Computer Algebra Systems

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.

(PhD Dissertation, Stanford Mathematics 2001)
What sort of geometric creation and performance can or cannot be supported in a writing technology that spans freehand sketching, manipulable graphics, text, symbolic computation and simulation? How do we work with and gain intuitions about supposedly abstract things like a Riemannian manifold or an infinite-dimensional function space by manipulating material marks using chalk, pencil or computer? How do we exteriorize thought using the technologies of writing and sketching when it is about processes or objects that could not be embedded in a naive space-time?

I analyze in detail what differential geometric work can or cannot be performed easily in a hybrid medium provided by a generalized multi-modal writing and sketching technology. The critical part of this project is informed by insights from literary and performance studies as well as the mathematical sciences. Focusing on differential geometry through the lens of practice, I examine a-linguistic semiotic usage, and through the lens of phenomenological investigation, I examine how geometers work with continuous, smooth, or infinite structures and processes.

This investigation promises both pragmatic as well as conceptual results. Concretely, this study provides a design for future technologies of mathematical writing that will better augment the working practices of differential geometers.  Conceptually, my proposed materialized phenomenological view of mathematics as a performance practice eliminates three problems in philosophy and in critical studies of symbolic systems: (1) how do we represent continuity and smoothness using discrete symbolic systems? (2) how do we represent infinite things or processes? (3) why do symbolic systems have explanatory or generative power? These conundrums have occupied philosophers of mathematics, and have driven some to propose strict finitist or radically fictionalist theories of mathematics. Moreover, approaches to their resolution underwrite all the technologies that serve mathematical practice today, so these problems fundamentally affect the design of technologies for the working mathematician as well.

There is, in France, a long-term tradition of research in the didactics of mathematics. In this paper, we revisit this tradition using, as a specific lens, the research carried out on the educational use of digital tools and resources for teaching and learning mathematics. We first briefly introduce this tradition and the three main theories at the base of it: the theory of didactical situations, the theory of conceptual fields, and the anthropological theory of the didactic. Then, considering three different technological lenses, i.e., dynamic geometry environments, computer algebra systems, and digital resources, we show how these theories and the long-term connection established in this tradition with the fields of cognitive ergonomics and computer sciences have influenced technological research and its outcomes. We also show that, conversely, didactic technological research has led to original and influential theoretical constructions, such as the instrumental approach and the documentational approach to didactics, and that it has contributed in a substantial way to the opening of this didactic tradition to other didactic cultures and other communities, beyond the didactic one. Keywords: anthropological theory of the didactic; computer algebra system; didactical engineering; didactic technological research; digital resources; documentational approach to didactics; dynamic geometry environment; instrumental approach to didactics; theory of conceptual fields; theory of didactical situations

In this paper, high school mathematics teachers’ views on computer-assisted mathematics instruction and on computer algebra systems (CAS) were explored in two stages, before and after they were trained on it. This study is based on the case study model of the qualitative research method. Semi-structured interview forms constitute the data of the study. Some codes and themes were established with the descriptive and content analyses of the data. The results show that teachers had positive opinions about computer-assisted mathematics instruction before they were trained, yet a majority of them was not able to use it in their classes for several reasons. However, it is seen that none of the teachers except for one knew about computer algebra systems (CAS). It is understood after the training that all the teachers wanted to use computer-assisted mathematics instruction in their classes, and thus they needed to use Mathematica, which is a computer algebra system, during the process. Likewise, while most of the teachers did not want their students to interact with computers or tablets in their classes before the computer-assisted mathematics instruction training, almost all of them had a favorable opinion of it after the training.

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In this paper we survey some of the developments with regards to technology-enhanced support of mathematics learning and teaching. Coverage is especially given on how to design systems that work effectively on the Web, particularly in the fields of economics and business studies. This is a rapidly evolving landscape in which general trends in e-learning have raised expectations of the electronic tools

Studies handling the lights and the colors as key components of computation attracted the scientists and engineers since these studies are potentially applicable to the signal processing through optical interconnections between electronic devices. Here, a novel optical computing model is proposed by modifying the known optical parallel logic gates, after employing the printed colors as the input/output data, instead of shadowgram images projected on a screen. The proposed approach allows spontaneous and parallel Boolean operations by simply overlaying the colors printed on films or duplicating the prints on papers. By setting a limited number of color prototypes with known CIELAB color coordinates, prediction of color changes due to duplicated color printing and/or film-on-paper overlaying of the printed colors was performed through Boolean “AND” operation and their experimental confirmation after actual color reading was also performed. Furthermore, possible applications of this CIELAB-coded printable logic gate system in natural computing and development of novel color barcodes were discussed.

Mathematics is one the most important subjects during school days of the world educational institutes. And handling mathematical objects at user’s facilities easily is much more important for students to learn not only philosophy of natural science but also some kinds of social science. Almost all the students had better study mathematics efficiently and effectively anytime and anywhere, i.e. Ubiquitous Learning. This study is one of our ambitious projects to realize mobile mathematical expression processing environment for scientific learning very easily and more comfortably. A part of our prototype system has been designed as simple-structured Lisp system and another one has been implemented together with GNUPLOT sub system. Now we have been customizing our prototype system into Smart Mobile phones in order to realize “Ubiquitous Learning.” With these facilities, our system can provide suitable services of web-based mathematical expression processing for higher education.

Abstract:- In this study, it is observed how a CAS programme can help student to increase their mathematical understanding. CAS have a lot of great properties that can be used for educational purposes. For example, you can produce 2D, 3D and even animated graphs. Symbolic calculations and some advanced mathematical level calculations can also be done. CAS is a name of a group of software such as Mathematica, Maple, Derive and Mathlab. In this study, Maple software has been chosen to design a discovery learning environment. A group of students has been directed to discover the relations among calculus concepts by using 3D graphs produced by Maple. A semi-structured interview has been employed in this study to collect data. At the end of this paper, mathematical acquirements of students have been discussed.

In the last two decades, a number of researchers have been engaged in the study of natural computing systems that employ physical, chemical, and biological properties as direct media for manifesting computations. Among such attempts, studies focusing on the use of lights as key computation components in particular have attracted the attention of researchers and engineers, since these studies are potentially applicable to signal processing through optical intercon-nections between electronic devices. Our research team has recently been engaged in the study of a novel color-based natural computing model. Our recent works included using CIELAB-coded colors on printed-paper to compute Boolean conjunctions (AND operations). In this study, we performed Boolean operations based on CIELAB-coded colors by placing color-printed films over aluminum-coated reflectors with and/or without color. The results of the operations were gathered by testing the color codes printed on the films for negation or highlighting. This type of CIELAB-based color computing has a wide range of potential applications, such as a method for security or access control to secured systems. Such applications could match paired color keys on which the arrays of color codes could be printed and optically computed.

TI-Nspire CAS technology by Texas Instruments, USA, is the only dynamic educational system in the world that enables teaching and learning of all levels of secondary school mathematics. In addition, it includes data collection application for main science disciplines. The software component of the system supports geometry, algebra, precalculus and calculus, statistics and data collection with probes. In the second article, readers learn about applications Lists & Spreadsheet and Data & Statistics to compare properties of box plots and histograms and to analyze several sets of data. Modeling of random processes is also discussed.

In this paper we survey some of the developments with regards to technology-enhanced support of mathematics learning and teaching. Coverage is especially given on how to design systems that work effectively on the Web, particularly in the fields of economics and business studies. This is a rapidly evolving landscape in which general trends in e-learning have raised expectations of the electronic tools to go beyond merely providing mechanical functions: nowadays the teacher expects them to help organise their teaching and even enhance their teaching methods. For the student, there is increasing focus on the learner experience, which seeks a wider range and availability of e-learning support tools together with greater ease of use. Advances in technology mean that options keep changing, setting many challenges for system developers, so this paper seeks to indicate the broad features in these changes, with particular regard to future planning. These general considerations are subsequently brought into focus in later sections of this paper by addressing the question of how e-learning systems may provide judicious support for the uniquely human abilities of solving problems. In this case visualisation is an excellent aid, but any tools provided need to draw on a wide range of online resources that are readily available so as to properly establish the learning context. Finally, we briefly introduce into this context two new systems for teaching mathematics: VisualEconoMath, an integrated environment comprising automated tools for deriving solutions to prescribed mathematical problems and How2SolveIt, which supports the quintessential human learning processes surrounding questions and answers, in this case between students and lecturers.

* Introduction
* Basic Calculations
* Parentheses, Brackets, and Braces
* Symbolic Vs. Numerical Computation
* Names and Capitalization; Basic Functions
* Algebraic Manipulation
* Plotting Graphs
* Variables
* Functions
-> Set (=) Vs. SetDelayed (:=)
-> Applying Functions with @ and //
-> Piece-wise Functions
* Equations
* List
* Calculus
* Solving DEs
* Exercises

, an open web-based tool to determine equations of geometric loci specified using Cabri Geometry and 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

Extending previous results we show that there are no Petrov type III space-times on which either the Weyl neutrino equation or Maxwell's equations satisfy Huygens' principle. We prove the result by using Maple's NPspinor package to convert the five-index necessary condition obtained by Alvarez and Wunsch to dyad form. The integrability conditions of the problem lead to a system of polynomial equations. We then apply Maple's grobner package to show that this system has no admissible solutions.

Abstract: This study compares the respective achievement of students in an integrated algebra course taught with two different types of handhelds over a period of one year. The experimental group was taught with TI-Nspire [TM] handhelds and was compared to the ...

Experiências francesas realizadas nos três últimos anos no ensino de matemática em ambientes complexos com calculadora - em especial com Tl-92 - permitem tirar algumas lições:
A introdução deste tipo de calculadoras não simplifica o trabalho do profes-sor, nem dos alunos;
- Requer uma nova organização do espaço de ensino e um novo estilo de gestão do tempo de aprendizagem. Nós devemos argumentar que, sob essas condições, trabalho em ambientes complexos com a calculadora pode levar a uma transformação das relações dos estudantes com a matemática (dando prioridade ao lado criativo do trabalho). Além disso, pode-se observar uma transformação das relações coletivas dos estudantes ao conhecimento (dando maior importância aos aspectos sociais) (Portuguese translation in 2015, revisão 2021)

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"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 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."

This is a short piece from the proceedings of the 23rd international conference on Applications of Computer Algebra. The main claim of this paper is that mathematics teachers need to adjust their epistemological stance in order to facilitate geogebra-based mathematical thinking.

We assess the current state of research in the application of computer aided formal reasoning to computer algebra, and argue that embedded verification support allows users to enjoy its benefits without wrestling with technicalities. We illustrate this claim by considering symbolic definite integration, and present a verifiable symbolic definite integral table look up: a system which matches a query comprising a definite integral with parameters and side conditions, against an entry in a verifiable table and uses a call to a library of lemmas about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. We present the full model of such a system as well as a description of our prototype implementation showing the efficacy of such a system: for example, the prototype is able to obtain correct answers in cases where computer algebra systems [CAS] do not. We extend upon Fateman's web-based table by including parametric limits of integration and queries with side conditions.

This study compares the respective achievement of students
in an integrated algebra course taught with two different types of handhelds over a period of one year. The experimental group was taught with TI-Nspire™ handhelds and was compared to the control group taught with TI-84 Plus graphing calculators. The teachers of each group received on-going professional development in the same format. Student achievement was measured via a midyear department test; Fall and Spring semester grades; and New York State Regents exam scores and passing rates. Results indicated that the group taught with the TI-Nspire outperformed the other group in all assessments, including passing rates on Regents but not on the Regents exam scores. Further analysis indicated that girls outperformed boys in an identical pattern. No significant differences in achievement by race were observed.

TI-Nspire CAS technology by Texas Instruments, USA, is the only dynamic educational system in the world that enables teaching and learning of all levels of secondary school mathematics. In addition, it includes data collection application for main science disciplines. The software component of the system supports geometry, algebra, precalculus and calculus, statistics and data collection with probes. In the first article, readers become acquainted with applications Graphs and Calculator using several examples of solving systems equations and inequalities, both graphically and algebraically.

There is, in France, a long-term tradition of research in the didactics of mathematics. In this paper, we revisit this tradition using, as a specific lens, the research carried out on the educational use of digital tools and resources for teaching and learning mathematics. We first briefly introduce this tradition and the three main theories at the base of it: the theory of didactical situations, the theory of conceptual fields, and the anthropological theory of the didactic. Then, considering three different technological lenses, i.e., dynamic geometry environments, computer algebra systems, and digital resources, we show how these theories and the long-term connection established in this tradition with the fields of cognitive ergonomics and computer sciences have influenced technological research and its outcomes. We also show that, conversely, didactic technological research has led to original and influential theoretical constructions, such as the instrumental approach and the d...

The article is devoted to the current problem of improving the level of students' professional competences. The main attention is paid to the preparation of teachers of Informatics and Mathematics. The article presents the list of students' competences that could be improved at the studying process of Numerical Analysis and ways of implementation e-learning for obtaining this educational goal.The paper describes the main components of e-learning course elaborated on the base of CLMS Moodle and corresponding learning activity of students: using learning materials elaborated on the base of Computer Algebra System Maple; filling glossaries of the course; problem solving; assessment of the level of competences, using tests.

The primary purpose of this study is to provide an immediate connection to analytics of biological functions and visualisation. We introduce a visual framework in the environment of a main computer algebra system (CAS), Mathematica, to picture variations and evolution schemes in gene expression. Our computational approach constructs snapshots for gene expression profiles, with the advantage of being selfevident, concise and clear. A variety of colours is employed, beyond the conventional heat map colours. We provide dynamic options to facilitate comparisons among treatments, several colour choices to perform reference gene selection tests and, thus reveal the variation of gene expression through time and/or among treatments, or genes. The performance of the programming techniques in selected case studies concerning genes involved in embryonic development of common sole (Solea solea), is presented. This work could assist researchers in biosciences with suggestions to specific gene expression mapping.