Not another computer algebra system: Highlighting wxMaxima in calculus

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.

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.

Promoting students’ graphical understanding of the calculus

Exploring Third-Order Chromatic Aberrations of Electron Lenses with Computer Algebra

What task should be played by symbolic mathematical computation facilities in Pure and applied Mathematics, scientific and engineering “problem solving environments”? Drawing upon standard facilities such as numerical and graphical libraries, symbolic computation should be useful for the creation and manipulation of mathematical models. Symbolic representation and manipulation can potentially play a central organizing role in PSEs since their more general object representation allows a program to deal with a wider range of computational issues. Symbolic processing power of software as MathCAD (MathCAD explorer version 8) is used to demonstrate the basics as well as its application dispersion of thermoelastic waves.

Examples of problems of motion stability solved with the help of computer algebra systems (CAS) are presented. The authors have experience in developing and applying problem-oriented systems of symbolic computations and software packages for solving problems of dynamics of multi-body systems. The algorithms under consideration are implemented (completely or partly) with the aid of up-to-date CAS. They are intended for inclusion in the ``Stability'' package of symbolic computation. Bibliography: 6 titles.

Maple is a computer algebra system and programming language that can be used for numerical computations, solving equations, manipulating symbolic expressions, plotting, and programming, just to name a few of the basics. APPL is, simply, a set of supplementary Maple commands and procedures that augments the existing computer algebra system. In effect, APPL takes the capabilities of Maple and turns it into a computer algebra system for computational probability. This chapter contains guidelines for using Maple, and discusses the Maple commands that are used in APPL programming. After reading this chapter, an APPL user will have the knowledge necessary to modify the APPL code to meet his or her particular needs. We will start with a discussion of basic numeric computation, then advance to defining variables, symbolic computations, functions, data types, solving equations, calculus and graphing. Then we will discuss the programming features of Maple that facilitate building the APPL language: loops, conditions, and procedures.

This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘ a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems- seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence – shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students.

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).

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.

If the use of a computer algebra system (CAS) is to be meaningful and have an impact on students, then it must be grounded in good pedagogy and have some clearly defined goals. It is the authors' belief that an important goal for teaching mathematics with the CAS is that courses be designed so that students can become active participants in their learning experience, planning the problem-solving strategies and carrying them out. The CAS becomes an important tool and a partner in this learning process. To this end, here the authors' have linked the use of the CAS to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.

This chapter reports the responses of 1088 students in Victoria, Australia. The Mathematics and Technology Attitudes Scale (MTAS) for secondary students was used to monitor five variables relevant to learning mathematics with technology: attitudes to learning mathematics with CAS, mathematics confidence, confidence with technology, and behavioural and affective engagement in mathematics learning. Principal Components Analysis, ANOVA with post hoc comparisons, and Structural Equation Modelling (SEM) techniques were used for the analysis of students’ responses. The findings revealed that there is a complex nexus of relationships between secondary mathematics students’ mathematics confidence, technology confidence, attitude to learning mathematics with CAS, affective engagement and behavioural engagement, achievement, CAS experience, gender, year level, and learning setting. Statistically significant differences were found between a number of the variables and overall students demonstrated higher levels of technology confidence compared to their attitudes to learning mathematics with CAS.

Mathematical models of biochemical reaction networks involve nonlinear differential equations with a number of parameters whose values are generally not known a priori. Analysis of such networks includes both analytical and numerical work. The analytical part of such analysis often requires manipulation with massive algebraic expressions that may become challenging due to the inherent nonlinearities and unknown parameters in the model. Computer algebra systems (CASs), which are also capable of performing calculations with symbols and to generate ready‐to‐use computer codes for numerical computations, are excellent tools in such analysis. The present work gives a brief description of how we can benefit from computer algebra to overcome such difficulties and combine symbolic and numeric computation techniques in such analysis with several illustrative biochemical reaction networks. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

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.

GLI (Geometric Locus Identifier), an open web-based tool to determine equations of geometric loci specified using Cabri Geometryand The Geometer's Sketchpad, is described. A geometric construction of a locus is uploaded to a Java Servlet server, where two computer algebra systems, CoCoA and Mathematica, following the Groebner basis method, compute the locus equation and its graph. Moreover, an OpenMath description of the geometric construction is given. GLI can be efficiently used in mathematics education, as a supplement of the locus functions of the standard dynamic geometry systems. The system is located at http://nash.sip.ucm.es/GLI/GLI.html .