Tuesday, October 6, 2009
Augmenting Human Intellect and Machine of the Year
For a short post on two more articles ("Augmenting Human Intellect" and "Machine of the Year"), see my course blog.
Sunday, October 4, 2009
As we may think
I am taking a PhD level course at my institution this year, called "Fragments". I've just started on the first reading list, and am happy to see that it has some historical texts included. The first one I've read is Vannevar Bush's "As We May Think" (Atlantic Monthly 1945).
"As We May Think" is a fascinating article, trying to figure out how technology may in future (as seen from the point of view of 1945) may help in the organisation of knowledge. The direction described is often very recognizable, even though Mr. Bush was obviously limited by the technologies he had knowledge of.
Particularly interesting, I think, was the "Memex" machine. This was an idea of an office desk in which was organized not only encyclopaedias and newspapers, but also personal notes, and they were connected in ways which made it possible to find them easily. However, from the point of view of the present "Web 2.0" phase of development, it is interesting to see that the information could be inserted by buying centrally produced information, by inserting your own information or by getting information from someone you knew. The idea of the single person contributing to the mass of knowledge available was not there (except, of course, by contributing to the encyclopaedias or newspapers).
The way in which we can all "instantly" contribute to the information structures was probably almost unthinkable at that time.
"As We May Think" is a fascinating article, trying to figure out how technology may in future (as seen from the point of view of 1945) may help in the organisation of knowledge. The direction described is often very recognizable, even though Mr. Bush was obviously limited by the technologies he had knowledge of.
Particularly interesting, I think, was the "Memex" machine. This was an idea of an office desk in which was organized not only encyclopaedias and newspapers, but also personal notes, and they were connected in ways which made it possible to find them easily. However, from the point of view of the present "Web 2.0" phase of development, it is interesting to see that the information could be inserted by buying centrally produced information, by inserting your own information or by getting information from someone you knew. The idea of the single person contributing to the mass of knowledge available was not there (except, of course, by contributing to the encyclopaedias or newspapers).
The way in which we can all "instantly" contribute to the information structures was probably almost unthinkable at that time.
Friday, September 11, 2009
PM's apology to codebreaker Alan Turing: we were inhumane
There's an interesting article in today's Guardian on a long overdue apology from the British government to gay mathematician Alan Turing, who committed suicide after being chemically castrated by the government.
The story of his life is a reminder of how mathematics can have an important role in society (in this case: in war) and how homophobia may blind people and governments even when faced with genius.
The story of his life is a reminder of how mathematics can have an important role in society (in this case: in war) and how homophobia may blind people and governments even when faced with genius.
Tuesday, September 1, 2009
Article: The emerging practice of a novice teacher: The roles of his school mathematics images
Jeppe Skott: The emerging practice of a novice teacher: The roles of his school mathematics images, Journal of Mathematics Teacher Education (2001) 4: 3-28.
From a project with four novice teachers including classroom observation and interviews, the author has chosen two episodes from one of the teachers, which sheds light on the connection between the teacher's "school mathematics images" (SMI) ("teachers' idiosyncratic priorities in relation to mathematics, mathematics as a school subject and the teaching and learning of mathematics in school") and their teaching.
The episodes can be interpreted to show that a teacher may very well do things during his teaching that clashes with his stated SMI, because other concerns are more pressing. In this article, one such concern was to build the confidence of a particular student - thus getting the right answer got more important than a full understanding of the process. On the other hand, the author worries that the actions in such situations may be copied to situations in which the more pressing concerns are not present.
The author also introduces another TLA (three letter acronym): CIP. A CIP is a "critical incident of practice", "an instance of [his] decision making in which multiple and possibly conflicting motives of his activity evolved; that were critical to his SMIs; and that were critical to the future development of the classroom interaction and for the students' learning opportunities".
From a project with four novice teachers including classroom observation and interviews, the author has chosen two episodes from one of the teachers, which sheds light on the connection between the teacher's "school mathematics images" (SMI) ("teachers' idiosyncratic priorities in relation to mathematics, mathematics as a school subject and the teaching and learning of mathematics in school") and their teaching.
The episodes can be interpreted to show that a teacher may very well do things during his teaching that clashes with his stated SMI, because other concerns are more pressing. In this article, one such concern was to build the confidence of a particular student - thus getting the right answer got more important than a full understanding of the process. On the other hand, the author worries that the actions in such situations may be copied to situations in which the more pressing concerns are not present.
The author also introduces another TLA (three letter acronym): CIP. A CIP is a "critical incident of practice", "an instance of [his] decision making in which multiple and possibly conflicting motives of his activity evolved; that were critical to his SMIs; and that were critical to the future development of the classroom interaction and for the students' learning opportunities".
Saturday, August 29, 2009
Article: Teachers' opinions about some teaching material involving history of mathematics
Barry J. Fraser and Anthony J. Koop: Teachers' opinions about some teaching material involving history of mathematics, International Journal of Mathematical education in science and technology (1978) 9: 147-151.
In this article, the authors describe research in which they have given 39 mathematics teachers access to two different kinds of teaching materials involving history of mathematics: a play about Thales and an article related to the history of conics. The teachers were asked to read the materials and then answer a questionnaire. I find this particularly interesting, because this could be seen as research on the teachers' attitudes - without the teachers being influenced by enthusiastic teacher-researchers.
Interestingly, teachers were quite positive to the materials, particularly the play. (A theory might be that a play is more instantly usable than an article?) However, a significant number of the teachers would nonetheless not use it in their own lessons. There are many who thinks that the play would take too much time, for instance.
Also interestingly, almost all teachers agreed that "materials like this are not readily available elsewhere". Of course, that situation has changed a bit since 1978, but it is still an obstacle to teachers' use of history of mathematics.
In this article, the authors describe research in which they have given 39 mathematics teachers access to two different kinds of teaching materials involving history of mathematics: a play about Thales and an article related to the history of conics. The teachers were asked to read the materials and then answer a questionnaire. I find this particularly interesting, because this could be seen as research on the teachers' attitudes - without the teachers being influenced by enthusiastic teacher-researchers.
Interestingly, teachers were quite positive to the materials, particularly the play. (A theory might be that a play is more instantly usable than an article?) However, a significant number of the teachers would nonetheless not use it in their own lessons. There are many who thinks that the play would take too much time, for instance.
Also interestingly, almost all teachers agreed that "materials like this are not readily available elsewhere". Of course, that situation has changed a bit since 1978, but it is still an obstacle to teachers' use of history of mathematics.
Wednesday, August 26, 2009
Article: Using the history of mathematics to induce changes in preservice teachers' beliefs and attitudes
Currently, I have been reading articles on history of mathematics and teachers' beliefs and attitudes. I will continue to blog about articles I read. I hope someone may find something interesting here - but must admit that I blog partly for my own sake, as a way of pushing me to try to state the essence of the articles I'm reading...
Charalambos, Panaoura and Philippou: Using the history of mathematics to induce changes in preservice teachers' beliefs and attitudes: insights from evaluating a teacher education program, Educational studies in mathematics (2009) 71: 161-180.
In this article, the authors have tried to trace and describe the development of 94 preservice teachers' beliefs and attitudes over a period of 2 years, during which they took two courses in history of mathematics.
It turned out that the students' formalist beliefs were intensified, while their Platonic and experimental beliefs were weakened. The students' attitudes towards mathematics got less positive during the period.
This does not, of course, mean that history of mathematics will necessarily have such dire consequences. Rather, it points to the importance of the contents and way of teaching. In this study, some students (in interviews) says that they are unable to draw connections between the contents of the courses and the mathematical content they need in their teaching career. Moreover, they found the mathematics difficult, which made the experience unpleasant. The value of seeing that mathematicians of the past had difficulties as well, was not seen. And importantly, the students did not see why the teachers had chosen the content - they were not told what the goals of the course were.
I would also have been interested to read about the students' attitudes towards the use of history of mathematics in their own teaching, but this is not covered in the article.
Anyway, the article is an interesting example of an evaluation that does not give the results one wished. It would be interesting to hear later if this evaluation lead to changes in the courses and whether further evaluations will be done later.
Charalambos, Panaoura and Philippou: Using the history of mathematics to induce changes in preservice teachers' beliefs and attitudes: insights from evaluating a teacher education program, Educational studies in mathematics (2009) 71: 161-180.
In this article, the authors have tried to trace and describe the development of 94 preservice teachers' beliefs and attitudes over a period of 2 years, during which they took two courses in history of mathematics.
It turned out that the students' formalist beliefs were intensified, while their Platonic and experimental beliefs were weakened. The students' attitudes towards mathematics got less positive during the period.
This does not, of course, mean that history of mathematics will necessarily have such dire consequences. Rather, it points to the importance of the contents and way of teaching. In this study, some students (in interviews) says that they are unable to draw connections between the contents of the courses and the mathematical content they need in their teaching career. Moreover, they found the mathematics difficult, which made the experience unpleasant. The value of seeing that mathematicians of the past had difficulties as well, was not seen. And importantly, the students did not see why the teachers had chosen the content - they were not told what the goals of the course were.
I would also have been interested to read about the students' attitudes towards the use of history of mathematics in their own teaching, but this is not covered in the article.
Anyway, the article is an interesting example of an evaluation that does not give the results one wished. It would be interesting to hear later if this evaluation lead to changes in the courses and whether further evaluations will be done later.
Sunday, August 23, 2009
Article: Didactics and History of Mathematics: Knowledge and Self-Knowledge
Michael N. Fried: Didactics and History of Mathematics: Knowledge and Self-Knowledge, Educational studies in mathematics (2007) 66: 202-223.
How to include history of mathematics in mathematics education in a way that is true to the history? This is an old question in the HPM community, and it is not resolved.
In this article, the author looks to Saussure and the theory of semiotics to argue that history of mathematics has an essential role to play in mathematics education. "The historian's and the working mathematician's ways of knowing are complementary."
First, he discusses what is the problem: "If, in the working mathematician's view, the otherness of a historical text is something illusory or merely superficial, the historical point of view is precisely the opposite. From tihs latter pole, a mathematical text is a cultural product, the product of a particular human being or group of human beings living in a particular time."
He gives a short (and welcome) introduction to Saussure, and describes two ways of knowing: synchronic and diachronic. Seen at one single point of time (synchronic), a language (or mathematics) seems static, and you can not see the social forces at play. When you look at a period of time (diachronic), you see how language (or mathematics) is evolving and ever-changing.
The author then gives an example: How Apollonius of Perga's Conics has been interpreted in different ways. Heath wrote that "[Apollonius'] method does not essentially differ from that of modern analytical geometry except that in Apollonius geometrical operations take the place of algebraic calculations" - an interpretation based squarely in the modern mathematics. The author of the article shows how Apollonius could rather give us an alternative way of viewing conics, which makes us more aware of our own modern view.
So what about mathematics education. The author argues that the object of mathematics education should not be only to learn mathematical methods, but "our self-knowledge as mathematical beings". As such, the history is essential to give another perspective and let us learn more about our own views.
How to include history of mathematics in mathematics education in a way that is true to the history? This is an old question in the HPM community, and it is not resolved.
In this article, the author looks to Saussure and the theory of semiotics to argue that history of mathematics has an essential role to play in mathematics education. "The historian's and the working mathematician's ways of knowing are complementary."
First, he discusses what is the problem: "If, in the working mathematician's view, the otherness of a historical text is something illusory or merely superficial, the historical point of view is precisely the opposite. From tihs latter pole, a mathematical text is a cultural product, the product of a particular human being or group of human beings living in a particular time."
He gives a short (and welcome) introduction to Saussure, and describes two ways of knowing: synchronic and diachronic. Seen at one single point of time (synchronic), a language (or mathematics) seems static, and you can not see the social forces at play. When you look at a period of time (diachronic), you see how language (or mathematics) is evolving and ever-changing.
The author then gives an example: How Apollonius of Perga's Conics has been interpreted in different ways. Heath wrote that "[Apollonius'] method does not essentially differ from that of modern analytical geometry except that in Apollonius geometrical operations take the place of algebraic calculations" - an interpretation based squarely in the modern mathematics. The author of the article shows how Apollonius could rather give us an alternative way of viewing conics, which makes us more aware of our own modern view.
So what about mathematics education. The author argues that the object of mathematics education should not be only to learn mathematical methods, but "our self-knowledge as mathematical beings". As such, the history is essential to give another perspective and let us learn more about our own views.
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