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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Definability of the symbol 0° and the general problem of boundary cases of mathematical concepts

100%
Semadeni Z.
Didactica Mathematicae
|
2006
|
tom 29
|
nr 01
EN
The article contains no abstract
2
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Replacing objects of mathematics by other objects with the same name

100%
Semadeni Z.
Didactica Mathematicae
|
2007
|
tom 30
EN
The purpose of the paper is to introduce and analyse the following conception: by an ontic shift $X \tilde \rightarrow X''$ we mean an (explicit or hidden) replacement of a mathematical object $X'$ called $X$ by a different object $X''$ which is also called $X$ and is to play the role of $X'$. If $X'$ and $X''$ are sets, then $X' \neq X''$ means that these sets are different in the usual sense. Otherwise $X'$ and $X''$ may stand for basic concepts of mathematics such as, e.g., point, straight line, natural number, addition of natural numbers, (standard) real number, set, ordered pair, and then $X' \neq X''$ is interpreted in terms of the Leibniz principle of indiscernibility. Several examples from secondary and college mathematics are discussed starting with the ontic shift $P \tilde \rightarrow (x, y)$, where a point $P$ of the plane is replaced by a pair $(x, y)$ of real numbers, also called a point; an angle $\varphi$ (thought of as a geometric figure) is replaced by its measure which is also denoted by $\varphi$; a number (natural, integer, etc.) is replaced by a sophisticated set; a real number a is replaced by the complex number $(a, 0)$, and so on. We distinguish three types of ontic shifts: (α) object $X'$ is replaced by $X''$, and $X'$ is discarded; (β) object $X'$ is replaced by a conglomerate $X' \coprod X''$ of two alternatively and exibly used objects $X'$ and $X''$; (γ) object $X'$ is replaced by a new object which is a mental synthesis $X' \& X''$ integrating features of $X'$ and $X''$. What is crucial in ontic shifts is the change of approach: $X'$ may be identified with $X''$ by declaring that: $X'$ is the same as $X''$. For example, "a function may be identified with a set of pairs" is replaced by "a function is the same as its graph", i.e., a function becomes a set of pairs. In turn, a set of pairs of real numbers may be identified with a geometric figure. If such ontic shifts were composed, a function such as, say, $x \mapsto \sin x$ would be the same as a geometric figure. Such arguments show that ontic shifts are often local in the sense that they apply only to a certain part of mathematics and composition of two identifications may be unacceptable. A point in $\mathbb{R}^3$ may be replaced by the corresponding vector and one may say: "A point is the same as a vector". Also a vector may be replaced by the corresponding translation and then we may say that "A vector is the same as a translation". This identification, however, is not transitive. To see that the statement "point is the same as translation" is unacceptable imagine a student who is asked "How do we define a translation of 3D-space?" and answers: "A translation is an arbitrary point of the space". The paper also deals with distinguishing between ontic shifts and metonymies and with the delicate border between two phenomena: (a) a change of the linguistic form, and (b) replacing one mathematical object by another. Although labelling different objects with the same name does not fit the stereotypical image of mathematics, ontic shifts are its significant feature. They should be used in an elastic way, with proper understanding of the concepts involved. However, if such shifts are introduced prematurely, they cause serious didactical troubles.
3
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Flattening of hierarchies of concepts, horizontal and vertical mathematizing and the ambiguity of the term ‘model’

100%
Semadeni Z.
Didactica Mathematicae
|
2003
|
tom 25
|
nr 01
EN
The article contains no abstract
4
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Arrow graphs and trees as iconic-enactive representations

100%
Semadeni Z.
Didactica Mathematicae
|
1992
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tom 14
|
nr 01
EN
The article contains no abstract
5
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Enactive and iconic representations in Bruner’s sense exemplified by representation of set-theoretical notions

100%
Semadeni Z.
Didactica Mathematicae
|
1982
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tom 1
|
nr 01
EN
The article contains no abstract
6
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

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100%
Semadeni Z.
Didactica Mathematicae
|
2005
|
tom 28
|
nr 01
EN
The article contains no abstract
7
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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States and actions on states as meaning-based aspects of mathematical concepts

100%
Semadeni Z.
Didactica Mathematicae
|
2004
|
tom 27
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nr 01
EN
The article contains no abstract
8
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Categories of thought in the history of science and in mathematics education

100%
Semadeni Z.
Didactica Mathematicae
|
2004
|
tom 27
|
nr 01
EN
The article contains no abstract
9
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

Meaning-based aspects and structural aspects of mathematical concepts

100%
Semadeni Z.
Didactica Mathematicae
|
2004
|
tom 27
|
nr 01
EN
The article contains no abstract
10
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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A conception of a web of ties between deep ideas a rid ties between their formal models

100%
Semadeni Z.
Didactica Mathematicae
|
2005
|
tom 28
|
nr 01
EN
The article contains no abstract
11
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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The triple nature of mathematics: deep ideas, surface forms and formal models

100%
Semadeni Z.
Didactica Mathematicae
|
2002
|
tom 24
|
nr 01
EN
The article contains no abstract
12
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Epistemological difficulties concerning the concepts: “ordered pair” and “function”

100%
Semadeni Z.
Didactica Mathematicae
|
2002
|
tom 24
|
nr 01
EN
The article contains no abstract
13
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

Identification of concepts, reductionism and equality in mathematics

100%
Semadeni Z.
Didactica Mathematicae
|
2002
|
tom 24
|
nr 01
EN
The article contains no abstract
14
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

The role of meaning in mathematical reasoning

100%
Semadeni Z.
Didactica Mathematicae
|
2002
|
tom 24
|
nr 01
EN
The article contains no abstract
15
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

Deep ideas of axjomatically defined mathematical structures

88%
Semadeni Z.
Didactica Mathematicae
|
2005
|
tom 28
|
nr 01
EN
The article contains no abstract
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