Automated Generation of User Guidance by Combining Computation and Deduction

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.Comment: In Proceedings THedu'11, arXiv:1202.453

THREAD: A programming environment for interactive planning-level robotics applications

THREAD programming language, which was developed to meet the needs of researchers in developing robotics applications that perform such tasks as grasp, trajectory design, sensor data analysis, and interfacing with external subsystems in order to perform servo-level control of manipulators and real time sensing is discussed. The philosophy behind THREAD, the issues which entered into its design, and the features of the language are discussed from the viewpoint of researchers who want to develop algorithms in a simulation environment, and from those who want to implement physical robotics systems. The detailed functions of the many complex robotics algorithms and tools which are part of the language are not explained, but an overall impression of their capability is given

Exploring the mathematics of motion through construction and collaboration

In this paper we give a detailed account of the design principles and construction of activities designed for learning about the relationships between position, velocity and acceleration, and corresponding kinematics graphs. Our approach is model-based, that is, it focuses attention on the idea that students constructed their own models – in the form of programs – to formalise and thus extend their existing knowledge. In these activities, students controlled the movement of objects in a programming environment, recording the motion data and plotting corresponding position-time and velocity-time graphs. They shared their findings on a specially-designed web-based collaboration system, and posted cross-site challenges to which others could react. We present learning episodes that provide evidence of students making discoveries about the relationships between different representations of motion. We conjecture that these discoveries arose from their activity in building models of motion and their participation in classroom and online communities

A Universal Machine for Biform Theory Graphs

Broadly speaking, there are two kinds of semantics-aware assistant systems for mathematics: proof assistants express the semantic in logic and emphasize deduction, and computer algebra systems express the semantics in programming languages and emphasize computation. Combining the complementary strengths of both approaches while mending their complementary weaknesses has been an important goal of the mechanized mathematics community for some time. We pick up on the idea of biform theories and interpret it in the MMTt/OMDoc framework which introduced the foundations-as-theories approach, and can thus represent both logics and programming languages as theories. This yields a formal, modular framework of biform theory graphs which mixes specifications and implementations sharing the module system and typing information. We present automated knowledge management work flows that interface to existing specification/programming tools and enable an OpenMath Machine, that operationalizes biform theories, evaluating expressions by exhaustively applying the implementations of the respective operators. We evaluate the new biform framework by adding implementations to the OpenMath standard content dictionaries.Comment: Conferences on Intelligent Computer Mathematics, CICM 2013 The final publication is available at http://link.springer.com

Implementing PRISMA/DB in an OOPL

PRISMA/DB is implemented in a parallel object-oriented language to gain insight in the usage of parallelism. This environment allows us to experiment with parallelism by simply changing the allocation of objects to the processors of the PRISMA machine. These objects are obtained by a strictly modular design of PRISMA/DB. Communication between the objects is required to cooperatively handle the various tasks, but it limits the potential for parallelism. From this approach, we hope to gain a better understanding of parallelism, which can be used to enhance the performance of PRISMA/DB.\ud The work reported in this document was conducted as part of the PRISMA project, a joint effort with Philips Research Eindhoven, partially supported by the Dutch "Stimuleringsprojectteam Informaticaonderzoek (SPIN)

Supporting students with learning disabilities to explore linear relationships using online learning objects

The study of linear relationships is foundational for mathematics teaching and learning. However, students’ abilities connect different representations of linear relationships have proven to be challenging. In response, a computer-based instructional sequence was designed to support students’ understanding of the connections among representations. In this paper we report on the affordances of this dynamic mode of representation specifically for students with learning disabilities. We outline four results identified by teachers as they implemented the online lessons