数学文化与儿童数学认知

数学文化以文化为视角.在人类文化发展的历史过程中审视和理解数学,为我们认识和理解儿童数学认知开辟了全新的方向.作为数学文化的重要内容的数学信念、数学价值观、数学典故、数学符号、数学法则以及实物工具等,在儿童数学认知的知识提取、理解与表征、记忆与策略、认知风格、情绪态度以及自我效能等方面起着重要的影响作用.     

Mathematical Intuition Vs. Mathematical Monsters*

Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.     

Mathematical Spandrels

The aim of this paper is to open a new front in the debate between platonism and nominalism by arguing that the degree of explanatory entanglement of mathematics in science is much more extensive than has been hitherto acknowledged. Even standard examples, such as the prime life cycles of periodical cicadas, involve a penumbra of mathematical features whose presence can only be explained using relatively sophisticated mathematics. I introduce the term ‘mathematical spandrel’ to describe these penumbral properties, and focus on the property that cicada period lengths are expressible as the sum of two perfect squares. I argue that mathematical spandrels pose a particular problem for nominalism because of the way in which they are entangled with scientific explanations.     

Mathematical Intelligence and Mathematical Creativity: A Causal Relationship

This study investigated the causal relationship between mathematical creativity and mathematical intelligence. Four hundred thirty-nine 8th-grade students, age ranged from 11 to 14 years, were included in the sample of this study by random cluster technique on which mathematical creativity and Hindi adaptation of mathematical intelligence test were administered with 4-month time lag. Cross-lagged panel analysis was used to analyze the data. The uncorrected cross-lagged correlations appeared to show no causal relation between mathematical creativity and mathematical intelligence. But after the correction the difference in the cross-lagged correlations was found to be small and does not give guarantee of unidirectional causal relation between these two constructs. It revealed that there is a mutually reinforcing (symmetric) relationship between mathematical intelligence and mathematical creativity, i.e., mathematical intelligence causes mathematical creativity and vice-versa.     

Mathematical Intuition and the Cognitive Roots of Mathematical Concepts

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.     

Mathematical Creativity and Mathematical Aptitude: A Cross-Lagged Panel Analysis

Cross-lagged panel correlation (CLPC) analysis has been used to identify causal relationships between mathematical creativity and mathematical aptitude. For this study, 480 8th standard students were selected through a random cluster technique from 9 intermediate and high schools of Varanasi, India. Mathematical creativity and mathematical aptitude tests were administered twice, 4 months apart. The CLPC analysis uncovered a significant relationship between mathematical creativity and mathematical aptitude. It appears that mathematical creativity was a cause of mathematical aptitude, rather than the converse (i.e., higher mathematical creativity leading to higher mathematical aptitude). Limitations and future directions are outlined.     

Against Mathematical Explanation

Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “explanation,” the term is not applicable to mathematics as it is in science.     

Mathematical leadership vision

This article is an analysis of a new type of leadership vision, the kind of vision that is becoming increasingly pervasive among leaders in the modern world. This vision appears to offer a new horizon, whereas, in fact it delivers to its target audience a finely tuned version of the already existing ambitions and aspirations of the target audience. The leader, with advisors, has examined the target audience and has used the results of extensive research and statistical methods concerning the group to form a picture of its members' lifestyles and values. On the basis of this information, the leader has built a "vision." The vision is intended to create an impression of a charismatic and transformational leader when, in fact, it is merely a response. The systemic, arithmetic, and statistical methods employed in this operation have led to the coining of the terms mathematical leader and mathematical vision.     

Against Mathematical Convenientism

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine–Putnam indispensability argument and Baker’s enhanced indispensability argument.     

Mathematical biophysics of conditioning

It is shown that from a somewhat more precise form of the fundamental postulates, used in a previous paper as a starting point for a system of mathematical biophysics of psychological phenomena, a mechanism can be derived, which represents the important features of conditioning.     

Mathematical Creativity: A Combination of Domain-general Creative and Domain-specific Mathematical Skills

Creativity is an understudied topic in elementary school mathematics research. Nevertheless, we argue that creativity plays an important role in mathematics, but that more research is needed to understand this relation. Therefore, this study aimed to investigate this relation, specifically between domain-general creativity, domain-specific mathematical creativity, and mathematical ability. Measures for these constructs were administered to 342 Dutch fourth graders. In order to examine the nature of the relation between creativity and mathematics, two competing models were tested, using Structural Equation Modeling. The results indicated that models in which general creativity and mathematical ability both predict mathematical creativity fitted the data better than models in which mathematical and general creativity predict mathematical ability. This study showed that both general creativity and mathematical ability are important to think creatively in mathematics.