Volumes/articles older than two years are open access except editorials which always are open access.

NOMAD
When does a difference make a difference? Teaching about language diversity in mathematics teacher education
Andrea Eikset and Tamsin Meaney
Abstract
There has been little attention in mathematics education research about how to include issues to do with language diversity in teacher education. This paper describes the process used by two teacher educators to examine their own practices of linking multilingual perspectives to mathematics education in their work with preservice teachers. By systematically analysing their discussion about a threehour, mathematicsteachereducation workshop on proportional thinking, the teacher educators were able to identify a series of Discourses. They considered that these Discourses underlay their decision making about how language diversity could be raised with preservice teachers. The results highlight the complexity connected to raising language diversity issues in mathematics teacher education. For example, deciding what challenging content should be provided to preservice teachers is a ected by the need to develop relationships with them as well as managing their learning. Joint re ection by the teacher educators was needed to ensure that the aim of challenging preservice teachers about how to deal with language diversity issues in mathematics classrooms could be achieved.Andrea Eikset
Andrea Eikset is an assistant professor working at Western Norway University of Applied Sciences in Bergen. Her academic background is in mathematics didactics, and her main research interests are mathematics in early childhood education, diversity and sustainable development. This is also connected to teacher educators own professional development.Tamsin Meaney
Tamsin Meaney is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. She has written journal articles on language diversity in mathematics education for twenty years.Skapad: 20181106 kl. 11:20

NOMAD
Talking about mathematics in two languages: Can parental views inform the development of digital games for young children?
Troels Lange and Tamsin Meaney
Abstract
In this article, the results are presented from a survey of parents’ views about the digital games that their young multilingual children play. Previous research has indicated that parents struggled to describe how their children were learning from playing digital games. The results from this study indicate that parents could provide information about the digital games and the mathematical language they invoked. This information could be useful in developing playful, digital games that support multilingual children to talk about mathematics. The survey also provides insights into the followup qualitative research studies that are needed to support the development of new digital games.Troels Lange
Troels Lange is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. He has a long standing interest in children ́s perceptions of mathematics education and societal issues that in uence the views of parents, teachers and politicians about mathematics education for young children.Tamsin Meaney
Tamsin Meaney is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. She has written journal articles on language diversity in mathematics education for twenty years.Skapad: 20181106 kl. 11:18

NOMAD
Polish parents’ views on mathematics activities at home and in Swedish preschools
Dorota Lembrér
Abstract
This article describes the results of a digital survey of 41 Polish immigrant parents’ views on mathematics activities at home and at preschool as parents’ views potentially provide a range of perspectives on mathematics activities for young children. Parents were asked to describe and justify their views about how children engage with mathematical ideas and nominate activities that children engage in at home and at preschool. When parents justi ed their views about young children and mathematics, they tended to align themselves with the norms and values of the Swedish preschool curriculum. The ndings suggest that parents, like children, are socialised into Swedish preschools. However, this alignment could limit possibilities for broadening perspectives about mathematics education in preschool, which could be available by incorporating input from immigrant parents’ di erent cultural and linguistic backgrounds.Dorota Lembrér
Dorota Lembrér is a doctoral student at Western Norway University of Applied Sciences in Bergen, Norway. She has a background as a preschool teacher and a lecturer at Malmö University in Sweden. Her main research interests are early childhood mathematics education and aspects of mathematics activities in preschool and home environments.Skapad: 20181106 kl. 11:16

NOMAD
Bishop Sámegillii – utfordringer ved oversetting av matematikkdidaktisk fagterminologi
Anne Birgitte Fyhn, Ellen J. Sara Eira, Ole Einar Hætta, Inga Anne Marit Juuso, Siv Ingrid Nordkild og Ellen Margrethe Skum
Sammenfatning
I artikkelen belyser vi kulturelle og språklige utfordringer som kan oppstå når teoretiske begreper fra matematikkdidaktisk forskning oversettes til et språk og en kultur som er ernt fra språket og kulturen der begrepene ble utviklet. I samarbeid med forskere oversatte lærere ved Guovdageainnu nuoraidskuvla/Kautokeino ungdomsskole et matematikkdidaktisk rammeverk fra engelsk til nordsamisk. Vi fokuserer på oversetting av aktiviteten locating som inngår i rammeverket. Locating omfatter grunnlaget for elevers utvikling av romlig forståelse. Vi bidrar også til diskusjonen om hvorvidt hovedområdene i matematikklæreplanen skal utformes som verb eller substantiver.Abstract
In this article we enlighten cultural and linguistic challenges that may occur when concepts from mathematics education are translated into a language and a culture that is far from the language and the culture where the concepts were developed. In cooperation with researchers, teachers at Guovdageainnu nuoraidskuvla/Kautokeino lower secondary school translated a mathematics education framework into North Sámi. We focus on translations of the activity locating, which is part of this framework. Locating constitutes the basis for students’ development of spatial understanding. We also contribute to the discussion about whether the national mathematics curriculum’s content areas should be verbs or nouns.Anne Birgitte Fyhn
Anne Birgitte Fyhn is professor in mathematics education at UiT, the Arctic University of Norway, Tromsø. She holds a professor II position at Sámi Allaskuvla/ Sámi University of Applied Sciences. Her main research interests are relations between mathematics education and culture.Ellen J. Sara Eira
Ellen J. Sara Eira is principal of Guovdageainnu nuoraidskuvla. She has taught mathematics in Sámi for 35 years and she taught Sámi at Sámi Allaskuvla/Sámi University of Applied Sciences for two years. Since 2010, she is mathematics sensor for the national compulsory school exam. From 1983–1998 she was member of the national exam group in Sámi at upper and lower secondary level.Ole Einar Hætta
Ole Einar Hætta is mathematics teacher at Guovdageainnu nuoraidskuvla. He has been sensor at the Sámi mathematics exam for the compulsory school for ten years. He was member of the 2017 national curriculum group that worked out core elements in mathematics. He is master student in mathematics education at UiT, the Arctic University of Norway.Inga Anne Marit Juuso
Inga Anne Marit Juuso is mathematics teacher at Guovdageainnu nuoraidskuvla. She also teaches duodji (Sámi handicraft). She is master student in mathematics education at UiT, the Arctic University of Norway.Siv Ingrid Nordkild
Siv Ingrid Nordkild is Ph.D. student at UiTThe Arctic University of Norway, Campus Tromsø. Her Ph.D. is based in mathematics education in the North Sámi area in Norway. Her research interests are culture based mathematics education and indigenous related issues in education. She is educated as an electrical engineer with Master’s degree in pedagogy.Ellen Margrethe Skum
Ellen Margrethe Skum is vice principal of Guovdageainnu nuoraidskuvla. She teaches Sámi language and duodji (Sámi handicraft). She uses her competencies in Sámi tradition and culture, among others from reindeer husbandry, in her teaching.Skapad: 20181106 kl. 11:12

NOMAD
En registeranalyse af centrale matematiske begreber i en grønlandsk kontekst
Mette Hjelmborg and Ane Fleischer
Sammenfatning
Denne artikel beskriver resultaterne af en registeranalyse af grønlandske matematiske begreber. Fokus for analysen er de karakteristiske træk, der opstår når man genererer et matematisk register i et polysyntetisk sprog. Vores teoretiske ramme er faglige registre og registerkontinuum i et systemisk funktionelt lingvistisk perspektiv. Metodisk foretages dokumentanalyser, der mættes af interviewsvar fra semistrukturerede interviews. Vores analyse indikerer, at der som oftest benyttes samtlige tre teknikker når grønlandske matematiktermer genereres: omskrivninger, nominaliseringer og metaforer. Det matematiske register knytter sig dermed i høj grad og på mange og varierede måder til det grønlandske dagligdagssprog.Abstract
This article describes the results of an analysis of the mathematical register in Greenland concerning the characteristics for developing terms for mathematical concepts in a polysynthetic language. The theoretical frame is registers, register continuum with respect to systemic functional linguistic. The analysis indicates that three techniques (gerunding, circumlocutions, and metaphors) are used simultaneously when generating new terms in Greenlandic. The mathematical concepts in Greenlandic thereby have a direct link to everyday meanings of similar concepts.Mette Hjelmborg
Mette Hjelmborg er lektor på UCL Erhvervsakademi og Professionshøjskole, læreruddannelsen på Fyn. De vigtigste interesser er: sprog og begrebsdannelse, Dansk som andetsprog, elever med særlige behov, læremiddelanalyse.Ane Fleischer
Ane Fleischer underviser på læreruddannelsen i Grønland, efteruddanner, lærebogsforfatter. De vigtigste interesser er: matematiklæring og især sprog og matematikbegreber på grønlandsk, elever med særlige behov.Skapad: 20181106 kl. 11:04

NOMAD
Culturally based mathematics tasks: a framework for designing tasks from traditional Kven artefacts and knowledge
Hilja L. Huru, AnnaKaisa Räisänen and Anita Movik Simensen
Abstract
This article discusses mathematical and cultural task design to support minority and endangered languages and cultures. More precisely, we propose a theoretical framework to design mathematical tasks for language immersion in mathematics for Kven students. Drawing on previous studies, we suggest that traditional tools have the potential to support the learning of mathematics, language, and culture. One challenge for endangered languages and cultures is that the younger generations may have lost connections with their traditional language and culture. We argue that the older generations can mediate authentic aspects of Kven culture to students, which then become historicalcultural authentic (HiCuA) aspects.Hilja L. Huru
Hilja L. Huru is a professor in mathematics at OsloMet – Oslo Metropolitan Univerity and UiT – The Arctic University og Norway. Her background is in pure mathematics with a PhD in noncommutative algebra connected to mathematical physics. Her research interest also includes multicultural classrooms with a focus on pressured minorities and indegenous mathematics.AnnaKaisa Räisänen
AnnaKaisa Räisänen is a language advisor at the Kainun institute – Kvensk institutt, Norway and a doctoral student at the University of Oulu, Finland. She is involved with language education programs for Kven language and leads Kven language nests projects in northern Norway. Her main research interests are sociolinguistics and language policy as well as language revitalization and language vitality.Anita Movik Simensen
Anita Movik Simensen is assistant professor of mathematics education at UiT – The Arctic University of Norway. Her research interests include mathematics education in inclusive classrooms, indigenous mathematics, and the use of natural outdoor learning environments as setting for young children’s (age 1–6) mathematical experiences.Skapad: 20181106 kl. 10:55

NOMAD
Newly and earlyimmigrated secondlanguage students’ knowledge of arithmetic syntax
Jöran Petersson
Abstract
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 orderofoperation rules and the associative and distributive laws of arithmetic operations. On these four test items, the mostrecently arrived students showed on average signi cantly 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 subcategories of secondlanguage students when it comes to teaching, assessment and research.Jöran Petersson
Jöran Petersson is a senior lecturer at Stockholm University, Sweden. He wrote his PhDdissertation on test achievement of second language students in the last year of compulsory school. Presently he is doing postdoctoral research on how foundational number sense appears in textbooks and homework in the rst year of compulsory school. Jöran also has an interest in mathematical modelling.Skapad: 20181106 kl. 10:52

NOMAD
The language factor – what exactly is it? Bilingual speakers of Russian and Finnish solving mathematical tasks
Maria Ahlholm and Päivi PortaankorvaKoivisto
Abstract
A lack of knowledge of the language of instruction is often believed to be the main reason for low achievement among students with an immigrant background. We regard language as a tripartite unit comprising aspects of concept formation, pragmatic language usage and the linguistic form. In this theoretical framework, we report two case studies of bilingual, Russian and Finnish speaking students’ explanations of their procedures while solving mathematical tasks. The students’ linguistic processing varied in terms of conceptualization, pragmatic meaningmaking and grammatical form. In a bilingual context, the labelling of concepts and meaningmaking through argumentation are simultaneously processed in two languages.Maria Ahlholm
Maria Ahlholm is a University lecturer (PhD) in the Faculty of Educational Sciences at the University of Helsinki, and an adjunct professor in Finnish language, especially applied linguistics. Ahlholm is interested in the multilingual socialisation process of emerging second language.Päivi PortaankorvaKoivisto
Päivi PortaankorvaKoivisto is a University lecturer (PhD) of mathematics didactics in the Faculty of Educational Sciences at the University of Helsinki. PortaankorvaKoivisto is interested in developing mathematics teacher education and teachers’ awareness of the language factor in teaching and learning mathematics.Skapad: 20181106 kl. 10:49

NOMAD
Developing mathematical reasoning by using questions in a multilingual mathematics classroom
Marie Sjöblom
Abstract
In this paper, students’ questions while working in small groups on mathematical problemsolving tasks are investigated. In order to improve students’ reasoning and communication abilities in mathematics, an intervention study was designed in a multilingual upper secondary mathematics classroom in Sweden. In their discussions students used Swedish, which was their second language and also the language of instruction. The changes in students’ ways of using questions across the three cycles of the intervention were analysed. The results showed how students over the cycles changed their ways of framing questions from looking for the correct answer towards clarifying other students’ meaning in order to understand each other’s reasoning. The implication from the study is that it is important to promote interactions between students rather than focusing on students’ need to develop their second language competencies.Marie Sjöblom
Marie Sjöblom is a PhDstudent in mathematics education at Malmö University. She is also a mathematics teacher, and work as senior lecturer with school development in Malmö, supporting teachers and school leaders on collegial learning processes. Key research interests are interaction in multilingual mathematics classrooms and educational design research.Skapad: 20181106 kl. 10:46

NOMAD
Identity formations as mathematical learners in the context of transition
Petra Svensson Källberg
Abstract
This paper explores the relation between discourses and identity formations as mathematical learners in a context of transition. The data consists of an interview with two 16 yearold immigrant girls, who were relocated when their school, in a multicultural and socioeconomically disadvantaged area in Sweden, was closed. The girls showed dynamic and unstable identities by drawing on di erent discourses. Social relational discourses, more than mathematical pedagogical discourses, governed their actions as learners of mathematics; enabling identities as noisy, unengaged, but able students in the old school, and as engaged and accepted, but also as strangers, in the new school.Petra Svensson Källberg
Petra Svensson Källberg has a doctoral degree in mathematics education from Stockholm University, the department of mathematics and science education. She works at Pedagogisk Inspiration, a department which works with school development and research in the muncipality of Malmö, Sweden. Her main research interests concern sociopolitical issues in mathematics education and are related to multilingual and multicultural issues in mathematics education.Skapad: 20181106 kl. 10:44

NOMAD
Fabrication of newlyarrived students as mathematical learners
Eva Norén and Petra Svensson Källberg
Abstract
As a response to recent laws on how to support newlyarrived students’ schooling, new policy texts have been released in Sweden. By analyzing policy texts we show how a particular kind of human, ”the newlyarrived student as a mathematical learner” is fabricated through discursive processes. We show how the policy texts are framed within an including discourse that encourages multiculingualism and views students’ mother tongue and backgrounds as resources. However, simultaneously the newlyarrived student is thought of, in a more excluding discourse, as being in need of rescue and as lacking the most valuable asset, the Swedish language.Eva Norén
Eva Norén is senior lecturer in mathematics education at the department of mathematics and science education, Stockholm University. Her main research interest is multilingual students in mathematics classrooms. She defended her PhD thesis, Flerspråkiga matematikklassrum [Multilingual mathematics classrooms] in 2010. She has also researched gender issues related to mathematics teaching and learning. Since 2017 she is involved in a development and research project on programming in subject didactics.Petra Svensson Källberg
Petra Svensson Källberg has a doctoral degree in mathematics education from Stockholm University, the department of mathematics and science education. She works at Pedagogisk Inspiration, a department which works with school development and research in the muncipality of Malmö, Sweden. Her main research interests concern sociopolitical issues in mathematics education and are related to multilingual and multicultural issues in mathematics education.Skapad: 20181106 kl. 10:41

NOMAD
Language diversity in mathematics education in the Nordic countries 2008–2018
Tamsin Meaney and Toril Eskeland Rangnes
Abstract
A paper in which the guest editors introduce this thematic issue of NOMAD.
“The aim of the thematic issue is to provide an overview of what was being done and from this to determine what still needed to be done on language diversity in mathematics classrooms and early childhood centres in the Nordic countries.”Tamsin Meaney
Tamsin Meaney is professor in mathematics education at Western Norway University of Applied Sciences in Bergen, Norway. She has written journal articles on language diversity in mathematics education for twenty years.Toril Eskeland Rangnes
Toril Eskeland Rangnes is associate professor in mathematics education at the Faculty of Education, Arts and Sports at Western Norway University of Applied Sciences, campus Bergen. Her main research interest are critical mathematics education, teacher professional development and language diversity in mathematics classrooms.Skapad: 20181106 kl. 10:35

NOMAD
Scrutinizing teacherlearner interactions on volume
Anita Tyskerud and Reidar Mosvold
Abstract
This study adds to research on volume and spatial reasoning by investigating teacherlearner 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.Anita Tyskerud
Anita Tyskerud is a PhD candidate in Educational Science, Department of Education and Sports Science, University of Stavanger, Norway. Her research interests are related to teachers’ professional development in mathematics and Lesson study.Reidar Mosvold
Reidar Mosvold is Professor of Mathematics Education at the Department of Education and Sports Science, University of Stavanger, Norway. His research interests are related to mathematics teaching and developing mathematics teachers.Skapad: 20180822 kl. 16:08

NOMAD
Læreres utbytte av kunnskap om hjernen
Jan Roksvold
Sammandrag
Konkrete klasseromsanvendelser av hjerneforskning har latt vente på seg. I denne oversiktsartikkelen undersøkes det potensielle utbyttet for lærerstudenter ved å kjenne til ulike temaer knytta til hjernens befatning med tall og aritmetikk – uavhengig av hvorvidt slike entilenanvendelser eksisterer eller kan eksistere. Av potensiell verdi for lærere framheves blant annet kunnskap om hvilke vanskeligheter et assosiativt minne forårsaker i forbindelse med aritmetiske tabeller. Med bakgrunn i moderne hjerneforskning belyses ei tallbehandling som kan deles opp i en medfødt ”tallsans” og et kultur og utdanningsavhengig eksakt tallsystem, hvordan ulike binæroperasjoner behandles på grunnleggende forskjellig vis av hjernen, og hvordan innlæringsstrategi kan påvirke lagringa av aritmetisk kunnskap. Temaer som barnets ”logaritmiske indre tallinje” og dyskalkuli blir også belyst. Jeg konkluderer med at denne typen kunnskap om hjernen vil utvide lærerstudentenes forståelse av det lærende barnet, og dermed kunne påvirke deres praksis.Abstract
Concrete applications of neuroscience to the classroom are yet to be confirmed. The topic of this research article is the potential gains to be had for trainee teachers in knowing about various topics concerning the brain’s processing of numbers and arithmetic – regardless of whether onetoone applications exist or can exist. Highlighted as potentially valuable to teachers, is knowledge about: the dichotomy between an inborn ”number sense” and a culturally and educationally dependant exact number system; how different binary operations are processed; how learning strategy can affect the encoding of arithmetic facts; the difficulties caused by an associative memory in relation to arithmetic tables; the child’s ”logarithmic inner number line”; dyscalculia, and the neuromyth pertaining to it. I conclude that this type of knowledge will expand trainee teachers’ understanding of the learning child, and thereby possibly in uence their practice.Jan Roksvold
Jan Roksvold er førsteamanuens i matematikkdidaktikk ved UiT Norges arktiske universitet. Hans forskningsinteresser omfatter anvendelse av funn fra kognitiv psykologi i matematikkundervisning, hjernens befatning med matematikk, samt bruk av matematikkhistorie og historiefortelling i undervisning.Skapad: 20180822 kl. 16:02

NOMAD
Disciplinary competence descriptions for external use
Jens Højgaard Jensen and Uffe Thomas Jankvist
Abstract
The article addresses the need for competence descriptions of disciplines as a means for fostering more productive communication between different disciplines and between the disciplines and their surroundings. It is argued that the usual competence descriptions devised for use within a discipline itself, e.g. in relation to teaching and learning of the discipline – socalled competence descriptions for internal use – are not the best means to achieve this. The same is true for the general, nondisciplinary competence descriptions. Instead, specially devised disciplinary competence descriptions for external use are called for. Our main illustration is a competence description of mathematics for external use devised so that it can support the dialogue about justi cation of mathematics education between the discipline’s practitioners and its recipients. This description for external use is counterposed with one for internal use i.e. that of the Danish KOM project. It is also counterposed with a competence description for external use for physics, taking into account the different justification problem of physics education. Together these two descriptions showcase how competence descriptions of disciplines for external use may support interdisciplinary collaboration and division of labor in the educational system.Jens Højgaard Jensen
Jens Højggard Jensen is associate professor in physics at Roskilde University. For many years he was involved in the governing of Roskilde University in various positions. Besides publications with more specific physics content he has published on university politics, general didactics, science didactics, and interdisciplinarity.Uffe Thomas Jankvist
Uffe Thomas Jankvist is professor with special responsibilities in mathematics education at the Danish School of Education, Aarhus University. He has published on the use of history of mathematics in mathematics education, technology in mathematics education, interdisciplinarity, and students’ learning difficulties in mathematics. Besides teaching and supervising future mathematics teacher educators at the Danish School of Education, he is also part of the Danish researchbased “maths counsellor” upper secondary teacher program at Roskilde University.Skapad: 20180822 kl. 15:56

NOMAD – 23(2), 2018
Volume 23, No 2, June 2018
eNOMAD
Jens Højgaard Jensen and Uffe Thomas Jankvist
Disciplinary competence descriptions for external useJan Roksvold
Læreres utbytte av kunnskap om hjernenAnita Tyskerud and Reidar Mosvold
Scrutinizing teacherlearner interactions on volumeSkapad: 20180822 kl. 15:48

NOMAD
The gap between school mathematics and university mathematics: prospective mathematics teachers’ conceptions and mathematical thinking
Jani Hannula
Abstract
In Finland, both prospective and inservice mathematics teachers report a discontinuity between universitylevel mathematics and mathematics taught at comprehensive and secondary school. In this study, ten prospective mathematics teachers (PMTs) were interviewed to examine their conceptions of the nature of this gap as well as their mathematical thinking. The study’s findings support research that has revealed difficulties experienced by PMTs in the secondary–tertiary transition and in connecting formal and informal components of mathematical thinking. Additionally, the study provides new insight into PMTs’ conceptions of teacher knowledge, such as the relationship between knowledge of advanced mathematics and the knowledge needed in teaching situations. The findings offer guidelines for further studies that could help the development of mathematics teacher education.Jani Hannula
Jani Hannula is a doctoral student at the University of Helsinki, Finland. He is involved with mathematics teacher education at the Department of Mathematics and Statistics. He has a background as a lecturer of mathematics and information technology at Helsinki Metropolia University of Applied Sciences. His main research interests are teacher knowledge and beliefs as well as cognitive aspects of mathematical thinking.Skapad: 20180320 kl. 15:50

NOMAD
Discourses in school algebra: the textbooks’ different views on algebra and the positioning of students
Kristina Palm Kaplan
Abstract
The purpose of this study is to understand the school algebra offered in Swedish mathematic textbooks for grade 8. Using a social semiotic perspective, textbook tasks are analysed with a method inspired by Systemic Functional Linguistics. Five school algebra discourses are identified: symbolic discourse, geometrical discourse, arithmetical discourse, (un)realistic discourse and the scientific discourse. It is argued that these offer different views on the nature of algebra and the positioning of students.Kristina Palm Kaplan
Kristina Palm Kaplan is a doctoral student at Uppsala University since September 2014. The main research interests are mathematics and language, especially algebra and social semiotics.Skapad: 20180320 kl. 15:47

NOMAD – 23(1), 2018
Volume 23, No 1, March 2018
eNOMAD
Anna Ida Säfström
Preschoolers exercising mathematical competenciesMagnus Fahlström and Lovisa Sumpter
A model for the role of the physical environment in mathematics educationKristina Palm Kaplan
Discourses in school algebra: the textbooks’ different views on algebra and the positioning of studentsSkapad: 20180320 kl. 12:27

NOMAD
Preschoolers exercising mathematical competencies
Anna Ida Säfström
Abstract
The mathematical ideas that emerge in children’s free and guided play can be both complex and sophisticated, and if they are linked to formal mathematics, they can be a powerful basis for mathematical development. To form such links, one needs knowledge of how children use and express these ideas. This is especially true in the intersection of arithmetic and geometry, where the intermingling of numerical and spatial concepts and skills is not yet fully understood. This study aims to gain understanding of children’s mathematical practices by describing the interplay of key mathematical ideas, and more specifically how young children exercise mathematical competencies in the intersection of early arithmetic and geometry. The results show that children can use spatial representations when reasoning about numbers, and that they are able to connect spatial and numerical structures. Furthermore, it is shown that children not only use and invent effective procedures, but also are able to explain, justify and evaluate such procedures.Anna Ida Säfström
Anna Ida Säfström is associate professor in mathematics education at Halmstad University. Her main research interests are mathematical competence, mathematics as conceptual fields, design research and teachers’ professional development.Skapad: 20180319 kl. 15:22

NOMAD
A model for the role of the physical environment in mathematics education
Magnus Fahlström and Lovisa Sumpter
Abstract
In this paper, we develop an analytical tool for the role of the physical environment in mathematics education. We do this by extending the didactical triangle with the physical environment as a fourth actor and test it in a review of literature concerning the physical environment and mathematics education. We find that one role played by the physical environment, in relation to mathematical content, is to portray the content in focus, such as geometry and scale. When focusing on teachers, students, and the interaction between them, the role of the physical environment appears to be a precondition, either positive (enabling) or negative (hindering). Many of the findings are valid for education in general as well, such as the importance of building status.Magnus Fahlström
Magnus Fahlström is PhDstudent in Microdata Analysis and a mathematics teacher educator at Dalarna University. Key research interests are physical school environment and mathematics education.Lovisa Sumpter
Lovisa Sumpter is Senior Lecturer and Associate Professor in Mathematics Education at Stockholm University. Key research interests are mathematical reasoning, affect and gender.Skapad: 20180319 kl. 15:16

NOMAD
Stimulating critical mathematical discussions in teacher education: use of indices such as the BMI as entry points
Suela Kacerja, Toril Eskeland Rangnes, Rune Herheim, Meinrad Pohl, Inger Elin Lilland and Ragnhild Hansen
Abstract
The main purpose of our research project is to gain insight into, and develop teaching on indices and their applications in society. In this paper, the focus is to present insights into teachers’ reflections when discussing the Body Mass Index (BMI). Skovsmose´s concept of mathemacy, and source criticism, are chosen as conceptual framework. The data analysed were collected in a numeracy across the curriculum class with practising teachers. The findings show that the practising teachers engaged in meaning making of the index formula, and they critically discussed how BMI is used in society and the role the BMI index can have in our lives. We gain insight into the potential of such an index for developing teachers’ awareness of the application of mathematics to the real world and the issues it raises, both for the teachers and for ourselves.Suela Kacerja
Suela Kacerja has a postdoc position in mathematics education at the Western Norway University of Applied Sciences. She has a background as mathematics teacher educator from Albania and Norway. Her research interests are: developing critical mathematics education possibilities in teacher education and in schools, in intersection with reallife contexts in learning mathematics, as well as preservice teachers’ reflections about their own practice.Toril Eskeland Rangnes
Toril Eskeland Rangnes is associate professor in mathematics education. She works at the Department of Teacher Education Study in Mathematics, Western Norway University of Applied Sciences. Rangnes has a background as primary school teacher, textbook author and editor for Tangenten. Her main research interests are critical mathematics education, teacher professional development and language diversity in mathematics classrooms.Rune Herheim
Rune Herheim is associate professor at Western Norway University of Applied Sciences. His research focuses on connections between communication qualities and learning in mathematics with a particular focus on argumentation and agency in reallife contexts and when students use digital learning tools. Herheim is the Editor in chief for Tangenten, a Norwegian journal on mathematics teaching.Meinrad Pohl
Meinrad Pohl is associate professor in history. He works at the Department of Social Science, Western Norway University of Applied Sciences, Bergen. His main research interests are early modern economic theory and economic policy, trade history and mining history.Inger Elin Lilland
Inger Elin Lilland is associate professor at the Western Norway University of Applied Sciences, where she works at the Department of Teacher Education Study in Mathematics. She has previous experience as mathematics teacher at the upper secondary school level. Her main research interests are critical mathematics education and mathematics teacher professional development.Ragnhild Hansen
Ragnhild Hansen is associate professor at the Department of Teacher Education Study in Mathematics at Western Norway University of Applied Sciences (HVL). She received her master and PhD degrees from the University of Bergen within applied mathematics. Hansen has a background in as a researcher in different modelling projects. Her main research interests are critical mathematics education and teacher professional development.Skapad: 20180104 kl. 00:00

NOMAD
Finnish entrylevel students’ views of teacher knowledge and the characteristics of a good mathematics teacher
Mervi A. Asikainen, Antti Viholainen, Mika Koponen and Pekka E. Hirvonen
Abstract
This paper reports a study of the views held by Finnish students at the start of their university studies concerning their understanding of the knowledge and characteristics of a good mathematics teacher. A total of 97 students following a basic university course responded to a questionnaire. The results showed that a knowledge of teaching mathematics was more often used to describe the good mathematics teacher than a knowledge of mathematics. According to the students’ views, mathematics teachers need to be able to take the level of understanding of individual students into account in their teaching. Good mathematics teachers were also considered to be skilled in explaining, simplifying and transforming mathematical contents for their students. A good mathematics teacher was often described by the respondents as a patient, clear, inspiring and consistent person. On the other hand, characteristics such as humorous, likeable, empathetic, or fair were seldom used in the students’ responses to describe a good mathematics teacher. Those respondents who planned to become teachers demonstrated a more learnercentred concept of a good mathematics teacher than did those who were aiming at some other subject specialist profession than that of teaching.Mervi A. Asikainen
Docent Mervi A. Asikainen is a senior lecturer at the UEF Department of Physics and Mathematics. Asikainen directs the UEF physics and mathematics education research group. Her current field of interest include teacher knowledge of mathematics and physics teachers, teaching and learning of physics in higher and secondary education, and researchbased development of STEM education.Antti Viholainen
Antti Viholainen is a senior lecturer in mathematics / mathematics education at University of Eastern Finland. His research areas are mathematical beliefs, mathematics teacher education, learning materials (textbooks etc.) in mathematics, and mathematical argumentation.Mika Koponen
Mika Koponen is a postdoctoral researcher at the University of Eastern Finland. He has used Mathematical Knowledge for Teaching (MKT) framework for evaluating and improving mathematics teacher education. In his dissertation study, he presented a novel approach for investigating teacher knowledge and its interconnections by making use of network analysis methods. His postdoctoral research continues from this work by focusing on how the components of teacher knowledge are interconnected.Pekka E. Hirvonen
Docent Pekka E. Hirvonen is a senior lecturer at the UEF Department of Physics and Mathematics. Hirvonen has published more than 30 peerreviewed articles in international journals, books, and proceedings.Skapad: 20180104 kl. 00:00

NOMAD
A study of students’ concept images of inverse functions in Ireland and Sweden
Sinéad Breen, Niclas Larson, Ann O’Shea and Kerstin Pettersson
Abstract
In this paper we focus on firstyear university students’ conceptions of inverse function. We present results from two projects, conducted in Ireland and Sweden respectively. In both countries, data were collected through questionnaires, as well as through student interviews in Sweden. We draw on the notion of concept image and describe the components of students’ evoked concept images. The students’ responses involved e.g. ”reflection”, ”reverse”, and concrete ”examples”, while just a few students gave explanations relating to the definition of inverse functions. We found that the conceptions of inverses as reflections and reverse processes are important and relatively independent of local factors, and the data seemed to suggest that a ”reverse” conception is linked to an appreciation of injectivity more than a ”reflection” conception.Sinéad Breen
Sinéad Breen holds a PhD in Mathematics (on Asymptotic Analysis) from Dublin City University and has recently returned there as an Assistant Professor in the School of Mathematical Sciences. She conducts research in mathematics education, her main interest being in the teaching and learning of mathematics at undergraduate level.Niclas Larson
Niclas Larson is an associate professor at the Department of Mathematical Sciences, University of Agder, Kristiansand, Norway. His research interest lies in the teaching and learning of mathematics at secondary or university level. Current projects, both comparative, deal with students’ understanding of proof by mathematical induction and student teachers’ explanations of solutions to linear equations respectively. His methodological and theoretical standpoints are varied and driven by current research questions.Ann O’Shea
Ann O’Shea is a Senior Lecturer in the Department of Mathematics and Statistics at the Maynooth University in Ireland. She received a PhD in Mathematics from the University of Notre Dame, Indiana in 1991. Currently her research interests lie in Mathematics Education, especially at undergraduate level.Kerstin Pettersson
Kerstin Pettersson is an associate professor at the Department of Mathematics and Science Education, Stockholm University, Sweden. Her research interests concern university students’ conceptions of threshold concepts. Current projects deal with students’ learning in small groups teaching and students’ understanding of proof by mathematical induction.Skapad: 20180104 kl. 00:00

NOMAD
Oral presentations as a tool for promoting metacognitive regulation in real analysis
Margrethe Naalsund and Joakim Skogholt
Abstract
Real Analysis is for many students their first proofbased mathematics course, and many find it challenging. This paper studies how oral presentations of mathematical problems for peers can contribute to students’ metacognitive reflections. The paper discusses several aspects tied to preparing for, and carrying out, oral presentations, that seem to spur important subcomponents of metacognitive regulation such as planning, monitoring, and evaluating. Thoughtful guidance from an expert encouraged the students to further monitor their cognition, and evaluate their arguments and cognitive processes when expressing their reasoning to their peers.Margrethe Naalsund
Margrethe Naalsund is associate professor in Mathematics Education. She works at Faculty of Science and Technology (Section for Learning and Teacher Education) at Norwegian University of Life Sciences (NMBU). Her main research interests are learning and teaching algebra at primary and secondary school, and learning and teaching real analysis at university level.Joakim Skogholt
Joakim Skogholt is PhDstudent in Mathematics. He works at Faculty of Science and Technology (Section for Applied Mathematics) at Norwegian University of Life Sciences (NMBU). His main research interests are applied linear algebra, and learning and teaching real analysis at university level.Skapad: 20180104 kl. 00:00

NOMAD
Mathematics lecturers’ views on the teaching of mathematical modelling
Stephanie TreffertThomas, Olov Viirman, Paul HernandezMartinez and Yuriy Rogovchenko
Abstract
The paper reports on the views and use of mathematical modelling (MM) in university mathematics courses in Norway from the perspective of lecturers. Our analysis includes a characterisation of MM views based on the modelling perspectives developed by Kaiser and Sriraman (2006). Through an online survey we aimed to identify the main perspectives held in higher education by mathematics lecturers and the underlying rationale for integrating (or not) MM in university courses. The results indicated that most respondents displayed a realistic perspective on MM when it came to their professional practice. There was a more varied response when it came to their views on MM in teaching. Regarding conditions influencing the use or nonuse of MM in teaching, these mainly concerned the mathematical content and the institutional practices.Stephanie TreffertThomas
Stephanie TreffertThomas is a lecturer at Loughborough University (UK) with experience of teaching mathematics at school level, tertiary (college) level and at university, mainly to engineering students. Her research interests are in university level mathematics teaching and learning using sociocultural educational theories. She has a particular interest in the mathematical teaching practices of lecturers, including the use of mathematical modelling in teaching.Olov Viirman
When the research reported on in this paper was conducted, Olov Viirman was a postdoctoral researcher within the MatRIC centre at the University of Agder, Norway. He has recently taken up a position as senior lecturer at the University of Gävle, Sweden. His research is in university mathematics education, mainly focusing on the discursive practices of lecturers and students, and on the teaching and learning of mathematics, for instance mathematical modelling, in other academic disciplines.Paul HernandezMartinez
Paul HernandezMartinez is a senior lecturer in the Department of Mathematics at Swinburne University of Technology, Australia, and a visiting fellow in the Mathematics Education Centre at Loughborough University, UK. His research is in postcompulsory Mathematics Education, where he uses sociocultural educational theories to investigate teachinglearning practices (e.g. Mathematical Modelling) that have the potential to develop in students rich mathematical meanings while at the same time create in them positive dispositions towards the subject.Yuriy Rogovchenko
Yuriy Rogovchenko is a Professor of Mathematics at the University of Agder, Kristiansand, Norway. His research interests include qualitative theory of ordinary, functional and impulsive differential equations, mathematical modelling, and mathematics education related to teaching and learning of differential equations and mathematical modelling at university level.Skapad: 20180104 kl. 00:00

NOMAD
Peer assessment of mathematical understanding using comparative judgement
Ian Jones and David Sirl
Abstract
It is relatively straightforward to assess procedural knowledge and difficult to assess conceptual understanding in mathematics. One reason is that conceptual understanding is better assessed using openended test questions that invite an unpredictable variety of responses that are difficult to mark. Recently a technique, called comparative judgement, has been developed that enables the reliable and valid scoring of openended tests. We applied this technique to the peer assessment of calculus on a firstyear mathematics module. We explored the reliability and criterion validity of the outcomes using psychometric methods and a survey of participants. We report evidence that the assessment activity was reliable and valid, and discuss the strengths and limitations, as well as the practical implications, of our findings.Ian Jones
Ian Jones obtained a PhD in Mathematics Education from the University of Warwick and is a Senior Lecturer in the Mathematics Education Centre at Loughborough University, UK. Prior to this he was a Royal Society Shuttleworth Education Research Fellow and taught in primary and secondary schools for ten years. His research interests are in school children’s learning of algebra and the assessment of procedural and conceptual understanding of mathematics.David Sirl
David Sirl is a Lecturer in the School of Mathematical Sciences at the University of Nottingham. He is enjoying spending some time working with education researchers to explore new ways of improving teaching and learning.Skapad: 20180104 kl. 00:00

NOMAD
Characterising undergraduate mathematics teaching across settings and countries: an analytical framework
Angeliki Mali and Georgia Petropoulou
Abstract
This paper explores the characteristics of teaching of a sample of university mathematics teachers in two countries, Greece and Great Britain, and in two settings, lectures and tutorials, seeking to identify a common ground for undergraduate mathematics teaching. Our observations of teaching and our sociocultural perspectives enabled us to develop a framework for a detailed description of the observed teaching. The description reveals categories of teaching actions, and the associated tools teachers use in selecting tasks for their students, providing comprehensive explanations, extending students’ mathematical thinking, or evaluating students’ mathematical meaning. The findings are across settings and countries in the direction of a profound understanding of undergraduate mathematics teaching.
Angeliki Mali
Angeliki Mali is a Postdoctoral Research Fellow in the School of Education at the University of Michigan. Prior to her fellowship, she was member of the Culture, Pedagogy and Identity group in the Mathematics Education Centre at Loughborough University, where she was awarded her PhD. She holds a BSc in Mathematics, and an MSc in Didactics and Methodology of Mathematics from the University of Athens in Greece. Her research focuses on university mathematics education. She has experience in teaching mathematics to students attending STEM programmes at university level.Georgia Petropoulou
Georgia Petropoulou is finishing her PhD in the Mathematics Department at the University of Athens, Greece. Her PhD is in mathematics education, focusing on university mathematics teaching for students’ learning needs. She has an MSc in Didactics and Methodology of Mathematics and a BSc in Mathematics, both awarded by the University of Athens. Her research interests are in university mathematics teaching practice and its development to meet students’ learning needs.Skapad: 20180104 kl. 00:00

NOMAD
Developing practice through research into university mathematics education
Simon Goodchild and Barbara Jaworski
Abstract
The paper provides a very brief outline review of research into some key issues that affect students’ performance in mathematics in higher education. Community of practice theory is used to frame and focus the discussion. Policies regarding the recruitment of students, institutional practices for grouping students and the cultures of teaching and learning mathematics are considered. The research reviewed provides a context for examining the contributions of the research reports included within this thematic issue of NOMAD. The reports address three themes: regular approaches adopted in teaching mathematics in higher education, innovative approaches to teaching and learning, with emphasis on student participation in the educational process, and the characteristics of mathematical knowledge students appropriate. The paper endorses calls for large scale studies, especially those which relate teaching approaches, both regular and innovative, to the qualities and characteristics of students’ learning. The absence of a single overarching theoretical framework that embraces all the studies is also perceived as an obstacle that interferes with scientific developments in the field of researching university mathematics education. However, the value of teachers researching their own practice and their students’ learning is argued to be crucial for developing knowledge ”in practice” and this underscores the value of the papers included in this issue of NOMAD, both for the authors and the inspiration of other higher education mathematics teachers who, it is hoped, will be inspired to engage in similar studies.Simon Goodchild
Simon Goodchild is Professor of Mathematics Education at the University of Agder, he is also leader of MatRIC, Centre for Research Innovation and Coordination of Mathematics Teaching. MatRIC is one of eight Norwegian centres for excellence in higher education. He has over two decades of experience of school classroom research and school mathematics teaching development. In his role leading MatRIC he is using and extending his experience of mathematics teaching development in the context of university mathematics education.Barbara Jaworski
Barbara Jaworski is Professor of Mathematics Education at Loughborough University and coordinates research, including a group of eight PhD research fellows, within MatRIC. She has held positions of Chair of the British Society for Research into Learning Mathematics; President of the Congress of European Researchers in Mathematics Education; and President of the International Group for the Psychology of Mathematics Education. She has been research mathematics teaching and teaching development for over three decades.Skapad: 20180104 kl. 00:00

NOMAD – 22(4), 2017
Volume 22, No 4, December 2017
Volume 22, No 4, December 2017
eNOMAD
Simon Goodchild and Barbara Jaworski
Developing practice through research into university mathematics educationAngeliki Mali and Georgia Petropoulou
Characterising undergraduate mathematics teaching across settings and countries: an analytical frameworkSuela Kacerja, Toril Eskeland Rangnes, Rune Herheim, Meinrad Pohl, Inger Elin Lilland and Ragnhild Hansen
Stimulating critical mathematical discussions in teacher education: use of indices such as the BMI as entry pointsMervi A. Asikainen, Antti Viholainen, Mika Koponen and Pekka E. Hirvonen
Finnish entrylevel students’ views of teacher knowledge and the characteristics of a good mathematics teacherSinéad Breen, Niclas Larson, Ann O’Shea and Kerstin Pettersson
A study of students’ concept images of inverse functions in Ireland and SwedenMargrethe Naalsund and Joakim Skogholt
Oral presentations as a tool for promoting metacognitive regulation in real analysisStephanie TreffertThomas, Olov Viirman, Paul HernandezMartinez and Yuriy Rogovchenko
Mathematics lecturers’ views on the teaching of mathematical modellingIan Jones and David Sirl
Peer assessment of mathematical understanding using comparative judgementBarbro Grevholm
BokanmälanInnehåll: JH
Skapad: 20180104 kl. 00:00

NOMAD
Analysing genomgång: a Swedish mathematics teaching lesson event
Paul Andrews and Niclas Larson
Abstract
In this paper, drawing on group interviews focused on Swedish upper secondary students’ perspectives on school mathematics, we analyse participants’ use of the noun genomgång. Loosely translated as a ”whole class event during which the teacher goes through something” and for which there is no English equivalent, the word was used by both interviewers and interviewees even when referring to di erent forms of whole class activity. Analyses identi ed four broad categories of genomgång based on their form and function. With respect to form, genomgångs were either transmissive or participative. With respect to function they were either instructional or problem solving.Paul Andrews
Paul Andrews is Professor of Mathematics Education at Stockholm University. His current research is focused on the development of foundational number sense in year one students in England and Sweden (a project funded by the Swedish Research Council); Cypriot, Norwegian and Swedish teacher education students’ understanding of linear equations; Norwegian and Swedish upper secondary students’ perspectives on the nature and purpose of school mathematics; and the extent to which PISA misreports Swedish students’ mathematical competence.Niclas Larson
Niclas Larson is an associate professor at the Department of Mathematical Sciences, University of Agder, Kristiansand, Norway. His research interest lies in the teaching and learning of mathematics at secondary or university level. Current projects deal with students’ understanding of proof by mathematical induction and the understanding of the concept of inverse function respectively. His methodological and theoretical standpoints are varied and driven by current research questions.Skapad: 20170921 kl. 01:00

NOMAD
The development of preservice teachers’ selfefficacy in teaching mathematics
Annette Hessen Bjerke
Abstract
Teacher efficacy has received much attention in the general field of educational research, but applications in mathematics teacher education are few. In order to deepen the understanding of the nature and development of selfefficacy in teaching mathematics (SETM) during teacher education, the study presented here followed over a period of two years preservice teachers (PSTs) preparing to teach primary school mathematics in Norway (grades 1–7, ages 6–13). Their developing SETM was investigated by means of an instrument designed to target the core activity of teaching mathematics: helping a generic child with mathematics tasks. A comparison of responses collected from 191 novice PSTs with those from the same cohort two years later (n = 103) shows a rise in SETM in the typical PST, and indicates the nature of the development of SETM during teacher education.Annette Hessen Bjerke
Annette Hessen Bjerke got her PhD degree in September 2017 and this article is a part of her thesis. She has worked as a teacher educator in mathematics at Oslo and Akershus University College since 2004, and is a textbook author in elementary school mathematics. Her research interest concerns how teacher education fosters future mathematics teachers.Skapad: 20170921 kl. 01:00

NOMAD
Research as praxis, en route theory/practice teacherresearcher collaboration: a selfstudy
Sharada Gade
Abstract
This paper relates to project related instructional interventions, conducted via teacherresearcher collaboration in a Grade Four mathematics classroom. Drawing upon cultural historical activity theory or CHAT perspectives, such conduct exemplies research as praxis. While CHAT perspectives argue for a theory/practice approach, enabling practitioners to act on their re exivity and address contradictions found in ongoing practice; research as praxis views practitioner re exivity as central to pursuing openly ideological work and practising in empirical inquiry what one preaches in theoretical formulations. Such pursuit led to our becoming stakeholders in each other’s professional practice and the conduct of interventions becoming the shared object of both teaching and research. In teacherresearcher collaboration realising expansive learning activity, it was possible to question modernist assumptions which view abstract theory as applicable to any concrete practice and take political action in dialectic with theory.Sharada Gade
Sharada Gade works at the intersection of three domains – culturalhistorical activity theory or CHAT perspectives, practitioner inquiry and mathematics education. After more than a decade of teaching at middle school grades in India and doctoral work at the University of Agder, Norway; Sharada has held postdoctoral fellowships at Homi Bhabha Centre for Science Education, Mumbai; Umeå Mathematics Education Research Centre, Sweden; The Graduate Centre, City University of New York and the Department of Education, University of Oxford.Skapad: 20170920 kl. 01:00

NOMAD
A review of the impact of formative assessment on student achievement in mathematics
Torulf Palm, Catarina Andersson, Erika Boström and Charlotta Vingsle
Abstract
Research reviews show that formative assessment has great potential for raising student achievement in general, but there is a need for reviews of formative assessment in individual subjects. This review examines its impact on student achievement in mathematics through an assessment of scientific journal articles published between 2005 and 2014 and indexed in Web of science. Through the use of search terms such as ”formative assessment”, ”assessment for learning” and ”selfregulated learning”, different approaches to formative assessment were included in the review. While varying in approach, they all share the defining characteristic of formative assessment: agents in the classroom collect evidence of student learning and, based on this information, adjust their teaching and/or learning. The results show positive relations between student achievement in mathematics and the ways of doing formative assessment included in the review.Torulf Palm
Torulf Palm is associate professor in pedagogical work and a member of Umeå Mathematics Education Research Centre (UMERC). He works at the Department of Science and Mathematics Education, Umeå University. His main research interests are formative assessment, teacher professional development and mathematics education.Catarina Andersson
Catarina Andersson is assistant professor in pedagogical work and a member of Umeå Mathematics Education Research Centre (UMERC). She works at the Department of Science and Mathematics Education, Umeå University. Catarina has a background as a primary teacher and special education teacher. Her main research interests are formative assessment, teacher professional development, special education and mathematics education.Erika Boström
Erika Boström is a PhD student in Mathematics Education and a member of Umeå Mathematics Education Research Centre (UMERC). She works at the Department of Science and Mathematics Education, Umeå University. Erika has a background as a teacher in mathematics and biology and has also worked with developing Swedish national tests in mathematics. Her research interests concern formative assessment, teacher professional development and mathematics education.Charlotta Vingsle
Charlotta Vingsle is a PhD student in pedagogical work and a member of Umeå Mathematics Education Research Centre (UMERC). She works at the Department of Science and Mathematics Education, Umeå University. Charlotta has a background as a primary teacher. Her research interests concern formative assessment, selfregulated learning and mathematics education.Skapad: 20170920 kl. 01:00

NOMAD – 22(3), 2017
Volume 22, No 3, September 2017
eNOMAD
Sharada Gade
Research as praxis, en route theory/practice teacherresearcher collaboration: a selfstudyTorulf Palm, Catarina Andersson, Erika Boström and Charlotta Vingsle
A review of the impact of formative assessment on student achievement in mathematicsAndreas Eckert
Theorizing the interactive nature of teaching mathematics: contributing to develop contributions as a metaphor for teachingAnnette Hessen Bjerke
The development of preservice teachers’ selfefficacy in teaching mathematicsPaul Andrews and Niclas Larson
Analysing genomgång: a Swedish mathematics teaching lesson eventInnehåll: JH
Skapad: 20170920 kl. 01:00

NOMAD
Theorizing the interactive nature of teaching mathematics: contributing to develop contributions as a metaphor for teaching
Andreas Eckert
Abstract
The teachers’ role in teacherstudent interaction in mathematics has received increased attention in recent years. One metaphor used to describe teaching in teacherstudent interaction is to describe teaching as a learning process itself, in terms of learning to develop learning. The aim of the present study is to contribute to the conceptualization and understanding of this view of teaching mathematics. This is done by introducing and elaborating on a new conceptual framework, describing teaching as Contributing to Develop Contributions (CDC). The CDC framework is constructed by combining the theory of symbolic interactionism with a complementing metaphor for learning; learning as contribution. The CDCframework is illustrated in the context of experimentationbased, interactive teaching of probability. The analysis shows how the CDCframework helps in coming to understand how teachers develop their own contributions to manipulate the negotiation of meaning of mathematics in the classroom and thereby also develops the students’ contributions. In the presented case we can see how CDC particularly helps in giving account of how a teacher develops her way of using symbols and indications and adjust her own interpretations during a whole class discussion where the teacher and students interpret the empirical results of a random generator. In addition, the analysis also illustrates how the framework draws our attention to how a teacher can contribute to the negation of meaning, and so, to students’ opportunities to learn, by making her own interpretations and ways of ascribing meaning to objects transparent to the students in the interaction.Andreas Eckert
Andreas Eckert is a doctoral student in mathematics education at Linnaeus University, Växjö, Sweden. His research interests include teacherstudent interaction in the mathematics classroom and teachers’ inpractice professional development.Skapad: 20170920 kl. 01:00

NOMAD
A tool for understanding pupils’ mathematical thinking
Hanna Viitala
Abstract
This article provides a tool for studying pupils’ mathematical thinking. Mathematical thinking is seen as a cognitive function that is highly influenced by affect and metalevel of mind. The situational problem solving behaviour is studied together with metacognition and affect which together with pupils’ view of mathematics form a dynamic construct that reveals pupils’ mathematical thinking. The case of Daniel is introduced to illustrate the dynamic nature of the framework.Hanna Viitala
Hanna Viitala is a PhD student at the University of Agder, Norway, and a mathematics teacher in a secondary school in Finland. She is interested in pupils’ mathematical thinking, problem solving, metacognition, affect, and mathematics learning.Skapad: 20170530 kl. 01:00

NOMAD – 22(2), 2017
Volume 22, No 2, March 2017
eNOMAD
Hanna Viitala
A tool for understanding pupils’ mathematical thinkingJöran Petersson
First and second language students’ achievement in mathematical content areasReidar Mosvold
Studier av undervisningskunnskap i matematikk: internasjonale trender og nordiske bidragHeidi Strømskag
Et miljø for algebraisk generalisering og dets innvirkning på studenters matematiske aktivitetHelena Johansson
Dependence between creative and noncreative mathematical reasoning in national physics testsInnehåll: JH
Skapad: 20170530 kl. 01:00

NOMAD
First and second language students’ achievement in mathematical content areas
Jöran Petersson
Abstract
This study compares Swedish first (n = 2 253) and second language (n = 248) students’ achievement in mathematical content areas specified by the TIMSSframework. Data on mathematics achievement from three national tests 2007–2009 in school year 9 are used. The present study found that the achievement difference between the mathematical content areas algebra and number was smaller for second language students than for first language students and this result holds with statistical significance (p = 0.016). The same holds for algebra versus data and chance (p = 0.00053). A hypothesis for further research is suggested; that students immigrating in late school years have contributed to the observed result by bringing experiences from other curricula into their new schooling.Jöran Petersson
Jöran Petersson is lecturer at Stockholm University and has a PhD in mathematics education. Jöran’s research is in the intersection of students having Swedish as a second language and students’ use of mathematical concepts. Moreover, he is interested in statistics education. He also has a master (licentiate) in mathematical systems theory and optimization from the Royal Institute of Technology, Stockholm and a diploma as upper secondary school mathematics and physics teacher from Linköping University.Skapad: 20170530 kl. 01:00

NOMAD
Studier av undervisningskunnskap i matematikk: internasjonale trender og nordiske bidrag
Reidar Mosvold
Sammandrag
De siste tiårene har forskere vist stadig mer interesse for den matematiske kunnskapen som er spesifikt knyttet til matematikkundervisningen. I denne artikkelen diskuteres nordiske bidrag til forskningen på dette feltet i lys av internasjonale trender. Diskusjonene bygger på resultater fra en litteraturstudie av 190 empiriske artikler som ble publisert i perioden 2006–2013. I tillegg trekkes her inn nordiske artikler etter 2013. Noen av disse studiene fokuserte på kunnskapens innhold og natur, andre fokuserte på hvordan denne kunnskapen kan utvikles, mens en tredje gruppe studier undersøkte hvordan lærerkunnskapen påvirker elevenes resultater og kvaliteten på undervisningen. Videre nordisk forskning på feltet kan blant annet bidra til styrking av teori og praksistilknytning.Abstract
In recent decades, researchers have shown an increasing interest in the mathematical knowledge that is specific to the work of teaching mathematics. In this article, Nordic contributions to this field are discussed in light of international research trends. The discussions draw upon results from a literature review of 190 empirical articles that were published in 2006–2013. In addition, Nordic studies that have been published after this are included in the discussion. Some of the studies focus on the nature and composition of this knowledge, other studies focus on the development of this knowledge, whereas a third group of studies focus on how teachers’ knowledge contributes to student learning and the quality of instruction. Further Nordic research in this field might contribute to strengthening theoretical perspectives and connections to practice.Reidar Mosvold
Reidar Mosvold er professor i matematikkdidaktikk ved Universitetet i Stavanger, Norge. Hans forskningsinteresser omfatter læreres undervisningskunnskap i matematikk, læreres oppfatninger, læreridentitet og diskursive perspektiver, samt bruk av matematikkens historie i undervisningen.Skapad: 20170530 kl. 01:00

NOMAD
Dependence between creative and noncreative mathematical reasoning in national physics tests
Helena Johansson
Abstract
It is known from previous studies that a focus on rote learning and procedural mathematical reasoning hamper students’ learning of mathematics. Since mathematics is an integral part of physics, it is assumed that mathematical reasoning also influences students’ success in physics. This paper aims to study how students’ ability to reason mathematically affects their success on different kinds of physics tasks. A descriptive statistical approach is adopted, which compares the ratio between conditional and unconditional probability to solve physics tasks requiring different kinds of mathematical reasoning. Tasks from eight Swedish national physics tests for upper secondary school, serve as a basis for the analysis. The result shows that if students succeed on tasks requiring creative mathematical reasoning, the probability to solve the other tasks on the same test increases. This increase is higher than if the students succeed on tasks not requiring creative mathematical reasoning. This result suggests that if students can reason mathematically creatively, they have the ability to use their knowledge in other novel situations and thus become more successful on tests.Helena Johansson
Helena Johansson has a PhD in Mathematics, specialising in Educational Sciences and is a postdoc at Mid Sweden University. Her research interests concern students’ mathematical reasoning and how this competence influences students’ learning in mathematics and in physics; and how natural language influences students’ learning of the symbolic language of mathematics.Skapad: 20170530 kl. 01:00

NOMAD
Et miljø for algebraisk generalisering og dets innvirkning på studenters matematiske aktivitet
Heidi Strømskag
Sammandrag
Denne artikkelen handler om hvordan egenskaper ved en didaktisk situasjon i matematikk påvirker studenters muligheter til å løse en algebraisk generaliseringsoppgave. Studien er gjennomført innenfor et lærerutdanningsprogram ved en høgskole, og datamaterialet består av matematikkoppgaven og et videoopptak av tre studenters samarbeid for å løse oppgaven. Transkripsjonen av videoopptaket er analysert ved den konstant komparative metoden, der teorien for didaktiske situasjoner i matematikk (TDS) er brukt for å forstå hvilke egenskaper ved den didaktiske situasjonen som begrenser studentenes muligheter for å løse oppgaven. Den observerte didaktiske situasjonen er en ordinær undervisningssituasjon i den forstand at den ikke er et resultat av didaktisk ingeniørvirksomhet basert på TDS. Resultatene fra analysen viser hvordan to faktorer skaper avstand mellom lærerens hensikt med den gitte matematikkoppgaven og studentenes aktivitet knyttet til oppgaven. Den ene faktoren handler om begrepet ”matematisk setning” som studentene tillegger en annen betydning enn den læreren legger til grunn; den andre faktoren handler om lærerens bruk av et generisk eksempel uten at de generelle egenskapene til eksemplet blir diskutert. Studien bidrar til innsikt i sammenhengen mellom et miljø for en adidaktisk situasjon og den matematikkunnskapen som studenter har mulighet for å utvikle i det aktuelle miljøet.Abstract
This article is about how features of a didactical situation in mathematics at a university college a ect students’ opportunity to solve an algebraic generalisation task. The study is conducted within a teacher education programme for primary and lower secondary education. The empirical material contains the mathematical task and a video recorded episode of three students’ collaborative engagement with the task. The transcription of the episode is analysed by the constant comparative method, where the theory of didactical situations in mathematics (TDS) is used to conceptualise what features of the didactical situation that constrain the students’ opportunity to solve the task. The observed didactical situation is a regular teaching situation in the sense that it is not a result of didactic engineering based on TDS. Analysis of the data shows how two factors create a gap between the teacher’s intention with the mathematical task and the students’ engagement with the task. The first factor is about the concept of ”mathematical sentence” of which the students have a different conception than intended by the teacher. The second factor is about the teacher’s use of a generic example without a discussion of its general properties. The study provides insight into the relationship between a milieu for an adidactical situation and the mathematical knowledge that the milieu enables the students to develop.Heidi Strømskag
Heidi Strømskag er førsteamanuensis i matematikkdidaktikk ved Institutt for matematiske fag, Norges teknisknaturvitenskapelige universitet. Hennes forskningsinteresser omfatter undervisning og læring av algebra, oppgavedesign i matematikk, og det didaktiske forholdet mellom læreren, studentene og spesielle deler av matematisk kunnskap, i undervisningssituasjoner der intensjonen er at studentene skal lære denne kunnskapen.Skapad: 20170530 kl. 01:00

NOMAD – 22(1), 2017
Volume 22, No 1, March 2017
eNOMAD
Catarina Andersson, Erika Boström and Torulf Palm
Formative assessment in Swedish mathematics classroom practiceAttila Szabo
Matematikundervisning för begåvade elever – en forskningsöversiktMary G. Billington and Egil Gabrielsen
The older the better? Are younger Norwegian adults losing ground on basic numeracy skills?Anna Pansell and Paul Andrews
The teaching of mathematical problemsolving in Swedish classrooms: a case study of one grade five teacher’s practiceEvaLena Erixon
Convergences and influences of discourses in an online professional development course for mathematics teachersInnehåll: JH
Skapad: 20170315 kl. 00:00

NOMAD
Formative assessment in Swedish mathematics classroom practice
Catarina Andersson, Erika Boström and Torulf Palm
Abstract
Research shows that substantial learning gains are possible through the use of formation assessment. However, little is known about Swedish mathematics teachers’ use of formative assessment, and thus about the possible value of professional development programmes. This study uses teacher interviews and classroom observations to examine the classroom practice of 38 randomly selected primary and secondary school teachers in a midsized Swedish municipality. A framework of formative assessment comprising one big idea and five Key strategies structured the analysis. The study identifies characteristics of current formative assessment practices. The results show that the teachers do use a variety of formative assessment activities, but also that there is much room for development towards a more effective formative classroom practice.Catarina Andersson
Catarina Andersson has a PhD in pedagogical work and is a member of Umeå Mathematics Education Research Center (UMERC). She works at the Department of Science and Mathematics Education, Umeå University. Catarina has a background as a primary teacher and special education teacher. Her main research interests are formative assessment, teacher professional development, special education and mathematics education.Erika Boström
Erika Boström is a PhD student in Mathematics Education and a member of Umeå Mathematics Education Research Centre (UMERC). She works at the Department of Science and Mathematics Education, Umeå University. Erika has a background as a teacher in mathematics and biology and has also worked with developing Swedish national tests in mathematics. Her main research interests are formative assessment, teacher professional development and mathematics education.Torulf Palm
Torulf Palm is associate professor in pedagogical work and a member of Umeå Mathematics Education Research Centre (UMERC). He works at the Department of Science and Mathematics Education, Umeå University. His main research interests are formative assessment, teacher professional development and mathematics education.Skapad: 20170315 kl. 00:00

NOMAD
Matematikundervisning för begåvade elever – en forskningsöversikt
Attila Szabo
Sammanfattning
Artikeln redovisar de huvudsakliga pedagogiska och organisatoriska metoder relaterade till begåvade elevers matematikundervisning som fokuseras i forskningslitteraturen – även könsskillnader, motivation och matematiskt begåvade elevers sociala situation i klassrummet diskuteras. Översikten visar att det finns åtgärder – t ex frivillig acceleration i ämnet där undervisningen är anpassad till elevens förkunskaper och kapacitet eller arbete med utmanande uppgifter i prestationshomogena grupper – som antas ha goda effekter på begåvade elevers kunskapsutveckling i matematik. Analysen visar också att det kan uppfattas som problematiskt att vara begåvad i matematik samt att begåvade flickor upplever vissa aspekter av matematikundervisningen annorlunda jämfört med motsvarande grupp pojkar.Abstract
The present article offers an overview of those main methodological and pedagogical approaches associated with gifted pupils’ education in mathematics which are focused in the research literature. Furthermore, the article discusses gender differences, motivation and some central aspects of mathematically gifted pupils’ social situation in the classroom. The analysis shows that there are some pedagogical and organizational approaches, e.g. voluntary acceleration where the teaching is adapted to the knowledge and the capacity of the participants or working with challenging mathematical problems in performancehomogenous groups, which may have good effects on gifted pupils’ mathematical achievement. The overview also indicates that mathematically gifted adolescents are facing difficulties in their social interaction and that gifted female and male pupils are experiencing certain aspects of their mathematics education differently.Attila Szabo
Attila Szabo är fil. lic. i matematikämnets didaktik och doktorand vid Stockholms Universitet. Hans forskningsintressen rör den matematiska förmågans struktur och det matematiska minnets uttryckssätt vid problemlösning hos högpresterande elever.Skapad: 20170315 kl. 00:00

NOMAD
The older the better? Are younger Norwegian adults losing ground on basic numeracy skills?
Mary G. Billington and Egil Gabrielsen
Abstract
Results from the OECD survey of adult skills, 2012 brought good tidings for Norway. The average numeracy score for the Norwegian adult population lies well over the OECD average. However, a closer look at the age skill profile shows that while older Norwegians score well over the OECD average for their age group, younger Norwegians score around the OECD average. Comparing these results to results from an earlier study of adult skills, conducted in 2003, suggests a downward trend in numeracy proficiency for the younger generation. We discuss recent school reforms as a possible cohort effect influencing this trend.Mary G. Billington
Mary G. Billington is working as a researcher at National Reading Centre, University of Stavanger and at the International Research Institute of Stavanger. Her current research interests are in adult numeracy and workplace learning. Billington has a PhD. in Mathematics Education.Egil Gabrielsen
Egil Gabrielsen (PhD) is docent at the National Reading Centre, University of Stavanger. He was national study manager for both the International Adult Literacy Study, 1998 (IALS) and the Adult Literacy and Life Skills survey, 2003 (ALL). Gabrielsen is involved in three studies based on data from the Program for International Assessment of Adult Competencies, 2012 (PIAAC).Skapad: 20170315 kl. 00:00

NOMAD
The teaching of mathematical problemsolving in Swedish classrooms: a case study of one grade five teacher’s practice
Anna Pansell and Paul Andrews
Abstract
In this paper we examine the teaching of mathematical problemsolving to grade five students of one wellregarded and experienced Swedish teacher, whom we call Mary. Working within a decentralised curriculum in which problemsolving is centrally placed, Mary is offered little systemic support in her professional decision making with respect to problemsolving instruction. Drawing on Lester’s and Schroeder’s descriptions of teaching for, about and through problemsolving, we draw on multiple sources of data, derived from interviews and videotaped lessons, to examine how Mary’s problemsolvingrelated teaching is constituted in relation to the weakly framed curriculum and the unregulated textbooks that on which she draws. The analyses indicate that Mary’s emphases are on teaching for and about problemsolving rather than through, although the ambiguities that can be identified throughout her practice with respect to goals, curricular aims and the means of their achievement can also be identified in the curricular documents from which she draws.Anna Pansell
Anna Pansell is a PhD student in Mathematics Education at Stockholm University. Her research interest is about understanding mathematics teachers and their teaching practice in relation to the classroom’s broader context. The PhD project is focused on one mathematics teacher in grade five and her decisionmaking in relation to the institutionell context within which she works.Paul Andrews
Paul Andrews is Professor of Mathematics Education at Stockholm University. His current research is focused on the development of foundational number sense in year one students in England and Sweden (a project funded by the Swedish Research Council), Swedish and Cypriot teacher education students’ understanding of linear equations, Norwegian and Swedish upper secondary students’ beliefs about the purpose of school mathematics, and examining the extent to which PISA misreports Swedish students’ mathematical competence.Skapad: 20170315 kl. 00:00

NOMAD
Convergences and influences of discourses in an online professional development course for mathematics teachers
EvaLena Erixon
Abstract
Despite the everincreasing number of online professional development (OPD) courses, few studies have examined online education for mathematics teachers. This article reports on a case study of discourses in an OPD course for mathematics teachers concerning the convergence and influence of discourses in course seminar discussions and in mathematics teaching in school when course participants are given the task of translating their insights into actual teaching, with a focus on the participants’ discussions of their own and one another’s videorecorded lessons. The analysis shows that there is a convergence of discourses in the seminars and in the school context related to a focus on concepts and everyday life connections. However, the study also suggests that there is a risk of students remaining outside in an ”everyday discourse”, in which knowledge of mathematics might be useful, but mathematics is discussed in imprecise and simplified terms.EvaLena Erixon
EvaLena Erixon works as a lecturer in mathematics education at Dalarna University. She is also a doctoral student at Örebro University with the professional development of mathematics teachers as research interest.Skapad: 20170315 kl. 00:00

NOMAD
Measuring temperature within the didaktic space of preschool
Ola Helenius, Maria L. Johansson, Troels Lange, Tamsin Meaney and Anna Wernberg
Abstract
The informal character of preschool mathematics, engaged in during children’s play, places complex requirements on preschool teachers. It also leads to challenges in developing appropriate analytical tools for researching teacher work. In this paper a framework, the ”didaktic space”, is described and used to analyse interactions between preschool teachers and children in relationship to mathematical learning situations. An interaction between a preschool teacher and a group of children about how to compare their temperatures is analysed, using this framework. The analysis focuses on how the teacher’s contributions as well as those of the children changed as the role of the mathematics changed. The paper discusses how the didaktic space offers a nuanced understanding of preschool mathematical situations, both to researchers and to teachers.
Ola Helenius
Ola Helenius has PhD in mathematics and is a researcher and deputy director at the National Center for Mathematics Education, University of Gothenburg. Some of his current research interests include neuroscience perspectives on learning mathematics and mathematics curriculum design research. Another interest is sociocultural perspectives on preschool mathematics and he is a member of the research group Young children’s mathematics.
Maria L. Johansson
Maria L. Johansson has a PhD in mathematics and one in mathematics education. Her PhD in mathematics education focused on issues related to mathematics in preschool, particularly to professional development of preschool teachers and childcare workers. She is an associate professor at Luleå University where she works in teacher education. Her research interests are wide ranging, covering both mathematics and mathematics education for a range of ages. She has also published extensively with the research group, Young children’s mathematics.
Troels Lange
Troels Lange has been a mathematics teacher educator for twenty years, first in Denmark, then Australia and Sweden. He currently works at Bergen University College in Norway and in 2016 became Professor in mathematics education. His research interests centre on how children experience learning mathematics, but he has published on a range of different issues regarding mathematics education. Since 2011, he is part of and publishes with the research group Young children’s mathematics.
Tamsin Meaney
Tamsin Meaney is professor of mathematics education at Bergen University College, having previously been professor at Malmö University. She has worked in teacher education in New Zealand, Australia, Sweden and Norway. Her research interests are varied but centre around the need for mathematics education to support social justice concerns. Whilst in Sweden she started the research group, Young children’s mathematics, with researchers from Norway, Sweden and USA.
Anna Wernberg
Anna Wernberg is associate professor in mathematics teacher education in the Faculty of Education and Society, Malmö University. She has previously worked at Kristianstad University and Borås University, also in Sweden. Her research interests focus on the use of Learning studies in mathematics classrooms for young children. She has also published extensively with other members of the research group Young children’s mathematics.
Skapad: 20161212 kl. 00:00

NOMAD
What characterises mathematical conversations in a Norwegian kindergarten?
Trude Fosse
Abstract
This paper presents a study of young children’s mathematical conversations in a Norwegian kindergarten and provides examples to illustrate some of their characteristics. Using points made in previous research about conversations and mathematics, an analysis of an interaction involving a group of children who are putting a toy together exemplifies what can and what cannot be considered a mathematical conversation. For a conversation to be considered mathematical, it is suggested that it must include not only references to mathematics, but also specific structural elements in the conversation, as well as valuing participants’ contributions and encouraging participants to reflect on the mathematical theme, thereby promoting learning.
Trude Fosse
Trude Fosse is assistant professor at the University of Bergen. Her research interests concern beginners education in mathematics, with a particular interest in mathematical conversations and how mathematics is handled in kindergartens and primary schools. Another interest concerns how the mathematics in teacher education programs can support students’ learning and teaching practices.
Skapad: 20161212 kl. 00:00