Mathematics Education

בשנים האחרונות הולכת ומתחזקת המגמה של הוראה מקוונת במוסדות להשכלה גבוהה, וכפועל יוצא מכך עולה הצורך לשנות, להתאים ולעדכן את הגישות החינוכיות המקובלות להוראה. מתוך אמונה שהבניית גוף הידע העדכני הרלוונטי צריכה להתבצע בסיוע המרצים עצמם, במחקר הנוכחי נעשה ניסיון לקבל תמונת מצב ראשונית בנוגע לקיומם של קורסים מקוונים במחלקות לחינוך מתמטי במכללות בישראל, וללמוד מקרוב על תפיסותיהם של המרצים באשר למגוון היבטים הקשורים להוראה וללמידה בסביבות מקוונות. במחקר השתתפו 73 מרצים, מתוכם 13 מרצים שכבר התנסו בהוראה מקוונת של קורסים מתמטיים ו-60 מרצים שטרם התנסו בכך. כל קבוצה של משתתפים השיבה על שאלון שכלל שאלות סגורות ופתוחות, שהותאם לניסיון שלה בהוראה מקוונת.

There is strong evidence in mathematics education literature that students benefit
extensively from the use of technology that allows for multiple representations of
mathematical concepts. The benefits include developing an advanced level of
mathematical thinking and conceptual understanding. The purpose of this study was
to investigate the impact of teaching limits and continuity topics in GeoGebrasupported environment on students’ conceptual understanding and attitudes toward
learning mathematics through technology. The sample consisted of 34 students
studying in a unique high school for gifted and talented students in Turkey. This
study followed a pre-test post-test controlled group design. Conceptual
understanding of the topics of limits and continuity was measured through openended questions while attitudes toward learning mathematics through technology
was measured using a Likert-type survey. The intervention was teaching with
GeoGebra in contrast to using traditional instruction in the control group. Data were
analyzed with an independent samples t-test on gain scores for control and
iv
experimental groups. In the conceptual understanding test, the gain scores of the
experimental group was found to be 1.33 standard deviations higher than that of the
control group on the average. This finding was evaluated noteworthy in terms of
previously-conducted research on the impact of GeoGebra. Furthermore, the study
found that student attitudes toward learning mathematics through technology
improved, as well. The researcher concluded that Geogebra may be an effective tool
for teaching calculus to gifted and talented students .

Com a difusão do uso da internet, integrando vários recursos e ferramentas que facilitam o processo ensino aprendizagem é necessário um olhar diferenciado para o novo paradigma educacional que a Web 2.0 propõe aos educadores do século XXI. Os professores devem ser capazes de criar materiais de ensino inovadores e que facilitam a aprendizagem significativa dos alunos. A internet disponibiliza ferramentas aos professores para que possam criar aulas e/ou projetos que estimulam a pesquisa científica e o trabalho em equipe, propiciando uma aprendizagem significativa e colaborativa. Dentre as variadas possibilidades que o ciberespaço nos apresenta, a Webquest mostra-se como uma metodologia acessível, no entanto, a sua elaboração constitui um desafio aos professores, pois, demanda empenho, capacidade critica e técnica. Este trabalho aponta um possível caminho para elaboração de projetos inovadores, fazendo uso da metodologia da Webquest, para o desenvolvimento da pesquisa cientifica em matemática no ensino fundamental e médio.

Palavras-chaves: Webquest;  Investigação; Metodologias de Ensino.

As cônicas constituem um tópico de grande relevância no desenvolvimento tecnológico moderno; porém, o seu estudo no ensino médio, na maioria das vezes, é feito sob um enfoque puramente analítico, descontextualizado e fragmentado. O Currículo de Matemática da Secretaria de Educação do Estado de São Paulo (SEE) contempla as cônicas inicialmente no último ano do ensino fundamental e sendo aprofundadas no ensino médio. E, ainda, muitas das vezes, o ensino das cônicas não acontece para a maioria dos alunos e, quando acontece, é de forma pouco profunda e restrita a um curto período de tempo, o que acarreta algum desprezo por parte dos alunos e, até mesmo, de alguns professores que, por falta de formação adequada, desconhecem a sua importância e utilidade. O quadro apresentado motivou a busca por uma alternativa diferenciada para o ensino das Cônicas, por meio da Teoria das Situações Didáticas, desenvolvida por Guy Brousseau, que indica a criação de situações didáticas que favoreçam a investigação do aluno aproximando-o do modo como é produzida a atividade científica.  Neste trabalho será apresentada uma situação didática que pode ser desenvolvida no terceiro ano do ensino médio. Essa foi proposta de modo a buscar um “fazer matemática” mais próximo da vida real do aluno, explorando construções e discutindo suas particularidades através do uso do software GeoGebra.

Palavras-Chave: Cônicas, Situações Didáticas, Recursos de Informática.

This study focuses on the constructions of equivalent triangles developed by students aged 12 to 15 years-old, using the tools provided by the well known educational software Cabri-Geometry II (Laborde, 1990). Twenty-five students participated in a learning experiment where they were asked to construct several triangles and to transform them into other equivalent triangles ‘in any possible way’. The analysis of the data shows that all the students were actively involved in this transformation task and they constructed strategies that fell into eight categories. Most of the students initially ignored the fact that two triangles can conserve their areas when their figures are altered. Despite this fact, all the students, while also exploiting the variety of the provided tools successively constructed equivalent triangles in more than one way, by expressing different pieces of knowledge they possessed. Most students viewed the conservation of the area of a triangle in a non-fragmented way in the context of the tools of Cabri-Geometry II. These tools provided the students with the challenge of viewing this concept in interrelation with the concepts of: regular polygons, area measurement using spatial units and area formulae. Some students also recognized the concept of conservation of area in families of equivalent triangles with common bases and equal altitudes. Hence, they all constructed a broader view of the concept of conservation of area regarding triangles. Students also used these tools in combination with the ‘drag mode’ and the ‘automatic tabulation of numerical data’ to explore the concepts of conservation of area in relation to the perimeter of the constructed triangles.

This paper presents the design and basic features of a computer microworld for learning the concept of volume, for use by both primary and secondary level education students. The design of this microworld is the result of a modelling process in which three models were constructed, namely: the learning model, the subject matter model and the learner model. The construction of the learning model was based on constructivist and social theories of learning interpreted within the context of the computer. The construction of the subject matter model was based on the two fundamental aspects that constitute the concept of volume, namely: conservation and measurement of volume. The construction of the learner model was based on the literature referring to student behaviour regarding the learning of the concept of volume. All the models above are in the form of hierarchical trees, where educational specifications were gradually transformed into operational specifications which were then used for the implementation of the software.
In the context of VOLUME, students can study any parallelepiped. The basic features of VOLUME are: a) the construction of any parallelepiped by the students, just by drawing its dimensions, b) the dynamic transformation of a parallelepiped into a plethora of other parallelepipeds of equal volume, c) the construction of a cubic-unit for volume-measurement by the students, simply by drawing its dimensions, d) the iteration of the constructed cubic-unit, e) the automatic projection of the constructed unit onto the figure of a parallelepiped, f) the automatic measurement of the volume of the parallelepiped under study, using as a unit the cubic-unit constructed by the student, g) the automatic measurement of the volume of the parallelepiped under study, using the volume formulae, and h) the automatic measurement of the length of the edges of a parallelepiped using the length of the volume-unit constructed by the student as the length unit. 
The learning activities that students can perform in the context of VOLUME are: a) the transformation of a parallelepiped into other parallelepipeds with equal volume, b) the comparison of volume for a variety of parallelepipeds, c) the investigation of the basic properties of a parallelepiped, e.g. the length of its edges, the area of its sides, its angles etc., and d) the investigation of equivalence regarding the volume of parallelepipeds.

In this issue of Math Planet we explore some very interesting subjects of linear algebra, such as application of Eigenvalues, and some consequences of the direct sum and direct product of matrices over a vector space. Also we publish the solutions of the problems that we suggested in the previous issue (Math Planet Issue 2: Analysis). We hope that you find interesting our work.

This study focuses on the design of activity-based interfaces suitable for the learning of geometrical concepts in the context of Cabri Geometry II (Laborde, 1990). This design has emerged from a field study with real students aiming at the formation of appropriate learning settings, including learning materials and activities concerning Eucleidian geometry.  Three different types of learning settings were tested: firstly, a combined setting consisting of Cabri-tools and paper and pencil, secondly, a hypermedia setting consisting of Word documents hyperlinked with Cabri constructions and thirdly a learning setting that was totally performed in the context of Cabri, by integrating the presentation, the performance and the evaluation of the activities proposed into specific ‘acivity-oriented’ interfaces. The analysis of the data presented better student performance in the last learning setting due to the fact that the ‘activity-based’ interfaces proposed helped students to avoid wasting their energy on unnecessary keystrokes and navigation and concentrate on solving the tasks at hand.

This study focuses on the role of Multiple Representation System (MRS) in the expression of inter-individual learning differences in students. More specifically, thirty students participated in a comparative learning experiment on the learning of the concept of area. This experiment was performed both in the traditional paper and pencil environment and in a computer microworld (The C.AR.ME. microworld; Kordaki & Potari, 1998) providing a variety of tools that support the expression of the concept of area in a variety of representation systems. Students were given the same tasks in both environments. The analysis of the data shows that students were encouraged by the MRS provided in the context of C.AR.ME to express their inter-individual learning differences in terms of the concept of area. In particular, students constructed 20 different types of solution strategies for the given tasks within the context of C.AR.ME in comparison with four types of solution strategies constructed while trying to solve these tasks in the paper and pencil environment. 

This study focuses on the potential of multiple-solution tasks in e-learning environments providing a variety of learning tools. This was presented through a multiple-solution-based example for the learning of the mathematical notion of angle in the context of the well known e-learning environment Cabri-Geometry II (Laborde, 1990) dedicated for the learning of geometrical concepts. An a-priori task analysis showed that a variety of solution strategies could be invented by the students to face this type of task. In fact, students can select among the provided tools the most appropriate to express their knowledge. In the integrated context of such tasks and tools, students can express both inter-individual and intra-individual differences in the learning concepts in focus. In addition, students can consolidate these concepts, integrate the different kinds of knowledge they possess, enhance their learning styles and aquire advanced problem-solving skills. 

This paper focuses on the potential of integration of Multiple Learning Activities in Interactive Constructions (MLA-ICs), appropriate to support student learning of a specific learning subject. In fact, these constructions could be transformed using appropriate macros to support a variety of learning activities, beginning with real-life activities and gradually moving to more sophisticated scientific activities. In addition, these constructions can be transformed in a way that supports student learning of a variety of related concepts. The idea, the architecture and the interface associated with MLA-ICs was the result of a modeling process including field studies and using real students. The general concept, the design, the architecture and the interface of MLA-ICs is presented through a specific example for the learning of a mathematical theorem - Thales’ theorem - within the context of tools from the well-known e-learning environment, Cabri-Geometry II (Laborde, 1990).