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Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students’ routines according to whether they are driven by social reward or by generating a substantiated narrative. Explorative routines in this theorisation may refer not to inquiry-based activity but to the result of a student’s routine moving from being process-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well as a teacher’s moves offer students different opportunities to engage in rituals, explorative routines and deritualisations. Through nuancing the space spanned by opportunities to engage in rituals and explorative routines respectively, we describe and contrast classroom practices in three lessons from three contexts. The lessons share a commonality in encouraging explorative routines as a starting point, yet being adapted towards ritual activity through decreased openings for student agentivity, fewer invitations for students’ own substantiations or both. We argue that such adaptations are driven by the teachers’ commitment to reach mathematical closure in a lesson, to balance considerations of the classroom community and individual students and to meet curricular requirements. Our model helps interrogate the nature and relevance of hybrids of explorative routines and rituals., TRACE

To get an overview of the characteristics of the studies in mathematics education research that explicitly state that they deal with implementation, we have conducted a systematic review. In this paper, we report on a subset of the identified studies from the review, dealing with large-scale professional development for teachers. For the subset of the 11 identified papers, we ask the question: What designs are used to support teachers to adopt new ideas in their practice and what dimensions of scaling are considered in the studies? To articulate design and dimensions of scaling, we draw on theoretical constructs from both mathematics education research and more general implementation research. Results indicate that the choice of facilitating strategy impacts the dimensions of scaling considered in the implementation.

Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students’ routines according to whether they are driven by socialreward or by generating a substantiated narrative. Explorative routines in this theorisation mayrefer not to inquiry-based activity but to the result of a student’s routine moving from beingprocess-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well asa teacher’s moves ofer students diferent opportunities to engage in rituals, explorative routines and deritualisations. Through nuancing the space spanned by opportunities to engage inrituals and explorative routines respectively, we describe and contrast classroom practices inthree lessons from three contexts. The lessons share a commonality in encouraging explorativeroutines as a starting point, yet being adapted towards ritual activity through decreased openings for student agentivity, fewer invitations for students’ own substantiations or both. We arguethat such adaptations are driven by the teachers’ commitment to reach mathematical closure ina lesson, to balance considerations of the classroom community and individual students and tomeet curricular requirements. Our model helps interrogate the nature and relevance of hybridsof explorative routines and rituals., TRACE

As with any programs in teacher education, Swedish mathematics teacher education is influenced by changing political winds, developments in Information and Communication Technology (ICT), culture, history, PISA results, research-based program designs, and a fair amount of passion. Content and outcomes are nationally determined and include the requirement of a strong research foundation, but this is often not how practcing techers work, which exerts its own pull on teacher education. The specific implementations of programs take different forms at the universities that offer mathematics teacher education. In order to provide a comprehensive yet meaningful ntroduction to both the current system and current practices, we describe the overall organization of Swedish mathematics teacher education, and then offer short cases of implemented programs. To ensure inclusivity, the various parts are written by mathematics educators from the respective institutions. In this way, both variation across mathematicas teacher education for diffrent grade levels and variation across different institutions working with the same national directives can be distinguished. Issues such as the academization of teacher education are problematized, as are other forces that constitute the crosscurrents in Swedish mathematics teacher education.

As with any programs in teacher education, Swedish mathematics teacher education is influenced by changing political winds, developments in Information and Communication Technology (ICT), culture, history, PISA results, research-based program designs, and a fair amount of passion. Content and outcomes are nationally determined and include the requirement of a strong research foundation, but this is often not how practcing techers work, which exerts its own pull on teacher education. The specific implementations of programs take different forms at the universities that offer mathematics teacher education. In order to provide a comprehensive yet meaningful ntroduction to both the current system and current practices, we describe the overall organization of Swedish mathematics teacher education, and then offer short cases of implemented programs. To ensure inclusivity, the various parts are written by mathematics educators from the respective institutions. In this way, both variation across mathematicas teacher education for diffrent grade levels and variation across different institutions working with the same national directives can be distinguished. Issues such as the academization of teacher education are problematized, as are other forces that constitute the crosscurrents in Swedish mathematics teacher education.

In this study, the learning encouraged by teaching activities in a small-group setting was investigated through the analysis of students’ responses to survey and interview questions. The results indicate that the students perceived an increased ability to communicate mathematics in written form, but to a lesser extent developed their ability to discuss mathematics and build conceptual understanding.

This combined interview and survey study explored the relationship between interview data and data from an inventory describing engineering students' ratings of their approaches to studying. Using the 18-item Approaches and Study Skills Inventory for Students (ASSIST) students were asked to rate their approaches to studying in relation to particular statements. A subsample of nine first-year engineering students participated in subsequent interviews exploring their experiences of studying and learning. The students' views were examined and interpreted into inventory scores which were compared to the students' actual ratings. The interviews confirmed the scales measured in the inventory and provided illustrations to them. While students who were extreme in either approach were easier to interpret, others provided a good example of the complex combination of approaches that can exhibit itself in one individual. The study illustrates how combined data sets can contribute to achieve a holistic understanding of student learning in its context.

This combined interview and survey study explored the relationship between interview data and data from an inventory describing engineering students’ ratings of their approaches to studying. Using the 18-item Approaches and Study Skills Inventory for Students (ASSIST) students were asked to rate their approaches to studying in relation to particular statements. A subsample of nine first-year engineering students participated in subsequent interviews exploring their experiences of studying and learning. The students’ views were examined and interpreted into inventory scores which were compared to the students’ actual ratings. The interviews confirmed the scales measured in the inventory and provided illustrations to them. While students who were extreme in either approach were easier to interpret, others provided a good example of the complex combination of approaches that can exhibit itself in one individual. The study illustrates how combined data sets can contribute to achieve a holistic understanding of student learning in its context., QC 20170911. QC 20191111

Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students' multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication.

In this paper we focus on first-year 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.

In this study of arithmetical reasoning, which extends earlier work, we explore what properties students, when working in pairs, discern in additive and multiplicative co-variation problems that help them to distinguish between problem types. Results showed that pairs who solved each problem appropriately discerned mathematically significant properties such as speed, starting time and distance. Pairs who over-used additive reasoning focused on the distance difference without considering speed. While speed is considered to be a difficult quantity, here it seems to help students distinguish between multiplicative and additive situations.

In a study investigating first-year engineering students’ approaches to learning and studying, a strategic approach to learning was shown to be the best predictor of academic achievement. A surface approach to studying was found to be a negative factor. A questionnaire was distributed to first-year students aiming for Master of Science in Engineering in either Computer Science or Media Technology in the beginning and the end of their first year of study. In the first instance 154 students (62 %) answered the questionnaire and 91 (35 %) participated on both occasions., QC 20131024

This study investigates the relationship between approaches to studying and course completion in two online preparatory university courses in math- ematics and computer programming. The students participating in the two courses are alike in age, gender, and approaches to learning. Four hundred and ninety-three students participating in these courses answered the short version of the Approaches and Study Skills Inventory for Students (ASSIST). Results show that students demonstrating a deep approach to learning in ei- ther course are more likely to complete. In the mathematics course, a com- bination of deep and strategic approaches correlates positively with course completion. In the programming course, students who demonstrate a surface approach are less likely to complete. These results are in line with the inten- tions of the course designers, but they also suggest ways to improve these courses. Furthermore, the study demonstrates that ASSIST can be used to evaluate course design., Cette recherche examine la relation entre les différentes approches envers les études et la réussite des cours adoptées par les étudiants de deux différents cours universitaires offerts en ligne : l’un en mathématiques, l’autre en programmation. Dans les deux cours, les participants présentent les mêmes caractéristiques quant à l’âge, au sexe et à l’attitude vis-à-vis des études. Quatre cent quatre-vingt-treize étudiants inscrits à ces cours ont répondu à la version abrégée d’une recherche sur les approches et les compétences d’études chez les étudiants intitulée « Approaches and Study Skills Inventory for Students » (ASSIST). Les résultats démontrent que les étudiants qui abordent l’apprentissage de manière sérieuse dans l’un ou l’autre des cours ont plus de chances de réussir. Dans le cours de mathématiques, une combinaison d’approches sérieuses et stratégiques est positive pour réussir le cours, tandis que les étudiants du cours de programmation qui adoptent une approche superficielle envers l’apprentissage présentent moins de chance de réussite. Bien que ces résultats soient conformes aux intentions des créateurs de ces cours, ils proposent aussi certaines améliorations possibles. L’étude démontre qu’on peut aussi utiliser ASSIST pour l’évaluation de la conception de cours., QC 20150420, Samverkansprojekt mellan NADA, MND och Institutionen för pedagogik och didaktik

In this study of university students’ discourse on the threshold concept of function, the aim was to examine what kind of changes of their mathematical discourse could be observed during a study year. Data was collected through interviews, questionnaires and observations during the students’ first year of mathematics courses and was analyzed using the commognitive framework of Sfard. The result shows substantial differences in the students’ discourse. The transformation of their discourse resulted in broader registers of words, mediators, narratives and routines. Students not transforming their discourse showed an unbalanced use of visual mediators and fewer connections between the interrelated features of the discourse.

In order to analyse the course design in two online preparatory university courses in mathematics and programming, a Swedish translation of the short version of the Approaches and Study Skills Inventory for Students (ASSIST) was distributed. Four hundred and ninety three students, representing 24 % response rate, participated. The result showed that the mathematics course discouraged students who adopt a deep/strategic approach, while the programming course discouraged students with a deep approach and encouraged students with a surface approach. The indications differences in course design in two online preparatory university courses in mathematics and programming, a Swedish translation of the short version of the Approaches and Study Skills Inventory for Students (ASSIST) was distributed to the students participating. The result showed that the mathematics course discouraged students who adopt a deep/strategic approach, while the programming course discouraged students with a deep approach and encouraged students with a surface approach. The indications may beare a contribution to the discussion on disciplinary differences in approaches to studying as well as in disposition of online courses., QC 20130619

A threshold concept is a ‘portal’ or a ‘conceptual gateway’ that leads to a previously inaccessible, and initially troublesome, way of thinking about something. A new way of understanding may thus emerge – a transformed view of the subject. The framework of threshold concepts has been used for some years in research of teaching and learning in higher education in several subjects but there are only few articles in mathematics education using this framework. The aim of this paper is to introduce threshold concepts into mathematics education. The result of searching papers in mathematics education using threshold concepts is presented. The need for more research using this framework to create a meeting point for mathematicians and educationalists and to improve students’ learning is pointed out.

I elevers strävan att utveckla förståelse inom ett område blir vissa begrepp mer avgörande än andra. Dessa begrepp kan benämnas tröskelbegrepp. De fungerar som en portal till ett tidigare onåbart och i början problematiskt sätt att tänka om någonting. De har en potential att öppna för förståelse av hela området. Exempel på tröskelbegrepp är funktion, gränsvärde, derivata och integral. I min avhandling har jag undersökt studenters begreppsuppfattningar för dessa begrepp. Resultaten visar att studenterna tolkar den matematik de möter på olika sätt, och att studenterna utnyttjar en dynamik mellan olika aspekter av sina begreppsuppfattningar. Denna dynamik utgör en viktig del i utvecklingen av förståelsen av tröskelbegreppen. Även i grundskolans matematik finns tröskelbegrepp, t.ex. bråkbegreppet. Kunskaper om dessa begrepp kan hjälpa oss att skapa goda lärsituationer för eleverna., Presentation på Matte/NO-biennette 2011, 29 januari, Stockholm.

Two case studies were carried out to explore students’ understanding of concepts in calculus. The first study explored engineering students’ understanding of limit and integral. The second study involved students within a mathematics programme, working on a challenging task including the concepts function and derivative, requiring proof by induction. Drawing on a theory of contextualisation data were analysed within a constructivist research framework following the principles of intentional analysis. The results reveal that the students in the mathematics programme expressed their understanding in a formal context where also intuitive ideas played an important role. The engineering students expressed their understanding in an algorithmic context, in which procedural knowledge was predominant. The studies display how students may have a potential for developing a formal understanding of a mathematical concept previously viewed within an algorithmic context.

This study explores the nature of university students' experiences of understanding the threshold concept ‘function'. Fifteen teacher students taking an introductory course in mathematics were at the beginning of the course asked to reflect in writing on the meaning of ‘function'. Subsequent interviews explored in greater detail individual students' conceptual understandings of this concept. At the end of the course, students were again asked to reflect in writing on the meaning of function and then to elaborate further on those written reflections in a qualitative research interview. In a previous study of students' understandings of limit and integral an intentional analysis of the interview transcripts underscored the crucial interplay between understandings developed within an algorithmic and conceptual context respectively. The present study seeks to develop this contextual framework for analysing students' understanding in mathematics by linking the analysis of students' emerging understandings of function to ongoing research on threshold concepts in higher education and more broadly to research on students' experiences of understanding disciplinary ways of thinking and practising.

Two empirical studies were carried out to explore students’ understandings of threshold concepts in calculus. The studies investigated engineering and mathematics students’ understandings of the concepts of function, limit, derivative and integral. The engineering students expressed their understanding in an algorithmic context, in which procedural knowledge was predominant. The students in the mathematics programme expressed their understanding in a formal context where also intuitive ideas played an important role. The results also point out the importance of contextual shifts in the development of conceptual understanding of function, limit, derivative and integral as threshold concepts.

I elevers strävan att utveckla förståelse inom ett område blir vissa begrepp mer avgörande än andra. Dessa begrepp kan benämnas tröskelbegrepp. De fungerar som en portal till ett tidigare onåbart och i början problematiskt sätt att tänka om någonting. De har en potential att öppna för förståelse av hela området. Exempel på tröskelbegrepp är funktion, gränsvärde, derivata och integral. I min avhandling har jag undersökt studenters begreppsuppfattningar för dessa begrepp. Resultaten visar att studenterna tolkar den matematik de möter på olika sätt, och att studenterna utnyttjar en dynamik mellan olika aspekter av sina begreppsuppfattningar. Denna dynamik utgör en viktig del i utvecklingen av förståelsen av tröskelbegreppen. Även i grundskolans matematik finns tröskelbegrepp, t.ex. bråkbegreppet. Kunskaper om dessa begrepp kan hjälpa oss att skapa goda lärsituationer för eleverna.

I studier av elevers begreppsuppfattningar fokuseras ofta elevers svårigheter och missuppfattningar, men det finns också forskare som menar att dessa ofullständiga eller felaktiga uppfattningar utgör en del av utvecklingsprocessen. Jag har i min forskning valt att lyfta fram den potential som finns i elevers agerande när de möter ett matematiskt material. Med det perspektiv jag har tagit vill jag visa hur elever använder de begreppsuppfattningar de har och hur de förmår utnyttja sina uppfattningar i ett matematiskt arbete. Genom att studera det sammanhang i vilket universitetsstudenter tolkar ett matematiskt material kan jag visa på en dynamik mellan olika aspekter av en begreppsuppfattning som aktualiseras under ett matematiskt arbete. Resultaten från två studier, en intervjustudie och en problemlösningsstudie, visar att dessa studenter, trots brister i sina begreppsuppfattningar, ändå förmår att utnyttja ett samspel mellan olika aspekter av matematiken på ett sätt som har stora likheter med en matematikers arbetssätt. I kapitlet presenteras de två studierna. Därefter presenteras de sammantagna slutsatserna från dessa studier. I det avslutande avsnittet diskuteras resultatens relevans för matematikundervisning på alla nivåer.

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.

Studiens syfte är att visa hur en växelverkan mellan intuitiva idéer och formella resonemang kan gestalta sig i en problemlösningsprocess. Studien visar att universitetsstudenter redan under sitt första år av matematikstudier förmår utnyttja en sådan växelverkan. En grupp studenter har arbetat med en analysuppgift som berör begreppen funktion och derivata samt inkluderar ett induktionsbevis. Studenterna utnyttjar i den kreativa processen intuitiva idéer och formella resonemang i ett dynamiskt samspel. Växlingarna har ett flertal funktioner: att kontrollera intuitiva uppfattningar, att skaffa nya utgångspunkter för problemlösningsprocessen, att ekonomisera resonemang och att driva arbetet vidare.

Focusing on the potentiality of students’ ways of treating a mathematicalmaterial this thesis aims to investigate how students use their conceptualunderstanding when working with mathematical tasks in calculus. Two casestudies were carried out to explore students’ understanding of thresholdconcepts. The first study, an interview study, explored engineering students’understanding of limit and integral. The second study, a problem solvingstudy, involved students within a mathematics programme, working on achallenging task including the concepts function and derivative, requiringproof by induction. Drawing on a theory of contextualisation data wereanalysed within a constructivist research framework following the principlesof intentional analysis. The results reveal that the students in themathematics programme expressed their understanding in a formal contextin which also intuitive ideas played an important role. They used intuitiveideas and formal reasoning in a dynamic interplay with several functions:to control intuitive ideas, to offer a new basis of reasoning, to reduce thecomplexity of the problem and to push the problem solving process forward.The engineering students expressed their conceptions in an algorithmiccontext, in which procedural knowledge was predominant and the operationsof the concepts were seen as defining features and a basis for understanding.However, faced with probing questions, the students appeared to shift to acontextualisation foregrounding ideas relating to conceptual dimensions ofcalculus. These contextual shifts display the transformative aspect ofthreshold concepts allowing the development of conceptions and students’awareness of ways of thinking and practising in mathematics.

The aim of this study is to illuminate the interplay between intuitive ideas and formal justification. In a perspective where we focus on the students’ competences we describe the interaction between intuitive ideas and formal structures, as they appear especially in relation to concept definitions and formal proofs. A group of university students have been working on a task in calculus. The task includes the concepts function and derivative and requires of them to use a proof by induction. The discussion between the members of the group has been analyzed in accordance with the principles of intentional analysis, a method by which we regard the students’ activities as intentional. This method of analysis makes the process of interpretation visible. It also makes it possible to make explicit the competences of the students’ and the mathematical content of their actions. In the study examples are given of the interplay between the intuitive ideas and the formal structures and also of the function for this interplay. The students have rich concept images and access to intuitive ideas relevant to the concepts brought to the 4 fore by the task. During the group discussion all components of a complete proof are included in the students’ reasoning. The students create a proof by induction which matches the ordinary pattern for such a proof, but they do not themselves regard it as a proof fitting into the ordinary scheme of argumentation, as they remember it from text-books and teaching. The students put heavy demands upon the formalization of their ideas and these demands are sometimes hampering the problem solving process, but they also encourage the students to expand their search for a solution to the problem at hand.

Let R be a commutative Noetherian one-dimensional domain such that each ideal in R can be generated by n elements. A strong n-generator in R is an element r which can be chosen as one of n generators for each ideal in which it is contained. If moreover r is contained in some ideal requiring n generators, then r is said to be a proper strong n-generator. We prove that if R is two-generated, then r is a strong two-generator if and only if r doesn't belong to Mˆ2 for any non-invertible maximal ideal M of R. Let S be an index set such that for each i in S, M(i) is a maximal ideal such that the localization of R at M(i) possesses an ideal requiring n generators. We show that there is a unique biggest M(i)-primary ideal I(i) in R requiring n generators. We prove that r is a strong n-generator if and only if r doesn't belong to I(i)M(i) for any i in S. Let R' be the integral closure of R. Suppose moreover that the conductor C of R' in R is non-zero. We prove that if n > 2, then r is a proper strong n-generator if and only if r belongs to the complement of I(i)M(i) in I(i) for some i in S. And if n = 2, then r is a proper strong two-generator if and only if r belongs to the complement of Mˆ2 in M for some non-invertible maximal ideal M of R or r belongs to some non-principal invertible maximal ideal of R. We prove that if C requires n generators, then R is n-generated and the extension of C to the localization of R at M(i) for some i in S requires n generators. Suppose moreover that R is a Cohen-Macaulay ring, R possesses a superficial element and R' is a local domain. Then (R,M) is a local domain. Let B(M) be the blowing-up ring of R at M. We give an algorithm for determining the conductor J of B(M) in R. We determine the reduction exponent of M for a semigroup ring and give a sufficient condition for J not being a power of M., Härtill fyra uppsatser.

Distributivity is considered to be essential for multiplicative understanding but difficult to learn. The difficulties may arise as overgeneralisations of addition strategies. Rectangular models emphasise the two-dimensionality of multiplication, separating it from addition and are suggested to support understanding of distributivity better compared to equal groups. Coincidently, studies report of students’ understanding of distributivity based on equal groups, leaving no consensus on equal groups’ suitability for understanding distributivity. In this paper we investigate how students can exploit equal groups to understand distributivity, by analysis of two students’ reasoning when they successfully explain distributivity construing the multiplication as heaps of sticks and bags of coins. The role of equal groups with respect to multiplicative understanding of distributivity is discussed in relation to previous ambiguous findings and to the extension of multiplication beyond integers, in which the equal groups model may be inappropriate.

Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students’ multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication.