This paper presents an empirical exploratory study involving 158 Swedish secondary school students (Years 7-11) asked to develop fair prescriptive mathematical models to rank real-life, multi-faceted heptathlon outcomes. A developed framework for identifying Characteristics of Fairness is used to investigate What types of models do the students develop? and What aspects of fairness in the context of heptathlon come to the fore in the students' models? Discussed and situated in relation to previous research, the result highlight and elaborate four identified types of models focusing on Event placement, Individual performance, Comparative performance and Baseline comparison. In particular the results show that students emphasised outcome fairness at the expense of process fairness. Based on the findings, implications and possible future lines of research and theoretical development are suggested.

This chapter discusses mathematical literacy in the context of the teaching and learning of statistics at the secondary level (Grades 7–9). This includes exemplifying how tasks and activities in statistics can be designed and implemented to support students in developing mathematical literacy using the models and modelling theoretical perspective and model development sequences. The chapter discusses literature on statistical literacy and related notions in the field of statistics education and links these to mathematical literacy more generally. Connections are made and highlighted between mathematical literacy and tenets of the models and modelling perspective as well as model development sequences. These connections are then illustrated as they come to the fore in an example of an activity from a research project focusing on the teaching and learning of statistics, which sheds light on the students’ opportunities to develop their mathematical literacy.

This chapter presents a literature review exploring how mathematical modelling as a vehicle in vocational mathematics education can be understood in terms of connecting vertical and horizontal mathematising, and what type of components of authenticity has been established to emphasise authentic learning opportunities. The results show six examples illustrating the diversity in how both vertical and horizontal mathematising can be achieved. In addition, the following key components for authentic learning to develop the teaching of mathematics at vocational education were identified: teamwork, identity, mathematics in artefacts and outputs as well as client and consumer perspective. The chapter ends with a discussion of how the mathematical modelling research community can advance if more research studies on mathematical modelling will be situated in vocational education.

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.

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 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.

Many students do not experience usefulness in mathematics. To address this problem, we offered them a mathematical tool, Sankey diagrams, which is a flow chart appearing in news media to visualize social, industrial or environmental processes. We carried out an Educational Design Research (EDR) to develop and evaluate lesson materials about contextualized Sankey diagrams. We tested these materials with a class of grade 8 students and evaluated these on the feasibility of students’ appropriation of the diagrams. In the lesson, we observed how students were able to read the Sankey diagrams, liked the societal processes visualized, yet did not fully grasp their mathematical properties. However, weeks later, the same students skillfully used Sankey diagrams in an unrelated project, which showed they needed time for their learning. Our contributions are that (1) grade 8 students can appropriate Sankey diagrams and use these in situations relevant to them, (2) design researchers should consider long-term learning effects beyond the experimental phase in EDR. We recommend educational designers to innovate curricula and introduce diagrams from news media to make students experience usefulness in school content.

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.

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 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.