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.     

Goodbye and farewell: Siegel vs. Feyerabend∗

The discussion between Searle and the Churchlands over whether or not symbolmanipulating computers generate semantics will be confronted both with the rulesceptical considerations of Kripke/Wittgenstein and with Wittgenstein's privatelanguage argument in order to show that the discussion focuses on the wrong place: meaning does not emerge in the brain. That a symbol means something should rather be conceived as a social fact, depending on a mutual imputation of linguistic competence of the participants of a linguistic practice to one another. The alternative picture will finally be applied to small children, animals, and computers as well.     

A first grade student’s exploration of variable and variable notation / Una alumna de primer grado explora las variables y su notación

Abstract

This paper presents a case study of a first grade student to illustrate the diversity of her understandings related to variables and variable notation. While prior research has documented secondary school students’ difficulties with variables and variable notation, we identify many productive understandings in this much younger student, leading us to question the prevailing argument that students might have difficulties with variables due mostly to their own limitations. We draw our data from a teaching experiment that explored functional relationships. Individual interviews were carried out with a subset of the students in the experiment prior to, as well as mid-way through and at the end of the experiment. This paper focuses on a set of three interviews with one of the first grade students. We illustrate the shifts that occurred in the student’s understandings about variables and variable notation across as well as within each of the three interviews.     

数学证明:数学真理的来源?

对于一个数学实在论者来说,一个特别急迫的任务是回答数学证明是如何建立起关于数学对象的真理性问题。本文批判性考察由Resnik所提出的认知解答,指出了其中不成立的前提。并且在此基础上提出我们自己的解答,它是基于由Oliver．G所提倡的方面(aspects)概念而来的社会——论辩方案。     

Good and Bad Years: An Index of American Social,Economic, and Political Threat (1920–1986)

Information provided by 196 U.S. history professors, indicating the degree to which they considered each of the years from 1920 to 1986 threatening to the established order and way of life in America, was pooled to form a social, economic, and political threat (SEPT) index. Interrater reliability was high, and substantial test-retest reliability was evident for a selected subsample over a 6-month period. The index significantly correlated with 11 objective indices of threat, including the suicide rate, unemployment rate, rise and fall of common stocks, and number of military men on active duty. Several studies involving threat and authoritarianism were replicated and in some instances extended with the SEPT index (McCann & Stewin, 1984; Sales, 1972, 1973). The pseudo-archival SEPT index has utility when relatively global estimates of prevailing threat are required for historiometric research testing a diversity of hypotheses gleaned from psychological, sociological, historical, and political science theories.     

Both non-symbolic and symbolic quantity processing are important for arithmetical computation but not for mathematical reasoning

This study investigated whether numerical processing was important for two types of mathematical competence: arithmetical computation and mathematical reasoning. Thousand eight hundred and fifty-seven Chinese primary school children in third through sixth grades took eight computerised tasks: numerical processing (numerosity comparison, digit comparison), arithmetical computation, number series completion, non-verbal matrix reasoning, mental rotation, choice reaction time, and word rhyming. Hierarchical regressions showed that both non-symbolic numerical processing (numerosity comparison) and symbolic numerical processing (digit comparison) were independent predictors of arithmetical computation but neither was a predictor of mathematical reasoning (assessed by number series completion). These findings suggest that the cognitive basis of mathematical performance varies depending on the type of mathematical competence measured.     

早期儿童数学学习与执行功能的关系

执行功能是个体对复杂的认知活动的自我调节和以明确目标为导向的活动过程, 对早期儿童的数学学习起着重要的作用。早期儿童数学学习与执行功能呈显著正相关, 执行功能是儿童数学学习的重要认知加工机制。早期儿童执行功能和数学学习之间存在着相互预测的关系, 执行功能可以预测数学成绩, 数学成绩可以预测执行功能。高质量的早期数学教育可能具有发展儿童执行功能和数学能力的双重价值。未来研究可以明确执行功能的界定和统一测量工具, 提供更可靠的证据证明早期儿童执行功能与数学能力的因果关系, 以及进一步探究语言、数学以及执行功能三者之间的关系。     

母亲评价与幼儿数学认知发展的关系

母亲对幼儿数学能力的评价是母亲教育观念的重要组成部分。研究运用问卷法与个别测查法,考察母亲对子女数学能力的评价与幼儿数学认知能力,并通过二者的一致程度揭示母亲评价其子女数学能力的准确性。研究发现,母亲对子女数学能力的评价存在明显的高估倾向;与母亲受教育水平及子女性别等因素不同,子女前期的数学成绩对母亲评价其数学能力的准确性存在显著影响;母亲对子女数学能力评价的准确性与子女数学认知发展水平存在显著的正相关,适当的高估对子女未来数学认知的发展最为有利     

Mathematical psychology: Prospects for the 21st century: A guest editorial

The twenty-first century is certainly in progress by now, but hardly well underway. Therefore, I will take that modest elasticity in concept as a frame for this essay. This frame will serve as background for some of my hopes and gripes about contemporary psychology and mathematical psychology’s place therein. It will also act as platform for earnest, if wistful thoughts about what might have (and perhaps can still) aid us in forwarding our agenda and what I see as some of the promising avenues for the future. I loosely structure the essay into a section about mathematical psychology in the context of psychology at large and then a section devoted to prospects within mathematical psychology proper. The essay can perhaps be considered as in a similar spirit, although differing in content, to previous editorial-like reviews of general or specific aspects of mathematical psychology such as [Estes, W. K. (1975). Some targets for mathematical psychology. Journal of Mathematical Psychology, 12, 263-282; Falmagne, J. C. (2005). Mathematical psychology: A perspective. Journal of Mathematical Psychology, 49, 436-439; Luce, R. D. (1997). Several unresolved conceptual problems of mathematical psychology. Journal of Mathematical Psychology, 41, 79-87] that have appeared in this journal.     

Conceptual engineering for mathematical concepts

In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.     

乡镇小学生的数学焦虑与数学成绩:数学自我效能感和数学元认知的链式中介作用

采用儿童数学焦虑量表、小学生数学学习自我效能感量表和小学生数学元认知问卷,对508名乡镇中、高年级小学生进行测量,并运用结构方程模型探讨数学焦虑影响数学成绩的内在作用机制。结果发现：（1）数学焦虑显著负向预测数学自我效能感、数学元认知和数学成绩,数学自我效能感显著正向预测数学元认知和数学成绩,数学元认知显著正向预测数学成绩;（2）在数学焦虑对数学成绩的预测中,数学自我效能感和数学元认知均发挥了部分中介作用;（3）数学自我效能感和数学元认知在数学焦虑和数学成绩之间起链式多重中介的作用。因此,数学焦虑除了直接作用于小学生的数学成绩,还可通过数学自我效能感或数学元认知间接影响数学成绩,而且可通过数学自我效能感进而通过数学元认知间接影响数学成绩。文章讨论了上述发现的理论及教育实践含义。     

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.     

小学儿童数学态度与数学成就的纵向联系：学业拖延和数学元认知的作用

为深入探讨小学儿童的数学态度、学业拖延、数学元认知与数学成就之间的纵向联系及内在作用机制,对515名三、五年级小学生进行为期半年的追踪研究。结果表明:(1)学业拖延在儿童的数学态度与数学成就之间发挥着即时和纵向中介作用;(2)不同水平数学元认知个体在纵向中介模型中“数学态度→数学成就”这一路径上存在差异。这意味着较积极的数学态度有利于减少小学儿童的学业拖延行为,进而提高其数学成就,而高数学元认知则能够监控和调节个体的数学态度,使其发挥积极作用,从而提高数学成就。该发现为有效促进儿童的数学学习提供了重要实践启示。     

Prospective Elementary Teachers Learning to Reason Flexibly with Sums and Differences: Number Sense Development Viewed Through the Lens of Collective Activity

I present a viable learning trajectory for prospective elementary teachers’ number sense development with a focus on whole-number place value, addition, and subtraction. I document a chronology of classroom mathematical practices in a Number and Operations course. The findings provide insights into prospective elementary teachers’ number sense development. These include the role of standard algorithms and their relationship to the evolution of classroom mathematical practices that involve reasoning flexibly about number composition, sums, and differences.     

数学焦虑研究的认知取向

数学焦虑一直是心理学界研究和讨论的热点问题。本文主要从认知的角度对数学焦虑进行解释,并以此揭示数学焦虑的心理机制和思维规律。     

加减文字题解决研究概述

加减文字题指应用加减法运算解答的简单数学应用题。基本类型有合并题、变化题和比较题。人们主要采用四种方法研究解题过程：解答问题、回忆和构造问题、建立计算机模型和眼动记录。过去研究发现语义类型、年龄、难以理解的词句、问题陈述的简约性、题材个人化、问题陈述结构、数量大小、未知集类型和解答问题的方式等因素显著影响解题过程。人们对解题过程提出了两种理论模型,一是数学知识应用模型,一是语言理解模型