A comparison of self-instructional checking procedures for remediating mathematical deficits

We examined the relative effectiveness of three procedures for teaching long multiplication/division to seven adolescents with learning disabilities: no-checking, end-checking, and multi-checking. During training, each subject was taught by modelling and imitation to verbalize self-instructions in the form of a strategy while solving the problems. The relative effects of the checking and no-checking procedures on accuracy and rate of problems completed were examined in an alternating treatments design. The best treatment was then given alone and a reversal was implemented six weeks later, followed by a return to the best treatment during a final phase. Irrespective of the procedure used, the subjects' accuracy improved while the rate of problems completed decreased. These effects were greatest with the multi-checking procedure for six of the seven subjects. Generalization to untaught problems of various levels of complexity occurred under all procedures. Though maintenance effects were seen during the follow-up, accuracy was generally higher and more reliable when the subject's best checking procedure was reinstated. It is suggested that error detection and correction were important for maintaining high levels of accuracy and that these operated differentially in the three procedures to produce the differing levels of accuracy. The role of other factors such as pre-skill knowledge, complexity of the problem and prior reinforcement history are also considered.     

Use them or lose them: Are manipulatives needed to assess numeracy and geometry performance in preschool?

In two studies, we investigated whether using three-dimensional (3D) manipulatives during assessment aided performance on a variety of preschool mathematics tasks compared to pictorial representations. On measures of children's understanding of counting and cardinality (n = 103), there was no difference in performance between manipulatives and pictures, with Bayes factors suggesting moderate evidence in favor of the null hypothesis. On a measure of children's shape identification (n = 93), there was no difference in performance between objects and pictures, with Bayes factors suggesting moderate evidence in favor of the null hypothesis. These results suggest flexibility in the materials that can be used during assessment. Pictures, or 2D renderings of 3D objects, which can be easily printed and reproduced, may be sufficient for assessing counting and shape knowledge without the need for more cumbersome concrete manipulatives.     

初中生数学元认知的发展轨迹：两年半追踪研究

采用问卷法对101名初中生在两年半间数学元认知的发展状况进行5次测试。利用潜类别增长模型等探讨初中生数学元认知的发展轨迹,并考察性别对数学元认知的影响。结果发现：(1)初中生数学元认知及各成分在初二表现出下降趋势。(2)初中生数学元认知的发展表现出三种类型,即高-缓慢下降组(32.67%)、中-显著下降组(54.46%)以及低-缓慢下降组(12.87%)。(3)与女生相比,男生有着更多的数学元认知知识和更高的数学元认知初始水平,且与低-缓慢下降组相比,男生比女生更有可能属于高-缓慢下降组。     

A Negationless Interpretation of Intuitionistic Theories. II

This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.     

A Set Theory with Support for Partial Functions

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.     

父母家校沟通与小学生参与数学课外补习的关系研究

基于某特大型城市学区658名小学四年级学生及其父母的调查数据,讨论小学生参与数学课外补习的影响因素,特别是家校沟通的作用。数据分析结果表明：在控制学生数学学业成绩与家庭社会经济地位的情况下,母亲的家校沟通行为可以显著预测四年级学生的数学课外补习参与情况,而未发现父亲的家校沟通行为的预测作用,研究结论印证了信息沟通在家庭教育决策中的重要作用。     

An Investigation of the Relationship between Age,Achievement, and Creativity in Mathematics

In this exploratory study, I investigate the relationship between age, knowledge, and creativity in mathematics, by looking at to what extent does grade level, controlled for mathematical achievement, influence mathematical creativity and what characterizes the relationship between grade level, mathematical achievement and mathematical creativity. This was accomplished in two steps. In the first part, 301 students, 184 grade eight students and 117 grade eleven students, were given a creative mathematics test. A 3 × 2 ANOVA indicates that the older students were more creative; however, there was a significant interaction effect between grade level and achievement in mathematics on mathematical creativity. In the second part, an inductive content analysis was performed on the solutions of high achievers in grade eleven and grade eight. The results indicate that high achievers in grade eight are more creative than high achievers in grade eleven, but the nature of the task mediates the relationship between creativity and knowledge.     

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.