Susana Carreira

Computer Spreadsheet and Investigative Activities: A Case Study of an Innovative Experience

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.

Youngsters solving mathematical problems with technology: the results and implications of the Problem@Web project

This book contributes to both mathematical problem solving and the communication of mathematics by students, and the role of personal and home technologies in learning beyond school. It does this by reporting on major results and implications of the Problem@Web project that investigated youngsters’ mathematical problem solving and, in particular, their use of digital technologies in tackling, and communicating the results of their problem solving, in environments beyond school. The book has two focuses: Mathematical problem solving skills and strategies, forms of representing and expressing mathematical thinking, technological-based solutions; and students´ and teachers´ perspectives on mathematics learning, especially school compared to beyond-school mathematics.

Students’ Modelling Routes in the Context of Object Manipulation and Experimentation in Mathematics

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.

Mathematical Modelling of Daily Life in Adult Education: Focusing on the Notion of Knowledge

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).

Looking Deeper into Modelling Processes: Studies with a Cognitive Perspective – Overview

As pointed out by Galbraith and Stillman (2001), the issue of the role played by extra-mathematical knowledge in modelling and applications activities is part of an agenda shared by researchers and educators who are interested in the specific nature and characteristics of modelling activities within educational settings. An important aspect of such mathematical activities is the actual structure and elements of the context situation that are embedded (either explicitly or implicitly) in the task formulation, thus opening significant room for assumptions, interpretations and considerations concerning the so-called extra-mathematical world. Nurturing a long-standing debate on the nature of mathematical models and on the types of knowledge and experiences activated by mathematical modelling, the six articles composing this chapter highlight in very different and yet strikingly coherent ways the paramount role of knowledge, besides mathematics, in illuminating students’ processes, decisions and understandings while working on modelling tasks.

From Acorns to Oak Trees: Charting Innovation Within Technology in Mathematics Education

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.

Mathematical Problem-Solving with Technology: An Overview of the Problem@Web Project

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.

Perspectives of Teachers on Youngsters Solving Mathematical Problems with Technology

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.

Digitally Expressing Conceptual Models of Geometrical Invariance

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.

Youngsters Solving Mathematical Problems with Technology: Summary and Implications

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.