Teachers Reporting on Dynamic Geometry Utilization Related to Reasoning Competency in Danish Lower Secondary School

Within educational research, there has been an increased interest in recent years to develop learners’ language awareness (LA) in the context of plurilingual education. Recommendations and intentions are given pointing to the benefits, but there are few concrete tools for how it can be carried out in practice. The article seeks to contribute to this field by presenting two theoretically and empirically embedded models, developed in the PE-LAL project, used as didactic tools for developing LA activities in the context of plurilingual education across educational levels. The article describes how the models were used in a collaborative work between a researcher and in-service language teachers in a primary and lower secondary school in Denmark and gives three examples of teaching activities for grades 1, 5, and 7, respectively. Based on qualitative content analysis of semi-structured interviews with selected teachers exploring the teachers’ experiences with and perceptions of using the models, the article problematises that despite willingness the teachers find it difficult to use the models themselves because of, e.g. the monolingual norm reflected in language curricula in Denmark and the teachers’ limited linguistic knowledge. The article concludes with linking plurilingual education to teachers’ continuing professional development and teacher cognition.

Physical activity that strengthens muscles and bones at least three times per week are recommended, but few adolescents meet this target. The aim of our study was to adapt and evaluate the feasibility and preliminary efficacy of the Resistance Training for Teens (RT4T) program in Danish lower secondary schools. Developed and evaluated in Australia, the aim of RT4T is to provide adolescents with competence, confidence, knowledge, and motivation to participate in resistance training. Translation and adaptation were based on the PRACTIS-guide and involved workshops with physical education teachers. Three 7th-grade classes and three 9th-grade classes were recruited for the feasibility study and followed the program over eight weeks. Participants completed a push-up test, a standing long jump test, and a beep-test before and after the intervention. In addition, they completed a survey about their self-efficacy, motivation, and resistance training competency. Four qualitative interviews were performed with participants and their physical education teachers. The participating teachers were motivated for the program, but they had difficulties providing students with appropriate feedback. Students were motivated by the equipment, games, and their improvements in fitness, but motivation declined during the program. A total of 57 students completed the physical tests and answered the survey. Resistance training self-efficacy improved significantly, while most other measures improved over time, without reaching significance. Overall, the RT4T was acceptable and feasible in lower secondary schools in Denmark, but adjustments need to be made to increase the reach and efficacy of the program.

This study investigated risk factors for school absenteeism in a sample of 983 children in elementary and lower secondary schools in Denmark, using administrative data on absenteeism measured in the year following risk factor measurement. Risk factors were measured by survey (children and teachers) and register data. Two methods of determining importance of risk factors were compared: individual risk factors versus four domains of risk factors (psychological problems, physical problems, school factors, and demographic and family factors). Significant individual risk factors were found in all four domains. When teacher-reports of the children’s psychological problems were used, psychological problems was the risk factor domain that predicted school absenteeism best and the school-related factors domain predicted worst. The results highlight the need to distinguish between single risk factors that identify groups of individuals with elevated risk of school absence and the detection of risk factor domains that better predict school absence.

What makes Dynamic Geometry Environment (DGE) a powerful mathematical knowledge acquisition microworld is its ability to visually make explicit the implicit dynamism of thinking about mathematical geometrical concepts. One of DGE’s powers is to equip us with the ability to retain the background of a geometrical configuration while we can selectively bring to the fore dynamically those parts of the whole configuration that interest us. That is, we can visually study the variation of an aspect of a DGE figure while keeping other aspects constant, hence anticipating the emergence of invariant patterns. The aim of this paper is to expound the epistemic value of variation of the Dragging tool in DGE in mathematical discovery. Functions of variation (contrast, separation, generalization, fusion) proposed in Marton’s theory of learning and awareness will be used as a framework to develop a discernment structure which can act as a lens to organize and interpret dragging explorations in DGE. Such a lens focuses very strongly on mathematical aspects of dragging in DGE and is used to re-interpret known dragging modalities (e.g., Arzarello et al.) in a potentially more mathematically-relevant way. The exposition will centre about a specific geometrical problem in which two dragging trajectories are mapped out, consequently resulting in a DGE theorem and a visual theorem. In doing so, a new spectral dragging strategy will be introduced that literally allows one to see the drag mode in action. A model for the lens of variation in the form of a discernment nest structure is proposed as a meta-tool to interpret dragging experiences or as a meta-language to relate different dragging analyses which consequently might give rise to pedagogical and epistemological implications.

The ability of a student to obtain, analyze, measure and compare the many instances of a mathematical proposition in a dynamic geometry environment gives him/her opportunities to make conjectures and test a proposition. These roles for dynamic geometry software are widely acknowledged as having the potential to enrich the teaching of geometry. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by ‘‘what if’’ and ‘‘what if not’’ questions, has the potential to promote links between empirical and deductive reasoning. A classic set of problems in Euclidean plane geometry consists of finding the path of a point that is subject to given constraints. Its systematic study goes back to a lost work (in two books) of Apollonius of Perga, Plane loci (Botana and Valcarce 2003). Except for the simplest loci, such as lines, circles and perhaps the conics, this subject is usually avoided in most geometry texts. This is due to the common difficulties faced when mentally visualizing various objects with different movements. With the emergence of dynamic geometry software, the locus problems have attracted new interest from researchers. Using the locus generation features of these software one can easily obtain the locus of a point under some constraints. Although there are some minor differences, different dynamic geometry software, behave in a similar way when obtaining loci. In general, two objects must be selected: the first one, usually a point called the driving point or mover, is bound to a path, whereas the other, the locus point, must depend somehow on the first one. Since the element

Students’ conceptions through the lens of a dynamic online geometry assessment platform

Digital tools have a potential to change significantly the form of mathematical learning taking place in classrooms, with research pointing to various affordances in comparison with physical tools such as paper-and-pencil environments. Nevertheless, there is a scarcity of research that has examined in-depth the interrelated roles these two types of tools fulfil in mathematics learning. This issue of inter-related roles is important because, when digital tools are incorporated into classrooms, students usually also have notebooks and worksheets within which they carry out actions complementary to their use of digital tools. In this article, we focus on the use of dynamic geometry environments (DGEs) in conjecturing and proving, and, in particular, we examine the interplay between students’ paper-and-pencil activity and their use of a DGE during the producing and proving of a generalisation of a statement. We analyse a series of lessons involving secondary school students (aged 14–15, Grade 9) and show that, while DGE use supported the students in generalising a statement, they were initially unable to prove the generalisation while using the DGE, but subsequently succeeded through their paper-and-pencil activity. Our research illustrates the affordance of paper-and-pencil environments to support students in working on different representations, and thus highlights how the interplay between paper-and-pencil activity and DGE use can be important for the progress of conjecturing and proving. We also show the roles taken by the teacher in supporting the students’ work, and point to the need for further research into the back-and-forth use of digital and physical tools.

This study concerns geometric prediction, a process of anticipation that has been identified as key in mathematical reasoning, and its possible constructive relationship with explorations within a Dynamic Geometry Environment (DGE). We frame this case study within Fischbein’s Theory of Figural Concepts and, to gain insight into a solver’s conceptual control over a geometrical figure, we introduce a set of analytical tools that include: the identification of the solver’s geometric predictions, theoretical and phenomenological evidence that s/he may seek for, and the dragging modalities s/he makes use of in the DGE. We present fine-grained analysis of data collected during a clinical interview as a high school student reasons about a geometrical task, first on paper-and-pencil, and then in a DGE. The results suggest that, indeed, the DGE exploration has the potential of strengthening the solver’s conceptual control, promoting its evolution toward theoretical control.

This article sheds light on a set of religious literacy practices which have so far received very little scientific attention, namely ‘quranic app practices’. Quranic app practices are conceptualized as ways of engaging with the Quran through religious apps on smartphones or tablets. The article is based primarily on an explorative group interview with four young people from a Year 8 class in a multilingual lower secondary school in Denmark. The youth have Somali and Afghan background, consider themselves practicing Muslims and can be described as ‘religious heritage learners of Arabic’ (Temples 2013). The article explores the quranic app practices of the young people as reported in the interview and shows how the young people use the quranic apps to engage with the sacred text of the Quran in different ways, with different objectives and drawing on different languages. Building on Rosowsky’s (2008, 2010, 2012, 2013, 2015) research on faith literacies, the analysis reveals that the young people have radically different understandings of that it entails to ‘read the Quran’ and that their quranic app practices vary greatly, but also that their choice of app is consistent with their stated objectives, priorities, preferences and practices. The different quranic app practices thus all appear meaningful, consistent and legitimate. Based on the analysis, the article calls for further investigation of young people’s quranic app practices.

This study was designed to investigate how 13-year-old students reason about 2-dimensional shapes while working with Dynamic Geometry Software (DGS).

Purpose – The purpose of this paper is to examine how students’ participation in an integrated school food program was related to the development of components of food and health-related action competence (F & HRAC). These components were understood to be the knowledge, insight, motivation, ownership and social skills that made students able to take action regarding food and health in everyday life. Design/methodology/approach – Research was undertaken as a single case study of the development of an integrated education and health program called LOMA-Local Food (LOMA) in a secondary school in Denmark. Qualitative methods were applied, including an action research component, where researcher and teachers examined how students developed action competence. The program was based on a whole school approach with the aim of improving F & HRAC. As a way to obtain this, students participated in planning, preparing, cooking and serving their own school food as integrated in curriculum. The study applied the Health Promoting Schools’ (HPS) conceptual framework and the I nvestigation, V ision, A ction and C hange (IVACE) approach. Findings – Students who participated in LOMA educational activities became motivated for developing a food F & HRAC, which included components such as knowledge, insight, motivation, ownership, action experience, commitment, cooperation and critical thinking. Students developed practical skills related to food and health, when they were cooking healthy school food together with professionals and peers. The study also points to the importance of capacity building among teachers. The IVACE matrix is suggested as a relevant tool for monitoring forms of participation that contributes to students’ development of F & HRAC. Practical implications – There were indications of how participation in LOMA contributed to students’ development of F & HRAC. The practical implication of this is that “setting” is very important for the success of food and health education initiatives. In this integrated approach the production kitchen and the dining hall are indispensable. Also the new organization of the school day and the introduction of a shared daily meal are important practical components for the improvement of the learning environment. The possibility of combining theory and practice seem conducive for students’ achievement of action competence. Social implications – The current study is an example of how the IVACE matrix can be applied in order to plan, conduct and evaluate LOMA educational activities, which could be considered as a contribution to the HPS scientific community. It would be useful for other schools that intend to apply the LOMA approach. However, more research is needed, where teachers, students, staff and other stakeholders collaborate in an action research process. This could promote students’ health and support other initiatives regarding public health, sustainable development and democracy. Originality/value – This research may have implications for the way that school food programs are developed and implemented if they are to make a contribution to students’ development of F & HRAC. Taking the political interest for research-based interventions into account, it is important that future strategies include teachers’ capacity building. Research is also needed regarding further development and test of the IVACE matrix as a method in participatory, health education approaches. This should be seen in combination with a renewed focus on integrated curricula models related to the on-going discussion on redesign of western school curricula.

The purpose of the article is to highlight the specifics and content of the civic component in the Danish school education system, aimed at developing the personality potential of students. The research exploited such methods as: analysis and generalization of scientific and pedagogical sources on the analyzed problem; methods of interpretation and generalization of the facts regarding the implementation and content of civic education in the Danish school system, which ensures the development of the personality potential of students; monographic method – for a logical presentation of the obtained results. It has been found that civic education in Denmark arose in the 19th century with the very system of formal education in the state with the aim of promoting the formation of Danish national identity based on national Christian ethics. It is emphasized that the concept of civic education is widely accepted and reflects the broader social concepts of Danish democracy as a common way of life and consensual decision-making, contributing to both the formation of national identity of young people and their belonging to the globalized world. It is noted that in the institutions of general secondary education in Denmark, in accordance with tradition and law, civic education, giving priority to civic knowledge and democratic values, is based on a complex combination of educational, interdisciplinary and extracurricular components. The content and purpose of the compulsory subject “Social Studies” in Danish schools is outlined. It has been established that the Danish educational tradition of education connects civic education with the main provisions of the German concept of Bildung, which understands education as the personal and cultural development of an individual. It was concluded that the integration of civic education in the Danish secondary school system, implemented through civic subjects, the study of cultural diversity and the formation of civic values, ensures the maintenance of the democratic foundations of Danish society in general and to some extent contributes to the development of the personal potential of students. The effectiveness of civic education in secondary schools in Denmark in modern conditions is determined by the preparation of students for conscious responsible activities for the benefit of their country.

This paper describes the problem solving behavior of two preservice teachers as they worked individually on three nonroutine geometry problems. A dynamic tool software, namely the Geometer’s Sketchpad, was used as a tool to facilitate inquiry in order to uncover and investigate the patterns of metacognitive processes. Schoenfeld’s (1981) model of episodes and executive decisions in mathematics problem solving was used to identify patterns of metacognitive processes in a dynamic geometry environment. During the reading, understanding, and analysis episodes, the participants engaged in monitoring behaviors such as sense making, drawing a diagram, and allocating potential resources and approaches that helped make productive decisions. During the exploring, planning, implementation, and verification episodes, the participants made decisions to access and consider knowledge and strategies, make and test conjectures, monitor the progress, and assess the productivity of activities and strategies and the correctness of an answer. Cognitive problem-solving actions not accompanied by appropriate metacognitive monitoring actions appeared to lead to unproductive efforts. Redirection and reorganizing of thinking in productive directions occurred when metacognitive actions guided the thinking and when affective behaviors were controlled.

In this chapter, I explore the use of dynamic, touchscreen technology for supporting the development of bilingual learners’ mathematical discourse during pair-work exploratory activities in calculus. This study emerged from increasing linguistic diversity in mathematics education and a new mathematics curriculum movement which emphasizes fluency of mathematical communication and the consistent use of technology for learning mathematics. In response to the changing needs and curriculum, this study investigates bilingual high school students’ communication when they interact with touchscreen dynamic geometry environments (DGEs) during calculus discussion and exploration. Specifically, I address bilingual learners’ linguistic and nonlinguistic means of communication in the activity with touchscreen DGEs and the mathematical competence demonstrated by the students in the activity. Using the theoretical framing of thinking-as-communicating, I provide a vignette with qualitative analyses of one pair of bilingual learners’ communication, focusing on the students’ language, gestures, and touchscreen-dragging actions with touchscreen DGEs during the mathematical activity. Results suggest that a multimodal lens for understanding bilingual learners’ mathematical thinking, along with the touchscreen interface and dynamic features of the DGEs, facilitated productive discussion and exploration about calculus for the students. This study raises implications for achieving equity in today’s increasingly multilingual and technological learning environments.

ABSTRACTBackground: For years researchers have been engaged in revealing the impact of the hidden curriculum in physical education (PE) on students’ participation and non-participation. The hidden PE curriculum encompasses the knowledge, the relations, the assumptions, the norms and the beliefs that students unconsciously and unintentionally learn through the process of education. As the hidden curriculum reinforces particular values and attitudes among students in a very subtle and often unnoticed fashion, it limits students’ possibilities for becoming aware of, and thus reporting, how the tacit messages communicated through the hidden curriculum impact on their position of participation and non-participation. Thus, in this article, we argue that examining students’ silences, that is the things students do not voice, is significant for the understanding of the impact of the hidden curriculum on students’ participation and non-participation in PE.Purposes: In this article, we aim to develop insight into students’ silences in order to elucidate how aspects of the hidden curriculum serve to reinforce some students’ non-participation in PE. Much attention has been devoted to particular values and attitudes unintentionally transmitted by teachers in PE. However, in this article, we examine how the everyday exchanges between the students themselves may also convey a hidden set of meanings, that impact on students’ actual experiences of the PE curriculum, and thus mitigate the intended effects of students’ participation.Research design: The backdrop for this article is a single-case study carried out in a multi-ethnic and co-educational secondary school in Denmark from January to December 2014. The article draws on material collected through focus group interviews with 7th grade students (including participant-diagrams filled out by students) along with observations of their PE classes. The observations took place once a week throughout the whole calendar year.Findings: In the article, we point to students’ intentional silences that are highly reflective of the normative expectations negotiated within the peer group. In addition, we show that the pressures toward social conformity have a direct impact on the positions of non-participation intentionally taken up by some of the less socially respected students in PE. These students were highly aware that how they behaved in PE and what they voiced in the interviews might have consequences for their peer group connections within PE and for their social reputation among peers outside of PE. In addition, we add to the current literature on student silence by pointing to a category of non-privileged silences. These silences revealed that a minor group of students were not aware of or had not recognized their position as non-participants in PE. Moreover, they appeared unable to imagine that things could be different and to voice a desire for change.Conclusions: We argue that our findings reveal critical aspects of students’ non-participation that would be difficult to access if we did not listen to, hear and attempt to understand students’ silences. In order to extend the knowledge base on students’ participation and non-participation in PE, we hope that this article may also encourage other researchers to let students’ silences breathe and speak.