Knowledge of future primary teachers for teaching mathematics: an international comparative study

This article reports the results of the Teacher Education and Development Study in Mathematics (TEDS-M) that are related to prospective primary teachers’ knowledge for teaching mathematics. TEDS-M was conducted under the auspices of the International Association for the Evaluation of Educational Achievement with additional support from the US. National Science Foundation and the participating countries. In 2008 more than 15,000 future primary teachers, enrolled in about 450 institutions that prepare future primary teachers, were surveyed. Two domains of knowledge for teaching mathematics were assessed using items that had been developed and validated in a cross-national field trial. Large differences in the structure of teacher preparation programs are reported. Differences in mathematical content knowledge (MCK) and mathematical pedagogical content knowledge (MPCK) were also observed both within and between programs and countries. Anchor points on the MCK and MPCK scales are used to describe qualitative characteristics of knowledge for teaching mathematics.     

The conceptualisation of mathematics competencies in the international teacher education study TEDS-M

The main aim of the international teacher education study Teacher Education and Development Study in Mathematics (TEDS-M), carried out under the auspices of the International Association for the Evaluation of Educational Achievement (IEA), was to understand how national policies and institutional practices influence the outcomes of mathematics teacher education. This paper reports on the definition of effective mathematics teacher education in TEDS-M, distinguishing between mathematics content knowledge and mathematics pedagogical content knowledge as essential cognitive components of mathematics teachers’ professional competencies. These competence facets were implemented as proficiency tests based on extensive coordination and validation processes by experts from all participating countries. International acceptance of the tests was accomplished whereas, by necessity, national specifications had to be left out, as is common in comparative large-scale assessments. In this paper, the nature of the TEDS-M tests for the primary study is analysed and commented on detail. The aims are to increase our understanding of mathematics content knowledge and mathematics pedagogical content knowledge, which are still fuzzy domains, to provide a substantive background for interpretations of the test results and to examine whether some educational traditions may be more accurately reflected in the test items than others. For this purpose, several items that have been released by the IEA are presented and elaborately analysed in order to substantiate the test design of TEDS-M. Our main conclusion is that the overall validity of the TEDS-M tests can be regarded as a given, but that readers have to be aware of limitations, amongst others from a continental European point of view.     

When an argument is the content: Rational number comprehension through conversions across registers

This article reframes previously identified misconceptions about repeating decimals by describing these misconceptions as limited understandings of how mathematics concepts are referenced. In particular, misconceptions about repeating decimals and their quotient of integer representations are recast as limited understandings of mathematics as a discipline that derives its content from representational systems and the denotations they provided. Under this framework, arguments (e.g., proofs) that convert repeating decimals to their quotient of integer representations provide content for “rational number,” which is represented in multiple ways, each offering distinct opportunities for mathematical activity. The notion of an argument as content is illustrated as arguments providing access to a concept. One Grade 8 student’s struggle with understanding rational number is used to illustrate this framework and its implications for teaching and learning.     

数学建模对高中生构建数学底层思维的作用及其教学实施建议

数学底层思维即用数学的眼光观察世界、用数学的思维分析世界以及用数学的语言表达世界,是人们面对自然和社会中纷繁多样的现象和问题时,所展现的自发的、不依赖监督的、融汇数学学科核心素养的思维方式.作为国家高中新课程标准中数学六大核心素养之一的数学建模,是培养学生数学底层思维的良好载体,对人才培养和社会发展均起到良好的促进作用.本文主要阐述了数学建模对高中生构建数学底层思维的作用,并结合教学实例给出教学实施建议.     

Learning from the Eastern and the Western debate: the case of mathematics teacher education

Comparative studies have gained significant influence in the last decades, and school systems of many countries have been revised referring to better results of other countries in international large-scale assessments. Authors of such studies commonly link their interpretations of the results to distinctions between “Eastern” and “Western” cultures, in particular with respect to the consistent and continuing outstanding performance of East Asian learners compared with their Western counterparts. One question is whether the same achievement pattern holds for future teachers and whether similar cultural differences may cause it. International Association for the Evaluation of Educational Achievement’s “Teacher Education and Development Study in Mathematics” (TEDS-M) was the first comparative study that focused on the outcomes of teacher education with standardised testing. In this paper—based on the TEDS-M results—commonalities and differences in the achievement of future teachers from Eastern and Western countries are explored and related to a cultural perspective. Cultural differences between Eastern and Western approaches concerning mathematics, mathematics education and mathematics teachers are analysed with respect to the achievement pattern. The paper closes with reflections on possible consequences concerning the development of teachers’ knowledge and teachers’ expertise in mathematics education.     

“Last curves not quite correct”: diagnostic competences of future teachers with regard to modelling and graphical representations

The article describes the results of a national enrichment to the six-country study Mathematics Teaching in the 21st century (MT21)—an international comparative study about the efficiency of teacher education. The enrichment focuses on the diagnostic competence of future mathematics teachers as sub-component of teachers’ professional competence for which the evaluation of students’ solutions of a modelling task about the course of a racetrack is demanded. In connection with two sub-facets of the diagnostic competence, namely the competence to recognise students’ misconceptions and the competence of criteria-guided assessment of students’ solutions, typical answer patterns are distinguished as well as the frequency of their occurrence with regard to future teachers’ phase of teacher education and the level of school teaching they are going to teach in.     

Explanation,motivation and question posing routines in university mathematics teachers' pedagogical discourse: a commognitive analysis

This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard's commognitive framework is used to investigate the teaching practices of seven Swedish university mathematics teachers on the topic of functions. The present paper looks at the discourse of mathematics teaching, presenting a categorization of the didactical routines into three categories – explanation, motivation and question posing routines. All of these are present in the discourses of all seven teachers, but within these general categories, a number of different sub-categories of routines are found, used in different ways and to different extent by the various teachers. The explanation routines include known mathematical facts, summary and repetition, different representations, everyday language, and concretization and metaphor; the motivation routines include reference to utility, the nature of mathematics, humour and result focus; and the question posing routines include control questions, asking for facts, enquiries and rhetorical questions. This categorization of question posing routines, for instance, complements those already found in the literature. In addition to providing a valuable insight into the teaching of functions at the university level, the categorizations presented in the study can also be useful for investigating the teaching of other mathematical topics.     

Preparing primary teachers in the United States: balancing selection and preparation

The recent movement in the United States toward common (national) mathematics standards brings to the fore issues of implementation which includes how teachers are prepared to teach. The Teacher Education and Development Study in Mathematics (TEDS-M) provide an international context for an in depth analysis of the US primary future teachers’ TEDS-M results. The US education system is decentralized with many aspects controlled by the states and many other aspects controlled by over 16,000 local school districts. Consequently, there is great variation in what secondary mathematics individuals have had before entering teacher preparation. In addition, as certification is a state regulation, teacher preparation programs also differ in what is provided to prospective teachers. Analyses reveal significant relationship between both of these to the mathematics content knowledge and mathematics pedagogical content knowledge of US future primary teachers. A two-stage hierarchical model is used to explore the relative contribution of these two factors.     

The preparation of primary mathematics teachers in Singapore: programs and outcomes from the TEDS-M study

This paper describes five aspects of primary mathematics teacher education in Singapore: (a) the teaching profession in Singapore, (b) the structure of pre-service teacher education programs offered by the National Institute of Education, (c) self-reports of Singapore future primary mathematics teachers about the opportunities to learn mathematics-related contents offered by these programs, based on the Teacher Education and Development Study in Mathematics (TEDS-M) survey, (d) the performance of these future teachers in mathematics content knowledge and mathematics pedagogical content knowledge assessed by the TEDS-M study, and (e) the relationships of opportunities to learn with this performance. The paper concludes with some suggestions about how to improve the quality of initial teacher preparation in the areas of recruitment and training.     

A model of professional competences in mathematics to update mathematical and didactic knowledge of teachers

This paper describes part of a research and development project carried out in public elementary schools. Its objective was to update the mathematical and didactic knowledge of teachers in two consecutive levels in urban and rural public schools of Region de Los Lagos and Region de Los Rios of southern Chile. To that effect, and by means of an advanced training project based on a professional competences model, didactic interventions based on types of problems and types of mathematical competences with analysis of contents and learning assessment were designed. The teachers’ competence regarding the didactic strategy used and its results, as well as the students’ learning achievements are specified. The project made possible to validate a strategy of lifelong improvement in mathematics, based on the professional competences of teachers and their didactic transposition in the classroom, as an alternative to consolidate learning in areas considered vulnerable in two regions of the country.     

Developing teachers’ and teaching researchers’ professional competence in mathematics through Chinese Lesson Study

This study examines co-learning of mathematics practicing teachers and mathematics teaching researchers through parallel lesson study in China. Two cases are illustrated and compared to highlight what practicing teachers and teaching researchers learned. The practicing teachers developed their competence in identifying instructional objectives, improving instructional process, selecting and sequencing mathematical tasks, and developing professional vision. The mathematics teaching researchers developed their professional competence in effectively carrying out teaching research activities, effectively mentoring teachers, and deepening the understanding of teaching.     

Searching for good mathematics instruction at primary school level valued in Taiwan

In this article, we aim to provide a glimpse of what is counted as good mathematics instruction from Taiwanese perspectives and of various approaches developed and used for achieving high-quality mathematics instruction. The characteristics of good mathematics instruction from Taiwanese perspectives were first collected and discussed from three types of information sources. Although the number of characteristics of good mathematics instruction may vary from one source to another, they can be generally organized in three phases including lesson design before instruction, classroom instruction during the lesson and activities after lesson. In addition to the general overview of mathematics classroom instruction valued in Taiwan, we also analyzed 92 lessons from six experienced teachers whose instructional practices were generally valued in local schools and counties. We identified and discussed the characteristics of their instructional practices in three themes: features of problems and their uses in classroom instruction, aspects of problem–solution discussion and reporting, and the discussion of solution methods. To identify and promote high-quality mathematics instruction, various approaches have been developed and used in Taiwan including the development and use of new textbooks and teachers’ guides, teaching contests, master teacher training program, and teacher professional development programs.     

A cross-national comparison of reform curricula in Korea and the US in terms of cognitive complexity: the case of fraction addition and subtraction

The overall level of conceptual understanding and mathematical proficiency of students has been a matter of increasing national interest in South Korea. Recently, a new edition of mathematics textbooks aligned with the amendment of the 7th national mathematics curriculum has become available for all elementary grade levels. To characterize the current reform efforts in South Korea, this study examined the quality of the mathematical problems in the current version of the Korean reform textbooks (KM 2) compared with the previous version (KM 1) and one representative US reform curriculum text (EM). Webb’s (Research monograph No. 18: Alignment of science and mathematics standards and assessments in four states. National Institute for Science Education, Madison, 1999) depth of knowledge framework and Son and Senk’s (Educ Stud Math 74(2):117–142, 2010) cognitive expectation feature were employed to examine the kind and level of students’ opportunities to learn along with the type of word problems presented in the three sets of materials. Analysis revealed that the KM 2 provided better opportunities for students to learn fraction addition and subtraction than the KM 1 in terms of the depth and breadth of cognitive complexity. However, there was little difference in addressing and developing the meaning of fraction addition and subtraction through word problems. Moreover, compared with the US reform curriculum materials, the KM 2 provided more problems requiring lower depth of knowledge levels than the US counterpart. Implications of these findings for curriculum developers, textbook and learning materials developers, teachers and future researchers are discussed.     

Conceptions of Turkish mathematics teachers about the effectiveness of classroom teaching

This study investigated how Turkish mathematics teachers evaluate the effectiveness of classroom teaching in terms of improving students’ mathematical proficiency. To this purpose, teachers were asked to evaluate a mathematics lesson as presented them in a vignette. By means of cluster analysis, the participants’ evaluations of the lesson were described in five thematic dimensions, which could be further assembled into two overriding categories: students’ understanding of the subject, and teachers’ classroom practices. The overall aim of the current paper is to propose a preliminary model of the framework that Turkish mathematics teachers use to evaluate a mathematics lesson.     

ICT integration in mathematics initial teacher training and its impact on visualization: the case of GeoGebra

This case study investigates the impact of the integration of information and communications technology (ICT) in mathematics visualization skills and initial teacher education programmes. It reports on the influence GeoGebra dynamic software use has on promoting mathematical learning at secondary school and on its impact on teachers’ conceptions about teaching and learning mathematics. This paper describes how GeoGebra-based dynamic applets – designed and used in an exploratory manner – promote mathematical processes such as conjectures. It also refers to the changes prospective teachers experience regarding the relevance visual dynamic representations acquire in teaching mathematics. This study observes a shift in school routines when incorporating technology into the mathematics classroom. Visualization appears as a basic competence associated to key mathematical processes. Implications of an early integration of ICT in mathematics initial teacher training and its impact on developing technological pedagogical content knowledge (TPCK) are drawn.     

Construction and Properties of Adhesive and Weak Adhesive High-Level Replacement Categories

As presented in Ehrig et al. (Fundamentals of Algebraic Graph Transformation, EATCS Monographs, Springer, 2006), adhesive high-level replacement (HLR) categories and systems are an adequate framework for several kinds of transformation systems based on the double pushout approach. Since (weak) adhesive HLR categories are closed under product, slice, coslice, comma and functor category constructions, it is possible to build new (weak) adhesive HLR categories from existing ones. But for the general results of transformation systems, as additional properties initial pushouts, binary coproducts compatible with a special morphism class and a pair factorization are needed to obtain the full theory. In this paper, we analyze under which conditions these additional properties are preserved by the categorical constructions in order to avoid checking these properties explicitly.      

Gender justice,human rights and measurement in the mathematics classroom

In line with international trends, the new South African mathematics curriculum implores mathematics educators to realize a pedagogy in their classrooms that is more practical, activity-oriented, and connected to their learners' lives. Drawing on data from a larger study that explores theory–practice relations in mathematics education, this paper shows how such progressive practices, when interpreted with respect to the teaching of measurement which required learners to use different measuring instruments for measuring the school grounds in learning about length and perimeter, were found to be deeply gendered. In two different contexts of an ‘African' township school and a predominantly ‘Indian' suburban school, girls in a grade 6 mathematics classroom faced direct sexism as they struggled to take the opportunity to participate in the activity and learn how to measure – an important mathematical competence and everyday knowledge and skill. The article analyses the data with reference to the human rights imperatives of the new national curricula and approaches to addressing disadvantage and discrimination for girls in mathematics classrooms.     

CAS in general mathematics education

A rational discussion of the use of Computer algebra systems (CAS) in mathematics teaching in general education needs an explicit image of (general) mathematics education, an explication of global perspectives and goals on mathematics teaching focusing on general education (chapter 1). The conception of general education according to the «ability of communication with experts» described in chapter 2 can be such an orientation for analysing, considering, classifying and assessing the didactical possibilities of using CAS. CAS are materialised mathematics allowing for more or less exhaustive outsourcing of operative (also symbolically) knowledge and skills to the machine. This frees up space of time as well as mental space for the development of those competences being in our view relevant for general mathematics education. In chapter 3 the idea of outsourcing and the role of CAS for it is discussed more detailed as well as consequences being possible for the CAS-supported teaching of mathematics. Beyond, CAS can be didactically used and reflected as a model of communication between (mathematical) experts and lay-persons (chapter 4). Chapter 5 outlines some research perspectives.     

Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie-Kieren Theory

The study reported here extends the work of Pirie and Kieren on the nature and growth of mathematical understanding. The research examines in detail a key aspect of their theory, the process of ‘folding back’, and develops a theoretical framework of categories and sub-categories that more fully describe the phenomenon. This paper presents an overview of this ‘framework for folding back’, illustrates it with extracts of video data and elaborates on its key features. The paper also considers the implications of the study for the teaching and learning of mathematics, and for future research in the field.     

Integration of Technology in the Design of Geometry Tasks with Cabri-Geometry

Beginning with the gap in France between the institutional support for the use of technology in mathematics teaching and its weak integration into teacher practice, this paper claims that integrating technology into teaching is a long process. The aim of the paper is to identify and analyse the steps in this integration using as an example the evolution over time (3 years) in the design of teaching scenarios based on Cabri-géomètre for high school students. The analysis indicates that the role played by the technology moved from being a visual amplifier or provider of data towards being an essential constituent of the meaning of tasks and as a consequence affected the conceptions of the mathematical objects that the students might construct.This revised version was published online in September 2005 with corrections to the Cover Date.