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This study involves two classes from different educational levels, namely 9th grade and university. Students in both contexts were given a modelling task that required the development of a hand biometrics recognition system, during which they performed experimentation and simulation. As aims of the study, we look for distinctions and commonalities between the models developed in the two classes and seek to know how simulation and experimentation influence students' production of meaning. The theoretical framework comprises the relationship between the modelling process and the prototyping process and adopts Peirce's pragmatic perspective on meaning. The research is of a qualitative nature, assuming the characteristics of a case study. The results reveal many commonalities between the modelling in the two contexts. Moreover, experimentation and simulation were relevant elements for the production of meaning by the students, which is endorsed by a pragmatic perspective on meaning. info:eu-repo/semantics/publishedVersion

Mathematical modelling has acquired relevance at all educational levels in the last decades since integrating this activity in instruction provides significant contexts for improving students' learning, including in linear algebra courses that have a notable presence in many undergraduate courses from different fields, including engineering and sciences. This paper reports a study aiming to characterise the distinct modelling routes performed by Costa Rican undergraduate students when solving a mathematical modelling task involving the concept of system of linear equations (SLE). In analysing those modelling routes, it was possible to identify their learning of linear algebra concepts and their modelling competencies as well as the associated difficulties that students faced. Data collection included participant observation, with audio recording of the students' discussions, their written work on the task, and digital files of their work with technology. The results show that non-linear routes are associated with a greater mobilisation of students' knowledge on SLE concepts and with their development of modelling competencies. The results also highlight the need to improve the students' competency of validating results, an important step that they did not take, and suggest the need to make technology relevant to the students' work on modelling tasks. University of Costa Rica info:eu-repo/semantics/publishedVersion

Based on a teaching intervention with modelling activities involving experimental work in 9th grade classes, the goal of this study is to find out how students estimate the credibility a modelling task setting when it integrates a hands-on experimental approach. The theoretical background is based on the concept of authenticity and its long tradition in mathematical modelling and inquiry-based learning and advances a path of discussion around the concept of credibility. This is done in close connection to the relevance of experiments and hands-on activities in mathematical modelling in line with a science, technology, engineering, and mathematics approach. The students were assigned the task of creating custom colour paint, responding to an order from a customer in a paint manufacturing company. The empirical data were collected from the observation of two classes and a questionnaire after completion of the task. The results indicate that students viewed the event as credible, as well as the goal of the task. They also considered the experimental work to be necessary and found the mathematical model obtained to be feasible. Moreover the students showed awareness of a distinction between their experiments and models and those developed by professionals. In short, the students ascribed credibility to the task setting and were able to acknowledge an approximation to reality in both the prototype created and the model built.

This innovative volume seeks to broaden the scope of research on mathematical problem solving in different educational environments. It brings together contributions not only from leading researchers, but also highlights collaborations with younger researchers to broadly explore mathematical problem-solving across many fields: mathematics education, psychology of education, technology education, mathematics popularization, and more. The volume’s three major themes - technology, creativity, and affect - represent key issues that are crucially embedded in the activity of problem solving in mathematics teaching and learning, both within the school setting and beyond the school. Through the book’s new pedagogical perspectives on these themes, it advances the field of research towards a more comprehensive approach on mathematical problem solving. Broadening the Scope of Research on Mathematical Problem Solving aims to provide a valuable resource for researchers and teachers interested in mathematical problem solving, as well as researchers and teachers interested in technology, creativity, and affect.

The study that is partially reported in this chapter has been developed, in theoretical and empirical terms, within the Problem@Web Project. The overall goal of the project was to study mathematical problem solving in a context outside the classroom, namely, web-based mathematical problem-solving competitions. One of the research foci of this project is the creativity manifested in the solutions of mathematical problems produced by the participating children. Thus, the project sought to analyse and examine the close and also complex relationship between creativity and problem solving. The perspectives that emphasize the development of creativity also look for tools that allow their assessment. We have considered, in a first approach, the components of creativity that are most often found in the research on divergent thinking, i.e. originality, fluency and flexibility. The next step was to find mathematical problem-solving dimensions, supported by the research in the field, which would lead to mathematical creativity as a combination of divergent thinking, convergent thinking and other abilities. An analytical tool that integrates the dimensions of originality, activation of mathematical knowledge and activation of forms of representation was then achieved. This tool is here applied to a sample of ten solutions to a problem of the mathematical competition SUB12. In combining a qualitative description and a numerical-graphical interpretation, the results highlight the three dimensions contributing to the overall score of mathematical creativity. They also illustrate the relevance of the ability to represent and express mathematical ideas involved in solving problems.

The mathematical competitions SUB12 and SUB14 are web-based inclusive competitions where multiple affective dimensions come to the fore in relation to mathematics and problem solving. The moderate mathematical challenges proposed to participants, independent of the official curriculum and intended to be mathematically rich problems, allow young students to find their own strategies and approaches and incite them to come up with ways of expressing their thinking. As part of our research focusing on the competitions SUB12 and SUB14, one goal is to obtain an outline of students’ attitudes in the context of their participation. For this, we adopted a three-dimensional model that addresses students’ views, students’ emotional dispositions, and students’ perception of their competence, to study the participants’ attitudes towards mathematical problem solving and mathematics. In this chapter, we present and analyse data from a questionnaire, which are complemented by the voice of the participants themselves, collected through interviews and from a pool of students’ messages arising from their online participation. Our results indicate that the three affective dimensions are closely interconnected. The global positive attitude towards the participation in the competition reveals emotional dispositions such as fun and enjoyment in solving problems, beliefs such as the view that they become more confident and learn more, and perceived self-competence, when they realize that they are capable of success with different levels of performance and knowledge. The mathematical competitions SUB12 and SUB14 provide fulfilling experiences to the young participants (in each of the three dimensions of the model adopted), which are linked to the will to solve and express mathematical problems, to a favourable view about mathematics, and to a perception of self-improvement.

This study offers a view on students' technology-based problem solving activity through the lens of a theoretical model which accounts for the relationship between mathematical and technological knowledge in successful problem solving. This study takes a qualitative approach building on the work of a 13-year-old girl as an exemplary case of the nature of young students' spontaneous mathematical problem solving with technology. The empirical data comprise digital records of her approaches to two problems from a web-based mathematical competition where she resorted to GeoGebra and an interview where she explains and describes her usual problem solving activity with this tool. Based on a proposed model for describing the processes of mathematical problem solving with technologies (MPST), the main results show that this student's solving and expressing the solution are held from the early and continuing interplay between mathematical skills and the perception of the affordances of the tool. The analytical model offers a clear picture of the type of actions that lead to the solution of each problem, revealing the student's ability to deal with mathematics and technology in problem solving. By acknowledging this as a case of a human-with-media in solving mathematical problems, the students' efficient way of merging technological and mathematical knowledge is portrayed in terms of her techno-mathematical fluency. project Problem@Web - Fundacao para a Ciencia e Tecnologia [PTDC/CPE-CED/101635/2008] Fundacao para a Ciencia e Tecnologia [SFRH/BD/73363/2010] info:eu-repo/semantics/publishedVersion

The final chapter summarises the overall findings of the Problem@Web project and considers the implications of the findings in terms of how the youngsters of today tackle mathematical problems and communicate their mathematical problem-solving. With data from the youngsters’ participation in two online mathematical problem-solving competitions that were characterised by moderately challenging problems, we found that the youngsters we studied had domain over a set of general-use digital tools and while they were less aware of digital resources with a stronger association with mathematics they were able to gain many capabilities by tackling the mathematical problems and seeking expeditious, appropriate and productive ways of expressing their mathematical thinking. In this respect, they were able to harness their technological skills while simultaneously developing and improving their capacity to create and use a range of mathematical representations. We explain this as co-action between the tool and the solver, with this interconnectedness leading to jointly developed technological skills and mathematical skills that result in the capacity of mathematical problem-solving with technology. Given the possibility of youngsters developing this capacity, a key issue is how this can be harnessed to promote the success of youngsters in mathematics in our digital era.

Over several years of the SUB12 and SUB14 online mathematical competitions, we became aware of the technological fluency of many of the young participants. We draw on quantitative data from a survey that was administered online by inviting all participants to respond. The data show how the participants describe themselves in terms of their experience with several digital tools. We have found that they feel comfortable with the use of text and presentation editors and know how to use several tools for writing, creating tables and constructing diagrams and visual representations. In contrast, they seem to be less capable with spreadsheets (especially as a mathematical tool) and dynamic geometry software. Some participants preferred to submit copies of their hand-written answers to the problems as scanned images or digital photos. In reporting the results of our survey, we present a selection of solutions covering a palette of examples that help to exemplify the skills and fluency of the competition participants. They unveil a particular trait of this mathematical problem-solving activity since these digital solutions bring together problem-solving and the expressing of mathematical thinking.

This chapter offers the perspectives of teachers on youngsters solving mathematical problems with technology during the SUB12 and SUB14 mathematics competitions. Drawing on a series of interviews with teachers who have supported the participation of their students over several editions of the competitions, we identified what they see as the competitions’ most significant features. The teachers spoke about the different kinds of support that are available to youngsters throughout the successive stages of the competitions, from the initial dissemination, to the online Qualifying phases, and lastly to the on-site Final. Based on their statements, the teachers say that they value the type of problems they characterise as challenging, real problems, appropriate for all students and useful as pedagogical resources. They make a distinction between such non-routine and extracurricular problems and the more school-like problems presented in mathematics textbooks. They are favourable to the use of technologies within the competitions, even when admitting initial difficulties that they nevertheless seemed to have overcome over the years. Some of these teachers enthusiastically describe how they sometimes integrated the competition problems into their class teaching and how they helped and encouraged students to use digital technologies for solving and expressing the solutions they submitted. The need to develop mathematical communication is seen as another challenge, and this, say the teachers, gave them the opportunity to explore different mathematical representations with their students. As a final point, several teachers highlighted the fact that youngsters’ participation in the competitions was a motivating factor, contributing to their enjoyment of mathematics and feelings of inclusion in a community gathering many youngsters, parents and teachers around mathematical challenges.

Today’s youngsters are growing up in an era of rapidly advancing digital technologies. While young people in this generation are undoubtedly active users of digital technologies, the issue of whether their digital competency levels are necessarily well developed is a topic of debate. This chapter provides an introduction to, and an overview of, the Problem@Web project, a project that grew out of our interest in understanding how Portuguese youngsters participated in two online mathematical problem-solving competitions. These online competitions have allowed youngsters in any suitable place, and at any suitable time, to engage themselves in tackling mathematical problems by utilising solving strategies with any digital tools that they have available. During the project, we analysed numerous problem solutions submitted throughout three editions of the competitions and interviewed a sample of young participants, mathematics teachers and youngsters’ parents and relatives. The chapter captures the contribution that the Problem@Web project makes to understand youngsters’ mathematical problem-solving with technology.

In this article we report and discuss a contextual problem solving task that was proposed to a class of 8th grade (13^14 -year-old) students. These students had been developing a reasonable experience in the use of the spreadsheet to model relations within contextual problems and chose to use this tool to solve the mentioned problem, engaging in the process of translating relations between variables and combining them in chained models, while working with fractions, multiples and expressions. We intend to highlight the role of the spreadsheet in students’ processes of variable identification and translation of the problem conditions, their numerical approaches to algebraic models and their experimental forms of finding solutions to equations.

By looking at the global context of two inclusive mathematical problem solving competitions, the Problem@Web Project intends to study young students’ beyond-school problem solving activity. The theoretical framework is aiming to integrate a perspective on problem solving that emphasises understanding and expressing thinking with a view on the representational practices connected to students’ digital mathematical performance. Two contextual problems involving motion are the basis for the analysis of students’ digital answers and an opportunity to look at the ways in which their conceptualisations emerge from a blend of pictorial and schematic digital representations.

The starting point of WG6 was a panel discussion on the nature of modelling and applications and its theoretical description. One central aspect, namely the distinction between applications and mod...

In our research, we aim to look at the notion of knowledge as it is elicited through mathematical modelling of daily life situations, within the context of adult education. In the school scenario of adult education, notions from situated cognition will be brought into play to examine the meaning of mathematisation and of mathematical modelling competence. The empirical data refer to a 2-month period of work on the theme of cookery, one that was chosen by the students. Data were collected in a school environment within the subject “Mathematics for Life”, a course in Adult Education, for certification of compulsory general education (i.e., 9th grade in regular school).

The present study is a classroom-based research where students develop mathematical modelling tasks that involve manipulating and experimenting with real objects. The research was developed in two 9th grade classes of students aged 14/15 years old. These students never had this kind of modelling activities in their mathematics classes before. Our purpose is to discuss the modelling routes produced by middle school students in an experimental mathematics environment – both from the point of view of realistic mathematics education and of the model-eliciting perspective.

This paper analyses an experience undertaken by a group of teachers, who introduced the computer in a 10th grade mathematics classroom for carrying out problem solving and investigational activities. The main question of interest is the discussion of significant issues and unexpected situations that emerged within this innovative process, specially regarding the reactions of the students. Using a case study methodology [1], the sources of data were interviews with the teachers, visits to the school, and the learning and dissemination materials produced.