‘Infinity Means it Goes on Forever’: Siblings' Informal Teaching of Mathematics

Sibling‐directed teaching of mathematical topics during naturalistic home interactions was investigated in 39 middle‐class sibling dyads at two time points. At time 1 (T1), siblings were 2 and 4 years of age, and at time 2 (T2), siblings were 4 and 6 years of age. Intentional sequences of sibling‐directed mathematical teaching were coded for (i) topics (e.g., number), (ii) contexts (e.g., play with materials/toys), and (iii) type of knowledge (conceptual and procedural). Siblings engaged in teaching number, geometry, and measurement at T1 and demonstrated preliminary evidence of teaching of grouping, relations, and operations at T2. Regarding context, at T1, mathematical teaching occurred most frequently during play with materials/toys, while at T2, games with rules were prominent. Teaching of conceptual or procedural knowledge varied over time and by topic and context. Findings are discussed in light of recent work on understanding children's mathematical knowledge as it develops in the informal family context. Copyright © 2015 John Wiley & Sons, Ltd.     

A Naturalistic Look into Maddy's Naturalistic Philosophy of Mathematics

This paper discusses Penelope Maddy's (b.1950) naturalistic philosophy of mathematics,which is one of the most influential forms of post-Quinean naturalism in the philosophy of mathematics.Two defining features of Maddy's theory,namely the methodological autonomy of mathematics and the equivalence of Thin Realism and Arealism,are analyzed,and some criticisms of them are posed from within the naturalistic line of thought itself.In the course of this analysis and criticism,the paper will also consider Maddy's objections to the Quinean Indispensability Argument,which are the starting point of her own version of naturalism.     

父母教育卷入与小学儿童数学焦虑的纵向联系:数学态度的中介作用

为探讨父母教育卷入与学龄期儿童数学焦虑之间的纵向联系及内在作用机制,从山东省聊城市两所普通小学选取1734名三、四年级学生,对其进行为期一年的追踪研究。结果发现：（1）T1父母教育卷入能够显著负向预测T2儿童数学焦虑,但T1儿童数学焦虑对T2父母教育卷入的预测不显著;（2）在同一时间点上,儿童数学态度在父母教育卷入与儿童数学焦虑之间发挥显著中介作用;（3）在不同时间点上,儿童数学态度的中介作用仍然成立,表明儿童数学态度的中介作用具有跨时间的稳定性。该结果强调了父母教育卷入对学龄期儿童数学焦虑变化的重要作用,亦为从父母教育卷入和数学态度角度降低儿童的数学焦虑水平提供了初步有力证据。     

Some Implications of a Behavioral Analysis of Verbal Behavior for Logic and Mathematics

The evident power and utility of the formal models of logic and mathematics pose a puzzle: Although such models are instances of verbal behavior, they are also essentialistic. But behavioral terms, and indeed all products of selection contingencies, are intrinsically variable and in this respect appear to be incommensurate with essentialism. A distinctive feature of verbal contingencies resolves this puzzle: The control of behavior by the nonverbal environment is often mediated by the verbal behavior of others, and behavior under control of verbal stimuli is blind to the intrinsic variability of the stimulating environment. Thus, words and sentences serve as filters of variability and thereby facilitate essentialistic model building and the formal structures of logic, mathematics, and science. Autoclitic frames, verbal chains interrupted by interchangeable variable terms, are ubiquitous in verbal behavior. Variable terms can be substituted in such frames almost without limit, a feature fundamental to formal models. Consequently, our fluency with autoclitic frames fosters generalization to formal models, which in turn permit deduction and other kinds of logical and mathematical inference.     

数学学业不良初中生的工作记忆特点：领域普遍性还是特殊性?

研究选取上海111名(男生48名, 女生63名, 平均年龄11.97岁)初中学生(数学学业不良学生55名与数学学业优秀学生56名), 根据国家数学课程标准, 将数学划分为数与代数、空间与几何两部分, 又从空间与几何领域中选取初一数学学业知识点—— 轴对称和中心对称图形, 分析在不同内容知识领域的解答过程中所涉及的工作记忆成分。结果发现：(1)数与代数学习需要中央执行系统、视觉-空间模板、语音环路三个成分的共同作用; 空间与几何学习主要受到视觉-空间模板和中央执行系统的影响, 但不存在语音环路的影响。(2)对轴对称与中心对称图形任务的成绩影响最大的是视觉-空间模板, 其次是中央执行系统, 语音环路对该任务作用不明显。两个研究说明工作记忆在初中不同年级的各类数学学业任务中具有不同的作用, 中央执行系统和视觉-空间模板相对更具有普遍性作用, 语音环路具有特殊性作用, 并且随着年龄增高, 视觉－空间模板对数学学业任务的作用更为突显。     

Maimon’s Theory of Differentials As The Elements of Intuitions

Abstract

Maimon’s theory of the differential has proved to be a rather enigmatic aspect of his philosophy. By drawing upon mathematical developments that had occurred earlier in the century and that, by virtue of the arguments presented in the Essay and comments elsewhere in his writing, I suggest Maimon would have been aware of, what I propose to offer in this paper is a study of the differential and the role that it plays in the Essay on Transcendental Philosophy (1790). In order to do so, this paper focuses upon Maimon’s criticism of the role played by mathematics in Kant’s philosophy, to which Maimon offers a Leibnizian solution based on the infinitesimal calculus. The main difficulties that Maimon has with Kant’s system, the second of which will be the focus of this paper, include the presumption of the existence of synthetic a priori judgments, i.e. the question quid facti, and the question of whether the fact of our use of a priori concepts in experience is justified, i.e. the question quid juris. Maimon deploys mathematics, specifically arithmetic, against Kant to show how it is possible to understand objects as having been constituted by the very relations between them, and he proposes an alternative solution to the question quid juris, which relies on the concept of the differential. However, despite these arguments, Maimon remains sceptical with respect to the question quid facti.     

Kant's Neoplatonism: Kant and Plato on Mathematical and Philosophical Method

Both Plato and Kant devote much attention and care to deliberating about their method of philosophizing. And, interestingly, both seek to expand and explain their view of philosophical method by one selfsame strategy: explaining the contrast between rational procedure in mathematics and in philosophy. Plato and Kant agree on a fundamental point of philosophical method that is at odds with the mathematico‐demonstrative methodology of philosophy found in Spinoza and present in Christian Wolff. Both reject the axiomatic approach with its insistence on fundamental truths postulated from the outset. Both alike insist that philosophizing—unlike mathematics—is an exercise in theorizing where the questions of basicness and foundations come into view only after the inquiry has gone on for a long, long time—and certainly not at its start.     

Relationships of Cognitive and Metacognitive Learning Strategies to Mathematics Achievement in Four High-Performing East Asian Education Systems

The authors examined the relationships of cognitive (i.e., memorization and elaboration) and metacognitive learning strategies (i.e., control strategies) to mathematics achievement among 15-year-old students in 4 high-performing East Asian education systems: Shanghai-China, Hong Kong-China, Korea, and Singapore. In all 4 East Asian education systems, memorization strategies were negatively associated with mathematics achievement, whereas control strategies were positively associated with mathematics achievement. However, the association between elaboration strategies and mathematics achievement was a mixed bag. In Shanghai-China and Korea, elaboration strategies were not associated with mathematics achievement. In Hong Kong-China and Singapore, on the other hand, elaboration strategies were negatively associated with mathematics achievement. Implications of these findings are briefly discussed.     

Dispositionality: Beyond The Biconditionals

I defend pretence hermeneutic fictionalism against the Autism Objection. The objection is this: since people with autism have no difficulty in engaging with mathematics even if they cannot pretend, it is not the case that engagement with mathematics involves pretence. I show that a previous response to the objection is inadequate as a defence of the kind of pretence hermeneutic fictionalism put forward as a semantic thesis about the discourse in question. I claim that a more general response to the Autism Objection is to deny the premise that people with autism cannot pretend. To motivate this response, I appeal to psychological studies suggesting that people with autism can understand pretence and they can pretend under certain conditions. Finally, I provide explanations for why it is the case that people with autism do not have a problem with engaging in mathematics whereas they have so much difficulty with other kinds of figurative language and pretence.