Mathematical modelling and activation – a study on a large class, a project-based task and students' flow

International audience; 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 val…

Authenticity in Extra-curricular Mathematics Activities: Researching Authenticity as a Social Construct

In this chapter I study authentic aspects in mathematics education, in particular with respect to mathematical modelling. I define ‘authenticity’ as a social construct, building on the French sociologist Emile Durkheim. For an aspect to be authentic, it needs to have: (1) an out-of-school origin and (2) a certification of originality. The study validates this definition, asking: what authentic aspects can be identified within mathematics education? Data were collected from the excursion Railway Timetable Dynamics. During the excursion secondary school students were exposed to research carried out by university mathematicians on behalf of the National Railway Company. The authentic aspects w…

Grade 8 students appropriating Sankey diagrams: The first cycle in an educational design research

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…

Using Social Simulations in Interdisciplinary Primary Education: An Expert Appraisal

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 paramet…

A commentary on the Special Issue “Innovations in measuring and fostering mathematical modelling competencies”

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 an…

Solving Rate of Change Tasks with a Graphing Calculator: a Case Study on Instrumental Genesis

In an increasing number of mathematics classes throughout the world, technology is being used for the teaching and learning of mathematics. But knowledge is limited about the long-term development of students’ mathematical thinking when learning mathematics with the use of technology. This article reports on the development of a student and the role of the graphing calculator (GC) in his learning about derivatives and instantaneous rate of change. This case is compelling, because the student is an intensive user of the GC and develops flexible problem-solving techniques – techniques which differ from those of his peers and from what he was taught in mathematics class. We used the framework …

Chapter 11: Affect and Mathematical Modeling Assessment: A Case Study on Engineering Students’ Experience of Challenge and Flow During a Compulsory Mathematical Modeling Task

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.

How much space for communication is there for a low achieving student in a heterogeneous group ?

International audience; This paper reports on a case study aiming to deepen our understanding of low achieving students' learning of algebra, in particular when they work with pattern problems. We observed one low achieving student, May, who participated and worked in three different heterogeneous settings. Data were analysed from a multimod-al perspective on key and regulating activities in the groups. The analysis revealed that May's contribution varied, depending on the composition of the groups, and that her contributions were influenced by regulating activities by peers and access to physical artefacts. The findings show that a low achieving student is able to generalise beyond her ari…

Video-based Word Problems or Modelling Projects—Classifying ICT-based Modelling Tasks

Mathematical modelling tasks increasingly feature the use of digital tools and media. In this chapter, we discuss the wide variety of these. Until now, classifications for modelling tasks did not consider the use of tools and media. Therefore, we developed a new classification for ICT-based modelling tasks. One class relates to mathematics; the others differentiate across (1) modelling aspects unrelated to tool and media, (2) the task context, (3) the digital tools and media (CAS, Wikipedia, type of feedback, etc.) and (4) students’ anticipated activities guided by task regulations, such as group work or time restrictions. The classification was validated with three example tasks. A visual …

“How Real People Really Need Mathematics in the Real World”—Authenticity in Mathematics Education

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 …

Exploring grade 9 students' assumption making when mathematizing

International audience; Making assumptions is a key activity in modelling. The present study aims to explore the variety of assumptions that lower secondary students make in this process. As theoretical basis for the data analysis, we used the modelling cycle by Blum and Leiss (2007) and framed a definition of assumptions. The study was carried out with grade 9 students. The results show three categories of assumptions: (1) parameter assumptions, (2) assumptions for the choice of the mathematical model, and (3) assumptions about task expectations. Assumptions from the first two categories assist students to use extra-mathematical knowledge to construct a mathematical model, while the third …

Connections of Science Capital and the Teaching and Learning of Mathematical Modelling: An Introduction

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 capit…

Assessment of Modelling in Mathematics Examination Papers: Ready-Made Models and Reproductive Mathematising

In the Netherlands modelling is integrated into the mathematics curriculum. This chapter describes a study of modelling characteristics in recent mathematics examination papers. Results show that the tasks convey the message that mathematics can be found in unexpected situations. In many tasks a situation and a ready-made model are given and students are asked to work with the model. For mathematising, two design principles were: to finalise the parameters of a given model, and to reconstruct a given mathematical model from verbal descriptions and diagrams of a situation. These formats were coined as mechanistic and reproductive mathematising respectively. These formats have been introduced…

The object-tool duality in mathematical modelling. A framework to analyze students’ appropriation of Sankey diagrams to model dynamic processes

Students often do not experience the relevance of learning mathematics. This paper reports on an exploratory case study, in which a class of grade 8 students (n=35) was introduced to Sankey diagrams. The aim was to explore to what extent these students could appropriate Sankey diagrams, meaning: they could describe these as objects in themselves and they could use them to model and visualize phenomena relevant to them. Based on Cultural-Historical Activity Theory, we developed an analytical construct defined as the object-tool duality, coordinating mathematics as a set of objects and as a set of tools. The analysis of students’ answers showed that they could use these diagrams as tools to v…

Long-Term Development of How Students Interpret a Model: Complementarity of Contexts and Mathematics

When students engage in rich mathematical modelling tasks, they have to handle real-world contexts and mathematics in chorus. This is not easy. In this chapter, contexts and mathematics are perceived as complementary, which means they can be integrated. Based on four types of approaches to modelling tasks (ambivalent, reality bound, mathematics bound or integrating), we used task-based interviews to study the development of students’ approaches while the students moved from grade 11 to 12. Our participants were ten Dutch students. We found that their approaches initially were either ambivalent, reality bound or mathematics bound. In subsequent interviews, the preference was maintained, and …

Task Contexts in Dutch Mathematics Education

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 betwe…

Students of Religion Studying Social Conflict Through Simulation and Modelling: An Exploration

Researchers at our university use modelling and simulation (M&S) to study religious conflicts, and we wanted to introduce undergraduate students of religion to this research approach. Hence, we started a three-year educational design research project to empirically study ways to introduce these students to M&S as a viable research method in their discipline. The research project will entail several iterations, which aim to have a feasible and effective design of lessons and a better understanding of the learning processes. The first iteration was exploratory and is reported here. For this exploration, we organised a seminar, which was videotaped for post hoc analysis. The seminar started wi…

“Why do I have to learn this?” A case study on students’ experiences of the relevance of mathematical modelling activities

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 pres…

Topic Study Group No. 39: Large Scale Assessment and Testing in Mathematics Education

Assessing Mathematizing Competences Through Multiple-Choice Tasks: Using Students’ Response Processes to Investigate Task Validity

In this chapter, we report on multiple-choice tasks for assessing mathematizing competences of grade 9 students. The task format is complex, consisting of two layers. In the first layer, students are asked to consider a holistic modelling problem. In the second layer, they are asked for an atomistic competence (making assumptions, assigning variables, etc.) related to the same modelling problem. We conducted a qualitative study to investigate the validity of these tasks based on students’ response processes. Eight students worked in pairs solving the tasks collaboratively. The results show that all students were able to handle the layered task format. They reflected meta-cognitively on the …

On Science Museums, Science Capital, and the Public Understanding of Mathematical Modelling

Students’ opportunities to learn informally (e.g. by watching documentaries, visiting museums) explain socio-economic inequities in school performances. To explore informal learning about mathematical modelling, I studied two science museums, as these are environments typically visited by middle-class families. I framed the study by using the notions science capital and the public understanding of mathematical modelling (PUMM) and explored how these are mediated in science museums. The research method entailed observations of displays, artefacts, and visitors. One science museum completely detached mathematics from its use value, whereas the other offered histories of how people used mathem…