Prototyping "Systems that Explain Themselves" for Education

Abstract

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

Lucas-Interpretation on Isabelle's Functions

“…Such adpation is concern of a dialogue-module: by use of pattern-matching on the next step a hint can be given in full or partially (from a formula or a tactic, see the example calculation on p.82), can be a list of options or can be idle (in case of a written exam). Such a dialogue-module is implemented as a stub in ISAC's prototype [22], but several field tests [35,33,34] already clarified a lot of requirements for such a module; realisation shall be in collaboration with experts in educational psychology -which exhibits a great advantage of LI over present principles of educational software: design and development in computer mathematics (writing LI-programs, mechanising deductive knowledge) are strictly separated from design and development of dialogues (extending prototypes like [18] and adopting educational theories).…”

“…LI aims at meeting these requirements for more than a decade, when Peter Lucas 3 shifted his interests from programming languages [24] to education. His specific contribution has been named after him, and now reveals the ingenuity of the original design, when the prototype has been migrated: a proprietary version [22] was migrated to Isabelle's function package [20,21] with surprising little effort.…”

Technologies for "Complete, Transparent & Interactive Models of Math" in Education

“…More and more knowledge mechanised in proof assistants addresses problems from STEM faculties, for instance Networks, Security, Economics or Probability Theory in Isabelle's Archive of Formal Proofs 14 . But the kind of explicit specification addressed in the paper is still missing; so specifications need to be added to existing knowledge -assuming to have knowledge associated with specifications covering all undergraduate mathematics seems not unrealistic; see for instance 15 .…”

“…Small field tests have been mentioned as successful in §1, but the tests also showed that usability in classroom requires considerable efforts to develop prototypes further to such a stage. A possible approach from the side of educational sciences could be to imagine "systems that explain themselves" announced in [15], somewhat naively from an educational point of view. From the side…”

“…In order to achieve a reliable source for educators, who want to be informed of these technologies and their relevance for mathematics education, the paper separates technical facts (already implemented in prototypes) and prospects (already in a concrete stage of planning for different software products) from foreseeable effects in education. Examples will be mostly taken from a prototype development [15] 8 where the author is involved and early field tests were successful [19,24,25].…”