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 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.

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.

Science teaching and particularly chemistry and physics teaching is subject to ongoing discussions concerning aims, goals and relation to modern society, as well as how to teach so that more students find physics interesting and meaningful. The need for further research on reasons why physics teaching is diverse and teaching shows different curriculum emphases have been put forward by several authors (cf. Belo, van Driel, van Veen, & Verloop, 2014; Johansson, Andersson, Salminen-Karlsson, & Elmgren, 2016). The general picture is that relationships between teachers views, curriculum emphases, classroom practices, problems and possible student shortcomings need further studies in order to generate more knowledge about the basis for teaching conditions helping students to gain knowledge and interest in physics.This study continues a line of research that has a special focus on the role of mathematics in physics teaching (cf. Redfors, 2015; 2018; Turşucu, Spandaw, Flipse, & de Vries, 2017). One strand of this research focus on students’ problems in transferring mathematical knowledge to new and applied situations during physics teaching (cf. Kuo, Hull, Gupta, & Elby, 2013). However, there is also research focusing not only problem-solving, but physics teaching in general, from the perspective of the role of mathematics skills among students, since this is viewed as important for physics learning (Angell, Lie, & Rohatgi, 2011; Uhden, Karam, Pietrocola, & Pospiech 2012; Redfors, 2015; 2018). 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 (Redfors 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 2018-2019 will be reported and discussed at the conference.

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 uppersecondary physics. A developed analytical framework from the pilot (Hansson, Hansson, Juter & Redfors 2015) is used to focus the analysis of the classroom communication on relations made (by teachers and students) between Reality – Theoretical models – Mathematics. Results from classroom studies during spring 2018 will be reported and discussed at the conference.

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 Swedish upper-secondary physics. A developed analytical framework (Hansson, Hansson, Juter, & Redfors, 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 will be reported on and discussed at the conference.