Adults' beliefs about children and mathematics: how important is it and how do children learn about it?

A series of studies was conducted which focused on US adults' beliefs about the relative importance of acquiring mathematical skills for preschool children and about how children acquire these skills. In Study 1, adults rated general information, reading and social skills as all being more important than mathematical skills. They also claimed that parents have the most influence on preschool children's learning regardless of content area. In Study 2, the parents of kindergarten children also rated reading, general information and social skills as all being more important than mathematics in preparing children for the first grade. The more important parents felt mathematics were, the more they reported engaging in a variety of mathematical-related activities with their children. However, the importance they placed on mathematics was not related to their child's actual mathematical performance. In summary, adults seem to value mathematics less than other skills in preparing young children to enter elementary school. © 1998 John Wiley & Sons, Ltd.     

Remarks on Penrose’s “New Argument”

It is commonly agreed that the well-known Lucas–Penrose arguments and even Penrose’s ‘new argument’ in [Penrose, R. (1994): Shadows of the Mind, Oxford University Press] are inconclusive. It is, perhaps, less clear exactly why at least the latter is inconclusive. This note continues the discussion in [Lindström, P. (2001): Penrose’s new argument, J. Philos. Logic 30, 241–250; Shapiro, S.(2003): Mechanism, truth, and Penrose’s new argument, J. Philos. Logic 32, 19–42] and elsewhere of this question.     

Informal versus formal mathematics

We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics. The author is Visiting Researcher at IEA/USP, Professor of Communications, Emeritus, at the Federal University in Rio de Janeiro, and a full member of the Brazilian Academy of Philosophy.     

Increasing Retention Without Increasing Study Time

ABSTRACT— Because people forget much of what they learn, students could benefit from learning strategies that yield long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions—the spacing effect —depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning.     

工作记忆在数学运算过程中的作用

Baddeley和Hitch(1974)提出的工作记忆模型被广泛地应用数学运算领域,但是已有研究还缺乏系统性。首先,当前的研究主要集中在加法和乘法两种运算上,较少涉及减法、除法和更为抽象的代数运算;第二,研究者对语音环路和中央执行系统的作用进行了较深入的研究,但常常忽略视空间模板的作用;第三,工作记忆在数学运算过程中的作用具有动态性,它受到一些外部因素、数学任务内在因素和个体认知因素（如认知策略）的影响;第四,几乎有关研究都肯定中央执行系统在数学运算过程中的重要作用,然而负荷于中央执行系统的次级任务所含成分或功能的复杂性,导致我们很难确定中央执行系统如何作用于数学运算。对这些问题的研究将是未来可能的研究方向     

Making Numbers Come to Life: Two Scoring Methods for Creativity in Aurora's Cartoon Numbers

Although creativity has long been recognized as an important aspect of mathematical thinking, both for the advancement of the field and in students' developing expertise in mathematics, assessments of student creativity in that domain have been limited in number and focus. This article presents an assessment developed for creativity that provides a score for mathematical creativity (MaC) in addition to a score for general creativity in the numeric domain, or what we might call numerical creativity (NuC). We developed different rating scales for each and then explored how each scoring method accounts for the students' mathematical/numerical and creative skills. The psychometric properties for both scoring approaches were examined. Each method was shown to reflect different relationships with other performance tests. In addition, it is proposed that MaC may provide useful insight into students' levels of adaptive expertise in mathematics, as reflected by their ability to apply mathematical knowledge (i.e., language, operations, concepts) to novel situations, representing an informative supplement to performance indicators of math achievement.     

Contrasting two models of academic self‐efficacy – domain‐specific versus cross‐domain – in children receiving and not receiving special instruction in mathematics

In social cognitive theory, self‐efficacy is domain‐specific. An alternative model, the cross‐domain influence model, would predict that self‐efficacy beliefs in one domain might influence performance in other domains. Research has also found that children who receive special instruction are not good at estimating their performance. The aim was to test two models of how self‐efficacy beliefs influence achievement, and to contrast children receiving special instruction in mathematics with normally‐achieving children. The participants were 73 fifth‐grade children who receive special instruction and 70 children who do not receive any special instruction. In year four and five, the children's skills in mathematics and reading were assessed by national curriculum tests, and in their fifth year, self‐efficacy in mathematics and reading were measured. Structural equation modeling showed that in domains where children do not receive special instruction in mathematics, self‐efficacy is a mediating variable between earlier and later achievement in the same domain. Achievement in mathematics was not mediated by self‐efficacy in mathematics for children who receive special instruction. For normal achieving children, earlier achievement in the language domain had an influence on later self‐efficacy in the mathematics domain, and self‐efficacy beliefs in different domains were correlated. Self‐efficacy is mostly domain specific, but may play a different role in academic performance depending on whether children receive special instruction. The results of the present study provided some support of the Cross‐Domain Influence Model for normal achieving children.     

The mathematics of sharing / Las matemáticas que realizan los niños al compartir

Abstract

The purpose of sharing is to construct equivalent sets, making it an ideal context for analysing important quantitative concepts such as counting, equivalence and cardinality. Two studies analysed how four- and five-year-olds shared blocks in equal sharing and reciprocity conditions and their number inferences about one set after counting the other. The researcher asked children to share double and single blocks between two characters. They succeeded more in building equivalent shares in an equal sharing than reciprocity condition. Most children who shared correctly also made appropriate number inferences. To examine whether perceptual cues helped children share the blocks, a second study used Canadian $1 and $2 coins. A double block is twice the size of a single, whereas there is no visual cue about the value relation between coins because they are the same size. Unexpectedly, children shared equally well with blocks and coins, and most children made number inferences.     

儿童执行功能与早期学业能力的关系:学习品质的中介作用

以上海市6所幼儿园245名3~6岁儿童为研究对象,分别采用头–脚–膝盖–肩膀(HTKS)任务、修订后的儿童行为评定量表(CBRS)及亚太学前儿童发展量表中的语言和前阅读分量表以及认知分量表考察执行功能、学习品质及早期语言和数学学业发展状况。结果表明:(1)执行功能对学习品质、早期语言能力和数学能力均有显著正向预测作用;(2)学习品质在执行功能和早期数学能力关系中起部分中介作用,但对执行功能和早期语言能力的关系无中介作用。     

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.