Knowledge Spaces and Learning Spaces

How to design automated procedures which (i) accurately assess the knowledge of a student, and (ii) efficiently provide advices for further study? To produce well-founded answers, Knowledge Space Theory relies on a combinatorial viewpoint on the assessment of knowledge, and thus departs from common, numerical evaluation. Its assessment procedures fundamentally differ from other current ones (such as those of S.A.T. and A.C.T.). They are adaptative (taking into account the possible correctness of previous answers from the student) and they produce an outcome which is far more informative than a crude numerical mark. This chapter recapitulates the main concepts underlying Knowledge Space Theory and its special case, Learning Space Theory. We begin by describing the combinatorial core of the theory, in the form of two basic axioms and the main ensuing results (most of which we give without proofs). In practical applications, learning spaces are huge combinatorial structures which may be difficult to manage. We outline methods providing efficient and comprehensive summaries of such large structures. We then describe the probabilistic part of the theory, especially the Markovian type processes which are instrumental in uncovering the knowledge states of individuals. In the guise of the ALEKS system, which includes a teaching component, these methods have been used by millions of students in schools and colleges, and by home schooled students. We summarize some of the results of these applications

Knowledge Spaces and the Completeness of Learning Strategies

We propose a theory of learning aimed to formalize some ideas underlying Coquand's game semantics and Krivine's realizability of classical logic. We introduce a notion of knowledge state together with a new topology, capturing finite positive and negative information that guides a learning strategy. We use a leading example to illustrate how non-constructive proofs lead to continuous and effective learning strategies over knowledge spaces, and prove that our learning semantics is sound and complete w.r.t. classical truth, as it is the case for Coquand's and Krivine's approaches

A note on complementary knowledge spaces

The pair $(Q, \mathscr{K})$ is a {\it knowledge space} if $\bigcup\mathscr{K}=Q$ and $\mathscr{K}$ is closed under union, where $Q$ is a nonempty set and $\mathscr{K}$ is a family of subsets of $Q$. A knowledge space $(Q, \mathscr{K})$ is called {\it complementary} if there exists a non-discrete knowledge space $(Q, \mathscr{L})$ such that the following (i) and (ii) satisfy: (i) for any $q\in Q$, there are finitely many $K_{1}, \cdots, K_{n}\in \mathscr{K}$ and $L_{1}, \cdots, L_{m}\in \mathscr{L}$ such that $$(\bigcap_{i=1}^{n}K_{i})\cap (\bigcap_{j=1}^{m}L_{j})=\{q\};$$ (ii) $\mathscr{K}\cap \mathscr{L}=\{\emptyset, Q\}$. In this paper, the existence of a complementary knowledge space for each knowledge space is proved, and a method of the construction of complementary finite knowledge spaces is given.Comment: 5 page

EPOS : evolving personal to organizational knowledge spaces

EPOS will leverage the user´s personal workspace with its manyfold native information structures to his personal knowledge space and in cooperation with other personal workspaces contribute to the organizational knowledge space which is represented in the organizational memory. This first milestone presents results from the project´s first year in the areas of the personal informational model, user observation for context elicitation, collaborative information retrieval and information visualization

Virtual knowledge spaces: A call for research

Davis (1989) authored the widely acclaimed book titled “Future Perfect” prescribing that, in a “future perfect, anyone in an anytime – anyplace mode would be able to communicate to anyone else in the world.” The year 2020 provided clear affirmation that the knowledge workforce of the future is poised to not only communicate anytime – anyplace, but to create workplace environments that thrive across time zones and unlimited virtual locations. Knowledge management (KM) is “a systematic and integrative process of coordinating organization-wide activities of acquiring, creating, storing, sharing, diffusing, and deploying knowledge by individuals and groups, in pursuit of major organizational goals” (Rastogi, 2000, p. 40). Information scientists and knowledge management scholars must reexamine models of organizational learning, competency development and organizational culture to harness the collective capability of not only a virtual workforce, but a virtual organization. The researchers’ “work in progress” poster presents a preliminary systematic literature review and offers guiding questions to scholars and scholar practitioners exploring this rich area of KM research in a virtual organization. The three primary research areas are organizational learning, knowledge archiving, and knowledge system modeling. The final systematic literature review will define the topic and will utilize scholarly research methodologies (e.g., Torocco, 2016) to critically analyze and synthesize existing knowledge management literature and present virtual workforce implications that give direction for future research. In this growing research area, this poster poses the questions: (1) What are the obstacles of storing and deploying knowledge in a virtual organization? (2) How does the virtual organization impact the social nature of knowledge (namely sharing and creation)? (3) How must knowledge systems evolve to accommodate a virtual workforce? Davis, S. (1989). Future Perfect. Reading, Massachusetts: Addison-Wesley. Rastogi, P. (2000) Knowledge management and intellectual capital — the new virtuous reality of competitiveness. Human Systems Management 19(1), 39 – 49. Torocco, R. (2016). Writing Integrative Reviews of Literature: Methods and Purposes. International Journal of Adult Vocational Education and Technology, 7(1), 62 – 70. doi: 10.4018/IJAVET.201607010

Knowledge Spaces and the Completeness of Learning Strategies

We propose a theory of learning aimed to formalize some ideas underlying Coquand\u27s game semantics and Krivine\u27s realizability of classical logic. We introduce a notion of knowledge state together with a new topology, capturing finite positive and negative information that guides a learning strategy. We use a leading example to illustrate how non-constructive proofs lead to continuous and effective learning strategies over knowledge spaces, and prove that our learning semantics is sound and complete w.r.t. classical truth, as it is the case for Coquand\u27s and Krivine\u27s approaches

Human exploration of complex knowledge spaces

Driven by need or curiosity, as humans we constantly act as information seekers. Whenever we work, study, play, we naturally look for information in spaces where pieces of our knowledge and culture are linked through semantic and logic relations. Nowadays, far from being just an abstraction, these information spaces are complex structures widespread and easily accessible via techno-systems: from the whole World Wide Web to the paramount example of Wikipedia. They are all information networks. How we move on these networks and how our learning experience could be made more efficient while exploring them are the key questions investigated in the present thesis. To this end concepts, tools and models from graph theory and complex systems analysis are borrowed to combine empirical observations of real behaviours of users in knowledge spaces with some theoretical findings of cognitive science research. It is investigated how the knowledge space structure can affect its own exploration in learning-type tasks, and how users do typically explore the information networks, when looking for information or following some learning paths. The research approach followed is exploratory and moves along three main lines of research. Enlarging a previous work in algorithmic education, the first contribution focuses on the topological properties of the information network and how they affect the \emph{efficiency} of a simulated learning exploration. To this end a general class of algorithms is introduced that, standing on well-established findings on educational scheduling, captures some of the behaviours of an individual moving in a knowledge space while learning. In exploring this space, learners move along connections, periodically revisiting some concepts, and sometimes jumping on very distant ones. To investigate the effect of networked information structures on the dynamics, both synthetic and real-world graphs are considered, such as subsections of Wikipedia and word-association graphs. The existence is revealed of optimal topological structures for the defined learning dynamics. They feature small-world and scale-free properties with a balance between the number of hubs and of the least connected items. Surprisingly the real-world networks analysed turn out to be close to optimality. To uncover the role of semantic content of the bit of information to be learned in a information-seeking tasks, empirical data on user traffic logs in the Wikipedia system are then considered. From these, and by means of first-order Markov chain models, some users paths over the encyclopaedia can be simulated and treated as proxies for the real paths. They are then analysed in an abstract semantic level, by mapping the individual pages into points of a semantic reduced space. Recurrent patterns along the walks emerge, even more evident when contrasted with paths originated in information-seeking goal oriented games, thus providing some hints about the unconstrained navigation of users while seeking for information. Still, different systems need to be considered to evaluate longer and more constrained and structured learning dynamics. This is the focus of the third line of investigation, in which learning paths are extracted from advances scientific textbooks and treated as they were walks suggested by their authors throughout an underlying knowledge space. Strategies to extract the paths from the textbooks are proposed, and some preliminary results are discussed on their statistical properties. Moreover, by taking advantages of the Wikipedia information network, the Kauffman theory of adjacent possible is formalized in a learning context, thus introducing the adjacent learnable to refer to the part of the knowledge space explorable by the reader as she learns new concepts by following the suggested learning path. Along this perspective, the paths are analysed as particular realizations of the knowledge space explorations, thus allowing to quantitatively contrast different approaches to education