Manuel Santos-Trigo

The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts

The significant development and use of digital technologies has opened up diverse routes for learners to construct and comprehend mathematical knowledge and to solve problems. This implies a revision of the pedagogical landscape in terms of the ways in which students engage in learning, and how understandings emerge. In this chapter we consider how the availability of digital technologies has allowed intended learning trajectories to be structured in particular forms and how these, coupled with the affordances of engaging mathematical tasks through digital pedagogical media, might shape the actual learning trajectories. The evolution of hypothetical learning trajectories is examined, while the transitions learners make when traversing these pathways are also considered. Particular instances are illustrated with examples in several settings.

Instructional Qualities of a Successful Mathematical Problem-Solving Class.

∗ A short version of this article will appear in a special issue of Research in Collegiate Mathematics Education. American Mathematical Society. Problem solving has been an important issue in mathematics education during the last 25 years. Research studies in this area have documented extensively what students show while working with mathematical problems (resources, beliefs, and cognitive and metacognitive strategies). However, there is little information about how mathematical problem‐solving activities should be implemented in regular classrooms. This paper shows what activities have been successful implemented by an expert during a mathematical problem‐solving course. It focuses on the identification of qualities of the problems used to promote the development of students strategies and values that reflect mathematical practice in the classroom. Specia’ attention is paid to the type of question that help students evaluate their own problem‐solving processes and the importance of allowing the students ...

Problem Solving in Mathematics Education

Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to foster problem solving approaches. In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use? (ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions? (iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?

Teachers' Use of Computational Tools to Construct and Explore Dynamic Mathematical Models.

To what extent does the use of computational tools offer teachers the possibility of constructing dynamic models to identify and explore diverse mathematical relations? What ways of reasoning or thinking about the problems emerge during the model construction process that involves the use of the tools? These research questions guided the development of a study that led us to document the process exhibited by high school teachers to model mathematical situations dynamically. In particular, there is evidence that the use of computational tools helped them identify and explore a set of mathematical relations or conjectures that appear throughout the interaction with the task. Thus, the participants had the opportunity of fostering an inquisitive approach to the process of model construction that led them to formulate conjectures or mathematical relations and to search for ways to support them.

The use of digital technology in finding multiple paths to solve and extend an equilateral triangle task

Mathematical tasks are crucial elements for teachers to orient, foster and assess students’ processes to comprehend and develop mathematical knowledge. During the process of working and solving a task, searching for or discussing multiple solution paths becomes a powerful strategy for students to engage in mathematical thinking. A simple task that involves the construction of an equilateral triangle is used to present and discuss multiple solution approaches that rely on a variety of concepts and ways of reasoning. To this end, the use of a Dynamic Geometry System (GeoGebra) became instrumental in constructing and exploring dynamic models of the task. These model explorations provided a means to generate novel mathematical results.

On the use of computational tools to promote students’ mathematical thinking

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. In this snapshot students employ dynamic geometry software to find great mathematical richness around a seemingly simple question about rectangles.Computer Math SnapshotsEditor: Uri WilenskyCenter for Connected Learning and Computer-Based Modeling Northwestern University, USA E-mail: uri@northwestern.edu

Connecting Dynamic Representations of Simple Mathematical Objects with the Construction and Exploration of Conic Sections.

Different technological artefacts may offer distinct opportunities for students to develop resources and strategies to formulate, comprehend and solve mathematical problems. In particular, the use of dynamic software becomes relevant to assemble geometric configurations that may help students reconstruct and examine mathematical relationships. In this process, students have the opportunity of formulating questions, making conjectures, presenting arguments and communicating results. This article illustrates that simple geometric configurations can be used to generate all conic sections studied in a regular course of analytic geometry. Observing and interpreting loci, searching for mathematical arguments (including empirical reasoning) and presenting results are problem-solving activities that seem to be enhanced with the use of dynamic software. During the development of the activities, students can exhibit a line of thinking in which they constantly reflect on the meaning and connections among mathematical concepts.

High School Teachers' Problem Solving Activities to Review and Extend Their Mathematical and Didactical Knowledge

Abstract The study documents the extent to which high school teachers reflect on their need to revise and extend their mathematical and practicing knowledge. In this context, teachers worked on a set of tasks as a part of an inquiring community that promoted the use of different computational tools in problem solving approaches. Results indicated that the teachers recognized that the use of the Cabri-Geometry software to construct dynamic representations of the problems became useful, not only to make sense of the problems statement, but also to identify and explore a set of mathematical relations. In addition, the use of other tools like hand-held calculators and spreadsheets offered them the opportunity to examine, contrast, and extend visual and graphic results to algebraic approaches.

The Use of Digital Technology to Frame and Foster Learners’ Problem-Solving Experiences

The purpose of this chapter is to analyze and discuss the extent to which the use of digital technology offers learners opportunities to understand and appropriate mathematical knowledge. We focus on discussing several examples in which the use of digital technology provides distinct affordances for learners to represent, explore, and solve mathematical tasks. In this context, looking for multiple ways to solve a task becomes a powerful strategy for learners to think of different concepts in problem-solving approaches. Thus, the use of a dynamic geometry system such as GeoGebra becomes important to represent and analyze tasks from visual, dynamic, and graphic approaches.

Digital Technologies and a Modeling Approach to Learn Mathematics and Develop Problem Solving Competencies

This study is framed within a conceptual approach that integrates modeling, problem solving, and the use of digital technologies perspectives in mathematical learning. It focuses on the use of a Dynamic Geometry System (GeoGebra) to construct mathematical models as a means to represent and explore mathematical relationships. In particular, we analyze and document what ways of reasoning high school students exhibit as a result of working on a mathematical task in problem solving sessions. Results show that the students rely on a set of technology affordances to dynamically visualize, represent and explore mathematical relations. In this process, the students’ discussions became relevant not only to explain their approaches; but also to contrast, and eventually refine, their initial models and ways of reasoning.