‘Experimental pedagogy’ in Germany,elaborated for mathematics – a case study in searching the roots of PME

PME, the International Group for the Psychology of Mathematics Education, was founded in 1976, at the Third International Congress on Mathematical Education in Karlsruhe, organised by the International Commission on Mathematics Instruction (ICMI). While PME is thus beyond coming of age and is reflecting its further orientation – due to the present “social turn” – the origins of investigating psychological aspects of mathematics learning have not yet been systematically studied. I am undertaking here a first such approach, concentrating on Germany, where the first pertinent monographs were published in 1913 and 1916. Different endeavours, focussing in particular on the notion of error, merged into the characteristic approach of ‘experimental pedagogy’. Given the key function of ICMI for founding PME, an additional aspect is whether the forerunner of ICMI: the Internationale Mathematische Unterrichtskommission (IMUK), founded in 1908, had an impact upon promoting research into the psychology of mathematics education. The pertinent research was effected by psychology; doing research themselves was still outside the horizon of mathematics educators. Perspectives of future research, in particular comparative ones, are outlined.     

Connecting theories in mathematics education: challenges and possibilities

This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary “themes”—integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the “limit” of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.     

Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

Lessons From the Past Look to the Future

Dr. Howden is now formally retired from a long career teaching mathematics in elementary school through graduate level, editing and coauthoring mathematics textbooks, serving as District Mathematics Coordinator K‐12 for the Albuquerque, New Mexico, Public Schools, writing numerous professional articles, and participating on the writing team of the Curriculum and Evaluation Standards for School Mathematics. She now enjoys long walks with her dog Quincy, “becoming increasingly aware, of mathematics all around us: patterns of leaf growth, distances between consecutive veins in leaves, kinds of ant hills, snake trails in the sand, bird call patterns.” She plans to relate many such examples to mathematical patterns in the book she is currently writing.     

Neuansätze und eine andere Sichtweise des mathematischen und naturwissenschaftlichen Unterrichts

In this paper, we discuss the question of how mathematics (in a typical manner) can contribute to general abilities aimed at at school, to general education and to the “Allgemeinbildung” of the pupils (especially of higher ages and in secondary schools). Our discussion concerns contributions of mathematics education in addition to providing mathematical literacy, technological aspects and all those concrete mathematical abilities necessary for “modern life”. Amongst others, the paper was motivated by the results of the international TIMS-studies (TIMSS) and—as well—by the discussions caused by the book of H. W. Heymann (1996) in Germany which, in many cases, had been held in a wrong way. Of course, the questions as well as some of our results are old ones, but they have to be discussed under new aspects from time to time, and they should be illustrated by concrete examples.     

Computational Literacy and “The Big Picture” Concerning Computers in Mathematics Education

This article develops some ideas concerning the “big picture” of how using computers might fundamentally change learning, with an emphasis on mathematics (and, more generally, STEM education). I develop the big-picture model of computation as a new literacy in some detail and with concrete examples of sixth grade students learning the mathematics of motion. The principles that define computational literacy also serve as an analytical framework to examine competitive big pictures, and I use them to consider the plausibility, power, and limitations of other important contemporary trends in computationally centered education, notably computational thinking and coding as a social movement. While both of these trends have much to recommend them, my analysis uncovers some implausible assumptions and counterproductive elements of those trends. I close my essay with some more practical and action-oriented advice to mathematics educators on how best to orient to the long-term trajectory (big picture) of improving mathematics education with computation.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

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.     

Exploring MLD in mathematics education: Ten years of research

Mathematical learning disabilities or difficulties (MLD) are an increasing source of educational inequalities. This article explores research about MLD in Mathematics Education over the past ten years. The methodology focuses on specific, validated keywords. These keywords are used to identify articles in leading journals in mathematics education. Our work makes several new contributions to the field of mathematics education, notably: a reusable methodology and keywords for a literature review; an exhaustive list of articles about MLD in leading mathematics education journals; a discussion of the definitions and features of MLD used in these articles; and a tool to classify research dealing with MLD (categories that characterize students with MLD). We also highlight some unexplored dimensions regarding MLD in Mathematics Education research.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

Books and Teaching Aids Received

Bachert , Russel E., Jr ., Hundreds of Ideas for Outdoor Education. Waters , John F., Hatchlings: The Lives of Baby Turtles. Cornett , Charles AND Cornett , Claudia , Reading Math. Greenlaw , James , History of Mathematics—18 Posters.     

问题驱动 多元表征的概念教学探究——对偶线性规划教学案例设计

数学与应用数学(师范)专业中的《运筹学》具有跨学科、实践性的课程特点,目标在于培养职前教师用数学方法解决实际问题的能力.结合义务教育阶段新课程标准中"四基"的提出这一背景,本文将以线性规划部分(运筹数学)对偶线性规划概念的引入这一知识模块为例,探讨通过问题串形式进行问题驱动、多元表征的概念教学过程.即遵循问题驱动—兴趣驱动—问题意识发展—提出和解决新问题,依据数学与外部联系、数学内部联系两条主线设计教学和学习,探索如何通过问题驱动、多元表征的结构化教学过程引导学生的学习方式发生改变,增强探究学习的动机,发展问题解决能力.课堂教学实践证明效果优于以往单一的讲授式教学法,一定程度上提高了学生的学业成绩、应用问题的兴趣和问题解决意识.     

The mathematics of war

During the Desert Storm, the Gulf war, it was possible to read in the newspapers words such as: “Inmathematical terms, was is becoming more and more electronically controlled and, as a result, it is moving away from the battlefield. Then, when war comes down to earth, it becomes bloody, it loses its mathematical asceticism” Reading the newspapers in those days, one had the impression that modern warfare is based on mathematics, as if it were not men but computers that decided where to carry out “surgical operations”. By contrast, the volume published a few years before the Gulf war conceived as a didactic unit to be used in schools with a guide for the teacher with the titleLa matematica della guerra (The Mathematics of War) published by Gruppo Abele in Turin begins with the words “Mathematics, like any other discipline, lends itself to building several paths towards education for peace”. The volume, written by a group of teachers belonging to an anti-violence organisation forming part of the “education for peace” project, highlights the power or ambiguitiy of mathematical models used to simulate war or conflict situations and demonstrates that in some cases the use of mathematics leads to a better understanding of the situation, but in other cases, the mathematical model itself can lead to conclusions which are either wrong or morally unacceptable.     

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM—The International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.     

Case study projects for college mathematics courses based on a particular function of two variables

Based on a sequence of number pairs, a recent paper (Mauch, E. and Shi, Y., 2005, Using a sequence of number pairs as an example in teaching mathematics, Mathematics and Computer Education, 39(3), 198–205) presented some interesting examples that can be used in teaching high school and college mathematics classes such as algebra, geometry, calculus, and linear algebra. In this paper, this study is generalized further to develop a few interesting case study proposals that can be used for student projects in college mathematics courses such as real functions, analytic geometry, and complex variables. In addition to using them in individual courses, these studies may also be combined to offer seminars or workshops to college mathematics students. Projects like these are likely to promote student interest and get students more involved in the learning process, and therefore make the learning process more effective.     

Felix Klein's “Erlanger Antrittsrede”: A Transcription with English Translation and Commentary

One of Felix Klein's leading interests was the role of mathematics education not only in the German universities but in the secondary schools as well. Klein played a leading role in the educational reform movements that flourished during the twenty-year period prior to World War I, and in 1908 he was elected President of the International Mathematics Instruction Commission. The “Erlanger Antrittsrede” of 1872, presented herein, gives a clear expression of Klein's views on mathematics education at the very beginning of his career. While previous writers, including Klein himself, have stressed the continuity between the Antrittsrede and his later views on mathematics education, the following commentary presents an analysis of the text together with external evidence supporting exactly the opposite conclusion.     

Learning from similarities and differences: a reflection on the potentials and constraints of cross-national studies in mathematics

Research in mathematics education that crosses national boundaries provides new insights into the development and improvement of the teaching and learning of mathematics. In particular, cross-national comparisons lead researchers to more explicit understanding of their own implicit theories about how teachers teach and how children learn mathematics in their local contexts as well as what is going on in school mathematics in other countries. Further, when researchers from multiple countries and regions study collaboratively aspects of teaching and learning of mathematics, the taken-for-granted familiar practices in the classroom can be questioned. Such cross-national comparisons provide opportunities for researchers and educators to probe typical dichotomies such as “high-performing” versus “low performing”, “teacher-centred versus student-centred”, or even “East versus West”, in searching for similarities and differences in educational policies and practices in different cultural contexts.     

The origins of mathematics education research in the UK: a tribute to Brian Griffiths

The meeting of the British Society for Research into Learning Mathematics held at the University of Southampton on 21st June 2008 was dedicated to the memory of Brian Griffiths, who had died earlier that month. At the meeting it was suggested that I write a paper tracing the development of research in mathematics education in the UK up to the writing of Mathematics: Society and Curricula, a book that Brian and I co-authored, published in 1974. This paper is dedicated to Brian's memory: I hope that it will be considered a fitting tribute to a great friend and colleague, and to an outstanding mathematician and mathematics educator.