Identifying national cultures of mathematics education: Analysis of cognitive demands and differential item functioning in TIMSS

Large-scale assessments of student competencies address rather broad constructs and use parsimonious, unidimensional measurement models. Differential item functioning (DIF) in certain subpopulations usually has been interpreted as error or bias. Recent work in educational measurement, however, assumes that DIF reflects the multidimensionality that is inherent in broad competency constructs and leads to differential achievement profiles. Thus, DIF parameters can be used to identify the relative strengths and weaknesses of certain student subpopulations. The present paper explores profiles of mathematical competencies in upper secondary students from six countries (Austria, France, Germany, Sweden, Switzerland, the US). DIF analyses are combined with analyses of the cognitive demands of test items based on psychological conceptualisations of mathematical problem solving. Experts judged the cognitive demands of TIMSS test items, and these demand ratings were correlated with DIF parameters. We expected that cultural framings and instructional traditions would lead to specific aspects of mathematical problem solving being fostered in classroom instruction, which should be reflected in differential item functioning in international comparative assessments. Results for the TIMSS mathematics test were in line with expectations about cultural and instructional traditions in mathematics education of the six countries.     

Use of comics to enhance students’ learning for the development of the twenty-first century competencies in the mathematics classroom

Abstract

This paper discusses the use of comics in teaching mathematics in the secondary mathematics classroom. We explicate how the use of comics in teaching mathematics can prepare students for the twenty-first century competencies. We developed an alternative teaching package using comics for two lower secondary mathematics topics. This alternative teaching package consists of (1) several sets of comic strips expounding all related mathematical concepts in a lively way; (2) tiered practice questions for learning reinforcement; and (3) a set of proposed lesson outlines with suggestions on how to use the comics for mathematics teaching. We also report how one of the teachers in our study used this teaching package in her mathematics lessons. Her lessons were video-recorded and 11 students were interviewed to help us understand how the mathematics comics lessons were enacted and the students’ perception of comics as instruction. We identified instances in which the teacher tweaked the provided resource to further enhance student learning and incorporated elements of the twenty-first century competencies during her lessons. Through selected student interviews, we also identified instances in which students commented on their gain from the new approach from the perspective of the twenty-first century competencies.     

Pedagogical Media Competencies of Preservice Teachers in Germany and the United States: A Comparative Analysis of Theory and Practice

Teachers need knowledge and skills to make effective use of the growing variety of media and technology texts and tools available for use in elementary and secondary education. For this reason, pedagogical media competencies are highly relevant for teachers’ professional development. The theory of media pedagogical competencies is first defined and then located in the context of the relevant scholarly literature from both Germany and the United States. We conducted a comparative analysis of German and U.S. pedagogical media competency models and report data on a survey of teacher education programs in Germany and the United States to identify the proportion of programs of study that include courses in media didactics, media and school reform, and media education. We consider the implications of the missing connections between the theoretical framework of pedagogical media competencies and the current practice of media pedagogical teacher training, revealing implications for further work needed to improve the integration of media in various school-related contexts.     

中国小学生解决国外数学试题分析

通过中国小学生解决国外数学试题的分析 ,探讨中国基础教育现状 ,为数学教学改革提供实证研究基础。我们从国外数学测试中筛选、改编了一套小学五年级数学试卷 ,在全国九个省、市做了测验。结果显示 ,中国学生在考察基础知识的项目中表现出色 ;但在考察利用已有的生活经验解决问题的项目中 ,存在着相当的问题。因此 ,中国数学教育在重视基础知识的同时 ,应该引导学生有意义地理解数学、灵活自主地选择数学方法解决问题。     

The development and validation of a framework for teaching competencies in higher education

In higher education, approaches to teaching are becoming more student-centred, which demands different teaching competencies. Therefore, it is necessary to have an adequate framework of teaching competencies that can be used for evaluation purposes. The weaknesses of the existing frameworks are that they do not pay attention to the person as teacher, they are too narrowly defined, they are not validated and they are not adjusted to modern approaches to teaching. The aim of this study was to develop and validate a framework of teaching competencies in higher education. A framework for teaching competencies was therefore constructed containing the following domains: The Person as Teacher, Expert on Content Knowledge, Facilitator of Learning Processes, Organiser and Scholar/Lifelong Learner. The framework was validated using a Delphi method. Educational experts (N = 63) were asked: ``How important are the following teaching competencies in each domain for an experienced teacher in higher education?'' A confirmatory factor analysis was conducted to assess the adequacy of the domains. After two rounds, the shift in ratings was minimal, so the results were considered stable. The response rate was 82%. From the original list with 134 items, 30 items were omitted because many experts failed to respond to them. The experts reached consensus on 61 out of the resulting 104 items (59%). A confirmatory factor analysis on the three best scoring items in each domain confirmed the model. A framework of teaching competencies was developed and validated that can be used as a starting point for teacher evaluation in higher education.     

Didactics: Swedish and Norwegian understandings compared by a Norwegian

A debate concerning didactics as a concept and research field has been conducted in Sweden during the 1980s. This article presents central points of view from this debate, and discusses them on the basis of a Norwegian understanding of didactics. From this perspective the author holds that the Swedish discussion of didactics may appear somewhat near‐sighted. The conclusion drawn is that the primary task of didactics must be the establishment of a basis for the attainment of more knowledge about, better understanding of, and deeper insight into education. Such a task is scarcely best solved through a fight over the right understanding of didactics. A better strategy would comprise efforts at fruitful dialogue and focus on research aimed at illuminating all aspects of education and its influencing factors.     

The Impact of Item Stem Format on the Dimensional Structure of Mathematics Assessments

Item stem formats can alter the cognitive complexity as well as the type of abilities required for solving mathematics items. Consequently, it is possible that item stem formats can affect the dimensional structure of mathematics assessments. This empirical study investigated the relationship between item stem format and the dimensionality of mathematics assessments. A sample of 671 sixth-grade students was given two forms of a mathematics assessment in which mathematical expression (ME) items and word problems (WP) were used to measure the same content. The effects of mathematical language and reading abilities in responding to ME and WP items were explored using unidimensional and multidimensional item response theory models. The results showed that WP and ME items appear to differ with regard to the underlying abilities required to answer these items. Hence, the multidimensional model fit the response data better than the unidimensional model. For the accurate assessment of mathematics achievement, students’ reading and mathematical language abilities should also be considered when implementing mathematics assessments with ME and WP items.     

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.     

Analyzing the use of history in mathematics education: issues and challenges around Balacheff’s cKȼ model

Part of the international reflection on the use of history in mathematics teaching consists in a quest of frameworks and models suitable for empirical studies. Following this demand, this paper explores the way Balacheff’s cK? model, a model taken from the didactics of mathematics, can be used in the analysis of learning at student level. In the first part of this paper, Balacheff’s cK? model (conceptions, knowledge, ?oncepts) is shortly presented, and in the second part, the relationship between the epistemological background of the model and the use of the history of mathematics is explored in order to show a possible suitableness. The third part addresses an example of a school activity (about ancient Indian geometry) in which the model is applied and the historical issues clarified. Questioning the role of problems both in the cK? model and in the use of history, the last part shows how a study at the students’ conception level enlightens the way in which historical elements can interact with contemporary mathematical learning.     

Teaching units as the integrating core of mathematics education

How to integrate mathematics, psychology, pedagogy and practical teaching within the didactics of mathematics in order to get unified specific theories and conceptions of mathematics teaching? This problem—relevant for theoretical and empirical studies in mathematics education as well as for teacher training—is considered in the present paper. The author suggests an approach which is based on teaching units (Unterrichtsbeispiele). Suitable teaching units incorporate mathematical, pedagogical, psychological and practical aspects in a natural way and therefore they are a unique tool for integration.     

数学建模视角下的新加坡小学数学书Learning Maths

新加坡数学在TIMSS和PISA两大测试中脱颖而出,备受国际数学教育界关注。新加坡数学练习书Learning Maths将新加坡数学教育的精髓数学建模融入其中,而且在解题方法、编排方式、学习效果检测、家庭教育、信息技术的应用等方面具有独特之处。通过对新加坡小学数学书Learning Maths的剖析,进一步挖掘新加坡数学教育的内涵,探讨其中可资借鉴之处。     

A Competency Model for Process Dynamics and Control and Its Use for Test Construction at University Level

This study examined students’ competencies in engineering education at the university level. First, we developed a competency model in one specific field of engineering: process dynamics and control. Then, the theoretical model was used as a frame to construct test items to measure students’ competencies comprehensively. In the empirical application, the reliability of the test was examined. The results confirmed the applicability of the theory-based test. Also, associations between students’ competencies and individual characteristics were explored. Students who showed higher competencies could be identified based on their individual characteristics, such as participation in self-assessments. Finally, we demonstrated how to apply the standards of theory-based testing to everyday routines at the university and also highlighted some challenging issues to be considered when implementing theory-based testing in practice. This approach could be valuable: Tests based on theoretically sound models allow for appropriate measurement of students’ competencies in higher education.     

A meta-analysis of comparative studies on Chinese and US students’ mathematics performance: Implications for mathematics education reform and research

US and China are reforming mathematics teaching by shifting from students’ attainment of facts and procedures toward development of competencies in reasoning, communication, connections, and problem solving, and application of these in real life contexts. Differences in students’ overall performance, curricula, and teachers’ knowledge and instruction between US and Eastern Asian countries are often used to support US reform with two obvious limitations. First, their performance has not been delineated into specific areas which raise questions about whether overall higher Asian mathematics performance over US is also evident in the specific US reform competencies. Second, Asians are often used as an indiscriminate group with inattention to different schooling and non-schooling factors between countries that might contribute differently to performances. This meta-analysis examines US and Chinese student mathematics performance studies and identifies the strengths and weaknesses in overall and specific competencies. It raises questions about theoretical assumptions, discusses limitations of research designs, and proposes research that may lead to a critical understanding of the quality of mathematics learning.     

Applied mathematical problem solving,modelling, applications,and links to other subjects — State,trends and issues in mathematics instruction

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.Written version of a Survey Lecture given jointly at the Sixth International Congress on Mathematical Education (ICME-6, Budapest 1988). A condensation was published in a volume with contributions from the ICME-6 Theme Groups on Problem Solving, Modelling and Applications and on Mathematics and Other Subjects (Blum/Niss/Huntley 1989, pp. 1–21).     

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.     

Reclaiming School Mathematics: The girls fight back

ABSTRACT In this paper I consider what appears to be an emerging feminist perspective within mathematics education that suggests that theories such as 'attribution theory' lay too much 'blame' upon girls and women for their underachievement in mathematics and not enough blame upon the wider school system. I attempt to extend this theoretical position further through the use of case study data from two schools. Interviews with underachieving girls are used to show the way in which girls link their underachievement, not to themselves, but to the type of mathematics that is widely taught in the UK, which they believe denies them access to understanding. An alternative model of mathematics teaching is described that is open and project-based and that may be able to eradicate underachievement and disaffection amongst girls.     

Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students’ problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.     

Primary Mathematics Teacher Education in Greece: Reality and Vision

This paper focuses on two main issues concerning the mathematics education of prospective primary school teachers in Greece: the integration of mathematics and pedagogy and the relation of theory to practice. In particular, specific teaching approaches are discussed concerning the problem of integration both in mathematics and in mathematics education courses. The problem of "theory – practice" is examined through an analysis of the kind of teaching practice in which the prospective teachers are involved. Finally, the constraints that the mathematics educators face and the impact of their work on the professional life of the teachers in the future are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

ARE TAIWANESE AND SINGAPOREAN FUTURE TEACHERS SIMILAR IN THEIR MATHEMATICS-RELATED TEACHING COMPETENCIES?

This study investigates the similarities and differences of mathematics-related teaching competencies between the future secondary school teachers of Taiwan and Singapore by using data from the international Teacher Education and Development Study in Mathematics (TEDS-M), organized by the International Association for the Evaluation of Educational Achievement. Taiwan and Singapore are identified as the only two Asian countries integrating the Chinese/Confucian tradition in their education regarding TEDS-M. The relative strengths and weaknesses of these two countries, as compared to other TEDS-M higher-achieving Western countries, such as Germany, Poland, Russia, Switzerland, and the USA, are examined after reselecting and reclassifying the achievement items based on Niss’ (2003) mathematics competence and Hsieh’s (Secondary Education, 63(3), 2012) mathematics teaching competence structures. This study shows that Singaporean future teachers are weaker in the competence of devising formal mathematical arguments and transforming heuristic ideas into valid proofs than Taiwanese future teachers. Taiwan and Singapore demonstrate relative strengths at primary, lower-, and upper-secondary levels in mathematics competencies but show relative weaknesses at the tertiary level than do other higher-achieving Western countries. Comparing these two countries, Taiwan shows relative strengths at upper-secondary and tertiary levels, and Singaporean strengths are at primary and lower-secondary levels. This distinction is primarily because of the thought-oriented category of mathematics competencies. This study also finds that Taiwan and Singapore belong to different clusters compared with the higher-achieving Western countries regarding their future teachers’ relative strengths and weaknesses in either mathematics or mathematics teaching competence.