'Solutioning': a model of students' problem-solving processes

The aim of this study was to generate a model (or theory) that explains students’ concerns as they tackle non-routine mathematical problems. This was achieved by using the grounded theory approach as suggested by Glaser and Strauss (1967) and further developed by Glaser (1978; 1992; 1998; 2001; 2003). The study took place in the context of a problem-solving course offered at the undergraduate level. As methods of data collection, the study made use of the problem-solving rubrics (or scripts) that students generated during the course. Other sources of data included interviews with the students and observations in class. The model generated as a result of this study suggests that problem solving can be seen as a four-stage process. The process was labelled ‘solutioning’ and is characterised by students trying to resolve the following concerns: Generating knowledge; Generating solutions; Validating the results, and Improving the results. The model also makes reference to pseudo-solutioning as an alternative approach to solutioning. During pseudo-solutioning, instead of trying to resolve the concerns listed above, students focus on trying to satisfy the academic requirement to submit an acceptable piece of work. Thus, pseudo-solutioning can be seen as an important variation to solutioning. After presenting the model of ‘solutioning’, the study provides an illustration of how it can be used to describe students’ processes. This is done in set of case studies in which three problem-solving processes are considered. The case studies provide a view of how the model developed fits the data and serves to highlight relevant patterns of behaviour observable as students solve problems. The case studies illustrate how the concepts suggested by the model can be used for explaining success and failure in the processes considered. This study contributes to the study of problem solving in mathematics education by providing a conceptualisation of what students do as they try to solve problems. The concepts that the model suggests are relevant for explaining how students resolve their main concerns as they tackle problems during the course. However, some of these concepts (e.g., ‘reducing complexity’, ‘blinding activities’, ‘transferring’) may also be of relevance to problem solving in other areas

'Solutioning' : a model of students' problem-solving processes

The aim of this study was to generate a model (or theory) that explains students’ concerns as they tackle non-routine mathematical problems. This was achieved by using the grounded theory approach as suggested by Glaser and Strauss (1967) and further developed by Glaser (1978; 1992; 1998; 2001; 2003). The study took place in the context of a problem-solving course offered at the undergraduate level. As methods of data collection, the study made use of the problem-solving rubrics (or scripts) that students generated during the course. Other sources of data included interviews with the students and observations in class. The model generated as a result of this study suggests that problem solving can be seen as a four-stage process. The process was labelled ‘solutioning’ and is characterised by students trying to resolve the following concerns: Generating knowledge; Generating solutions; Validating the results, and Improving the results. The model also makes reference to pseudo-solutioning as an alternative approach to solutioning. During pseudo-solutioning, instead of trying to resolve the concerns listed above, students focus on trying to satisfy the academic requirement to submit an acceptable piece of work. Thus, pseudo-solutioning can be seen as an important variation to solutioning. After presenting the model of ‘solutioning’, the study provides an illustration of how it can be used to describe students’ processes. This is done in set of case studies in which three problem-solving processes are considered. The case studies provide a view of how the model developed fits the data and serves to highlight relevant patterns of behaviour observable as students solve problems. The case studies illustrate how the concepts suggested by the model can be used for explaining success and failure in the processes considered. This study contributes to the study of problem solving in mathematics education by providing a conceptualisation of what students do as they try to solve problems. The concepts that the model suggests are relevant for explaining how students resolve their main concerns as they tackle problems during the course. However, some of these concepts (e.g., ‘reducing complexity’, ‘blinding activities’, ‘transferring’) may also be of relevance to problem solving in other areas.EThOS - Electronic Theses Online ServiceConsejo Nacional de Ciencia y Tecnología (Mexico) (CONACYT)Overseas Research Student Awards Scheme (ORSAS)GBUnited Kingdo

Applying science of learning in education: Infusing psychological science into the curriculum

The field of specialization known as the science of learning is not, in fact, one field. Science of learning is a term that serves as an umbrella for many lines of research, theory, and application. A term with an even wider reach is Learning Sciences (Sawyer, 2006). The present book represents a sliver, albeit a substantial one, of the scholarship on the science of learning and its application in educational settings (Science of Instruction, Mayer 2011). Although much, but not all, of what is presented in this book is focused on learning in college and university settings, teachers of all academic levels may find the recommendations made by chapter authors of service. The overarching theme of this book is on the interplay between the science of learning, the science of instruction, and the science of assessment (Mayer, 2011). The science of learning is a systematic and empirical approach to understanding how people learn. More formally, Mayer (2011) defined the science of learning as the “scientific study of how people learn” (p. 3). The science of instruction (Mayer 2011), informed in part by the science of learning, is also on display throughout the book. Mayer defined the science of instruction as the “scientific study of how to help people learn” (p. 3). Finally, the assessment of student learning (e.g., learning, remembering, transferring knowledge) during and after instruction helps us determine the effectiveness of our instructional methods. Mayer defined the science of assessment as the “scientific study of how to determine what people know” (p.3). Most of the research and applications presented in this book are completed within a science of learning framework. Researchers first conducted research to understand how people learn in certain controlled contexts (i.e., in the laboratory) and then they, or others, began to consider how these understandings could be applied in educational settings. Work on the cognitive load theory of learning, which is discussed in depth in several chapters of this book (e.g., Chew; Lee and Kalyuga; Mayer; Renkl), provides an excellent example that documents how science of learning has led to valuable work on the science of instruction. Most of the work described in this book is based on theory and research in cognitive psychology. We might have selected other topics (and, thus, other authors) that have their research base in behavior analysis, computational modeling and computer science, neuroscience, etc. We made the selections we did because the work of our authors ties together nicely and seemed to us to have direct applicability in academic settings

Tendencias actuales en investigación en matemáticas y afecto

En esta ponencia se presenta una síntesis de investigaciones realizadas en Matemáticas y afecto, especialmente, en éstas dos últimas décadas. Se describe el estado de la cuestión respecto al desarrollo de marcos teóricos y metodológicos y a propuestas de programas de actualización didáctica para profesores y alumnos, dando prioridad a cuestiones abiertas que pueden plantear investigaciones futuras. Se ponen de relieve aspectos a repensar y a avanzar en este campo de investigación referidos: a conceptualización, a la interacción cognición y afecto en los procesos de pensamiento matemático, a identidad y afecto en el contexto del desarrollo profesional del profesor y a propuestas de articulación entre teoría y práctica que hagan posible que esta temática llegue de forma operativa al aula

A case study of a high achiever's learning of physical science.

Thesis (M.Ed.)-University of Natal, Durban, 2002.This is a case study of the learning of physical science of a high achiever, selected on the assumption that instruction in learning strategies and styles used by successful learners may improve learning effectiveness of less successful learners. Operating in an interpretive paradigm, qualitative data was gathered by participant observation aimed at sensing the complexities of the case. A rich, holistic description is given, enabling readers to form naturalistic generalisations of their own. The data corpus spans three years and is composed of audio-recorded lessons and interviews, field notes and written material. Data collection, analysis and interpretation were done in an inductive, cyclic manner, guided by research questions about learning strategies used by the learner, instructional strategies used by the teacher, and the roles played by intrinsic factors, practical work and problem solving, in contributing to effective learning of physical science by the high achiever. The study implies that effective learning, even by the highly intelligent, involves struggle and requires the use of a variety of strategies. This fits a constructivist, rather than transmissionist, view of learning, and thus supports learner-centered transformations in South African education. The learner is interpreted to be intrinsically motivated by interest and a high regard for knowledge precision, elegance, and transferability, to use a large number of learning strategies, particularly while solving open-ended problems and performing practical investigations, in order to come to a deep understanding of physical science. The study suggests that teaching children how to learn, particularly by addressing their outlook on learning and introducing them to a variety of strategies, should be an aim of physical science instruction, and that interesting, open-ended, learner-centered tasks should be used in attempts to induce self-regulated learning

Longitudinal investigation of the curricular effect: An analysis of student learning outcomes from the LieCal Project in the United States

In this article, we present the results from a longitudinal examination of the impact of a Standards-based or reform mathematics curriculum (called CMP) and traditional mathematics curricula (called non-CMP) on students’ learning of algebra using various outcome measures. Findings include the following: (1) students did not sacrifice basic mathematical skills if they are taught using a Standards-based or reform mathematics curriculum like CMP; (2) African American students experienced greater gain in symbol manipulation when they used a traditional curriculum; (3) the use of either the CMP or a non-CMP curriculum improved the mathematics achievement of all students, including students of color; (4) the use of CMP contributed to significantly higher problem-solving growth for all ethnic groups; and (5) a high level of conceptual emphasis in a classroom improved the students’ ability to represent problem situations. (However, the level of conceptual emphasis bears no relation to students’ problem solving or symbol manipulation skills.

Paradigms for the design of multimedia learning environments in engineering

The starting point for this research was the belief that interactive multimedia learning environments represent a significant evolution in computer based learning and therefore their design requires a re-examination of the underlying principles of learning and knowledge representation. Current multimedia learning environments (MLEs) can be seen as descendants of the earlier technologies of computer-aided learning (CAL), intelligent tutoring systems (ITS) and videodisc-based learning systems. As such they can benefit from much of the wisdom which emerged from those technologies. However, multimedia can be distinguished from earlier technologies by its much greater facility in bringing to the learner high levels of interaction with and control over still and moving image, animation, sound and graphics. Our intuition tells us that this facility has the potential to create learning environments which are not merely substitutes for "live" teaching, but which are capable of elucidating complex conceptual knowledge in ways which have not previously been possible. If the potential of interactive multimedia for learning is to be properly exploited then it needs to be better understood. MLEs should not just be regarded as a slicker version of CAL, ITS or videodisc but a new technology requiring a reinterpretation of the existing theories of learning and knowledge representation. The work described in this thesis aims to contribute to a better understanding of the ways in which MLEs can aid learning. A knowledge engineering approach was taken to the design of a MLE for civil engineers. This involved analysing in detail the knowledge content of the learning domain in terms of different paradigms of human learning and knowledge representation. From this basis, a design strategy was developed which matched the nature of the domain knowledge to the most appropriate delivery techniques. The Cognitive Apprenticeship Model (CAM) was shown to be able to support the integration and presentation of the different categories of knowledge in a coherent instructional framework. It is concluded that this approach is helpful in enabling designers of multimedia systems both to capture and to present a rich picture of the domain. The focus of the thesis is concentrated on the domain of Civil Engineering and the learning of concepts and design skills within that domain. However, much of it could be extended to other highly visual domains such as mechanical engineering. Many of the points can also be seen to be much more widely relevant to the design of any MLE.Engineering and Physical Sciences Research Counci

Problem solving in primary mathematics : a national survey and an in-depth analysis

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN033659 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Investigating and representing inquiry in a college mathematics course

Recent calls by the National Research Council and the National Science Foundation have stressed the need for excellence in undergraduate mathematics and science education with emphasis placed on inquiry learning. The purposes of this qualitative study include (1) the examination of the pursuit of inquiry in two collegiate mathematics classrooms incorporating methods of mathematical modeling and (2) the generation of a quantitative representation of characteristics of an inquiry environment;Instructors and students in two classes of laboratory-based calculus for life sciences majors were observed. To capture descriptions of the environments and students\u27 mathematical modeling skills, the classes surrounding three science investigations were audio or video recorded; interviews were conducted with one instructor and six students in the researcher\u27s class; and copies of students\u27 lab reports were obtained. Transcripts were coded using various scales to produce graphical images of the classroom environments;The data were used to describe and document the effects of both classroom environments. Instructors\u27 goals and time factors influenced the development of inquiry, mathematical modeling, symbol and language use, and the amount of reflection. In both classes when time was of minimal concern, the class pursued students\u27 questions, developed students\u27 modeling methods and notation, and consistently and frequently linked the mathematics and science contexts. When pressured by time to cover specific mathematical topics, the class pursued instructors\u27 questions and methods and less frequently linked the mathematics and science contexts. Most students in both classes retained a process conception of mathematical modeling as they could apply the developed methods but relied on instructor prompts to relate the mathematics and science contexts;The pictorial representations of the classroom environments illustrated that both classes had periods reflecting a constructivist inquiry environment. The graphs highlighted the classes\u27 implementation of multiple cycles of inquiry, periods of consistency and inconsistency in connecting the mathematics and science, and intervals in which students\u27 or instructor\u27s ideas dominated discussion. Class observations suggested that the pictures lacked clarity in identifying periods of agreement or disagreement of the resonating concepts of students and instructors. Recommendations are made for future examination and representation of inquiry environments

Development and Implementation of Computer Assisted Instruction Programme for Instruction Programme for Instruction in Geometry

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