Eye gaze patterns reveal how we reason about fractions

ABSTRACT

Fractions are defined by numerical relationships, and comparing two fractions’ magnitudes requires within-fraction (holistic) and/or between-fraction (componential) relational comparisons. To better understand how individuals spontaneously reason about fractions, we collected eye-tracking data while they performed a fraction comparison task with conditions that promoted or obstructed different types of comparisons. We found evidence for both componential and holistic processing in this mixed-pairs task, consistent with the hybrid theory of fraction representation. Additionally, making within-fraction eye movements on trials that promoted a between-fraction comparison strategy was associated with slower responses. Finally, participants who performed better on a non-numerical test of reasoning took longer to respond to the most difficult fraction trials, which suggests that those who had greater facility with non-numerical reasoning attended more to numerical relationships. These findings extend prior research and support the continued investigation into the mechanistic links between numerical and non-numerical reasoning.     

Three Dual Ontologies

In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aim of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world is made up of things, or properties, or facts, may be no more than a matter of how we look at it.     

小学生视觉-空间表征类型和数学问题解决的研究

本研究考察并比较了四至六年级儿童的视觉－空间表征策略、数学问题解决和空间视觉化能力。结果表明：五、六年级儿童的解题正确率、使用图式表征策略的程度显著高于四年级儿童;使用图像表征策略的程度各年级无显著差异。将数学问题分成三个难度等级,发现年级差异主要表现在难度等级1的题目上。另外,六年级儿童的空间视觉化能力显著高于四年级儿童。     

儿童早期数学认知能力的结构及其特点

从北京市10所幼儿园中选取234名3、4岁的儿童为被试,采用个别测查的方法对儿童早期数学认知能力的结构及其特点进行了考察。经验证性因素分析发现：(1)儿童早期数学认知能力的结构模型是合理的,可接受的,具有较好的构想效度,具体讲,数、计算、测量、空间／几何和模式认知能力五个维度共同解释着儿童早期的数学认知能力;(2)不同年龄儿童早期数学认知能力的结构具有稳定性,但是结构模型并不完全一致,某些项目的解释率有所不同;(3)男、女儿童早期数学认知能力的结构模型具有一致性。     

Quantum Mechanics and "Song of Myself": Getting a Grip on Reality

Most recent writing linking science and literature has concerned itself with challenges to the epistemological status of scientific knowledge in an attempt to demonstrate its contingency, arguing in the more radical efforts that the structures of science are no more than useful fictions. This essay also includes an epistemological comparison between science and literature, but instead of making grand or meta–statements about the nature of knowing generally in the two fields, mine is a much narrower aim. My exploration entails two tasks. First, I provide a close–up look at a particular type of experiment, called the delayed–choice experiment, which clearly reveals the strangeness of the quantum world. In connection with this experiment, I discuss wave functions—mathematical expressions used by physicists to describe quantum behavior and predict the outcome of experiments involving quanta. Second, I look at Walt Whitman's “Song of Myself” focusing on the meaning of the “self” in the poem. My aim is to treat the object of study in each field as a “text” and to assert and demonstrate a parallel in the strategies of thought and response between physicists (“readers”) pondering the meaning and status of a wave function and poem readers pondering the meaning and status of the poem's self. In Whitman's “Song” we find an attempt to understand complex aspects of human experience that are said to transcend ordinary reality, an effort for which I believe there are parallels in the attempts of modern physicists to understand complex, nonintuitive aspects of the subatomic world. While not making the kind of broad claims eschewed above, I do suggest that this focused study has interesting implications since both the wave function and the poem's self force their respective sets of “readers” to confront questions of ultimacy—to consider, that is, epistemological and ontological issues of more than passing interest to students of science as well as those of metaphysics and theology.     

Ricoeur versus Taylor on Language and Narrative

Abstract: Although Ricoeur and Taylor are often grouped together, their conceptions of language, literature, and practical reason are very different. The first half of this essay focuses on Ricoeur's theory of triple mimesis and narrative, showing how his attempt to synthesize Kant, Husserl, and structuralism results in a formalism that blocks out the ontological, hermeneutical, and historical dimensions of literature and practical reason. The second half of the essay develops Taylor's ontological conception of public imagination and illustrates the dynamics of this conception of language and interpretation by showing how literary works debate with each other over language, subjectivity, and reference. Narrative does not just order; it argues.     

Das Ontologische Dilemma der Normativen Ethik

The Ontological Dilemma of Normative Ethics. This paper pursues two goals. The first is to show that normative ethics is confronted with the following dilemma: to be coherent, this discipline is ontologically committed to acknowledge the existence of objective values, but, to be scientifically respectable, it is committed to repudiate such values. The second goal is to assess the possible solutions to this dilemma. To this end, the following strategies are discussed: Kant’s constructive objectivism, Jürgen Habermas’ “epistemic ersatzism”, Franz von Kutschera’s “confirmation pragmatism”, and David Brink’s “objectivist tour de force”. The paper’s conclusion is that the dilemma cannot be solved because it rests on a clash of intuitions none of which can be given up. This revised version was published online in August 2006 with corrections to the Cover Date.     

Should One Be A Left or A Right Sellarsian? (And is There Really Such A Choice?)

The followers of Wilfrid Sellars are often divided into “right” and “left” Sellarsians, according to whether they believe, in Mark Lance's words, that “linguistic roles constitutive of meaning and captured by dot quoted words are ‘normative all the way down.’” The present article anatomizes this division and argues that it is not easy to give it a nontrivial sense. In particular, the article argues that it is not really possible to construe it as a controversy related to ontology, and goes on to argue that it is also not easy to construe it as one concerning the translatability of the normative idiom into the non‐normative one. The conclusion is that the only coherent interpretation of this disagreement is as a disagreement about the possibility and desirability of assuming a standpoint “inside” our linguistic practices.     

Predictive model for early math skills based on structural equations

Early math skills are determined by higher cognitive processes that are particularly important for acquiring and developing skills during a child's early education. Such processes could be a critical target for identifying students at risk for math learning difficulties. Few studies have considered the use of a structural equation method to rationalize these relations. Participating in this study were 207 preschool students ages 59 to 72 months, 108 boys and 99 girls. Performance with respect to early math skills, early literacy, general intelligence, working memory, and short‐term memory was assessed. A structural equation model explaining 64.3% of the variance in early math skills was applied. Early literacy exhibited the highest statistical significance (β = 0.443, p < 0.05), followed by intelligence (β = 0.286, p < 0.05), working memory (β = 0.220, p < 0.05), and short‐term memory (β = 0.213, p < 0.05). Correlations between the independent variables were also significant (p < 0.05). According to the results, cognitive variables should be included in remedial intervention programs.     

Ontology of early visual content

The main goal of the paper is to sketch an ontological model of visual content at the low- and medium-level of visual processing, relying on psychological conceptions of vision. It is argued that influential cognitive models contain assumptions concerning “objects of content,” that is, objects whose presence is a necessary condition of the adequacy of visual representations. Subsequently, the structure of considered objects of content is presented, and its development through the perceptual process is described. In addition, during the course of the article I present some connections between analytic metaphysics and the ontology of visual content.