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A task about a cube; or, on generalization in 3D

100%
Jureczko J.
Didactica Mathematicae
|
2016
|
tom 38
EN
The necessity of teaching such an activity as generalization was considered by Z. Krygowska in the ‘70s (Krygowska, 1977). The formation of this ability requires an adequate selection of non-stereotypical tasks, for which the algorithm is unknown to the person solving the task, ones in which a student is forced to search for their own method of solving the task on the basis of their knowledge. In literature, there are known studies concerning the process of generalization of students of different ages which use, at most, 2D visual patterns. However, the author still did not find any research based on tasks which examine the process of generalization in 3D. In this paper, results will be shown of using a task concerning a cube carried out in a diverse group of students from middle school, high school, and university.
2
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O roli prac dyplomowych z dydaktyki matematyki w kształceniu przyszłych nauczycieli matematyki

88%
Major J., Olik-Pawlik B., Ratusiński T., Zaręba L.
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
|
2017
|
tom 9
111-138
EN
In the training process of students – future teachers of mathematics, an important role is played by, among others, participation in the diploma seminar, during which the student (being at the final stage of studies)is tasked with preparing a bachelor’s or master’s thesis. This dissertation may be purely mathematical in nature or refer to problems in didactics of mathematics. The article aims to develop and illustrate some thoughts from previous works of the authors (Zaręba, 2009; Major, Olik-Pawlik, Ratusiński, Zaręba, 2016), pointing to the legitimacy of the preparation of teacher diploma theses in the field of didactics of mathematics by students.
3
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Isomorphism as generalization tool (in combinatorics)

88%
Krpec R.
Didactica Mathematicae
|
2016
|
tom 38
EN
The article deals with constructing the discovery of the number of combinations. It stems from the approach of genetic constructivism, which works with the generic model theory, and its research goal was to develop the stages of isolated models, particularly using the phenomenon of isomorphism. Children aged 11–13 participated in the experiment during their mathematics club, in which they worked in groups at first and together at the blackboard afterwards. The results come from analyzing the experiment protocol and a video recording of part of the experiment. In this study, the authors focus primarily on a detailed analysis of individual sub-phases of isolated models under the generic model theory. The experiment highlights the importance of isomorphism as a generalization tool. The authors have identified the obstacles on the way to advanced sub-phases of isolated models. Another contribution is the division of the fourth sub-phase into two separate sub-phases, which has been achieved using the method of atomic analysis.
4
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Probabilities of discrepancy between minima of cross-validation, Vapnik bounds and true risks

63%
Klęsk P.
International Journal of Applied Mathematics and Computer Science
|
2010
|
tom 20
|
nr 3
525-544
EN
Two known approaches to complexity selection are taken under consideration: n-fold cross-validation and structural risk minimization. Obviously, in either approach, a discrepancy between the indicated optimal complexity (indicated as the minimum of a generalization error estimate or a bound) and the genuine minimum of unknown true risks is possible. In the paper, this problem is posed in a novel quantitative way. We state and prove theorems demonstrating how one can calculate pessimistic probabilities of discrepancy between these minima for given for given conditions of an experiment. The probabilities are calculated in terms of all relevant constants: the sample size, the number of cross-validation folds, the capacity of the set of approximating functions and bounds on this set. We report experiments carried out to validate the results.
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