心算的加工机制：来自认知神经科学的研究

心算加工分编码（表征）、运算（或提取）和反应三个阶段,这三个阶段相互影响。不同输入形式的数字表征在顶叶的不同区域完成。算术知识提取主要与左脑顶内沟有关,但当心算变得更复杂时而需要具体运算时,左脑额叶下部出现明显激活。所有与心算有关的脑区涉及大脑前额皮层和颞顶枕联合皮层的综合作用,并总体表现为左脑优势,但估算、珠心算以及某些具有特殊心算能力的人的心算还依赖视空间表征,这与右脑额顶区和楔前叶的活动有关     

"不可或缺性论证"与反实在论数学哲学

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. __________ Translated from Zhexue Yanjiu 哲学研究 (Philosophical Researches), 2006, (8): 74–83     

Erkenntnistheorie der Zahldefinition und philosophische Grundlegung der Arithmetik unter Bezugnahme auf einen Vergleich von Gottlob Freges Logizismus und platonischer Philosophie (Syrian, Theon von Smyrna u.a.)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. This revised version was published online in August 2006 with corrections to the Cover Date.     

Anxiety-induced miscalculations,more than differential inhibition of intuition,explain the gender gap in cognitive reflection

The Cognitive Reflection Test (CRT) is among the most common and well-known instruments for measuring the propensity to engage reflective processing, in the context of the dual-process theory of high-level cognition. There is robust evidence that men perform better than women on this test—but we should be wary to conclude that men are more likely to engage in reflective processing than women. We consider several possible loci for the gender difference in CRT performance, and use mathematical modeling to show, across two studies, that the gender difference in CRT performance is more likely due to women making more mathematical mistakes (partially explained by their greater mathematics anxiety) than due to women being less likely to engage reflective processing. As a result, we argue that we need to use gender-equivalent variants of the CRT, both to improve the quality of our instruments and to fulfill our social responsibility as scientists.     

EFFECTS OF HIGH-PREFERENCE SINGLE-DIGIT MATHEMATICS PROBLEM COMPLETION ON MULTIPLE-DIGIT MATHEMATICS PROBLEM PERFORMANCE

The purpose of this study was to examine the effects of a sequence of three single-digit (1 digit × 1 digit) multiplication problems on the latency to initiate multiple-digit (3 digit × 3 digit) multiplication problems for 2 students in an alternative education school. Data showed that (a) during the preference assessment, both students selected the single-digit problems in a majority of the sessions, and (b) intervention resulted in a decrease in latency between problems for both students. Results are discussed in relation to using high-preference sequences to promote behavioral momentum in academic content areas.     

The Nctm Standards and the Philosophy of Mathematics

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy.     

CONTROL OF ARITHMETIC ERRORS USING INFORMATIONAL FEEDBACK AND GRAPHING1

The effect of informational feedback and graphing on reducing the number of arithmetic worksheet errors was investigated. The present study replicated and extended earlier findings with other populations (Journal of Applied Behavior Analysis, 1974, 7 , 547–555; 1970, 3 , 1–4; 1970, 3 , 235–240) to a first-grade classroom. Ten first-grade pupils (four males and six females) served in an ABAB design. During the feedback-only phase, subjects were provided informational feedback, in the form of a written number, on the number of errors made on individual arithmetic worksheets. The feedback-only phase lasted seven days and was followed by the feedback-plus-graphing phase, during which subjects graphed the number (of errors written at the top of the individual worksheet) daily on individual graphs on their desks. The feedback-plus-graphing phase lasted 10 days and was followed by a reversal to feedback only for 10 days. The final phase was a replication of the feedback-plus-graphing phase. All subjects showed a change in number of worksheet errors, in the predicted direction, during the feedback-plus-graphing phases. An overall mean difference of ?2.66 was found to be statistically significant (P < 0.01), using a Wilcoxson Matched-Pairs Sign Test. The results were interpreted as being empirical support for earlier findings in other populations. In addition, the present findings represented a successful extension of feedback and graphing interventions to the management of academic behaviors in a first-grade classroom.     

REDUCING TIME LIMITS: A MEANS TO INCREASE BEHAVIOR OF RETARDATES1

A common assumption in special education is that temporal limits for a task should be expanded so that ample time is provided for completing the work. This study describes the opposite strategy of restricting temporal limits to augment academic performance. Three educable retarded children received token reinforcement contingent on the number of correct math problems answered during daily sessions. A reversal design was used to assess the effects of an abrupt reduction in time limits (20-5-20 min) and a graduated sequence of reductions (20-15-10-5-20 min). The graduated sequence resulted in rate increases of correct responding ranging from 125% to 266% and these gains endured when temporal limits were again expanded. In contrast, the abrupt shift produced interfering emotional behaviors and rate decreases in academic performance of 25% to 80%. The findings indicate that systematically restricting temporal limits for an academic task can further enhance the performance of slow learners already maintained by a token system.