In this methodological paper we discuss the development and testing of a framework for analysing and visualizing the overall structure of the content and tasks in mathe-matics textbooks. The aim of the framework is to provide a lens on mathematics textbooks at the secondary level that is mathematics topic- and content-area-unbiased to facilitate an as objective as possible comparisons between both (a) different contents, topics, and tasks within a textbook; and (b) contents, topics, and tasks across different series of textbooks or within a textbook series. The paper explores the results of the framework applied to four mathematical textbook chapters visualized in terms of so-called hierarchy charts. Inter-rater coding reliability with respect to a developed coding manual, and future development and applications of the framework is also discussed.

This paper investigates secondary pre-service teachers’ strategies of sampling. The work of eight groups of pre-service secondary teachers asked to devise and imple- ment a sampling strategy to answer two questions about three given populations of different sizes are analyzed. The result presented highlights the models devised and used by the groups. The paper especially discusses the pre-service teachers’ models of samplings in term of how the number of samples and sample size used were decided on in the model as fixed, interval-static, or dynamic.

This bookprovides readers with an overview of recent international research and developments in the teaching and learning of modelling and applications from a variety of theoretical and practical perspectives. There is a strong focus on pedagogical issues for teaching and learning of modelling as well as research into teaching and practice. The teaching of applications of mathematics and mathematical modelling from the early years through primary and secondary school and at tertiary level is rising in prominence in many parts of the world commensurate with an ever-increasing usage of mathematics in business, the environment, industry and everyday life. The authors are all members of the International Community of Teachers of Mathematical Modelling and Applications and important researchers in mathematics education and mathematics. The book will be of interest to teachers, practitioners and researchers in universities, polytechnics, teacher education, curriculum and policy.

In this paper we report on the use of a pre-post-test design to study pre-service secondary mathematics teachers’ mathematical content knowledge in statistics before and after their first university course in statistics. The results show that the participants were successful in the pre-test on items related to sampling, probabi- lity and the general logic of making formal statistical inferences, but struggled with items concerning distributions. Comparing the pre- and post-test reveals an increas- ing average of the participants’ scores in most statistical areas, but that topics like informal statistical inferences and distributions remain challenging for the majority of the participants.

In this paper Sfard's theory of commognition is used to investigate how two students work collaboratively to develop mathematical models in a natural school setting. The students' communication is analysed from both an insider and an outsider perspective to recreate the construction of their developed models in terms of realization trees. The analysis shows how the students draw on different discourses and that the models evolved through activities of recognition and negotiation. In particular, the students formulated their mathematical models using already known mathematical models. However, these latter models were tacit models in that they were not explicitly mentioned or discussed during the developmental process. To facilitate students’ capability to productively engage in mathematical modelling, the result of this study points to the need renegotiate the learning-teaching agreement so that it incorporates activities offering the students opportunities to practice negotiation and recognition.

In this paper we use the framework of cultural affordances to analyze students’ work in an activity using a commercial mobile game app for smartphones and tablets that provided a context and simulation of a pandemic outbreak. The written work carried out by 22 upper secondary students working in pairs is analyzed, with a focus on the extent to which the students’ answers and produced graphs illustrated i) the students’ gameplay, ii) the context of the phenomena (the development and spread of a pandemic), or iii) the mathematical characteristics of the graphs. The results highlight the challenges of exploiting, leveraging, and aligning students’ gameplay with curricula learning goals in what can be seen as a clash of norms and behavior associated with the game and gameplay on the one hand, and the teaching and learning of mathematics on the other.

This chapter analyses and discusses nine groups of upper secondary students’ work on the question “How many red books are there in the library?”. The students devised and implemented a plan, collected data, calculated estimates and reflected on what aspects and factors might have influenced their results caused by their adopted strategy. The analysis of the students’ work focused on reconstructing and categorizing the models the students devised and implemented, as well as the sources and types of variability that the activity elicited. The results show how the central statistical idea of variability is manifested in the models developed and implemented by the students, and how these can be further explored and applied as central and bearing ideas for organizing a whole section of statistics at the upper secondary level.

This paper elaborates on the theoretical conceptualization of a so-called dual integrated modeling approach to the teaching and learning of modeling and the teaching and learning of mathematics through modeling. Departing from a sociocultural perspective on teaching, the models and modeling perspective, and using the previously developed theoretical constructs of the Teaching Triad and the Expanded Mediated Triangle, we discuss modeling in terms of (i) micro modeling in which modeling is used as a vehicle to teach and learning other curricula content; and (ii) macro modeling in which modeling is a self-standing mathematics curricula goal. We illustrate this dual integrated approach by analyzing and discussing a lower secondary teachers’ implementation of a modeling activity focusing on a simple statistical investigation and measures of central tendency in statistics.

Making inferences about unknown populations is central in statistical reasoning. However, little attention has been paid to empirical investigations of how and why students develop sampling models when investigating a categorical variable whose values are nominal. This paper reports on an intervention that draws on the models and modeling perspective, where 25 pre-service teachers were asked to develop a sampling model that could be used to make inferences about the number of different colored beads and the distribution of different colored beads in different sized populations. Using a thematic analysis, three main results about the characteristics of the students’ models of sampling and inference with nominal variables were identified: to catch all the low frequency colors in the population; not to overestimate the low frequency colors in the population; and, to formalize the relationships used in making inferences. These results highlight several issues about students’ understanding of the relationship between sample representativeness and sample variability and its consequences for making inferences.

Based on an investigation of research literature on the role of mathematics within the context of STEM education and twenty-first century skills, this explorative study presents and analyses 19 pre-service teachers’ lesson proposals for innovative STEM activities in secondary mathematics classrooms. Drawing on a categorisation of twenty-first century skills and key concepts from Realistic Mathematics Education and Basil Bernstein’s writings, the analysis indicated a use of detailed instructions regarding what students should do but unspecified criteria for their expected knowledge productions. Rather than serving as a digital tool for problem solving, programming was used for the purpose of generalising students’ conceptual knowledge in mathematics. At the same time a focus on modelling and applications in the STEM activities tended to weaken the disciplinary character of mathematics. The learning of various twenty-first century skills was promoted to different extents. The study raises a discussion of affordances and constraints regarding students’ access to mathematical knowledge through different modes of integration of mathematics and other STEM subjects aiming to support the development of twenty-first century skills.

A new national curriculum has recently been implemented in the Swedish upper secondaryschool where one of the goals to be taught is modelling ability. This paperpresents a content analysis of 14 ”new” mathematical textbooks with the aim to investigatehow the notion of mathematical modelling is presented. An analytic scheme isdeveloped to identify mathematical modelling in the textbooks and to analyse modellingtasks and instructions. Results of the analysis show that there exist a variety ofboth explicit and implicit descriptions, which imply for teachers to be attentive tocomplement the textbooks with other material.

Mathematical modelling is one competence assessed in the Swedish national course tests in mathematics (NCT) in upper secondary school. This paper presents, with empirical data from the NCT, a content analysis about what aspects of mathematical modelling are assessed and how it is assessed. The main conclusion is that only fragments (intra-mathematical aspects) of the modelling process are being assessed.

This chapter presents pilot study with aim to characterise the potential connections and synergies between statistics and modelling education. By developing a methodological approach of applying the software NVivo and grounded theory, we analysed the 17 ICTMA books published to date and the books from ICME-6 and the 14th ICMI study. The results show synergies in terms of the identified themes: teach and learn statistics/modelling; ICT; curriculum; and theory.  The context units analysed frequently provide suggestions for how to teach statistics using modelling approach, but seldom is the relationship between mathematical and statistical modelling from a theoretical point of view discussed.  The discussion brings forward some limitations of the methodological approach applied and suggestions for how to overcome these to develop a more robust methodology.

This chapter presents a discussion about similarities and differences between working with mathematical modelling in ‘school’ and mathematical modelling as a ‘professional task’ in the workplace based on empirical and theoretical research studies. Issues discussed concern goals; technology; division of labour, communication and collaboration; model construction, including the application and adaption of pre-defined models; projects; and risks involved in using the models. Based on this discussion and examples from innovative teaching practices, approaches to simulate modelling as a ‘professional activity’ in educational settings are explored and exemplified with a role play activity.

The official curriculum guidelines for upper secondary school in Sweden emphasise the use of mathematical models and mathematical modelling in mathematics education. However, no explicit definitions or descriptions of the notions are given in the curriculum. This licentiate thesis is an exploratory study which investigates teachers’ and students’ conceptions of the notion of mathematical modelling as well as their attitudes and experiences of working with mathematical modelling in mathematics classrooms. One experience of mathematical modelling that faces both students and teachers which is investigated is the national course tests in mathematics.

The thesis includes five papers and a preamble, where the papers are summarised, analysed, and discussed. Both quantitative and qualitative methods are being used in the thesis and theoretical aspects concerning mathematical modelling and conceptions are examined.

The results indicate that mathematical modelling plays a minor role in the investigated mathematics classrooms. The students as well as the teachers were not familiar with the notion of mathematical modelling. Only 23% of the 381 students and 50 % of the 18 teachers had heard the notion before participating in the study. Both teachers and students participating in this study expressed a variety of different interpretations of the notion of mathematical modelling. Negative attitudes were expressed by the students as well as by some of the teachers concerning mathematical modelling. These negative attitudes may present obstacles for implementing mathematical modelling in the upper secondary mathematics classroom. However, these negative attitudes are related to the used test items, which may have had a negative impact on the research, especially, as the test items only test parts of the modelling process. One dominant conception found among the teachers was that mathematical modelling is related to physics or chemistry. The conclusion made from the investigation about national course tests in mathematics course D, is that there is a lack of holistic assessment of mathematical modelling. Intra-mathematical aspects of mathematical modelling are put in favour for extra-mathematical aspects.

Researchers argue that if we want develop students’ modelling competency, than modelling has to be explicitly used and practised in the mathematics classrooms. However, for the Swedish upper secondary school this study concludes that this is not the case. A suggestion for future research is to focus on mathematical modelling in teacher education and design studies of incorporation of modelling activities into mathematics classrooms.

This paper presents a critical review of literature investigating assessment of mathematical modelling. Written tests as described in the reviewed papers draw on an atomistic view on modelling competencies while studies adopting more holistic approaches are rare. To assess the quality of students' work with mathematical models, an elaborated view on the meaning of quality of mathematical models seems to be needed.

The relevance of using mathematics in and for out-of-school activities is one main argument for teaching mathematics in education. Mathematical modelling is considered as a bridge between the mathematics learned and taught in schools and the mathematics used at the workplace and in society and it is also a central notion in the present Swedish mathematical syllabus for upper secondary school. This doctoral thesis reports on students’, teachers’ and modelling experts’ experiences of, learning, teaching and working with mathematical modelling in and out of school settings and their interpretations of the notion of mathematical modelling.

The thesis includes five papers and a preamble, where the papers are summarised, analysed, and discussed. Different methods are being used in the thesis such as video analysis of students’ collaboration working with modelling problem, interview investigations with teachers and expert modellers, content analysis of textbooks and literature review of modelling assessment. Theoretical aspects concerning mathematical modelling and the didactic transposition of modelling are examined.

The results presented in this thesis provide a fragmented picture of the didactic transposition of mathematical modelling in school mathematics in Sweden. There are significant differences in how modellers, teachers and students work with modelling in different practices in terms of the goal with the modelling activity, the risks involved in using the models, the use of technology, division of labour and the construction of mathematical models. However, there are also similarities identified described as important aspects of modelling work in the different practices, such as communication, collaboration, projects, and the use of applying and adapting pre-defined models. Students, teachers and modellers expressed a variety of descriptions of what modelling means. The variety of descriptions in the workplace is not surprising, since their working approaches are quite different, but it makes the notion difficult to transpose into school practise. Questions raised are if it is unrealistic to search for a general definition and if it is really necessary to have a general definition. The consequence, for anyone how uses the notion, is to always be explicit with the meaning.

An implication for teaching is that modelling as it shows in the workplace can never be fully ‘mapped’ in the mathematical classroom. However, it may be possible to ‘simulate’ such activity. Working with mathematical modelling in projects is suggested to simulate workplace activities, which include collaboration and communication between different participants. The modelling problems may for example involve economic and environmental decisions, to prepare students to be critically aware of the use of mathematics in private life and in society, where many decisions are based on mathematical models.

This paper presents a critical review of literature investigating assessment of mathematical modelling. Written tests, projects, hands-on tests, portfolio and contests are modes of modelling assessment identified in this study. The written tests found in the reviewed papers draw on an atomistic view on modelling competencies, whereas projects are described to assess a more holistic modelling competence but obstacles regarding reliability of assessing projects are identified. The outcome of this investigation also indicates that the criteria used in frameworks or modes of assessment seldom are derived from a theoretical analysis, but more often based on ad hoc constructions, experience from assessment situations or empirical studies of students work. Finally, this study suggests that an elaborated view on the meaning of quality of mathematical models is needed in order to assess the quality of students work with mathematical models.

This paper is the result of a comparative literature study aimed at identifying relevant historical mathematical contexts that may be applicable to present day mathematics education. Five Swedish textbooks in algebra from the period around the year 1800 are scrutinized. A content analysis and a comparison between seven categories shows that there exist several historical features that may be used in teaching. These have the potential to serve both teachers and learners of mathematics.

This paper analyses 263 Swedish upper secondary students’ descriptions of the notions of mathematical models and modelling using a coding scheme inspired by grounded theory, resulting in 34 coding categories. These are reduced and brought together in seven broader categories capturing different dimensions of the students’ expressed views of mathematical models and modelling. It is found that the students prominently connect mathematical models and modelling with aspects of problem solving and with applying mathematics in extra-mathematical situations. Mathematical models are also equated with mathematical formulae and modelling is considered to be a constructive and creative activity.

This paper presents a literature review of the notions of mathematical models and modelling from three different theoretical perspectives in order to analyse students’ descriptions in a future empirical study. Similarities and differences between the perspectives are discussed and suggestions regarding how to use the different perspectives and examples of possible research questions are given.

Researchers argue that teachers’ conceptions (beliefs) about mathematical modelling have an impact on the low integration of modelling activities into the mathematics classroom. This paper presents a case study of 18 teachers, investigating their conceptions about mathematical modelling and their experiences of working with modelling activities in Swedish upper secondary school. The results, based on an analysis with a grounded theory inspired coding strategy, indicate that the teachers’ conceptions of the notion mathematical modelling relates to designing a mathematical model based on a situation. It is also concluded that the teachers have only minor experience of the notion of mathematical modelling in the mathematics class, but in physics class modelling is used as a common activity.

Researchers argue that teachers’ conceptions (beliefs) about mathematical modelling have an impact on the low integration of modelling activities into the mathematics classroom. This paper presents a case study of 18 teachers, investigating their conceptions about mathematical modelling and their experiences of working with modelling activities in Swedish upper secondary school. The results, based on an analysis with a grounded theory inspired coding strategy, indicate that the teachers’ conceptions of the notion mathematical modelling relates to designing a mathematical model based on a situation. It is also concluded that the teachers have minor experience of the notion of mathematical modelling in mathematics class, but in physics class modelling is used as a common activity. Overall the teachers in this study seem not to give priority to integrate mathematical modelling into their everyday mathematics teaching. One reason may be related to teachers’ conceptions about mathematics where more than every other teacher expressed that some of the modelling items discussed during the interviews were not considered to be mathematics.

This chapter adapts the concept, science capital, to explore mathematics teachers’ possibilities to accumulate science capital, with the aim to identify teachers’ potentials and constraints regarding their use of realistic modelling activities within mathematics education. An analysis of transcripts indicated that three interviewed teachers’ scientific preferences and intentions for establishing a teaching practice connected to the world beyond school are high. While the interviewees presented examples of how modelling activities can be realised in the education system, obstacles for progress are also identified in the analysis referring to lacking financial resources and social network of people working with science.

This paper presents a discussion of some characteristic features about the role and place of mathematics education in vocational education. Based on examples from the Swedish vocational educational system, research studies and curricula, it is concluded that central roles of vocational mathematics education include the connection of informal and formal mathematics as well as facilitation of the use of technology such as artefacts, ICT, and pen and paper. To smoothen the transition from school to workplace the place of vocational mathematics education relates to different learning environments and activities that connect the classroom and workplaces. For creating a starting point for discussions about how vocational mathematics education might be organized analogies are used. They act as tools to visualize and pin point characteristic features that make impact on teaching to prepare individuals for future work and life.

This paper reports on the first results from a study investigating Swedish upper secondary students’ (11th – 12th grade) mathematical modelling competency. Using non-parametric statistical methods the data from 381 students are analysed and the students’ modelling competency is described in terms of seven subcompetencies. Possible factors affecting the students’ mathematical competency such as attitudes toward modelling, previous experiences, last taken mathematics course, grade, class and gender were also investigated.

This chapter presents the design and results from the first iteration of a classroom activity using the context and simulation provided by a commercial game app for smartphones and tablets. The aim was to study students’ experiences of an intervention entailing the game app in order to inform a more mature design with the long-term goal of being able to develop principles to design and implement modelling activities using game apps. An analysis focusing on the interplay between the designed intervention environment and students’ work from two upper secondary classes resulted in information in terms of affordances to inform the redesign. In addition, students experienced the activity as interesting and engaging, but with significant gender differences. The results inform a discussion of the role of new technology and simulation within classroom teaching and learning of mathematical modelling.

This study investigates the potential of engaging secondary pre-service teachers in a team-teaching approach called Socratic lectures to develop their communicational skills. An intervention was designed centered around a workshop sandwiched between a pre- and a post-presentation on natural logarithmic and derivatives to an audience of undergraduate students. A quantitative and qualitative analysis of 155 pre- and 92 documented post-experiences of the undergraduate students’ shows that the pre-service teachers’ presentations and communicating skills developed positively over the course of the intervention. The accompanying statistical analysis of Likert evaluation items shows asignificant increase of multiple communicational aspects of the pre-service teachers, and the line-by-line analysis of written evaluations support this claim regarding the pre-service teachers’interactivity. Issues about generalizability and future research is also discussed.

Educational research literature on mathematical modelling is extensive. However, not much attention has been paid to empirical investigations of its scholarly knowledge from the perspective of didactic transposition processes. This paper reports from an interview study of mathematical modelling activities involving nine professional model constructors. The research question was: How can mathematical modelling by professional mathematical model constructors be characterised? The analysis of our interview data inspired by the coding procedure of grounded theory led us to the description of three main types of modelling activities as a characterisation of mathematical modelling as a professional task. In data-generated modelling the models are developed principally from quantitative data drawing on no or only some assumed knowledge of the system being modelled, while in theory-generated modelling the models are developed based on established theory. In the third activity, model-generated modelling, the development of new models is based on already established models. For all types, the use of computer support and communication between clients, constructors and other experts are central aspects. Finally, the three types of modelling activities are related to existing theoretical descriptions of mathematical modelling and the relevance of the study for mathematical modelling in education is discussed.

From an explorative study on how professionals engage in mathematical modelling in their work, three main types of activities were identified, i.e. empirical, theoretical and applicational modelling. Based on the outcomes of the study, some issues related to mathematical modelling in education are discussed.

The notion of mathematical literacy has gained momentum internationally during recent time through the influence of the Programme of International Student Assessment and national concerns about the ability of citizens to use mathematics effectively in   in person, civic and work life. Accordingly, it is to be expected that these concerns should be reflected in relevant curriculum documents. This chapter presents a content analysis of a sample of 12 national curricula documents in relation to mathematical literacy. The analysis shows that there does not appear to be general agreement about the definition and of mathematical literacy within the analysed documents. We conclude the chapter by considering the implications for practice of the identified variations in understandings of mathematical literacy internationally.

This article presents an analysis of a team-teaching approach, applied in two learning spaces: a regular mathematics classroom; and a hairdressing salon at an upper secondary vocational education and training (VET) school. A mathematics teacher and a VET teacher jointly developed, planned and carried out the teaching activities in these two learning spaces. The overall goal was to prepare their 15 students for professional life. Observations made from the two lessons were analysed with the aim of identifying the extent to which the outcome of the team-teaching approach is dependent on the choice of the learning space. Drawing on Engestroms activity theory and research literature on learning spaces, our results indicate that tools, norms, division of labour and community differ significantly within the two learning spaces. The environment of the salon appeared to be more effective in promoting discussion, and encouraging self-confidence and identity-making, in comparison with the mathematics classroom, and there was a more visible interaction between mathematics and vocational subjects. This raises questions about the optimal design of the applied team-teaching approach and about whether the mathematics classroom can be considered a suitable learning space for facilitating students learning of workplace mathematics.

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.

Forskning visar att elever har svårigheter att rekontextualisera sina matematikkun- skaper från skolkontexten till yrkeslivskontexten om matematikundervisningen saknar koppling till yrkeslivet. I artikeln analyseras, utifrån Bernsteins teorier, intervjuer med elever på tre yrkesprogram som deltagit i en interventionsstudie med målet att knyta matematiken närmare elevernas yrkesinriktning. Resultaten tyder på matema- tikämnet har en stark klassifikation och inramning som är svår att förändra, men om klassifikation och inramning förändras mot ett mer ämnesintegrerat arbetssätt verkar elevernas förmåga att rekontextualisera sina matematikkunskaper förbättras. Undervisningsmiljö och användandet av autentiska problem diskuteras som centrala faktorer för att detta ska ske.

The modelling cycle is a theoretical construct frequently applied in research studies on mathematical modelling. On the one hand, the modelling cycle highlights essential aspects of modelling, which makes it a tool for conceptualizing. On the other hand, the modeling cycle can be used as a research tool for analysis of students’ work. In the latter case, it has the limitation of primarily yielding results of a cognitive nature. We sought ways to include other aspects to analyze, such as metacognitive strategies, tool use, and social norms. These aspects support and change the cognitive activities involved in mathematical modelling practice. Rather than the standard modelling cycle, we propose an enriched modelling cycle with overarching layers for analysis of results. The enriched modelling cycle is a wider theoretical framework with interacting dimensions that affect the phases in the modelling cycle. We discuss potentials and challenges of this framework for new research studies.

This chapter is an introduction to the theoretical concept of science capital. It also serves as an introduction to the next three chapters of this book. These ensuing chapters all connect mathematical modelling education to science capital. In short, science capital is a set of resources that offer people advantages within scientific contexts. For example, a friend who works in research can be a resource to better understand the gist of science. Not all people have such friends, and the underlying sociological theory of Bourdieu explains how inequities are caused by some people having better access to science capital than others. In this chapter, we explain how the concept of science capital can be used in research on mathematical modelling education. This analytic use of science capital will be enacted in the studies described in the ensuing three chapters.