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Any ball or small stone that is thrown follows a certain path. Many of us have experienced throwing objects towards a specific goal many times. If we throw a ball to someone or against something, the ball will follow a certain path, a visualized motion. Hopefully you can imagine a motion path if you are asked to throw something towards a target some 10 meters away. Somehow we can imagine the height and path in order to get the ball (or small stone) in the basket or what we are aiming at. If we want to shoot a basketball to go into the basket, we consider elevation angle and initial velocity also. Most of us humans do that without thinking about it. [ABSTRACT FROM AUTHOR]

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]

Relativistic time deals with concepts as spacetime and time dilation. I will try to explain this as clearly as possible. To explain the concept of spacetime is not at all impossible, although it is four-dimensional. But that means that we define it using four numbers. If we describe us through age, weight, height, and IQ, that would be a fourdimensional representation of us. [ABSTRACT FROM AUTHOR]

This is an article about the methodology of substitution in problem solving of mathematics. The operative use of substitution depends on two things: in an equation or situation in which variables appear, a substitution is a change of variable, and second, the change of variable is effective only if the change of variable makes the expression easier to understand and makes it possible to solve the problem. Substitution will be successful if the original equation has a symmetry or especial property that is possible to explore, and the proficiency in using the method of substitution depends on this. [ABSTRACT FROM AUTHOR]

Friction occurs early in our lives. We rub our hands and stamp our feet to get them warm. Another effect of friction is the use of winter tires at cars, new bicycle tires, or special shoes for icy surface. This article discusses how to present the way from friction to air resistance through GeoGebra applets. The approach has been analyzed using the theory of technology as amplifier and reorganizer, theory of variation, and the notion of internal and external representations. The dragging feature of GeoGebra enables students to experience variation and make conjectures. Furthermore, students' static internal representations will be enhanced to dynamic external representations, which can play an important role in students' learning. Three aspects of fidelity have been discussed with regard to the use of GeoGebra based applets and how such applets need to be critically designed and assessed in order to support students' learning. [ABSTRACT FROM AUTHOR]

This article presents two different classroom situations in which two pairs of students, one in grade 2 and the other in grade 6, attempted mathematically challenging problems through GeoGebra applets. Their reasoning, conjecture making, and argumentation is analyzed using Pea's (1985, 1987) theory of technology as amplifier and reorganizer and the notion of internal and external representations. GeoGebra played the role of amplifier and reorganizer in enabling students' explorations and led them to make conjectures. Furthermore, it was observed that students' static internal representations were enhanced to more dynamic external representations and this played a significant role in developing students' thinking. The three aspects of fidelity have been discussed with regard to the use of GeoGebra based applets and how such applets need to be critically designed and assessed in order to support students' learning. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

The work in the Comenius Network project Developing Quality in Mathematics Education II (DQME II) has a main focus on development and evaluation of modelling tasks. One reason is the gap between what mathematical modelling is and what is taught in mathematical classrooms. This article deals with one modelling task and focuses on how two teachers handle this task in their classrooms. Initially, the notion of a teacher being the manager of the learning process is elaborated. Using criteria developed from taking this perspective, we analyse classroom sequences to determine the nature of “teaching like a manager” and the actions that are classroom evidence for working in this way. Conclusions include recommendations for how to realize “acting like a manager” in mathematics classrooms. [ABSTRACT FROM AUTHOR]

The study reported in this article deals with the observed actions of prospective Swedish mathematics teachers as they were working with a modeling situation. These prospective teachers were preparing to teach in Grades 4 to9 or in the gymnasium (Grades 10 to 12) and were students in a course in mathematical modeling. The larger study of which this study was a part focused on these students' understanding of modeling and how they related mathematical models to the real world. This article also attempts to illustrate how mathematics is intertwined with many other subjects, in this case medicine. [ABSTRACT FROM AUTHOR]

Information about several papers discussed at the Seventh Congress of the European Society for Research in Mathematics (CERME7) is presented. One paper cited the theoretical description of modelling and applications. Another paper described the concept of problem solving as the method of handling pre-mathematised word problems.