Journal of Computer and Mathematical Sciences, Vol.7(3), 122-129, March 2016 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319 - 8133 (Online)

Integrating New Technologies and Tools in Teaching and Learning of Mathematics: An Overview

Momin Fasiyoddin Inayat1 and Shaikh Naeem Hamid2

1Department of Mathematics, Milliya Arts, Science and Management Science College, Beed, Maharashtra -431122, INDIA. email:[email protected] 2Department of Computer Science, Milliya Arts, Science and Management Science College, Beed, Maharashtra -431122, INDIA. email:[email protected]

(Received on: March 26, 2016)

ABSTRACT

Mathematics has remained a difficult and unpopular subject for most of the students. This is despite its importance almost in all careers, especially in the science and technology. The difficulties in learning mathematics might be due to the teaching methods employed by the educators. With the advancements in the Information Communication Technologies (ICTs), the nature of teaching and learning in mathematics is expected to change. The use of new technologies in the classrooms makes higher level mathematical activities accessible to students and also makes learning more fun, interesting, and more effective. Thus, technology can enhance students learning process by presenting content graphically, symbolically and numerically without spending extra time to calculate the complex computational problems by hand. In this paper we present an overview of the latest technologies used by the educators to make teaching and learning in mathematics more effective, student-centric and dynamic.

Keywords: Technology, ICT, Teaching, Learning, Mathematics.

1. INTRODUCTION

The developments in technology and tools in the past century have brought changes that transformed education. Mathematics education has also seen some of the most drastic changes in recent past. Mathematics as a subject has remained difficult and unpopular for most of the students, despite its importance in almost all careers, especially in the science,

March, 2016 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Momin Fasiyoddin Inayat, et al., J. Comp. & Math. Sci. Vol.7 (3), 122-129 (2016) 123 engineering and technology fields. The mathematics makes bad memories for most of the generations which studied with rules, the compass and repeated exercises. The emergence of new technologies in education has changed the teaching and learning terrain of mathematics. Students need to understand the significant concepts of mathematics and comprehend the meaning and relatedness of these concepts so that they can apply these to problems in their everyday lives. Since some students are gifted and there are still many students who struggle in learning mathematics. Using technology in teaching and learning would improve students understanding of basic mathematics concepts and improves the way mathematics should be taught. This adds a new perspective in mathematics teaching from traditional approach to teaching with Information Communication Technology.1 Today most educators support students to actively involve in constructing their own understanding of mathematical ideas by using the latest and emerging technologies.2 The US National Council of Teachers of Mathematics (NCTM), one of the largest mathematics teacher’s organization in their position statement claims that: “Technology is an essential tool for learning mathematics in the 21st century and all schools must ensure that all their students have access to technology.”3 Most of the school and colleges now offers ICT facilities in teaching and learning of mathematics. The digital technology has been used in mathematics classrooms since the introduction of simple four-function calculators in the 1970s. Since then, modern digital technologies including computers with increasingly sophisticated software, more advanced graphics calculators integrating graphical and symbolic manipulation, statistical and dynamic geometry packages, and virtual learning environments offered by web-based applications have innovated the teaching and learning process of mathematics.

2. BENEFITS OF INTEGRATING ICT IN MATHEMATICS

Research evidences tell that integrating ICT into the mathematics teaching has enormous benefits. Here we present some of the benefits:

2.1 Immediate feedback to the student’s efforts: When using ICT tools students gets instant feedback to their efforts as opposed to using the traditional methods where feedback takes time.4

2.2 Motivation, Interaction and Cooperation: The use of ICT tools especially multimedia tools provides a good environment for pupils to work in groups and have an interaction on a given task. This ultimately helps to generate a motivation and collaboration among pupils.5

2.3 Improved Skills: The use of technology in mathematics education helps students in following skills6:  Use of software like drill and practice can help young children to develop skill such as counting and sorting.  Proper use of computer games can also help in developing mathematical skills of young children.

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 With the use of technology students can work in collaboration and communicate their understanding and knowledge in different ways with others, this will ultimately improve their communication skills.

2.4 Active participation: The interactive technology used in mathematics helps students to become active partners in the learning process through experimentation, demonstration, exploration, and calculation. This helps students in the process of developing their understanding of a topic.7

2.5 Integrate theory and practice into one: Subject like discrete mathematics is necessary for a computer scientist and software engineer as it is centered on correctness, logic and algorithm. But students find it rigorous and challenging as it need reasoning and logic both at the same time. Wu. Xiuguo8 in his study found that teaching discrete mathematics with experiments using technology can integrate theory and practice into one and enhance the comprehension of discrete mathematics to the students.

2.6 Teaching Mathematics Better and Teaching Better Mathematics: Integrating technology into the classroom can improve mathematics teaching and also introduce better mathematics.9 With the use of technology teachers can focus more on developing ideas, exploring consequences, justifying solutions, and understanding connections which is the heart of mathematics.10

3. ICT TOOLS IN TEACHING AND LEARNING MATHEMATICS

There are many ICT tools available which are readily being applied in teaching and learning of mathematics in schools and colleges. The ICT tools include hardware tools such as OHP, LCD Projectors, Handheld devices, advanced calculators, PDAs etc. There are enormous software tools for effective teaching and learning of mathematics, some of these are graphing tools, dynamic geometry software, systems, spreadsheets, and online tools for learning. In this paper we only present an overview of the most popular tools and Medias used for the effective teaching and learning of mathematics.

3.1 Dynamic graphing tools

Dynamic graphing tools are the effective tools for visualization and representation of relationships between entities in readable, scalable and effective diagrammatic form. There are so many dynamic graphing tools available for effective visualization of relationships between entities. Here we provide details of some of the popular dynamic graphing tools available for building the dynamic graphs:

i. Google Charts: Google Charts are simple to use, powerful and free interactive tools for creating visualization of data for browsers and mobile devices. Google Charts are perfect tools to visualize data on websites. The chart gallery of Google Charts provides number of template chart types including simple line charts to complex hierarchical tree maps.

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ii. D3.js: D3.js is an effective JavaScript for visualization and manipulation of documents based on data. It allows tagging arbitrary data to Document Object Model (DOM), and then applies data-driven transformations to the document. For example, D3 can be used to create an HTML table from an array of numbers or an interactive SVG bar chart with smooth transitions and interactions.

iii. amCharts: amCharts is an advanced library for data visualization. It includes number of charting options from simple line, bar, column, area, step, step without risers to more advanced charts.

iv. GraphStream: GraphStream is a Java library for generating, importing, exporting and visualizing the dynamic graphs. GraphStream provides several classes of graphs that allow modeling of directed and undirected graphs, 1-graphs or p-graphs. It also allows to store any data attribute on the graph elements i.e. numbers, string, or any objects. It can be customized with CSS style sheets for graph visualization.

3.2 Dynamic Geometry Software (DGS)

Dynamic Geometry Software is an effective mathematical tool for interactive representation and manipulation of geometric objects. One of the characteristic features of such programs is to build a geometric model of objects, like points, lines, circle etc., with the dependencies that relate these objects to each other.11 Dynamic geometry technology can be used to maximize students learning in geometry and enabling them to visualize geometry concepts being studied. Here is the summary of some of the popular dynamic geometry packages available worldwide.

Table 1: Summary of Dynamic Geometry Software

Software License Uses/Notes It includes tools for interactive geometry, algebra, statistics, and for exploring Euclidean geometry. This package has got number of awards for its excellence in teaching GeoGebra Proprietary and learning, it includes, National Technology Leadership Award 2010, MERLOT Award for Exemplary Online Learning Resources. Geometry Dynamic drawing and automated geometry proving and discovery. Free Expert (GEX) .a.R. Free Interactive Geometry Software in Euclidean and Non-Euclidean Geometry. Software package for dynamic geometry in 3D. It won a German Government award of Archimedis outstanding educational software in 2007. It is used to trace the movement of point, line, Geo 3D segments and circles and generate locus of lines and surfaces. Geometer’s Popular commercial interactive geometry software for algebra, calculus and other areas. Proprietary Sketchpad Interactive Online Geometry Software and analysis of Euclidean, Geometry, Spherical Cindrella 2.0 Proprietary Geometry, Hyperbolic Geometry. Includes Physics simulation engine and scripting language. Professional Geometry learning software built using live geometry engine. Helps visualize Tabula Proprietary and solve geometry problems. Commercial Interactive Geometry Software for teaching and learning of Geometry and Cabri Proprietary Trigonometry. GAViewer Free Interactive Geometric Algebra program with OPENGL visualization.

March, 2016 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 126 Momin Fasiyoddin Inayat, et al., J. Comp. & Math. Sci. Vol.7 (3), 122-129 (2016) 3.3 Computer Algebra Systems(CAS)

A is a software package used in manipulation of mathematical expressions in order to automate the algebraic manipulation tasks which are tedious and sometimes difficult to solve manually. The basic difference between Computer Algebra Systems and a traditional calculator is the ability of CAS to deal with symbolically instead of numerically. The specific capabilities and uses of these systems vary greatly from system to system, yet the purpose remains same i.e. manipulation of symbolic equations. Computer algebra systems may be divided in two classes: the specialized and general purpose CAS. The specialized CAS is designed to deal with group theory, number theory, or teaching elementary mathematics. The general purpose CAS is helpful for users who need to manipulate mathematical expressions in any scientific field.

Table 2: Summary of Dynamic Algebra Systems

Software License Uses/Notes Mathematica Proprietary General Purpose CAS Proprietary General Purpose CAS Proprietary Numerical Software with some CAS capabilities General Purpose CAS originally specialized in group theory. Works with elements of Magma Proprietary algebraic structures rather than non typed math expressions. Proprietary The oldest general purpose CAS. Wolfram Online CAS with step by step solution. Proprietary Aplha Fermat Proprietary Specialized CAS for resultant computation and linear algebra with polynomial entries. Specialized CAS for group theory. It provides a programming language for implementing GAP Free algebraic algorithms. Free General Purpose CAS PARI/GP Free Specialized CAS for number theory. Mathomatic Free , calculus, complex numbers and polynomial manipulations. Free Specialized CAS for algebraic geometry and commutative algebra. Magnus Free Specialized CAS for group theory. SciLab Free Matlab alternative Specialized CAS for polynomial computation, with special emphasis on commutative and Free non commutative algebra, algebraic geometry and singularity theory. SAGE Free Numerical computation, Statistics and Image Processing. KANT/KASH Free Specialized CAS for algebraic number theory. COCOA Free Specialized CAS for commutative algebra. Axiom Free General Purpose CAS

3.4 Numerical Analysis and Statistical tools

Numerical analysis tools are the computer programs used to design, analyze and implement the algorithms for getting the numerical solution to the problems containing continuous variables. Statistical tools are used to analyze large volume of data and derive some meaning out of it for further analysis and decision making.

March, 2016 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Momin Fasiyoddin Inayat, et al., J. Comp. & Math. Sci. Vol.7 (3), 122-129 (2016) 127

Table 3: Summary of Numerical Analysis and Statistical Tools

Software License Uses/ Notes

It is a high-level language primarily designed for numerical computations. It provides solutions GNU Open for linear and nonlinear problems, other numerical experiments. It also provides data visualization Octave Source and manipulation. It is similar to Matlab. R is a language for statistical computing and graphics. It provides wide range of statistical operations like linear and nonlinear modeling, classical statistical tests, time-series analysis, R Free classification, clustering etc and also provides graphing techniques. It provides a well designed publication plots including mathematical symbols and formulae. ROOT is a modular scientific framework. It provides all the capabilities to deal with statistical ROOT Free analysis, huge data processing, visualization and storage. It can be integrated with other languages like python and R. MaxStat is a easy to use statistical software. It can be used to perform more than 100 statistical tests and makes easy the interpretation result. It also provides a facility to create high-quality MaxStat Proprietary graphs for visualization. It supports linear and nonlinear regression, multivariate analysis, correlation and time series analysis. The Lite version MaxStat is free for non-commercial use. It is powerful tools for visual quantitative modeling. It offers a list of statistical functions for Analytica Proprietary analyzing sample data. Also provides built in graphics to visualize uncertainties using probability band, density functions and cumulative functions. GAUSS is a fast programming language for mathematical and statistical analysis. It is GAUSS Proprietary widely used by engineers, scientists, biometricians, statisticians, and financial analysts. It provides an easy to learn, powerful and versatile learning environment. It provides a menu driven interface for novice and a powerful command language interface for more experienced user. It is used for vast range of statistical operations (basic statistics, GenStat Proprietary microarray analysis, time series, regression, spatial analysis.) GenStats Discovery Edition is free for non-commercial use. It is an interactive tool used for both education in statistics and simulation and solution of Stastical Free statistical problems. A GUI is provided to easily understand complex statistical relations. Lab MATLAB is a multi-paradigm numerical computing environment. It is also used as a fourth generation programming language. Matlab allows matrix manipulations, function and data Matlab Proprietary plotting, creating user interfaces etc. It is primarily designed for numerical computation but optional toolbox can include MuPAD symbolic engine for symbolic computing, simulink for simulation etc.

3.5 WME (Web-based Mathematics Education)

WME is a modern online distributed system for mathematics education. The WME system provides an extended set of facilities for effective online teaching and learning. Web- base learning can help in extending the reach of education and also expand its impact and influence. WME provides classroom ready lessons which are well planned, effective, easy to use, interoperable and interesting. Lesson pages are designed such that students can understand and explore their mathematical knowledge through hands-on activities.12 WME is a complete learning infrastructure focusing on mathematics education and integrating lessons, special manipulative, assessment and evaluation facility, teacher-student interaction environment for effective and interesting teaching and learning of mathematics as a subject. WME architecture contains number of active components for effective delivery of services to the students, it includes:

March, 2016 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 128 Momin Fasiyoddin Inayat, et al., J. Comp. & Math. Sci. Vol.7 (3), 122-129 (2016)

Figure 1: Working of WME12

 Content-Markup: Mathematics education contents are prepared using specially designed MeML language and these tags can be combined with HTML and MathML elements for effective content preparation.  Back-end Support: MeML interpreter process MeML pages at server side which dynamically transform each MeML page in to regular document suitable for standard browser. The MeML interpreter invokes WML server for building the web pages delivered to the client as specified by MeML tags to obtain embedded contents. The resultant pages contains the appropriate client-side programming (java applets, CSS, , DHTML, XML) to provide support for dynamic and interactive learning. So, MeML interpreter and WME server constitute a back-end for WME.  Front-end Support: on the client side any with MathML support can be used with some additional features that may be added through java applets, CSS, javascript programming, and browser plug-ins. The IBM publishing group has specially designed a browser plug-in Techexplorer that format and display Tex/LaTeX encoded documents.

3.6 Spreadsheets

Spreadsheets were invented in the 1970’s and since then used as a tool for mathematical, scientific, financial and statistical calculations. It allows entering numerical and textual data in rows and columns for performing various operations. The use of formula commands in spreadsheets gives them the power of programming, allowing many financial and statistical projections to be made. Due to all these capabilities the spreadsheets have become an important part of many curriculums at different levels of education. They have been mainly used in mathematics education as a tool for students to understand the mathematical concepts such as exploring functions, patterns, probability, and statistics, mathematical modeling, and geometric transformations. The use of spreadsheets in mathematics education has number benefits as: 1. Complex calculations can be performed with ease and quickly. 2. Calculations can be automatically updated by just changing the variables in the previously defined formulas. 3. A good visualization of data can be presented by creating graph charts from the range of data. 4. ‘what..if’ and goal seek tools can be used effectively for probability analysis.

March, 2016 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Momin Fasiyoddin Inayat, et al., J. Comp. & Math. Sci. Vol.7 (3), 122-129 (2016) 129 4. CONCLUSION

This paper considered the role of latest technologies in the teaching and learning of mathematics. The paper also provides the number of benefits of integrating technology in teaching and learning of mathematics as evident from various researches. Review of various research papers tells that integrating ICT in teaching and learning of mathematics has greater impact on complete education system. The ICT tools have changed the way mathematics has been taught and brought new dimensions in the process of teaching and learning. This paper mainly focuses on the technological tools available for effective teaching and learning of mathematics.

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March, 2016 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org