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  • 1
    Electronic Resource
    Electronic Resource
    Factors affecting probabilistic judgements in children and adolescents (1991)
    Springer
    Educational studies in mathematics 22 (1991), S. 523-549 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Six hundred and eighteen pupils, enrolled in elementary and junior-high-school classes (Pisa, Italy) were asked to solve a number of probability problems. The main aim of the investigation has been to obtain a better understanding of the origins and nature of some probabilistic intuitive obstacles. A linguistic factor has been identified: It appears that for many children, the concept of “certain events’ is more difficult to comprehend than that of “possible events”. It has been found that even adolescents have difficulties in detaching the mathematical structure from the practical embodiment of the stochastic situation. In problems where numbers intervene, the magnitude of the numbers considered has an effect on their probability: bigger numbers are more likely to be obtained than smaller ones. Many children seem to be unable to solve probability questions, because of their inability to consider the rational structure of a hazard situation: “chance” is, by itself, an equalizing factor of probabilities. Positive intuitive capacities have also been identified: some problems referring to compound events are better solved when addressed in a general form than when addressed in a particular way.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00312714
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The theory of figural concepts (1993)
    Springer
    Educational studies in mathematics 24 (1993), S. 139-162 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The main thesis of the present paper is that geometry deals with mental entities (the so-called geometrical figures) which possess simultaneously conceptual and figural characters. A geometrical sphere, for instance, is an abstract ideal, formally determinable entity, like every genuine concept. At the same time, it possesses figural properties, first of all a certain shape. The ideality, the absolute perfection of a geometrical sphere cannot be found in reality. In this symbiosis between concept and figure, as it is revealed in geometrical entities, it is the image component which stimulates new directions of thought, but there are the logical, conceptual constraints which control the formal rigour of the process. We have called the geometrical figuresfigural concepts because of their double nature. The paper analyzes the internal tensions which may appear in figural concepts because of this double nature, development aspects and didactical implications.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01273689
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    The concept of irrational numbers in high-school students and prospective teachers (1995)
    Springer
    Educational studies in mathematics 29 (1995), S. 29-44 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It has been assumed, on historical and psychological grounds, that the concept of irrational numbers faces two major intuitive obstacles: a) the difficulty to accept that two magnitudes (two line segments) may be incommensurable (no common unit may be found); and b) the difficulty to accept that the set of rational numbers, though everywhere dense, does not cover all the points in an interval: one has to consider also the more “rich” infinity of irrational points. In order to assess the presence and the effects of these obstacles, three groups of subjects were investigated: students in grades 9 and 10 and prospective teachers. The results did not confirm these hypotheses. Many students are ignorant when asked to classify various numbers (rational, irrational, real) but only a small part of the subjects manifest genuine intuitive biases. It has been concluded that such erroneous intuitions (a common unit can always be found by indefinitely decreasing it and “in an interval it is impossible to have twodifferent infinite sets of points [or numbers]”) have not a primitive nature. They imply a certain intellectual development.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01273899
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Choice of operation in verbal arithmetic problems: the effects of number size, problem structure and context (1984)
    Springer
    Educational studies in mathematics 15 (1984), S. 129-147 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract 12-and 13-year-olds were tested with two types of tasks to test their understanding of applications of the multiplication and division of positive numbers: (i) writing down calculations required to solve verbal problems, and (ii) making up stories to fit given calculations. Selected pupils were interviewed to investigate further the thinking processes involved. The results indicate (a) the pervasive nature of certain numerical misconceptions, (b) the effects of structural differences among the items; particularly whether multiplication can be conceived as repeated addition or not, and whether division has the structure of partition, quotition or rate, (c) specific effects of context attributable to such aspects as relative familiarity, and (d) various interactions between these three sets of factors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00305893
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Intuitions and Schemata in Mathematical Reasoning (1999)
    Springer
    Educational studies in mathematics 38 (1999), S. 11-50 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper is an attampt to analyze the relationship between intuitions and structural schemata. Intuitions are defined as cognitions which appear subjectively to be self-evident, immediate, certain, global, coercive. Structural schemata are behavioral-mental devices which make possible the assimilation and interpretation of information and the adequate reactions to various stimuli. Structural schemata are characterized by their general relevance for the adaptive behavior. The main thesis of the paper is that intuitions are generally based on structural schemata. The transition from schemata to intuitions is achieved by a particular process of compression described in the paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1003488222875
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    The Mathematical Concept of set and the 'Collection' Model (1998)
    Springer
    Educational studies in mathematics 37 (1998), S. 1-22 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Analysing the various misconceptions held by students with regard to the mathematical set concept, the authors hypothesized that these misunderstandings may be explained by the initial ‘collection’ model. Even after learning the formal properties of a set in the mathematical sense, the students are still influenced in their reactions by the collection representation, which acts ‘from behind the scenes’ as a tacit model. If the mathematical concept is not continually reinforced through systematic use, it is the initial figural interpretation which will replace, as an effect of time, the formal one. The findings confirmed this hypothesis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1003421206945
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Defining in Classroom Activities (1997)
    Springer
    Educational studies in mathematics 34 (1997), S. 219-248 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper discusses some aspects concerning the defining process in geometrical context, in the reference frame of the theory of ‘figural concepts’. The discussion will consider two different, but not antithetical, points of view. On the one hand, the problem of definitions will be considered in the general context of geometrical reasoning; on the other hand, the problem of definition will be considered an educational problem and consequently, analysed in the context of school activities. An introductory discussion focuses on definitions from the point of view of both Mathematics and education. The core of the paper concerns the analysis of some examples taken from a teaching experiment at the 6th grade level. The interaction between figural and conceptual aspects of geometrical reasoning emerges from the dynamic of collective discussions: the contributions of different voices in the discussion allows conflicts to appear and draw toward a harmony between figural and conceptual components. A basic role is played by the intervention of the teacher in guiding the discussion and mediating the defining process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1002985109323
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    Schemata and Intuitions in Combinatorial Reasoning (1997)
    Springer
    Educational studies in mathematics 34 (1997), S. 27-47 
    Details
    ISSN: 1573-0816
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem that inspired the present research refers to the relationships between schemata and intuitions. These two mental categories share a number of common properties: ontogenetic stability, adaptive flexibility, internal consistency, coerciveness and generality. Schemata are defined following the Piagetian line of thought, either as programs for processing and interpreting information or as programs for designing and performing adaptive reactions. Intuitions are defined in the present article as global, immediate cognitions. On the basis of previous findings (Fischbein et al., 1996; Siegler, 1979; Wilkening, 1980; Wilkening & Anderson, 1982), our main hypothesis was that intuitions are always based on certain structural schemata. In the present research this hypothesis was checked with regard to combinatorial problems (permutations, arrangements with and without replacement, combinations). It was found that intuitions, even when expressed as instantaneous guesses, are; in fact, manipulated'behind the scenes' (correctly or incorrectly) by schemata. This implies that, in order to influence, didactically, students' intuitions, those schemata on which these intuitions are based should be identified and acted upon.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1002914202652
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    Paper (German National Licenses)
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