Computing Locus Equations for Standard Dynamic Geometry Environments

Owing to its superb drawing utilities and symbolic computation capabilities, computer algebra system (CAS) has become one of the most preferred tools for drawing scientific artworks. However, according to the result of our questionnaire survey, many collegiate mathematics teachers in Japan feel difficulty in effectively using the graphics generated by CAS in printed materials. For instance, quite a number of mathematics teachers in universities are using TeX as their daily tool for editing high-quality mathematical documents, but the use of graphics generated by CAS in TeX documents seems not to be so easy task for usual TeX users. This difficulty seems to prevent many teachers from using CAS graphics. To fill this gap, some dynamic geometry softwares (DGS) and others have been equipped with the capability to export TeX-readable code of their graphical output. Also the authors have developed a macro package KETpic designed for CAS to supply high-quality graphics in TeX documents. In this paper, we will propose a scheme for demonstrating and improving the effect of CAS use in mathematics education. Through this scheme, it is shown that we can greatly improve the educational effect of our mathematics class materials by appropriately using the precise 2D-graphics generated by CAS and KETpic.

This article introduces and explains a computer algebra system (CAS) wxMaxima for Calculus teaching and learning at the tertiary level. The didactic reasoning behind this approach is the need to implement an element of technology into classrooms to enhance students' understanding of Calculus concepts. For many mathematics educators who have been using CAS, this material is of great interest, particularly for secondary teachers and university instructors who plan to introduce an alternative CAS into their classrooms. By highlighting both the strengths and limitations of the software, we hope that it will stimulate further debate not only among mathematics educators and software users but also also among symbolic computation and software developers.

With the growing significance of technology-based education and the advancements in Computer Algebra and Dynamic Geometry systems, automated tools for interactive learning have attracted significant attention. Nonetheless, the scarcity of interactive open-source online tools for geometric construction problems that offer feedback has hindered their wider implementation within the educational system. The primary reason for this absence is that geometric construction problems necessitate a constructed object as a solution rather than a numerical one. This feature renders the evaluation of geometric construction problems to be particularly challenging for machine evaluation (solutions cannot be copied into answer boxes) and time-consuming for teacher evaluation. In this work, the focus will be on triangle construction problems within the broader topic of geometric constructions. Emphasis will be placed on methods for creating interactive triangle construction applets in the widely popular GeoGebra software. The necessary steps will be outlined to enable mathematics teachers to create interactive GeoGebra applets with automatic verification that offer feedback for a diverse range of triangle construction problems.

Advantages and disadvantages of the use of Digital Technologies (DT) and especially of Computer Algebra Systems (CAS) in mathematics lessons are worldwide discussed controversially. Many empirical studies show the benefit of the use of DT in classrooms and there are also many useful examples concerning their use. However, despite these inspiring results and the countless ideas, classroom suggestions, lesson plans and research reports, the use of DT—and especially CAS—has not succeeded, as many had expected during the last decades see Hoyles & Lagrange, (2010). The thesis of this article is that we have not been able to convince teachers, lecturers at university and parents of the benefit of CAS in the classrooms in a sufficient way. What are the arguments that justify the use of CAS in the classroom? The article gives examples of a fruitful use of CAS with regard to the generally accepted goals or standards of mathematics education—like fostering students’ abilities in problem solving, modelling, proving or communicating—and to the subjects taught in high school. The basis of the argumentation is a competence model which classifies the relation between contents or topics: sequences and limits, functions and equations; representations of DT or CAS: static isolated, static multiple, dynamic isolated and dynamic multiple representations; and classroom activities: calculate, consult, control, communicate and discover.

3D-Loci Discovery (3D-LD), a web-based system for automatic discovery of geometric loci in the space, is described. Its basic engine is based on the Groebner basis method. 3D-LD can be used to easily (re)discover many loci-related geometric surfaces, and it has allowed us to generalize a recent extension of Simson-Steiner’s theorems due to Guzmán. Being accessible through the web, 3D-LD is platform independent, and can be linked with other graphical tools. It uses the symbolic capabilities of Mathematica and CoCoA, by means of the recently distributed application webMathematica, a Java servlet technology allowing the generation of dynamic web content. A simple description of 3D-LD and some examples of use are available at http://rosalia.uvigo.es/sdge/web/3D/.

In this chapter we present an experimental activity conducted with eight teachers in several high-schools in the South of Italy. The activity deals with the study of some geometric loci and is based on the use of a Dynamic Geometry System (DGS) on the one hand and on properties of geometric transformations on the other hand. Here we explain the motivations at the base of our activity and show the contents, the modalities of building up the activity and the results of the teaching experiment.

In mathematics research and education, harmonious use of high-quality mathematical expressions and mathematically precise artwork is desirable. To produce them, an environment in which several kinds of software (including editing tools, computing tools, and high-quality graphics generators) interact in a collaborative way, is needed. To accomplish this, we have developed Cindy, a plug-in for the popular dynamic geometry software Cinderella, to help mathematicians generate high-quality graphical images. Cindy also enables us to bind the high-performance graphics capability of Cinderella with the powerful computing capability of various computer algebra systems. In this paper, we will show some mathematical artwork to illustrate how the collaborative use of $$\hbox{\LaTeX}$$ , Cinderella, and computer algebra systems via Cindy can enhance the power and convenience of each other. Since all the related software are freely accessible, Cindy should be especially valuable for educational purposes.

This article describes a new algorithm to compute a free resolution of an ideal of (or module over) a commutative ring R combining the Koszul complex with Groebner basis methods. The algorithm computes the resolution of an ideal I via the resolutions of a sequence of subideals \({{(0)=I_0 \subset\ldots\subset I_r = I}}\) differing by one generator each time. The article discusses special orderings and criteria applicable to this algorithm and gives some timings based on an implementation within the computer algebra system Singular.

A natural language for implementing algebraically Expert Systems

The paper was originally motivated by the request to accentuate the meaningful contribution of inequalities in Mathematics Education. Additionally nationwide approved competences such as estimating come to the fore when organizing mathematical contents along some central Big Ideas. Not least the integration of computers enriches the reasonable discussion of inequalities by modern well accepted methodological principles. The freeware MAXIMA is used as Computer Algebra System (CAS) representatively.

While computational technologies are transforming the professional practice of mathematics, as yet they have had little impact on school mathematics. This pioneering text develops a theorized analysis of why this is and what can be done to address it. It examines the particular case of symbolic calculators (equipped with computer algebra systems) in secondary education. Drawing on a substantial program of French innovation and research, as well as closely related studies from Australia and the Netherlands, it provides rich illustrations of the many aspects of technology integration, and of the ways in which these are shaped at different levels of the educational institution. This text offers the first English-language exposition of how an innovative synthesis of the theories of instrumentation and didactics can be used to illuminate the complexities of technology integration. It offers important guidance for policy and practice through its analysis of the central role of the teacher and its identification of key principles for effective didactical design and management. These distinctive features make this book essential reading for researchers, teacher educators, and graduate students in mathematics education and technology in education, as well as for teachers of mathematics at upper-secondary and university levels. This is a revised, English-language edition of D. Guin & L. Trouche (Eds.) (2002) Calculatrices symboliques. Transformer un outil en un instrument de travail mathematique: un probleme didactique (Editions La Pensee Sauvage, Grenoble).

There is an increasing use of and rapid advancements in technologies in education. In particular, there is a take-up of advanced calculators (graphing calculators and calculators with computer algebra system [CAS]) in senior secondary high-stakes examinations. Each year, tens of thousands of students in Australia, Singapore, and other regions, use advanced calculators in mathematics examinations that directly affect their entry into tertiary education. Thus, it was pertinent to investigate the ways in which students use these technologies when learning mathematics, and the factors that influenced them. This study was built upon a bricolage of theories from various fields such as mathematics education, educational technology, learning theories, learning styles, and gender studies. From a review of the literature in the different fields, it was hypothesised that factors such as students’ gender, beliefs about and attitudes toward mathematics and technology, approaches to studying mathematics, and learning preferences, influence their ways of using technology. A mixed methods research design was used, consisting of two parts: a quantitative large scale online survey of 964 Singaporean and 176 Victorian senior secondary mathematics students, and a qualitative small scale study of nine students in a Singaporean school. There were several main results found in this study: (i) How students interact with advanced calculators was influenced by their beliefs about and attitudes toward mathematics and their approaches to studying mathematics. Students’ ways of knowing and learning mathematics best explained their ways of interacting with the calculator. (ii) Students' learning preferences were situated, and the use of advanced calculators was associated with visual and kinesthetic modes and a preference to work cooperatively. (iii) There were regional differences (in favour of Victorian students) in how students use the advanced calculators and their attitudes toward calculators. (iv) There were gender differences (in favour of males) in students' attitudes toward mathematics and advanced calculators, and how they used the calculators. Although there were more student factors with gender differences in the Singaporean than the Victorian data, the effect sizes were larger in the Victorian sample. The findings have implications for mathematics teaching practice, in that deep understanding, intrinsic interest, beliefs about the creative, inter-connectedness and contextual aspects of mathematics are associated with using calculators as a partner and collaborator for learning. Additionally, students tend to employ visual and kinaesthetic modes when learning how to use the calculators. Students were also found to have a preference for working on the calculator cooperatively with friends. Therefore, teachers can employ methods such as encouraging students to try out the calculator keys and to work cooperatively in groups when learning how to use the calculators. Another significant contribution of this study was in the use of Facebook to recruit participants. More research can be done on the participation and response rates associated with this recruitment method. Further research can also be conducted to translate the implications of the study into practical classroom strategies and to investigate longitudinally the dynamic interplay between these factors and students’ interactions with the advanced calculators.

The article is devoted to analyzing and justifying the importance of integrating computer algebra systems (CAS), particularly Maple, into teacher education for pre-service mathematics teachers. The key aspects of utilizing such systems for developing STEM competencies, enhancing research skills, integrating interdisciplinary knowledge, and creating an innovative learning environment are considered. The document highlights the role of Maple in transforming traditional mathematics education into a digital and STEM-oriented paradigm, emphasizing practical approaches to designing educational activities. This study aims to explore the role of Maple-based computer tools in the context of training pre-service mathematics teachers to implement STEM education while studying a discrete mathematics course. The research methods used in the article include theoretical analysis, systematization, and generalization of data from scientific, methodological, and specialized literature. The article addresses the training of pre-service mathematics teachers to implement STEM education by designing their professional activities using the Maple computer mathematics system. The feasibility of integrating Maple into the professional training of teachers as a means of forming digital, research, and interdisciplinary competencies is substantiated. A project activity model is proposed, based on the principles of the STEM approach and utilizing Maple as a tool for modeling, visualization, and analytical processing of mathematical problems. The use of Maple in the training of mathematics teachers promotes the integration of knowledge from related disciplines, characteristic of STEM, and ensures the readiness of future teachers to utilize digital technologies in organizing collaborative learning and project work within the STEM education framework. This aligns with modern trends in mathematics education, which place special emphasis on innovative methods and technologies.

The article discusses three empirical examples of Computer Algebra System (CAS) use in Danish upper secondary school mathematics class with a recent change of teacher. All examples lead to didactical problems surrounding the situation and unclear expectations between teacher and students, involving loss of students’ mathematical skills and confidence, loss of global mathematical perspective, and the students losing sight of the mathematical objects in question. The article is the result of collaboration between two mathematics education researchers and an upper secondary school mathematics teacher, who experienced severe difficulties when taking over a class from another teacher. CAS was experienced as a crucial part of and reason for these difficulties. As a means for investigating the potential reasons behind the difficulties, a selection of constructs from the Theory of Didactical Situations (TDS) is applied. In particular, it is observed that unclear contractual relations about the role of CAS bring with them misguided winning strategies and metacognitive shifts, eventually causing the students to ‘lose the game’.

Relying on the students' cultural background to teach mathematics may be a strong incitement for the learning process. This background can include artistic creations, items from the news, among others. Because of the strong presence of space related news (in particular the almost simultaneous launching of three spacecrafts towards Mars), we present activities in a technology-rich environment, using both a Dynamic Geometry System and a Computer Algebra System. We start from a simplified model of planetary orbits and building new knowledge on plane curves. Interesting byproducts are the discovery of the strong influence of changing numerical approximations, non – trivial aspects of the study of periodicity, the plotting of interesting curves and their connections with existing pieces of visual art, including architecture and mandalas. The educative aspect of these activities is also discussed and is presented from the point of view of STEAM Education.