Mathematics draws a wider perspective on the issues addressed and some of the constructs used in ... more Mathematics draws a wider perspective on the issues addressed and some of the constructs used in research in Realistic Mathematics Education (RME). In particular, it tries to show that while the problems addressed existed within the world-wide arena of mathematics education and were not unique to the Dutch educational system, the methods used at the Freudenthal Institute to address them were uniquely adapted to that system yet foreshadowed developments in the wider field of mathematics education. The predictive aspects of mathematizing, didactizing, and guided reinvention, in which models-of become models-for on various levels, resonate with trends in mathematics education in recent years, including those promoted by the National Council of Teachers of Mathematics in the USA. Research methodologies, too, have broadened to include more humanistic qualitative methods. Developmental research as epitomized in the RME tradition makes the distinction between quantitative and qualitative research obsolete, because there is no restriction on research methods that may be useful in investigating how to improve the teaching and learning of mathematics, and in the designing of mathematics curricula. Thus some aspects of this research resonate with what have come to be known as multitiered teaching experiments. However, in RME there is also a special content-oriented didactical approach that harmonizes with an emphasis on didactics (rather than pedagogy) in several other European countries. Some implications are drawn for future research directions.
Evidence from two research projects, one with high school students and the other with graduate st... more Evidence from two research projects, one with high school students and the other with graduate students in an Ethnomathematics course, is presented to suggest that the beliefs students hold about the nature of mathematics both enables and constrains their ability to construct conceptual bridges between familiar everyday practices and mathematical concepts taught in school or university. Using a semiotic theoretical
... Research literature (eg, Presmeg, 2006) supports the assertion that understanding of mathemat... more ... Research literature (eg, Presmeg, 2006) supports the assertion that understanding of mathematics is strongly related to the ability to use visual and analytic thinking. ... Journal of Mathematical Behavior, 17, 197-218. Presmeg, NC (2006). ...
In the literature, problem posing is claimed to be important in learning mathematics. This study ... more In the literature, problem posing is claimed to be important in learning mathematics. This study investigated US and Chinese high school students’ attitudes and abilities in posing mathematical problems. All of the participants were taking advanced mathematics in high school. A mathematics content test and a mathematical problem-posing test were administered to the students. The mathematical content test was adapted from the National Assessment of Educational Progress for 12th graders. The problem-posing test included three situations, namely a free problem-posing situation, a semi-structured problem-posing situation, and a structured problem-posing situation. Students who scored 39 or above out of 50 points were interviewed. During the interviews, the majority of the students reported that they had not had any prior experience in posing mathematical problems. Many students did not have a specific strategy for posing problems, and many had difficulty explaining their problem-posing processes. Most of the US students for various reasons said that problem posing was important in mathematics. Most Chinese students said that problem posing was not important in high school learning because of college entrance examinations.
Our field, our baby field that is brand new in comparison with the millennia for which mathematic... more Our field, our baby field that is brand new in comparison with the millennia for which mathematics has existed as a discipline, has seen some dramatic changes in its half-century of being a field in its own right, with its own journals and conferences. We have come a long way, even since the early 1980s, when "illuminative evaluation" was slowly replacing or, initially at least, supplementing the psychometric experiments that used "subjects" ( people) who were being taught mathematics. Before that period, in the old paradigm, no research that did not aim for objectivity by means of carefully controlled experiments and statistical analysis was considered scientific in our field. In connection with the research methods of this period, Krutetskii (1976) gave a pungent critique: It is hard to understand how theory or practice can be enriched by, for instance, the research of Kennedy, who computed, for 130 mathematically gifted adolescents, their scores on different kinds of test and studied the correlations between them, finding that in some cases it was significant and in others not. The process of solution did not interest the investigator. But what rich material could be provided by the process of mathematical thinking in 130 mathematically able adolescents! (p. 14) Krutetskii's interview methods, in Soviet Russia, were in many ways a precursor to the qualitative methodologies that followed this early period. Slowly, the qualitative research paradigm gained credence. After all, we are dealing with human beings in their teaching and learning of mathematics, with all the complexities and uncertainties that that fact implies! Even Krutetskii (1976), aware as he was of individual differences, wrote of "perfect teaching methods" (p. 6), terminology that we might use more circumspectly today. With regard to useful and believable research (rather than reliable and valid experiments), initial crude attempts at quality control became strengthened. Thus, triangulation of various types (Stake 1995) was needed to ensure that research results and insights reported more than merely the researcher's opinions. We learned to go back and ask the mathematics teachers and their stu-U. Kortenkamp et al. (eds.), Early Mathematics Learning,
Building on Michael Otte’s insights regarding the roles of icon, index, and symbol in mathematica... more Building on Michael Otte’s insights regarding the roles of icon, index, and symbol in mathematical signification, definitions of these categories of representation are explored in terms of metaphors and metonymies. A nested model of signs, based on Peirce’s triadic formulation, is described, along with his trichotomic distinction among interpretants that are intentional, effectual, and communicational (leading to the commens). The
... Various definitions of visualization have been used in the research literature. For us, visua... more ... Various definitions of visualization have been used in the research literature. For us, visualization involves both external and internal representations (or images), and thus following Presmeg (200624. ... View all references) and Presmeg (200624. Presmeg, NC 2006. ...
... what matters, without any necessary connections to sensory experience, even if such experienc... more ... what matters, without any necessary connections to sensory experience, even if such experience is essential because it is the only way that we can write down or communicate the form to others (in representations—or better, inscriptions, the term that I prefer, eg, Presmeg 2006 ...
There have been assertions in recent literature on visualization that students are reluctant to v... more There have been assertions in recent literature on visualization that students are reluctant to visualize when doing mathematics, particularly at the high school and college levels. While there is ...
... the various registers (Duval, 1999) of school trigonometry (Presmeg, 2006b). ... of such topi... more ... the various registers (Duval, 1999) of school trigonometry (Presmeg, 2006b). ... of such topics as ''Stu-dents' improper proportional reasoning'' (Modestou and Gagatsis, 2007), or ''Exploring young children's geometri-cal strategies'' (Gagatsis, Sriraman, Elia, & Modestou, 2006). ...
adaptation, distribution and reproduction in any medium or format, as long as you give appropriat... more adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
International Group for the Psychology of Mathematics Education, Jul 1, 2005
As part of a larger investigation of preservice teachers' use of, and movement amongst, various m... more As part of a larger investigation of preservice teachers' use of, and movement amongst, various modes of representing exponential relationships, this report focuses on one case study, that of Mike, whose facility in moving amongst representational registers was not matched by conceptual understanding of the underlying mathematical ideas as he attempted to solve algebraic problems involving exponential relationships. Mike's case casts doubt on the theoretical assumption that students who can move fluently amongst various inscriptions representing the same concept have of necessity attained conceptual knowledge of the relationships involved.
Mathematics draws a wider perspective on the issues addressed and some of the constructs used in ... more Mathematics draws a wider perspective on the issues addressed and some of the constructs used in research in Realistic Mathematics Education (RME). In particular, it tries to show that while the problems addressed existed within the world-wide arena of mathematics education and were not unique to the Dutch educational system, the methods used at the Freudenthal Institute to address them were uniquely adapted to that system yet foreshadowed developments in the wider field of mathematics education. The predictive aspects of mathematizing, didactizing, and guided reinvention, in which models-of become models-for on various levels, resonate with trends in mathematics education in recent years, including those promoted by the National Council of Teachers of Mathematics in the USA. Research methodologies, too, have broadened to include more humanistic qualitative methods. Developmental research as epitomized in the RME tradition makes the distinction between quantitative and qualitative research obsolete, because there is no restriction on research methods that may be useful in investigating how to improve the teaching and learning of mathematics, and in the designing of mathematics curricula. Thus some aspects of this research resonate with what have come to be known as multitiered teaching experiments. However, in RME there is also a special content-oriented didactical approach that harmonizes with an emphasis on didactics (rather than pedagogy) in several other European countries. Some implications are drawn for future research directions.
Evidence from two research projects, one with high school students and the other with graduate st... more Evidence from two research projects, one with high school students and the other with graduate students in an Ethnomathematics course, is presented to suggest that the beliefs students hold about the nature of mathematics both enables and constrains their ability to construct conceptual bridges between familiar everyday practices and mathematical concepts taught in school or university. Using a semiotic theoretical
... Research literature (eg, Presmeg, 2006) supports the assertion that understanding of mathemat... more ... Research literature (eg, Presmeg, 2006) supports the assertion that understanding of mathematics is strongly related to the ability to use visual and analytic thinking. ... Journal of Mathematical Behavior, 17, 197-218. Presmeg, NC (2006). ...
In the literature, problem posing is claimed to be important in learning mathematics. This study ... more In the literature, problem posing is claimed to be important in learning mathematics. This study investigated US and Chinese high school students’ attitudes and abilities in posing mathematical problems. All of the participants were taking advanced mathematics in high school. A mathematics content test and a mathematical problem-posing test were administered to the students. The mathematical content test was adapted from the National Assessment of Educational Progress for 12th graders. The problem-posing test included three situations, namely a free problem-posing situation, a semi-structured problem-posing situation, and a structured problem-posing situation. Students who scored 39 or above out of 50 points were interviewed. During the interviews, the majority of the students reported that they had not had any prior experience in posing mathematical problems. Many students did not have a specific strategy for posing problems, and many had difficulty explaining their problem-posing processes. Most of the US students for various reasons said that problem posing was important in mathematics. Most Chinese students said that problem posing was not important in high school learning because of college entrance examinations.
Our field, our baby field that is brand new in comparison with the millennia for which mathematic... more Our field, our baby field that is brand new in comparison with the millennia for which mathematics has existed as a discipline, has seen some dramatic changes in its half-century of being a field in its own right, with its own journals and conferences. We have come a long way, even since the early 1980s, when "illuminative evaluation" was slowly replacing or, initially at least, supplementing the psychometric experiments that used "subjects" ( people) who were being taught mathematics. Before that period, in the old paradigm, no research that did not aim for objectivity by means of carefully controlled experiments and statistical analysis was considered scientific in our field. In connection with the research methods of this period, Krutetskii (1976) gave a pungent critique: It is hard to understand how theory or practice can be enriched by, for instance, the research of Kennedy, who computed, for 130 mathematically gifted adolescents, their scores on different kinds of test and studied the correlations between them, finding that in some cases it was significant and in others not. The process of solution did not interest the investigator. But what rich material could be provided by the process of mathematical thinking in 130 mathematically able adolescents! (p. 14) Krutetskii's interview methods, in Soviet Russia, were in many ways a precursor to the qualitative methodologies that followed this early period. Slowly, the qualitative research paradigm gained credence. After all, we are dealing with human beings in their teaching and learning of mathematics, with all the complexities and uncertainties that that fact implies! Even Krutetskii (1976), aware as he was of individual differences, wrote of "perfect teaching methods" (p. 6), terminology that we might use more circumspectly today. With regard to useful and believable research (rather than reliable and valid experiments), initial crude attempts at quality control became strengthened. Thus, triangulation of various types (Stake 1995) was needed to ensure that research results and insights reported more than merely the researcher's opinions. We learned to go back and ask the mathematics teachers and their stu-U. Kortenkamp et al. (eds.), Early Mathematics Learning,
Building on Michael Otte’s insights regarding the roles of icon, index, and symbol in mathematica... more Building on Michael Otte’s insights regarding the roles of icon, index, and symbol in mathematical signification, definitions of these categories of representation are explored in terms of metaphors and metonymies. A nested model of signs, based on Peirce’s triadic formulation, is described, along with his trichotomic distinction among interpretants that are intentional, effectual, and communicational (leading to the commens). The
... Various definitions of visualization have been used in the research literature. For us, visua... more ... Various definitions of visualization have been used in the research literature. For us, visualization involves both external and internal representations (or images), and thus following Presmeg (200624. ... View all references) and Presmeg (200624. Presmeg, NC 2006. ...
... what matters, without any necessary connections to sensory experience, even if such experienc... more ... what matters, without any necessary connections to sensory experience, even if such experience is essential because it is the only way that we can write down or communicate the form to others (in representations—or better, inscriptions, the term that I prefer, eg, Presmeg 2006 ...
There have been assertions in recent literature on visualization that students are reluctant to v... more There have been assertions in recent literature on visualization that students are reluctant to visualize when doing mathematics, particularly at the high school and college levels. While there is ...
... the various registers (Duval, 1999) of school trigonometry (Presmeg, 2006b). ... of such topi... more ... the various registers (Duval, 1999) of school trigonometry (Presmeg, 2006b). ... of such topics as ''Stu-dents' improper proportional reasoning'' (Modestou and Gagatsis, 2007), or ''Exploring young children's geometri-cal strategies'' (Gagatsis, Sriraman, Elia, & Modestou, 2006). ...
adaptation, distribution and reproduction in any medium or format, as long as you give appropriat... more adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
International Group for the Psychology of Mathematics Education, Jul 1, 2005
As part of a larger investigation of preservice teachers' use of, and movement amongst, various m... more As part of a larger investigation of preservice teachers' use of, and movement amongst, various modes of representing exponential relationships, this report focuses on one case study, that of Mike, whose facility in moving amongst representational registers was not matched by conceptual understanding of the underlying mathematical ideas as he attempted to solve algebraic problems involving exponential relationships. Mike's case casts doubt on the theoretical assumption that students who can move fluently amongst various inscriptions representing the same concept have of necessity attained conceptual knowledge of the relationships involved.
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