılardan baktığında nesneyi veya nesneleri farklı görmesi ve bu farklılığı ifade etmesidir. Uzamsal yetenekle ilgili öğretim programlarında çeşitli konular yer almaktadır. 7. Sınıf öğretim programı incelendiğinde “cisimlerin farklı yönlerden görünümleri” konusunun olduğu görülmektedir. Konunun kazanımlarından biri “Üç boyutlu cisimlerin farklı yönlerden iki boyutlu görünümlerini çizer” şeklinde olup açıklamasında “Eş küplerden oluşturulmuş yapılar ve bilinen geometrik cisimler kullanılır, çizim için uygun kareli kâğıtlar kullanılır, yapıların farklı yönlerden görünümlerinin ilişkilendirilmesi istenir (ön-arka ve sağ-sol görüntülerinin simetrik olması gibi),uygun bilgi ve iletişim teknolojileriyle etkileşimli çalışmalara yer verilebilir” ifadeleri yer almaktadır. Buradan hareketle bu çalışmada Geogebra yazılımı ile 10 adet birim küpün farklı dizilimlerinden oluşan bir materyal hazırlanarak, yazılımın ilgili komutları yardımıyla materyalin üstten, alttan, sağdan, soldan, önden ve arkadan görünümlerinin incelenmesi amaçlanmıştır. Konu ve kazanımlara yönelik dinamik ortamlarda tasarlanan bu tür materyallerin öğrencilerin uzamsal yeteneklerine olumlu etkisinin olabileceği ön görülmektedir.
Anahtar kelimeler: GeoGebra, geometrik cisimler, farklı yönlerden görünüm

It has become a fact that fluency and competency in utilizing the advancement of technology, specifically the computer and the internet is one way that could help in facilitating learning in mathematics. This study investigated the effects of Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS) in teaching Mathematics. This was conducted in Zamboanga del Sur National High School (ZSNHS) during the third grading period of the school year 2015-2016. The study compared the achievement and attitude towards Mathematics between those students taught with DGS and CAS and those students who were taught without DGS and CAS. A quasi-experimental pretest-posttest control group design was used with two groups which was matched according to their mathematical level. An achievement test was developed and administered to assess the students' achievement in Mathematics. Aiken's attitude scale was also used to assess the students' attitude towards Mathematics. The data were treated with One-Way Analysis of Covariance (ANCOVA) and t-test at 0.05 level of significance. The study revealed a significant difference between the mathematics achievement between students taught with DGS and CAS and those who were taught without it. The integration of DGS and CAS in Mathematics lessons leads to better academic achievement result compared with that of the conventional method of teaching. Moreover, there was no significant difference on the attitude level of the students between the two groups.

1. Menghargai dan menghayati ajaran agama yang dianutnya. 2. Menunjukkan perilaku jujur, disiplin, tanggung jawab, peduli (toleran, gotong royong), santun, dan percaya diri dalam berinteraksi secara efektif dengan lingkungan sosial dan alam dalam jangkauan pergaulan dan keberadaannya. 3. Memahami dan menerapkan pengetahuan (faktual, konseptual, dan prosedural) berdasarkan rasa ingin tahunya tentang ilmu pengetahuan, teknologi, seni, budaya terkait fenomena dan kejadian tampak mata. 4. Mengolah, menyaji dan menalar dalam ranah konkret (menggunakan, mengurai, merangkai, memodifikasi, dan membuat) dan ranah abstrak (menulis, membaca, menghitung, menggambar, dan mengarang) sesuai dengan yang dipelajari di sekolah dan sumber lain yang sama dalam sudut pandang/teori. B. Kompetensi Dasar dan Indikator Pencapaian Kompetensi Kompetensi Dasar Indikator Pencapaian Kompetensi 3.7 Menjelaskan sudut pusat, sudut keliling, panjang busur, dan luas juring lingkaran, serta hubungannya 1. Membedakan sudut pusat dan bukan sudut pusat 2. Membedakan sudut keliling dan bukan sudut keliling 49 3. Menentukan hubungan sudut keliling dan sudut pusat yang menghadap busur yang sama 4. Menentukan hubungan panjang busur, keliling dan sudut pusat lingkaran 5. Menentukan hubungan luas juring, luas daerah, dan sudut pusat lingkaran 6. Menjustifikasi hubungan sudut pusat dan sudut keliling yang menghadap busur yang sama 7. Menjustifikasi hubungan panjang busur, keliling dan sudut pusat lingkaran 8. Menjustifikasi hubungan luas juring, luas daerah dan sudut pusat lingkaran 9. Menyusun pertanyaan dan penyelesaiannya tentang hubungan sudut pusat dan sudut keliling yang menghadap busur yang sama 10. Menyusun pertanyaan dan penyelesaiannya tentang hubungan panjang busur, keliling dan sudut pusat lingkaran 11. Menyusun pertanyaan dan penyelesaiannya tentang hubungan luas juring, luas daerah dan sudut pusat lingkaran 4.7 Menyelesaikan masalah yang berkaitan dengan sudut pusat, sudut keliling, panjang busur, dan luas juring lingkaran, serta hubungannya 1. Menerapkan hubungan sudut pusat dan sudut keliling yang menghadap busur yang sama dalam menyelesaikan masalah 2. Menerapkan hubungan panjang busur, keliling dan sudut pusat lingkaran 3. Menerapkan hubungan luas juring, luas daerah dan sudut pusat lingkaran 50

Matematiksel modelleme ile teknolojinin entegrasyonu ve modelleme sürecine teknolojinin sağladığı avantajlar günümüzün hızlı gelişen toplumunda daha önemli hale gelmektedir. Teknolojinin matematiksel modelleme sürecini nasıl etkilediği ve daha etkili nasıl kullanılabileceğine ilişkin çalışmalar önem taşımaktadır. Bu çalışmanın amacı, matematiksel modelleme sürecinde GeoGebra'nın nasıl kullanılabileceğini örneklemektir. Çalışmada, matematiksel modellemeye uygun olarak tasarlanan bir problemin çözüm sürecinde GeoGebra'dan yararlanılmış ve GeoGebra'nın süreçteki kullanım amaçları açıklanmıştır. Araştırmacılar tarafından tasarlanmış Boy-Ayak Uzunluğu probleminin çözümü, yedi basamaklı matematiksel modelleme süreci dikkate alınarak gerçekleştirilmiştir. Çalışma ile öğretmenlerin matematiksel modelleme ile GeoGebra'yı derslerinde nasıl kullanabilecekleri örneklenmeye çalışılmıştır. GeoGebra'nın modelleme becerilerinin ortaya çıkarılmasında ve geliştirilmesinde katkı sağlayacağı ve işlemler içinde kaybolmayı önleyerek daha fazla kavramsal ve matematiksel düşüncenin ortaya çıkarılmasını sağlayacağı düşünülmektedir.

The latest official national curriculum published by the Turkish Ministry of Education, now formally requires high-school mathematics teachers to actively incorporate computer software in their teaching. The primary purpose of this study is to demonstrate the development of an inquiry based learning unit especially geared for high school mathematics students and teachers for the general concept of integral calculus. The main theme chosen as a case for this proposed inquiry unit, is on volumes of solids of revolution of real life daily objects. As a result, the primary purpose will provide a report about a practical example of using pedagogically driven mathematics software, 3D digital modeling coupled with hands-on real life examples all embedded in a constructivist learning environment. A secondary goal of this study is to expose pre-service high-school mathematics teachers to this inquiry unit as a teacher and student. Finally, the shared experiences about the interconnected nature of knowledge construction through a double “lens”, that is as a teacher and students through collaboratively building, testing and reflecting on their learning process will be reported.

Keywords: Mathematic Education, Inquiry, Modeling, Teacher Education

La noción de función está considerada dentro de las más importantes de la matemática y en particular, de la matemática escolar, llegando a ser considerada como un “concepto unificador en matemática”. Sin embargo, muchos estudiantes a nivel escolar, y también en educación superior, tienen muchas dificultades para su aprendizaje. Motivados por la importancia de esta noción y las dificultades en su aprendizaje, en especial aquellas relacionadas con las características de los objetos matemáticos al no poder acceder a ellos de igual forma como los objetos de otras disciplinas, es que nos resulta conveniente realizar este trabajo y abordarlo desde la Teoría de Registros de Representación Semiótica propuesta por Raymond Duval. Se revisó el programa de estudio y los textos escolares de matemática y, adicionalmente, se averiguó sobre las concepciones de los profesores, para ver si existían elementos de esta teoría y, a partir de la información reunida, se diseñó una propuesta didáctica que incluía actividades en GeoGebra, para facilitar los aprendizajes de la noción de función.

It is necessary to get involved actively in the process of acquiring knowledge to learn mathematical contents successfully. The theory of constructivism emphasizes student's individual activity in which he/she creates his/her own knowledge through his/her own experience, while the teacher prepares and shapes the environment for learning and encourages and guides students in their independent learning. The aim of this approach is for students to learn how to learn, and Polya's problem-solving model provides an answer how to perform in practice -in 4 steps: understanding the problem, devising a plan, carrying out the plan and reflection.

The aim of the present study is to conceptualize the approaches displayed for validation of model and thought aprocesses provided in mathematical modeling process performed in technology-aided learning environment.
The participants of this grounded theory study were nineteen secondary school mathematics student teachers.
The data gathered from the video recordings received while participants were solving the given problems, the
written responses related to the solution of the problems, the GeoGebra solution files and the observation notes
taken by the researchers in the process of problem solving. The constant comparative analysis based on open,
axial, and selective coding methods were used for data analysis through the grounded theory. The five sub-
steps were emerged from data analyzing related to the process of the validation of model. It was determined
that these steps were covered by discussing the unexpected results in real-life situations, comparing the real
life results with the estimation based on the experiences or measurements, the problem data, the situations
given in video and pictures and the decision-making about the adequacy of the model. In this study, it was seen
that the technology-aided environment enriched the cognitive processes in validating step. With the help of this
research’s results, it was constructed a detailed description related to the cognitive processes which can be
displayed for validating of model in mathematical modeling. It was considered that the study would bring a different perspective to the researchers towards the process of mathematical modeling.