
NOMAD 23(34), 2018
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 23(34), 2018
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 23(2), 2018
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 23(2), 2018
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 23(2), 2018
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
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Jens Højgaard Jensen and Uffe Thomas Jankvist
Disciplinary competence descriptions for external use
[PDF]Jan Roksvold
Læreres utbytte av kunnskap om hjernen
[PDF]Anita Tyskerud and Reidar Mosvold
Scrutinizing teacherlearner interactions on volume
[PDF]Skapad: 20180822 kl. 15:48

NOMAD 23(1), 2018
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 23(1), 2018
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
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Anna Ida Säfström
Preschoolers exercising mathematical competencies
[PDF]Magnus Fahlström and Lovisa Sumpter
A model for the role of the physical environment in mathematics education
[PDF]Kristina Palm Kaplan
Discourses in school algebra: the textbooks’ different views on algebra and the positioning of students
[PDF]Jani Hannula
The gap between school mathematics and university mathematics: prospective mathematics teachers’ conceptions and mathematical thinking
[PDF]Skapad: 20180320 kl. 12:27

NOMAD 23(1), 2018
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 23(1), 2018
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 22(4), 2017
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
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 22(4), 2017
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 22(4), 2017
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 22(4), 2017
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 22(4), 2017
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 – 22(4), 2017
Volume 22, No 4, December 2017
Volume 22, No 4, December 2017
eNOMAD
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Simon Goodchild and Barbara Jaworski
Developing practice through research into university mathematics education
Angeliki Mali and Georgia Petropoulou
Suela Kacerja, Toril Eskeland Rangnes, Rune Herheim, Meinrad Pohl, Inger Elin Lilland and Ragnhild Hansen
Mervi A. Asikainen, Antti Viholainen, Mika Koponen and Pekka E. Hirvonen
Sinéad Breen, Niclas Larson, Ann O’Shea and Kerstin Pettersson
A study of students’ concept images of inverse functions in Ireland and Sweden
Margrethe Naalsund and Joakim Skogholt
Oral presentations as a tool for promoting metacognitive regulation in real analysis
Stephanie TreffertThomas, Olov Viirman, Paul HernandezMartinez and Yuriy Rogovchenko
Mathematics lecturers’ views on the teaching of mathematical modelling
Ian Jones and David Sirl
Peer assessment of mathematical understanding using comparative judgement
Barbro Grevholm
Innehåll: JH
Skapad: 20180104 kl. 00:00

NOMAD 22(4), 2017
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 22(4), 2017
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 22(3), 2017
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 22(3), 2017
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 22(3), 2017
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 – 22(3), 2017
Volume 22, No 3, September 2017
eNOMAD
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Sharada Gade
Research as praxis, en route theory/practice teacherresearcher collaboration: a selfstudy
Torulf Palm, Catarina Andersson, Erika Boström and Charlotta Vingsle
A review of the impact of formative assessment on student achievement in mathematics
Andreas Eckert
Annette Hessen Bjerke
The development of preservice teachers’ selfefficacy in teaching mathematics
Paul Andrews and Niclas Larson
Analysing genomgång: a Swedish mathematics teaching lesson event
Innehåll: JH
Skapad: 20170920 kl. 01:00

NOMAD 22(3), 2017
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
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 22(2), 2017
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
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 – 22(2), 2017
Volume 22, No 2, March 2017
eNOMAD
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Hanna Viitala
A tool for understanding pupils’ mathematical thinking
Jöran Petersson
First and second language students’ achievement in mathematical content areas
Reidar Mosvold
Studier av undervisningskunnskap i matematikk: internasjonale trender og nordiske bidrag
Heidi Strømskag
Et miljø for algebraisk generalisering og dets innvirkning på studenters matematiske aktivitet
Helena Johansson
Dependence between creative and noncreative mathematical reasoning in national physics tests
Innehåll: JH
Skapad: 20170530 kl. 01:00

NOMAD 22(2), 2017
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 22(2), 2017
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(2), 2017
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 – 22(1), 2017
Volume 22, No 1, March 2017
eNOMAD
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Catarina Andersson, Erika Boström and Torulf Palm
Formative assessment in Swedish mathematics classroom practice
Attila Szabo
Matematikundervisning för begåvade elever – en forskningsöversikt
Mary G. Billington and Egil Gabrielsen
The older the better? Are younger Norwegian adults losing ground on basic numeracy skills?
Anna Pansell and Paul Andrews
EvaLena Erixon
Innehåll: JH
Skapad: 20170315 kl. 00:00

NOMAD 22(1), 2017
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 22(1), 2017
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 22(1), 2017
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 22(1), 2017
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 22(1), 2017
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 21(4), 2016
Locating learning of toddlers in the individual/society and mind/body divides
Tamsin Meaney
Abstract
In this paper, theories about learning are discussed in relationship to toddlers locating themselves in space. This is important given the current debate about the mathematics that very young children can or should learn. Regardless of whether learning is seen as an individual or a social activity, learning theories of Piaget and Vygotsky emphasise the cognitive nature of learning and the need for linguistic reflektion on that learning. In order not to situate toddlers as deficient or irrational because they do not express their reflections linguistically, I suggest that learning should be considered as problem solving that occurs both with the mind and the body. Using examples of toddlers engaging in spatial explorations, I illustrate how learning mathematics can be reconceptualised in this way.
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.
Skapad: 20161212 kl. 00:00

NOMAD 21(4), 2016
Professional development in early mathematics: effects of an intervention based on learning trajectories on teachers’ practices
Julie Sarama, Douglas H. Clements, Christopher B. Wolfe and Mary Elaine Spitler
Abstract
We evaluated the effects of a researchbased model for scaling up educational interventions on teachers’ practices in preschool mathematics. The original participants were from 106 classrooms for 4yearolds in two distal city districts serving lowresource communities, with 42 schools randomly assigned to one of three groups, of which the two treatment groups were the same throughout preschool (thus, there were 72 treatment classrooms). The intervention, a professional development program based on young children’s mathematical learning trajectories, had a substantial positive effect on teachers’ instructional practices, some of which mediated student outcomes. Teachers also demonstrated sustained levels of fidelity as long as six years after the end of the intervention. Notable is these teachers’ encouragement and support for discussions of mathematics and their use of formative assessment. Finally, teachers taught the curriculum with increasing fidelity over the following six years without support from the project.
Julie Sarama
Julie Sarama is Kennedy Endowed Chair in Innovative Learning Technologies, CoExecutive Director of the Marsico Institute, and Professor at the Morgridge College of Education, University of Denver. Her research interests include young children’s development of mathematical concepts and competencies, implementation and scaleup of educational reform, professional development models and their influence on student learning, and implementation and effects of software environments.
Douglas H. Clements
Douglas H. Clements is Kennedy Endowed Chair in Early Childhood Learning; Executive Director, Marsico Institute; and Professor at the Morgridge College of Education, University of Denver. His research interests include the learning and teaching of early mathematics and computer applications in mathematics education, creating, using, and evaluating researchbased curricula and taking successful curricula to scale using technologies and learning trajectories.
Christopher B. Wolfe
Christopher B. Wolfe is an Assistant Professor of Psychology at Saint Leo University. His research interests include measurement, program evaluation, and early educational development.
Mary Elaine Spitler
Mary Elaine Spitler is an independent researcher with interests in early childhood education and professional development.
Skapad: 20161212 kl. 00:00

NOMAD – 21(4), 2016
Tidigare nummerPrevious issues
ArrayNummer/Issue
Volume 21, No 4, December 2016
Tamsin Meaney
Locating learning of toddlers in the individual/society and mind/body divides
Julie Sarama, Douglas H. Clements, Christopher B. Wolfe and Mary Elaine Spitler
Elin Reikerås
PerEinar Sæbbe and Reidar Mosvold
Jorryt van Bommel and Hanna Palmér
Young children exploring probability – with focus on their documentations
Camilla Björklund and Wolmet Barendregt
Teachers’ pedagogical mathematical awareness in diverse childagegroups
Trude Fosse
What characterises mathematical conversations in a Norwegian kindergarten?
Ola Helenius, Maria L. Johansson, Troels Lange, Tamsin Meaney and Anna Wernberg
Measuring temperature within the didaktic space of preschool
Hanna Palmér and Camilla Björklund
Different perspectives on possible – desirable – plausible mathematics learning in preschool
Innehåll: JH
Skapad: 20161212 kl. 00:00

NOMAD 21(4), 2016
Central skills in toddlers’ and preschoolers’ mathematical development, observed in play and everyday activities
Elin Reikerås
Abstract
In good mathematical development, it is important to master some central skills at the kindergarten age. Being aware of such skills for each child is useful for facilitating children’s learning and development. The present study examines how kindergarten children master central mathematical skills in toddler age and preschool age. The staff in kindergartens collected data by using structured observation on the basis of observational material The Mathematics, the Individual and the Environments (MIO). The areas examined were Mathematical language, Logical reasoning, Shape and space, Pattern and order, Counting and series of numbers and Enumeration. The children were observed in play and everyday activities in kindergarten in two threemonth periods when they were 2½ years of age (toddlers, n = 1003) and two year later when they were 4 ½ (preschoolers, n = 744). The results show a large dispersion in the children’s skills as toddlers, but as preschoolers, most of the children have a high level of mastery. The percentage of mastery at each of the observation times is reported and discussed in light of earlier findings. In our study, the children had slower development in the numerical area than was found in other research. Reflections about how the Norwegian kindergarten tradition and the method used to collect the data influence the results are included.
Elin Reikerås
Elin Reikerås is associate professor in special needs education at the National centre for reading education and research, University of Stavanger, Norway. Her research interests are on children’s early mathematical development, the relations between early mathematics and other development areas as language, motorlife skills and social skills as well as the relations between arithmetics and reading.
Skapad: 20161212 kl. 00:00

NOMAD 21(4), 2016
Initiating a conceptualization of the professional work of teaching mathematics in kindergarten in terms of discourse
PerEinar Sæbbe and Reidar Mosvold
Abstract
This study investigates a Norwegian kindergarten teacher’s work of teaching mathematics in an everyday activity involving play with LegoTM bricks. Analysis of the kinder garten teacher’s discourse identifies questioning and affirmation as two core components. We suggest that these are core discursive practices in the work of teaching mathematics in kindergarten. In identifying these practices, a theory of thinking as communicating is applied to investigate what makes these discursive acts of teaching mathematical. Further research is called for to continue investigating kindergarten teachers’ discursive practices in teaching and to continue the efforts to conceptualize the work of teaching mathematics in kindergarten in terms of discourse.
PerEinar Sæbbe
PerEinar Sæbbe is PhD candidate in Educational Science, Department of Early Childhood Education, University of Stavanger, Norway. His research interests are questions related to kindergarten children’s learnings of mathematics, kindergarten teachers ́ professional knowledge in mathematics and general pedagogy in early childhood education and care.
Reidar Mosvold
Reidar Mosvold is Associate Professor of mathematics education at the University of Stavanger, Norway. His research interests are related to the mathematical knowledge required to carry out the work of teaching mathematics at all levels, teacher beliefs, teacher identity and discursive perspectives, as well as use of history of mathematics in mathematics education.
Skapad: 20161212 kl. 00:00

NOMAD 21(4), 2016
Young children exploring probability – with focus on their documentations
Jorryt van Bommel and Hanna Palmér
Abstract
This article reports on an intervention where possibilities and limitations with problemsolving as a basis for mathematics education in preschool class were studied. In the article we explore how 50 children use nonguided documentation when working with a problemsolving task about probability. The results show that the task was possible to work with for these young children, and in the followup interviews many of the children seemed familiar with the mathematical concepts used, as well as with a relevant sample space. The children’s nonguided documentation showed a diver sity of strategies and contributed positively to their exploration of probability, both during the lesson and in the final discussions.
Jorryt van Bommel
Jorryt van Bommel is Senior lecturer in Mathematics education at Karlstad University. Her research is focused on teacher’s and student teacher’s professional development, as well as mathematics teaching and learning in preschool class, primary and secondary school. Ongoing research is focused on mathematics teacher’s collective work on social media and problem solving in preschool class with a special interest in the role of digital technology.
Hanna Palmér
Hanna Palmér is Senior lecturer in Mathematics education at Linnaeus University. Her research is focused on primary school teacher’s professional identity development, as well as mathematics teaching and learning in preschool, preschool class and primary school. Ongoing research is focused on problem solving, entrepreneurial teaching and learning in mathematics and young children’s learning of mathematics through digital technology.
Skapad: 20161212 kl. 00:00

NOMAD 21(4), 2016
Teachers’ pedagogical mathematical awareness in diverse childagegroups
Camilla Björklund and Wolmet Barendregt
Abstract
In this study we investigate preschool teachers’ attention to number sense, number sequence, geometrical shapes and patterns, and their pedagogical awareness regarding these areas in their work with younger and older preschool children. The specific research questions are: Are there differences in teachers’ attention to the different mathematical content areas depending on the age group they are teaching? And if mathematical content areas show differences between age groups, what pedagogical awareness levels constitute these differences? The study shows that the frequency of working with mathematical content is higher for all areas regarding older children, but teachers’ choice of mathematical content area is not necessarily depending on the age of the children. Teachers’ frequency of observing and engaging children in mathematics is lower among those working with younger children but the difference is in general quite small. Significant and substantial differences appear in teachers’ attention to children’s mathematizing initiatives and problematizing number sequences, geometrical shapes and patterns.
Camilla Björklund
Camilla Björklund is Associate Professor in Education at the University of Gothenburg. She is involved in preschool teacher education and researches young children’s mathematical learning and the teaching of mathematics in preschool.
Wolmet Barendregt
Wolmet Barendregt is Associate Professor in Interaction Design at the University of Gothenburg. She is involved in teacher education and researches young children’s use of mathematics games as well as other learning technologies.
Skapad: 20161212 kl. 00:00

NOMAD 21(4), 2016
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

NOMAD 21(4), 2016
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 21(4), 2016
Different perspectives on possible – desirable – plausible mathematics learning in preschool
Hanna Palmér and Camilla Björklund
Abstract
This article addresses the question of what is considered possible – desirable – plausible in preschool mathematics. On the one hand, there is a growing consensus that preschool mathematics matters, on the other hand, there are different opinions about how it should be designed and what constitutes an appropriate content. In the article we provide an overview of similarities and differences found in eight articles published in a thematic issue of Nomad on preschool mathematics. The overview is based on Bernstein’s notions vertical and horizontal discourses, and how content for learning is described as basic or advanced mathematics. The aim is not to evaluate or rate the articles but to illustrate diversity regarding possible – desirable – plausible in current research of preschool mathematics.
Hanna Palmér
Hanna Palmér is senior lecturer in Mathematics Education at Linnaeus University in Växjö. Her research is focused on mathematics teaching and learning in preschool, preschool class and primary school.
Camilla Björklund
Camilla Björklund is Associate Professor in Education at the University of Gothenburg. She is involved in preschool teacher education and researches young children’s mathematical learning and the teaching of mathematics in preschool.
Skapad: 20161212 kl. 00:00

NOMAD 21(3), 2016
Mathematics teachers’ knowledgesharing on the Internet: pedagogical message in instruction materials
Yvonne Liljekvist
Abstract
This article reports on a study of teachershared documents containing matematical tasks published on the Internet. The aim was to identify the goals, metods and pedagogical justifications presented in the documents and what was needed to solve the tasks. Content analysis was used to define their pedagogical message. The results show that the documents mainly involve content goals for younger pupils that are not consistent with the explicit descriptions. The conceptual goals are communicated to a great extent, but are not supported by task features. The reasons for why the tasks given are expected to lead to a certain goal are very often implicit, and, as a result, the content of the documents and the quality of the tasks are somewhat unclear to other teachers.
Yvonne Liljekvist
Yvonne Liljekvist is senior lecturer in mathematics education at the Department of Mathematics and Computer Science at Karlstad University, Sweden. One of her research interests is mathematics teachers’ use of Internet to improve their own teaching and to support colleagues. Liljekvist and her colleagues have an ongoing research project in this area that examines how teachers use social network sites to develop their pedagogical content knowledge.
Skapad: 20160922 kl. 01:00

NOMAD – 21(3), 2016
Tidigare nummerPrevious issues
ArrayNummer/Issue
Volume 21, No 3, September 2016
Yvonne Liljekvist
Evangelia Triantafyllou, Morten Misfeldt and Olga Timcenko
Anneli Dyrvold
The role of semiotic resources when reading and solving mathematics tasks
Janne Fauskanger og Reidar Mosvold
Lærerarbeidets matematiske undervisningsoppgaver
Shipra Sacheva, Marit Hvalsøe Schou and Andreas Lindenskov Tamborg
Reflections on having participated at the PreNORMA workshop
Innehåll: JH
Skapad: 20160922 kl. 01:00