bluuebunny/crossref_metadata_2025 huggingface dataset:
Education of lower secondary mathematics teachers in Denmark and France
Authors: Winsløw Carl , Durand-Guerrier Viviane
Published: November 2024

This paper presents a comparative study of two surprisingly different systems of preparing teachers for lower secondary level teaching of mathematics, namely those of Denmark and France. We first describe these systems differences succinctly. The main part of the paper reports on a qualitative study of how final year teacher students in the two countries handle two hypothetical situations of mathematics teaching. Then we discuss how the findings could be related to the systems of formation described first.
Mathematics education: a key for success in a globalised world?
Authors: Editors The
Published: November 2024

Ledare /Editorial
An agenda for mathematics education in the decade of education for sustainable development
Authors: Yasukawa Keiko
Published: November 2024

The United Nations declared 2005 to 2014 as the Decade of education for sustainable development. This presents an opportune moment for mathematics educators and mathematics education researchers to reflect about the effectiveness that mathematics education has had in creating citizens for a sustainable future. There is an important distinction between education about sustainable development, and education for sustainable development; the latter is the more important, but also the more difficult and challenging. The paper examines some of the obstacles that mathematics educators face in educating for sustainable development, and identifies the need for some radical alternatives. These alternatives will need to challenge the dominant discourses that shape identities of both learners and teachers.
Posing problems using Cabri
Authors: Engström Lil , Lingefjärd Thomas
Published: November 2024

The purpose of Engström’s research was to investigate in which ways and to what extent three different teachers, one in Switzerland and two in Sweden, used a specific dynamical geometry software, Cabri Géomètre, in upper secondary school.
The method consisted mainly of field notes and audio recording during observations in classrooms. The field research was then analysed and evaluated according to the following questions: a) How do teachers pose problems and b) how do teachers make students’ experiences useful when Cabri Géomètre is accessible? The result showed that one important factor enabling to challenge the pupils in learning mathematics when using Cabri Géomètre is the way the teacher poses the problems or the questions. The word ’challenging’ means among other things, that the pupils continue asking themselves questions so that there will be continuous learning, not limited to finding just one correct answer.
Review of empirical studies in HPM2004 & ESU4: Empirical research in the field of using history in mathematics education
Authors: Jankvist Uffe Thomas
Published: November 2024

This paper discusses empirical studies in the proceedings HPM2004 & ESU4. More precisely the paper deals with four of the more clear-cut empirical studies. These are the contributions by B. Smestad, C. Tzanakis & M. Kourkoulos, W.-S. Horng, and Y.-W. Su. These contributions are first presented and then later discussed in the context of whether their purpose of involving the history of mathematics in mathematics education is to promote the learning of mathematics or if it is to bring about aspects of mathematics which are not normally part of the teaching and learning agenda, e.g. cultural or social aspects of mathematics and its history – ’history as a tool’ or ’history as a goal’. The papers and their purposes for involving history are then related to a Nordic case, namely the official regulations for the Danish upper secondary mathematics programme for involving the history of mathematics. In the end the need for empirical research studies in the field of using history in mathematics education is discussed as well as further perspectives for the community regarding such studies or the lack of them.
Matematikdidaktik – nordiskt samarbete i historisk belysning
Authors: Johansson Bengt
Published: November 2024

I denna artikel ges en översiktlig beskrivning av hur nordiskt samarbete i ma­ tematikämnets didaktik växt fram, speciellt under de senaste 30 åren. Samarbetet har utvecklats vid matematiklärarmöten och i samband med det omfattande in­ ternationella reformarbete som startade kring 1960. Tidskrifter om matematik och matematikundervisning har haft en särkilt viktig roll i denna utveckling. Efter en nedgång under 1970­talet och början av 80­talet har det nordiska samarbetet åter börjat växa i styrka. Senaste exemplen på detta är denna tidskrift och ett allt inten­ sivare samarbete mellan forskare och lärarutbildare i matematikämnets didaktik.
Det var ikke meningen! - om kommunikation i matematikundervisningen
Authors: Alrø Helle , Skovsmose Ole
Published: November 2024

Artiklen beskriver og analyserer et eksempel på matematikundervisning i folkesko- len. Forfatterne beskæftiger sig specielt med spørgsmålet: Hvad sker der med kom- munikationen, når formålet med undervisningen er skjult for eleverne? Analysen relateres til en fortolkning af læring som handling, idet begreber som "elevintentio- ner" og "undergrundsintentioner"præsenteres og diskuteres.
Matematiken i den finska grundskolan. Attityder och kunskaper
Authors: Kupari Pekka
Published: November 2024

En omfattande utvärdering av den finska grundskolan, den s k "Evaluering 90", företogs år 1990 och 1991. I matematik undersöktes grundskolelevernas kunska- per i och attityder till matematik i årskurserna 4, 6 och 9. Ett jämförelsematerial från en motsvarande studie 1979 användes för undersökning av i vilken mån kunska- perna och attityderna hade förändrats fram till den senare utvärderingen. På lågsta- diet (åk 1-6 i Finland) var elevernas matematikattityder mycket positiva. Matematik var ett av de mest omtyckta ämnena i skolan. Emellertid sjönk ämnets popularitet upp genom årskurserna och matematiken upplevdes också som ett allt svårare ämne. Både på låg- och högstadiet tyckte eleverna att matematiken och studierna i ämnet var viktiga med tanke på de framtida möjligheterna till framgång i arbetslivet. Med avseende på kunskaperna visade undersökningsresultaten att nivån i mate- matik delvis hade förbättrats, delvis var oförändrad i jämförelse med situationen 1979. Totalt sett var nivån i årskurs 4 betydligt högre 1990, men i de övriga årskur- serna var nivån densamma som tidigare. Den fruktade nivåsänkningen på de mål- och innehållsområden inom högstadiet som var gemensamma för de båda utvärderingarna hade inte inträffat, trots att nivågrupperingssystemet hade slopats och timantalet i matematik hade minskats med en veckotimme före den andra ut- värderingen. Mycket positivt var det också att eleverna i årskurs 9 förbättrade sina prestationer just på det område som man satsat på under 80-talet, nämligen på området för tillämpning och praktiskt nyttiggörande av matematiken.
Scrutinizing teacher-learner interactions on volume
Authors: Tyskerud Anita , Mosvold Reidar
Published: November 2024

This study adds to research on volume and spatial reasoning by investigating teacher-learner interactions in the context of Lesson study. Our analysis illustrates how the mathematical object of volume is realized, and what metarules of discourse that can be observed over two iterations of a research lesson. The study unpacks the mathematical work of teaching volume in terms of discourse, and shows how an undesirable and unexpected result from the first research lesson can be attributed to the communicational work of teaching rather than to lack of skills among students.
Language diversity in mathematics education in the Nordic countries 2008–2018
Authors: Meaney Tamsin , Eskeland Rangnes Toril
Published: November 2024

In 2008, the Nordic Studies in Mathematics Education (Nomad) journal published a thematic issue on Bringing focus to mathematics education in multicultural and multilingual settings. It had four articles with contributors from Australia, Sweden, Denmark and New Zealand ...
Newly- and early-immigrated second-language students’ knowledge of arithmetic syntax
Authors: Petersson Jöran
Published: November 2024

The present study investigated how 259 Swedish, grade 9 students, of whom 90 had an immigrant background, achieved on twelve written test items in the content area of number. Four of the twelve test items required good knowledge of arithmetic syntax, such as when it was appropriate to apply order-of-operation rules and the associative and distributive laws of arithmetic operations. On these four test items, the most-recently arrived students showed on average significantly more knowledge than the students who had immigrated when they were younger and had participated in Swedish schools for longer periods of time. The outcome suggests that these two groups of immigrant students in later school years should be considered as separate sub-categories of second-language students when it comes to teaching, assessment and research.
Investigating the fit of a model for students’ understanding of fractions in a Norwegian context
Authors: Gustavsen Trond Stølen , Imenes Olav Gravir
Published: November 2024

To capture the complexity of students’ understanding of fractions, a model linking part-whole to the subconstructs ratio, operator, quotient and measure has been proposed. We ask if this model is compatible with students’ achievements in a Norwegian context. Responses from 638 students were analysed using structural equation modelling (SEM), and a good fit of the model was obtained after removing the ratio subconstruct. In particular, part-whole is seen to be important for operator, quotient and measure. Using qualitative analysis of interviews, we found reasoning associated with ratio, with a weak link to the part-whole subconstruct.
Authority in students’ peer collaboration in statistics: an empirical study based on inferentialism
Authors: Seidouvy Abdel , Helenius Ola , Schindler Maike
Published: November 2024

Students’ peer collaboration efforts in mathematics and statistics is a topic that has increasingly gained attention in research. In any collaboration, authority relations play a role for how meaning is constituted: Whenever things are discussed and decisions are made, authority is involved in a sense that some arguments or persons may be more convincing and powerful than others. In this article, we investigate how authority changes dynamically in type and in distribution as groups of fifth grade students collaborate in data generation processes. We identify and categorize authority using an epistemological framework, which is based on the philosophical theory of inferentialism. The results show that the three different types of authority described in inferentialism are all identifiable in students’ collaborative work. We also find and categorize further types of authority connected to the statistics group work, some of which are hardly addressed in previous research.
University students’ general and specific beliefs about infinity, division by zero and denseness of the number line
Authors: Juter Kristina
Published: November 2024

A study of university students’ beliefs about infinity and related concepts, e.g. division by zero and denseness of the number line, was conducted. The concepts were chosen for the students’ proven cognitive challenge in coping with them, and part of the study was to analyze individual beliefs of the different concepts in relation to each other. A questionnaire was designed to discover relationships between preservice teachers’ and technology students’ beliefs. Particular foci in the study were general and specific perspectives of the concepts and admission requirements for the programs. The results show incoherence with respect to general and specific representations of aspects concerning denseness of the number line, and also show that admission requirements are significant when it comes to validity of beliefs about division by zero.
Tilgange til tidlig algebra
Authors: Kaas Thomas
Published: November 2024

Undervisning af 6–12 årige i algebra og algebraisk tænkning har under betegnelsen ”tidlig algebra” gradvist etableret sig som forskningsområde og undervisningspraksis i et stigende antal lande. På basis af et hermeneutisk inspireret litteraturstudie karakteriserer denne artikel de indholdsmæssige tilgange til undervisning i tidlig algebra, som forskningslitteratur i perioden 1995–2017 giver. Analysen har resulteret i et rammeværk for tilgange til tidlig algebraundervisning, som præsenteres og diskuteres. Artiklen konkluderer bl.a., at undervisning i tidlig algebra typisk tager afsæt i elevers arbejde med tal, kvantiteter og/eller funktionelle sammenhænge, og at elevers algebraiske tænkning søges udviklet gennem aktiviteter, der enten har opdagelser af generelle egenskaber og egenskaber eller ræsonnementer vedrørende ukendte talstørrelser som fokus. Forskelle mellem de forskellige tilgange vedrører desuden den rolle, som repræsentationer og kontekster har i undervisningen.
Finnish students’ flexibility and its relation to speed and accuracy in equation solving
Authors: Hästö Peter , Palkki Riikka
Published: November 2024

A total of 266 Finnish students participated in a flexible equation solving test. By flexibility we understand the knowledge of multiple strategies and ability to choose the most mathematically appropriate strategy for a given task. Here we focus on the first aspect, namely knowledge of appropriate alternative, so-called innovative strategies. The test measured students’ capacity and inclination for producing innovative strategies. We consider the relationship between these measures and students’ speed and accuracy in solving equations. We find that students with high capacity for innovation have high speed and accuracy. On the other hand, some low capacity students had high speed or accuracy whereas others had low. Inclination for innovation is not related to speed or accuracy.
Materialisering av algebraiska uttryck i helklassdiskussioner med lärandemodeller som medierande redskap i årskurs 1 och 5
Authors: Eriksson Inger , Wettergren Sanna , Fred Jenny , Nordin Anna-Karin , Nyman Martin , Tambour Torbjörn
Published: November 2024

Syftet med denna artikel är att beskriva och diskutera vilka funktioner lärandemodeller kan ha för att främja yngre elevers kollektiva diskussioner om algebraiska uttryck. Artikeln bygger på data från ett designforskningsprojekt baserat på Davydovs principer för lärandeverksamhet, bestående av videofilmade forskningslektioner i årskurs 1 och 5. Analysen fokuserar på vad som skapar förutsättningar för helklassdiskussioner om algebraiska uttryck, hur de drivs framåt och kvalificeras samt vilka funktioner lärandemodeller kan ha för elevernas utforskande av matematiska strukturer och relationer i algebraiska uttryck. Resultatet indikerar att lärandemodeller som medierande redskap gör det möjligt för eleverna att föra kreativa och reflekterande diskussioner om algebraiska uttryck och deras komponenter.
Avtryck i problemlösnings- undervisning – en fenomenografisk studie
Authors: Sjölander Nordin Camilla
Published: November 2024

I denna artikel beskrivs hur nio lärare för åk 4–6 och 7–9 erfar sin problemlösnings-undervisning och kollegiala lärande två år efter att de deltagit i Matematiklyftet, som var en nationell fortbildningssatsning i Sverige under 2012–2016. Studien tar en fenomenografisk ansats, och visar en variation i hur lärare erfar sin problemlösnings-undervisning beroende på om de fortsatt att arbeta i en organiserad form av kollegialt lärande eller inte. Bilden studien ger är att det inte främst är lärarnas ämnesdidaktiska kunskaper, som påverkar hur de erfar problemlösningsundervisningen, utan att det istället är de organisatoriska faktorerna som spelar stor roll.
Meanings of decision-making in probability and statistics: comparing Chilean and Danish upper secondary school curricula
Authors: Elicer Raimundo
Published: November 2024

In this article I investigate the roles given to decision-making in probability and statistics upper secondary school curriculum, in a comparative study between Chile and Denmark. Drawing upon Fairclough’s model for Critical discourse analysis, I analyse selected official curricular texts as examples of broader discursive practices. In particular, I focus on the positioning of social actors and legitimation strategies. Present discourses position students as active decision makers, though in the Chilean case this is only evident in more recent texts. The teaching and learning of probability and statistics are legitimised through the appeal to political, professional and educational authorities, and through narratives of necessity for rational, grounded, evidence-based decisions. In this aspect Danish texts are more open to complex social and political decision processes. The analysis illustrates a common search for linking mathematics education to democratic involvement in social and political decision-making, but failing to specify the relevance of probability and statistics beyond the individual psychological scope.
Understanding the relationship between length and area when changing the size of a two-dimensional geometric figure
Authors: Svanteson Wester Jenny , Kullberg Angelika
Published: November 2024

When scaling up or down two-dimensional geometric figures, students tend to believe that if the lengths are doubled, the area is doubled as well. Although a lot of effort has been made to study and overcome this illusion of linearity, previous research reports that the illusion often remains after teaching. We add to this research by studying students’ experiences of the relationship between change in length and change in area when enlarging or reducing two-dimensional geometric figures, identified in a learning study aimed at finding powerful ways of teaching scale to 14-year-old students. The aim of this study is to contribute to a deeper understanding of students’ experiences of the relationship and how it can be taught. Teaching the change in length and the change in area simultaneously was found to be one key to students’ learning.
Swedish primary teacher education students’ perspectives on linear equations
Authors: Andrews Paul
Published: November 2024

Linear equations, connecting arithmetic to the symbolism of formal mathematics, represent a key topic of mathematics. However, the understanding primary teacher education students bring to their studies has been rarely examined. In this study, students were invited to explain in writing how an unannotated solution to x + 5 = 4x – 1 had been conceptualised by the hidden solver. Data, coded against an iteratively derived framework, showed that most students were familiar with linear equations, able to articulate an objective for equation solving and offer solution strategies, typically based on either doing the same to both sides, swapping the side swapping the sign or both.
Conservative and transformative changes in algebra in Swedish lower secondary textbooks 1995–2015
Authors: Palm Kaplan Kristina , Prytz Johan
Published: November 2024

The present study examines textbook algebra tasks in an attempt to understand how textbooks change in a reform of lower secondary school algebra. Changes in 1557 textbook tasks for year 8 are described in terms of algebraic activities and school algebra discourses. The tasks were taken from textbooks published before and after a new syllabus was introduced in Sweden in 2011. The results show that the new syllabus’ focus on mathematical competences was not stressed in the textbooks and that the greatest change was an increase in word problems connected to everyday situations. It is suggested that, in this reform, textbooks have been conservative and transformative in relation to the syllabus.
Fostering an intimate interplay between research and practice: Danish ”maths counsellors” for upper secondary school
Authors: Jankvist Uffe Thomas , Niss Mogens
Published: November 2024

The gap between research and practice is a well-known problem (and topic) in educational research, and not least in mathematics education research. This empirically based article discusses the effects of a particular attempt to foster a closer connection between research and practice by involving mathematics education research findings in the activities of selected Danish upper secondary school mathematics teachers, who (have) take(n) part in a research-based so-called ”maths counsellors” in-service teacher programme. A key aspect of the programme is to make the activation of research findings a mere necessity for significant aspects of the maths counsellors’ practice. To illustrate the teachers’ research-based work, the article presents six characteristic examples from the implementation of the programme, i.e. authentic examples of how prospective maths counsellors have identified students with mathematics specific learning difficulties, have diagnosed the nature of these difficulties, and how they have designed interventions to help the students overcome them. A discussion of how these activities draw upon and are grounded in mathematics education research findings serve as a basis for evaluating the ”model” behind this further education programme.
Formative assessment – from the view of special education teachers in mathematics
Authors: Andersson Catarina
Published: November 2024

The potential of using formative assessment is well demonstrated, but studies about the use of formative assessment from a special education perspective are lacking. This study adds to this gap by investigating the view of formative assessment in a group of 39 special education teachers in mathematics (SETMs) who had learned about formative assessment within the SETM-program 2–6 years earlier. Five respondent interviews were used to design a questionnaire answered by the rest of the group. The SETMs had perceived formative assessment beneficial and useful in all their common sub-responsibilities and reported experiences of benefits as well as challenges. The article discusses the importance of reaching an inclusive formative assessment practice in mathematics education.
Exercise design in mathematics textbooks: the case of subtraction
Authors: Norberg Malin
Published: November 2024

Mathematics textbooks are teaching tools used by most students studying mathematicsworldwide.Inthisdescriptivetextbookanalysis,allSwedishmathematics textbooks for year 1, both digital and printed, were mapped out according to the resources used for communication. For delimitation, a focus on subtraction as an arithmetic operation was chosen. The result shows large differences between the 17 textbook series, concerning both the type of subtraction exercises offered and the use of different resources for communication and learning, such as writing, images, and mathematical symbols. Digital textbooks were largely similar to the printed textbooks, except for one tablet-based textbook. Altogether, the study shows that the choice of mathematics textbooks affects how subtraction is presented to students and, by extension, the learning situations students encounter when working with mathematics textbooks.
Att utmana barns taluppfattning i en matematikaktivitet i förskolan
Authors: Johansson Susanne , Björklund Camilla , Kultti Anne
Published: November 2024

Syftet med denna studie är att visa vad som görs möjligt att erfara om tal i en delningsaktivitet med två treåriga barn i förskola. Den specifika frågeställningen är: Vilka aspekter av tal framträder i en delningsaktivitet mellan lärare och treåringar? I en variationsteoretisk analys av observationer där lärare och barn samspelar i delningsaktiviteten visas hur innebörder av tal som är nödvändiga för att utveckla taluppfattning synliggörs i interaktionen. Analysen visar en progression i hur tal som avgränsad mängd, som en samling av objekt, som del-helhetsrelation samt representationer framträder och hur dessa aspekter blir möjliga att urskilja genom den interaktion som uppstår mellan lärare och barn, där gestaltningar som fingermönster och språk utgör resurser för lärandet.
Fostering mathematical reasoning in inquiry-based teaching – the role of cognitive conflicts
Authors: Larsen Dorte Moeskær , Misfeldt Morten
Published: November 2024

Students’ independent mathematical inquiry is often endorsed as a valuable teaching method. In this article, we scrutinise in what ways these independent situations entail the students’ development of mathematical reasoning. We study the cognitive conflict in one fifth-grade class participating in an inquiry-based intervention study. The findings indicate that cognitive conflicts can support the students’ reasoning processes and that the environment has an important role in retaining the conflicting positioning by making the cognitive conflicts available for discussion and scrutiny. The students’ processes of resolving cognitive conflicts are stretched over time and involve different routes and exploring approaches and understandings.
Opportunities to learn ambitious mathematics teaching from co-planning instruction
Authors: Fauskanger Janne , Bjuland Raymond
Published: November 2024

This study explores ambitious teaching practices teachers have opportunities to learn when co-planning instruction as part of their professional development. An analytical framework associated with Sociocultural discourse analysis is applied to identify utterances (dialogue moves) in the co-planning sessions that are essential in helping the teachers to develop their reasoning together. The findings reveal that the participants work on the ambitious practices of predicting student responses, representing these responses and aiming towards the goal for the lesson when co-planning to introduce the distributive property of multiplication to their students. Dialogue moves in the reasoned dialogues such as expressing shared ideas and agreements, providing arguments and challenging each other’s ideas are found to be essential for providing the teachers with opportunities to learn to predict student responses, to represent these responses and to aim towards the learning goal for the lesson.
New mathematics teachers’ learning when participating in induction on mathematics education – a case study of two lower secondary teachers
Authors: Krog Skott Charlotte , Laursen Falkenberg Louise , Redder Honoré Ida
Published: November 2024

Newly qualified mathematics teachers often face teaching problems during their initial teaching years. We report on a study in which we designed an induction programme in mathematics education to support new teachers in tackling these problems. Using a social practice perspective, we investigate what and how two new lower secondary teachers learn by participating in the programme. We show that, while one teacher exceeds our expectations for learning, the other learns very little. We also show that the two teachers’ learning is reflexively related to school-local and broader contexts, which helps to explain the differences in their learning.
Gérard Vergnaud in action
Authors: Helenius Ola , Ahl Linda Marie
Published: November 2024

We have contemplated what we could add to the description of the legacy of Gérard Vergnaud, and have realized that we have at least one expe- rience that is probably not shared by many researchers in mathematics education ...
Full-body interaction in young children’s modelling of counting-based addition
Authors: Bjørnebye Morten
Published: November 2024

This study explores characteristics of kindergarteners’ talk, use of tools, and full-body interaction in the modelling of counting-based addition from the perspective of embodied cognition. Ten 4- and 5-year-olds participated in a 5-week outdoor embodied intervention for learning the min strategy (e.g. count on from the largest addend ”four, five, six” in 2 + 4). Video analyses of individual testing showed that strategy efficiency was associated with fluency in the embodied interaction, use of mental representations and a structural awareness of numerical relations. Inefficiency was shown to be related to extensive offloading of the additive thinking to fingers and/ or dice to keep track of counted units. An unexpected finding was the inclusion of expressive body movements (e.g. rotation, rhythm, force, and tempo) while modelling their counting strategies. The study contributes to educational research on body-based learning in mathematics by revealing patterns of young children’s physical modelling of arithmetic.
Transition between discourses – portraying teaching practices in collegial discussions
Authors: Kaufmann Odd Tore , Ryve Andreas
Published: November 2024

This paper investigates how teachers portray their own teaching practices while reflecting on them and discussing them in collegial discussions. Analysing data from eight groups with a total of 59 teachers, this study investigates how teachers portray their teaching practices and draw upon different discourses to represent their role as a teacher. This analysis finds that teachers describe different teaching practices in different lesson phases and draw upon different discourses in doing so. From this study emerges an eclectic, pragmatic teacher who rather comfortably navigates between different discourses to create a new, blended discourse.
Equivalence in multilanguage mathematics assessment
Authors: Theens Frithjof , Bergqvist Ewa , Österholm Magnus
Published: November 2024

When mathematics tasks are used in multilanguage assessments, it is necessary that the task versions in the different languages are equivalent. The purpose of this study is to deepen the knowledge on different aspects of equivalence for mathematics tasks in multilanguage assessment. We analyze mathematics tasks from PISA 2012 given to students in English, German and Swedish. To measure formal equivalence, we examine three linguistic features of the task texts and compare between language versions. To measure functional equivalence, a Differential item functioning (DIF) analysis is conducted. In addition, we examine statistically if there is a relation between DIF and the differences regarding linguistic features. The results show that there is both DIF and differences regarding the linguistic features between different language versions for several PISA tasks. However, we found no statistical relation between the two phenomena.
Teachers’ learning of ambitious mathematics teaching as changes in pedagogical discourse
Authors: Wæge Kjersti , Svingen Olaug Ellen Lona
Published: November 2024

This study investigates changes in teachers’ pedagogical discourse as they participate in a professional development project. Fourteen Norwegian elementary-school teachers collaborate in learning cycles where the overarching aim is to learn ambitious mathematics teaching. Data from interviews with two groups of teachers are analysed. The findings reveal significant changes in the teachers’ pedagogical discourse: valuing new teaching practices, talking about more specific aspects of ambitious teaching and making references to student learning. The findings also reveal a shift in the teachers’ pedagogical discourse around struggling students.
The problematic equal sign
Authors: Opsal Hilde , Topphol Arne Kåre
Published: November 2024

In this study, we investigated how 2,544 Norwegian students in the fifth, sixth, eighth and ninth grades of different schools answered a missing addend task that required a relational understanding of the equal sign. Only 50 % of the students correctly solved the task (30 % in grade 5 and 80 % in grade 9). We then selected the students who managed to correctly solve two tasks that did not have an explicit equal sign but required an assessment of equality. The percentage of correct answers in the missing addend task increased to 71 %, but even among those students who successfully handled the concept of equality, a substantial portion did not solve the equal sign task correctly. Our results indicate that equality and the equal sign cannot be treated as equivalent concepts.
Didactical usefulness of interactive mathematical maps – designing activities supporting prospective teachers’ learning
Authors: Vinerean Mirela , Liljekvist Yvonne , Brandl Matthias , Przybilla Johannes
Published: November 2024

Prospective teachers often have difficulty in linking school mathematics and university course content, which manifests itself as a lack of their understanding and the significance of university course content. This double discontinuity is experienced by future teachers in their transition from high school pupil to university student and then from university student to their school teaching career. Thus, it is necessary to improve university teaching and teacher education to try to bridge these ”gaps”. Using the educational context of a geometry course in the teacher education of upper secondary teachers, we explore the technical implementation and usefulness of the components of interactive mathematical maps. Such maps comprise a supplementing didactical tool that shows the interrelations between mathematical discoveries and the development of particular mathematical content – starting from an initial historical problem situated in time. The research findings showed the map in its current format to be perceived as useful and mostly easy to use. Further, the map seemed to promote both a process-oriented and an application-oriented approach as well as favourable beliefs, such as mathematics being an emerging science promoting a view of doing mathematics, in which an open error culture can be established.
Preservice and inservice teachers’ views on digital tools for diverse learners in mathematics education
Authors: Meaney Tamsin , Huru Hilja L. , Kvivesen Mona
Published: November 2024

Although teachers are expected to use digital tools in their mathematics teaching in many countries, little is known about preservice and inservice teachers’ digital competence, especially in relationship to specific groups of school students. Results from a survey of 394 preservice teachers and 61 inservice teachers, at three Norwegian institutions, provide information on how they considered different digital tools would support differentiated teaching, related to a student’s mathematical progress, and in multilingual classes. The results suggest that preservice and inservice teachers evaluated similarly the usefulness of different digital tools for differentiated mathematics teaching and in multilingual classrooms. However, for the majority of tools, the standard deviations indicate that the responses were somewhat spread, suggesting uncertainty in how they could use digital tools to support specific groups of students.
Proof as an explanation of dynamic geometry generated conjectures – task design in a toolbox puzzle approach
Authors: Højsted Ingi Heinesen
Published: November 2024

In the frame of a design-based research project, this paper presents analysis of Danish grade 8 students working together to prove conjectures, which they formulated based on guided explorations in a dynamic geometry environment. A systematic account of the rationale and hypothesis behind the task design is described in the form of objectives, hypotheses, and choices, which are then evaluated in light of the analysis of data. The case indicates that the designed task can bridge a connection between conjecturing activities in dynamic geometry environments and deductive reasoning. The students manage to explain theoretically, what is initially empirically evident for them in their exploration in the dynamic geometry environment. The proving activity seems to make sense for the students, as a way of explaining ”why” the conjecture is true. The Theory of Semiotic Mediation frames the design of the study and the data analysis.
Analysing mathematical programming schemes using different lenses
Authors: Borg Andreas , Fahlgren Maria
Published: November 2024

The use of programming in mathematics education is undergoing a renaissance and, in this paper, we analyse students’ handling of programming in mathematics using the Instrumental approach as a theoretical lens. We are especially interested in analysing the development of mental schemes using two analytical frameworks which are compared and contrasted according the idea of networking theories. The study illustrates that the frameworks’ detail of richness can have both advantages and disadvantages and that one of the frameworks are more customed to be applied when analysing students’ instrumental genesis concerning the use of a programming environment as a mathematical artefact.
Facilitating exploratory talk through mathematical programming problems
Authors: Munthe Morten
Published: November 2024

With several Nordic countries implementing programming into their curricula, there is a need for research into the combination of mathematical learning and programming. With tasks being the main ”thing to do” in the mathematics classroom and exploratory talk being closely linked to learning this article investigates what contri-butes to and what hinders exploratory talk when working on mathematical programming problems. The data collection comprises of recordings of students working on tasks in a mathematics classroom in Norway. The findings indicate that programming can facilitate mathematical talk in the classroom, that programming is best implemented to facilitate in-depth learning, and that adversities are present in both instances.
Challenges in developing a TPACK survey for preservice mathematics teachers in the Norwegian context
Authors: Gautam Ramesh , Jakobsen Arne
Published: November 2024

This study explores challenges when adapting and further developing a subject-specific survey based on the Technological pedagogical content knowledge (TPACK) framework in the Norwegian context. Two hundred forty-four preservice teachers from eight institutions across Norway participated in the survey. The most prominent challenges during the adaptation and further development of the survey were related to survey item composition and factor validation due to contextual differences and unclear boundaries between different TPACK domains. After evaluating the reliability and validity, a six-factor survey with 30 items is developed. The implications of this study and direction for further improvement are suggested.
Primary school students’ engagement in the formative feedback process in mathematics
Authors: Green Jenny
Published: November 2024

Engaging students in formative feedback practices is a significant aspect of mathematics teaching, but students do not always take an active part in the process. Factors contributing to the lack of engagement could be the design of the learning context and the nature of the subject. In this study, the engagement of Grade 2 students in the formative feedback process, focusing on their mathematics reasoning, is analysed to gain knowledge about the relationship between the learning context and students’ engagement in the formative feedback practice. Students’ engagement was observed during a feedback event, followed by stimulated recall sessions and semi-structured interviews, and later analysed and discussed in terms of socio-mathematical norms. The study revealed a main focus on performance and producing correct answers. In addition, different views of the purpose of the formative feedback event were identified as norm conflicts, acting as barriers for the involvement of students in the formative feedback practice.
Standardized mathematics testing in Sweden: the legacy of Frits Wigforss
Authors: Johansson Bengt , Kilpatrick Jeremy
Published: November 2024

Developed more than 50 years ago, the Swedish system of standardized testing as a means of moderating marks (or grades) is about to be replaced by a criterion- referencedmeasurementscheme. Theprincipaldeveloperoftheoriginalsystem, Frits Wigforss, was a psychologist and mathematics educator who understood the complex issues raised by any marking system and who attempted to use testing not to replace but to improve teachers' judgment A close examination of the history of standardized mathematics testing in Sweden reveals the magnitude of Wigforss's contribution as well as its subsequent eclipse by the elevation of measurement technique over mathematical substance and a serious absence of attention to the educational and social consequences of changes in the system.
Problembaserad matematikundervisning och demokrati
Authors: Johansson Bengt
Published: November 2024

Ledare/Editorial
Open ended problem solving in geometry
Authors: Borgersen Hans Erik
Published: November 2024

This article is a documentation of a project on open ended problem solving in geometry at Agder College. The project is part of a geometry course and is based on co-operation in small groups. Why we did it, how we did it, how it developed, and what came out of it is the main core of the article. Our conclusions are:

– It is possible to have students experience the whole process of doing mathematics.

– Geometry is a rich resource for finding problems to do so.
– It is possible to create learning environments where students co-operate and give each other support to go through the process together.
– It is possible to do this within ordinary courses in the first year at college/university.
– There has been observable progress on process writing and evaluation. The students' attempt to generalize and formulate new problems are not satisfactory.The emphasis on process writing and evaluation seems to have an effect on the students' motivation and understanding of doing mathematics.
– The students experience a great need for encouragement and support in the process of doing mathematics.
– Co-operation in small groups is highly recommended by the students.
'Informal activity' in mathematics instruction
Authors: Christiansen Iben Maj
Published: November 2024

On the basis of an analysis of portions of a modelling course in Danish high school, the existence of 'informal activity' is asserted. This activity can contain reflections on the mathematical activity performed when 'on task', but also allows the students to distance themselves from the content and school-tasks in gene- ral. Informal activity may thereby contain a potential for critique which is not rea- lized due to isolation in the informal activity. To give insight on the possibility for change, the formation of discourses and their relation to the setting is discussed.
Challenges to the preparation of teachers of mathematics
Authors: Niss Mogens
Published: November 2024

This article attempts a personal answer to the following two connected questions: "What are the most important problems in the preparation of mathematics teachers in view of the tasks, functions, and roles they have or ought to have in contemporary mathematics teaching?", and "What specific challenges to the preparation of teachers of mathematics do these problems generate?" These questions are posed as general ones with no particular reference to any specific country.
The answers which this article offers to these questions are based on three "axioms" concerning the job of a mathematics teacher and on a characterisation of the qualities which the "ideal, but unfortunately Utopian, mathematics teacher" ought to possess. From this point of departure six major current problems in the preparation, roles and functioning of mathematics teachers are identified: (i) work- ing and living conditions; (ii) qualitative and quantitative recruitment; (Hi) pre- and in-service preparation and education; (iv) teachers' knowledge, beliefs and perceptions of mathematics; (v) teachers' knowledge, beliefs and perceptions of mathematics education and of the tasks, roles, and positions of mathematics teachers; (vi) professional development.
These problems are discussed in some detail and the article concludes by suggesting points which deserve particular attention in efforts to develop and improve the education of mathematics teachers: knowledge and perceptions concerning mathematical content and processes (we are talking about teachers of MATHEMATICS, not just about teachers); the need for enthusiastic and committed TEACHERS rather than people who are somewhat knowledgeable in mathematics and happen to teach it; the need for frameworks and environments for continuing professional growth and development.
This is a slightly revised manuscript of a plenary lecture given by the author at the ICMI-China Regional Conference on Mathematics Education, Shanghai, 16-20 August 1994.
What on earth is a straight line? What can we learn from an epistemological episode in eighth grade?
Authors: Paasonen Johannes
Published: November 2024

A detailed account of a spontaneous discussion in eighth grade of the nature of geometrical objects is given and related to constructivistic learning theory. The pupils were quite eager and able to discuss such subjects as infinity and more generally the process of abstraction inherent in geometrical concepts. Discussions of this type in class are advocated in order to encourage pupils to problematize concepts and procedures often regarded as self-evident, to spot and discuss pupils' misconceptions, and thus to attempt to raise the level of abstraction in mathematical communication and concept formation in class.
Korta bokanmälningar
Authors: Emanuelsson Göran , Johansson Bengt
Published: November 2024

Det kommer en ström av nya böcker som behandlar olika delar av vårt vetenskapsområde. Produktionen har nog aldrig varit så omfattande som under det senaste året. Vi ger här några exempel på nya böcker från tre av de många förlag som satsar stort på matematikdidaktisk litteratur.
Undervisning i problemløsningsstrategier
Authors: Alseth Bjørnar
Published: November 2024

Artiklen presenterer i første del et rammeverk som dekker sentrale aspekter ved matematisk tenkning. Begrepene som blir beskrevet er: basiskunnskap, strategier, metakognisjon og holdninger. I tillegg blir forholdet mellom dem diskutert. Artiklens andre del er viet strategiundervisning. Søkelyset rettes mot en modell for slik undervisning; hvordan strategier kan undervises ved hjelp av denne modellen, men også hvordan de øvrige komponentene i rammeverket bør stimuleres for å nå målet: økt matematisk kompetanse gjennom rasjonell bruk av strategier. Artiklen konkluderer med at dette rammeverket og denne modellen for undervisning danner et troverdig teoretisk grunnlag for et videre empirisk arbeid med undervisning av strategier.
Korta litteraturanmälningar
Authors: Emanuelsson Göran
Published: November 2024

Korta litteraturanmälningar
The second issue in English
Authors: Johansson Bengt , Emanuelsson Göran
Published: November 2024

This issue focuses on mathematics education as a scientific field of research.
Mathematics education as theoretical knowledge
Authors: Firsov Victor
Published: November 2024

Didactics of mathematics (DM) is the theoretical part of our knowledge in mathe- matics education. Its connections with the mathematical sciences and with school mathematics determine the independent character of DM as a scientific discipline. Features of school mathematics such as its unique aims, the highly abstract nature and hierarchical construction of the material to be studied, and the varied kinds of educational activities lead to its specific character, situating it among the school subjects and making accepted theoretical conclusions exclusively applicable to mathematics education. The social character of DM generates the approaches to constructing categories within the discipline (inexact and "inaccurate" conceptions, exclusion principles in formulating absolute statements, plausible reasoning, diversity of proofs) and promotes the selection of adequate methods of research that are not typical of positivistic science (expert review, discussion, pedagogical experiment). The applied character of DM determines an appropriate methodology of research and efficient ways of overcoming contradictions. The concrete practice of education (partly in the form of experimental tests) gives teachers an opportunity to use "inaccurate" empirical methods, reasonable considerations, intuitive choice, and so forth. Some practical advice is given for discussing and conducting research projects.