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The aim of this study is to understand how students' mathematical thinking is activated and nurtured in solving a modeling problem, where the problem situation involves the design of a system. From a STEM integrated perspective, 9(th) grade students worked on a modeling task aiming to create an identification system based on hand biometrics. The theoretical framework proposes a conceptualization of the interplay between the mathematical modeling process, from a cognitive perspective, and the engineering design process. Central ideas refer to the cyclical nature of both processes and to the sub-processes involved in them. The empirical data were collected in two design-based research cycles with different 9(th) grade classes. The data from the groups' audio and video recording and the students' productions were analyzed under a directed qualitative content analysis informed by theory. The results showed a global pattern in the students' thinking in solving a design system problem. The overlapping and interplay between the mathematical modeling and the design process was a prominent characteristic of students' thinking. The modeling cycle was mirrored by a design cycle, with both running in parallel. System thinking pushed and drove students' mathematical thinking, from the system requirements to the prototype validation. info:eu-repo/semantics/publishedVersion

Neste artigo apresentamos a temática da avaliação, a partir da análise de um processo investigativo junto a estudantes de economia, de uma universidade pública, em que se reflete sobre a percepção que estes graduandos têm sobre avaliação e aprendizagem. O estudo caracteriza-se como uma pesquisa qualitativa, a investigação foi desenvolvida com trinta estudantes que estão no 6° semestre e os dados produzidos a partir de um questionário, com perguntas fechadas e abertas, foram analisados a partir da Análise Textual Discursiva, de Moraes e Galiazzi (2016). A pesquisa apresenta como aporte teórico algumas ideias de autores como Cid e Fialho (2012), Fernandes (2005), Pacheco (2012), Santiago (2012), William (2009), entre outros. Por meio deste estudo, evidenciamos que a maioria dos estudantes considera que o principal propósito da avaliação é de medir o seu desempenho e na atribuição de uma classificação, revelando pouco interesse na qualidade da aprendizagem, o que na concepção dos participantes é o mais importante.Palavras-chave: Avaliação; Aprendizagem; Metacognição. ABSTRACT: In this article the topic evaluation is presented, through the analysis of an investigative process with economics’ students, from a public university, in which we reflect about the perception these graduates have on evaluation and learning. The study is characterized as a qualitative research, the investigation was developed with thirty students who are in the 6th semester and the data produced through a questionnaire, with closed-ended and open-ended questions, were analyzed through the Discursive Textual Analysis, from Moraes and Galiazzi (2016). The research presents as theoretical contributions some ideas from authors as Cid & Fialho (2012), Fernandes (2005), Pacheco (2012), Santiago (2012), William (2009), among others. Through this study, it’s evident that most of the students consider the main purpose of evaluation to be measuring their performance and attributing a classification, revealing little interest in the learning quality, which in the participants’ conception it’s the most important.Keywords: Assessment; Learning; Metacognition.

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.

Resumo Assumindo a criatividade como um potencial que pode ser desenvolvido em todos os indivíduos, o objetivo deste estudo consiste em descrever, caracterizar e compreender a criatividade matemática manifestada nas respostas produzidas por alunos do Ensino Básico (10 a 12 anos) num campeonato de resolução de problemas de carácter inclusivo realizado através da Internet. A metodologia é de natureza interpretativa, utilizando a análise de conteúdo num conjunto de 10 resoluções a um dado problema. A análise apoia-se na aplicação de um referencial de criatividade matemática baseado nos parâmetros de originalidade, fluência e flexibilidade, adaptados à resolução de problemas matemáticos. Os resultados revelam diversas características da criatividade matemática nas produções dos participantes do campeonato, sobressaindo a primazia da originalidade que é reforçada pela capacidade de representação e de mobilização de conhecimento matemático. Apesar de ser variável a presença das três dimensões, o seu efeito combinado é determinante na individualização de resoluções matematicamente criativas.

Este artigo discute a atividade de resolução de problemas de Matemática com tecnologias no âmbito de um campeonato extraescolar a distância. O estudo visa descrever os aspetos subjacentes à utilização simultânea de conhecimento matemático e tecnológico nessa atividade, com foco no desenvolvimento de modelos conceptuais que conduzem às soluções dos problemas. A investigação, de natureza qualitativa, centra-se na análise de um conjunto de produções digitais elaboradas pelos participantes com recurso ao GeoGebra. Os dados mostram que o software é usado de forma livre e voluntária para estruturar e amplificar o pensamento matemático, influenciando os processos de resolução. Para além de suportar a materialização das situações problemáticas e favorecer a transformação de ideias estáticas em ideias dinâmicas, faz emergir matematizações que sustentam modelos conceptuais bastante distintos. Os aspetos subjacentes à existência de diferentes modos de resolver-e-exprimir um dado problema com o GeoGebra são explicados em termos da fluência tecno-matemática de cada participante., This paper addresses a mathematical problem-solving activity with technology in the context of an online beyond school competition. The study aims to describe the underlying aspects of simultaneously resorting to mathematical and technological knowledge in such an activity, focusing on the development of conceptual models for the solutions of problems. The research, following a qualitative approach, is based on the analysis of a set of digital solutions produced with GeoGebra. Main results show that the software is used freely and willingly to structure and amplify the mathematical thinking, influencing the solving and expressing processes. Besides supporting the materialization of the problematic situations and promoting the transformation of static ideas into dynamic ideas, it drives mathematization processes that support quite different conceptual models. The fundamental features that lead to different ways of solving-and-expressing a given problem with GeoGebra are explained in terms of the techno-mathematical fluency of each participant.

In Portugal, tuberculosis (TB) continues to be a major public health problem. The phenomenon of migration has a growing impact on the local epidemiology of TB. Aim: Retrospective assessment of TB cases in the immigrant population, based on their socio-demographic, risk factors and clinical features at an urban Chest Disease Center, in Portugal, over a period of 11 years. Results: Of a total of 524 new cases of TB reported between 2008 and 2018, 108 (20.6%) were immigrants. Seventy-eight (72.2%) were males, with a mean age of 45 years. The frequency of TB cases was higher in the 35-44 age group. It has been found that the immigrants were native mainly from Angola (37.9%) and Cape Verde (15.7%). Seventy-seven immigrants (75%) lived in Portugal for more than 5 years. From all evaluated cases, 80 corresponded to pulmonary tuberculosis; 62.5% with a positive sputum acid-fast bacilli smear and 92.5% were confirmed cases. In 53 patients (49%), the immunodeficiency virus infection was the main comorbidity associated (13.9%). The classic antibiotic treatment scheme (Isoniazid + Rifampin + Ethambutol + Pyrazinamide) was used in 86 patients. Eighty antibiograms were performed, 72 did not show any type of resistance. Direct Observed Therapy has been used in 90 patients (83.3%) and most of the patients (88.9%) completed the treatment. Conclusions: The high incidence rates of TB among immigrant population are mostly a reflection of the incidence rates in their countries of origin. Host countries have a responsibility to develop a deeper knowledge about health care needs of immigrants in general and particularly for TB, because of the impact of international migration on local TB epidemiology.

The article offers a reflection stemming from the elaboration of a doctoral research project, which involves the students enrolled in the first year of the Integrated Technical Secondary Education, at the Instituto Federal de Educação, Ciência e Tecnologia do Mato Grosso (IFMT). Aiming to illustrate the main research study, we present a section of a qualitative pilot study carried out in the mathematics subject. The broader study involves students at risk of failure in mathematics and seeks to bring the students’ voice into the discussion of their failure in the discipline. The main purpose of the study is to understand how first-year students experience situations of failure and difficulty in mathematics that put them at risk of dropout and/or retention. It is expected to contribute to the improvement of mathematics learning, to a better understanding of students’ evasion and retention, and to the design of policies and actions that may improve the quality of technical education and promote its democratization.

This chapter focuses on the affordances of the spreadsheet in solving a complex word problem in light of the multiple conceptual models that students produce in their approaches to the solution with the digital medium. The empirical study carried out examines, through a qualitative analysis, a set of resolutions produced by a class of 8th grade students to a word problem involving multiple variables and conditions. The aim is to appreciate how the way in which the students take advantage of the spreadsheet can be related to their approaches to a problem that is algebraically translated into a system of simultaneous equations. We found that the spreadsheet offered students diverse ways of formulating relational conditions in solving a problem for which they had not yet learned the formal algebraic method. The affordances of the spreadsheet were fundamental for the success of the students in solving the problem. It provided them a representational medium for capturing the structure of the situation and it suggested ways of conceptually modelling the several conditions presented. In particular, the spreadsheet opened up opportunities for assigning meaning to and expressing the multiple conditions (equations and inequalities) that structure the problem.

Research has long been using analytical tools to describe the processes students engage in when solving non-routine mathematical problems. In this chapter we describe and discuss our progress on devising and implementing an analytical tool that aims to account for the use of technological tools by combining a mathematical problem solving model with a digital problem solving framework. By means of the Mathematical Problem Solving with Technology model (MPST) we report the case of Marco using technologies for solving two problems from a beyond school competition. Results show that Marco’s choice of the tools is grounded on an explicit knowledge of their affordances and how they enhance his mathematical thinking, mainly by triggering visual approaches that support the development of conceptual models for solving-and-expressing the solutions to the problems.

O presente estudo buscou aprofundar os conhecimentos sobre avaliação da aprendizagem e conhecer as percepções dos alunos sobre este processo. Procurou-se identificar e analisar as possíveis relações entre as práticas de avaliação e os seus efeitos na vida escolar dos alunos, e também refletir sobre os diversos instrumentos avaliativos que possibilitam efetividade nos processos de ensino e de aprendizagem. Como instrumento de coleta de dados foi utilizado um questionário impresso com a pergunta: “Quando ouve falar de avaliação, qual a primeira ideia que te vem á cabeça?”. Foram abordados alunos de três turmas da mesma série de diferentes Cursos Técnicos Integrados ao Ensino Médio, nas aulas de Matemática. As respostas dos alunos demonstraram uma variedade de sentimentos/posturas e possibilitou a análise dos dados por meio da criação de seis categorias: Sentimentos; Preocupação com a recuperação; Dificuldade em calcular; Cobrança dos pais; Esquecimento e distração; Avaliação da aprendizagem.

Background: \ud We have previously shown that raised p-S6K levels correlate with resistance to chemotherapy in ovarian cancer. We hypothesised that inhibiting p-S6K signalling with the dual m-TORC1/2 inhibitor in patients receiving weekly paclitaxel could improve outcomes in such patients.\ud \ud Patients and methods: \ud In dose escalation, weekly paclitaxel (80 mg/m2) was given 6/7 weeks in combination with two intermittent schedules of vistusertib (dosing starting on the day of paclitaxel): schedule A, vistusertib dosed bd for 3 consecutive days per week (3/7 days) and schedule B, vistusertib dosed bd for 2 consecutive days per week (2/7 days). After establishing a recommended phase II dose (RP2D), expansion cohorts in high-grade serous ovarian cancer (HGSOC) and squamous non-small-cell lung cancer (sqNSCLC) were explored in 25 and 40 patients, respectively.\ud \ud Results: \ud The dose-escalation arms comprised 22 patients with advanced solid tumours. The dose-limiting toxicities were fatigue and mucositis in schedule A and rash in schedule B. On the basis of toxicity and pharmacokinetic (PK) and pharmacodynamic (PD) evaluations, the RP2D was established as 80 mg/m2 paclitaxel with 50 mg vistusertib bd 3/7 days for 6/7 weeks. In the HGSOC expansion, RECIST and GCIG CA125 response rates were 13/25 (52%) and 16/25 (64%), respectively, with median progression-free survival (mPFS) of 5.8 months (95% CI: 3.28–18.54). The RP2D was not well tolerated in the SqNSCLC expansion, but toxicities were manageable after the daily vistusertib dose was reduced to 25 mg bd for the following 23 patients. The RECIST response rate in this group was 8/23 (35%), and the mPFS was 5.8 months (95% CI: 2.76–21.25).\ud \ud Discussion: \ud In this phase I trial, we report a highly active and well-tolerated combination of vistusertib, administered as an intermittent schedule with weekly paclitaxel, in patients with HGSOC and SqNSCLC.\ud \ud Clinical trial registration: \ud ClinicialTrials.gov identifier: CNCT02193633.

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The purpose of this study is to analyze the use of the problematizing methodology as an evaluation process in the health area. The study field used comprised five students from am elderly care worker vocational course. The observed results describe the difficulty to incorporate the problematizing methodology as an evaluation process in an already structured curriculum. It was found that students still carry a stigma when inquired about evaluations and infer about punitive situations. It was observed that the students understand the importance of the process and made use of it while performing their own learning. It is considered that the methodology used allow to subjectively measure student learning by including their own discussions and previous knowledge established in the classroom.

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.

Across the world, problem-solving remains a key element of the mathematics curriculum and a central aim of mathematics teaching and learning. In order to embrace innovative ways of thinking and working, mathematical problem-solving in school classrooms is being challenged by the increasingly widespread use of digital technologies, the promotion of creativity, and the recognition of the affective and aesthetic dimensions. As an introduction to this edited collection, this chapter provides a brief overview of trends in research on mathematical problem-solving before outlining the central themes of the book: technology in mathematical problem-solving, creativity in mathematical problem-solving, and affect and aesthetics in mathematical problem-solving. The final section of the book makes evident the need to investigate in a comprehensive way how technology, creativity, and affect and aesthetics in mathematical problem-solving are being brought together.

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.

All over the world, several organizations have nurtured the development of students’ problem-solving abilities by organizing competitions and tournaments of different kinds. This is the case of the Mathematical Competitions SUB12 and SUB14, promoted by the University of Algarve, addressing students from grades 5 to 8 (10–14 year olds) in the south of Portugal. To each proposed problem, participants are required to explain their problem-solving process and find ways to express their thinking. They may use any of the digital tools they have available and they find useful for solving a given problem. Our research has uncovered the aptitudes of young competitors in taking advantage of everyday digital tools and its representational expressiveness to give form and substance to their reasoning and strategies. Another emerging aspect is the apparent existence of different degrees of robustness of the solutions submitted, mainly in terms of the strategies that competitors develop, with a particular technological tool, to solve the problems. In this chapter, we are taking a selection of solutions submitted to two problems, in which competitors resort to GeoGebra, in one case, and to Excel, in the other. We offer a proposal for identifying levels of sophistication and robustness of technology-based solutions to the problems, according to the characteristics of the tool use and its connection to the conceptual models underlying students’ thinking on the 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

Technology has created an expectation in all levels of education that requires us to understand how we can harness its potential for improving the depth and quality of mathematical learning. It is highly unlikely that there is a universal recipe or formula for how technology should be used that would satisfy every context or culture, but there have been recurring trends in the process of designing and implementing such innovative environments. By considering the papers included in proceedings of the past International Conferences on Technology in Mathematics Teaching (ICTMT), this chapter aims to highlight how a few key innovations have been seeded and taken root within this community. We begin by describing the ways in which innovation has been presented at ICTMT conferences with a view to exploring this from the perspectives of technology designers, researchers and teachers/lecturers from all levels of education. Given the extensive literature on this topic, it is not feasible to carry out a comprehensive survey of the complete literature base, however it is anticipated that the analysis of key ICTMT papers will be sufficient to present an informative and insightful picture and highlight some important knowledge and experience that has been elicited and disseminated.

Resumo: Introdução: A doença pulmonar obstrutiva crónica (DPOC) está associada a várias comorbilidades, contudo, a prevalência das mesmas varia entre os estudos. Objetivo: Determinar a prevalência das diversas comorbilidades em doentes com DPOC estádio IV do The Global Initiative for Chronic Obstructive Lung Disease (GOLD) 2010, seguidos em regime de ambulatório, num hospital universitário. Métodos: Foi concebido e aplicado um questionário com o objetivo de caracterizar a DPOC e suas comorbilidades. Os dados foram completados por consulta do processo clínico. Resultados: Foram incluídos 89 doentes (87% do género masculino), com média etária de 68 anos, 79% dos quais ex-fumadores. O valor de volume expiratório máximo por segundo (VEMS) médio foi de 38% do previsto e todos os doentes apresentavam insuficiência respiratória crónica. Trinta e cinco doentes (39%) apresentavam exacerbações frequentes.Trinta e sete doentes (42%) tinham apresentado pelo menos um internamento por exacerbação da sua doença respiratória no ano anterior e 66 doentes (74%) nos últimos 5 anos.A maioria dos doentes (97%) apresentava pelo menos uma comorbilidade, com uma média de 4 comorbilidades por doente e um índice de Charlson médio de 2.As comorbilidades mais frequentes foram doenças cardiovasculares (69%), patologia osteoarticular (51%), disfunção erétil (48%), síndrome da apneia do sono (43%), dislipidémia (35%), cataratas (31%), refluxo gastro-esofágico (29%) e diabetes (20%).Os exacerbadores frequentes apresentaram um risco aumentado de terem 2 ou mais comorbilidades (Odds ratio de 5), bem como uma maior prevalência de refluxo gastro-esofágico (p = 0,006) e um maior número de internamentos no último ano e nos 5 anos anteriores (p

Introduction: Chronic Obstructive Pulmonary Disease (COPD) is associated with several co-morbidities, however their prevalence varies from one study to another. Aim: To determine the prevalence of several co-morbidities in patients with COPD severity score GOLD 4 (The Global Initiative for Chronic Obstructive Lung Disease, 2010) followed in ambulatory care, in a University Hospital. Methods: A questionnaire was designed and carried out in order to characterize COPD and its co-morbidities. Clinical files were consulted in order to complete the data. Results: 89 patients (87% male) with a mean age of 68 years old, of which 79% were ex-smokers, were included. The average value of FEV1 (forced expiratory volume in one second) was 38% of the expected values and all the patients presented chronic respiratory failure. Thirty-five patients (39%) were frequent exacerbators.Thirty-seven patients (42%) had been hospitalized at least once due to exacerbation of their respiratory disease in the previous year, and 66 patients (74%) hospitalized in the previous five years.Most of the patients (97%) presented at least one comorbidity, with an average of 4 co-morbidities per patient and an average Charlson index of 2.The most frequent co-morbidities were cardiovascular diseases (69%), osteoarticular pathology (51%), erectile dysfunction (48%), sleep apnoea syndrome (43%) dyslipidaemia (35%), cataracts (31%), gastroesophageal reflux (29%) and diabetes (20%).Frequent exacerbators presented an increased risk of having two or more co-morbidities (Odds Ratio of 5), as well as a higher prevalence of gastroesophageal reflux (p = 0, 0006) and more hospitalizations in the last year and in the previous 5 years (p

O presente trabalho é embasado em estudos sobre formação continuada de professores para a utilização das tecnologias. A investigação desenvolveu-se com uma professora integrante de um curso de formação continuada para a utilização de tablets nas aulas de Matemática. A proposta do curso, de uma formação ancorada na prática, incluiu não apenas as sessões presenciais, mas também um período considerável de integração do conhecimento teórico à prática de sala de aula da professora. Essa ação apoiou-se no mentoring, apresentado por Amado (2007) como uma relação interpessoal, baseada na confiança e no apoio, que se desenvolveu por meio do acompanhamento da formadora/pesquisadora, tanto nas sessões do curso, quanto no planejamento e nas aulas em que a professora utilizou as tecnologias com os seus alunos. A sequência didática da formação embasou-se no modelo TPACK (Conhecimento Tecnológico e Pedagógico do Conteúdo), apresentado por Koehler e Mishra (2006; 2007; 2009) que consideram fundamental aos professores o desenvolvimento desse conhecimento para que possam integrar de modo eficiente as tecnologias nas suas aulas. Consistiu em uma tentativa de formação mais abrangente, que fosse além da instrução técnica e envolvesse o desenvolvimento individual, profissional e institucional da carreira docente. A pesquisa teve como objetivo verificar a trajetória de desenvolvimento da professora na utilização das tecnologias e se desenvolveu por meio de uma abordagem qualitativa caracterizada como um estudo de caso. Os resultados identificaram que entre a pesquisadora e a pesquisada se desenvolveu uma relação de mentoring. Também apontaram que a proposta contribuiu no desenvolvimento profissional da professora.

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.

Given that solving mathematical problems entails developing ways of thinking and expressing thoughts about challenging situations where a mathematical approach is appropriate, this chapter unveils a theoretical framework that aims to guide a better interpretation of students’ capability to solve mathematical problems with digital technologies, in the context of online mathematical competitions. The main purpose is to provide a way of understanding how students find effective and productive ways of thinking about the problem and how they achieve the solution and communicate it mathematically, based on the digital resources available. By discussing several theoretical tools and constructs, a theoretical stance is developed to conceptualise problem-solving as a synchronous process of mathematisation and of expressing mathematical thinking in which digital tools play a key role. This theorisation draws on the role of external representations and discusses how a digital-mathematical discourse is used to express the development of the conceptual models underlying the solution. In this conceptualisation, a symbiotic relation between the individual and the digital tools used in problem-solving and expressing is postulated and outlined: the inseparability of humans and media sustains the idea that students and tools are agents performing knowledge in co-action, while approaching mathematical problems. Looking at the solution to a problem is seeing a fusion of the solver’s knowledge and the tool’s built-in knowledge, rather than an aggregate of both or a complementarity between them.

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 chapter, we describe and analyse a number of examples of 7th and 8th graders showing diverse ways of expressing their mathematical thinking in solving algebraic word problems with a spreadsheet. Different youngsters’ approaches to situations where quantity variation is involved are characterised. The problems require finding an unknown value under a set of conditions that frame a problem situation. The use of the spreadsheet is thoroughly examined with the aim of highlighting the nature of problem-solving and expressing in the digital tool context as compared to the formal algebraic method; moreover, the ways in which students take advantage of the tool (being guided by and also guiding the spreadsheet distinctive forms of organising and performing variation in columns and cells) are important indicators of their algebraic thinking within the problem-solving activity. Finally, we pay attention to indicators of “co-action” in students’ work on the spreadsheet as it tends to be more related to structuring solutions by means of creating variable-columns than with tentative ways of generating inputs in recipient cells.

This chapter develops around two fundamental ideas, namely, that (1) the perception of the affordances of a certain digital tool is essential to solving mathematical problems with that particular technology and that (2) the activity thus undertaken stimulates different mathematising processes which, in turn, result in different conceptual models. Looking thoroughly, from an interpretative perspective, at four solutions to a particular geometry problem from participants who decided to use dynamic geometry software at some point of their solving activity, our main purpose is to illustrate the ways in which the same tool affords different approaches to the problem in terms of the conceptual models developed for studying and justifying the invariance of the area of a triangle. Their different ways of dealing with the tool and with mathematical knowledge are interpreted as instances of students-with-media engaged in a “solving-with-dynamic-geometry-software” activity, enclosing a range of procedures brought forth by the symbioses between the affordances of the dynamic geometry software and the youngsters’ aptitudes. The analysis shows that different people solving the same problem with the same digital media and recognising a relatively similar set of affordances of the tool produce different digital solutions, but they also generate qualitatively different conceptual models, in this case, for the invariance of the area.

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 central question addressed in this article concerns the ways in which applied problem situations provide distinctive conditions to support the production of meaning and the understanding of mathematical topics. In theoretical terms, a first approach is rooted in C. S. Peirce's perspective on semiotic mediation, and it represents a standpoint from which the notion of interpretation is taken as essential to learning. A second route explores metaphorical thinking and undertakes the position according to which human understanding is metaphorical in its own nature. The connection between the two perspectives becomes a fundamental issue, and it entails the conception of some hybrid constructs. Finally, the analysis of empirical data suggests that the activity on applied situations, as it fosters metaphorical thinking, offer students' reasoning a double anchoring (or a duplication of references) for mathematical concepts.