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Anton Eliëns, Zsofia Ruttkay

*familiar with mathematics* in order
to better understand the visual effects of complex phenomena,
e.g.

*processing* community,
that has arisen out of a long-standing tradition of what may
be characterized as *aesthetics by numbers*, pioneered
at MIT, *processing*, *flash* and other graphical
frameworks with stunning works, available in many online galleries
all over the web, see for example Jared Tarbell's
Computation Gallery.
For (future) game designers and developers, knowledge of mathematics
needs no argument, as exemplefied by the many sites about game
physics and math for game designers.

- highly visually oriented, they see and learn by trying out things rather than consulting traditional textbooks;
- proficient with using computers (games) at early age,
- less interested in science, and particularly, in mathematics, and do not perform well in these respects in many Western societies

- games to practice routine tasks
- manipulative virtual environments for
*try-out* - simulations for making conjectures
- strategic games -- manipulative & combinatorial
- visualization of structure(s) -- patterns of nature
- explorations in 2D and 3D geometry
- generative (visual) art

*heart of mathematics*, some directly related to mathematical aspects of design, such as generative algorithms, geometry, perspective and noise, but others of a more distinct abstract nature, to make our students familiar with the reasoning underlying mathematics.

Mr. E's knapsack

Mathematician Mr. E is traveling the world with his knapsack full of (mathematical) knowledge. Exploring the world, he meets famous colleagues like Euclid, Euler, Gauss, Newton, Einstein, Gödel, Leibniz, etc. He has to collaborate - learning as well as teaching - to proceed, get goods, get a regenerating cup of coffee, and sometimes a place to stay. He uses his problem solving skills to discover the world, and to unravel it's Big Mystery. Mr. E's journey is taking him around the globe and through the history of science and mathematics.

Mr. E's knapsack is meant to be an adventure game in historic and scientific setting, which combines the attractiveness of a fantasy world as in games like Zelda and Star Wars, with education, by providing a context that is on the one hand fictional, but on the other hand provides scientific knowledge and historic information. It lets the character learn 'real' uses of mathematical items, since the player has to actively use such items somewhere else in the game, e.g. pick up a theorem at Euler and use it to solve a problem posed by Gauss.

This brief scenario stems from a group of mathematics (master) students of the University of Amsterdam, and originated in the first place from the wish to get a more clear idea on how all the mathematics concepts learned fit together, and in the second place to create an environment to teach mathematics in an integrated way, supplementing ordinary textbooks, ranging from high-school students to advanced master-level students. It is also meant to play a role in their work for De Praktijk, a company that develops new concepts for natural science education, primarily targeting high-school students.

An adventure game, that we take as a convenient metaphor for an online mathematics textbook, provides flexible access to a variety of illustrations and exercises dependent on the skill level of the students, and allows for extensions by student-created content both visual (e.g. generated plants) and intellectual (e.g. a question raised for the future players). In addition interactive video may be used to provide the necessary historical and societal context, [ Eliens et al. (2008)] .When Java applets were introduced about a decade ago, there was a general enthusiasm among educators, at least in CS departments, that their topics - mathematics, programming languages - could be presented in an interactive way. However, although Java has been adopted as the programming language of choice at many universities, the early vision(s) of interactive education has not been realized, despite the increasingly large collection of interactive examples of mathematics, physics, and related disciplines on the Web. ## MATHS ON THE WEB -- FRAMEWORKS AND TOOLS

When we look at how, for example, Wikipedia has become a trusted source of information on mathematics, we may attribute this to the lack of a coherent framework for incorporating interactive math examples. Such a framework, apparently, is more easily provided for (hyper) textual information, or images and video, as testified by the large amounts of user-contributed content in sites such as flickr and youtube.

Is it possible to create a similar repository for samples of interactive mathematics, and, on a more modest scale, to formulate guidelines to enforce a coherent approach in both (visual) style and (mathematical) content? Although, clearly, we strongly endorse the visually-oriented programming-by-example paradigm supported by the processing environment, we will in a later phase of the curriculum also introduce other programming environments, in particular javascript/php, flex/as3, C++ with suitable libraries, as well as one or more of the many game engines and game development platforms such as Unity3D and the Half Life 2 SDK. Where the choice for processing was motivated by both ease-of-use and ease-of-learning, the other platforms and environments are primarily motivated by reasons of deployment, that is the means available by our (target) audience. In our efforts to decide on a first programming language, we found that processing libraries are available for both javascript and (Open Frameworks) C++, enabling a direct transfer of skills and knowledge. when using

In developing our Creative Technology curriculum, we discussed computer programming platform options ranging over C++, flash/actionscript and, of course, Java. Finally, as indicated before, we have chosen for processing, as a language to teach programming, but foremost as a platform for exploring both mathematical and visual ideas in an exploratory fashion using computational means, suited for educating creative engineers. processingto teach (introductory) programming. Adobe flex/as3 would provide a strong alternative, not only with regard to popularity of the flash player, but also looking at issues of efficiency and support for graphics. For example, the effects that can be obtained using pixelbender shaders, using relatively simple mathematical formulae, are simply stunning. However, we found that the flash CS3/4 tool would rather confound our methods of teaching. Also, the many examples made by code-artists, as they are called in theprocessingcommunity, seem to fit the targets we set our students better than the bewildering amount and variety of flash examples.Mathematics, by nature, is an abstract and mental discipline. The formal notations and the among mathematicians agreed rules of deriving proofs form the basis of the traditional pure mathematical textbooks, reminiscent e.g. of the style of the Bourbaki school, ## MOTIVATIONAL ISSUES -- THE HEART(S) OF MATHEMATICS

[ Aubin (1997)] . While reading such a pure and formal work may be considered as the way of coding and transferring mathematical knowledge, and provide high intellectual (even, aesthetical) pleasure for the professional, it does not raise the interest or provide an entry to grasp the very idea and the essence of the topic, neither the (exploratory) road of discovery to it, in an individual or cultural-historical context. However, accounts on mathematical discoveries are rich in visual representations, enlightening analogical stories, inspirations gained by observation of (natural) patterns,[ Chalmers & Cunningham (2002)] , and as such an excellent topic for a visually appealing adventure game.A loosely coupled collection of game components, organized as an adventure game, is a excellent means to avoid the linear format of traditional textbooks, and by allowing students to contribute content, a unique facility to learn mathematics by exploration, as well as, actively, constructing (computational) samples themselves. A detailed description of the architecture of such a system, however, is outside the scope of this paper, and, clearly, an issue for further research. Perhaps more important than the possible technical solutions is to indicate what type of interaction we wish to achieve, and how guidelines with respect to visual style may be enforced. ## guidelines for a maths adventure game

As for interaction, interesting examples are provided by a collection of physics-driven construction game(s), where vehicles must be assembled that will then either roll to their destination or crash halfway, dependent on the skill of the player, and, roughly based on the same principles, the Magic Pen game where the player can simply draw shapes, that then get a life dependent on the physical laws at work. Both games, in our mind, however, are lacking in (aesthetic) visual style and, frankly, too childish. An interesting example of a game where game play is determined by drawing is MijnNaamIsHaas, meant to teach children vocabulary (in Dutch). Dependent on the vocabulary learnt so far, the game reacts on the childs drawing, intelligently, by generating additional content. Using a simple line style, with simple visual effects mimicking child-like drawing, this game may act as a reference for the aesthetics of our mathematics adventure game, where intelligent drawing tools obviously may play an important role in interaction, and applying well-established techniques of code-generation, be used to store code-fragments in a database of game components. ## CONCLUSIONS AND FUTURE WORK

It is apparent, when browsing through resources, how many applications lack aesthetic appeal and design wit, usually because the author is a computer scientists or mathematician not equipped with the skills, or the environment he/she used to create the content does not provide good support for the visual design.

From an educational, or if you will, methodological point of view, games and interactive CG may be used: ## methodology

In our assessment, the above list also indicates the big jumps in challenges to develop the kind of application - starting from stand-alone puzzles via explorative environments to full-fledged interactive course materials. The challenges encompass both technological/design and mathematical/pedagogical aspects.

- as a means to provide illustrations or trigger interest
- in assignments to exercise routine tasks
- to enhance the curriculum (additional explorations)
- as the major theme and medium to teach mathematics
- (and) to learn algorithms and programming

By setting our ourselves the goal of a comprehensive mathematics adventure game, we not only hope to improve our teaching of mathematics, but also find appropriate styles and patterns that may guide the development of math (related) games and applications, [ Björk & Holopainen (2005)] .

In conclusion, answering the somewhat provocative question in the title of our paper, no, (math)games will not replace(traditional)math education, but (may) augment it with challenges and explorations, supported by the new developments in technology.## ACKNOWLEDGEMENT(S)

The MrE's Knapsack game concept was developed by Sebastiaan Eliëns and Jacobien Carstens, students from the University of Amsterdam.## REFERENCES

[Bourbaki]Aubin D. (1997), - The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Science in Context, 10. 1997, pp. 297-342.
[Patterns]Björk S. and Holopainen J. (2005), Patterns in Game Design, Charles River Media[Education]Chalmers A. and Cunningham S. 92002), - New Media and Future Education, Proc. East-West-Vision 2002. pp. 67-74.
[Teaching]Duchowski A.T. and Davis T.A. (2007), - Teaching Algorithms and Data Structures through Graphics, Proc. of Eurographics 2007, Education Papers.
[XIMPEL]Eliens A., Huurdeman H., van de Watering M., - Bhikharie S.V. (2008), XIMPEL Interactive Video -- between narrative(s) and game play, In Proc. GAME-ON 2008, Valencia, Spain
[Replay]Eliens A. and Ruttkay Zs. (2009), - Record, Replay & Reflect -- a framework for serious gameplay, In Proc. EUROMEDIA 2009, Brugge (Belgium)
[VideoGame]Gee J.P. (2003) ,What video games have to teach us about learning and literacy, Palgrave Macmillan[Fun]Lundgren S. (2006), Facets of Fun -- On the Design of Computer Augmented Entertainment Artifacts, Dissertation: Chalmers University of Technology[Animation]Peters K. (2006), Animation: Making Things Move!, Foundation Actionscript 3.0, Wrox Press[Processing]Reas C. and Fry B. (2008), Processing: A Programming Handbook for Visual Designers and Artists, MIT Press[Sangaku]Ruttkay Zs. (2008), - A Sangaku Revived, BRIDGES 2008, Leeuwarden, Tarquin Books, UK, 2008. pp. 155-162
[Math]Ruttkay Zs. and Eliens A. (2009), - From Puzzles to Interactive Textbooks -- CG in Maths Education, in preparation
[Visual]Terzidis K. (2008), Algorithms for Visual Design Using the Processing Language, MIT Press(PhD) is lecturer and coordinator of multimedia at the VU University Amsterdam, and was recently appointed at the University Twente as professor creative technology / new media. He has been closely collaborating with Zs\'ofia Ruttkay and is experienced in web-based interactive media, interactive video, and the application of such technologies in serious games. ## Anton Eliens

(PhD) is Assoc. Prof. at the University of Twente and headed the Creative Technology Working Group. Trained as a mathematician, she has expertise in creating styled multimodal behaviour and communication strategies for virtual humans. She also has a strong interest in educational game. Since september 2009, she is professor at the Moholy Academy of Arts, Budapest, Hungary ## Zs\'ofia Ruttkay

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(C) Æliens 04/09/2009

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