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State of the art

this text is now incorporated into the ParB document... please proceed from Intergeo page.

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The state of Interactive Geometry

This project is driven by European leaders in interactive geometry softwares. We are going to define what we mean by interactive, or dynamic geometry. Pictures in a textbook, diagrams on a black-board, sketches on a paper sheet, are very useful pedagogical tools to illustrate and support teaching or research in mathematics and other scientific fields. We will call this still geometry. Interactive geometry is to still geometry what cinema is to photography. It allows for a deeper exploration of a concept: by changing the unrelevant parameters, the student grasps the invariant properties of the notion which is illustrated and develops an intuition regarding the behavior of the objects he manipulates. Through a feedback loop, abstract objects become familiar. The interactive geometry we are talking about relies on computers and softwares. These softwares allow the manipulation and the visualization of a construction (a figure). The user manipulates the figure, through the keyboard, the mouse or a tracking device, in order to change some free parameters entering the figure, like the position of a control point. The construction is then changed accordingly. For example the control point drives the kinetic of a chemical reaction and the figures reflect the concentrations of the different chemical substances. In nowadays computer systems, these control points such as sliders, knobs, handles, scrollbars, mousewheels and other cursors have universally pervaded every part of the user interaction. Contemporary users are used to the fact that sliding a virtual cursor is going to explore some hidden dimensions, unveiling new features of the object she manipulates, the basic example being the forward/backward buttons on a multimedia player, controlling a virtual time that otherwise flows constantly. Interactive geometry is intended to manipulate scientific data relying on a hierarchical construction. To return to our example in chemistry, the reaction will be driven by a given differential equation, where some parameters will enter, attached to a cursor controlled by the user, that will allow a graphical representation of the reaction to be displayed. The figure encodes not only the graphical illustration (a curve here, a disk there) but as well the relations between the different entities that are drawn. Of course the main entities and relations in interactive geometry are of geometrical type, you will find triangles, circles, lines and points, that are barycenters, tangent, secant, with given angles and distances. But it is much more general than antique greek geometry, you can have functions, derivatives, colors, random variables, all sorts of constructs that allow you to visualize and manipulate concepts that arise in all sorts of contexts, in particular outside mathematics. A movie player could be indeed considered in the scope of interactive geometry! But this project focusses on constructs of a more abstract and scientific nature than motion pictures of the real world. A figure can be used in two main modes:

  • It can be used in the classroom by the professor, in complement to a classical black-board face to face teaching, in order to illustrate a concept. It is more precise and lively than a simple chalk diagram: such a diagram can only be drawn for a set of parameters. At best the professor sketches, with some hand-waving, the changes that would occur if the parameters would have been chosen differently. With a dynamic geometry figure on the contrary, the student sees the whole picture, not only a still version of a frozen set of parameters. Therefore only the invariant properties of the figure are identified, for example the three bisectors of a triangle all meet at its circumcenter, whatever the shape of the triangle.
  • It can be embedded into (or attached to) a document, that it illustrates, in a manner similar to a picture in a text. The reader is invited to manipulate the figure that is referred to in the text. The cognitive experience is much more intimate than in the previous case: It is one thing to see a figure evolve, half guessing which parameters were changed by the professor, it is another one to actually do it yourself interactively. This situation occurs in a eLearning setting or as a complement to a face to face learning, in the mixed mode (also known as blended) learning.

As we said, pictures are of great importance in teaching, it helps making abstract notions more concrete, a course involving no drawing to fix things down is likely to be lost on the students. An interactive geometry figure adds a dynamic side to the learning experience. But the real pedagogical value lies not in the production of a moving picture, but in the interaction between the student and the figure: It is important to realize that an interactive geometry figure is of some help, in face to face teaching, to attract the attention of the students or even to have them understand a notion. But it is our belief that an interactive geometry figure manipulated in a face to face course should always be available to the students for latter self-appropriation. Because it is through manipulation that a notion really ���sinks in���. It is easy to misunderstand the students bewilderment at a moving picture as the revelation of an enlightening truth when in fact the core complexity of what is really going on is blurred by nice pictures. Interactive geometry aims at helping the student build new mental pictures through activity and reaction to feedback. The mental picture is tested against the figure, it helps pushing aside the wrong impressions, that are part of the normal process of learning, pelting the hastened conclusions with unexpected situations exhibiting them to be false. Because the teacher is not in the same mental state as the student, he can not foresee all the possible dead-ends she can go through in learning, she has to explore them herself. The pedagogical value of interactive geometry can be summarized in the following motto:

I hear and I forget.
I see and I remember.
I do and I understand.
(from Selected Sayings, Confucius, K'ung Fu-tzu, K'ung-tzu, Kong Zi, Kong Qiu, Zhong Ni Circa 551 - 479)

This being said, an interactive geometry figure can be used in many different situations:

Course Material
A pedagogical material is illustrated by a figure, whether in face to face mode, the figure being projected on a screen, the course being given orally, or in eLearning mode, the figure being embedded inside or attached to an electronic document.
The student is faced with an online exercise involving a figure, with some randomly chosen parameters, and she has to answer a question that requires some manipulation of the figure.
  • The answer can be read from the figure, like the value of a certain quantity, and finding it involves some manipulation, or
  • the answer itself resides in the manipulation, like the positionning of an existing object in a specific situation, or the creation of a new object with specific properties.

The software probes the figure and assess the result against the expected answer.

To simulate stimulates, visualizing a situation which is not fully understood yet helps the researcher to discover properties that were not apparent from the bare equations. Interactive geometry is part of the growing field of experimental mathematics.

Building on Good Practices of e-Learning Technology

Great expertise has emerged in e-learning technologies lately. Achievements such as the learning object repositories grouping thousands of course-material resources in a readily accessible form are among the most visible ones: the paradigm of a shared space where resources can be published and annotated in order to be searched for is at the heart of them. Repositories such as ARIADNE, MERLOT, see also geometry, or EducaNext are such.

The development of these and other repository projects has seen the characterization of the learning object metadata standard (see [IEEE-LOM], and [LOM-resarch], a specification describing all the possible attributes one can assign to a learning-resource and which can be searched for in a repository. This standard is accepted by everyone and gives great freedom by allowing applications to specify sub-sets and extensions called application profiles.

Learning object repositories such as MERLOT and EducaNext complement the metadata tagging with evaluations of the resources. This practice is precious since the resources found on such a repositories are expected to be re-used by several and be polished for such. A basis for this re-use to happen, and for mutual enhancements to the resources to be applied, is the complete characterization of the usage, modification, and publication rights associated to each resource. GNU Edu, is one such learning-object-repository: experience has proved that, based on such a clear license, authors are easily willing to contribute back by enhancements to resources.

??? where to put something about learning-design ??

Quality monitoring has been explored extensively for learning resources in such projects as the European Quality Observatory and keeps being studied, for example in e-Quality. The EQO has recently published the first results of a large scale survey on quality in e-learning (see [EQO-survey]) which concludes, among others, that learners need to take an essential part to the quality process.

Learning objects, since they were originally named, are considered to be any possible resource. This generality allows great flexibility but, at the same time, ignore any introspection into the learning objects which are considered to be only a set of downloadble files. As a result quality monitoring for such resources can only be done manually. The current evaluation schemes of MERLOT and EducaNext are ratings of peer-reviews and user-comments.

Publishing and Sharing Interactive Geometry

In less generality that "any learning object", the possibility to assess automatically the achievements of the learners using the resources are greater: this has been understood by such standards as IMS QTI or ADL SCORM which allow a report of the learning activity to a host platform. This platform is, for example, a learning management system where teachers can, later, see reports.

No current geometry software we know of can, however, export exercises in a SCORM compliant way although the wish is there ??. Based on an extensions of such an API, it is clear that even simple explorative interactive geometry animations can report the achievements of the learner and that such reports can be used to assess the capability of the learners at use of this animation to attain the goals intended by the authors.

Currently, interactive geometry animations are often published embedded within a web-page. This publication is intended for consumption, most of the time and rarely allows further development of the animations. Only a publication with source with a well documented permission allows it which this is often neglected. The lack of such makes every recipient of such as a consumer.

Even in source format, interactive geometry animations remain documents created by and for a precise application: only the same application in the same version or a later one is candidate to open it. Little has been done in the direction of exchanging between two. (?? true ??).

The advent of semantic encodings of mathematical objects such as OpenMath or Content MathML has paved the way to promote the specification and encoding of a common set of symbols to encode geometrical constructions: based on such, commercial computer-algebra-systems have made it possible to copy and paste from a web-page into their system while keeping the semantic of the mathematical object. The set of tools based on these encodings is steadily growing ranging from queriable databases to many forms of visual rendering tools. To name a few, large projects such as MONET, MOWGLI, LeActiveMath, and Mizar have paved the way in this direction. ?? more about OpenMath ?? might make sense... a coverage of some tools might be sensible.

As noted by Robert Miner in his presentation "Robert Miner, Design Science: Trends in MathML Usage", the trend for semantic markup is being felt stronger every year by customers who wish, among others, to avoid the "vendor lock in". ?? good conclusion ?? might frighten some ??


Some are not incorporated... should they ?

seems to be much about learning-design, I am not sure appropriate
E-learning Quality Observatory
seems to be a somewhat old view of the quality monitoring and checking... the survey Online Survey "Use and Distribution of Quality in European E-Learning" - First Results is probably quite appropriate to quote.
Merlot LO repo:geometry
this major, widely used learning-object repository is a good example of a classical LO-repo and presents well the limitations of such, especially some poorness in metadata. It does not at all address the issue of international communication between educational communities.
this is another major learning object repository, European this time, can now be accessed anonymously. They seem to do barely more than spontaneous review. Merlot, in comparison, seems to assign review. Their check for metadata is... a... "please do it complete"!
probably the eldse Learning-Object repository... actually a closed community of universities... is quite big but... closed... does not do review (anymore), I heard.
a very large computer-science LO repository... a bit too large maybe... includes an amount of DBLP and such proceedings... does not have "person entities".
a learning object repository with automatic conversion from open format to other open format (OOo, HTML, PDF, TeX)
a resource server and catalog, uses LOM-compliant metadata to enhance search efficiency; additionally, its educational items convey a matching between resources from various countries and educational systems, but doing a similar job in terms of educational purpose. As it is free software and technically well designed, GNU Edu can be easily customized.
CETIS glossary
a good overview source about all these standards around e-learning standards... What's SCORM or IMS... you get it there... Overall, the web-site has valuable articles and news as well.
Quality Foundation
the descendant (for some) of EQO.. an interesting new spot to see the current state of the art in terms of quality validation.
Database of EU-funded projects outputs

Some links about metadata:

Some links to standardisation bodies:

Some specially math-oriented links:

  • MathForum: a repository of english-speaking math resources on the web pretty rich
  • the geometry junkyard of David Eppstein
  • PlanetMath
  •  ?? JOMA
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