# ZDM: the international journal on mathematics education

Published by Springer Verlag

Online ISSN: 1863-9704

Published by Springer Verlag

Online ISSN: 1863-9704

Publications

Article

In this article, we review the brain and cognitive processes underlying the development of arithmetic skills. This review focuses primarily on the development of arithmetic skills in children, but it also summarizes relevant findings from adults for which a larger body of research currently exists. We integrate relevant findings and theories from experimental psychology and cognitive neuroscience. We describe the functional neuroanatomy of cognitive processes that influence and facilitate arithmetic skill development, including calculation, retrieval, strategy use, decision making, as well as working memory and attention. Building on recent findings from functional brain imaging studies, we describe the role of distributed brain regions in the development of mathematical skills. We highlight neurodevelopmental models that go beyond the parietal cortex role in basic number processing, in favor of multiple neural systems and pathways involved in mathematical information processing. From this viewpoint, we outline areas for future study that may help to bridge the gap between the cognitive neuroscience of arithmetic skill development and educational practice.

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Article

In this article we present, illustrate, test and refine a framework developed by Galbraith, Stillman, Brown and Edwards (2006)
for identifying student blockages whilst undertaking modelling tasks during transitions in the modelling process. The framework
was developed with 14~15 year old students who were engaging in their first experiences of modelling at the secondary level.
ZDM-ClassificationC70-M10

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Article

Despite some studies by the historian Wilhelm Lorey, Edmund Külp is rather unknown today. His role in the development of mathematics
and mathematics teaching in the nineteenth century, however, deserves closer attention. Having been the director of the höhere
Gewerbe—und Realschule in Darmstadt, he can be counted among the founders of the Technische Hochschule Darmstadt. Moreover,
he had been, still at the Realschule, the mathematics teacher of Georg Cantor. Recently detected documents concerning Külp’s
mathematical formation in Brussels by A. Quetelet permit revealing insights into the evolution of Külp’s mathematical ideas
and of his views on the context of mathematics in Germany. The contribution presents extracts from these documents (in French)
and analyses them. Furthermore, the paper discusses possible influences exerted by Külp on Cantor.

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Article

In the 1970s significant research was conducted concerning the development of methods for teaching mathematics. The most outstanding
of these projects, led by the late Tamás Varga, and which had a major influence on teaching mathematics in Hungary, was called
OPI. This project comprised research based on experiments aiming at the complete renewal of methods and content in mathematics
teaching. In 1978 a centralized and compulsory new curriculum was introduced that was based on the results of the Varga’s
research. In the following decade development aimed at adopting and realizing the research results within practice. Research
mainly aimed at examining the effects of the newly introduced curriculum by looking into the development of children’s problem-solving
skills. Other research was associated with international studies such as SIMS, TIMMS, and PISA. Additional research and development
into different aspects of problem solving, summarized here, was conducted by various research groups around the country.

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Article

Problem solving was a major focus of mathematics education research in the US from the mid-1970s though the late 1980s. By
the mid-1990s research under the banner of “problem solving” was seen less frequently as the field’s attention turned to other
areas. However, research in those areas did incorporate some ideas from the problem solving research, and that work continues
to evolve in important ways. In curricular terms, the problem solving research of the 1970s and 1980s (see, e.g., Lester in
J Res Math Educ, 25(6), 660–675, 1994, and Schoenfeld in Handbook for research on mathematics teaching and learning, MacMillan, New York, pp 334–370, 1992, for reviews) gave birth to the “reform” or “standards-based” curriculum movement. New curricula embodying ideas from the
research were created in the 1990s and began to enter the marketplace. These curricula were controversial. Despite evidence
that they tend to produce positive results, they may well fall victim to the “math wars” as the “back to basics” movement
in the US is revitalized.

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Article

Delivered as the closing session, this talk was supposed to be an opening towards the future, both of children and humankind.
There will be problems and challenges. How is math involved? Math has permanent values, recognized by other scientists, and
also stable notions, some of them going back to the ancient Greeks. However math is in a perpetual motion. Old notions get
a new look, new notions appear, as well as new relations with other sciences, international relations, including developing
countries, new trends and a new conception of mathematical sciences. Math teaching should express both permanence and mobility
of the subject, utility and beauty. Informatics, probability and statistics, geometry and all kinds of computing are subjects
of reports under preparation, for a long term view of math education.

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Article

Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (by comparison) unusual gender
differences in mathematics: Iceland was the only country in which the mathematics gender gap favored girls. When data were
broken down and analyzed, the Icelandic gender gap appeared statistically significant only in the rural areas of Iceland,
suggesting a question about differences in rural and urban educational communities. In the 2007 qualitative research study
reported in this paper, the authors interviewed 19 students from rural and urban Iceland who participated in PISA 2003 in
order to investigate these differences and to identify factors that contributed to gender differences in mathematics learning.
Students were asked to talk about their mathematical experiences, their thoughts about the PISA results, and their ideas about
the reasons behind the PISA 2003 results. The data were transcribed, coded, and analyzed using techniques from analytic induction
in order to build themes and to present both male and female student perspectives on the Icelandic anomaly. Strikingly, youth
in the interviews focused on social and societal factors concerning education in general rather then on their mathematics
education.

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Article

This paper highlights the gender factor in the Trends in Mathematics and Science Study in 2003 (TIMSS-2003) of eight participating
countries from the Asia-Pacific region: Chinese Taipei, Hong Kong-SAR, Indonesia, Japan, Republic of Korea, Malaysia, Singapore,
and the Philippines (Chinese Taipei and Hong Kong-SAR will be referred to as countries in this paper.) in mathematics. The
focus is on gender-related data encompassing the overall performance of students from the participating countries and their
performance in the content and cognitive domains. At grade 8 level, the gender difference in the overall performance of students
mirrored the international average and favored girls in all countries of the Asia-Pacific region (except in Japan and the
Republic of Korea). The regional data also showed that, in general, the Philippines and Singapore can be considered to be
at one extreme with gender differences favoring girls in both content and cognitive domains, and the Republic of Korea and
to some extent Japan are at the other extreme with the gender differences favoring boys. At grade 4 level, girls from the
Philippines and Singapore also performed better than the boys whereas the boys from Japan and Chinese Taipei did slightly
better than the girls.

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Article

Article

This article deals with the activity of example generation as a special case of problem solving. We asked university students in the scientific-technological area to produce examples (which may exist or not) of mathematical objects fulfilling given requirements. For the analysis of students’ performances, we have developed a model that attempts to grasp the nature of the different stages in the solving process. The discussion of the findings allows us to outline specificities and educational potentialities of example generation activities.

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Article

The question of the order of theorems in geometry teaching is very important and it was one of the central issues in the early 20th Century in England. Employing ideas from the methodological framework proposed by Schubring (1987), the order of theorems in the geometry textbooks written by Godfrey and Siddons is analysed within their pedagogy and social context. The main foci for this analysis are Elementary Geometry (1903) and A Shorter Geometry (1912), which were widely used in secondary schools at that time. The theorems in these textbooks were arranged differently from those of Euclid's Elements. Godfrey claimed the order was organised from an general educational point of view. In A Shorter Geometry, flexibility concerning the order of theorems was recognised as a revision from Elementary Geometry. The analysis presented in this paper provides us with information about teaching practice at that time, for example that teachers might still be bound by examinations after 1903, and helps us to understand important aspects of dealing on the order of theorems in geometry teaching.

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Article

In this article, we present results of an empirical study with 500 German students of grades 7 and 8. The study focussed on
students' mathematics achievement and their interest in mathematics as well as on the relation between these two constructs.
In particular, the results show that the development of an individual student's achievement between grade 7 and grade 8 depends
on the achievement level of the specific classroom and therefore on the specific mathematics instruction Interest in mathematics
could be regarded a predictor for mathematics achievement Moreover, our findings suggest that the students show hardly any
fear of mathematics independent of their achievement level.
In diesem Beitrag wird über eine empirische Studie mit über 500 Schülerinnen und Schülern der Jahrgangsstufen 7 und 8 berichtet.
Im Fokus stehen dabei die fachlichen Leistungen und das fachspezifische Interesse der Jugendlichen sowie der Zusammenhang
zwischen diesen beiden Konstrukten Die Ergebnisse zeigen unter anderem dass die individuelle Leistungsentwicklung von Jahrgansstufe
7 zu 8 abhängig von der Klassenebene und damit vom Unterricht ist. Interesse und Leistung korrelieren erwartungsgemäß. Zudem
ist auffällig, dass die Schülerinnen und Schüler unabhängig von ihrer individuellen Leistung kaum Angst vor dem Fach Mathematik
haben.
ZDM-ClassificationC23-D53-E53

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Article

We start introducing some aspects of the theoretical framework: the Anthropological Theory of Didactics (ATD). Then, we consider
on the research domain commonly known as “modelling and applications” and briefly describe its evolution using the ATD as
an analytical tool. We propose a reformulation of the modelling processes from the point of view of the ATD, which is useful
to identify new educational phenomena and to propose and tackle new research problems. Finally, we focus on the problem of
the connection of school mathematics. The reformulation of the modelling processes emerges as a didactic tool to tackle this
research problem. We work on the problem of the articulation of the study of functional relationships in Secondary Education
and present a teaching proposal designed to reduce the disconnection in the study of functional relationships in Spanish Secondary
Education.
ZDM-ClassificationD20-D30-F80-I24-M14

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Article

This paper aims to examine key characteristics of exemplary mathematics instruction in Japanese classrooms. The selected findings
of large-scale international studies of classroom practices in mathematics are reviewed for discussing the uniqueness of how
Japanese teachers structure and deliver their lessons and what Japanese teachers value in their instruction from a teacher’s
perspective. Then an analysis of post-lesson video-stimulated interviews with 60 students in three “well-taught” eighth-grade
mathematics classrooms in Tokyo is reported to explore the learners’ views on what constitutes a “good” mathematics lesson.
The co-constructed nature of quality mathematics instruction that focus on the role of students’ thinking in the classroom
is discussed by recasting the characteristics of how lessons are structured and delivered and what experienced teachers tend
to value in their instruction from the learner’s perspective. Valuing students’ thinking as necessary elements to be incorporated
into the development of a lesson is the key to the approach taken by Japanese teachers to develop and maintain quality mathematics
instruction.

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Article

Problem solving has been a main focus in mathematics education for several decades, yet it seems that its definition and classroom
implementation are far from being consensual. We explore the views and approaches of a small community: the project leaders
of five elementary mathematics curriculum development projects in Israel, working within a centralized system, which dictates
the syllabus. We describe and analyze their views along six categories: What are problems? What are not problems? Classification
of problems, problem solving and individual differences, the ratio of problem solving tasks to other tasks in the project,
and the role of heuristics and metacognition in teaching problem solving. We describe, exemplify, interpret and discuss the
(few) points of convergence and the many different approaches. Finally, we reflect on the possible role of research in settling
those differences. We speculate that our analysis and results go beyond the local and the idiosyncratic.

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Article

The reconstruction of pupils modelling processes can be found in many empirical studies within the literature on modelling.
The empirical differentiations of the phases, which includes putting statements and actions of the pupils in the right phase,
has not been reconstructed from a cognitive psychological point of view on a micro level thus far: In this article different
modelling cycles are discussed with attention to distinctions in the various phases. The «modelling cycle under cognitive
psychological aspects» is specifically emphasized in contrast to the other cycles. On the basis of the results of the COM2-project (Cognitive psychological analysis of modelling processes in mathematics lessons, Borromeo Ferri) the phases of the
modelling processare described empirically. Some difficulties in the process of distinguishing the various phases are also pointed out.
Die Rekonstruktion von Modellierungsprozessen bei Lernenden ist in vielen empirischen Studien innerhalb der Literatur zum
Modellieren zu finden. Die empirische Unterscheidung der Phasen, was die Einordnung von Aussagen und Handlungen der Lernenden
miteinbezieht, wurde bisher noch nicht aus kognitionspsychologischer Sicht auf einer Mikroebene rekonstruiert. In diesem Artikel
wird nach einer Übersicht ausgewählter Modellierungskreisläufe der «Modellierungskreislauf unter kognitionspsychologischen
Aspekten im Vergleich hervorgehoben. Somit liegt der Fokus auf einer kognitionspsychologischen Perspektive hinsichtlich des
Modellierungskreislaufes. Auf der Basis von Ergebnissen des KOM2-Projekt (Kognitionspsychologische Analysen von Modellierungsprozessen im Mathematikunterricht, Borromeo Ferri) werden die
Phasen empirisch beschrieben und auch Schwierigkeiten bei der Einordnung in diese Phasen verdeutlicht.
ZDM-ClassificationC30-D10

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Article

This paper is an attempt to paint a picture of problem solving in Chinese mathematics education, where problem solving has
been viewed both as an instructional goal and as an instructional approach. In discussing problem-solving research from four
perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based.
We also describe several problem-solving activities in the Chinese classroom, including “one problem multiple solutions,”
“multiple problems one solution,” and “one problem multiple changes.” Unfortunately, there are no empirical investigations
that document the actual effectiveness and reasons for the effectiveness of those problem-solving activities. Nevertheless,
these problem-solving activities should be useful references for helping students make sense of mathematics.

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Article

We present a research work about an innovative national teacher training program in France: the Pairform@nce program, designed
to sustain ICT integration. We study here training for secondary school teachers, whose objective is to foster the development
of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments.
We adopt the theoretical background of the documentational approach to didactics. We focus on the interactions between teachers and resources: teachers’ professional knowledge influences these interactions, which at the same time yield knowledge evolutions, a twofold
process that we conceptualise as a documentational genesis. We followed in particular the work of a team of trainees; drawing on the data collected, we analyse their professional development,
related with the training. We observe intertwined evolutions and stabilities, consistent with ongoing geneses.
KeywordsCommunity of practice–Documentational approach–Dynamic geometry environment (DGE)–Inquiry-based learning and teaching–Mathematics teacher education–Training path

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Article

In this contribution a method for the treatment of analytic geometrical problems is introduced which integrates three-dimensional
computer graphics and computer algebra. At this a new computer graphics tool is used which has been developed for the visualization
of the corresponding spatial configurations and for the graphical solution of spatial analytic problems. (The virtual sphere
device is used for flexible and individual visualization). The tool is suitable both for the demonstration by the teacher
and for the interactive work of the students. The computer algebraic treatment is motivated by giving an explanation for the
computer graphical solution. Firstly the general algebraic problem is solved and graphically illustrated, after this the general
solution is numerically specified. The procedure of analytic solution can be illustrated also computer-graphically.
In diesem Beitrag wird eine Methode für das Behandlung dreidimensionaler analytisch-geometrischer Aufgaben vorgestellt, welche
dreidimensionale Computergrafik und Computeralgebra integriert. Dabei wird ein neues Grafikwerkzeug verwendet, das für die
Visualisierung und die grafische Lösung solcher Aufgaben entwickelt worden ist. (In diesem Werkzeug dient das Virtual-Sphere-Device
der flexiblen und individuellen Visualisierung.) Dieses Werkzeug eignet sich sowohl für die Demonstration durch den Lehrer
als auch für das interaktive Arbeiten der Schüler. Die computeralgebraische Behandlung der Aufgabe is motiviert durch die
Frage, was unter anderem berechnet werden muss, um die grafische Lösung auf der Oberfläche des Grafiksystems zu erhalten.
Deshalb wird zuerst die betreffende allgemeine Aufgabe gelöst, um durch Eingabe konkreter Daten die Lösung einer speziellen
Aufgabe zu erhalten Der analytische Lösungsweg lässt sich auch computergrafisch veranschaulichen.

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Article

This article deals with the interpretation of motion Cartesian graphs by Grade 8 students. Drawing on a sociocultural theoretical
framework, it pays attention to the discursive and semiotic process through which the students attempt to make sense of graphs.
The students’ interpretative processes are investigated through the theoretical construct of knowledge objectification and
the configuration of mathematical signs, gestures, and words they resort to in order to achieve higher levels of conceptualization.
Fine-grained video and discourse analyses offer an overview of the manner in which the students’ interpretations evolve into
more condensed versions through the effect of what is called in the article “semiotic contractions” and “iconic orchestrations.”

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Article

One important factor in the failure to learn arithmetic in the normal way is an endogenous core deficit in the sense of number.
This has been associated with low numeracy in general (e.g. Halberda et al. in Nature 455:665–668, 2008) and with dyscalculia more specifically (e.g. Landerl et al. in Cognition 93:99–125, 2004). Here, we describe straightforward ways of identifying this deficit, and offer some new ways of strengthening the sense
of number using learning technologies.

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Article

In the 1990s, handheld technology allowed overcoming infrastructural limitations that had hindered until then the integration
of ICT in mathematics education. In this paper, we reflect on this integration of handheld technology from a personal perspective,
as well as on the lessons to be learnt from it. The main lesson in our opinion concerns the growing awareness that students’
mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on instrumentation
and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion,
extrapolation of these lessons to a future with mobile multi-functional handheld technology leads to the issues of connectivity
and in- and out-of-school collaborative work as major issues for future research.
KeywordsMathematics education–Handheld technology–Instrumentalisation–Instrumentation–Orchestration

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Article

This is the first in a series of two papers whose goal is to contribute to the debate on a pair of questions: (1) What is
the mathematics that we should teach in school? (2) How should we teach it? This paper addresses the first question, and the
second paper, to appear in the next issue of ZDM, addresses the second question. The two questions are addressed from a particular
theoretical framework, called DNR-based instruction in mathematics. The discussions in the current paper are instantiated
mainly in proof-related contexts. The paper offers a definition of mathematics as a union of two categories of knowledge:
ways of understanding and ways of thinking. The latter are generalizations of the notions, proof and proof scheme, respectively.
The paper also discusses cognitive-epistemological and curricular implications of this definition, focusing mainly on the
inevitable production of narrow or faulty mathematical knowledge and the asymmetry in educators’ attention to ways of understanding
and ways of thinking.

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Article

In this report we present the results of a teaching study introducing the concept “power function” using a graphing calculator.
The focus of our attention is on the development of the understanding of 15–16 year-old mathematics students. In the centre
of our interest is their learning through graphs of power functions by discovering the properties of graphs. Our report presents
the mathematical and social constructivist background together with a new deliberately constructivist approach beginning the
teaching experiment with an open question. The students' cognitive and intuitive strategies and their attitudes towards computer
algebra are described.
In diesem Beitrag präsentieren wir die Ergebnisse einer Unterrichtsstudie zum Begriff der Potenzfunktionen mittels Präsentation
der “Potenzblume” beim Gebrauch eines graphikfähigen Taschenrechners. Hauptaugenmerk liegt auf der Entwicklung des Verständnisses
der 15–16-jährigen Schüler durch das Entdecken der Eigenschaften von Graphen bei Potenzfunktionen. Die Studie beruht auf einer
offenen Unterrichtsgestaltung, deren theoretischer Hintergrund der Soziale Konstruktivismus bildet. Die kognitiven und intuitiven
Strategien der Schüler zusammen mit ihrer Haltung zur Computeralgebra werden präsentiert.

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Article

Traditional mathematics assessments often fail to identify students who can powerfully and effectively apply mathematics to
real-world problems, and many students who excel on traditional assessments often struggle to implement their mathematical
knowledge in real-world settings (Lesh & Sriraman, 2005a). This study employs multi-tier design-based research methodologies
to explore this phenomenon from a models and modeling perspective. At the researcher level, a Model Eliciting Activity MEA)
was developed as a means to measure student performance on a complex real-world task. Student performance data on this activity
and on traditional pre- and post-tests were collected from approximately 200 students enrolled in a second semester calculus
course in the Science and Engineering department of the University of Southern Denmark during the winter of 2005. The researchers
then used the student solutions to the MEA to develop tools for capturing and assessing the strengths and weaknesses of the
mathematical models present in these solutions. Performance on the MEA, pre- and post-test were then analyzed both quantitatively
and qualitatively to identify trends in the subgroups corresponding to those described by lesh and Sriraman.
ZDM-ClassificationM15-D65-C80

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Article

The role of computers in elementary school math classrooms is still being determined. Although computers are promised effective visual tools to promote independent work and study; many educators neglect to use them. Since there are varying points of view, individual teachers generally decide whether to incorporate computers into their methods.Purpose: My experiment analyzes and quantifies the value of computers in elementary school math classrooms.Method: Over a course of 11 weeks, my first grade class worked with the teaching software “Mathematikus 1” (Lorenz, 2000). Using both interpersonal and video observation, I completed written evaluations of each pair of my students' will and ability to cooperate, communicate and independently solve mathematical problems.Conclusion: My results show that it is generally beneficial to use computers in elementary school math lessons. However, some elements of said software leave room for improvement.

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Article

In this paper, details of student difficulties in understanding the concept of acceleration and the mathematical and physical/intuitive
sources of these are delineated by utilizing the teaching experiment methodology. As a result of the study, two anchoring
analogies are proposed that can be used as a diagnostic tool for students’ alternative conceptions. These can be used in teaching
to highlight the peculiarity of acceleration concept. This study portrays how seeing acceleration as ‘rate of change’ of a
quantity (velocity) and recognizing the consequences of such a definition are hindered in certain ways which in turn negatively
affect learning the concept of force. This is also an example that illustrates that a rather “simple” mathematical concept
(i.e., rate of change) for the expert can become a complex phenomenon when embedded in a physical concept (i.e., acceleration)
which is consistently found to be as a misconception among learners at various levels that is widely occurring and very resistant
to change.
KeywordsRate of change-Acceleration-Force and motion-Misconception-Conceptual change-Analogy-Teaching experiment

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Article

Mathematics plays a dominant role in today's world. Although not everyone will become a mathematical expert, from an educational
point of view, it is key for everyone to acquire a certain level of mathematical literacy, which allows reflecting and assessing
mathematical processes important in every day live. Therefore the goal has to be to open perspectives and experiences beyond
a mechanical and tight appearance of the subject. In this article a framework for the integration of reflection and assessment
in the teaching practice is developed. An illustration through concrete examples is given.
Mathematik ist in unserer Welt auf vielfältige Weise präsent. Was an dieser Mathematik müssen allgemeingebildete Lai/inn/en
verstehen, um mathematisch mundig zu sein. Aus der Bildungsperspektive mathematischer Mündigkeit werden Reflektieren und Beurteilen
von Mathematik als wichtige Tätigkeiten im Unterricht begründet Es wird ein Rahmenkonzept für Reflexion im Unterricht ausgearbeitet
und an Beispielen für den Unterricht konkretisiert. Mit dem Ansatz ist die Hoffnung verbunden, das oft starre und mechanistische
Bild von Mathematik schrittweise in Richtung eines diskursiven Mathematikbildes wandeln zu können
ZDM-ClassificationD20-D30-C20-C30-C60

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Article

The focus of this study was to investigate primary school students’ achievement in the domain of measurement. We analyzed
a large-scale data set (N=6,638) from German third and fourth graders (8- to 10-year-olds). These data were collected in 2007 within the framework
of the ESMaG (Evaluation of the Standards in Mathematics in Primary School) project carried out by the Institute for Educational
Quality Improvement (IQB) at Humboldt University, Berlin, Germany. The data were interpreted using a classification scheme
based on a conceptual–procedural distinction in measurement competence. The analyses with this classification revealed that
grade, gender, and in particular figural reasoning ability are significantly related to overall measurement competence as
well as on the sub-competencies of Instrumental knowledge and Measurement sense. The paper concludes with a discussion of
the implications of the findings of this study for teaching and assessing measurement.
KeywordsMathematical competence–Measurement–Gender–Grade–Figural reasoning ability

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Article

The relationship between practised monitoring activities and performance, especially in mathematics was examined within three
nested studies. The first study deals with problems of faulty term rewritings submitted to three groups of subjects—10th to
13th graders, differing in their mathematical performance—whose task was to find the mistakes. Moreover, a questionnaire on
the practice and appreciation of monitoring activities was developed. The third study, first, repeats the first study with
a similar population and secondly adds interviews with some of the subjects while solving additional items concerning faulty
term rewritings. Studies 1 and 3 show similar success in finding mistakes and in the replies to the questionnaire within the
various groups. Furthermore, the third study points up that the subject’s answers do neither predict the practised monitoring
nor the success in the test. However, the success correlates significantly with the practised monitoring. For a deeper understanding
concerning the role of metacognition in explaining performance, the second study examined two of the groups who had already
been involved in the first study. These were assigned some problems of a matrices test as used in cognitive psychology. While
trying to solve the problem, their eye movements were recorded by means of an eye-tracker. Afterwards they had to justify
their solutions in an interview. The eye movements were analysed, the verbal comments classified. Again, the groups differ
in their problem solving success, dependant on the quality of the monitoring practised. Altogether, the results of the three
studies elucidate the importance of practised metacognitive monitoring activities not only for success in school algebra,
but furthermore the ability and the willingness to do it is deeper anchored in a person than just a trained behaviour for
school algebra.
KeywordsMetacognition–School algebra–Achievement–Figural matrices tests–Eye-tracker study

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Article

Several models have been developed in order to categorize the numerous expressions that people use in order to describe their
emotional experiences. The focus of the present study is on one of these theoretical classifications proposed by Pekrun (1992)
specifically concerning emotions which are directly related to learning and achievement in mathematics. In his model, emotions
are classified according to their valence (positive vs. negative) and their level of activation. In testing the assumptions
of this model, we investigated students' enjoyment, anxiety, anger and boredom experienced before, during, and after the completion
of a math test. Correspondence analyses which were used to generate a graphical illustration of structural interrelationships
between these emotions provide empirical support for the theoretical classification.
Vielfältige Modelle wurden bislang entwickelt, um die zahlreichen Ausdrücke zu kategorisieren, die zur Beschreibung emotionalen
Erlebens verwendet werden. Der Fokus dieses Beitrags liegt auf einer theoretischen Klassifizierungen von Pekrun (1992), die
insbesondere auf Emotionen im mathematischen Lern- und Leistungskontext bezogen sind. Demnach werden Emotionen entsprechend
ihrer Valenz (positiv vs. negativ) und ihrer Art der Aktivierung klassifiziert. Beim Testen der Modellannahmen untersuchten
wir die von Schülerinnen und Schülern selbstberichteten Emotionen Freude, Angst, Ärger und Langeweile vor, während und nach
der Bearbeitung eines mathematischen Leistungstests. Zur Auswertung wird die Korrespondenzanalyse verwendet, um eine graphische
Abbildung der strukturellen Verbindungen zwischen emotionen und dem Leistungsniveau in dem mathematischen Test zu geneieren.
Dabei soll insbesondere ein empirischer Beleg für die theoretische Klassifikation des emotionalen erlebens während der Bearbeitung
eines mathematischen Leistungstests erbracht werden.
ZDM-ClassifikationC20-C40-D60

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Article

In November 2004 Germany's largest federal state North-Rhine-Westphalia for the first time carried out central tests in the
subjects German, English and Mathematics in grade 9 with about 210.000 students participating. One of the main goals in assessing
students' performance was to improve of teaching. This imposed certain requirements on the construction of tasks and on the
feedback of results. In this article we present —referring to the specific experience from the development of the mathematics
test—concepts and requirements for comparative assessment that is intended to support desirable changes in teaching practice.
Im November 2004 wurden in Deutschlands bevölkerungsreichstem Bundesland, Nordrhein-Westfalen, erstmals Lernstandserhebungen
in der Jahrgangstufe 9 in den Fächem Deutsch, Englisch und Mathematik durchgeführt. Bei diesem Projekt bearbeiteten 210.000
Schülerinnen und Schüler zeitgleich zentral gestellte Aufgaben. Neben vielen weiteren administrativ formulierten Zielen stand
dabei vor allem die Unterrichtsentwicklung im Vordergrund. Dieses Ziel stellt besondere Anforderungen an die verwendeten Aufgaben
und die Rückmeldung von Ergebnissen an die Schulen. In unserem Beitrag werden wir, ausgehend von der Entwicklungsarbeit und
den Erfahrungen mit den Lernstandserhebungen sowie deren Analyse, konzeptionelle Überlegungen für und konkrete Anforderungen
an vergleichende Leistungsmessung formulieren, die eine fachdidaktisch wünschenswerte Unterrichtsentwicklung unterstützen
soll.
ZDM-ClassificationB13-C73-D63-D73

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Article

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural
versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored
between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural
knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural
versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs
about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs
about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination
performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.
KeywordsEpistemic beliefs–Knowledge representations–Statistics knowledge

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Article

This article compares the opportunity to learn length measurement in the USA and Singapore as revealed in the close analysis
of some of their written elementary curriculum materials. Written curricula strongly influence students’ learning of mathematics,
without completely determining it. The Trends in Third International Mathematics and Science study 2007 showed the relatively
low performance of the US and Singapore fourth graders in measurement, which was attributed in part to the learning opportunities
provided to the students. We examined and coded all instances of length measurement in three different US curricula and one
Singapore curriculum through Grade 3, using a very detailed scheme that identified particular elements of conceptual, procedural
and conventional knowledge and the textual forms that present this knowledge. Results show strong emphasis on measurement
procedures, across all grades and curricula, in both countries. However, in numerous ways, the Singapore curriculum is more
focused, organizationally, procedurally and conceptually. US curricula provide more diverse access to conceptual knowledge
where Singapore materials focus on independent work involving procedures, within and across grade levels. Limitations of the
curricula in both countries are discussed.
KeywordsLength–Opportunity to learn–Textbook comparison–Spatial measurement–Curriculum analysis–US curricula–Singapore curriculum

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Article

This paper starts from some observations about Presmeg’s paper ‘Mathematics education research embracing arts and sciences’
also published in this issue. The main topics discussed here are disciplinary boundaries, method and, briefly, certainty and
trust. Specific interdisciplinary examples of work come from the history of mathematics (Diophantus’s Arithmetica), from linguistics (hedging, in relation to Toulmin’s argumentation scheme and Peirce’s notion of abduction) and from contemporary
poetry and poetics.

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Article

Cross-national research studies such as the Program for International Student Assessment and the Third International Mathematics
and Science Study (TIMSS) have contributed much to our understandings regarding country differences in student achievement
in mathematics, especially at the primary (elementary) and lower secondary (middle school) levels. TIMSS, especially, has
demonstrated the central role that the concept of opportunity to learn plays in understanding cross-national differences in
achievement Schmidt et al., (Why schools matter: A cross-national comparison of curriculum and learning
2001). The curricular expectations of a nation and the actual content exposure that is delivered to students by teachers were
found to be among the most salient features of schooling related to academic performance. The other feature that emerges in
these studies is the importance of the teacher. The professional competence of the teacher which includes substantive knowledge
regarding formal mathematics, mathematics pedagogy and general pedagogy is suggested as being significant—not just in understanding
cross-national differences but also in other studies as well (Hill et al. in Am Educ Res J 42(2):371–406, 2005). Mathematics Teaching in the 21st Century (MT21) is a small, six-country study that collected data on future lower secondary teachers in their last year of preparation.
One of the findings noted in the first report of that study was that the opportunities future teachers experienced as part
of their formal education varied across the six countries (Schmidt et al. in The preparation gap: Teacher education for middle school mathematics in six countries, 2007). This variation in opportunity to learn (OTL) existed in course work related to formal mathematics, mathematics pedagogy
and general pedagogy. It appears from these initial results that OTL not only is important in understanding K-12 student learning
but it is also likely important in understanding the knowledge base of the teachers who teach them which then has the potential
to influence student learning as well. This study using the same MT21 data examines in greater detail the configuration of
the educational opportunities future teachers had during their teacher education in some 34 institutions across the six countries.

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Article

Acting and thinking are strongly interconnected activities. This paper proposes an approach to mathematical concepts from
the angle of hands-on acting. In the process of learning, special emphasis is put on the reflection of the own actions, enabling
learners to act consciously. An illustration is presented in the area number representation and extensions of number fields.
Using didactical materials, processes of mathematical acting are stimulated and reflected. Mathematical concepts are jointly
developed with the learners, trying to address shortcomings from own experiences. This is accompanied by reflection processes
that make conscious to learners the rationale of mathematical approaches and the creation of mathematical concepts. Teaching
mathematics following this approach does intent to contribute to the development of decision-making and responsibility capabilities
of learners.
ZDM-ClassificationB23-C33-D33-D83

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Article

Circa1895, James M. Baldwin introduced a powerful view regarding Darwinian Evolution. Baldwin suggested that behavioral flexibility
could play a role in amplifying natural selection because this ability enables individuals to modify the environment of natural
selection affecting the fate of future generations. In this view, behavior can affect evolution but, and this is crucial,
without claiming that responses to environmental demands acquired during one’s lifetime could be passed directly to one’s
offspring. In the present paper, we want to use this view as a guiding metaphor to cast light on understanding how students and teachers can utilize the environment of digital technologies to scaffold
their activities. We present examples of activities from geometry and algebra in high school settings that illustrate the
potential role that certain technologies can have in transforming classroom interaction and work.

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Article

Based on empirical data from a study of pre-service teachers engaged in non-routine mathematics problem solving, a five-phase
model is proposed to describe the range of cognitive and metacognitive approaches used. The five phases are engagement, transformation-formulation,
implementation, evaluation and internalization, with each phase being described in terms of sub-categories. The model caters
for a variety of pathways that can be adopted during any problem-solving process by recognizing that the path between these
five phases is neither linear nor unidirectional.
KeywordsMathematical problem solving–Metacognition–Cognition–Reflection–Pre-service teachers

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