**EVLM Teacher's Guide**

Over the last decade advances in computer technology have taken place at an ever increasing rate. These advances have been on two fronts: the price of computer technology has reduced significantly with an equally significant increase in the power of computer hardware. As a consequence, most students throughout Europe either own, or, through their university, have access to, high specification personal computing. This presents teachers with opportunities for introducing new ways of teaching their students. These new opportunities are particularly attractive in the subject of mathematics. There are a number of ways in which the power of computing can be harnessed to enhance the teaching and learning of mathematics.

The first is with regard to visualisation. Many mathematical concepts are made much more readily accessible and understandable when they are visualised. The graphical capabilities of computers provide a way in which such visualisations can be presented to students very easily. Indeed, it is a straightforward matter for students to generate their own visualisations.

A second advantage is the possibility of removing the distraction of large amounts of manipulation. There is evidence that learners sometimes lose sight of the key concepts they are trying to learn when large amounts of manipulation are associated with them. Completing the manipulation can become the learner's goal rather than understanding the mathematical principles. When used effectively, technology can remove this distraction and allow users to concentrate on understanding the concepts.

A third benefit of using technology is that it makes it possible for students to tackle much larger scale (and therefore more realistic) problems. Only the bravest student would try to solve 6 simultaneous linear equations armed only with a pencil and paper. But, with computer technology to call on, a problem with a hundred or more variables is a fairly routine challenge.

These three benefits can all be realised, to some extent at least, by the use of standard software such as the humble spreadsheet. In addition, however, there are a range of specialist applications that utilise the power of contemporary computers to present mathematics in a far more coherent way. This Guide will concentrate on introducing those which are relevant to educational use. Many of these packages have been specifically designed for educational purposes. Other packages were originally designed to be used by professional mathematicians who need the support of high powered symbolic manipulation or large scale computation in order to perform their work. Whilst not specifically designed for educational purposes, these packages can easily be integrated into educational programs.

In an ideal world, all teachers would have access to all of the available mathematical software packages in order to make their own comparisons. In practice, apart from those which are offered by the developers for free, it is financially viable to use only one or two. In order to provide some guidance for prospective users, the merits of a range of software packages are discussed and brief examples of syntax and screenshots given to provide a taster for the reader.

Chapter 2 discusses the change in the abilities of new intakes in engineering at university level, and why more emphasis should be placed on the use of computers for demonstrating mathematical concepts. Some personal experiences of the author are used for suggesting how improvements may be implemented.

*GeoGebra* is a free interactive software package which has been available since 2001. The project was originally started at the University of Salzburg by Markus Hohenwarter, and is now continuing at Florida Atlantic University. The software has received several European awards for excellence. It is written in Java and available from www.geogebra.org. As its name suggests, *GeoGebra* is primarily for the demonstration of geometric and algebraic ideas, but it can also be used for finding derivatives and integrals of functions. Apart from having the advantage of being free, it is also easy to use, and allows users to contribute their ideas and upload material to the *GeoGebra* website. There is a growing community of *GeoGebra* users who regularly contribute resources to the site. Chapter 3 takes the reader through the download, installation and launch procedures of *GeoGebra* then introduces the various control buttons on the *GeoGebra* toolbar. Chapter 4 demonstrates design of simple applets to produce straight line and quadratic graphs.

The computer algebra package *Derive* was developed in the 1970s and has gained a strong position in schools in many countries, notably Austria. Although *Derive* has recently been superseded by *TI-Nspire* as Texas Instruments' software of choice for graphic calculators, it is still in use in many schools and universities. The origins of *Derive* date back to the 1970s and a Honolulu based software company. The company was purchased by Texas Instruments in 1999 in order to integrate the software with their range of calculators. Although less comprehensive than larger software packages such as *Maple*, *Mathematica* and *MATLAB*, *Derive* is still capable of carrying out symbolic algebraic manipulations and can handle factorisation of large numbers with ease. Indeed, the fact that *Derive* is less comprehensive than some of the other packages can be a strength in terms of its use as an introduction to the ideas of computer algebra. Chapter 5 provides the reader with a quick guide to *Derive*, demonstrating its ease of use and highlighting some of the many useful functions available. Examples are given using the graphing facility and simplification of rational algebraic expressions.

Chapter 6 introduces *Calculus WIZ* - a powerful, interactive calculus tutorial. *Calculus WIZ* uses the calculus power of *Mathematica* to provide relatively inexpensive but comprehensive software to support calculus teaching up to undergraduate level. This stand-alone edition comes with a specially customized mathematics engine based on *Mathematica* technology. Also based on *Mathematica* technology, *Mathematical Explorer* combines text, graphics, and formulas in an easy-to-use notebook interface that is completely interactive, making the user a participant in mathematical ideas. *Mathematical Explorer* allows the user to explore many interesting questions about both physical and abstract phenomena and to gain insight by computation and visualization. A database containing biographical information about key mathematicians in history is also supplied. This enables teachers to present mathematics as ideas created by real people in a historical context. Chapter 6 provides a brief introduction to both *Calculus Wiz* and *Mathematical Explorer*, demonstrating through a series of screenshots the ease of use which is common to both software packages.

*Mathematica CalcCenter* is calculation software that combines powerful computational abilities with a simple and intuitive user interface. It costs less than half the price of the standard *Mathematica* package. The key feature of *CalcCenter* is the *'InstantCalculator'*, a form based method of specifying a *Mathematica* operation. The most innovative aspect of the *InstantCalculator* is that it produces *Mathematica* output, though this is normally invisible to the user unless he or she wants to display the resulting script. Through a combination of screenshots and worked examples, Chapter 7 introduces the main features of *Mathematica CalcCenter*. The main differences between *Mathematica CalcCenter* and *Mathematica* are highlighted, allowing the reader to decide which package is more appropriate for his or her needs

*Mathematica* is a very powerful mathematical tool aimed at the professional and research mathematician. It can perform an extremely large range of mathematical operations - far in excess of those ever likely to be required by the average undergraduate. As a consequence, it is an expensive piece of software. However, there are a number of derivatives of *Mathematica* that operate in the same manner as the parent software but have more restricted capabilities - and hence a lower, more affordable price.

*webMathematica* adds interactive calculations and visualization to a website by integrating Mathematica with the latest web server technology. As such, it allows users who are unfamiliar with Mathematica to carry out complex calculations without requiring any knowledge of the underlying software. Developers do not have to worry about session management and error recovery. *webMathematica* takes care of all aspects of development, allowing developers of a website to concentrate on solutions, not the implementation details. Chapter 8 presents a comprehensive overview of the capabilities of webMathematica, allowing the reader who is developing their own website to make the decision as to whether to incorporate it into their design. In Chapter 9, a demonstration is given of some basic worked examples which are suitable for a classroom situation, using *webMathematica*.

*Waterloo Maple *is a Canadian software company, based in Waterloo, Ontario. Its most famous product is the*Maple *Computer Algebra System*. Maple* has become one of the standard Computer Algebra Systems (CAS) and is readily available at many universities throughout the world, as well as being a standard tool in much of industry and finance sectors. Chapter 10 provides a brief introduction to *Maple, *covering basic ideas and demonstrating its ability to work with rational and irrational numbers. Screenshots are provided to familiarise the reader with the use of the classic *Maple* worksheet.

Chapter 11 offers advice to existing users of CAS on how best to utilise their software packages. Some familiarity with *Maple* and/or *Mathematica* is assumed, although the chapter can easily be read as a taster for either package. Although specifically addressing the problem of teaching multivariable calculus at undergraduate level, the general ideas put forward can be applied to any course where the use of CAS is being considered.

*MATLAB* (short for *MAT*rix *LAB*oratory) is a numerical software package. First marketed by The Maths Works in 1984, *MATLAB* has since gone on to be one of the most popular mathematical packages, both in education and industry, with over 1 million users. *MATLAB* allows easy matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages. Although it specializes in numerical computing, an optional toolbox interfaces with the Maple symbolic engine, allowing it to be part of a full CAS. The latest version of *MATLAB* is R2008a, released on 1 March 2008. Chapter 12 provides an introduction to *MATLAB* by demonstrating a series of command line inputs with their respective outputs, some of which produce graphed functions. The examples given are based around matrix algebra, an area of mathematics in which *MATLAB* is particularly suited and easy to use.

For many years mathematical notation has been difficult to reproduce effectively on the internet. HTML has relied on the conversion of equations to images which are then inserted into documents on-line. This has resulted for many years in off-centre mathematical notation which, when printed, is reproduced at a lower resolution compared with surrounding print. The recent introduction of *MathML* (Mathematical Markup Language) has radically improved this situation, allowing the accurate reproduction of mathematical documents on the internet. Chapter 13 provides a comprehensive introduction to *MathML 2.0*, outlining the reasoning behind its introduction and the language used. For those interested in writing their own documents in *MathML* the chapter includes several tables of elements, functions etc, and some worked examples of creating equations.

For those authors who wish to publish mathematical documents on-line using MathML but who would prefer to use a more intuitive approach which is closer in form to Microsoft Equation Editor, SciWriter is an inexpensive solution. *Sciwriter* enables the writer to produce elegant mathematical documents quickly and easily through the use of maths toolbars and/or keyboard shortcuts. Chapter 14 demonstrates these facilities through screenshots, and also highlights the facility to cut and paste formulae written in Sciwriter to either *Maple* or *Mathematica*.

*Mathematica* was conceived by Stephen Wolfram, developed by a team led by himself, and first introduced in 1988. Marketed by Wolfram Research, the latest release is Version 6.0.2 on February 25, 2008. *Mathematica* is a Computer Algebra System which can handle both symbolic and numerical calculations, possessing as it does a language which supports both functional and procedural programming. *Mathematica* has more than a million users, including all of the world's major universities. Chapter 15 gives a general overview of *Mathematica's* capabilities, showing such features as graphics and palettes. Chapter 16 introduces the reader to a number of the mathematical commands which are available, followed by a number of tutorials which cover the areas of calculus, series and linear algebra. The two chapters aim to provide users who are unfamiliar with *Mathematica* some insight into the ease with which it can be used for teaching complex mathematical concepts.