Bridging knowing and proving in mathematics An essay from a didactical perspective

Text of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006International audienceThe learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory

Early Geometrical Thinking in the Environment of Patterns, Mosaics and Isometries

This book discusses the learning and teaching of geometry, with a special focus on kindergarten and primary education. It examines important new trends and developments in research and practice, and emphasizes theoretical, empirical and developmental issues. Further, it discusses various topics, including curriculum studies and implementation, spatial abilities and geometric reasoning, as well as the psychological roots of geometrical thinking and teacher preparation in geometry education. It considers these issues from historical, epistemological, cognitive semiotic and educational points of view in the context of students' difficulties and the design of teaching and curricula

El Uso de Las Letras como Fuente de Errores de Estudiantes Universitarios en la Resolución de Tareas Algebraicas

La presente investigación es un estudio realizado con 194 estudiantes del Centro Universitario de la Costa Sur, en Autlán, México, cuyo objetivo es analizar los errores más comunes que los alumnos de primer semestre presentan en las producciones, al operar con los distintos significados que pueden tener las letras en álgebra y con base a esos resultados, establecer su ubicación dentro de alguna de las cuatro categorías de entendimiento en el uso y significado de las letras en álgebra que propone Küchemann (1980). Los resultados muestran que más de la mitad de los estudiantes de este nivel educativo no manifiestan dificultades al evaluar las letras, manejarlas como objetos o considerar su presencia, sin embargo, sí revelan deficiencias en el discernimiento para comprender el uso y significado de las letras como incógnitas de valor especifico, números generalizados y como variables

Research on Teaching and Learning Mathematics at the Tertiary Level:State-of-the-art and Looking Ahead

This topical survey focuses on research in tertiary mathematics education, a field that has experienced considerable growth over the last 10 years. Drawing on the most recent journal publication as well as the latest advances from recent high quality conference proceedings, our review culls out the following five emergent areas of interest: mathematics teaching at the tertiary level; the role of mathematics in other disciplines; textbooks, assessment and students’ studying practices; transition to the tertiary level; and theoretical-methodological advances. We conclude the survey with a discussion of some potential ways forward for future research in this new and rapidly developing domain of inquiry

New insights into the genetic etiology of Alzheimer's disease and related dementias

Characterization of the genetic landscape of Alzheimer's disease (AD) and related dementias (ADD) provides a unique opportunity for a better understanding of the associated pathophysiological processes. We performed a two-stage genome-wide association study totaling 111,326 clinically diagnosed/'proxy' AD cases and 677,663 controls. We found 75 risk loci, of which 42 were new at the time of analysis. Pathway enrichment analyses confirmed the involvement of amyloid/tau pathways and highlighted microglia implication. Gene prioritization in the new loci identified 31 genes that were suggestive of new genetically associated processes, including the tumor necrosis factor alpha pathway through the linear ubiquitin chain assembly complex. We also built a new genetic risk score associated with the risk of future AD/dementia or progression from mild cognitive impairment to AD/dementia. The improvement in prediction led to a 1.6- to 1.9-fold increase in AD risk from the lowest to the highest decile, in addition to effects of age and the APOE ε4 allele

Research On and Activities For Mathematically Gifted Students

This Topical Survey offers a brief overview of the current state of research on and activities for mathematically gifted students around the world. This is of interest to a broad readership, including educational researchers, research mathematicians, mathematics teachers, teacher educators, curriculum designers, doctoral students, and other stakeholders. It first discusses research concerning the nature of mathematical giftedness, including theoretical frameworks and methodologies that are helpful in identifying and/or creating mathematically gifted students, which is described in this section. It also focuses on research on and the development of mathematical talent and innovation in students, including connections between cognitive, social and affective aspects of mathematically gifted students. Exemplary teaching and learning practices, curricula and a variety of programs that contribute to the development of mathematical talent, gifts, and passion are described as well as the pedagogy and mathematics content suitable for educating pre-service and in-service teachers of mathematically gifted students. The final section provides a brief summary of the paper along with suggestions for the research, activities, and resources that should be available to support mathematically gifted students and their teachers, parents, and other stakeholders