Nélia Amado

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.

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).

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.

Youngsters Solving Mathematical Problems with Technology: Their Experiences and Productions

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.

Digitally Expressing Algebraic Thinking in Quantity Variation

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.

Theoretical Perspectives on Youngsters Solving Mathematical Problems with Technology

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.

Digitally Expressing Co-variation in a Motion Problem

Co-variational reasoning has received particular attention from researchers and mathematics educators because it is considered of paramount importance for the understanding of concepts such as variable, function, rate of change, derivative, etc. Some of the critical issues that have been identified in several studies consist of the difficulty in interpreting the simultaneous variation of two quantities, particularly in overcoming coordination problems of two variables changing in tandem. A relevant question in the study of co-variational reasoning concerns representing the joint variation of quantities and performing translations between different representations. Problems of motion involving variation over time are strongly linked to the concept of co-variation and require the ability to translate a dynamic situation by means of mostly static representations. Those problems require the construction of a conceptual model that, in some way, visually contains dynamism. In taking solving and expressing as a unit of analysis and focusing on the ways in which commonly available digital technologies are used by youngsters as tools in problem-solving, we analyse the approaches used by the participants in SUB14 to a motion problem. Some surprising results of the content analysis of over 200 answers indicate that the textual/descriptive form of presenting a model of the situation had a clear dominance. The use of tabular representations along with pictorial/figurative content was also present in a high percentage of solutions. Furthermore the use of digital media was decisive in producing visuality, i.e. ways of depicting the displacement with time (quasi-dynamic representations).