Papers by Manuel Santos-Trigo
Trends and Developments of Mathematical Problem-Solving Research to Update and Support the Use of Digital Technologies in Post-confinement Learning Spaces
Revista mexicana de bachillerato a distancia, Aug 31, 2023
En este video se destaca la necesidad de trabajar la matemática de forma contextualizada, ligada ... more En este video se destaca la necesidad de trabajar la matemática de forma contextualizada, ligada al entorno de los estudiantes, así como la conveniencia de que cuente con diagnósticos y la explicación de conceptos fundamentales. Se señala la relevancia de que los estudiantes exploren otras formas de llegar a la solución de los problemas y de que se apoyen en la tecnología. A modo de ejemplo de la implementación de estos puntos, se presenta un desarrollo dentro del CINVESTAV para UNICEF que incluye también el uso de herramientas diversas de evaluación y de recursos digitales, como las plataformas.
The Importance of Tasks and the Use of Digital Technologies Affordances in Mathematical Problem-Solving Approaches
Springer eBooks, 2022
ICME-13 topical surveys, 2016
changes made are indicated. The images or other third party material in this book are included in... more changes made are indicated. The images or other third party material in this book are included in the work's Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work's Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Can routine problems be transformed into non-routine problems?
Teaching Mathematics and Its Applications, Sep 1, 1998
Revisiting a Geometric Approach to the Study of Word Problems Through the Use of Digital Technologies
Social Science Research Network, 2020
What type of reasoning do high school students develop and exhibit when they use a Dynamic Geomet... more What type of reasoning do high school students develop and exhibit when they use a Dynamic Geometry System (GeoGebra) to work on word problems? We report a research study framed under problem-solving principles in which high school students relied on GeoGebra affordances to represent, explore and solve word problems. In this process, they recognized that all families of word problems involve a ratio of quantities or parameters embedded in problem statements that could be represented and explored geometrically. This exploration led them to identify relationships associated with those parameters that were analysed via finding loci and expressing their parametrization to solve word problems both geometrically and algebraically.
Research in mathematics education, 2019
It has been widely recognized that problem-solving activities are crucial in developing and learn... more It has been widely recognized that problem-solving activities are crucial in developing and learning mathematics. Indeed, it is common to structure and frame both mathematical curriculum and learning environments through problemsolving activities. Currently, significant developments of digital technologies are shaping both students' social interaction and ways of learning mathematics and solving problems. What types of strategies, representations, and resources emerge and are important in problem-solving approaches that rely on and foster the use of a Dynamic Geometry System affordances? The aim of this chapter is to analyze and discuss on how the use of a Dynamic Geometry System (GeoGebra) provides affordances to develop a geometric reasoning as a mean to work and solve mathematical problems. In this process, it becomes important to think of and represent problem statements and concepts geometrically, to construct dynamic models of problems, to trace and examine loci of particular objects, to analyze particular and general cases, and to communicate results.
Digital Technologies and a Modeling Approach to Learn Mathematics and Develop Problem Solving Competencies
Communications in computer and information science, 2016
This study is framed within a conceptual approach that integrates modeling, problem solving, and ... more 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.
The Coordinated Use of Digital Technologies in Learning Environments
Communications in computer and information science, 2014
The purpose of this paper is to outline a framework to reflect on the extent to which the coordin... more The purpose of this paper is to outline a framework to reflect on the extent to which the coordinated use of several digital technologies offers learners diverse opportunities to construct mathematical knowledge. An example is used to argue that different technologies could provide distinct affordances for learners to represent, explore, and solve mathematical tasks. Thus, YouTube videos are used to foster problem formulation and communication, the use of GeoGebra becomes important to represent and analyse the task from visual, dynamic, and graphic approaches. In addition, online resources (WolframAlpha and Wikipedia) are used to deal with algebraic operations involved in the algebraic model of the task, and to introduce information needed to review or comprehend embedded concepts. Finally, in order for the tools to be instrumental for students, it is important to take into account information related to the learners’ tool appropriation in which they transform an artefact into a problem-solving tool.
The Use of Digital Technology in Extending Mathematical Problem Solving Reasoning
Communications in computer and information science, 2015
The incorporation of digital technologies (both multiple purpose and mathematics action technolog... more The incorporation of digital technologies (both multiple purpose and mathematics action technologies) in mathematical learning environments can foster and extend discussions among learners and teachers even beyond class time. That is, learners not only keep reflecting on mathematical ideas or problems; but also they can review or consult related online resources. Similarly, the use of dynamic geometry systems provides affordances to construct dynamic models of tasks where learners can analyse how objects move within the configuration and formulate mathematical relations. In this report, we discuss three exemplars to characterize ways in which the use of technology extends mathematical reasoning in problem solving approaches. This information becomes important for teachers to value and frame the incorporation of technology in learning environments. At the end, some limitations of this approach are discussed.
Fostering and supporting the coordinated use of digital technologies in mathematics learning
International Journal of Learning Technology, 2015
ABSTRACT
International Journal of Computers for Mathematical Learning, Dec 1, 2006
This column will publish short (from just a few paragraphs to ten or so pages), lively and intrig... more 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.
Processes of Comprehension Shown by High School Teachers in Solving Optimization Problems with the Use of Technology
The aim of the Discussion Group is to capture the sense of the community, not only from experts a... more The aim of the Discussion Group is to capture the sense of the community, not only from experts about standardized testing. Reflection should provide a report to consider implications to global policies. How standardized testing favour math education aims and how much sensitive is to diversity? A regular view is “Standardized tests are needed because they can provide an amount of information and evidence of validity. Of course there can be incorrect interpretations from using any test, but these can be reduced if the quality of the test has the attributes that are associated with standardized assessments” A critical approach is to highlight “Limited Scope of standardized tests in school maths, because these tests undermine abilities to conjecture and to encourage open problems in class. Standardized testing devalues abilities to collaborate, to take risks, and to engage in real-world experience; failing to the mission of schooling: the pursuit of happiness and justice of all. In thi...
On the Use of Technology to Represent and Explore Mathematical Objects or Problems Dynamically
Mathematics and Computer Education, Apr 1, 2008
(ProQuest: ... denotes formulae omitted.) Questions and answers are more important than a profess... more (ProQuest: ... denotes formulae omitted.) Questions and answers are more important than a professor's repetition of what can be found in books or what can be put in lecture notes, duplicated, and passed out to students [4, p. 97]. The development of mathematical ideas can be traced and explained in terms of the type of representations used to deal with problems or mathematical objects. Indeed, "much of the history of mathematics is about creating and refining representational systems, and much of the teaching of mathematics is about students learning to work with them and solve problems with them" (Lesh, Landau, & Hamilton, 1993, cited in Goldin, & Shteingold, 2001, p 4). Similarly, Schoenfeld (1985) suggests that students' construction or understanding of mathematical ideas and their ways to solve problems depend on the types of representations they use to recognize and examine mathematical relations. For example, the number 46656 can be represented as 2^sup 6^ × 3^sup 6^, 36^sup 3^, 216^sup 2^ and, 12 × 888 + 12 × 3000. Each representation makes visible mathematical properties that involve prime factorization, perfect cube, perfect square, or multiple of 12. Hence, the significant development and use of computational tools, such as Computer Algebra Systems (CAS), dynamic software, and spreadsheets may facilitate students' ways to represent and operate mathematical objects embedded in problems. As a result, the use of the tools can help students identify and examine mathematical properties and diverse representations of the same object or problem. How can students use particular tools to represent and solve mathematical problems? What types of representations of problems may be useful for students to examine mathematical relations? To what extent does the students' use of a particular tool shape and favor a particular way of thinking about problems? Discussing these type of questions may shed light on the extent to which students' use of dynamic geometry software, like Cabri or Geometer's Sketchpad, helps them identify and explore mathematical relations or conjectures. A relevant feature in using dynamic software is that in order for students to represent mathematical objects or problems, they need to think of the problems in terms of mathematical properties. For instance, to draw a rectangle, students need to identify properties associated with this figure, such as pairs of parallel or perpendicular sides to help them choose proper commands to represent the figure. With the use of the software, students can also build dynamic geometric configurations formed by simple figures (lines, segments, triangles, perpendicular bisectors, etc.) that may be used as points of departure to identify mathematical conjectures or relationships. From this perspective, dynamic representations of problems seem to be an important tool that allows students to reconstruct or develop some mathematical relations. That is, by moving objects within the representations, students may have the opportunity to observe relations that lead them to make conjectures and to present various arguments to support them. Presenting an argument is a way for students to remove or overcome doubts about the validity of a conjecture. Indeed, "a person ceases to consider an assertion to be a conjecture and views it to be a fact once the person becomes certain of its truth" (Harel, in press). How can students construct a particular geometric configuration that can be used as a starting point to formulate conjectures or mathematical relationships? There may be different ways for students to assemble a particular configuration that lead them to identify relationships. For instance, students can start by drawing some initial objects, like points, segments, lines, triangles, rectangles, circles, etc., and then use them to build a configuration. Throughout the construction of that configuration, students can direct their attention to the behavior of some elements while moving objects within the configuration to observe invariants, loci, or mathematical relationships. …
A conceptual framework to structure remote learning scenarios: a digital wall as a reflective tool for students to develop mathematics problem-solving competencies
International Journal of Learning Technology
This study documents the extent to which university students utilize diverse representations and ... more This study documents the extent to which university students utilize diverse representations and mathematical processes to interpret and respond to a set of questions that involves fundamental concepts in the study of differential equations. Results indicate that students’ idea to solve a differential equation is reduced to the application of proper solution methods to a certain type of equation differential expressions. Thus, instructional activities should promote the students’ use of several representation systems in which they can reflect on the various aspects associated with the concept itself, the solution methods, procedures, and the corresponding meaning and connections among those representations.Este estudio muestra hasta qué punto los estudiantes de universidad utilizan diferentes procesos y representaciones matemáticas para interpretar y responder a un grupo de cuestiones que incluyen conceptos fundamentales relacionados con el estudio de las ecuaciones diferenciales. L...
Resumo 3 Pretendo neste texto-conferência, problematizar algumas concepções e práticas de produçã... more Resumo 3 Pretendo neste texto-conferência, problematizar algumas concepções e práticas de produção e aprendizagem de conhecimentos docentes em relação à prática de ensi-nar e aprender matemática. Para desenvolver essa problematização, tomarei como ferramenta de análise os conceitos de comunidade de prática (Lave & Wenger, 1991) e de aprendizagem social e situada (Wenger, 2001 e Lave, 2001) e destacarei e diferenciarei duas comunidades básicas no processo de produção/aprendizagem de conhecimentos docentes: a acadêmica e a profissional e suas respectivas práticas sociais. A partir desses pressupostos, analisaremos e discutiremos o fracasso da perspectiva acadêmica, baseada na racionalidade técnica e ainda muito presente nas institui-ções brasileiras e americanas, de investigar as práticas profissionais de ensinar e aprender e de propor melhorias para o ensino da matemática. A seguir, pretendo apresentar e descrever, algumas práticas emergentes de pes-quisa em comunidades investigativa...
Mathematical Problem Solving and the Use of Digital Technologies
Mathematical Problem Solving, 2019
The goal in this chapter is to analyze and document ways in which the use of digital technologies... more The goal in this chapter is to analyze and document ways in which the use of digital technologies provides affordances for teachers/students to develop knowledge and solve mathematical problems. What types of representations and problem explorations appear and characterize a technology enhanced problem-solving approach? What does the systematic and coordinated use of digital technologies bring to the students’ process of delving into problem statements, to the ways of representing and exploring tasks, and to the development of their problem-solving competencies? Four groups of problems are examined to illustrate what strategies and ways of reasoning emerge and are enhanced in a technological problem-solving approach. Focusing on the reconstruction of figures that often appear in problem statements, the transformation of a routine problem into an investigation task, the representation and exploration of a variation phenomenon task, and the construction of a dynamic configuration to p...
Delving into Conceptual Frameworks: Problem Solving, Representations, and Models and Modeling Perspectives
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Papers by Manuel Santos-Trigo