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Democratic access to powerful mathematical ideas
 In L. D. English (Ed.), Handbook of international research in mathematics education. Directions for the 21st Century
, 2002
"... Abstract. The emergence of the informational society creates the paradoxes of inclusion and citizenship, which call into question any simple interpretation of the meaning of “democratic access to powerful mathematical ideas”. In exploring this thesis we put forward ways of understanding what “powerf ..."
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Abstract. The emergence of the informational society creates the paradoxes of inclusion and citizenship, which call into question any simple interpretation of the meaning of “democratic access to powerful mathematical ideas”. In exploring this thesis we put forward ways of understanding what “powerful mathematical ideas” represent logically, psychologically, culturally and sociologically. As a way of tackling the issues of democratic access to these ideas, we elaborate on three arenas of mathematics education practices where it is possible to build a meaningful participation to committed political action, namely the classroom, school organization, and society both locally and globally. To conclude we explore the potentialities of the space of investigation into democratic access to powerful mathematical ideas defined by the four interpretations of “powerful ” and by the three arenas of democratic access. We point to the necessity of covering this whole space of research in order to give a full picture of the complexity of mathematics education in our current informational society. Carlos had to move out of his home. His mother seems to be worried. She lost her job and the money she made through great effort to pay for the small house is in the hands of the bank. Carlos, a tenth grade student, is one of the many Colombian youngsters who will finish high school at the beginning of the 21 st Century. Many of these students seem to be confused about their future. Teachers insist on the importance of schooling and learning, especially mathematics. Yet how could that help in their actual situation? On the other side of the world, in Denmark, Nicolai got seriously sick after eating a home
Empowerment in Mathematics Education
 Retrieved April 15, 2002, from the World Wide Web: http://www.ex.ac.uk/~PErnest/pome15/empowerment.htm
, 2000
"... this paper I explore the meaning of empowerment in the teaching and learning of mathematics. The main part of the paper is devoted to distinguishing three different but complementary meanings of empowerment concerning mathematics: mathematical, social and epistemological empowerment. Mathematical ..."
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this paper I explore the meaning of empowerment in the teaching and learning of mathematics. The main part of the paper is devoted to distinguishing three different but complementary meanings of empowerment concerning mathematics: mathematical, social and epistemological empowerment. Mathematical empowerment concerns gaining the power to use mathematical knowledge and skills in school mathematics
The Characteristics of Mathematical Creativity
"... Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics, a qualitative study ..."
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Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics, a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts and to verify the theory driven hypotheses. The results indicate that, in general, the mathematicians’ creative processes followed the fourstage Gestalt model of preparationincubationilluminationverification. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity Mathematical creativity ensures the growth of the field of mathematics as a whole. The constant increase in the number of journals devoted to mathematics research bears evidence to the growth of mathematics. Yet what lies at the essence of this growth, the creativity of the mathematician, has not been the subject of much research. It is usually the case that most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation (Ervynck, 1991). The earliest known attempt to study mathematical creativity was an extensive questionnaire published in the French periodical L'Enseigement Mathematique (1902). This questionnaire and a lecture on creativity given by the renowned 20th century mathematician Henri Poincaré to the Societé de Psychologie inspired his colleague Jacques Hadamard, another prominent 20th century mathematician, to investigate the psychology of mathematical creativity
Education about and through technology. In search of more appropriate pedagogical approaches to technology education
, 2001
"... This research thesis aimed to deepen understanding about the nature of technology and its possible
correspondence to the constructivist notion of learning. Since technology education is a relatively new
subject area in general education and still in an emerging phase in various countries, it provide ..."
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This research thesis aimed to deepen understanding about the nature of technology and its possible
correspondence to the constructivist notion of learning. Since technology education is a relatively new
subject area in general education and still in an emerging phase in various countries, it provided some
interesting opportunities to take into account the latest developments in educational psychology in relation
to the development of teaching technology. Moreover, this thesis aimed at finding ways for technology
education to provide possibilities to learning environments where the nature of technology could be
integrated effectively into the current notion of children as active agents in their learning processes.
The thesis was based on two Case Studies. Both of the Case Studies were carried out on the primary
school level. The overall purpose of Case Study I was to consider automation technology and its teaching
as a subjectmatter area in developing technology education in Finland. In Case Study II the purpose was
to explore the influences of sociocultural interaction on children's thinking and actions in prescribed and
open problemsolving situations while they were technologically creating a particular product which used
sound for a chosen purpose. Case Study II also involved English schoolchildren.
Teaching methods throughout the thesis were based on the assumption that constructivistdriven, open,
and creative problem solving, as well as childrencentered approaches, are especially suitable for
technology education. This assumption arises from the notions that innovation and problem solving are
important in technological processes and that technology has usually emerged as a response to human needs
and wants. Consequently, design briefs were developed to provide open, childrencentered problem solving
based on the acute needs found in the children's own living environment.
In both of the Case Studies multiple data collection procedures were applied. In Case Study I data were
collected by means of group observations documented in videotaped recordings, written field notes and
project files saved by the students. Moreover, In Case Study II data were collected in terms of photographs
of the pupils' final outcomes, including pupils' design folders and product evaluations, the teacher's
teaching notes, teacher's lesson evaluation notes, the researcher's field notes based on observations and a
questionnaire.
The methodological perspective in both of the Case Studies was qualitative in nature and grounded on
inductive and interpretative databased analysis. The analysis employed an open search for categories,
concepts and patterns emerging from the data. The inductive interpretative analysis process enabled the
results to be framed as empirical assertions. In addition to the assertions the results of Case Study I detailed
content classifications of the substance in the focus were included as well. The assertions and the
classifications were supported by evidentiary examples taken from the data. The supporting examples were
interpreted from the viewpoint of the research problems.
The results of the thesis suggested that in technology education it is important for children to be able to
work and learn in a way that fosters open problem solving with innovation and divergent thinking. In
technology education the design briefs and task allocations should be open enough to allow the children to
explore their own living environment in order to find problems that need to be solved. Actually, in
technology education, according to the nature of technology, there should not be right answers to the posed
questions, but rather appropriate solutions to emerging problems. Moreover, teaching methods adjusted
according to the nature of technology ensure naturally that the children are treated as active, intentional and
goaldirected humans whose activities are driven by human volition.
An Analysis of Errors Made in the Solution of Simple Linear
"... This is an investigation into the errors made by pupils when solving simple linear equations. ..."
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This is an investigation into the errors made by pupils when solving simple linear equations.
An Analysis of Thought Processes during Simplification of an
"... this report to identify and categorise all the specific errors made, but out of the 180 pupils' work: 70 accomplished few correct steps, 44 simply cancelled the original numerator and denominator x 2 , and stopped ..."
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this report to identify and categorise all the specific errors made, but out of the 180 pupils' work: 70 accomplished few correct steps, 44 simply cancelled the original numerator and denominator x 2 , and stopped
THE TEACHING OF TRADITIONAL STANDARD ALGORITHMS FOR THE FOUR ARITHMETIC OPERATIONS VERSUS THE USE OF PUPILS ' OWN METHODS
"... Abstract: In this article I discuss some reasons why it might be advantageous to let pupils use their own methods for computation instead of teaching them the traditional standard algorithms for the four arithmetic operations. Research on this issue is described, especially a project following pupil ..."
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Abstract: In this article I discuss some reasons why it might be advantageous to let pupils use their own methods for computation instead of teaching them the traditional standard algorithms for the four arithmetic operations. Research on this issue is described, especially a project following pupils from their second to their fifth school year. In this, the pupils were not taught the standard algorithms at all, they had to resort to inventing their own methods for all computations, and these methods were discussed in groups or in the whole class. The article ends with a discussion of pros and cons of the ideas that are put forward.
MATHEMATICS AND THEIR EPISTEMOLOGIES AND THE LEARNING OF MATHEMATICS
"... Abstract: This paper reports on the results of a study of the epistemologies of seventy research mathematicians utilising a model containing five categories, sociocultural relatedness, aesthetics, intuition, thinking style and connectivities. The perspectives of the mathematicians demonstrate extre ..."
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Abstract: This paper reports on the results of a study of the epistemologies of seventy research mathematicians utilising a model containing five categories, sociocultural relatedness, aesthetics, intuition, thinking style and connectivities. The perspectives of the mathematicians demonstrate extreme variability from one to another but certain persistent themes carry important messages for mathematics education. In particular, although mathematicians research very differently, their pervasive absolutist view of mathematical knowledge is not matched by their stories of how they come to know, nor of how they think about mathematics.
A FIBONACCI GENERALIZATION: A LAKATOSIAN EXAMPLE
, 1997
"... This paper presents a generalization of the Fibonacci series which roughly followed a Lakatosian heuristic. Using this example, some general comments will be made regarding the processes of discovery and invention in mathematics, and its relevance to the history and philosophy of mathematics. ..."
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This paper presents a generalization of the Fibonacci series which roughly followed a Lakatosian heuristic. Using this example, some general comments will be made regarding the processes of discovery and invention in mathematics, and its relevance to the history and philosophy of mathematics.