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Mathematical modelling emphasizes the connection between mathematics and reality -- still, tasks are often exclusively introduced inside the classroom. The paper examines the potential of different task settings for mathematical modelling with real objects: outdoors at the real object itself, with photographs and with a 3D model representation. It is the aim of the study to analyze how far the mathematical modelling steps of students solving the tasks differ in comparison to the settings and representations. In a qualitative study, 19 lower secondary school students worked on tasks of all three settings in a Latin square design. Their working processes in the settings are compared with a special focus on the modelling steps Simplifying and Structuring, as well as Mathematizing. The analysis by means of activity diagrams and a qualitative content analysis shows that both steps are particularly relevant when students work with real objects -- independent from the three settings. Still, differences in the actual activities could be observed in the students' discussion on the appropriateness of a model and in dealing with inaccuracies at the real object. In addition, the process of data collection shows different procedures depending on the setting which presents each of them as an enrichment for the acquisition of modelling skills.

Mathematical modelling competencies have become a prominent construct in research on the teaching and learning of mathematical modelling and its applications in recent decades; however, current research is diverse, proposing different theoretical frameworks and a variety of research designs for the measurement and fostering of modelling competencies. The study described in this paper was a systematic literature review of the literature on modelling competencies published over the past two decades. Based on a full-text analysis of 75 peer-reviewed studies indexed in renowned databases and published in English, the study revealed the dominance of an analytical, bottom-up approach for conceptualizing modelling competencies and distinguishing a variety of sub-competencies. Furthermore, the analysis showed the great richness of methods for measuring modelling competencies, although a focus on (non-standardized) tests prevailed. Concerning design and offering for fostering modelling competencies, the majority of the papers reported training strategies for modelling courses. Overall, the current literature review pointed out the necessity for further theoretical work on conceptualizing mathematical modelling competencies while highlighting the richness of developed empirical approaches and their implementation at various educational levels.

This paper reports on empirical results about the influence of two different teaching designs on the development of tertiary students' modelling competency and attitudes towards modelling. A total of 144 first year engineering students were exposed to a diagnostic entrance test, a modelling unit consisting of five lessons with ten tasks, enframed by a pre- and a post-test, and at the end a questionnaire on attitudes towards mathematical modelling. Similar to the German DISUM study, in the modelling unit, one group of participants followed an independence-oriented teaching style, aiming at a balance between students' independent work and teacher's guidance, while two other groups were taught according to a more traditional teacher-guided style. Linear mixed regression models were used to compare pre- and post-test results. The results show that all groups had significant learning progress, although there is much room for further improvement, and that the group taught according to the independence-oriented design had the biggest competency growth. In addition, this group exhibited more positive attitudes than the other groups in five of six attitudinal aspects.

Teaching mathematical modeling is a demanding task. Thus, fostering teachers' competencies in this regard is an essential component of teacher education. Recent conceptualizations of teachers' competencies include situation-specific skills based on the concept of noticing, which is of particular interest for the spontaneous reactions needed when teaching mathematical modeling. The study described in this paper aims to analyze the development of a video-based instrument for measuring teachers' noticing competencies within a mathematical modeling context and obtain evidence for the validity of the instrument. Three kinds of validity are examined in three different studies: content validity, elemental validity and construct validity. Indicators for content validity could be found through different expert ratings and implementation with the target group, where participants were able to perceive all relevant aspects. The qualitative analysis of participants' reasoning, which is consistent with the coded level, indicates elemental validity. Moreover, the results of the confirmatory factor analysis suggest construct validity with one overall factor of noticing competence within a mathematical modeling context. Taken together, these studies imply a satisfactory validity of the video-based instrument.

Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers' professional mathematical modeling competencies in three different regions--Germany, Mainland China, and Hong Kong--where educational and cultural traditions differ, including the role of mathematical modeling. In total, 232 pre-service mathematics teachers from the three regions completed a modeling task covering mathematics content knowledge (MCK) of modeling and mathematical pedagogical content knowledge (MPCK) of modeling. The results show that pre-service teachers from Germany demonstrated the strongest MCK and MPCK of mathematical modeling; by contrast, pre-service mathematics teachers from Mainland China and Hong Kong demonstrated relatively weaker MCK and MPCK of mathematical modeling. MCK and MPCK of mathematical modeling were also found to be unevenly developed at different competence levels for the three regions. These differences may be attributed to the history of mathematical modeling in mathematics curricula, teacher education, and teaching culture in these three regions.

Modelling competencies are currently included in numerous curricula worldwide and are generally accepted as a complex, process-oriented construct. Therefore, effective measurement should include multiple dimensions, like the sub-competencies required throughout the modelling process. Departing from the characteristics of modelling problems as open and often underdetermined real-world problems, we propose to enrich the current conceptualisation of mathematical modelling competencies by including creativity, which plays an important role in numerous phases of the mathematical modelling process but has scarcely been considered in modelling discourse. In the study described in this paper, a new instrument for the evaluation of this enriched construct has been developed and implemented. The modelling competencies incorporating creativity of the students were evaluated based on the adequacy of the models and the modelling processes proposed, and the appropriateness and completeness of the approaches were evaluated in detail. Adapting measurement approaches for creativity that have been developed in the problem-solving discourse, certain criteria of creativity were selected to evaluate the creativity of the students' approaches in tackling modelling problems--namely, usefulness, fluency, and originality. The empirical study was conducted among 107 Chinese students at the upper secondary school level, who attended a modelling camp and independently solved three complex modelling problems. The results reveal significant correlations between fluency and originality in students' performances across all tasks; however, the relationships between usefulness and the other two creativity aspects were not consistent. Overall, the results of the study support the importance of the inclusion of creativity in the construct of modelling competencies.

One can identify at least three different types of relationships between mathematics and crises. First, "mathematics can picture a crisis." This is in accordance with the classic interpretation of mathematical modelling, which highlights that a mathematical model provides a representation of a piece of reality, a reality that could be a critical situation such as, for instance, a pandemic. Second, "mathematics can constitute a crisis," meaning that mathematics can form an intrinsic part of the very dynamics of a crisis. This phenomenon can be illustrated by the economic crises that spread around the world in 2008. Third, "mathematics can format a crisis." This final formulation refers to a situation where a mathematical reading of a crisis brings about ways of acting in the critical situation that might be adequate, but also counterproductive, if not catastrophic. This is illustrated with reference to the potential crises due to climate changes. As a conclusion, the paper addresses the "politics of crises," which refers to the power that can be acted out through a crisis discourse in which mathematics may come to play a deplorable role.

We report a study on mathematical literacy with special emphasis on health literacy. In particular, we identify and characterize the mathematical competencies that a citizen needs to interpret the official information on the COVID-19 pandemic as experienced in Mexico. To achieve this, we analyze the daily reports on the pandemic issued by the Mexican Ministry of Health, using the framework of mathematical competencies as a theoretical lens. Our results show that there are five necessary competencies to interpret the official information: mathematical communication, mathematical representation, mathematical symbols and formalism, mathematical modeling, and mathematical reasoning competency. After characterizing and illustrating these competencies, we close the paper with a reflection on the importance of mathematical literacy in a pandemic world.

Contemporary scholars describe mathematical modeling as a transformation of a real-world problem to a mathematical problem and back again. This paper treats a critical issue in the modeling process: how modelers determine if the transformation from the real world to mathematics was carried out well. I present an empirically derived typology of validating activities explaining how validating functions to ensure a mathematical model will yield a reasonably accurate prediction. The typology arose from analysis of four engineering undergraduates' production of 276 instances of validating. The nuances of validating suggest that creating and maintaining relationships between reality and mathematics is more complex than a transformation and that we should afford a more prominent role to validation in the modeling process.

Classroom data modeling involves posing questions, identifying attributes of phenomena, measuring and structuring these attributes, and then composing, revising, and communicating the outcomes. Selecting attributes is a fundamental component of data modeling, and the considerations made when selecting attributes is the focus of this paper. A teaching experiment involving 2 teacher educators and 25 pre-service teachers (PSTs) was carried out with 24 young children (5-6-year-olds) as part of a 4-day data modeling investigation. Although perceptual features of the data influenced initial approaches to attribute selection, considerations of the problem situation influenced a shift from the perceptual and towards consideration of attributes such as taxonomy, habitat, behavior, and diet. Expertise in the data context (animal kingdom) and ability to collaborate and negotiate within groups supported children in their ability to switch attributes, attend to multiple situations presented by the problem, and modify and extend their categorizations of data.

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.

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence--namely, in Toulmin's sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants ("a priori, empirical, institutional and evaluative"). Our approach to analysing teacher arguments proposes an adaptation of Freeman's classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal.

The integration of history into educational practice can lead to the development of activities through the use of genetic "moments" in the history of mathematics. In the present paper, we utilize Oresme's genetic ideas--developed during the fourteenth century, including ideas on the velocity-time graphical representation as well as geometric transformations and reconfigurations--to develop mathematical models that can be employed for the solution of problems relating to linear motion. The representation of distance covered as the area of the figure between the graph of velocity and the time axis employed in these activities, leads on naturally to the study of problems on motion by means of functions, as well as allowing for the use of tools (concepts and propositions) from Euclidean geometry of relevance to such problems. By employing simple geometric transformations, equivalent real life problems are obtained which lead, in turn, to a simple classification of all linear motion-related problems. When applied to a wider range of motion problems, this approach prepares the way for the introduction of basic Calculus concepts (such as integral, derivative and their interrelation); in fact, we would argue that it could be beneficial to teach the basic concepts and results of Calculus from an early grade by employing natural extensions of the teaching methods considered in this paper.

This paper indicates that prospective teachers' familiarity with theoretical models of students' ways of thinking may contribute to their mathematical subject matter knowledge. This study introduces the intuitive rules theory to address the intuitive, "same sides-same angles" solutions that prospective teachers of secondary school mathematics come up with, and the proficiency they acquired during the course "Psychological aspects of mathematics education". The paper illustrates how drawing participants' attention to their own erroneous applications of "same sides-same angles" ideas to hexagons, challenged and developed their mathematical knowledge.

This paper describes university students' grasp of inflection points. The participants were asked what inflection points are, to mark inflection points on graphs, to judge the validity of related statements, and to find inflection points by investigating (1) a function, (2) the derivative, and (3) the graph of the derivative. We found four erroneous images of inflection points: (1) f ' (x) = 0 as a necessary condition, (2) f ' (x) is not equal to 0 as a necessary condition, (3) f " (x) = 0 as a sufficient condition, and (4) the location of "a peak point, where the graph bends" as an inflection point. We use the lenses of Fischbein, Tall, and Vinner and Duval's frameworks to analyze students' errors that were rooted in mathematical and in real-life contexts.

This paper reviews the present state, recent trends, and prospective lines of development concerning applied problem solving, modeling, and their respective applications. Four major trends are scrutinized with respect to curriculum inclusion: a widened spectrum of arguments, an increased universality, an increased consolidation, and an extended computer utilization. (JJK)

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin's (Educational Studies in Mathematics 67:59-76, "2008") model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the proof's high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory. (Contains 5 footnotes.)

In this paper, a focus is made on defining processes at stake in an unfamiliar situation coming from discrete mathematics which brings surprising mathematical results. The epistemological framework of Lakatos is questioned and used for the design and the analysis of the situation. The cognitive background of Vergnaud's approach enriches the study of freshmen's processes at university. The mathematical analysis and the results specifically underscore the in-action definitions and the zero-definitions and highlight the need of similar mathematical experiences in education, particularly focused on defining processes and the exploration of a research problem.

There is an international trend in mathematics education of antiauthoritarianism, emphasising cooperative learning and students' autonomous search for knowledge. In Sweden, this is framed by "students' independent work", characterised by only limited amounts of teacher-student and student-student interaction and communication and a low level of teacher responsibility for knowledge generation. Research investigating occurrence and effects of different dimensions of instructional responsibility and authority in classroom practise, however, often takes the very presence of instructional responsibility for granted. In this paper, an alternative model for description and analysis of instructional practice is proposed, which challenges the potential of the traditional division into teacher- versus student-centred modes of instruction. Theoretical starting points for conceptualising and defining mathematics classroom practices focus on instructional responsibility both for knowledge generation and for how this responsibility would be expressed in instructional practice. It was also investigated how this model corresponds with data from TIMSS 2003. The empirical results supported the possibility of adopting this alternative perspective in modes of instruction. (Contains 2 tables, 2 figures and 1 footnote.)

This paper reports a part of a study on the construction of mathematical meanings in terms of development of semiotic systems (gestures, speech in oral and written form, drawings) in a Vygotskian framework, where artefacts are used as tools of semiotic mediation. It describes a teaching experiment on perspective drawing at primary school (fourth to fifth grade classes), starting from a concrete experience with a Durer's glass to the interpretation of a new artefact. We analyse the long term process of appropriation of the mathematical model of perspective drawing (visual pyramid) through the development of gestures, speech and drawings under the teacher's guidance.

In recent years several mathematics education researchers have attempted to analyse students' arguments using a restricted form of Toulmina's ["The Uses of Argument," Cambridge University Press, UK, 1958] argumentation scheme. In this paper we report data from task-based interviews conducted with highly talented postgraduate mathematics students, and argue that a superior categorisation of genuine mathematical argumentation is provided by the use of Toulmina's full scheme. In particular, we suggest that modal qualifiers play an important and previously unrecognised role in mathematical argumentation, and that one of the goals of instruction should be to develop students' abilities to appropriately match up warrant-types with modal qualifiers.

This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics Education's instructional design theory to infer what the teacher might understand to effectively implement her instructional goals and, more importantly, support student learning. By coordinating these perspectives, we developed a working framework for analyzing the teacher's classroom practice retrospectively. For example, we illustrate how the mathematical possibilities related to one student's question might inform the teacher's decisions as she initiates shifts in students' self-generated models. Additionally, we illustrate how the teacher's decision to capitalize on particular students' models contributes in part to the kinds of mathematical ideas that can be explored and the connections students can make among those ideas. More generally, we explore the utility of coordinating these two perspectives to understand the landscape of ideas that teachers might traverse to align their practices with reform recommendations in the United States.

This paper examines 6th grade children's local conceptual development and mathematization processes as they worked a comprehensive mathematical modeling problem (creating a consumer guide for deciding the best snack chip) over several class periods. The children and their teachers were participating in a 3-year longitudinal teaching experiment in which sequences of mathematical modeling problems were implemented from the 5th grade (10 years of age) though to the 7th grade. In contrast to traditional problem solving, mathematical modeling requires children to generate and develop their own mathematical ideas and processes, and to form systems of relationships that are generalizable and reusable. Reported here is a detailed analysis of the iterative cycles of development of one group of children as they worked the problem, followed by a summary of the mathematization processes displayed by all groups. Children's critical reflections on their models are also reported. The results show how children can independently develop constructs and processes through meaningful problem solving. Children's development included creating systems for operationally defining constructs; selecting, categorizing, and ranking factors; quantifying quantitative and qualitative data; and transforming quantities.

How do teachers conceptualize and deal with context of mathematics word problems in their teaching? This question is discussed based on a study of 14 experienced teachers at the elementary, junior high and senior high school levels. Bruner's notions of paradigmatic and narrative modes of knowing formed the basis of analysis of data from sources that include interviews and classroom observations. The findings highlight the teachers' conceptions of problem context and teaching approaches for each of these modes of knowing. All of the teachers used the paradigmatic mode in their teaching but with different depth and most engaged in some form of the narrative mode to create a classroom environment that was motivational for students to learn word problems. The paper also highlights characteristics of these two modes as they relate directly to word problems and discusses implications for instruction, learning and teacher development.

Many mathematics departments usually teach a variety of courses for students from different science departments and even from different faculties. These "service" courses are usually taught in the same way as the courses for mathematics major students. However, in science, because of the need to better analyse and interpret experimental data and the increased use of mathematical tools in chemistry and geology textbooks, it is becoming necessary to teach these science students quantitative skills beyond the scope of first year mathematics courses. This paper describes the design of a one-semester second year mathematics course, mainly for chemistry and geology students, with three specific objectives: to develop students' ability to quantitatively analyse problems arising in their own field, to illustrate the great utility of mathematical models to provide answers to key chemistry and geology problems, to develop students' appreciation of the diversity of mathematical approaches potentially useful in the chemical and geological sciences.

This paper deals with a teaching approach aimed to help students become aware of targeted techniques of significance in problem solving. The teaching approach is to present a series of tasks that all can be solved by applying the same technique. Two levels of prompting are used; first for the students to realize solutions without necessarily being cognizant of the technique, second for them to perform further mathematical modeling that should highlight the similarities in solution shared by all the tasks. In the fieldwork, a teaching sequence based on this approach is implemented for a technique involving enumeration via constructing a bijection. Certain factors in the students' behavior suggested that their realization of the technique at the end was not as secure as desired. Some modifications of the teaching sequence are proposed to counter these factors.

The focus is on the metacognitive awareness of ten high-achieving high school pupils in mathematics in Denmark and England and their understanding of their cognitive learning processes and strategies. Mainly unstructured focus group interviews investigate how they explain that they learn a mathematical concept that is new to them. I develop the "CULTIS model for analysis" (Consciousness, Unconsciousness, Language, Tacit, Individual, Social), which consists of six themes in which various psychological theories of learning are expressed. The model uses the theories of Ernest, von Glasersfeld, Hadamard, Krutetskii, Mason, Piaget, Polya, Sfard, Skemp, and Vygotsky. The model is used to sort the pupils' explanations and it also functions as a tool with which to compare the pupils' explanations with the various theories. I conclude that the pupils can talk about their learning in their own words and each pupil refers to elements that the researcher associates with different theories. Half the pupils furthermore explain that how they are used to learning, or being taught, influences how they later on are able to learn. I also discuss metacognition and whether this can be an aid to enhance learning.

In this paper we present three cases of instructional design that illustrates both horizontal didactising, the activity of using already established principles to design instruction, and vertical didactising the activity of developing new tools and principles for instructional design. The first case illustrates horizontal didactising by elaborating how the constructs chains of signification and models were used to design an instructional sequence involving linear growth. The second and third cases illustrate vertical didactising by developing argumentation analyses and generative listening, respectively, as instructional design tools. In the second case, argumentation analyses emerge as a tool that other designers can use to anticipate the quality of conversations that can occur as students engage in tasks prior to implementing the instructional sequence. The third case develops the notion of generative listening as a conceptual tool within the context of designing differential equations instruction to gain insights into what are, for students, experientially-real starting points that are mathematical in nature and to provide inspirations for revisions to instructional sequences.

This paper is the "Discussion Document" for a forthcoming ICMI Study on "Applications and Modelling in Mathematics Education." As will be well-known, from time to time ICMI (the International Commission on Mathematical Instruction) mounts specific studies in order to investigate, both in depth and in detail, particular fields of interest in mathematics education. The purpose of this Discussion Document is to raise some important issues related to the theory and practice of teaching and learning mathematical modelling and applications, and in particular to stimulate reactions and contributions to these issues and to the topic of applications and modelling as a whole (see Section 4). Based on these reactions and contributions, a limited number (approximately 75) of participants will be invited to a conference (the "Study Conference") which is to take place in February 2004 in Dortmund (Germany). Finally, using the contributions to this conference, a book will be produced (the "Study Volume") whose content will reflect the state-of-the-art in the topic of applications and modelling in mathematics education and suggest directions for future developments in research and practice. The authors of this Discussion Document are the members of the "International Programme Committee" for this ICMI Study. The committee consists of 14 people from 12 countries, listed at the end of Section 4. The structure of the Document is as follows. In Section 1, we identify some reasons why it seems appropriate to hold a study on applications and modelling. Section 2 sets a conceptual framework for the theme of this Study, and Section 3 contains a selection of important issues, challenges and questions related to this theme. In Section 4 we describe possible modes and ways of reacting to the Discussion Document, and in the final Section 5 we provide a short bibliography relevant to the theme of this Study.

Proposes a framework for discussing the nature of the relationships between contexts and the formation of mathematical knowledge through model and field-of-experience concepts. Supports and illustrates this framework with references to curricular innovation and educational research. (Contains 31 references.) (Author/ASK)

An alternative explanation is presented for the pattern of relative difficulty of arithmetical word problems to that suggested in a 1983 article. It is argued, that for many problems, much of the symbolic manipulation is accomplished externally, not internally. (MNS)