Maintaining Accurate Math Responses In Elementary School Students: The Effects Of Delayed Intermittent Reinforcement And Programming Common Stimuli.

This study examined the effect of delayed reinforcement on digits completed by elementary school children and the effect of programming stimuli common to reinforcement conditions on the maintenance of their performance. Participants exhibited similar levels of responding during intermittent and continuous reinforcement. Responding continued for a number of sessions at similar levels during a maintenance phase that included stimuli present during delayed reinforcement.     

Ideas and processes in mathematics: A course on history and philosophy of mathematics

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice mathematics teachers.     

Hilbert,logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).     

Why Euclid’s geometry brooked no doubt: J. H. Lambert on certainty and the existence of models

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid’s fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid’s in justification. Contrary to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid’s theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these “models” are constructed. To understand Lambert’s view of postulates and the doubt they answer, I examine his criticism of Christian Wolff’s views. I argue that Lambert’s view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification.     

Children’s mapping between symbolic and nonsymbolic representations of number

When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children’s mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children’s mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children’s mathematical development.     

工作记忆对小学三年级学生解决比较问题的影响

研究工作记忆与小学生解决比较问题成绩之间的关系。实验1运用双任务作业研究一致问题和不一致问题的工作记忆负荷是否有显著差别,被试为34名小学三年级学生。实验2运用工作记忆测验研究成功解题者与不成功解题者的工作记忆容量是否有显著差别,被试为37名小学三年级学生。结果表明;(1)小学生解决比较问题的成绩受问题类型的影响,他们在一致问题上的成绩显著优于不一致问题。(2)一致问题和不一致问题的工作记忆负荷水平不同,不一致问题的工作记忆负荷大于一致问题。(3)成功解题者的工作记忆容量大于不成功解题者的工作记忆容量。本研究结果说明工作记忆对小学生解决比较问题有重要影响。     

基于空间分割的手写输入系统的用户绩效模型

为定量估计与提高基于空间分割的手写输入系统用户绩效 ,运用Fitts定律及离散变量的数学期望等方法推导了用户绩效的数学模型。实验验证了该模型在具体界面中能较好地拟合用户在 8次 (8× 2 4字 )训练后的实际操作绩效。应用该模型可较好地拟合用户使用手写系统完成抄写任务的时间、推算输入系统某些参数的最佳设置值 ;并发现当输入系统的其它参数为定值时 ,缩短用户单字手写时间会比缩短系统识别时间能更有效地提高用户绩效。     

The Perceiver's Share: Realism, Scepticism, and Response Dependence

Abstract: Response‐dispositional (RD) properties are standardly defined as those that involve an object's appearing thus or thus to some perceptually well‐equipped observer under specified epistemic conditions. The paradigm instance is that of colour or other such Lockean “secondary qualities”, as distinct from those—like shape and size—that pertain to the object itself, quite apart from anyone's perception. This idea has lately been thought to offer a promising alternative to the deadlocked dispute between hard‐line ‘metaphysical’ realists and subjectivists, projectivists, social constructivists, or hard‐line anti‐realists. A chief source text is Plato's Euthyphro, where the issue is posed in ethical terms: do the gods infallibly approve virtuous acts on account of their divine moral omniscience or are virtuous acts just those the gods approve? Among the areas proposed as amenable to an RD approach are epistemology, ethics, political theory, and philosophy of mathematics. It is claimed that by making due allowance for the involvement of normalised or optimised human responses one can steer a course between the twin poles of an objectivist realism that places truth beyond our cognitive grasp and an epistemic conception that confines truth within the limits of humanly attainable proof, knowledge, or verification. Here I argue—on the contrary—that RD approaches can be shown to offer nothing more than a variant of the same old realist versus anti‐realist dilemma. That is, they work out either as a trivial (tautologous) claim that ‘truth’ simply equates with ‘best judgement’ in the ideal (quasi‐objective) limit or as the claim—advanced by anti‐realists like Michael Dummett—that we cannot form any adequate conception of objective (recognition‐transcendent) truths. After looking at this issue in various contexts of debate, I conclude that one useful (if pyrrhic) outcome is to demonstrate the non‐availability of any middle‐ground stance. We are left with the strictly unavoidable choice between a realist or objectivist approach and one that assimilates truth to the consensus of accredited best opinion. This latter amounts to a roundabout, elaborately qualified version of the anti‐realist case.     

Gender Differences in Interest and Knowledge Acquisition: The United States,Taiwan, and Japan

The relationship between interest and knowledge was investigated in a representative sample of 11th grade students from cultures that differ in the strength of their gender-role stereotypes and their endorsement of effort-based versus interest-based learning. Among 11th graders from the United States (N = 1052), Taiwan (N = 1475), and Japan (N = 1119), boys preferred science, math, and sports, whereas girls preferred language arts, music, and art. General information scores were comparable across the three locations; however, boys consistently outscored girls. Gender and interest in science independently predicted general information scores, whereas gender and interest in math independently predicted mathematics scores. Cultural variations in the strength of the relationship between gender, interest, and scores indicate that specific socialization practices can minimize or exaggerate these gender differences.     

Mathematics and conceptual analysis

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light.