Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -it is as provocative as the idea of building educational software upon technology from computer theorem proving. In spite of recent success stories like the proofs of the Four Colour Theorem or the Kepler Conjecture, mechanised proof is still considered somewhat esoteric by mainstream mathematics.This paper describes the process of prototyping in the ISAC project from a technical perspective. This perspective depends on two moving targets: On the one side the rapidly increasing power and coverage of computer theorem provers and their user interfaces, and on the other side potential users: What can students and teachers request from educational systems based on technology and concepts from computer theorem proving, now and then?By the way of describing the process of prototyping the first comprehensive survey on the state of the ISAC prototype is given as a side effect, made precise by pointers to the code and by citation of all contributing theses. 1 In this paper TP abbreviates the academic discipline as well as the products this discipline develops, proof assistants and automated provers frequently included in the former.2 https://en.wikipedia.org/wiki/Dines_Bjoerner 3 https://de.wikipedia.org/wiki/Peter_Lucas_(Informatiker) 4 https://en.wikipedia.org/wiki/Bruno_Buchberger 5
A new generation of educational mathematics software is being shaped in ThEdu 1 and other academic communities on the side of computer mathematics. Respective concepts and technologies have been clarified to an extent, which calls for cooperation with educational sciences in order to optimise the new generation's impact on educational practice. The paper addresses educational scientists who want to examine specific software features and estimate respective effects in STEM education at universities and subsequently at high-school.The key features are characterised as a "complete, transparent and interactive model of mathematics", which offers interactive experience in all relevant aspects in doing mathematics. Interaction uses several layers of formal languages: the language of terms, of specifications, of proofs and of program language, which are connected by Lucas-Interpretation providing "next-step-guidance" as well as providing prover power to check user input.So this paper is structured from the point of view of computer mathematics and thus cannot give a serious description of effects on educational practice -this is up to collaboration with educational science; such collaboration is prepared by a series of questions, some of which are biased towards software usability (and mainly to be solved by computer mathematicians) and some of which are biased towards genuine research in educational sciences. 9 https://isabelle.in.tum.de/website-Isabelle2018/dist/library/HOL/index.html 10 https://www.isa-afp.org/
This paper discusses plans for joint work in order to gain early feedback from the community. Three lines of work pursued independently so far shall be joined: (1) narrowing the gap between declarative program specification and program generation already working in Isabelle, (2) reusing work, which embedded an input-response-loop resembling Computer Algebra Systems (CAS) into HOL Light, and (3) reconstructing an experimental language for applied mathematics by exploiting established as well as emerging features of Isabelle/Isar.These plans have to be seen as part of a variety of highly active research areas -on "integration of the deduction and the computational power" of Computer Theorem Proving (CTP) and CAS respectively (Calculemus), on "innovative theoretical and technological solutions for content-based systems" (MKM), on "Programming Languages for Mechanized Mathematics Systems" (PLMMS), just to cite from some related interest groups.Facing the abundant variety of approaches, of intermediate results and of ongoing developments, and taking under consideration the many difficulties in integrating such approaches, we pursue pragmatic goals:Design a component indispensable for working engineers, a programming language for engineering applications. Use Isabelle for an experimental embedding of the language, which is useful at least in engineering education as soon as possible.
Herewith, a fairly old concept is published for the first time and named "Lucas Interpretation". This has been implemented in a prototype, which has been proved useful in educational practice and has gained academic relevance with an emerging generation of educational mathematics assistants (EMA) based on Computer Theorem Proving (CTP). Automated Theorem Proving (ATP), i.e. deduction, is the most reliable technology used to check user input. However ATP is inherently weak in automatically generating solutions for arbitrary problems in applied mathematics. This weakness is crucial for EMAs: when ATP checks user input as incorrect and the learner gets stuck then the system should be able to suggest possible next steps. The key idea of Lucas Interpretation is to compute the steps of a calculation following a program written in a novel CTP-based programming language, i.e. computation provides the next steps. User guidance is generated by combining deduction and computation: the latter is performed by a specific language interpreter, which works like a debugger and hands over control to the learner at breakpoints, i.e. tactics generating the steps of calculation. The interpreter also builds up logical contexts providing ATP with the data required for checking user input, thus combining computation and deduction. The paper describes the concepts underlying Lucas Interpretation so that open questions can adequately be addressed, and prerequisites for further work are provided
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