The natural mathematics of behavior analysis

Models that generate event records have very general scope regarding the dimensions of the target behavior that we measure. From a set of predicted event records, we can generate predictions for any dependent variable that we could compute from the event records of our subjects. In this sense, models that generate event records permit us a freely multivariate analysis. To explore this proposition, we conducted a multivariate examination of Catania's Operant Reserve on single VI schedules in transition using a Markov Chain Monte Carlo scheme for Approximate Bayesian Computation. Although we found systematic deviations between our implementation of Catania's Operant Reserve and our observed data (e.g., mismatches in the shape of the interresponse time distributions), the general approach that we have demonstrated represents an avenue for modelling behavior that transcends the typical constraints of algebraic models.     

Semiotic systems,culture and early mathematical knowledge / Sistemas semióticos,cultura y conocimiento matemático temprano

Abstract

In this introduction to the issue Semiotic tools in early mathematical knowledge, we offer a theoretical perspective that provides meaning to the different contributions. We begin by reviewing three perspectives bearing an important influence on the study of mathematical development in childhood (the Piagetian, the information processing and the neonativist perspectives). Next, we underscore the need for developmental studies in this domain to consider the semiotic component, and hence the socio-cultural component, in depth. The contributions that form this volume — a theoretical article, five papers and four short reports of empirical studies and a review of an instructional approach — illustrate different ways of including the semiotic component in the study of children’s mathematical knowledge.     

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.     

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.     

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.     

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。     

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.     

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.