Pauline Vos

• Berget, Ingeborg Katrin Lid & Vos, Pauline (2024). Kva er matematisk modellering? Tangenten - Tidsskrift for matematikkundervisning. ISSN 0802-8192. 35(2), s. 2–8. Vis sammendrag

  Kva er matematisk modellering? Kvifor matematisk modellering? Korleis ser ei modelleringsoppgåve ut? Er ein modell det same som ein representasjon? Modellering har vore med i norske læreplanar i fleire tiår. Ikkje alle lærarar har hatt om matematisk modellering i eiga utdanning, noko som kan føre til usikkerheit knytt til korleis ein bør undervise om det (Berget, 2023). Gjennom denne artikkelen vil vi gi nokre svar på desse spørsmåla. Målet er å få fram mangfaldet i korleis ein kan forstå omgrepet, og at mange ulike elevaktivitetar kan inngå som arbeid med matematisk modellering

• Vos, Pauline; Wiik, Anders & Hernandez Martinez, Paul (2024). “Imagine, maths is used anywhere, and we don’t get to know this”—upper secondary students and the relevance of advanced mathematics. Frontiers in Education. ISSN 2504-284X. 9. doi: 10.3389/feduc.2024.1338205.
• Frejd, Peter & Vos, Pauline (2024). The spirit of mathematical modeling – a philosophical study on the occasion of 50 years of mathematical modeling education. The Mathematics Enthusiast. ISSN 1551-3440. 21(1-2), s. 269–300. doi: 10.54870/1551-3440.1626. Fulltekst i vitenarkiv
• Eriksen, Stig & Vos, Pauline (2023). Formal intentions in a new mathematics curriculum versus example tasks published before the standardized examinations–do they align? I Drijvers, Paul; Csapodi, Csaba; Palmer, Hanna; Gosztonyi, Katalin & Kónya, Eszter (Red.), Proceedings of the thirteenth Congress of the European Society for Research in Mathematics Education (CERME13). European Society for Research in Mathematics Education. ISSN 978-963-7031-04-5. s. 3948–3955. Vis sammendrag

  In 2020, a new Norwegian mathematics curriculum formally defined five competencies as pillars for mathematics learning, such as exploring, reasoning, representing, generalizing, or modelling. Shortly after the curriculum’s launch, example tasks were published to show what type of tasks could come in the national standardized examinations. Within our theoretical framework, we interpret both the described competencies and the example tasks as resources in a discourse on an intended curriculum. We studied their alignment, namely, the extent to which the competencies were needed to solve the example tasks. We developed a scoring analysis with good inter-rater reliability. We found that the competencies were mainly needed for tasks for the theoretical stream and hardly for practical streams. Also, modelling was virtually absent. The first published example tasks communicated the ideals of the intended curriculum better than those published later.

• Vos, Pauline & Bolstad, Oda Heidi (2023). Using preservice teachers’ perceptions of a project-based statistics course to investigate knowledge creation in collaborative settings. I Drijvers, Paul; Csapodi, Csaba; Palmer, Hanna; Gosztonyi, Katalin & Kónya, Eszter (Red.), Proceedings of the thirteenth Congress of the European Society for Research in Mathematics Education (CERME13). European Society for Research in Mathematics Education. ISSN 978-963-7031-04-5. s. 5062–5069. Vis sammendrag

  In project-based courses, students learn new knowledge through ‘knowledge creation in collaborative settings’. We studied how personal, social and cultural aspects interact in such learning situations. Our empirical basis was a project-based course in statistics in primary teacher education. Participants (n=21) wrote reflective, post-course reports, in which we identified five ‘drivers’ of their learning. These were: (1) awareness of adaptable teaching conventions; (2) discussing with and explicating to others; (3) competencies relevant later, (4) knowledge of how statistics is used in society; (5) knowledge about oneself. The drivers connected the personal, social and cultural aspects in different ways. For instance, personal learning connected to the social through the group work, and to the cultural, such as teaching conventions and the societal role of statistics. These cross-connections were powerful drivers of learning and can be used for designing other courses.

• Eriksen, Stig & Vos, Pauline (2022). Kjerneelementer og eksempeloppgaver. Tangenten - Tidsskrift for matematikkundervisning. ISSN 0802-8192. 33(3), s. 45–51.
• Vos, Pauline & Frejd, Peter (2022). The modelling cycle as analytic research tool and how it can be enriched beyond the cognitive dimension. I Hodgen, Jeremy; Geraniou, Eirini; Bolondi, Giorgio; Feretti, Federica & Ferretti, Federica (Red.), Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education (CERME12). European Society for Research in Mathematics Education. ISSN 979-1-22-102537-8. s. 1–8. doi: https:/hal.archives-ouvertes.fr/hal-03759063.
• Vos, Pauline & Frejd, Peter (2022). Grade 8 students appropriating Sankey diagrams: The first cycle in an educational design research. Journal on Mathematics Education (JME). ISSN 2087-8885. 13(2), s. 289–306. doi: 10.22342/jme.v13i2.pp289-306. Fulltekst i vitenarkiv
• Frejd, Peter & Vos, Pauline (2021). A commentary on the Special Issue “Innovations in measuring and fostering mathematical modelling competencies”. Educational Studies in Mathematics. ISSN 0013-1954. 109, s. 455–468. doi: 10.1007/s10649-021-10122-4. Fulltekst i vitenarkiv Vis sammendrag

  This is a commentary on the ESM 2021 Special Issue on Innovations in Measuring and Fostering Mathematical Modelling Competencies. We have grouped the ten studies into three themes: competencies, fostering, and measuring. The first theme and the papers therein provide a platform to discuss the cognitivist backgrounds to the different conceptualizations of mathematical modelling competencies, based on the modelling cycle. We suggest theoretical widening through a competence continuum and enriching of the modelling cycle with overarching, analytic dimensions for creativity, tool use, metacognition, and so forth. The second theme and the papers therein showcase innovative ideas on fostering and on the definition and analysis thereof. These reveal the need for a social turn in modelling research in order to capture aspects of student collaboration and agency, as well as tensions in fostering when tasks are derived from real-world scenarios, but socio-mathematical norms come from the (pure) mathematics classroom. The third theme, measuring, and the papers therein offer insights into the challenges of positivist research that aims to develop innovative measurement instruments that are both reliable and valid, particularly in light of student group work, cultural background, and other socio-cultural aspects. Drawing on the three discussions, we go on to make recommendations for further research.

• Greefrath, Gilbert & Vos, Pauline (2021). Video-based word problems or modelling projects – classifying ICT-based modelling tasks. I Leung, Frederick Koon Shing; Stillman, Gloria A.; Kaiser, Gabriele & Wong, Ka Lok (Red.), Mathematical modelling education in East and West. Springer. ISSN 9783030669959. s. 489–500. doi: 10.1007/978-3-030-66996-6_41.
• Radmehr, Farzad & Vos, Pauline (2020). Issues and challenges of 21st century assessment in mathematics education. I Leite, Laurinda (Red.), Science and mathematics education for 21st century citizens: challenges and ways forward. Nova Science Publishers. ISSN 978-1-53618-334-4. s. 437–462. Vis sammendrag

  In the 20th century, assessment in mathematics classrooms mainly focused on knowledge with low cognitive demands, in particular on testing students on memorized calculation procedures. In the 21st century, we need to focus more on higher order thinking (HOT), which should also be reflected in mathematics assessment. This will require quite an overhaul, not only in assessment, but also of instruction as teachers will need to carefully look at how to best prepare students for assessment formats that activate HOT. In this chapter, we will explain that assessment tasks activating HOT focus on inquiry and creativity rather than on reproduction and imitation. To assist with analysing and re-designing tasks, we explain the usefulness of the Revised Bloom’s Taxonomy as a framework to conceptualize cognitive processes (e.g., summarizing, comparing, explaining, exploring, analysing, critiquing, and creating). We will validate the framework through four exemplary types of tasks: problem-posing tasks, puzzle-problems, modelling tasks based on realistic questions, and cloze paragraphs; and we will discuss how assessment tasks can be designed or re-designed based on the framework. We will review some challenges in the research and development of mathematics assessment for the 21st century, present research-based evidence on how assessment tasks activating HOT in mathematics can be accessible tasks that lead to a better conceptual understanding of mathematics, and finally, we discuss issues of reliability and validity.

• Vos, Pauline; Hernandez-Martinez, Paul & Frejd, Peter (2020). Connections of science capital and the teaching and learning of mathematical modelling. I Stillman, Gloria A.; Kaiser, Gabriele & Lampen, Christine Erna (Red.), Mathematical Modelling Education and Sense-making. International Perspectives on the Teaching and Learning of Mathematical Modelling. Springer Nature. ISSN 978-3-030-37672-7. s. 33–38. doi: 10.1007/978-3-030-37673-4_3.
• Vos, Pauline (2020). On science museums, science capital and the public understanding of mathematical modelling. I Stillman, Gloria A.; Kaiser, Gabriele & Lampen, Christine Erna (Red.), Mathematical Modelling Education and Sense-making. International Perspectives on the Teaching and Learning of Mathematical Modelling. Springer Nature. ISSN 978-3-030-37672-7. s. 63–73. doi: 10.1007/978-3-030-37673-4_6. Fulltekst i vitenarkiv
• Poudel, Amrit Bahadur; Vos, Pauline & Shults, F. LeRon (2020). Students of Religion Studying Social Conflict through Simulation and Modelling - An Exploration. I Verhagen, Harko; Borit, Melania; Bravo, Giangiacomo & Wijermans, Nanda (Red.), Advances in Social Simulation. Looking in the Mirror. Springer Nature. ISSN 978-3-030-34126-8. s. 379–383. doi: 10.1007/978-3-030-34127-5_37.
• Vos, Pauline; Dallas, Markos; Poudel, Amrit Bahadur & Shults, F. LeRon (2020). Using Social Simulations in Interdisciplinary Primary Education - an Expert Appraisal. I Verhagen, Harko; Borit, Melania; Bravo, Giangiacomo & Wijermans, Nanda (Red.), Advances in Social Simulation. Looking in the Mirror. Springer Nature. ISSN 978-3-030-34126-8. s. 451–456. doi: doi:%2010.1007/978-3-030-34127-5_45.
• Vos, Pauline & Frejd, Peter (2020). The object-tool duality in mathematical modelling - a framework to analyze students’ appropriation of Sankey diagrams to model dynamic processes. Avances de Investigación en Educación Matemática (AIEM). 17(17), s. 52–66. doi: 10.35763/aiem.v0i17.305. Fulltekst i vitenarkiv
• Vos, Pauline (2020). Task contexts in Dutch mathematics education. I VAN DEN HEUVEL, M P (Red.), National Reflections on the Netherlands Didactics of Mathematics; Teaching and Learning in the Context of Realistic Mathematics Education. Springer. ISSN 978-3-030-33823-7. s. 31–53. doi: 10.1007/978-3-030-33824-4_3. Fulltekst i vitenarkiv Vis sammendrag

  This chapter offers a description of task contexts in mathematics education in the Netherlands. International comparative studies show that the Dutch average percentage of mathematics tasks with real-life connections per lesson exceeds any other country by far. This tradition goes back more than 500 years, when the earliest mathematics textbooks in the Dutch language consisted entirely of tasks, in which mathematics was put to use in commercial contexts. In this chapter characteristics of contexts in mathematics tasks in the Netherlands are studied. Underlying frame is the notion of usefulness, which is a perception by students on future practices outside school. A distinction is made between bare tasks (without contexts), tasks with mathematical contexts (e.g. matchstick pattern problems), dressed-up tasks (hiding a mathematical question), tasks with realistic contexts and questions that are useful within the context, and tasks with authentic contexts. The empirical part of this chapter is an analysis of a mathematics textbook chapter and a sample of examination papers. The analysis shows that Dutch mathematics education contains many links to real-life, which is not just verbally presented, but also visually with drawings, photos, diagrams and other visualisations. The contexts are drawn from a wide spectrum of areas in real-life, reflecting that mathematics can be found anywhere in society. The examinations contain more authentic aspects than the textbook, and the higher level examinations have more authentic aspects than the lower level examinations. Nevertheless contexts can be artificial and the posed questions would not be asked by actors within the context. Task contexts often come from recreational or professional practices, demonstrating to students the usefulness of mathematics in their future lives beyond school.

• Poudel, Amrit Bahadur; Vos, Pauline & Shults, F. LeRon (2019). Students of Development Studies learning about modelling and simulations as a research approach in their discipline. I Jankvist, Uffe Thomas; Van den Heuvel-Panhuizen, Marja & Veldhuis, Michiel (Red.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. European Society for Research in Mathematics Education. ISSN 978-90-73346-75-8. s. 1256–1263. Vis sammendrag

  Researchers in the social sciences are increasingly using modelling and simulation (M&S) as a research approach. They create virtual worlds to discover relations across variables, and to test theories and potential policies. We introduced this research approach to students in the department of Development Studies at our university. The goal was to investigate the way in which such students can gain meta-knowledge about M&S-based research, that is, general knowledge about its nature and rationale. We organized a seminar to introduce the research approach and illustrated it with a simulation of the behaviour of agents with varying levels of tolerance towards their outgroup neighbours (based on Schelling’s segregation model). We analysed students’ interactions through a socio-cultural lens. Students were able to gain meta-knowledge about M&S-based research, which they judged as useful for their future as professionals when working on development projects.

• Wiik, Anders & Vos, Pauline (2019). “I want a high-educated job that pays well and is fun”; secondary students’ reasons for selecting the Advanced Mathematics course. I Jankvist, Uffe Thomas; Van den Heuvel-Panhuizen, Marja & Veldhuis, Michiel (Red.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. European Society for Research in Mathematics Education. ISSN 978-90-73346-75-8. s. 1573–1580. Vis sammendrag

  Many students experience the content of the mathematics classroom irrelevant to their lives. At the same time, achievement in mathematics is widely regarded as an important part of general education. In this paper, we report on a study regarding students’ relevance beliefs. For this, we network Engeström’s cultural-historical activity theory with Skovsmose’s theory on foregrounds, and introduce the terms ‘realized’ and ‘emergent’ activity systems. Students provided a wide variety of beliefs, in which the belief of mathematics as door opener to higher education and key to options in life was leading. The belief in the exchange-value of mathematics overshadowed beliefs on the use-value of school mathematics (for other school subjects, for undefined jobs or for mental training). We recommend teachers and textbook authors to include more information on the relevance of mathematics in future studies and jobs.

• Gjesteland, Thomas & Vos, Pauline (2019). Affect and mathematical modeling assessment – A case study on students’ experience of challenge and flow during a compulsory mathematical modeling task by engineering students. I Chamberlin, Scott A. & Sriraman, Bharath (Red.), Affect in Mathematical Modeling. Springer Publishing Company. ISSN 978-3-030-04431-2. s. 257–272. doi: 10.1007/978-3-030-04432-9-16. Fulltekst i vitenarkiv Vis sammendrag

  This chapter describes a study on engineering students’ affect while working on the Tracker Project Task, a group assessment task that asks students (1) to use digital tools (the camera in their smart phones and free tracker software) to capture the movement of an object, (2) to mathematically model that movement, and (3) to create a poster reporting on the video analysis of the movement. We applied an activity-based conceptualization of affect in mathematics (“do you like this activity?”), which differs from a subject-based conceptualization of affect (“do you like mathematics?”). A subject-based conceptualization has two drawbacks: (1) it does not distinguish among different aspects of mathematics, and (2) it draws in students’ bias and beliefs from earlier, often bad experiences of poor mathematics teaching. We found an activity-based operationalization of affect by using the concepts of challenge and flow. Flow is a state of absorption, in which people forget about time and experience feelings of happiness. We assessed n ¼ 346 students through the Tracker Project Task. To study affect, we developed an instrument of 10 items (Likert-type) to measure students’ experience of challenge and flow. We administered the survey through a web-based platform yielding a high response rate (n ¼ 239, 69%) and good reliability (Cronbach’s Alpha: 0,795). The results revealed that three out of five students experienced challenge and flow, which expresses students’ positive affect regarding a mathematical assessment activity. This can be ascribed to, on the one hand, the activity and the instrument not clearly being related to mathematics, and thus not being tainted by students’ earlier negative experiences with mathematics. On the other hand the Tracker Project Task had characteristics that can bring about flow: being open, offering ample time to submit the product, being accessible to all students (low floor), but also enabling the better students to challenge themselves further (high ceiling). Such characteristics may be better feasible within mathematical modeling assessment than canonical mathematics assessment.

• Wiik, Anders & Vos, Pauline (2018). Hvorfor lære om logaritmer? Tangenten - Tidsskrift for matematikkundervisning. ISSN 0802-8192. 29(4), s. 7–12.
• Vos, Pauline (2018). “How Real People Really Need Mathematics in the Real World” — Authenticity in Mathematics Education. Education Sciences. ISSN 2227-7102. 8(4), s. 1–14. doi: 10.3390/educsci8040195. Fulltekst i vitenarkiv Vis sammendrag

  This paper discusses authenticity from the perspective of mathematics education. Often, school mathematics offers students inauthentic word problems, which don’t show the authentic usefulness of mathematics in real life. In some tasks, authentic aspects are combined with inauthentic ones (e.g., an authentic context, but the question is artificial and different from what people within that context would ask). Several studies show that students are more motivated by authentic questions than by authentic contexts. Embedding these findings, I discuss issues associated with defining authenticity in education. A first issue is that philosophers use the term to characterize a person’s existential expressions (e.g., being true to oneself), whereas in education, we use the term for learning environments, artefacts, etc. Second, some researchers define authentic learning environments according to criteria (being open to different approaches, simulate a real-life activity, etc.), but I will illustrate that inauthentic activities can comply with such criteria as well. Alternatively, I suggest using the term for separate aspects in a learning environment (contexts, questions, etc.), and define authenticity as a social construct rather than as a subjective perception. In this way, a community (teachers, students, out-of-school experts) can reach agreement on the nature of this characteristic. For an aspect to be authentic, it needs to have: (1) an out-of-school origin and (2) a certification of originality (e.g., by bringing artifacts physically into a classroom or by testimony of an expert). This approach is illustrated by a study on students’ project work during an excursion to a mathematics research workplace.

• Rensaa, Ragnhild Johanne & Vos, Pauline (2018). Interpreting teaching for conceptual and for procedural knowledge in a teaching video about linear algebra. Skrifter från Svensk förening för matematikdidaktisk forskning. ISSN 1651-3274. 12, s. 109–118. Fulltekst i vitenarkiv Vis sammendrag

  The aim of this study is to investigate teaching videos about mathematics, seeking to uncover research-based foundations for their quality. By drawing on the notions of procedural and conceptual knowledge, the research was operationalized by asking professionals in undergraduate mathematics education (n=18) to interpret sections of a teaching video. The video dealt with a topic in linear algebra. The results indicate rather divergent interpretations of conceptual knowledge. This can hinder a reliable evaluation of teaching in terms of aiming for conceptual or procedural knowledge. It is recommended that the notions should be carefully used, defined and explained when used to evaluate the quality of teaching videos in particular, or of teacher’s explanations in classrooms in general.

• Vos, Pauline & Roorda, Gerrit (2018). Students’ readiness to appropriate the derivative - meta-knowledge as support for the ZPD. Proceedings of the International Groups for the Psychology of Mathematics Education. ISSN 0771-100X. 4, s. 387–394.
• Gjesteland, Thomas; Vos, Pauline & Wold, Margrethe (2018). Mathematical modelling and activation – a study on a large class, a project-based task and students’ flow. International Network for Didactic Research in University Mathematics : INDRUM. ISSN 2496-1027. s. 175–184. Vis sammendrag

  We studied how engineering students in a large class (n=346) can be activated by a project-based task, in which they have to model mathematically the motion of an object. The students had to throw an object, use (1) their smart phones for filming, and (2) tracker software for capturing the motion. Through a poster, they had to report their video analysis. We framed activation through the concept of flow, which is a state of being fully absorbed by an activity. We administered a web-based questionnaire (response rate 69%). The results show that such a project-based task is feasible with >300 students and activated them: three out of five experienced flow. Also, we validated the theory that for experiencing flow, a task must be perceived as challenging and that one’s skills should match that challenge.

• Gjesteland, Thomas; Vos, Pauline & Wold, Margrethe (2017). Students experiencing flow in a Physics lab task using mobile phones and free software. I Tveiten, Oddgeir (Red.), Smart Universities: Education´s Digital Future. Logos Verlag Berlin. ISSN 978-3-8325-4595-6. s. 37–44.
• Hernandez-Martinez, Paul & Vos, Pauline (2017). “Why do I have to learn this?” A case study on students’ experiences of the relevance of mathematical modelling activities. ZDM: Mathematics Education. ISSN 1863-9690. 50(1-2), s. 245–257. doi: 10.1007/s11858-017-0904-2. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we explore how students can experience the relevance of mathematical modelling activities. In the literature we found that relevance is a connection among several issues (relevance of what? to whom? according to whom? and to what end?). We framed this concept in terms of Cultural-Historical Activity Theory (CHAT), a theory for analysing how individuals engage in activities within social environments. We designed modelling activities within a mathematics course for engineering students: there were ample mathematical modelling tasks, a guest lecture by an employee from an engine company who used mathematical modelling in his job, and a group work modelling assessment with a presentation to the whole group. After the course, we interviewed ten students with a wide range of final grades in the course. We analysed the interview data in light of the theoretical framing of the concept of relevance. Our analysis showed that, generally, students experienced the modelling activities as relevant, and that they imagined themselves working in professional practices for which mathematics is relevant. However, doing mathematics was also judged as being relevant only to obtain grades, leave school and enter professions for which mathematics might not be needed. We offer recommendations for making mathematics education more relevant to more students.

• Kim, Rae Young; Suurtamm, Christine; Silver, Edward; Ufer, Stefan & Vos, Pauline (2017). Topic Study Group No. 39: Large Scale Assessment and Testing in Mathematics Education. I Kaiser, Gabriele (Red.), Proceedings of the 13th International Congress on Mathematical Education ICME-13. Springer. ISSN 978-3-319-62596-6. s. 565–569. doi: 10.1007/978-3-319-62597-3_66.
• Djepaxhija, Brikena; Vos, Pauline & Fuglestad, Anne Berit (2017). Assessing Mathematizing Competences Through Multiple-Choice Tasks: Using Students’ Response Processes to Investigate Task Validity. I Stillman, Gloria A.; Blum, Werner & Kaiser, Gabriele (Red.), Mathematical Modelling and Applications - Crossing and Researching Boundaries in Mathematics Education. Springer. ISSN 978-3-319-62967-4. s. 601–611. doi: 10.1007/978-3-319-62968-1_50.
• Vos, Pauline & Roorda, Gerrit (2017). Long-Term Development of How Students Interpret a Model: Complementarity of Contexts and Mathematics. I Stillman, Gloria A.; Blum, Werner & Kaiser, Gabriele (Red.), Mathematical Modelling and Applications - Crossing and Researching Boundaries in Mathematics Education. Springer. ISSN 978-3-319-62967-4. s. 479–489. doi: 10.1007/978-3-319-62968-1_40.
• Baartman, Desire & Vos, Pauline (2017). Analyse av deleksamen for grunnskolelærerutdanningen. Tangenten - Tidsskrift for matematikkundervisning. ISSN 0802-8192. 28(1), s. 41–48. Fulltekst i vitenarkiv
• Hogstad, Ninni Marie; Isabwe, Ghislain Maurice Norbert & Vos, Pauline (2016). Engineering students' use of visualization to communicate about representations and applications in a technological environment. International Network for Didactic Research in University Mathematics : INDRUM. ISSN 2496-1027. s. 211–220.

Se alle arbeider i Cristin

• Berget, Ingeborg Katrin Lid & Vos, Pauline (2024). Elefanten i klasserommet. Tangenten - Tidsskrift for matematikkundervisning. ISSN 0802-8192. 35(2), s. 66–66.
• Vos, Pauline (2024). Modellering som matematisk verktøykasse. Tangenten - Tidsskrift for matematikkundervisning. ISSN 0802-8192. 35(2), s. 1–1. Vis sammendrag

  Læreplanverket LK20 har fokuset på modellering i matematikkundervisningen. Dermed blir modellering blant det viktigste faglige innholdet elevene skal arbeide med i matematikk i grunnskoler og videregående skoler. Tangenten støtter opp under dette med et nytt temanummer. Bakgrunnen er at matematikk i økende grad blir en verktøykasse for å beskrive og løse problemer i utenformatematiske områder. Mange fagfolkene arbeider ikke med matematikk som et isolert fag, men de bruker matematikk som verktøy innen utenformatematiske områder. Den fortløpende matematiseringen i samfunnet fører til at mange elever kommer til å møte, å bruke og å lage beskrivelser av virkeligheten i matematisk språk. Det er nettopp hva matematisk modellering handler seg om. Modellering skjer i sammenheng med andre aktiviteter og dermed gir anledning til å løse problemer fra virkeligheten ved hjelp av matematikk. Å lage en matematisk beskrivelse av virkeligheten er dermed ikke et formål i seg selv, men et nyttig verktøy i besvarelsen av utenformatematiske spørsmål. Dermed kan modellering gi elevene innsikt i relevansen av faget i hverdagslivet og i yrkeslivet og vise hvorfor de må lære matematikk.

• Vos, Pauline (2023). Introduction to the Seminar on societal perspectives in research on mathematics education .
• Eriksen, Stig & Vos, Pauline (2023). Formal intentions in a new mathematics curriculum versus example tasks published before the standardized examinations – do they align? .
• Vos, Pauline & Bolstad, Oda Heidi (2023). Using preservice teachers’ perceptions of a project-based statistics course to investigate knowledge creation in collaborative settings.
• Eriksen, Stig & Vos, Pauline (2023). Formal intentions in a new mathematics curriculum versus example tasks published before the standardized examinations–do they align?
• Vos, Pauline & Bolstad, Oda Heidi (2023). Using preservice teachers’ perceptions of a project-based statistics course to investigate knowledge creation in collaborative settings.
• Vos, Pauline (2023). How a quality newspaper teaches about mathematical modelling. Vis sammendrag

  In public discourses (e.g., social media, tv), there are many negative tones about mathematics, yet an increasing mention of mathematical models. For instance, the recent price fluctuations of electricity make many people watch line and bar graphs on a daily basis. As part of a larger study on the public understanding of mathematical modelling (PUMM), I will present preliminary results from a study on a quality newspaper and how the journalists foster a certain type of knowledge about MM among the general public. MM is portrayed in authentic situations where problems need solving (e.g., pandemics, transport bottlenecks). Modelling is described as omnipresent in society, as humanized, relational and intentional (e.g., professional modellers work for authorities, collaborate with non-mathematical professionals to improve services or products), intentional, inaccurate, incomplete, and occasionally harmful. This yields recommendations for mathematics education.

• Vos, Pauline (2022). The Public Understanding of Mathematical Modelling (PUMM).
• Vos, Pauline (2022). Prosjekt-basert læring i en student-aktiv lærer-utdanning .
• Vos, Pauline (2022). Out of the mathematics classroom – towards a better understanding of informal and implicit learning of mathematics. Vis sammendrag

  Mathematics is not always learnt with help of teachers. Without any teachers or textbooks, Newton and Leibnitz learnt the topic of ‘the derivative’. More than 40 000 years ago, the person who made notches on the Lebombo bone to count upto 29 probably had no mathematics teacher either. Also, in our daily lives, we see children making spontaneous mathematical discoveries without help of adults or peers. In this presentation, I will present examples from history, from children’s play, from workplaces, and from daily life, which yield characteristics of informal and implicit learning of mathematics. These have to do with needs, tools, norms and a culture that give space to collaboration, agency, creativity, curiosity, and persistence. As a foretaste to the presentation, participants may already ponder on the question: how do weather forecasts or COVID-19 diagrams in the media contribute to out-of-school mathematical learning?

• Vos, Pauline (2022). The modelling cycle as analytic research tool and how it can be enriched beyond the cognitive dimension.
• Vos, Pauline (2022). Math Walks in mobile learning - summary of some research.
• Garshol, Lenka; Drange, Eli-Marie D.; Palmer, Helen Suzanne; Vos, Pauline & Baartman, Desire (2021). Actionbound seminar.
• Vos, Pauline (2021). Project-based Learning (PjBL) within mathematics education (and in my teaching).
• Vos, Pauline & Omobude, Eric Oziegbe (2021). Sankey diagrams in the mathematics class. Mathematics Teacher. ISSN 0025-5769.
• Eriksen, Stig & Vos, Pauline (2021). Evaluating example tasks against the new core elements for grade 11 in Norway – what do tasks communicate? .
• Vos, Pauline & Frejd, Peter (2021). Grade 8 students creating Sankey diagrams to model, visualize and communicate complex processes .
• Vos, Pauline & Omobude, Eric Oziegbe (2021). Sankey-diagrammen in de wiskundeles [Sankey diagrams in the mathematics classroom]. Euclides. ISSN 0165-0343. 96(7), s. 4–7.
• Vos, Pauline (2021). Lineær algebra - hjemmeeksamen.
• Vos, Pauline (2020). Sankeydiagrammen in de les [Sankeydiagrams in the classroom].
• Baartman, Desire & Vos, Pauline (2020). Video: Brusetablletter (oppløsningsevne)- Inquiry basert læring i prkasis. Trinn 5.
• Baartman, Desire & Vos, Pauline (2020). Video: Fire kuber - Inquiry basert læring i praksis. Trinn 4. Tverrfaglig: mattematikk, design, samfunnsfag.
• Vos, Pauline (2020). Sankey-diagrammen in de klas - data visualiseren en processen modelleren [Sankey diagrams in the classroom - visualising data and modelling processes]. Vis sammendrag

  We gaan hier in Noorwegen met tweedeklassers op excursie naar bedrijven, bijv de afvalverwerking. We koppelen deze excursies tegenwoordig aan lessen over Sankey-diagrammen voor het modelleren van de industriële processen. Sankeydiagrammen zijn stroomdiagrammen waarbij de breedte van de stroom proportioneel is met de kwantiteit. Ons tooltje is SankeyMATIC ( http://sankeymatic.com/ ). In deze lezing zal ik enkele ervaringen delen.

• Baartman, Desire & Vos, Pauline (2020). Designing videos for teacher education on inquiry-based learning - Investigating and exploring design principles .
• Vos, Pauline (2020). Visualisering av miljø- eller industriprosesser med 8. klasseelever.
• Vos, Pauline & Frejd, Peter (2019). Grade 8 students learning to visualize and mathematically model industrial processes through Sankey diagrams.
• Vos, Pauline (2019). Educational Design Research - a research method for changing curricula, teaching methods, and assessment. Vis sammendrag

  The world is changing rapidly whilst universities stick to conventional, large-scale lectures and tests that look like the test from the previous years. Yet, graduates need to be prepared for the 21st century job market, meaning: they should be flexible, reflective and creative problem-solvers who can use a wide range of resources when working on a problem, when making decisions, and when communicating. In order to change curricula, teaching methods, and assessment, we will need to intervene into routine practices and to challenge conventions. This is complex, but can be done in a scientifically sound, and scholarly reflective way. In this presentation, I will argue that educational designs, teaching interventions and pedagogical out-of-the-box thinking can, and should be part of academic research practices. In my own research, I focus on inquiry-based, interdisciplinary tasks (e.g. combining social sciences and mathematics), in which digital tools and authentic resources (e.g. newspaper clippings, annual reports from industries) play a vital role. I will shortly sketch three exemplary cases.

• Wiik, Anders & Vos, Pauline (2019). Å lese værmeldingene i norske aviser (1945-2015): Hvilke matematiske ferdigheter forventes?
• Vos, Pauline (2019). Reactor at 90% Seminar of PhD candidate - What makes a good PhD thesis?
• Hernandez-Martinez, Paul & Vos, Pauline (2019). Conceptualising the modelling cycle from a socio-cultural learning perspective.
• Greefrath, Gilbert & Vos, Pauline (2019). Video word problems, mathematizing applets, or modelling projects – categorizing the wide area of ‘modelling tasks with digital tools’.
• Vos, Pauline & Frejd, Peter (2019). Grade 8 students learning to visualize and mathematically model industrial processes through Sankey diagrams.
• Vos, Pauline & Kubens, Valerie (2019). Bare ett skolefag forbindes med hat og angst. [Avis]. Fædrelandsvennen.
• Vos, Pauline & Dean, Evert (2019). Connecting school mathematics to the world of work.
• Rogovchenko, Yuriy & Vos, Pauline (2019). MERGA Strand 2 - The teaching & learning of Mathematics in non-mathematical practices.
• Vos, Pauline (2019). What makes a good PhD thesis?
• Vos, Pauline (2019). Didactical tools for Learning Mathematics. Vis sammendrag

  In this presentation, I will not discuss digital tools for learning mathematics (geogebra, Kikora, etc.), but rather I will take a tool-perspective on mathematical concepts (proportions, algebraic equations, logarithms, etc). Mathematical concepts are not just pure ‘ideas’ but also tools that assist people to solve real-life problems. This is known as the concept-tool duality. For instance, we can look at how mathematics is used in non-mathematical areas (in demography, in computer science, in politics) and ask how this usage can be turned didactical. Will the learning of mathematics become more conceptual, more engaging … or more mechanistic and procedural? I will discuss two mathematics-didactical tools that give students security and motivation, rather than alienation with abstract symbols and strange theorems. If time allows, I'll show a third one, too, of which we are presently researching with grade 8 pupils.

• Poudel, Amrit Bahadur; Vos, Pauline & Shults, F. LeRon (2019). Students of Development Studies learning about modelling and simulations as a research approach in their discipline. Vis sammendrag

  Researchers in the social sciences are increasingly using modelling and simulation (M&S) as a research approach. They create virtual worlds to discover relations across variables, and to test theories and potential policies. We introduced this research approach to students in the department of Development Studies at our university. The goal was to investigate the way in which such students can gain meta-knowledge about M&S-based research, that is, general knowledge about its nature and rationale. We organized a seminar to introduce the research approach and illustrated it with a simulation of the behaviour of agents with varying levels of tolerance towards their outgroup neighbours (based on Schelling’s segregation model). We analysed students’ interactions through a socio-cultural lens. Students were able to gain meta-knowledge about M&S-based research, which they judged as useful for their future as professionals when working on development projects.

• Vos, Pauline (2018). Response to the First year Seminar by Ida Maria Landgärds.
• Simensen, Anita Movik & Vos, Pauline (2018). Matematiske læringsmuligheter for elever som presterer lavt i matematikk.
• Vos, Pauline (2018). Ordformel som bro mellom uformelt algebraisk resonnement og symbolsk algebra.
• Poudel, Amrit Bahadur; Vos, Pauline & Shults, F. LeRon (2018). Students of religion studying social conflict through simulation and modelling - An Exploration. .
• Vos, Pauline (2018). Mathematics as a tool (to solve real-life problems) and mathematics with tools.
• Vos, Pauline; Dallas, Markos; Poudel, Amrit Bahadur & Shults, F. LeRon (2018). Using Social Simulations in Interdisciplinary Primary Education - an Expert Appraisal. Vis sammendrag

  Many people recognize that teaching basic skills in primary schools (reading, writing and arithmetic) is no longer sufficient for pupils in the digital age. Therefore, governments now increasingly ask schools to add other skills (oral, digital), and to create connections between subjects (e.g. use mathematics in history lessons). In this study, we explored how social simulations can be used in primary education to meet these new goals. We conducted an expert appraisal (a qualitative Delphi method) with four experts specializing in innovating primary education. We selected three simulations that were freely available on the web, relevant for pupils’ lives, and had a limited number of parameters. They dealt with segregation, gossip spread and population dynamics. We asked the experts to critically discuss these. Afterwards, we analyzed the videotaped discussions in terms of affordances and constraints. The results showed that the affordances of social simulations include their broad appeal to students and their capacity to help users explore relevant concerns through an integrative approach (e.g. interpreting graphs, reasoning with parameters, predicting). Also, the experts warned that social simulations can touch on ethical issues that might be stressful for some pupils. If well-orchestrated, the use of social simulations has great potential to fulfill the new primary school goals.

• Vos, Pauline (2018). The role of reference to reality in mathematics in society, teacher education and school instruction in Norway.
• Vos, Pauline (2018). A Fermi Problem on the Varodd Bridge Kristiansand, Norway.
• Poudel, Amrit Bahadur; Vos, Pauline & Shults, F. LeRon (2018). Students of Religion studying social conflicts through simulation and modelling - a pilot study. Proceedings of the International Groups for the Psychology of Mathematics Education. ISSN 0771-100X. 5, s. 140–140.
• Vos, Pauline (2018). Task contexts in mathematics education.
• Vos, Pauline (2017). Mixed Methods and Design Research.
• Gjesteland, Thomas; Vos, Pauline & Wold, Margrethe (2017). Students experiencing flow in a Physics lab task using mobile phones and free software.
• Vos, Pauline (2017). Wordformula as a bridge between informal algebraic reasoning and symbolic algebra.
• Gjesteland, Thomas & Vos, Pauline (2017). Assessment of Physics laboratory skills of engineering students – project-based assessment with mobile phones and free tracker software.
• Vos, Pauline (2017). The usefulness and challenges of visual-numeric literacy.
• Vos, Pauline (2017). The Study of Visualizations in Research on Mathematics Education.
• Vos, Pauline (2017). Learning about Simulations as a Research Approach (LaSiRA) - a flanking project to MODRN.
• Klock, Heiner; Siller, Hans Stefan & Vos, Pauline (2017). Students' Solutions During the Modelling of Social Interactions by Means of a Segregation Simulation.
• Vos, Pauline; Hernandez-Martinez, Paul & Frejd, Peter (2017). Symposium on Connecting Science Capital to the Public Understanding of Mathematical Modelling.
• Gjesteland, Thomas & Vos, Pauline (2017). Modelling motion by engineering students - getting into a flow.
• Vos, Pauline (2017). Connections of Science Capital and the Teaching/Learning of Mathematical Modelling.
• Vos, Pauline (2017). Am I an effective supervisor?
• Gjesteland, Thomas & Vos, Pauline (2017). The Physics Lab experiment: Students Experiencing Flow.
• Vos, Pauline (2017). The modelling course, new tasks.
• Rensaa, Ragnhild Johanne & Vos, Pauline (2017). Interpreting teaching for conceptual and for procedural knowledge in a teaching video about linear algebra.
• Hogstad, Ninni Marie; Isabwe, Ghislain Maurice Norbert & Vos, Pauline (2017). A digital tool for applying integrals in a kinematic simulation: A perspective on instrumental genesis, epistemic value and semiotic potential.
• Gjesteland, Thomas; Vos, Pauline & Wold, Margrethe (2016). Students Experiencing Flow in a Physic Laboratory Task using Mobile Phones and Free Software.
• Djepaxhija, Brikena; Vos, Pauline & Fuglestad, Anne Berit (2016). The PISA scoring guidelines and their prescriptions for interpreting a problem situation.
• Vos, Pauline (2016). Digital Technology Support in Teaching Processes.

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