Hélia Jacinto

Techno-mathematical fluency (TmF) is the ability to coordinate mathematical knowledge with technological means—digital and non-digital—to solve mathematical problems and express solutions, by recognising affordances, selecting appropriate tools and data, and integrating them with mathematical ideas in iterative cycles of exploration and integration. It goes beyond instrumental tool use to encompass reasoning, modelling, representation, and communication mediated by technologies, and functions as a form of expertise important for both students’ learning and teachers’ professional practice.

This study examines the relationship between problem solving (PS) and problem posing (PP) in mathematics education, focusing on 3rd-graders’ ability to create problems in Geometry and Measurement. A theoretical framework was adopted to assess the evolution of the problems posed, which considers mathematical and linguistic complexity. The study also compared posed and solved problems, for a deeper understanding of the connections between PS and PP skills. Conducted over eight lessons, it combined curriculum-aligned PS and PP tasks. Students’ written work was analyzed qualitatively. Findings show substantial improvements in PP skills, including better mathematical language, grammar and expression, alignment to task prompt, and problem solvability. Initially, students replicated structures from solved problems but gradually showed more creativity and autonomy. The study highlights how task design and the interplay between PS and PP enhance mathematical competencies and offer a viable approach to integrating PP as a learning objective in primary mathematics education.

The COVID-19 pandemic and the requirement for social distancing led to the closure of extracurricular activities that usually involve teamwork and collaboration, such as math clubs. Research on the design and effectiveness of extracurricular mathematical activities that aim to promote student interest and improve mathematical skills is limited, particularly in these challenging times. This exploratory case study examines an online after-school program, the Tecn@Mat Club, aimed at promoting middle grade students’ ability in solving mathematical problems with digital technologies during the pandemic. Based on document analysis, observation of the club sessions, questionnaires, and interviews with the participants (students and a mathematics education expert), this research documents the design and implementation of the Tecn@Mat Club, as well as its participants’ experiences and perspectives regarding the activity in which they engaged. The main results show the feasibility of adapting in-person, after-school math activities for an online setting, engaging middle grade students in mathematical problem-solving with technology by providing moderate mathematical challenges and promoting collaborative work. Results of a SWOT analysis (the acronym for strengths, weaknesses, opportunities, threats) allow for the identification of the key features to preserve and challenges to overcome in future replications of the club. Limitations of this study and future research directions are discussed.

This paper presents the results of an exploratory case study that examined a veteran secondary teacher’s knowledge for teaching non-routine mathematical problem-solving with digital technologies. Data was collected through the observation of a veteran mathematics teacher, in real time, solving a mathematical problem with the digital tools of his choice. The descriptive model Mathematical Problem-Solving with Technology (MPST) was used to analyse the teacher’s utterances and actions while solving a mathematical problem, with GeoGebra and a spreadsheet, and expressing his reasoning also with technology. Our findings reveal the complexity of expert problem-solving with technology, through regulation processes and several micro-cycles that mainly involve the processes integrate and explore. Mathematical problem solving with technology entails relevant mathematical knowledge as well as knowledge about the mathematical affordances of the digital tools available, and the ability to combine them to develop a conceptual model of the solution to the problem. Thus, the teacher’s techno-mathematical fluency seems crucial to successfully solve the problem and express the reasoning with technology. Based on the findings, we discuss the teachers’ knowledge for teaching mathematical problem-solving with technology as including a particular kind of proficiency, techno–mathematical fluency for solving-and-expressing problems with technology. The limitations of the study, further research topics and implications for professional development and teacher education programmes are discussed.

Information about the paper titled "THE USE OF TECHNOLOGIES IN MATHEMATICS TEACHING AND LEARNING IN PORTUGAL: TEACHERS’ PRE-PANDEMIC VIEWS" at IATED Digital Library

This study aims at understanding the role of the tools chosen throughout the processes of solving a non-routine mathematical problem and communicating its solution. In assuming that problem-solving is a synchronous activity of mathematization and expression of mathematical thinking we take our proposed Mathematical Problem Solving with Technology (MPST) model to analyze the processes of solving-and-expressing-problems. Resorting to qualitative methods for data collection and analysis, we report on the case of an 8th grader working on a covariation problem to examine the role that paper-and-pencil and digital tools play in the development of a conceptual model of the situation. We found that the resources used throughout the solving-and-expressing activity influenced the depth of the conceptual model developed, within a process of progressive mathematization. Whereas paper-and-pencil led to the emergence of a conceptual model based on exploring particular cases, the digital transformation of the solution was triggered by the process of communicating its mathematical justification and expanded the previous model. Moreover, the complexity of this activity is evidenced by its multiple sequences of processes. Finally, the integration process seems crucial as the concomitant use of technological and mathematical resources precedes major advancements in the expansion of the conceptual model.

The research on deductive reasoning in mathematics education has been predominantly associated with the study of proof; consequently, there is a lack of studies on logical reasoning per se, especially with young children. Analytical reasoning problems are adequate tasks to engage the solver in deductive reasoning, as they require rule checking and option elimination, for which chains of inferences based on premises and rules are accomplished. Focusing on the solutions of children aged 10–12 to an analytical reasoning problem proposed in two separate settings—a web-based problem-solving competition and mathematics classes—this study aims to find out what forms of deductive reasoning they undertake and how they express that reasoning. This was done through a qualitative content analysis encompassing 384 solutions by children participating in a beyond-school competition and 102 solutions given by students in their mathematics classes. The results showed that four different types of deductive reasoning models were produced in the two venues. Moreover, several representational resources were found in the children’s solutions. Overall, it may be concluded that moderately complex analytical reasoning tasks can be taken into regular mathematics classes to support and nurture young children’s diverse deductive reasoning models.