Developing Tools for Teaching Mathematics Using Mathematica 3.0
"Training In Mathematics and Applications: New Method and Approach" 
T.I.M.A.N.M.A. project (*)
"Leonardo da Vinci" European Community Programme

(*) This project is partially supported by European Community within the LEONARDO DA VINCI Programme;
the partnership constituted for T.I.M.A.N.M.A. realisation  includes
Consortium CRMPA (Co-ordinator), Salerno - Italy,
Middlesex University, London - United Kingdom,
Addlink Software Cientifico, Barcellona - Spain,
High School ITG "R. Di Palo", Salerno - Italy.

Saverio Salerno, R.Cavaliere, A.Cavallone, V.Martuscelli
Consorzio C.R.M.P.A.- Centro di Ricerca in Matematica Pura ed Applicata,
c/o Dipartimento di Ingegneria dell'Informazione e Matematica Applicata,
Università degli Studi di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy
E-mail:salerno@ponza.dia.unisa.it

Abstract

In this work we intend to introduce the main idea of the T.I.M.A.N.M.A. [1] project, about the development of applications addressed to mathematics didactic using a system of symbolic, numeric and graphics elaboration as Mathematica 3.0.

Introduction

It's well known that mathematics learning is still now a crux in the scholastic career of many students because of objective troubles connected with the subject and didactic methods required to learn it.
In our work we want to propose a innovative approach to didactic of mathematics from point of view of method and instruments employed to the interaction among teacher and students

Innovation and functional requirements

The innovative aspects in the applications to learn mathematics by computers can be divide in two groups:
■ Layout of contents: all the theoretical concepts are explained by inductive approach, i.e., starting from easy applications and practise examples, you get to the formalism of mathematics theory; furthermore these subjects  are introduced through multimedia support and so they are easily understood in a customisable way and in according to user's profile.
■ Instrument-users interaction (user friendly): user has an additional tool which allows him to help to do some interactive exercise, using a powerful instruments of symbolic computation and formal abstraction as Mathematica 3.0

Now we can define the functional requirement of the applications, which can be summarise as follows.
As regards the contents we have:
■ possibility of arranging contents with examples, applications, study cases, exercises, historical outlines in order to explain better theoretical concepts
■ possibility of adding different types of information as for example images, sounds, films, interactive and scientific experiments to prove theoretical concepts studied during the lessons 
■ possibility to take information arbitrarily choosing the navigation through hyperlink and then following cognitive ways, according to user's profile
As regards the interaction among instrument and students we can:
■ insert of suitable functions to make exercises both for amount and for troubles customisable; that 's means users have lots of routines which are always able to introduce new exercises about topics shown in the theory and to help them to choose solving procedure
■ insert of functions for evaluation of the level of learning according to exercise results: in substance these functions can evaluate the nature of mistakes and then show users the best way to go on
■ add automatic resolution instruments with a step-to-step decomposition of the problem in such a way as to help users to understand the solving procedure. Furthermore these instruments use to give some hint about the exercises presented by users

State of the art

In literature there are lots of projects, about didactic of mathematics using Mathematica 3.0 (in the previous version); shortly we can say that there are two approach to the development of such applications.
One of these embeds engine of Mathematica 3.0 (the Kernel) with other software development instruments, used to create Use Graphics Interface so it joins two different and complementary potentiality, the power of symbolic computation and formal abstraction to manipulate scientific data (Kernel) and the flexible and easy understanding of GUI arranged ad hoc using software as Visual Basic or Visual C++. An example of this strategy is represented by Hypermedia Mathematics Learning Environment (HMLE) [2] developed by Tampere University of Technology Department of Mathematics, which presents an interface developed  in C/C++ connected with Mathematica via MathLink.
The second only uses Mathematics environment, with the Kernel used for the computation and the FrontEnd used for the content's introduction; such approach allows to work in a only environment and to simplify the development. An example of this approach is the Transitional Mathematics Project (TMP ) [3], [4], developed by London Imperial College, including essentially a pre-course for the use of Mathematica and then a course on some mathematics subjects.
Clearly both approaches above, present advantages and disadvantages, in according to the applications. The last version of Mathematica 3.0 allows us to resolve better the problem connected with didactic, remembering the remarkable strain of Wolfram to have introduced multimedia aspects in the new FrontEnd.

The platform Mathematica 3.0 - a singular and innovative platform

This platform connected with additional tools as MathLivePro, MathLink and others is the better choice for implementation of innovative applications addressed both to mathematics didactic and to scientific subjects. In fact the new potentialities given by Mathematica 3.0, especially flexible, complete, and interactive introduction of topics, allow to build complex and complete tools for a better understanding of mathematics, satisfying all the requirement introduced above. The introduction of hotword, hyperlink, active bottom, executing commands in FrontEnd (these are only a part of) well satisfies who want to develop advanced graphics interfaces for theoretical introduction, interactive applications, examples and exercises, by symbolic, numeric and graphics calculus of the new Kernel which has both a bigger amount of alghoritms and best computational performances.

An example of development: T.I.M.A.N.M.A. project prototype

In our project, we present a course of Linear Algebra realised with Mathematica 3.0, using hypertext features. This course has theoretical lessons and exercise sections, connected between them, i.e. we can shift from theoretical lesson to the exercises and vice versa when user has some laps about the theory solving exercises. In this way we create an interaction among users and instrument. The instrument is able to evaluate the level of learning through analysis of mistakes made during the exercise phase, and then to find user laps.

General framework of Exercise Session

Some example of Exercise Session

When user start an Exercise Session, appear on the screen a ToolBar Palette with some usefull command, like the following :

Back	Save	Chapter' s Index Exercise	 Guide	Functions

With this button the user can do exercise choosing what kind of exercise resolve (Exercise button) or what kind of function use (Functions button); for example Exercise button open another Palette with a table of all exercise available in the current chapter of Courseware and  Functions button open another Palette with all functions available in the Exercise Session.

Exercise Palette

Functions Palette

An example :

Example[6]

In the following example we call the function Exercise[1], and system generate a new exercise about sum of vectors. In this case user can choose to carry-out calculations and then verify its solution calling the function Verify[solution] or it can view the resolving method shown by system, step-by-step, clicking on the How you can solve the exercise button.

Exercise[1]

Conclusions

To develop applications about scientific topics and didactic learning, new release of Mathematica  gives all the specifications needed to build user-friendly interfaces, allowing to make hyperlink within the same notebook or others of them, Palette with active bottom, multimedia objects embedded OLE links, etc.
At the end we can assert that by the innovative features of the FrontEnd we can develop didactic tools which allow to introduce new interactive methodology in such a way as to improve the learning and to get over the abstraction of the mathematics' idea.

About Authors

Prof. Saverio Salerno 
Professor of Mathematical Analisys, Faculty of Engineering - University of Salerno Italy
Director of Consortium Center of Research in Pure and Applied Mathematics (CRMPA)
Via Ponte don Melillo, 84084 Fisciano (Salerno) - Italia

Dott. Roberto Cavaliere
Mathematica developer, Consortium Center of Research in Pure and Applied Mathematics (CRMPA)

Dott((.)^ssa) Antonella Cavallone
Mathematica developer, Consortium Center of Research in Pure and Applied Mathematics (CRMPA)

Dott((.)^ssa) Valeria Martuscelli
Mathematica developer, Consortium Center of Research in Pure and Applied Mathematics (CRMPA)