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Thermodynamic processes are often presented in so called P-V diagrams and the processes are often isobaric, isochoric, isothermal, and adiabatic. The purpose of the qualitative research reported here was to explore students' reasoning and interpretation of P-V diagrams that were presented by the help of GeoGebra. The research group had 15 students and the control group had 12 students. The control group were not given any dynamical explanation of the PV diagrams but were taught with static images in the same text book as the research group. The thermodynamics in Physics 1 at the Swedish gymnasium level is normally done within 2 weeks. We allocated 3 weeks and 10 hours for the thermodynamics to be taught and then gave all 27 students a short test. At the end of the Physics 1 course, we also gave a test with two questions from the thermodynamics. It seems that the students who were given the opportunity to dynamically change the content of the PV diagram benefit more from the teaching compared to the control group.

The subject of mathematics is important for many other subjects, including physics. The questions we ask in physics often need a mathematical translation to be graphically interpreted and transformed. Students learn physics when interacting with physical questions; questions from physics also give insights and discoveries in mathematical problems. In physics we often deal with questions that benefit from being illustrated with mathematical tools. A scientific theory (in physics or in another science) is an isomorphism between physical reality and a mathematical model.

In understanding upper secondary school students' interpretations of information in symbolic representations of a distance-time-relation, little attention has been paid to the analysis of the condition of the conceptual development related to utterances. Understanding this better can help improve the teaching of attribute and information in symbolic representations of different phenomena. Two theoretical perspectives have been used to conduct the analysis: Tall and Vinner's theoretical perspectives on learning and Gray's & Talls's theory of three mathematical worlds. The findings provide evidence that a detailed analyse of student's utterances show difference in quality related to student's interpretations of a distance-time relation. The qualities were related to student's concept images of functions and derivatives.

This is an article about abstraction, generalization, and the beauty of mathematics. We claim that abstraction and generalization in of itself may very well be a beauty of the human mind. The fact that we humans continue to explore and expand mathematics is truly beautiful and remarkable. Many years ago, our ancestors understood that seven stones, seven fish, and seven trees represent in some sense one entity, in this case all united by the number seven. That in itself is an abstraction. We will discuss abstraction and the different beautiful properties of mathematics in relation to some examples, where we describe the connection between algebra and geometry. Modern technological tools enable and encourage us to perform mathematical investigations of a kind that was once considered difficult.

In understanding upper secondary school students' interpretations of information in graphical representations of a distance--time graph and an ECG graph, little attention has been paid to the analysis of the condition of the conceptual development related to their utterances. Understanding this better can help improve the teaching of interpretations of information in graphical representations of different situations. This paper integrates results from 2 studies and 3 theoretical perspectives: Tall and Vinner's theoretical perspectives on learning, Chi's ontological perspectives on conceptual development and Friel's theoretical perspectives on interpretation of graphical information. The findings provide evidence to support the conjecture that iconic interpretations could be stimulated and generated as a result of student categorisation of a distance--time graph as an event, when in fact the graph is being used to describe and communicate a process. The outcome further indicates that students found a resemblance between the ECG diagram and the periodic function of f(x) = sin(x).

One of my last assignments at the university of Gothenburg was to teach a sequence of three seminars in programming for prospective teachers (n=37). The three seminars are given in the introduction of this manuscript. Since this was a course for prospective upper secondary teacher of mathematics, it was decided that it should be a course in programming for learning mathematics. This manuscript is a research article but also a manuscript about programming in GeoGebra, Python, and Wolfram Alpha. The examples I shared with my students led me to write a book about programming. See the reference list. [ABSTRACT FROM AUTHOR]