Michael N. Fried | Ben Gurion University of the Negev (original) (raw)
Books by Michael N. Fried
Papers by Michael N. Fried
This panel considers theoretical rather than practical frameworks for the alignment of history of... more This panel considers theoretical rather than practical frameworks for the alignment of history of mathematics and mathematics education. Naturally, such a theoretical framework will have practical implications, but its main task is to set out a certain line of questioning. First of all, it must also ask, even if only implicitly, what it really means to be a theoretical framework in the first place. But, principally, it must ask the ends and the value, and, ultimately, the meaning of history of mathematics in mathematics education. Though all the authors agree that in any theoretical framework, history of mathematics, as such, must be taken as the starting point. That said, no claim is made that there must be a single theoretical framework. Nevertheless, each of the three parts of this paper emphasize similar themes. One of these is the relationship between readers and texts: what do readers bring to texts? How are they affected by the texts? How are their own mathematics selves shap...
The College Mathematics Journal, 2010
Abstract Children often incorrectly reduce fractions by canceling common digits instead of common... more Abstract Children often incorrectly reduce fractions by canceling common digits instead of common factors. There are cases, however, in which this incorrect method leads to correct results. Instances such as 16/64 and 19/95 are well-known. In this paper, we consider ...
International Journal of Emerging Technologies in Learning (iJET)
This study reports on findings of two design cycles of augmented reality environment intended to ... more This study reports on findings of two design cycles of augmented reality environment intended to engage high school students in covariational reasoning. The study used a designed-based research method to develop and improve the learning environment. In this report, we present the initial design and discuss how it promoted students' engagement at elementary levels of covariation. Following this first cycle, we introduced a redesigned learning environment. We provide evidence of how the new design in the second cycle promoted students' engagement at advanced levels of covariational reasoning. Six groups of three 15- to 17-year-old students participated in the research. Using AR headsets, each group carried out two activities well-suited, in principle, to covariational reasoning. The students' interactions were video-recorded, and the theory of semiotic representation was used to analyze the degree of their engagement in covariational reasoning. The design emphasized multip...
Research in Mathematics Education, 2014
ZDM – Mathematics Education
The Mathematical Gazette
On his website dedicated to questions and investigations arising out of dynamic geometry technolo... more On his website dedicated to questions and investigations arising out of dynamic geometry technology, Michael de Villiers has a series called Geometry Loci Doodling [1]. These are locus problems connected to the centroids of cyclic quadrilaterals – ‘centroids’ in the plural, for there are three different kinds of centroid depending whether one understands the quadrilateral in terms of its vertices, perimeter or area. The corresponding centroids are the point-mass centroid, the perimeter-centroid, and the lamina-centroid. In each case, de Villiers keeps three vertices of the quadrilateral fixed on the circumcircle, and then traces the locus of the different centroids as the fourth point moves round the circle. In this paper, I shall take a brief look at the point-mass centroid and then a lingering view of the lamina-centroid.
The Journal of Mathematical Behavior
Oxford Research Encyclopedia of Classics, 2019
The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated inten... more The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated intensely in Greek mathematics. The most famous work on the subject was the Conics, in eight books by Apollonius of Perga, but conics were also studied earlier by Euclid and Archimedes, among others. Conic sections were important not only for purely mathematical endeavors such as the problem of doubling the cube, but also in other scientific matters such as burning mirrors and sundials. How the ancient theory of conics is to be understood also played a role in the general development of the historiography of Greek mathematics.
Augmented Reality in Educational Settings, 2019
Humanistic Mathematics Network Journal, 2004
The phrase, 'Mathematics for All', has a certain ambiguity. It may be taken as an exhortation-'Ma... more The phrase, 'Mathematics for All', has a certain ambiguity. It may be taken as an exhortation-'Mathematics for all!', or as an indication of a special kind of mathematics, 'mathematics-for-all'. These two ways of reading the phrase can be translated into two closely related questions: 1) Should mathematics really be for all? and 2) Are there aspects of mathematics, or parts of mathematics, truly appropriate for all? Since one question inevitably leads to the other, the ambiguity of 'mathematics for all' reveals only that there are two sides to this one coin. We shall begin, then, with the first question and conclude with the second. I. Asking whether mathematics should be taught to everyone presupposes that mathematics is something everyone is able to do. As we all too painfully know, there are plenty of people happy to confess that mathematics was something they never could do. There are also, at the other extreme, those sanguine pedagogues who earnestly believe that one can learn anything if only properly taught. The truth, of course, lies somewhere in between. For no one can doubt that people have limits. But, people also have potential, and they can learn more than they think and they have learned more than often they are willing to say. Thus, defining the limits of students' ability, finding ways to teach students what they can learn, and developing methods of
Mathematical Thinking and Learning
Educational Studies in Mathematics, 2020
The Mathematics Enthusiast, 2010
Apollonius of Perga's Conica, 2001
This panel considers theoretical rather than practical frameworks for the alignment of history of... more This panel considers theoretical rather than practical frameworks for the alignment of history of mathematics and mathematics education. Naturally, such a theoretical framework will have practical implications, but its main task is to set out a certain line of questioning. First of all, it must also ask, even if only implicitly, what it really means to be a theoretical framework in the first place. But, principally, it must ask the ends and the value, and, ultimately, the meaning of history of mathematics in mathematics education. Though all the authors agree that in any theoretical framework, history of mathematics, as such, must be taken as the starting point. That said, no claim is made that there must be a single theoretical framework. Nevertheless, each of the three parts of this paper emphasize similar themes. One of these is the relationship between readers and texts: what do readers bring to texts? How are they affected by the texts? How are their own mathematics selves shap...
The College Mathematics Journal, 2010
Abstract Children often incorrectly reduce fractions by canceling common digits instead of common... more Abstract Children often incorrectly reduce fractions by canceling common digits instead of common factors. There are cases, however, in which this incorrect method leads to correct results. Instances such as 16/64 and 19/95 are well-known. In this paper, we consider ...
International Journal of Emerging Technologies in Learning (iJET)
This study reports on findings of two design cycles of augmented reality environment intended to ... more This study reports on findings of two design cycles of augmented reality environment intended to engage high school students in covariational reasoning. The study used a designed-based research method to develop and improve the learning environment. In this report, we present the initial design and discuss how it promoted students' engagement at elementary levels of covariation. Following this first cycle, we introduced a redesigned learning environment. We provide evidence of how the new design in the second cycle promoted students' engagement at advanced levels of covariational reasoning. Six groups of three 15- to 17-year-old students participated in the research. Using AR headsets, each group carried out two activities well-suited, in principle, to covariational reasoning. The students' interactions were video-recorded, and the theory of semiotic representation was used to analyze the degree of their engagement in covariational reasoning. The design emphasized multip...
Research in Mathematics Education, 2014
ZDM – Mathematics Education
The Mathematical Gazette
On his website dedicated to questions and investigations arising out of dynamic geometry technolo... more On his website dedicated to questions and investigations arising out of dynamic geometry technology, Michael de Villiers has a series called Geometry Loci Doodling [1]. These are locus problems connected to the centroids of cyclic quadrilaterals – ‘centroids’ in the plural, for there are three different kinds of centroid depending whether one understands the quadrilateral in terms of its vertices, perimeter or area. The corresponding centroids are the point-mass centroid, the perimeter-centroid, and the lamina-centroid. In each case, de Villiers keeps three vertices of the quadrilateral fixed on the circumcircle, and then traces the locus of the different centroids as the fourth point moves round the circle. In this paper, I shall take a brief look at the point-mass centroid and then a lingering view of the lamina-centroid.
The Journal of Mathematical Behavior
Oxford Research Encyclopedia of Classics, 2019
The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated inten... more The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated intensely in Greek mathematics. The most famous work on the subject was the Conics, in eight books by Apollonius of Perga, but conics were also studied earlier by Euclid and Archimedes, among others. Conic sections were important not only for purely mathematical endeavors such as the problem of doubling the cube, but also in other scientific matters such as burning mirrors and sundials. How the ancient theory of conics is to be understood also played a role in the general development of the historiography of Greek mathematics.
Augmented Reality in Educational Settings, 2019
Humanistic Mathematics Network Journal, 2004
The phrase, 'Mathematics for All', has a certain ambiguity. It may be taken as an exhortation-'Ma... more The phrase, 'Mathematics for All', has a certain ambiguity. It may be taken as an exhortation-'Mathematics for all!', or as an indication of a special kind of mathematics, 'mathematics-for-all'. These two ways of reading the phrase can be translated into two closely related questions: 1) Should mathematics really be for all? and 2) Are there aspects of mathematics, or parts of mathematics, truly appropriate for all? Since one question inevitably leads to the other, the ambiguity of 'mathematics for all' reveals only that there are two sides to this one coin. We shall begin, then, with the first question and conclude with the second. I. Asking whether mathematics should be taught to everyone presupposes that mathematics is something everyone is able to do. As we all too painfully know, there are plenty of people happy to confess that mathematics was something they never could do. There are also, at the other extreme, those sanguine pedagogues who earnestly believe that one can learn anything if only properly taught. The truth, of course, lies somewhere in between. For no one can doubt that people have limits. But, people also have potential, and they can learn more than they think and they have learned more than often they are willing to say. Thus, defining the limits of students' ability, finding ways to teach students what they can learn, and developing methods of
Mathematical Thinking and Learning
Educational Studies in Mathematics, 2020
The Mathematics Enthusiast, 2010
Apollonius of Perga's Conica, 2001
Mathematical Thinking and Learning, 2021
The premise of dialectical materialism is, we recall: ‘It is not men’s consciousness that determi... more The premise of dialectical materialism is, we recall: ‘It is not men’s consciousness that determines their existence, but, on the contrary, their social existence that determines their consciousnes...
Mathematics Classrooms in Twelve Countries, 2006
Apollonius of Perga's Conica, 2001
ABSTRACT: The words “public” and “private” are often employed to describe one aspect or another o... more ABSTRACT: The words “public” and “private” are often employed to describe one aspect or another of mathematical life. This, perhaps, is not surprising given the importance of the public/private dichotomy in so many other aspects of our lives—and it may be that its importance in mathematics education derives from that wider context. With that in mind, this paper looks at notions of public and private and their specific relevance to mathematics education. The public domain is one governed by the needs of communication: it is the realm of signs and norms. The private domain is a realm of reflection and inwardness, of idiosyncratic meanings, of freedom and spontaneity. In the paper, I maintain that our mathematical selves dwell in both domains, that an utterly public self is as untenable as an utterly private self. The implication is that mathematics education must see itself as mediating between students’ public and the private mathematical selves—an effort, I believe, that is aided an...
The Encyclopedia of Ancient History, 2012
As with so many mathematicians of the Hellenistic period, little about the life of Apollonios of ... more As with so many mathematicians of the Hellenistic period, little about the life of Apollonios of Perge is known and, with the exception of his great work on conics, the Conica (Konikon), only hints of the true breadth of his mathematical activities survive. Keywords: biography; history of science; medicine and technology; science
