Mathematics ability and related skills in preschoolers born very preterm

Children born very preterm (VPT) are at risk for academic, behavioral, and/or emotional problems. Mathematics is a particular weakness and better understanding of the relationship between preterm birth and early mathematics ability is needed, particularly as early as possible to aid in early intervention. Preschoolers born VPT (n = 58) and those born full term (FT; n = 29) were administered a large battery of measures within 6 months of beginning kindergarten. A multiple-mediation model was utilized to characterize the difference in skills underlying mathematics ability between groups. Children born VPT performed significantly worse than FT-born children on a measure of mathematics ability as well as full-scale IQ, verbal skills, visual–motor integration, phonological awareness, phonological working memory, motor skills, and executive functioning. Mathematics was significantly correlated with verbal skills, visual–motor integration, phonological processing, and motor skills across both groups. When entered into the mediation model, verbal skills, visual–motor integration, and phonological awareness were significant mediators of the group differences. This analysis provides insights into the pre-academic skills that are weak in preschoolers born VPT and their relationship to mathematics. It is important to identify children who will have difficulties as early as possible, particularly for VPT children who are at higher risk for academic difficulties. Therefore, this model may be used in evaluating VPT children for emerging difficulties as well as an indicator that if other weaknesses are found, an assessment of mathematics should be conducted.     

The Relations Between Patterning,Executive Function,and Mathematics

Patterning, or the ability to understand patterns, is a skill commonly taught to young children as part of school mathematics curricula. It seems likely that some aspects of executive function, such as cognitive flexibility, inhibition, and working memory, may be expressed in the patterning abilities of children. The primary objective of the present study was to examine the relationship between patterning and executive functioning for first grade children. In addition, the relations between patterning, executive functioning, mathematics, and reading were examined. The results showed that patterning was significantly related to cognitive flexibility and working memory, but not to inhibition. Patterning, cognitive flexibility, and working memory were significantly related to mathematical skills. Only patterning and working memory were significantly related to reading. Regression analyses and structural equation modeling both showed that patterning had effects on both reading and mathematics measures, and that the effects of cognitive flexibility were entirely mediated by patterning. Working memory had independent effects on reading and mathematics, and also effects moderated by patterning. In sum, these findings suggest that cognitive flexibility and working memory are related to patterning and express their effects on reading and mathematics in whole or in part through patterning.     

中学生社会支持对数学学习坚持性的影响:数学自我效能感的中介作用

以北京市通州区一所普通中学的228名初一和初二学生为被试,采用领悟社会支持量表、数学自我效能感和数学学习坚持性问卷,考察在中学生数学学习中不同来源的社会支持对数学学习坚持性的影响,并检验数学自我效能感在其中的中介作用。结果发现:(1)社会支持中的教师支持和同伴支持能够显著正向预测数学学习坚持性水平,而父母支持的预测作用不显著;(2)数学自我效能感在同伴支持和数学学习坚持性之间起完全中介作用,在教师支持和数学学习坚持性之间起部分中介作用。     

学习困难的ERP研究

事件相关脑电位（Event—related potentials,ERPs）是研究心理学和认知神经科学的重要技术手段,将该技术用于学习困难的脑机制的研究,有助于发现学习困难的神经机制。研究表明,学习困难者的P300波幅较小,潜伏期较长。学习困难者的MMN波幅比控制组小,在信息的自动加工方面存在缺陷。在单词命名任务中,学习困难者的N400较小,而且不同类型学习困难者的波幅和潜伏期存在明显差异。     

Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency (LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8] and [9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8] and [9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbert?s program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B∧¬B and falsifies many arbitrarily complex instances of non-contradiction principle ¬(A∧¬A). Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACIarithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.     

Structuralism as a philosophy of mathematical practice

This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. I wish to thank Colin McLarty as well as the anonymous referees for helpful comments on earlier versions of this paper.     

中国苏州与美国15岁学生数学学习特征比较

本研究首次采用"学生能力国际评估计划"(PISA)的学生问卷,对苏州市504名15岁学生的数学学习心理及特征进行了调查研究,并与美国学生数据作比较.调查表明:中国苏州大多数学生在数学学习上具有较强的竞争意识与合作意识,他们会运用多种学习策略;那些对数学有更大兴趣和更高动机的学生,有更积极的自我概念和更少的焦虑体验.中美两国学生的数学学习特征性别差异显著.     

The ontological commitments of inconsistent theories

In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer to the last question: there are times when it is legitimate to believe in inconsistent objects.     

Is ZF a hack?: Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics

This paper presents Automath encodings (which are also valid in LF/λP) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.

The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church's higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo's extended calculus of constructions, and Martin-Löf's predicative type theory) and one foundation based on category theory.

The conclusions of this paper are that the simplest system is type theory (the calculus of constructions), but that type theories that know about serious mathematics are not simple at all. In that case the set theories are the simplest. If one looks at the number of concepts needed to explain such a system, then higher order logic is the simplest, with twenty-five concepts. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.

(The full Automath sources of the contexts described in this paper are one the web at http://www.cs.ru.nl/~freek/zfc-etc/.)