The aim of this three-year study is to further contribute to the understanding of how relations between Reality – Theoretical models – Mathematics are communicated in different kinds of instructional situations (lectures, problem solving and labwork) in Swedish upper-secondary physics. A developed analytical framework from the pilot (Authors, 2015; 2019) is used to focus the analysis of the classroom communication on relations made (by teachers and students) between Reality – Theoretical models – Mathematics. The framework, results from an online survey to Swedish upper-secondary teachers on views of physics, mathematics and physics teaching, and results from classroom studies at upper secondary school during 2018 will be reported and discussed at the conference.

This chapter describes a case study of the role of mathematics in physics textbooks and in associated teacher led lessons. The theoretical framework (Hansson et al. 2015) used in the analysis focuses on relations communicated between three entities: Theoretical models, Mathematics, and Reality. Previously the framework has been used for analysing classroom situations. In this chapter, the framework is further developed and refined, and for the first time used to analyse physics textbooks. The case study described here is a synchronised analysis of a physics textbook and associated classroom communication during teacher led lessons, and contributes with an in-depth description of relations made between Theoretical models, Mathematicsand Reality. With the starting point in this case we discuss future uses of the analysis framework. We also raise questions for further research concerning how physics textbooks support and not support a meaningful physics teaching with respect to the role of mathematics and how relations between Theoretical models, Mathematics, and Reality are communicated.

This article addresses physics teachers’ views about physics teaching in upper-secondary school. Their views have been investigated nationwide through a web-based questionnaire. The questionnaire has been developed based on several published instruments and is part of an ongoing project on the role of mathematics in physics teaching at upper-secondary school. The selected part of the results from the analysis of the questionnaire reported on here cross-correlate physics teachers’ views about aims of physics teaching with their view of physics classroom activities, and perceived hindrances in the teaching of physics. 379 teachers responded to the questionnaire (45% response rate). The result indicates that teachers with a high agreement with a Fundamental Physics curriculum emphasis regarded mathematics as a problem for physics teaching, whereas teachers with high agreement with the curriculum emphases Physics, Technology and Societyor Knowledge Development in Physics did not do so. This means that teachers with a main focus on fundamental theories and concepts believe that mathematics is a problem to a higher extent than teachers with main focus on the role of physics in society and applied aspects or physics knowledge development do.  Consequences for teaching and further research are discussed. 

Matematik är ett viktigt verktyg för fysiken och matematiken sägs varafysikens språk. Tidigare forskning visar dock att elever ägnar mycket tid åt matematisk formelmanipulation medan mindre tid och kraft läggs på att relatera fysikens teoretiska modeller och begrepp till verk- ligheten. Syftet med forskningsprojektet vi beskriver här, är att för- djupa vår förståelse av matematikens roll i fysikundervisningen gene- rellt. Vi studerar därför matematikens roll i såväl problemlösningssitu- ationer som lärarledda genomgångar och laborativa moment. Pro- jektet kommer att ge förutsättningar för en ökad förståelse av matema- tikens roll i olika typer av fysikundervisning och för att identifiera i vilka situationer som kommunikationen visar på att matematiken ut- gör hinder eller möjligheter för fysiklärandet. Genom att identifiera så- dana tillfällen öppnas också möjligheten att arbeta för att bryta oöns- kade och stimulera önskade kommunikationsmönster och förstå hur matematiken kan användas på ett konstruktivt sätt i fysikundervis- ningen. Slutsatserna från projektet kommer därför att kunna användas i lärarutbildning, lärarfortbildning och av läromedelsförfattare, liksom av fysiklärare som vill arbeta för att utveckla sin undervisning.

This article discusses the role of mathematics during physics lessons in upper-secondary school. Mathematics is an inherent part of theoretical models in physics and makes powerful predictions of natural phenomena possible. Ability to use both theoretical models and mathematics is central in physics. This paper takes as a starting point that the relations made during physics lessons between the three entities Reality, Theoretical models and Mathematics are of the outmost importance. A framework has been developed to sustain analyses of the communication during physics lessons. The study described in this article has explored the role of mathematics for physics teaching and learning in upper-secondary school during different kinds of physics lessons (lectures, problem solving and labwork). Observations are from three physics classes (in total 7 lessons) led by one teacher. The developed analytical framework is described together with results from the analysis of the 7 lessons. The results show that there are some relations made by students and teacher between theoretical models and reality, but the bulk of the discussion in the classroom is concerning the relation between theoretical models and mathematics. The results reported on here indicate that this also holds true for all the investigated organisational forms lectures, problem solving in groups and labwork.

This study adds to research on the use of mathematics in physics classrooms at upper secondary school. The aim is to look closer into what types of transfer do the teacher and textbook set up for the pupils with respect to ways of reasoning from other physics contexts as well as from mathematics. The frame for analysis is an analytical model based on relations made between Reality, Theoretical models and Mathematics (Redfors, Hansson, Hansson & Juter, 2016). Horizontal and vertical transfer is defined as mappings of new information to an activated known structure and as the creation of a new structure in the learner’s mind, respectively (Rebello, Cui, Benett, Zollman & Ozimek, 2007). Transfer occurs within mathematics and physics and also between the topics.We will focus on a physics lecture (40 min, video recorded) in a 3rd year class. When reasoning movement of charged particles in electric fields the teacher stresses hori- zontal transfer from mechanics and projectile motion. The procedure used is focused on analysing movement in “x direction” and “y direction” separately, not explicitly relating movement to the field direction. Whereas the argumentation in the textbook is based on movement in relation to the existence of a field direction. When considering velocity, the main focus is in both cases on a framework where the components of velocity is central.The tangent of a curve is a notion the students in the present study are quite familiar with from their courses in mathematics, which makes an opportunity for transfer from a mathematics context to help understanding physics. However, the notion of tangent is not used in the textbook or by the teacher in relation to velocity. Using the vector concept in this way would require students and teachers to perform a vertical transfer. This has been shown hard for both students and teachers. However, introducing this way of reasoning had made use of an opportunity for structural use of mathematics – an opportunity overlooked by both teacher and textbook.

We outline a framework to study the use of mathematics in physics classrooms. The framework focuses on the relations made between Reality, Theoretical models and Mathematics. In this paper the analyses of one teacher and her 3rd year classes at secondary school are presented. The results show that phenomena in reality are often used as a short prelude to put focus on the relationship theoretical model and mathematics. Mathematics is generally used in an instrumental way to handle various formulas without further insight or discussion of the related models or their relation to reality. There is a lack of varied communication with a structural use of mathematics, i.e., mathematics used to support reasoning in relation to a theoretical model, highlighting the meaning of concepts and models in the studied classrooms. 

This study adds to research on the use of mathematics in physics classrooms at upper secondary school. The aim is to look closer into what types of transfer do the teacher and textbook set up for the pupils with respect to ways of reasoning from other physics contexts as well as from mathematics. The frame for analysis is an analytical model based on relations made between Reality, Theoretical models and Mathematics (Redfors, Hansson, Hansson & Juter, 2016). Horizontal and vertical transfer is defined as mappings of new information to an activated known structure and as the creation of a new structure in the learner’s mind, respectively (Rebello, Cui, Benett, Zollman & Ozimek, 2007). Transfer occurs within mathematics and physics and also between the topics.We will focus on a physics lecture (40 min, video recorded) in a 3rd year class. When reasoning movement of charged particles in electric fields the teacher stresses hori- zontal transfer from mechanics and projectile motion. The procedure used is focused on analysing movement in “x direction” and “y direction” separately, not explicitly relating movement to the field direction. Whereas the argumentation in the textbook is based on movement in relation to the existence of a field direction. When considering velocity, the main focus is in both cases on a framework where the components of velocity is central.The tangent of a curve is a notion the students in the present study are quite familiar with from their courses in mathematics, which makes an opportunity for transfer from a mathematics context to help understanding physics. However, the notion of tangent is not used in the textbook or by the teacher in relation to velocity. Using the vector concept in this way would require students and teachers to perform a vertical transfer. This has been shown hard for both students and teachers. However, introducing this way of reasoning had made use of an opportunity for structural use of mathematics – anopportunity overlooked by both teacher and textbook.

This study concerns the teaching of drug calculations in nursing education. It is part of a larger study and focuses on the first year of a three-year nursing program when the students are introduced to drug calculations. The students who attended the first year on the program was divided into smaller groups. We followed one group where the lecture and problem-solving session was video recorded.It is well known that drug calculations are a critical component in nursing practice. Nurses need to do drug calculations correctly and as part of their nursing education must take a drug calculation test obtaining no errors in the results. However, in spite of drug calculation tests many adverse events occur in nursing practice (e.g., Røykenes & Larsen, 2010). Studies of nursing practice show that mathematics enters practices in a rich variety of ways and that it is not advisable to avoid the complexity of a situation by only using standard methods to capture its visible arithmetic and teach it (Coben & Weeks, 2014). To restrict the teaching to an elementary use of mathematics will not cover all the knowledge that is actually relevant to practice. In routine use, mathe- matical reasoning can be almost invisible and many artefacts in the nursing profession often depends on this invisibility. But at times nurses will need to understand under- lying mathematical models to sort out what is happening or what has gone wrong (Pozzi, Noss & Hoyles, 1998).The results of the current study show that the teaching of first-year students did not support conceptual understanding of mathematics including discussions about mathe- matical reasoning or relevant mathematical concepts. Instead, the students were ad- vised to forget their previous mathematical skills – in particular if they felt insecure about mathematics – and apply “safe” methods with a strong focus on instrumental use. For example, in drug dose calculations a triangular arrangement of dosage (d), concentration (c) and volume (v) was used in relation to the “formula” d=cv, instead of reasoning about how to solve an equation. Discussions about the use of  mathematicsand underlying models were absent in the teaching.

En studie av studenternas uppfattning av derivator och integraler på lärarutbildningsprogrammet mot år 4-9 i grundskolan presenteras. Hur påverkas studenternas begreppsuppfattning av undervisningen? Undersökningar före och efter undervisningen som består av olika moment som föreläsningar, datorövningar, räkneövningar, gruppövningar, projektarbete och annat redovisas.

Students’ pre-knowledge and conceptual development in analysis were investigated at a teacher education program to reveal what pre-knowledge endured and how the students perceived the concepts a year after the course had ended. Questionnaires and interviews were used to collect data. Two students’ results are presented in more detail in the article. The study was cognitively framed with the influence of situated theories to take as many aspects of concept development into account as possible. The students showed numerous connections between concepts, but they were often unable to discern valid links from invalid links. The perceived richness from many connections causes unjustifiably strong self-confidence which prevents further work with the concept. A tool for classification of the students’ connections between concepts resulted from the analysis.

Students learning limits of functions perceive and treat limits differently. A study on students’ conceptual development of limits of functions was conducted at a Swedish university (Juter, 2006). The results imply differences in high achieving and low achieving students’ work with limits, but also a lack of differences at some points. The students’ developments and abilities were studied in terms of concept images (Tall & Vinner, 1981) in the sense that their actions, such as problem solving and reasoning, were considered traces of their mental representations of concepts. A concept image of a concept comprises all mental representations of that concept and is linked to related concepts in a web. As could be expected, high achieving students’ abstraction abilities were more developed than other students’. The former group were to a much higher degree than the latter able to link theory to problem solving and explain the meaning of, for example, the limit definition. The students were studied during a semester and there were similarities of the high achieving students’ developments with the historical development of limits that the other students did not reveal. Several similarities were linked to abstraction and formality. Students with positive attitudes to mathematics in general were better limit problem solvers. Most of the high achieving students thought that they had control over the concept of limits, but many of the low achieving students also claimed to have control even if that was not the case. An unjustifiably strong self confidence can prevent students from further work on erroneous or incomplete parts of their concept images. There were no clear patterns of students’ mental representations of limits as exact values or approximations, limits as objects or processes, and limits as attainable or unattainable for functions. Of the 15 students interviewed, only two showed a coherent trace of their concept images. Both students were high achievers. The lack of patterns in all students’ concept images, particularly in the high achievers’, points to the complex nature of limits and the challenge to teach and learn limits.

Most concepts taught in a basic analysis course at university are somewhat familiar to the students. They use their representations from prior experiences to form new knowledge about the concepts. Fifteen students’ were examined at the beginning of a basic university course in analysis. Their pre-knowledge was sometimes vague and even wrong, often due to intuitive perceptions

This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.

This study was conducted to reveal how students at university level justify their solutions to tasks with various degrees of difficulty. The study is part of a larger study of students' concept formation of limits. The mathematical area is limits of functions. The study was carried out at a Swedish university at the first level of mathematics. The results are, however, applicable to other countries as well since students meet similar challenges in their learning of limits. I have, in discussions with some Australian mathematics teachers at university level, found out that the topics taught in basic mathematics courses in Australia are similar to Swedish courses. Two groups of students taking the same course in successive semesters have been solving tasks. Their solutions are categorised here and analysed to create a picture of how students reason about limits.

Students at a Swedish university were subjects in a study about learning limits of functions. The students' perceptions were investigated in terms of traces of concept images through interviews and problem solving. The results imply that most students' foundations were not sufficiently strong for them to understand the concept of limit well enough to be able to form coherent concept images. The traces of the students' concept images reveal confusion about different features of the limit concept.

Avhandlingen består av sex artiklar som föregås av en kappa. Kappan sammanfattar de teorier som använts i artiklarna, empiridelen samt resultat och slutsatser. Fem av artiklarna är publicerade i tidskrifter eller accepterade för publikation i tidskrifter och den återstående är accepterad för publicering i en konferensrapport.

Gränsvärden är väldigt centrala i matematisk analys och därför viktiga att förstå för studenter som ska syssla med matematik. Avhandlingen handlar om studenter som läser sin första matematiktermin på universitetsnivå. Två studentgrupper har undersökts med avseende på hur de hanterar gränsvärden av funktioner under 10 respektive 20 veckor. Utländsk litteratur har visat att detta begrepps komplexitet försvårar studenternas inlärning. Undersökningen genomfördes för att ytterligare klargöra hur studenterna resonerar kring gränsvärden, speciellt i Sverige där liknande undersökningar inte gjorts tidigare.

Undersökningen genomfördes år 2002 med en grupp på våren och en på hösten. Studenterna fick tre enkäter med gränsvärdesproblem och attitydfrågor om matematik. Femton studenter från den andra gruppen intervjuades vid två tillfällen vardera. Dessutom gjordes fältanteckningar. De olika sätten att samla in data gav en bred bild av verksamheten i klassrummen och föreläsningssalen samt hur studenterna resonerade i olika situationer.

Undersökningen visade att kontinuiteten i studenternas inlärningssituation brutits på flera nivåer och skapat stora problem för studenterna. Det fanns till exempel klyftor mellan gymnasiets hantering av gränsvärden och universitetets hantering av gränsvärden och mellan teoretisk hantering av gränsvärden och problemlösning. Studenterna var bra på att lösa standardproblem, men de hade svårigheter att koppla den formella teorin till problemen de löste och att lösa problem som var formulerade något annorlunda än studenterna var vana vid. Resultaten visar på att det behövs större variation i studenternas inlärning så de blir varse om sina egna uppfattningar och kan korrigera det som är felaktigt. Många studenter tyckte själva att de hade kontroll över begreppet trots att undersökningen visade att de inte hade det.

Students’ beliefs about division by zero and numbers on the number line were studied through explanations of the concepts in questionnaires and interviews during their teacher education to become primary school teachers in the years 4–6. The concepts were chosen for students’ proven cognitive challenges in coping with them, with the aim to add to the existing knowledge in terms of specific and general explanation types. General and specific parts of the students’ concept images were contradictory in several cases and the examples used for explaining were often based on other mathematical structures than the ones explained, e.g. 2/1 instead of 2/0 or a finite decimal recitation instead of an infinite one.

Statistik är ett matematikinnehåll som inbjuder till såväl tematiskt arbete som ämnesintegrerat. Redan i statistikens historiska barndom insåg man vikten av presentationen. I den här artikeln som främst riktas till åk 4-6 diskuteras olika uttrycksformer i undervisningen om statistik.

The main aim of this article is to discuss the attitudes to mathematics of students taking a basic mathematics course at a Swedish university, and to explore possible links between how well such students manage to solve tasks about limits of functions and their attitudes. Two groups, each of about a hundred students, were investigated using questionnaires, field notes and interviews. From the results presented a connection can be inferred between students’ attitudes to mathematics and their ability to solve limit tasks. Students with positive attitudes perform better in solving limit problems. The educational implications of these findings are also discussed.

Students’ learning developments of limits were studied in a calculus course. Their actions, such as problem solving and reasoning, were considered traces of their mental representations of concepts and were used to describe the developments during a semester. Several students went through the course with a vague conception of limits which in some cases was wrong. A higher awareness about their mental representations’ reliability is required.

University students’ conceptions of differentiability, continuity and relations between the concepts were studied to reveal their choices of representations and their strategies to justify their relational claims. Questionnaires and interviews were used to collect data (questionnaires in the part presented here). The results were analysed and categorized through a framework based on Skemp’s (1976) definitions of relational and instrumental understanding, and Tall’s (2004) three worlds of mathematics. The students showed ambiguous representations opposing their own statements in some cases. The most common feature among the students to describe a continuous function was incorrect implying a need to develop the students’ concept images in that area.

Mathematical concepts are mentally represented differently depending on individual, context and existing conceptions of related concepts among other things. The present paper reports on a study of students‘ representations in analysis with an emphasis on the types of representations and the links they have between their representations. The data collection was designed to evoke different parts of the students‘ concept images and also to return to the concepts several times over time at every data collection session. The results show that formal and intuitive representations in combination are rare. The number of links between concepts is not in itself a measure of the quality of the concept image, as there is a vast number of erroneous links misleading the students to think they understand the concepts.

The aim of the study reported in this paper is to investigate how students understand continuity and differentiability during and after a calculus course. The students’ choices of representations, both claimed and acted, were also studied. The study is part of a larger study of four student groups taking a calculus course. 207 students answered a questionnaire during the course and of them, 11 were interviewed after the course (the ones in this paper). Answers in questionnaires and interviews were categorised and compared. All students who preferred formal theoretical representations, and only those students, were able to produce formal proofs. The students’ stated and acted preferencesof representations were quite coherent, with only a few inconsistencies.

Physics teachers at upper secondary school indirectly teach mathematics in their physics classes through their teaching strategies and preferred ways of using mathematics. Their views of physics and mathematics are important for the way they depict mathematics to the students. A web-questionnaire was administered to Swedish physics teachers. Part of the questions investigated views of mathematics, i.e. as a means for application, as a schema, as a formal construct or as processes. Mathematics as a means for application was the dominant opinion. Students’ lack of knowledge in mathematics was regarded as a problem to many of the teachers, and particularly problem solving and modelling. Students’ conceptual and relevance proficiencies in mathematics were less problematical.

A physics lecture and a mathematics lecture, by the same teacher and partly the same students, were studied at upper secondary school. Both lectures covered ordinary differential equations. The main aim of the present paper was to investigate the teacher’s different and similar ways to handle related mathematical content in the two school subjects. The findings show a structural use of mathematics with an analytical approach in mathematics and an applied approach in relation to formulas in physics. This study is part of a larger study about mathematics in physics education funded by the Swedish research council.

We compared five groups of students to investigate the effects of ”Just-in-time teaching” (JiTT), a method designed to both help students keep up with the often fast pace of undergraduate calculus and to deepen their learning. In total, 137 students participated in the study. The outcome is discussed in terms of conceptual and procedural knowledge in relation to examination and other assessment tasks. We observed an improvement on the assessed items and a shift in study habits.

Students’ mathematics teacher identity is formed in various settings. A study with 45 pre-service students in their first year of education was conducted as part of alongitudinal study of year 4-6 mathematics teachers’ identity formation, to study the development during their education in terms of mathematical knowledge, pupils’ learning and the teacher role. Questionnaires and interviews were used to collect data. The result shows that many students were reluctant to use mathematics and had conceptions that may mislead pupils. The students’ learning focus was less on pupils’ learning than mathematics and teacher role, but theirideal teacher focused on pupils’ learning.

This study aims at exploring processes of flexibility and coordination among acts of visualization and analysis in students’ attempt to reach a general formula for a three-dimensional pattern generalizing task.

The investigation draws on a case-study analysis of two 15-year-old girls working together on a task in which they are asked to calculate the number of blocks in a three-dimensional tower of different heights. The students’ activity was video- and audio-taped, fully transcribed and lasted for 50 min.

The analysis discloses several instances of how the students were linking acts of visualization and analysis to reach a general formula. However, regarding flexibility, we found that it was more natural for the students to change visual format than to change analytical position and direction in their attempts to generalize the three-dimensional pattern of the task in a closed formula.