Efraim Fischbein

The Theory of Figural Concepts

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.

The Evolution with Age of Probabilistic, Intuitively Based Misconceptions.

The purpose of this research was to investigate the evolution, with age, of probabilistic, intuitively based misconceptions. We hypothesized, on the basis of previous research with infinity concepts, that these misconceptions would stabilize during the emergence of the formal operation period. The responses to probability problems of students in Grades 5, 7, 9, and 11 and of prospective teachers indicated, contrary to our hypothesis, that some misconceptions grew stronger with age, whereas others grew weaker. Only one misconception investigated was stable across ages. An attempt was made to find a theoretical explanation for this rather strange and complex situation.

Does the Teaching of Probability Improve Probabilistic Intuitions

The paper analyzes the effects of a teaching programme in probability devised for junior high school pupils (grades 5, 6 and 7). It was found that most of the notions were too difficult for the fifth grade pupils. In contrast, about 60–70% of the sixth graders and about 80–90% of the seventh graders were able to understand and use correctly most of the concepts contained in the programme. It was also found that, as an indirect effect the course on probability had a beneficial effect on some intuitively based misconceptions of the subjects, like: the “representiveness” effect; the positive recency effect; the notion of “a lucky choice”; the superstitious belief in the possibility of influencing the course of events by some particular behaviour.

Intuitions and Schemata in Mathematical Reasoning

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.

Factors Affecting Probabilistic Judgements in Children and Adolescents.

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.

Defining in Classroom Activities.

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.

Tacit Models and Infinity.

The paper analyses several examples of tacit influences exerted by mental models on the interpretation of various mathematical concepts in the domain of actual infinity. The influences of the respective tacit models, being generally uncontrolled consciously, may lead to erroneous interpretations, to contradictions and paradoxes. The paper deals especially with the unconscious effect of the figural-pictorial models of statements related to the infinite sets of geometrical points (on a segment, a square, or a cube) related to the concepts of function and derivative and to the spatial interpretation of time and motion in Zenos paradoxes.

Is It Possible to Measure the Intuitive Acceptance of a Mathematical Statement

The main purpose of the present research was to check the possibility of measuring the feeling of “intuitive acceptance”, experienced by a subject when he offers an intuitive solution to a problem. It was postulated that two dimensions have to be considered and combined: The level of confidence and the degree of obviousness. Almost all the questions asked referred to the notion of infinity. The subjects were pupils belonging to grades 8 and 9. Three main categories of problematic situations have been identified:(a)Problems which got high percentages of correct solutions and high levels of intuitive acceptance.(b)Problems which got two types of contradictory solutions, each of them being accepted with moderate intuitiveness.(c)Problems which got low frequencies of correct solutions and high frequencies of typical incorrect solutions, the second category presenting higher levels of intuitive acceptance than the first (counter-intuitive problematic situations).

The concept of irrational numbers in high-school students and prospective teachers

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.

The psychological structure of naive impetus conceptions

Abstract The present paper analyses, on the basis of questionnaires and interviews, the various factors affecting the naive impetus interpretations in tenth and eleventh grade students. It has been found that, according to naive subjects, the action of impetus (even in the absence of any external force) depends on the shape, the weight and function of the moving body. A certain variety of naive impetus interpretations has also been found: the Marchian type‐‐the impetus is self‐expending; the Buridanian type‐‐the impetus is of a permanent nature; the transition to the Newtonian conception.