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The role of visualisation in the process of generalisation

100%

Zaręba L.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

2015

tom 7

111-128

In my paper, I discuss the results of individual observations of pupils aged 13-14. These children undertook an attempt to solve problems which were designed in order to bring about an inductive type of generalizing. The main aim of the study was to classify typical methods of proceeding, which represent the pupils’ ways of reasoning and to determine what influence a particular method of proceeding may have on the final outcome of their work. These results indicate the general pupils’ strategies of generalizing. Presumably, visual thinking produces a positive effect on the pupils’ process of generalizing and abilities to describe regularity with the aid of a letter symbol.

Automation for Dependently Typed Functional Programming

70%

Wilson S. , Fleuriot J. , Smaill A.

2010

tom Vol. 102, nr 2

209-228

Writing dependently typed functional programs that capture non-trivial program properties is difficult in current systems due to lack of proof automation. We identify proof patterns that occur when programming with dependent types and detail how automating such patterns allow us to work more comfortably with types that capture, for example, membership, ordering and nonlinear arithmetic properties. We describe the role of the rippling heuristic, both for inductive and non-inductive proofs, and generalisation in providing such automation. We then discuss an implementation of our ideas in Coq with practical examples of dependently typed programs, that capture useful program properties, which can be verified automatically. We demonstrate that our proof automation is generic in that it can provide support for working with theorems involving user-defined functions and inductive data types.

K filozofickým předpokladům hledání výchovy k autenticitě. Úvaha o antinomiích a abstrakcích v pedagogickém myšlení

60%

Strouhal M.

Studia paedagogica

2013

nr 2-3

11-29

The paper examines the problematic nature of making generalisation ambivalent or, in other words, abstraction in pedagogical consequences. The paper adds to the discussion of good and evil in education by answering two questions. The fi rst question stems from the antinomous nature of educational aims (i.e. education is to servethe society but also to develop an individual): can educational antinomies be eliminated or is education an antinomous activity and hence it is necessary to take into account its ambivalence? The second question inquires to which extent do we understand what it means to be an authentic personality and the degree to which we can educate for authenticity. The paper proposes Kierkegaard’s and Blondel’s motive of authenticity as a partial way out of contradictions which result from the mentioned antinomies. The paper also shows that looking for education for authenticity is complicated by attempts at formulating a generally acceptable principle of education and that education suffers the most when it forgets about its antinomous nature. For its attempts to avoid contradictions lead to unacceptable abstractions and formalism. The paper then introduces the problem of making generalisation ambivalent in relation to educational competencies and concludes with describing the irrevocable yet restorative nature of antinomies.