小学生数学示例学习的信息加工过程实验研究

该研究试图让小学生通过示例学习的方式学习小数乘小数知识。口语报告分析及后测表明,小学四年级学生已具备示例学习的能力。实验还通过口语报告的个案分析,揭示示例学习的信息加工过程,并探讨了经过改进的口语报告分析法的应用问题。     

数学日记对数学学习有效性的实验研究

本研究对高中数学学科进行实验研究,以探讨数学日记对主观数学学习效能感与客观数学成绩的影响.结果显示数学日记能显著提高学生的数学学习效能感,有助于数学学习成绩的提高,尤其对数学差生更具有显著作用;同时数学日记还能使学生对数学学习的信念与数学学习的实效协调起来,获得一致性效应.结论是数学日记能使学生形成一种有效性的学习.     

The Utility of Brief Experimental Analysis and Extended Intervention Analysis in Selecting Effective Mathematics Interventions

The present study evaluated the utility of brief experimental analysis (BEA) in predicting effective interventions for increasing the math fluency of 3 elementary students identified as having math skill deficits. Baseline data were collected followed by implementation of a BEA consisting of the following interventions: cover, copy, and compare, taped problems (TP), and math to mastery (MTM). An extended analysis phase using an alternating treatments design compared all 3 interventions against the results of the BEA. Two follow-up measurements were taken 5 days and 15 days after termination of the extended intervention analysis phase. Results indicated the BEA correctly predicted the most effective intervention for enhancing math fluency for all 3 students. Comparison of the intervention conditions revealed the MTM intervention to be the most effective intervention for 2 of the 3 students, while the TP intervention was the most efficient for 2 of the 3 students.     

An Experimental Analysis of Mathematics Instructional Components: Examining the Effects of Student-Selected Versus Empirically-Selected Interventions

Within the context of an experimental analysis, the current study examined the effects of student- and empirically-selected interventions on the mathematics computational fluency of three elementary-aged students. For all of the participants, the empirically-selected intervention resulted in higher levels of computational fluency then the student-selected intervention. These results suggest that empirically-selected interventions may enhance the mathematics computational fluency of students experiencing mathematics problems. However, individual responsiveness to empirically-selected interventions suggests the importance of conducting brief experimental analyses to determine the most efficacious intervention. The implication of these results for intervention selection and incorporating choice-making opportunities into academic interventions are discussed.     

Mathematics of forgiveness

This study was aimed at determining the integration rule--summation or averaging--underlying the forgiveness schema. The main reason for distinguishing between these structures is that they have very different practical implications regarding the influence of various factors specific to each case on the propensity to forgive. In a summative model, the impact of the different factors and the direction of the effects are constant. For example, the presence of apologies always is a positive element even when these apologies assume a very weak form. By contrast, in an averaging model, the apologies can be a positive or a negative element depending on the current level of propensity to forgive and the form of the apologies. Two experiments were conducted using the functional theory of cognition framework. Experiment 1 applied the missing information test. Experiment 2 applied the credibility of information test. In both experiments, clear evidence favored a summative rule for judging willingness to forgive from circumstantial information such as presence or absence of intent, presence of absence of apologies, and degree of cancellation of consequences.     

Visualizations in Mathematics

In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.

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Boltzmann on Mathematics

Boltzmann’s lectures on natural philosophy point out how the principles of mathematics are both an improvement on traditional philosophy and also serve as a necessary foundation of physics or what the English call “Natura Philosophy”, a title which he will retain for his own lectures. We start with lecture #3 and the mathematical contents of his lectures plus a few philosophical comments. Because of the length of the lectures as a whole we can only give the main points of each but organized into a coherent study. Behind his mathematics stands his support of Darwinian evolution interpreted in a partly Lamarckian way. He also supported non-Euclidean geometry. Much of Boltzmann’s analysis of mathematics is an attempt to refute Kant’s static a priori categories and his identification of space with “non-sensuous intuition”. Boltzmann’s strong attention toward discreteness in mathematics can be seen throughout the lectures. Part II of this paper will touch on the historical background of atomism and focus on the discrete way of thinking with which Boltzmann approaches problems in mathematics and beyond. Part III briefly points out how Boltzmann related mathematics and discreteness to music. This revised version was published online in June 2006 with corrections to the Cover Date.     

Theism and Mathematics

The author investigates the connection between God and mathematics, and argues (1) that the “unreasonable effectiveness of mathematics” makes much better sense from the perspective of theism than from that of naturalism, (2) that the accessibility (to us human beings) of advanced mathematics is much more likely given theism than given naturalism, (3) that the existence of sets, numbers, functions and the like fits in much better with theism than with naturalism, and (4) that the alleged epistemological obstacles to knowledge of mathematics offered by the abstract character of numbers, sets, etc., disappear from the point of view of theism.     

The Objectivity of Mathematics

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.     

Non-Language Thinking in Mathematics

After a brief outline of the topic of non-language thinking in mathematics the central phenomenological tool in this concern is established, i.e. the eidetic method. The special form of eidetic method in mathematical proving is implicit variation and this procedure entails three rules that are established in a simple geometrical example. Then the difficulties and the merits of analogical thinking in mathematics are discussed in different aspects. On the background of a new phenomenological understanding of the performance of non-language thinking in mathematics the well-known theses of B. L. van der Waerden that mathematical thinking to a great extent proceeds without the use of language is discussed in a new light.     

How to Apply Mathematics

This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.     

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.     

Mathematics and Program Explanations

Aidan Lyon has recently argued that some mathematical explanations of empirical facts can be understood as program explanations. I present three objections to his argument.     

The Necessity of Mathematics

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.