Aspects of the Nature and State of Research in Mathematics Education

This paper offers an outline and a characterisation of the didactics of mathematics, alias the science of mathematics education, as a scientific and scholarly discipline, and discusses why its endeavours should be of interest to research mathematicians (and other mathematics professionals). It further presents and discusses a number of major, rather aggregate findings in the discipline, including the astonishing complexity of mathematical learning, the key role of domain specificity, obstacles produced by the process-object duality, students' alienation from proof and proving, and the marvels and pitfalls of information technology in mathematics education.This revised version was published online in September 2005 with corrections to the Cover Date.     

On mathematics education and culture: Critical afterthoughts

This paper centers on the May 1988 Special Issue of Educational Studies in Mathematics devoted to mathematics education and culture, which is now available in book form. It questions some of the current tenets about the meaning and significance of the concept culture for a theory of mathematics education. More generally, it also argues against the cooping up of scientific problems by dividing the mathematics education community into small circles of experts which behave like a peculiar breed of mutual benefit societies, without giving due attention to the needs of scientific democracy and the simple pursuit of truth. Ultimately, it calls for an open scientific debate, unfettered by moral and ideological prejudices and, whenever necessary, disrespectful of fashionable notions. In this essay, all these points are tackled in close relation to a thorough — and unusually long — review of the book.     

高师数学教育中存在的问题与改革思路

高师数学教育是数学教师职前教育的基础和主要形式.当前高师数学教育中存在的主要问题：师范性未能很好突出,课程设置缺乏科学的指导,高等数学知识与初等数学教学脱节,实践环节薄弱,对数学教育研究不够.新一轮高师数学教育的改革思路：正确理解高师数学专业的培养目标,设置科学的课程结构,改进教学方法,强化高等数学与初等数学的联系,强化数学应用性和数学思想方法教学,强化教学技能训练和实践环节,加强数学教育研究.     

The Role of Intuitive Rules in Science and Mathematics Education

During the last two decades many researchers in mathematics and science education have studied students’ conceptions and ways of reasoning in mathematics and science. Most of this research is content‐specific. It was found that students hold alternative ideas that are not always compatible with those accepted in science. It was suggested that in the process of learning science or mathematics, students should restructure their specific conceptions to make them conform to currently accepted scientific ideas. In our work in mathematics and science education it became apparent that some of the alternative conceptions in science and mathematics are based on the same intuitive rules. We have so far identified two such rules: “More of A, more of B”, and “Subdivision processes can always be repeated”. The first rule is reflected in subjects’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.) in all tasks related to intensive quantities (density, temperature, concentration, etc.) and in all tasks related to infinite quantities. The second rule is observed in students’, preservice and inservice teachers’ responses to tasks related to successive division of material and geometrical objects and in seriation tasks. In this paper, we describe and discuss these rules and their relevance to science and mathematics education.     

普特南数学竞赛的历史与启示——一项实证研究

调查研究1938-1978年普特南数学竞赛优胜者--普特南会员的职业发展情况发现:大多数(76.4%)普特南会员都会获得博士学位,并且获得的博士学位几乎全部都是理工科博士学位,特别是数学(含统计学)博士学位;大多数(77.0%)普特南会员都会选择科学研究和教学作为自己的职业,特别是从事理工类科学研究和教学工作;多数(60.9%)普特南会员都会选择成为一名数学家,在高等院校或科研机构从事数学教学和研究工作;普特南会员的性别差异极为显著,40年间没有一位女性成为普特南会员.数学竞赛是识别和选拔数学与科学研究人才的有效途径.     

Mathématique et affectivité

Summary This article sums up a survey carried out on pupils of 15 to 18 years of age to study the relations existing between mathematics and affectivity. Pupils were considered according to their sex and whether they were in literary or scientific streams.This work has been carried out according to a clinical method based on 60 recorded interviews and also according to an experimental method based on 600 sets of questions. Its objective was to show in which way mathematics can be considered as a support for libidinal cathexis used by the unconscious through various mechanisms.This research work has been published in: Mathématique et Affectivité, Édition Stock, Paris.     

Creativity, Mathematizing, and Didactizing: Leen Streefland's work continues

This reaction to the papers in this PME Special Issue of Educational Studies in Mathematics draws a wider perspective on the issues addressed and some of the constructs used in research in Realistic Mathematics Education (RME). In particular, it tries to show that while the problems addressed existed within the world-wide arena of mathematics education and were not unique to the Dutch educational system, the methods used at the Freudenthal Institute to address them were uniquely adapted to that system yet foreshadowed developments in the wider field of mathematics education. The predictive aspects of mathematizing, didactizing, and guided reinvention, in which models-of become models-for on various levels, resonate with trends in mathematics education in recent years, including those promoted by the National Council of Teachers of Mathematics in the USA. Research methodologies, too, have broadened to include more humanistic qualitative methods. Developmental research as epitomized in the RME tradition makes the distinction between quantitative and qualitative research obsolete, because there is no restriction on research methods that may be useful in investigating how to improve the teaching and learning of mathematics, and in the designing of mathematics curricula. Thus some aspects of this research resonate with what have come to be known as multitiered teaching experiments. However, in RME there is also a special content-oriented didactical approach that harmonizes with an emphasis on didactics (rather than pedagogy)in several other European countries. Some implications are drawn for future research directions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Whose STEM? Disrupting the Gender Crisis Within STEM

This article challenges implicit understandings of scientific inquiry and gender within contemporary responses to the underrepresentation of women in science, technology, engineering, and mathematics (STEM). Failing to recognize the gendered history of science, and thus STEM disciplines, we argue that much research and curricular interventions are overly invested in simply providing women institutional access to or creating feminized environments within STEM. Such shortsighted analyses lead to processes we term gender washing and painting pink, both of which diminish the possibilities for meaningful engagement with gendered ontologies within scientific inquiry. Working from important and diverse strands of feminist theory, we suggest that serious considerations of women and STEM must include a framework that critically engages the entanglement of gender and science. We conclude that a focused and intentional analysis of gender and scientific inquiry, that opposes constructions of STEM as fixed entities to simply be imparted or made accessible to women, has important implications for understanding science as a (post) human endeavor.     

Intuitive Interference in Probabilistic Reasoning

One theoretical framework which addresses students’ conceptions and reasoning processes in mathematics and science education is the intuitive rules theory. According to this theory, students’ reasoning is affected by intuitive rules when they solve a wide variety of conceptually non-related mathematical and scientific tasks that share some common external features. In this paper, we explore the cognitive processes related to the intuitive rule more A–more B and discuss issues related to overcoming its interference. We focused on the context of probability using a computerized “Probability Reasoning – Reaction Time Test.” We compared the accuracy and reaction times of responses that are in line with this intuitive rule to those that are counter-intuitive among high-school students. We also studied the effect of the level of mathematics instruction on participants’ responses. The results indicate that correct responses in line with the intuitive rule are more accurate and shorter than correct, counter-intuitive ones. Regarding the level of mathematics instruction, the only significant difference was in the percentage of correct responses to the counter-intuitive condition. Students with a high level of mathematics instruction had significantly more correct responses. These findings could contribute to designing innovative ways of assisting students in overcoming the interference of the intuitive rules.     

Collaborative learning with scientific approach and multiple intelligence: Its impact toward math learning achievement

Abstract

This study examined the effect of collaborative learning with a scientific approach (SA) on the learning achievement in mathematics viewed from the students’ multiple intelligences. This study compared two types of collaborative learning models, Three Steps Interview (TSI), and Think Pair Share (TPS). The participants of this study were grade 8 students of Secondary Schools in Sukoharjo, Central Java, Indonesia (N?=?262). This study was a quasi-experimental study with a 3?×?3 factorial design. The results of the study can be concluded as follows. (1) TSI with a scientific approach got better math learning achievement than TPS with a scientific approach and classical with a scientific approach. Besides, TPS with a scientific approach got better learning achievement than classical with a scientific approach. (2) Students with logical-mathematical intelligence got better math learning achievement than students with linguistic intelligence and students with interpersonal intelligence, students with linguistic intelligence got better math learning achievement than students with interpersonal intelligence.     

Establishing an Explanatory Model for Mathematics Identity

This article empirically tests a previously developed theoretical framework for mathematics identity based on students’ beliefs. The study employs data from more than 9,000 college calculus students across the United States to build a robust structural equation model. While it is generally thought that students’ beliefs about their own competence in mathematics directly impact their identity as a “math person,” findings indicate that students’ self‐perceptions related to competence and performance have an indirect effect on their mathematics identity, primarily by association with students’ interest and external recognition in mathematics. Thus, the model indicates that students’ competence and performance beliefs are not sufficient for their mathematics identity development, and it highlights the roles of interest and recognition.     

The Future of the Discipline? Mathematics and the <Emphasis Type="Italic">Undergraduate Ambassadors Scheme</Emphasis>

The crisis in UK mathematics education, both in schools and universities, has been widely reported. Currently, 24% of full-time mathematics teachers in English secondary education have `weak’ or `nil’ qualifications in the subject, 31% of secondary mathematics teachers are now over the age of 50, and it is estimated that England alone is short of 3500 qualified mathematics teachers. Furthermore, studies suggest that UK mathematics undergraduates often lack confidence in several key skills essential for teaching, and the availability of attractive careers for mathematics graduates impacts upon the numbers entering the profession, thus risking a spiral of decline. This paper recounts the development of a unit of study in the Faculty of Mathematical Studies at the University of Southampton, England, in conjunction with a new national initiative entitled the Undergraduate Ambassadors Scheme (UAS). This aims to address the crisis in UK mathematics education through the training and placement of final year undergraduates as teaching assistants in local schools, with a view to promoting mathematics as a university choice to the broadest possible constituency, whilst simultaneously developing those key skills in which undergraduates regularly feel least confident, and facilitating an insight into teaching as a career option. The paper also provides a detailed evaluation of this new initiative, together with the recent comments of students who, having completed the UAS unit, have progressed subsequently to postgraduate teacher education.     

Locating mathematics within post-16 vocational education in England

The political importance of mathematics in post-16 education is clear. Far less clear is how mathematics does and should relate to vocational education. Successive mathematics curricula (e.g. core skills, key skills) have been developed in England with vocational learners in mind. Meanwhile, general mathematics qualifications remain largely disconnected from vocational learning. Following a brief historical survey of mathematics within vocational education, the paper presents findings from a nested case study of student groups in three large Further Education colleges in England. The primary unit of analysis herein is student groups learning Functional Mathematics in two vocational areas: construction and hairdressing. We show how approaches to organising teaching, developing connected curricula and classroom pedagogy tend to isolate or integrate mathematics from/with the vocational experience. Integrated approaches are shown to impact positively on student engagement and attitudes to learning mathematics. The paper concludes by discussing the potential impact of academic qualifications displacing vocationally relevant mathematics.     

Linking mathematics with engineering applications at an early stage – implementation,experimental set-up and evaluation of a pilot project

Too difficult, too abstract, too theoretical – many first-year engineering students complain about their mathematics courses. The project MathePraxis aims to resolve this disaffection. It links mathematical methods as they are taught in the first semesters with practical problems from engineering applications – and thereby shall give first-year engineering students a vivid and convincing impression of where they will need mathematics in their later working life. But since real applications usually require more than basic mathematics and first-year engineering students typically are not experienced with construction, mensuration and the use of engineering software, such an approach is hard to realise. In this article, we show that it is possible. We report on the implementation of MathePraxis at Ruhr-Universität Bochum. We describe the set-up and the implementation of a course on designing a mass damper which combines basic mathematical techniques with an impressive experiment. In an accompanying evaluation, we have examined the students' motivation relating to mathematics. This opens up new perspectives how to address the need for a more practically oriented mathematical education in engineering sciences.     

Mathematics and Mathematicians at Sapienza University in Rome (XVII–XVIII Century)

This article introduces some data regarding the teaching of mathematics in La Sapienza in the 17th century, with particular reference to the discipline’s role in the statutes, the lecturers, the courses’ programmes, the interest that Popes took in it. Specifically, it will focus on the changes that occured at the end of the 17th century, with regards to the development of the discipline and the improvement of a “scientific culture” in the city of the Pope.     

Educating the Young Mathematician: The Twentieth Century and Beyond

Educational programs for young children emerged reasonably early in the history of the United States of America. The movements of Child-Centered Education, the Nursery School, the Project Method, Curriculum Reform, and contemporary research have all influenced mathematics in early childhood education. The Froebelian kindergarten and the Montessori Casa die Bambini (Children’s House) included approaches to teaching mathematics. This article reviews the history of mathematics education in relation to the history of early childhood education from the turn of the twentieth century. It also discusses how research in mathematics education attempted to gain its own identity. Throughout history, researchers have identified mathematics issues and addressed them, defining the field, and generating a cadre of mathematics researchers.     

Assessing mathematics within advanced school science qualifications

Following sustained discussion regarding the relationship between advanced mathematics and science learning in England, the government has pursued a reform agenda in which mathematics is embedded in national, high stakes A-level science qualifications and their assessments for 18-year-olds. For example, A-level Chemistry must incorporate the assessment of relevant mathematics for at least 20% of the qualification. Other sciences have different mandated percentages. This embedding policy is running in parallel with an adding policy that is encouraging all young people to include the study of mathematics to 18. In this paper, we present a detailed scrutiny of the published sample assessment materials in the new A-level Physics, Chemistry and Biology qualifications in order to consider what the impact of this policy move might be for the teaching and learning of mathematics, its applications in upper secondary school advanced science studies and the implications in the transition to mathematically-demanding undergraduate studies.     

The Relation Between Executive Functions,Fine Motor Skills,and Basic Numerical Skills and Their Relevance for Later Mathematics Achievement

Research Findings: This study examined the interrelations between fine motor skills, executive functions, and basic numerical skills in kindergarten as well as their predictive value for mathematics achievement in 2nd grade in a sample of 136 children. Structural equation modeling techniques were used to uncover the unique predictive value and mediation of 4 predictors. The results indicated that each of these factors made a unique contribution to the prediction of later mathematics achievement; when estimated simultaneously, basic numerical skills were the strongest predictor, which suggests that domain-specific factors have a greater impact on mathematics achievement than domain-general factors. A strong direct and indirect predictor was updating. Nevertheless, indirect effects of fine motor skills and an inhibition/switching factor indicated that domain-general skills have a direct impact on early domain-specific precursors and through them an indirect effect on mathematics achievement. Practice or Policy: These findings suggest that the relationship between motor skills, executive functions, and mathematical skills is more complex in its nature. Therefore, to achieve the best outcome, all skills should be promoted if a child has difficulties with mathematics.