Plato, Pascal, and the dynamics of personal knowledge

Abstract educational practices are to be based on proven scientific knowledge, not least because the function science has to perform in human culture consists of unifying practical skills and general beliefs, the episteme and the techne (Amsterdamski, 1975, pp. 43–44). Now, modern societies first of all presuppose regular and standardized ways of organizing both our concepts and our institutions. The explanatory schemata resulting from this standardization tend to destroy individualism and enchantment. But mathematics education is in fact the only place in which to treat the human subject’s relationship with mathematics. And that is what mathematics education is all about: make the human subject grow intellectually and as a person by means of mathematics. At first sight, mathematics, in its formal guise, seems the opposite of philosophy, because philosophy constructs concepts (meanings), whereas mathematics deals with extensions of concepts (sets). We shall, however, turn this problem into an instrument, using the complementarity of intensions and extensions of theoretical terms as our main device for discussing the relationship between philosophy and mathematics education. The complementarity of the “how” and the “what” of our representations outlines, in fact, the terrain on which epistemology and education are to meet.     

Remediating Computational Deficits at Third Grade: A Randomized Field Trial

The major purposes of this study were to assess the efficacy of tutoring to remediate 3rd-grade computational deficits and to explore whether remediation is differentially efficacious depending on whether students experience mathematics difficulty alone or concomitantly with reading difficulty. At 2 sites, 127 students were stratified on mathematics difficulty status and randomly assigned to 4 conditions: word recognition (control) tutoring or 1 of 3 computation tutoring conditions: fact retrieval, procedural computation and computational estimation, and combined (fact retrieval + procedural computation and computational estimation). Results revealed that fact retrieval tutoring enhanced fact retrieval skill, and procedural computation and computational estimation tutoring (whether in isolation or combined with fact retrieval tutoring) enhanced computational estimation skill. Remediation was not differentially efficacious as a function of students' mathematics difficulty status.     

Remediating Computational Deficits at Third Grade: A Randomized Field Trial

Abstract

The major purposes of this study were to assess the efficacy of tutoring to remediate 3rd-grade computational deficits and to explore whether remediation is differentially efficacious depending on whether students experience mathematics difficulty alone or concomitantly with reading difficulty. At 2 sites, 127 students were stratified on mathematics difficulty status and randomly assigned to 4 conditions: word recognition (control) tutoring or 1 of 3 computation tutoring conditions: fact retrieval, procedural computation and computational estimation, and combined (fact retrieval + procedural computation and computational estimation). Results revealed that fact retrieval tutoring enhanced fact retrieval skill, and procedural computation and computational estimation tutoring (whether in isolation or combined with fact retrieval tutoring) enhanced computational estimation skill. Remediation was not differentially efficacious as a function of students' mathematics difficulty status.     

数学教育价值观的嬗变与重构

数学教育价值观的形成是历史和文化的产物,历史上存在着科学主义和人文主义两种不同的数学教育价值观。事实上,数学在为人类社会创造物质财富的同时也丰富了人的精神世界。当代数学教育的价值是多元的,数学教育的价值不仅在于推进数学知识的应用,还在于为人类文明传承着一种独特的思维方式和理性精神。     

Didactics and History of Mathematics: Knowledge and Self-Knowledge

The basic assumption of this paper is that mathematics and history of mathematics are both forms of knowledge and, therefore, represent different ways of knowing. This was also the basic assumption of Fried (2001) who maintained that these ways of knowing imply different conceptual and methodological commitments, which, in turn, lead to a conflict between the commitments of mathematics education and history of mathematics. But that conclusion was far too peremptory. The present paper, by contrast, takes the position, relying in part on Saussurean semiotics, that the historian's and working mathematician's ways of knowing are complementary. Recognizing this fact, it is argued, brings us to a deeper understanding of ourselves as creatures that do mathematics. This understanding, which is a kind of mathematical self-knowledge, is then proposed as an alternative commitment for mathematics education. In light of that commitment, history of mathematics assumes an essential role in mathematics education both as a subject and as a mediator between the aforementioned ways of knowing.     

Mathematics Anxiety In Preservice Elementary School Teachers

Mathematics anxiety is a condition that exists in many children and adults. Studies (Bulmahn & Young, 1982, Kelly & Tomhave, 1985) have indicated that about 10% of all preservice elementary school teachers have mathematics anxiety. The author verified this statistic in a research study conducted by Basta & Unglaub (1994). In fact, with the particular sample used in the 1994 study (79 preservice elementary school teachers) the number of high mathematics anxious rose to 11.4% based on the Mathematics Anxiety Rating Scale (Suinn, 1972).     

The mathematics skills of children with reading difficulties

Although many children with reading difficulty (RD) are reported to struggle with mathematics, little research has empirically investigated whether this is the case for different types of RD. This study examined the mathematics skills of third graders with one of two types of RD: dyslexia (n = 18) or specific reading comprehension difficulty (n = 22), as contrasted to a comparison group (n = 247). Children's performance on arithmetic fact fluency, operations, and applied problems was assessed using standardized measures. The results indicated that children with dyslexia experienced particular difficulty with arithmetic fact fluency and operations: they were 5.60 times and 8.54 times more likely than other children to experience deficits in fact fluency and operations, respectively. Our findings related to arithmetic fact fluency were more consistent with domain-general explanations of the co-morbidity between RD and mathematics difficulty, whereas our findings related to operations were more consistent with domain-specific accounts.     

Hating school, loving mathematics: On the ideological function of critique and reform in mathematics education

Students’ engagement with fictions in the form of “word problems” plays an important role in classroom practice as well as in theories of mathematical learning. Drawing on the Dutch historian Johan Huizinga and the Austrian philosopher Robert Pfaller, I show that this activity can be seen as a form of play or game, where it is pretended that mathematics is useful in real life in a way that it is not. With Pfaller, I argue that play can take hold of the imagination of the players, infusing everyday life with meaning borrowed from the imagery of the play and that these effects are more powerful when the play is forced and takes an institutionalized form. I show that mathematics education does in fact have these characteristics, including sophisticated mechanisms for translating in-game performance (test scores) to real-life goods (grades and examinations). A central theme of the article is the perceived discrepancy between mathematics education as it is, and how it supposedly could and should be in light of the properties of mathematics. The analysis implies that this gap actually is an effect of play and thus an inherent property of mathematics education itself.     

Imagining the mathematician: young people talking about popular representations of maths

This paper makes both a critical analysis of some popular cultural texts about mathematics and mathematicians, and explores the ways in which young people deploy the discourses produced in these texts. We argue that there are particular (and sometimes contradictory) meanings and discourses about mathematics that circulate in popular culture, that young people use these as resources in identity making as (non-)mathematicians, negotiating their meaning in ways that are not always predictable, and that their influence on young people is diffuse and nevertheless important. The paper discusses the discourses that prevail in some of the popular cultural images of mathematics and mathematicians that came up in our research. We show how mathematics is represented as a secret language, while mathematicians are often mad, mostly male and almost invariably white. We then explore how young people negotiate these discourses, positioning themselves in relation to mathematics. Here we draw attention to the fact that both those who continue with mathematics after it ceases to be compulsory and those who do not, deploy similar images of mathematics and mathematicians. What is different is how they respond to and negotiate these images.     

探索法教学的动态模型与控制策略

由于近30年来系统论、控制论、信息论等现代学科的诞生，为数学教育本身的认识提供了一个全新的视角，并提供了新思维和近方法。研究证明，数学教育过程是一个闭环控制系统。为此，可以建立起数学教育系统的动态模型。     

Fact retrieval deficits in low achieving children and children with mathematical learning disability

Using 4 years of mathematics achievement scores, groups of typically achieving children (n = 101) and low achieving children with mild (LA-mild fact retrieval; n = 97) and severe (LA-severe fact retrieval; n = 18) fact retrieval deficits and mathematically learning disabled children (MLD; n = 15) were identified. Multilevel models contrasted developing retrieval competence from second to fourth grade with developing competence in executing arithmetic procedures, in fluency of processing quantities represented by Arabic numerals and sets of objects, and in representing quantity on a number line. The retrieval deficits of LA-severe fact retrieval children were at least as debilitating as those of the children with MLD and showed less across-grade improvement. The deficits were characterized by the retrieval of counting string associates while attempting to remember addition facts, suggesting poor inhibition of irrelevant information during the retrieval process. This suggests a very specific form of working memory deficit, one that is not captured by many typically used working memory tasks. Moreover, these deficits were not related to procedural competence or performance on the other mathematical tasks, nor were they related to verbal or nonverbal intelligence, reading ability, or speed of processing, nor would they be identifiable with standard untimed mathematics achievement tests.     

Between the Academic Mathematics and the Mathematics Education Worlds

Change is always difficult, and there is no great doubt that teachers need time to come to terms with it. This fact is, however, too often forgotten. In the spirit of my earlier work, this paper is shaped by an action research perspective. It provides some insights into the learning experiences of a group of eleven experienced secondary mathematics teacher, who were enrolled in a Perspectives on Mathematics Education two semesters course, within the context of a Masters on Mathematics Teaching programme, held at a Department of Mathematics, in a Portuguese University. The first part of the paper highlights the conflicting pressures and stresses suffered, during the first semester course, by the participating teachers. Confrontation with new ideas about both mathematics and mathematics education, as well as work overload, had a damaging impact on the teachers’ self‐confidence and morale. The second part of the paper covers the second semester course by addressing three fundamental questions for teachers, which aimed at helping the students bridge the academic mathematics and the mathematics education worlds. Finally, brief scenarios of three participating teachers’ professional development throughout the course are discussed in order to illustrate the challenges they had to face and the possibilities the course (and the Masters programme) offered to promote individual change.     

Contribution of mathematics in-service training course to the professional development of elementary school teachers in Israel

The present study aimed to explore the contribution of a mathematics in-service training course to elementary school teachers (1st–6th grades) in Israel. The study was conducted among 449 educators. They were required to respond to background questions. Moreover, they were asked to indicate their expectations from the in-service training course and, at its end, point out to what extent they benefitted from that course. The research findings illustrate that educators who teach mathematics at elementary school and attended the course are generally women in their 40s, holding a BEd degree and a teaching certificate not in mathematics, with an average 13-year seniority. The participating teachers indicated their wish to enrich their didactic knowledge in order to acquire varied tools for teaching mathematics to the entire pupil population as well as to gifted pupils and pupils with learning difficulties. Nevertheless, their demand to expand their mathematics knowledge was very limited. Based on the fact that most teachers have no mathematics education, this is a surprising finding as, in order to be a good teacher, one must be versed not only in Pedagogical Content Knowledge but also in Subject Matter Knowledge.     

What Could be More Practical than Good Research?

These seem to be very special times for mathematics education. The public interest in the topic has never been greater. Probably the most prominent among the occurrences that occasioned this recent leap in popularity are international comparative studies such as TIMSS and PISA. The fact that, in spite of the ongoing efforts toward reform in mathematics education, many countries found the results of the international measurements of their students’ achievements rather disappointing led the ICME 10 Program Committee to create the Survey Team on Relations between Mathematics Education Research and Practice. The team, coordinated by the author of this talk, and including Aline Robert from France, Ole Skovsmose from Denmark, Yoshihiko Hashimoto from Japan, and Gelsa Knijnik from Brazil, was invited to reflect on the question of how research has been informing the practice of mathematics education over the last decade. Following the invitation, the Survey Team turned to the members of the mathematics education community asking them to answer three queries about their own work: (1) How would you describe the essence of your work in mathematics education over the last 5 years or so? (2) During this period, to what extent was your work stirred and influenced by the current state of mathematics education in your country and/or in the world? (3) Do you think that the work done by you and by your colleagues over the last five years or so had, or is going to have, an actual impact on the practice of mathematics education? Analysis of the 74 responses received from all over the world revealed several interesting trends. This article is the text of the ICME plenary address in which the author presented an “executive summary” of the findings.     

现代数学与大众关系浅析

就现代数学与不被大众所理解的事实进行了原因探析，从现代数学发展的状态、公众所具有的数学素质、数学家的任务与主要动力等方面讨论了现代数学与大众的关系．同时，对应用数学在现代数学中的地位、数学界的评价标准及其数学的艺术性等问题进行了剖析．     

ICT as a Change Agent of Mathematics Teaching in Swedish Secondary School

There is a common assumption that computers will change the conditions for mathematics teaching. In this article the author discusses how the computer as a change-agent may influence the conditions, methods and results in everyday mathematics teaching. The empirical material is collected through interviews with eighteen teachers in lower secondary school. The author has also participated in all computer-aided lessons given by two teachers during one year. That means 700 possible computer-aided lessons. Teaching of mathematics seems to have such a strong tradition that the computer as a change-agent is relatively weak. The fact is that the computer is assimilated into an old tradition of methods and contents. A great deal of the computer-aided lessons give attention to drilling pupils with different types of drill-program where they can learn mathematical procedures. In some lessons laborative work is pursued with the intention that the pupils computer-aided learn mathematical concepts.     

数学教学中应重视数学应用意识及能力培养

在数学教学中,应通过数学史料简介,使学生认识数学源于实际,从而激发学生的学习兴趣;应认清数学知识的实用性,使学生认识到学习数学的必要性;应通过数学建模训练,培养学生的数学应用能力。课堂教学中要让学生体验数学在解决实际问题中的作用,激发学生数学应用意识,培养并提高数学应用能力。     

Constructivism,mathematics and mathematics education

Learning theories such as behaviourism, Piagetian theories and cognitive psychology, have been dominant influences in education this century. This article discusses and supports the recent claim that Constructivism is an alternative paradigm, that has rich and significant consequences for mathematics education. In the United States there is a growing body of published research that claims to demonstrate the distinct nature of the implications of this view. There are, however, many critics who maintain that this is not the case, and that the research is within the current paradigm of cognitive psychology. The nature and tone of the dispute certainly at times appears to describe a paradigm shift in the Kuhnian model. In an attempt to analyse the meaning of Constructivism as a learning theory, and its implications for mathematics education, the use of the term by the intuitionist philosophers of mathematics is compared and contrasted. In particular, it is proposed that Constructivism in learning theory does not bring with it the same ontological commitment as the Intuitionists' use of the term, and that it is in fact a relativist thesis. Some of the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge are discussed here.     

Mathematical knowledge and the problem of proof

Every proof is faced with the requirement of proving that the proof is correct, and the proof of the correctness of the proof again meets the same requirement and the proof of the correctness of the correctness of the proof also, etc. In order to escape from an infinite regress into which one is led one has to come down with a purely algorithmic criterion for correctness or to claim that thinking is identical with its subject matter. Whence the preference of number and more generally of conceptualism in pure mathematics. Conceptualism is a kind of nominalism that does not give a realist understanding of mathematics (note that Platonism is not an opponent of nominalism as some seem to believe). The paper presents some examples and reflections intending to hint at the role of formal thought in the process of knowledge growth. It argues that there is no division of labor according to which certain modes of human cognition are associated with certain tasks and certain cognitive roles exclusively. In this connection, the paper claims that the subject matter of mathematical activity is represented within the system of activity by many different means. Mathematics differs in fact from logic in as much as a principle of heterogeneity or of flexible means-objects-relationships is valid. Formalization in contrast brings forward a principle of homogeneity — that like follows like. Every subject matter requires principles homogeneous with itself. The paper tries to draw some conclusions from this difference with respect to the role of formalization within human cognitive development.     

数学问题情境对数学问题解决影响的实验研究

本研究以测试材料为基础，运用教育统计方法分析和探讨了数学问题情境对数学问题解决的影响。