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 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. 

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.

The purpose of this paper is to examine preservice teachers’ conceptions of the function concept as well as their conceptions of the significance of functions in mathematics. A further purpose is to study the effects of an intervention regarding the function concept. Two groups of preservice teachers who are specializing in mathematics and science are participating in the study. The findings indicate changes, firstly in the preservice teachers’ view on the function concept related to the intervention, and secondly with respect to different themes of relevance in their reasoning on the significance and presence of functions in various contexts.

This paper examines preservice teacher's views on the function concept. It includes two groups of preservice teachers as they consider the presence of functions in school mathematics. The findings imply that preservice teachers will give issues related to the concept of function low priority in their teaching, and that they require additional support to develop their views on functions. The study illustrates that discussion about fundamental concepts should be an essential component in the preparation of preservice teachers. It also illustrates that reflection on the presence of functions impacts preservice teachers' reasoning, and serves as a starting point for valuable didactical discussions, which are related to subject matter knowledge, pedagogical content knowledge and curricular knowledge.

This research examines one high achieving preservice teacher’s conceptions of the function concept and her line of reasoning as she after a calculus course tries to make clear what constitutes a function. A group of twenty-five preservice teachers specializing in mathematics and science participated in the study. One preservice teacher, called Emma, who completed the calculus course with “high pass” is the research subject for the case study in this presentation. The concepts of function and equation are closely linked for the preservice teacher in question. In the advancement of the preservice teacher’s reasoning on functions, feedback from evoked concept images to the concept definition is central, and leads her to a more developed understanding of what constitutes a function. The findings of the research call attention to one component of importance in the development of a concept image, namely, the feedback-process from evoked concept images to the concept definition. Based on the case study results, preservice teachers seemingly need to be exposed to problem formulations concerning the relationships of the function concept to other mathematical concepts (this conclusion is also supported by Hansson, 2006). It was made clear during the interview with Emma that she is not experienced in working with such problems. During an individual’s reasoning, the concept image will almost always be evoked, whereas the concept definition will remain inactive or even be forgotten (Tall & Vinner, 1981). The study indicates that one significant component in the development of preservice teachers’ understanding of the function concept could be to stimulate a feedback of evoked concept images to the concept definition in various contexts, as when Emma comes to a better understanding in the interview. The results of the study also support Vinner’s (1991, 1992) suggestion that this type of reconnection is primarily possible with problems that are not of the standard variety.

References

Hansson, Ö. (2006). Studying the views of preservice teachers on the concept of function(LTU doctoral thesis series No. 2006:22). Luleå: Luleå University of Technology. Tall, D., & Vinner, S. (1981). Concept images and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D.Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer.Vinner, S. (1992). The function concept as prototype for problems in mathematics. In G.Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 195-213). Washington, DC: Mathematical Association of America.

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.

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.

This chapter discusses a framework for analysing the role of mathematics during physics lessons in upper-secondary school. It takes as a starting point that relations made during physics lessons between Reality, Theoretical models and Mathematics are of the outmost importance. The framework was developed to analyse the communication during physics lessons. It was developed during a pilot study exploring the role of mathematics for physics teaching and learning in upper-secondary school during different kinds of physics lessons (lectures, problem solving and labwork). In the overall project observations have been made in three physics classes (in total 7 lessons) led by one teacher. Here the developed analytical framework is described together with selected results from one class (3 lessons) showing how the framework could be used. This chapter describes and discusses the uses of the framework and shows how results on students and teachers usages of links between the three entities Reality, Theoretical models and Mathematics can be brought to the forefront in an analysis of complex physics teaching situations. And, how the framework can be used to analyse the different organisational forms: lectures, problem solving in groups and labwork.

Syftet med vårt projekt är att fördjupa förståelsen av matematikens roll i fysikundervisningen. I fysiken är matematiken ett viktigt verktyg – man pratar om matematiken som fysikens språk. Samtidigt upplevs matematik som ett hinder för meningsfullt lärande i fysik. Elever kan bli så upptagna av matematiska operationer att fysikaliska modellers innebörd och relation till de verkliga fenomen de beskriver hamnar i bakgrunden. Vi har studerat hur relationer mellan verklighet, teoretiska modeller och matematik kommuniceras i olika undervisningssituationer i svensk gymnasiefysik. Resultaten visar på samband mellan fysiklärares syn på syftet med fysikundervisningen och matematikens roll i deras klassrum.

This paper discusses the role of mathematics in physics teaching andlearning. The intention and strength of science is to describe what happens in the realworld by organizing explanations through theories and theoretical models. It has beenwidely suggested that learning in science is influenced by views about the nature ofscientific knowledge. Furthermore, mathematics is an inherent part of theories andmakes powerful predictions of natural phenomena possible. Ability to use mathematicsto argue and prove results is central in physics. Students have been found tostruggle with physics problems where they need to relate mathematically formulatedtheoretical models to the real world. This project has explored the role of mathematicsfor physics teaching and learning in upper-secondary school through investigations ofrelations made during physics lessons (lectures, problem solving and lab-work)between The real world – Theoretical models – Mathematics. A preliminary analysisindicate that there are some connections made by students and teacher betweentheories and the real world, but the bulk of the discussion in the classroom is concerningthe relation between theoretical models and mathematics, which is in linewith earlier research on problem solving. The preliminary results reported on hereindicate that this also holds true for all the investigated organisational forms lectures,problem solving in groups and lab-work. A finalized and extended analysis will bepublished elsewhere.

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.