Different moments in the participatory stage of the secondary students’ abstraction of mathematical conceptions

This study provides support to the characteristics of participatory and anticipatory stages in secondary school pupils’ abstraction of mathematical conceptions. We carried out clinical task-based interviews with 71 secondary-school pupils to obtain evidence of the different constructed mathematical conceptions (Participatory Stage) and how they were used (Anticipatory Stage). We distinguish two moments in the Participatory Stage based on the coordination of information from particular cases by activity-effect reflection which, in some cases, lead to a change of focus enabling secondary-school pupils to achieve a reorganization of their knowledge. We argue that (a) the capacity of perceiving regularities in sets of particular cases is a characteristic of activity-effect reflection in the abstraction of mathematical conceptions in secondary school, and (b) the coordination of information by pupils provides opportunities for changing the attention-focus from the particular results to the structure of properties.

Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Visualising mathematical knowledge and understanding

Contemporary mathematics education attempts to instil within learners the conceptualization of mathematics as a highly organized and inter-connected set of ideas. To support this, a means to graphically represent this organization of ideas is presented which reflects the cognitive mechanisms that shape a learner’s understanding. This organisation of information may then be analysed, with the view to informing the design of mathematics instruction in face-to-face and/or computer-mediated learning environments. However, this analysis requires significant work to develop both theory and practice.

A teoria de Dienes no ensino de transformação de medidas de comprimento, área e volume no curso de pedagogia

The study was aimed to test a teaching module on the Zoltan Paul Dienes theory, focusing on the content: The transformation of measurements: length, areas and volumes. The study based on constructivist theory consisted in a methodological intervention with students of the 7th period of the Course of Pedagogy, in Central Campus, Federal University of Rio Grande do Norte (UFRN). A preliminary study with 40 students called diagnostic evaluation found that students did not understand the concept of measurements transformation and its processing steps. The latter was performed only with the help of the table of measurements transformation with no understanding of the content. He applied a pretest, a set of activities and a post-test. The latter was used as a tool for evaluation of the student learning process. The answers of these ones were evaluated according to the concept of reflective abstraction of Jean Piaget, one of the authors who influenced the Dienes theory

Ideias de estudantes sobre a origem da Terra e da vida e suas relações com o desenvolvimento cognitivo: um estudo psicogenético

Pós-graduação em Educação - FFC

As relações entre a construção da abstração reflexionante e o conhecimento social: um estudo psicogenético

This article aims to investigate the construction of reflective abstraction and relate it to the construction of social knowledge. So, two methodological tools were applied to 60 students aged between ten and 16 years: a) evidence of the construction of surfaces and perimeters b) clinical interview about the origins of Earth and life. The main results show that participants were presented at more elementary levels of constuction of reflective abstraction and contruction of social knowledge despite the advanced age and schooling. Statistical analysis revealed a highly significant relationship between the constuction of abstraction and social knowledge.

Características de la tematización del esquema de límite de una función

Esta investigación estudia las diferentes estructuras subyacentes en el esquema de límite de una función observadas en 23 estudiantes de Bachillerato situados en el nivel Trans del desarrollo del esquema de límite de una función. El esquema de límite de una función se caracterizó en términos de la habilidad que los estudiantes manifestaron en la construcción de la concepción dinámica del límite mediante la coordinación de los procesos de aproximación en el dominio y en el rango, diferenciando aquellas en las que las aproximaciones laterales coinciden de las que no coinciden. Nuestros resultados sugieren que los estudiantes construyen diferentes estructuras subyacentes al esquema debido a las relaciones que establecen entre el límite de una función en un punto y su representación gráfica que permiten identificar características del esquema tematizado del límite de una función.

Un experimento de enseñanza sobre el límite de una función. Factores determinantes en una trayectoria de aprendizaje

El objetivo del estudio fue identificar características de la construcción del significado de límite de una función en estudiantes de bachillerato (16-17 años). Se diseñó un experimento de enseñanza utilizando una descomposición genética (APOE) del concepto de límite de una función integrando recursos informáticos. Se usó el constructo “Reflexión sobre la Relación Actividad-Efecto” (Simon, Tzur, Heinz y Kinzel, 2004) como una particularización de la abstracción reflexiva para identificar factores que configuran la Trayectoria de Aprendizaje. Los resultados indican que la trayectoria está determinada por la coordinación de las aproximaciones en el dominio y en el rango en diferentes tipos de funciones.

Perfiles de estudiantes en la comprensión de la aproximación al área de una superficie bajo una curva

El objetivo de este estudio es identificar perfiles de estudiantes en la forma en que aproximan el área de la superficie bajo una curva. Los estudiantes participaron en un experimento de enseñanza dirigido a la construcción del concepto de integral definida partiendo de la idea de aproximación al área de una superficie. Las tareas fueron diseñadas atendiendo a una trayectoria hipotética de aprendizaje considerando las fases de construcción de un concepto apoyadas en la abstracción reflexiva. El análisis de las respuestas, teniendo en cuenta los distintos momentos del proceso de abstracción reflexiva, permitió identificar tres perfiles en la comprensión de la aproximación al área de una superficie bajo una curva.

Rock, paper, scissors : reflective practice for design process in the novice landscape architect

We describe a pedagogical approach that addresses challenges in design education for novices. These include an inability to frame new problems and limited-to-no design capability or domain knowledge. Such challenges can reduce student engagement with design practice, cause derivative design solutions as well as the inappropriate simplification of design assignments and assessment criteria by educators. We argue that a curriculum that develops the student’s design process will enable them to deal with the uncertain and dynamic situations that characterise design. We describe how this may be achieved and explain our pedagogical approach in terms of methods from Reflective Practice and theories of abstraction and creativity. We present a landscape architecture unit, recently taught, as an example. It constitutes design exercises that require little domain or design expertise to support the development of conceptual thinking and a design rationale. We show how this approach (a) leveraged the novice’s existing spatial and thinking skills while (b) retaining contextually-rich design situations. Examples of the design exercises taught are described along with samples of student work. The assessment rationale is also presented and explained. Finally, we conclude by reflecting on how this approach relates to innovation, sustainability and other disciplines.

Navigating the terrain of lived experiences: the value of lifeworld existentials for reflective analysis

In qualitative research we are often presented with a tension between having open and fluid interviews to support staying true to the lived experiences of participants and achieving a level of abstraction from the data in order to uncover the essential structures and meanings of that particular lived experience. A way of resolving this tension is through the application of theoretical frameworks. Van Manen’s four lifeworld existentials offer a lens through which to explore and navigate disparate interview data and uncover the essences of lived experience, without imposing categories upon the data itself. Drawing on a study exploring the lived experiences of childless women, we explore the process and principles in operationalising the existentials and discuss the potential implications for analysis and findings. The article demonstrates how Van Manen’s lifeworld existentials present us with a holistic and valuable method for reflective practice, in coming to understand lived experience.

Forms, Foundations, and the Ethos of Foremost: Zen Practice, Epistemology, and Ontology in Reflective Knowledge Organization

In reflecting on the practice of knowledge organization, we tacitly or explicitly root our conceptions of work and its value in some epistemic and ontological foundation. Zen Buddhist philosophy offers a unique set of conceptions vis-à-vis organizing, indexing, and describing documents.When we engage in knowledge organization, we are setting our mind to work with an intention. We intend to make some sort of intervention. We then create a form a realization of an abstraction (like classes or terms) [1], we do this from a foundation of some set of beliefs (epistemology, ontology, and ethics), and because we have to make decisions about what to privilege, we need to decide what is foremost in our minds. We must ask what is the most important thing?Form, foundation, and the ethos of foremost require evoke in our reflection on work number of ethical, epistemic, and ontological concerns that ripple throughout our conceptions of space, “good work”, aesthetics, and moral mandate [2,3]. We reflect on this.