MAWEN-Project

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Mathematics is a genuine European discipline of thought with a history of almost 3.000 years, the first summit in Ancient Greece. Mathematics appears not only the immediate prerequisite for development of our technologies, there are also patterns of mathematical thinking deeply embedded in European access to the world -- both reasons for European hegemony during almost 300 years.

Now time has come to reconsider all in one and to understand principles of mathematics in a broad sense reaching a wide audience. Mathematics has been called science for mechanisation of thinking, so the mechanical nature of mathematics needs to be understood and even more: experienced. This can nowadays be achieved by software analogous to chess software, governed by explicit rules and interactive experimentation.

A novel "Mathematical Working Environment" (MAWEN) shall be developed by international efforts pursuing these goals.

Stepwise calculations like playing chess
Mathematics is considered a difficult subject at schools. Th actual state of the art in computer mathematics, however, opens completely new ways of interactive learning -- like learning with a chess computer: There two partners play move by move, where both adhere to clear rules:



The chess computer checks the moves of the player for conformance to the rules. Moreover, the player is free to choose particular moves. Depending on the "difficulty level" the player will run into difficulties sooner or later. Then the player can ask for a next move. In case the player's chess position is hopeless, he can return to a previous one and try a variant.

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Mathematics is calculating (with symbols) and justifying the steps of calculation -- in justification computer mathematics is superior to chess computers: the rules (of formal logics) are as clear as the rules of chess, but the stratgies are simpler and easier to look through:



Here the student ("player") applies the rule (a.c)/(b.c) = a/b  in a wrong way, the computer will notify. If the student does not know how to proceed, he can request an appropriate rule. The computer decides (depending on the "difficulty level") whether to present the whole rule or only a part, whether to present a list of rules, or whether a rule is just applied and the resulting formula output.

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Since each step has a formal justification, collaboration of several students in a calculation-"game" becomes feasible and interesting. Even sighted and blind students can collaborate this way, as shown in the next section.

Blind and sighted people collaborate
Blind humans replace vision by developing other capabilities. They also cultivate another life style ("no sports" etc). Mathematics could be a replacement for sports, or even open up new career opportunities. Sadly, facts are against this: despite great efforts of software developers for "accessibility", mathematical formulas mostly are "dead" graphics .

On the other hand, in most countries modern pedagogy includes blind pupils in lessons for sighted contemporaries ("inclusive" education). This is welcome by all - but not in mathematics lessons: there visually impaired pupils lose the overview quickly as soon as formulas become complicated, and get separate support from remedial teachers. But with the software described above pupils (and later students) need not be separated, they even can collaborate on one and the same computer:



Sighted and blind students collaborating on a computer use different tools for pointing at elements of a formula: Sighted students look at the screen and point with the mouse, visually impaired students read (line by line!) from the Braille and point by use of function keys.

Blind students are forced to a structure-oriented access to formulas, while sighted ones prefer "intuition". But intuition on mathematical formulas occasionally mocks, for instance in Fig.3: While students prefer to grasp the structure by intuition (and the vertical alignment), blind ones need to comprehend complicated formulas by "sub-terms", for instance identifying nominator and denominator of the fraction separately; thus they are not so much endangered by errors like the above one.

As soon as software like MAWEN consistently cultivates structure-oriented mathematics, one will observe, whether sighted students can learn from blind ones - in the opposite direction as usual.

The MAWEN software is versatilely applicable
The MAWEN software is based on concepts and technologies from "proof assistants", a fundamental difference to all other educational software for general mathematics and thus versatile far beyond what was possible so far, see the next section below.


 * Collaborative und inclusive learning in schools as described in the two two previous sections. There are already successful field tests 1 2 3.


 * Mathematics interesting for people of any age becomes accessible within and outside of educational institutions via internet. How many latecomers will try how much their judgement is justified when saying "I was never good in mathematics, nevertheless I am successful"?


 * Games and competitions about skills in applying mathematical rules, in selection of mathematical methods, in formal describing problems, etc. This is possible in small learning groups as well as on an international scale.


 * Specific mathematics groups for interested people in educational institutions as well as independent learners can be attracted by the MAWEN software. Why not offer such groups and courses in youth camps or clubs? This works already for blind addressees.


 * Evaluation of knowledge within and of educational institutions can be mechanised beyond multiple choice tests (prerequisites are well defined by knowledge bases in "proof assistants"), see section below.

Accomplished and ongoing work in the MAWEN project
The strengths of the software mentioned above partially concern features which are not yet realised. Thus here follow detailed descriptions what has been accomplished so far and what is planned for the future. Both descriptions will be kept up to date here.

In computer mathematics
R&D already done as well as under construction is in the hands of the developer team of the proof assistant Isabelle and the chief technologist in the team.

Isabelle is a well established product based on a well-defined logical base . During recent years Isabelle attracted attention from hundresds of mathematicians all around the world, who step by step implement their specific expertise specific expertise and this way mechanise more and more of mathematics. Mechanisation not only concerns definitions and theorems, but also each step of proofs -- in a human readable format:



''The figure shows Isabelle with the hitherto mainly used interface jEdit: The proof of the "Chinese Remainder Theorem" known from number theory is proved automatically ("by auto") by use of cleverly formulated lemmas. At position "hence" the "proof state" has to be resolved as shown in the window at the bottom. Further knowledge required for the proof are to be found in the theories listed at the right.''

Isabelle's system architecture exploits multi-core hardware for efficient asynchronous communication between proof engine and user interface . The current user interface is Isabelle/jEdit (in Fig.3) , but Isabelle/PIDE ("prover IDE") is designed generic such, that other user interfaces can be supported as well. Actually Isabelle/VSCode is under construction as the next generation user interface; this is relevant for accessibility and HCI.

Isabelle/Isar is not only input language for interactive proofs but offers powerful toos for defining other input languages . Thus no principal difficulties are exptected, when implementing input languages simplified for educational purpuse like "structured derivations" . An early example for this input language is:



This input language is rather close to what students at faculties of technology (and schools!) are used to see. The educational software development as described below takes advantage of generic Isabelle/PIDE.

Open R&D tasks for the MAWEN project:
 * 1) Although basic functionality (state and preview panel, HTML preview of theories, etc) is implemented already, much efforts are required to reach the state of the art given by Isabelle/jEdit.
 * 2) Further functionality concerns various views for "outline", "debugger", etc, which shall use teh VSCode "extension APIs"or methods from HTML/CSS/JS Methoden in order to handle the generic webviews of Chromium in VSCode.
 * 3) Isabelle's term structure needs to exactly be transported from the proof engine to the frontend in order to present formulas as shown in Fig.3; this is a novel requirement raised by educational software development.
 * 4) Educational software development also extends requirements of interactivity: while in proofs little help can be expected for a next step, in an educational system usability depends on a system's ability to propose a next step in solving a certain mathematical problem. This requirement challenges Isabelle/PIDE's architecture.

Part of the tasks above concern general development of Isabelle/VSCode, which can be accomplished independently from MAWEN.

In accessibility, HCD and field tests
This research area in the MAWEN project is covered by two universities, the Johannes Kepler University of Linz and the University of Applied Sciences of Hagenberg.

Accessibility is covered by the Institut Integriert Studieren (IIS) of University of Linz. IIS works on accessibility, particularly of software in mathematics and specifically on accessibility of formula editors: . At IIS the activities on accessibility in mathematics joined with development of educational software for mathematics for many years. IIS employs one of the very few academics graduated in a formal discipline, who is born blind; this person researches accessibility for decades and is the ideal consultant and alpha-tester for MAWEN.

Open R&D tasks for the MAWEN project:
 * 1) Determine the Braille codes used in specific countries (which easily use Isabelle/MAWEN's formulas and subterms given as character strings).
 * 2) Review and improve accessibility of the views in Isabelle/VSCode as well as in Isabelle/MAWEN.
 * 3) Make graphical presentations of theory dependencies accessible; these are not trees but DAGs (directed acyclic graphs), where no previous work on accessibility seems to exist.
 * 4) R&D on the question, how far visual patterns of proofs or formulas (trees with indentation, syntax highlighting, etc) can be approximated by auditive patterns.

Human Centered Design (HCD) is covered by the research center in Hagenberg. There is manifold expertise in usability-engineering and very specific experience form projects like Welding Interaction in Future Industry and Human-Centered Workplace 4 Industry, where interaction between machine and impaired humans are successfully dealt with. A dedicated team in the research center is prepared for collaboration with teachers and students in inclusive classes.

Field tests are planned in collaboration with schools, which in turn manage collaboration with other schools covering almost all of Austria: These "Zentralschulen" are ready to manage field tests, which include teachers and students into requirements analysis. This does not only serve developers with precise requirements capture, but also students and teachers: The former get first experiences in requirements-engineering ("Why behaves the computer differently from what we expect?" Because our request was not precise enought? Or because there are technical constraints?) beyong gestures on the mobile. And the latter get an offer to develop competences in e-learning design.
 * Michael Reitter-Landesschule in Linz for Upper Austria
 * Odilien-Institut in Graz for Styria
 * Bundes-Blindeninstitut in Vienna for Vienna and Lower Austria.

Open R&D tasks for the MAWEN project, in particular for HCD, will become active as soon as the MAWEN software is applied in real educational settings and feedback loops between developers and users become effective. The following methods shall be employed:
 * 1) Thinking Aloud Method
 * 2) Analysis of successful collaboration
 * 3) Awareness for activities of collaborators.

In development of educational software
The ISAC projekt is engaged in prototyping educational software based on concepts and technologies of Isabelle for a long time . Prototyping follows three main points: (1) ISAC re-uses the simplified proof language of "structured derivations" and extends it with next step guidance such that a stundent can request a next step from the system in case he or she does not know one . (2) The proof language is extended by formal specification; this is a requirement in education of engineers and this allows to decompose problems into sub-problems. (3) The powerful system resulting from Pt.(1) and Pt.(2) requires control: The system should neither overwhelm a student with problems too difficult, nor should it bother a student with too much help .

The prototypes developed so far have been explored in field test at different kinds of schools at different levels of knowledge . Why is the ISAC system not in wide spread usage? For a simple reason: Students need a handy formula editor supporting editing comparable easy as using paper & pencil. Such an editor did not yet appear in the Java world.

Since the accessibility of Isabelle/VSCode is known, integration of ISAC into Isabelle is pushed and concern of ongoing work. For the formula editor an intermediate goal is addressed: Isabelle's one-dimensional term representation (with subscripts, superscrips, etc plus semantic information) perfectly fits the Braille. This is sufficient for logics and mathematics for eight graders. Since Chromium is a "rich client" in contrary to jEdit, a two-dimensional formula editor appears as a realisable final goal.

Open R&D tasks for the MAWEN project:
 * 1) Continue ongoing integration of ISAC into Isabelle. Accessibility will be a by-product of re-using Isabelle/VSCode.
 * 2) Implement an accessible formula editor mit read- and write-access on sub-terms.
 * 3) Implement ISAC-specific views in Isabelle/VSCode/MAWEN.
 * 4) Implement ISAC-specific interaktions in Isabelle/PIDE (see Pt.4 of R&D-tasks in MAWEN-Project-RU).
 * 5) Design and implement a dialog module together with a user model.
 * 6) Implement mathematics knowledge (problem patterns, algorithms, rule sets, term orders, etc) specific to ISAC.

Participate in the MAWEN project
Participation is possible on various ways and different levels, welcome are institutions and groups as well as individuals with specific expertise and interests.

Frontend development for mathematical software
This part of development concerns a wide range of experties, which is partitioned below such, that different interest are addressed:

(1) ISAC input language is (and will be further) implemented using definitional tools of Isabelle/Isar. These tools cope with syntax (parsers, etc.) as well as with semantics and integrate ISAC's mathematics engine (heavily re-using Isabelle) into Isabelle/PIDE, see here.

(2) System architecture is centered around Isabelle/PIDE (written in SML and Scala) and is under construction here for decades. Isabelle/PIDE needs adaptations for Isabelle/VSCode and for Isabelle/MAWEN. The latter extends interactivity such, that also adaptation of Node.js (as part of Electron) might be concerned.

(3) VScode/HTML/CSS/JS finally creates the user interface with different views, for Isabelle/VSCode an outline view and a debugger view, for Isabelle/MAWEN a view on problem patterns, on algorithms solving problems, etc. For that purpose VSCode APIs are available, to be complemented by HTML/CSS/JS-tools. Accessibility is inherited from the Chromium browser.

This part of development also comprises authoring tools.

Userinterface design for educational software
The userinterface Isabelle/VSCode/MAWEN combined with the mathematics engine of Isabelle/MAWEN will result in a powerful educational tool, which raises exciting challenges of completely novel kinds.

Dialogues between student and system can be seen as "dialogues on an equal base": both partners collaboratively contribute to the construction of a solution to a given problem. Meanwhile a "dialog guide" ensures, that students are neither overwhelmed nor bored -- and this requires a "user model" of hitherto unrivalled depth.

Concerns of accessibility will accompany interaction design. Specific attention will be paid to interactions on mathematical formulas and respective subterms, as described above.

Authoring of interaktive learning modules
In order to enable "next step guidance" (where the system is able to propose a next step within a problem solution) specific mathematics knowledge is required according to ISAC's concepts: types of mathematical problems, algorithms solving particular problems, rule sets, etc.

This kind of "mathematics authoring" is done within Isabelle/Isar/MAWEN and requires specific knowledge about computer mathematics. Another kind of authoring concerns the "user guide" (learning strategies, et) and "user model" (level of knowledge, et) addressed above

These kinds of authoring by course designers and textbook authors will leas to "massive open online courses (MOOCs)". These shall be open such, that teachers interested in modifying the MOOCs for the needs of their students can do so; of course, this shall include blind teachers.

Introduction of learning modules to schools
Isabelle/MAWEN will be an ideal tool in self-study for sighted and for blind students of any age and on levels from eight-graders to academic mathematics at engineering faculties.

In order to accomplish effectiveness of Isablle/MAWEN and wide-spread understanding of mechanistic aspects in mathematics, also introduction to schools shall be supported in an organised way. The various educational systems and cultural mentalities in different countries will be sustainably served only, if teachers and experts in didactics are involved. International cooperation can also improve mutual understanding at this point.

Specific support for blind children
Blind pupils cannot follow mathematics lessons as soon as formula become more complicated. In Austria they get specific support outside of lessons from remedial teachers in order to enable them to stay in their class. Software like Isabelle/MAWEN shall improve this situation considerably as described above.

In the medium term we expect variants of Isabelle/MAWEN on mobiles, since modern browser technologies are ready to present their content either on desktops or on mobiles. Together with Isabelle/MAWEN's affinity to games this could lead to meaningful gaming, in particular for visually impaired people who cultivate a life style without sports in general. As as indicated above, blind people could develop better skills in such games than sighted people.

Interest groups in youth camps
In Europ there are variouls summer camps for youth, for blind ones, too. Such camps are run with various cultural an sportive activities. Why not offer interest groups for mathematis, or gaming groups with handheld Isabelle/MAWEN, since this is close to games as described above?

And eventually international summer camps are not only occasions for crossing language borders, for enjoying cultural activities and sports together, but also discussing in-depth views, attitudes, aspirations and concerns of materialism and socialism mentioned in the introduction, backed by experiences from Isabelle/MAWEN?

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