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Revisiting Problem Posing: Versuch einer konzeptuellen Ordnung
Abstract
Im vorliegenden Beitrag wird der Versuch einer konzeptuellen Ordnung unterschiedlicher Problem-Posing-Tätigkeiten vorgenommen. Dieses Konzept verbindet drei Konstrukte aus der Forschung zum Problem Posing, Problemlösen und der Psychologie: (1) Problemstellen als Generieren neuer oder Reformulieren gegebener Probleme, (2) das Aufwerfen von Aufgaben auf dem Spektrum zwischen Routineaufgaben und Problemen und (3) metakognitives Verhalten beim Problem Posing. Diese Dimensionen werden zunächst theoretisch konzeptuali- siert und schließlich operationalisiert. Anschließend wird die Anwendung dieser Dimensionen qualitativ anhand empirischer Studien zum Problem Posing demonstriert.
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This article aims to develop a framework for the characterisation of problem-posing activities. The framework links three theoretical constructs from research on problem posing, problem solving, and psychology: (1) problem posing as an activity of generating new or reformulating given problems, (2) emerging tasks on the spectrum between routine and non-routine problems, and (3) metacognitive behaviour in problem-posing processes. These dimensions are first conceptualised theoretically. Afterward, the application of these conceptualised dimensions is demonstrated qualitatively using empirical studies on problem posing. Finally, the framework is applied to characterise problem-posing activities within systematically gathered articles from high-ranked journals on mathematics education to identify focal points and under-represented activities in research on problem posing.
Existing studies have quantitatively evidenced the relatedness between problem posing and problem solving, as well as the magnitude of this relationship. However, the nature and features of this relationship need further qualitative exploration. This paper focuses on exploring the interactions, i.e., mutual effects and supports, between problem posing and problem solving. More specifically, this paper analyzes the forms of interactions that happened between these two activities, the ways that those interactions supported prospective primary teachers’ conceptual understanding, and the difficulties that prospective teachers encountered while engaged in alternating problem-posing and problem-solving activities. The results indicate that problem posing contributes to problem-solving effectiveness while problem solving supports participants in posing more reasonable problems. Finally, multiple difficulties that demonstrate prospective primary teachers’ misunderstanding with fractions and their operations provide insight for teacher educators to design problem-posing tasks involving fractions.
While a wide range of approaches and tools have been used to study children’s creativity in school contexts, less emphasis has been placed on revealing students’ creativity at university level. The present paper is focused on defining a tool that provides information about mathematical creativity of prospective mathematics teachers in problem-posing situations. To characterize individual differences, a method to determine the geometry-problem-posing cognitive style of a student was developed. This method consists of analyzing the student’s products (i.e. the posed problems) based on three criteria. The first of these is concerned with the validity of the student’s proposals, and two bi-polar criteria detect the student’s personal manner in the heuristics of addressing the task: Geometric Nature (GN) of the posed problems (characterized by two opposite features: qualitative versus metric), and Conceptual Dispersion (CD) of the posed problems (characterized by two opposite features: structured versus entropic). Our data converge on the fact that cognitive flexibility—a basic indicator of creativity—inversely correlates with a style that has dominance in metric GN and structured CD, showing that the detected cognitive style may be a good predictor of students’ mathematical creativity.
This paper is concerned with organizational principles of a pool of familiar problems of expert problem posers and the ways by which they are utilized for creating new problems. The presented case of Leo is part of a multiple-case study with expert problem posers for mathematics competitions. We present and inductively analyze the data collected in a reflective interview and in a clinical task-based interview with Leo. In the first interview, Leo was asked to share with us the stories behind some problems posed by him in the past. In the second interview, he was asked to pose a new competition problem in a thinking-aloud mode. We found that Leo’s pool of familiar problems is organized in classes according to certain nesting ideas. Furthermore, these nesting ideas serve him in posing problems that, ideally, are perceived by Leo as novel and surprising not only to potential solvers, but also to himself. Because of the lack of empirical research on experts in mathematical problem posing, the findings are discussed in light of research on experts in problem solving and on novices in mathematical problem posing.
As an introduction to the special issue on problem posing, the paper presents a brief overview of the research done on this topic in mathematics education. Starting from this overview, the authors acknowledge important issues that need to be taken into account in the developing field of problem posing and identify new directions of research, some of which are addressed by the collection of the papers included in this volume.
This paper proposes a framework for understanding people's theories about their own cognition. Metacognitive theories are defined broadly as systematic frameworks used to explain and direct cognition, metacognitive knowledge, and regulatory skills. We distinguish tacit, informal, and formal metacognitive theories and discuss critical differences among them using criteria borrowed from the developmental literature. We also consider the origin and development of these theories, as well as implications for educational research and practice.
This article focuses on the construction, description and testing of a theoretical model of problem posing. We operationalize procesess that are frequently described in problem solving and problem posing literature in order to generate a model. We name these processes editing quantitative information, their meanings or relationships, selecting quantitative information, comprehending and organizing quantitative information by giving it meaning or creating relations between provided information, and translating quantitative information from one form to another. The validity and the applicability of the model is empirically tested using five problem-posing tests with 143 6 th grade students in Cyprus. The nalysis shows that three different categories of students can be identified. Category 1 students are able to respond only to the comprehension tasks. Category 2 students are able to respond to both the comprehension and translation tasks, while Category 3 students are able to respond to all types of tasks. The results of the study also show that students are more successful in first posing problems that involve comprehending processes, then translation processes and finally editing and selecting processes.
Updated and expanded, this second edition satisfies the same philosophical objective as the first -- to show the importance of problem posing. Although interest in mathematical problem solving increased during the past decade, problem posing remained relatively ignored. The Art of Problem Posing draws attention to this equally important act and is the innovator in the field. Special features include: •an exploration ofthe logical relationship between problem posing and problem solving •a special chapter devoted to teaching problem posing as a separate course •sketches, drawings, diagrams, and cartoons that illustrate the schemes proposed a special section on writing in mathematics. © 1990 by Stephen I. Brown and Marion I. Walter. All rights reserved.
Teachers are at the heart of implementing any educational innovation or improvement. One critical need is to investigate how teachers learn to use problem posing to teach mathematics in the classroom. This article conceptualizes issues about mathematical problem posing (MPP) and about learning to teach through problem posing. It highlights significant recent findings on using problem posing for teachers’ learning. It discusses methodological issues about problem-posing research, as well as the future directions of research on learning to teach through problem posing. The international perspective provides a broad view of ways to integrate MPP in classrooms in general and the use of problem posing for teacher professional development in particular.
This one-year study involved designing and implementing a problem-posing program for fifth-grade children. A framework developed for the study encompassed three main components: (a) children's recognition and utilisation of problem structures, (b) their perceptions of, and preferences for, different problem types, and (c) their development of diverse mathematical thinking. One of the aims of the study was to investigate the extent to which children's number sense and novel problem-solving skills govern their problem-posing abilities in routine and nonroutine situations. To this end, children who displayed different patterns of achievement in these two domains were selected to participate in the 10-week activity program. Problem-posing interviews with each child were conducted prior to, and after the program, with the progress of individual children tracked during the course of the program. Overall, the children who participated in the program appeared to show substantial developments in each of the program components, in contrast to those who did not participate.
Studies suggest that young children are quite limited in their knowledge about cognitive phenomena—or in their metacognition—and do relatively little monitoring of their own memory, comprehension, and other cognitive enterprises. Metacognitive knowledge is one's stored knowledge or beliefs about oneself and others as cognitive agents, about tasks, about actions or strategies, and about how all these interact to affect the outcomes of any sort of intellectual enterprise. Metacognitive experiences are conscious cognitive or affective experiences that occur during the enterprise and concern any aspect of it—often, how well it is going. Research is needed to describe and explain spontaneous developmental acquisitions in this area and find effective ways of teaching metacognitive knowledge and cognitive monitoring skills. (9 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Children's understanding of their own cognitive skills, or metacognition, has been hypothesized to play a major role in learning and development. In this study, we examine the developing relation between children's metacognition and reading comprehension. Children in third- and fifth-grade classes were given an experimental curriculum, Informed Strategies for Learning (ISL), designed to increase their awareness and use of effective reading strategies. In both grades, children in experimental classes made significant gains in metacognition and the use of reading strategies compared with children in control classes. The multivariate profiles of reading skills derived from the developmental analyses helped to identify subgroups of children who responded differently to the metacognitive instruction. Although there were specific aptitude-by-treatment interactions, there was a general trend for metacognition and strategic reading to become more congruent from 8 to 10 years of age. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
This paper reports on observational approaches developed within a UK study to the identification and assessment of metacognition
and self-regulation in young children in the 3–5year age range. It is argued that the development of observational tools,
although containing methodological difficulties, allows us to make more valid assessments of children’s metacognitive and
self-regulatory abilities in this age group. The analysis of 582 metacognitive or self-regulatory videotaped ‘events’ is described,
including the development of a coding framework identifying verbal and non-verbal indicators. The construction of an observational
instrument, the Children’s Independent Learning Development (CHILD 3–5) checklist, is also reported together with evidence
of the reliability with which it can be used by classroom teachers and early indications of its external validity as a measure
of metacognition and self-regulation in young children. Given the educational significance of children’s development of metacognitive
and self-regulatory skills, it is argued that the development of such an instrument is potentially highly beneficial. The
establishment of the metacognitive and self-regulatory capabilities of young children by means of the kinds of observational
tools developed within this study also has clear and significant implications for models and theories of metacognition and
self-regulation. The paper concludes with a discussion of these implications.
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