## Sunday, October 18, 2009

### HPM Newsletter 72

The HPM Newsletter (of which I'm the editor together with Chris Weeks), has posted its 72nd issue at the HPM website.

## 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.

## Thursday, August 20, 2009

### Article: On the argument of simplicity in Elements and schoolbooks of Geometry

Evelyne Barbin: On the argument of simplicity in Elements and schoolbooks of Geometry, Educational studies in mathematics (2007) 66: 225-242.

We all want to present the mathematics in a simple way to our students, don't we? Sure we do, but what do we mean by "simple" exactly? In this article, the author very interestingly traces the concept of "simplicity" through several editions of Euclid's Elements as well as in some recent French textbooks. And it turns out that "simplicity" is not as simple as we thought...

Proclus, in his commentary to Euclid, proposed two "orders of simplicity": simplicity of a figure and simplicity of theorems. "A figure is more complex than another if it is obtained from the former with additions of lines or circles." A theorem is simpler than another if it comes earlier in the order of logical deduction.

Peletier preferred clearness of proofs rather than simplicity of figures.

Arnauld sought to follow the "natural order". "It is contrary to this natural order to prove, as Euclid does, propositions on perpendicular straight lines or on parallel straight lines (simple things) using triangles (compound things)." Therefore, the whole book was ordered according to the natural order. A consequence of this was that "perpendicular lines [were] studied without using angles and angles [were] studied without using triangles."

Hoüel wanted simplicity of "principles and proofs". He defined "the straight line by the motion of a point, and the plane by the motion of a straight line. Angles are defined by the ideas of motion and direction." (I find this last point particularly interesting, as it has been a problem in Norwegian textbooks that angles have been defined as static things only, without any dynamic dimension.)

Then, of course, in the 20th century, the notion of sets was regarded as "the simplest of all the notions." This had absurd consequences; "In the Mathématiques, classe de sixième, directed by Mauguin (1977), for eleven-year old pupils, the angle is defined as an equivalence class."

Finally, in a textbook from 1996, the angle is introduced with drawings of fans which are opened to different degrees. This quite good idea is however partly destroyed by a confusing use of colors.

I have just given a tiny idea of the contents of the article, of course. The author shows how these different ideas of simplicity is determining the structure of the books. Choose your notion of simplicity, and the structure will have to change. She ends with a reference to Descartes:

We all want to present the mathematics in a simple way to our students, don't we? Sure we do, but what do we mean by "simple" exactly? In this article, the author very interestingly traces the concept of "simplicity" through several editions of Euclid's Elements as well as in some recent French textbooks. And it turns out that "simplicity" is not as simple as we thought...

Proclus, in his commentary to Euclid, proposed two "orders of simplicity": simplicity of a figure and simplicity of theorems. "A figure is more complex than another if it is obtained from the former with additions of lines or circles." A theorem is simpler than another if it comes earlier in the order of logical deduction.

Peletier preferred clearness of proofs rather than simplicity of figures.

Arnauld sought to follow the "natural order". "It is contrary to this natural order to prove, as Euclid does, propositions on perpendicular straight lines or on parallel straight lines (simple things) using triangles (compound things)." Therefore, the whole book was ordered according to the natural order. A consequence of this was that "perpendicular lines [were] studied without using angles and angles [were] studied without using triangles."

Hoüel wanted simplicity of "principles and proofs". He defined "the straight line by the motion of a point, and the plane by the motion of a straight line. Angles are defined by the ideas of motion and direction." (I find this last point particularly interesting, as it has been a problem in Norwegian textbooks that angles have been defined as static things only, without any dynamic dimension.)

Then, of course, in the 20th century, the notion of sets was regarded as "the simplest of all the notions." This had absurd consequences; "In the Mathématiques, classe de sixième, directed by Mauguin (1977), for eleven-year old pupils, the angle is defined as an equivalence class."

Finally, in a textbook from 1996, the angle is introduced with drawings of fans which are opened to different degrees. This quite good idea is however partly destroyed by a confusing use of colors.

I have just given a tiny idea of the contents of the article, of course. The author shows how these different ideas of simplicity is determining the structure of the books. Choose your notion of simplicity, and the structure will have to change. She ends with a reference to Descartes:

"As Descartes writes, arithmetic and geometry are more certain than the other sciences because their object is "so pure and so simple". [...] History of mathematics invites us to come back and to work with the notion of simplicity to construct the teaching of geometry."

## Monday, August 17, 2009

### Article: Mathematical history, philosophy and education

Michael Otte: Mathematical history, philosophy and education, Educational studies in mathematics (2007) 66: 243-255.

Combining history and philosophy of mathematics, this is not an easy read, and I will not claim to have grasped even the main points in the first reading.

It does start out with describing a point of view (which the author of this article does not share): "Within this context, it is frequently claimed, by mathematicians in particular, that mathematics has no history worth knowing. The newest state of the art of mathematics has taken up and reformulated in modern terms whatever appeared as worthwhile during its history."

By looking at some topics from the history of mathematics, he ends up in this conclusion: "Mathematical ideas that appear extremely abstract and difficult at first sight become understandable from a historical perspective only. The transformation of processes into structures with which we have dealt here is quite instructive in this respect. History of mathematics occupies itself describing processes of growth and development, whereas philosophy of mathematics is concerned with questions of justification. Both play an essential role within the educational context."

On the way from the beginning to the conclusion, the author looks at the view of numbers throughout history as well as the theory of integration from Cauchy to Lebesgue. The second of these was particularly interesting to me. Topology was one of my favorite courses in university, and it is interesting to see the development of topology from this perspective.

Combining history and philosophy of mathematics, this is not an easy read, and I will not claim to have grasped even the main points in the first reading.

It does start out with describing a point of view (which the author of this article does not share): "Within this context, it is frequently claimed, by mathematicians in particular, that mathematics has no history worth knowing. The newest state of the art of mathematics has taken up and reformulated in modern terms whatever appeared as worthwhile during its history."

By looking at some topics from the history of mathematics, he ends up in this conclusion: "Mathematical ideas that appear extremely abstract and difficult at first sight become understandable from a historical perspective only. The transformation of processes into structures with which we have dealt here is quite instructive in this respect. History of mathematics occupies itself describing processes of growth and development, whereas philosophy of mathematics is concerned with questions of justification. Both play an essential role within the educational context."

On the way from the beginning to the conclusion, the author looks at the view of numbers throughout history as well as the theory of integration from Cauchy to Lebesgue. The second of these was particularly interesting to me. Topology was one of my favorite courses in university, and it is interesting to see the development of topology from this perspective.

## Friday, August 14, 2009

### Article: The roles of Mesopotamian bronze age mathematics tool...

Jens Høyrup: The roles of Mesopotamian bronze age mathematics tool for state formation and administration - carrier of teachers' professional intellectual autonomy, Educational studies in mathematics (2007) 66: 111-129.

It is a privilege to read this article by Høyrup, no doubt based on insights gained through decades of studies. If you want a very short introduction to Mesopotamian mathematics, this is a good place to start.

He tells of how tokens were placed in clay containers for accounting purposes, that later impressions were made on the surface to make it possible to "read" the information without breaking the containers, and how even later, the tokens themselves were dropped, no longer being of any significance. He describes the notation developed.

Later, mathematics developed as a means of ensuring "just measure" (and the author is quick to point out that mathematical justice can also be cruel).

From Shuruppak (-2600), examples of school texts have been found. An example of an "exercise": "A silo containing 40x60 gur ('tuns') of grain, each of 8x60 sila ('litres') is distributed in portions of 7 sila per worker." The answer is 164,571 workers, with a remainder of 3 sila, but the numbers included are surely not from a practical situation. (Already at that time, unrealistic numbers were used in mathematics exercises...)

In 2074, an administrative reform were carried out bringing my thoughts to harsh central planning regimes such as Mao's China or North Korea. Workers were organized in troups, and the overseers were responsible for their unit's performance, with preset goals (read the details in the article). Mathematics was necessary to keep the system running.

The author spends some time on the text "BM 13901". The translation goes as the following:

The explanation (in geometrical terms) is fascinating.

It is a remarkable article with insights I will make sure to bring to my classrooms.

It is a privilege to read this article by Høyrup, no doubt based on insights gained through decades of studies. If you want a very short introduction to Mesopotamian mathematics, this is a good place to start.

He tells of how tokens were placed in clay containers for accounting purposes, that later impressions were made on the surface to make it possible to "read" the information without breaking the containers, and how even later, the tokens themselves were dropped, no longer being of any significance. He describes the notation developed.

Later, mathematics developed as a means of ensuring "just measure" (and the author is quick to point out that mathematical justice can also be cruel).

From Shuruppak (-2600), examples of school texts have been found. An example of an "exercise": "A silo containing 40x60 gur ('tuns') of grain, each of 8x60 sila ('litres') is distributed in portions of 7 sila per worker." The answer is 164,571 workers, with a remainder of 3 sila, but the numbers included are surely not from a practical situation. (Already at that time, unrealistic numbers were used in mathematics exercises...)

In 2074, an administrative reform were carried out bringing my thoughts to harsh central planning regimes such as Mao's China or North Korea. Workers were organized in troups, and the overseers were responsible for their unit's performance, with preset goals (read the details in the article). Mathematics was necessary to keep the system running.

The author spends some time on the text "BM 13901". The translation goes as the following:

I have heaped the surface and my confrontation: it is 3/4. 1 the projection you posit, half-part of 1 you break, make 1/2 and 1/2 hold, 1/4 and 3/4 you join: alongside 1, 1 is equilateral. 1/2 which you have made hold from the body of 1 you tear out: 1/2 is the confrontation.

The explanation (in geometrical terms) is fascinating.

It is a remarkable article with insights I will make sure to bring to my classrooms.

## Wednesday, August 12, 2009

### Article: Introduction: The topos of meaning or the encounter between past and present

Luis Radford, Fulvia Furinghetti and Victor Katz: Introduction: The topos of meaning or the encounter between past and present, Educational studies in mathematics (2007) 66: 111-129.

This article works as a foreword for the special issue. In addition to outlining the contents of the issue, it also argues for the role of history of mathematics in mathematics education.

"Learning by doing" is a well-known slogan in modern pedagogy, and extreme constructivists will argue that a pupil will have to construct all knowledge on his experiences. (I'm not saying that Dewey himself had this view.) The authors quote Leontiev, who said that

The authors continue: "The very possibility of learning rests in our capability of immersing ourselves - in idiosyncratic, critical and reflective ways - in the conceptual historical riches deposited in, and continuously modified by, social practices."

And studying the history of mathematics is one way of being immersed in this while asking questions we might not ask when we are faced with modern methods.

This article works as a foreword for the special issue. In addition to outlining the contents of the issue, it also argues for the role of history of mathematics in mathematics education.

"Learning by doing" is a well-known slogan in modern pedagogy, and extreme constructivists will argue that a pupil will have to construct all knowledge on his experiences. (I'm not saying that Dewey himself had this view.) The authors quote Leontiev, who said that

"No one's personal experience, no matter how rich it might be, can result in thinking logically, abstractly and mathematically, and in individually establishing a system of ideas. To do this, one would need not just one lifetime, but thousands."

The authors continue: "The very possibility of learning rests in our capability of immersing ourselves - in idiosyncratic, critical and reflective ways - in the conceptual historical riches deposited in, and continuously modified by, social practices."

And studying the history of mathematics is one way of being immersed in this while asking questions we might not ask when we are faced with modern methods.

## Sunday, August 9, 2009

### Article: Learning to listen: from historical sources to classroom practice

Abraham Arcavi and Masami Isoda: Learning to listen: from historical sources to classroom practice, Educational studies in mathematics (2007) 66: 111-129.

For prospective teachers, it is important to learn to listen to their students. There are several challenges to this, however, and the main idea of this article is that maybe prospective teachers can learn to listen by "listening" to historical texts. As when listening to students, "listening" to a historical text involves taking another person's perspective. The historical text, however, has a certain authority that makes it harder to ignore - while teachers may ignore students' attempts and instead just teach them "the right way".

The authors make a distinction between "evaluative listening" and "attentive listening". I surely recognize this distinction - at times, I will notice that I'm not exactly listening to what the student is saying, instead I'm listening FOR something particular. I'm waiting for the student to say what I want to hear instead of trying to make sense of what he is actually saying. In teaching based on constructivism, of course attentive listening is necessary.

The particular examples of history of mathematics that the authors used in their teaching, are from Egyptian mathematics, for instance the Egyptian method of multiplication. I agree that this is a good example to use (see also an earlier post on this (in Norwegian).

I also agree with the authors that as well as a collection of good ideas for use in teacher education, we also need good examples of students' work as a next step in "learning to listen", after working on the history of mathematics.

For prospective teachers, it is important to learn to listen to their students. There are several challenges to this, however, and the main idea of this article is that maybe prospective teachers can learn to listen by "listening" to historical texts. As when listening to students, "listening" to a historical text involves taking another person's perspective. The historical text, however, has a certain authority that makes it harder to ignore - while teachers may ignore students' attempts and instead just teach them "the right way".

The authors make a distinction between "evaluative listening" and "attentive listening". I surely recognize this distinction - at times, I will notice that I'm not exactly listening to what the student is saying, instead I'm listening FOR something particular. I'm waiting for the student to say what I want to hear instead of trying to make sense of what he is actually saying. In teaching based on constructivism, of course attentive listening is necessary.

The particular examples of history of mathematics that the authors used in their teaching, are from Egyptian mathematics, for instance the Egyptian method of multiplication. I agree that this is a good example to use (see also an earlier post on this (in Norwegian).

I also agree with the authors that as well as a collection of good ideas for use in teacher education, we also need good examples of students' work as a next step in "learning to listen", after working on the history of mathematics.

## Thursday, August 6, 2009

### Article: Teacher education through the history of mathematics

Fulvia Furinghetti: Teacher education through the history of mathematics, Educational studies in mathematics (2007) 66: 131-143.

Teacher students have a tendency to reproduce the teaching styles that they have themselves experienced when they were pupils. And then, when they start in their first job as teachers, they will have a tendency to learn from their older colleagues. These two tendencies have a conservative effect on teaching.

In this article, the author discusses how history of mathematics can be a part of teacher education in a way that will work against the first tendency mentioned above. "Prospective teachers need a context allowing them to look at the topics they will teach in a different manner", she argues.

The history of mathematics was not taught as a goal in itself, but rather as "a mediator of knowledge for teaching". The students had "difficulties in considering ways of teaching a given mathematical topic other than the way that they have seen it in their school days", but by working on a teaching segment on algebra and the history of algebra in that context, they were helped to find new ways.

Teacher students have a tendency to reproduce the teaching styles that they have themselves experienced when they were pupils. And then, when they start in their first job as teachers, they will have a tendency to learn from their older colleagues. These two tendencies have a conservative effect on teaching.

In this article, the author discusses how history of mathematics can be a part of teacher education in a way that will work against the first tendency mentioned above. "Prospective teachers need a context allowing them to look at the topics they will teach in a different manner", she argues.

The history of mathematics was not taught as a goal in itself, but rather as "a mediator of knowledge for teaching". The students had "difficulties in considering ways of teaching a given mathematical topic other than the way that they have seen it in their school days", but by working on a teaching segment on algebra and the history of algebra in that context, they were helped to find new ways.

## Monday, August 3, 2009

### Article: Stages in the History of Algebra with Implications for Teaching

Victor J. Katz (with Bill Barton): Stages in the History of Algebra with Implications for Teaching, Educational studies in mathematics (2007) 66: 185-201.

In this article, Victor Katz gives "highlights" from the history of algebra. He does not spend time on the well-known three stages in the expression of algebra: the rhetorical stage, the syncopated stage, and the symbolic stage. Instead, he looks at four conceptual stages: the geometric stage, the static equation-solving stage, the dynamic function stage and the abstract stage.

The by far most long-lasting of these stages (so far) was the geometric stage. Euclid's algebra was geometrical, but so was Babylonian algebra from about 4000 years ago. Al-Khwarizmi (about 825) still justified the methods by geometrical means, but the reader was supposed to learn the algorithm without needing recourse to the geometry. For Katz, Al-Khwarizmi marks the move to the sttatic equation-solving stage.

Although Sharaf al-Din (died 1213) were using methods that could have been the start of the dynamic function stage, it instead had to wait until the early 1600s to take hold. With Fermat and Descartes, and later Newton, algebra moved from being mostly concerned with solving equations to be a method for determining curves, for instance.

Then, during the 1800s mathematicians worked on more general concepts, such as the group (Galois and Cayley are important names in that development).

Katz asks whether these stages should have pedagogical implications. Should geometric figures play a bigger part in the beginning of work on algebra? Should we work more on equations before introducing functions? And so on.

As usual, there are no easy answers when it comes to pedagogical issues, but at least it seems a good idea to know the history before designing the teaching of the future.

In this article, Victor Katz gives "highlights" from the history of algebra. He does not spend time on the well-known three stages in the expression of algebra: the rhetorical stage, the syncopated stage, and the symbolic stage. Instead, he looks at four conceptual stages: the geometric stage, the static equation-solving stage, the dynamic function stage and the abstract stage.

The by far most long-lasting of these stages (so far) was the geometric stage. Euclid's algebra was geometrical, but so was Babylonian algebra from about 4000 years ago. Al-Khwarizmi (about 825) still justified the methods by geometrical means, but the reader was supposed to learn the algorithm without needing recourse to the geometry. For Katz, Al-Khwarizmi marks the move to the sttatic equation-solving stage.

Although Sharaf al-Din (died 1213) were using methods that could have been the start of the dynamic function stage, it instead had to wait until the early 1600s to take hold. With Fermat and Descartes, and later Newton, algebra moved from being mostly concerned with solving equations to be a method for determining curves, for instance.

Then, during the 1800s mathematicians worked on more general concepts, such as the group (Galois and Cayley are important names in that development).

Katz asks whether these stages should have pedagogical implications. Should geometric figures play a bigger part in the beginning of work on algebra? Should we work more on equations before introducing functions? And so on.

As usual, there are no easy answers when it comes to pedagogical issues, but at least it seems a good idea to know the history before designing the teaching of the future.

## Friday, July 31, 2009

### Article: The notion of historical "parallelism" revisited: historical evolution and students' conception of the order relation on the number line

Yannis Thomaidis and Constantinos Tzanakis: The notion of historical "parallelism" revisited: historical evolution and students' conception of the order relation on the number line, Educational studies in mathematics (2007) 66: 165-183.

This article combines historical study with classroom study, to examine the "relation between historical evolution of mathematical concepts and the process of their teaching and learning". They wanted to look at both whether there is a "parallelism" here and what such a parallel "between a creative mathematician of the past and a student learning mathematics in a modern classromm" might consist of.

The mathematical topic considered is the order relation on the number line - that is (for instance); is -4 greater than or smaller than -2? In section 2 of the article, the authors trace the development of mathematicians' thought on this. For instance, Newton wrote that "the greatest negative [root]" was the one "most remote [from zero]", that is -4 > -2. Also, Bolzano used the notation e<±1 for which we would write -1< e< 1.

The study of students posed three questions, of which the first one was:

"What are the solutions of the inequality x² > 9 when x ∈ ℜ?"

Lots of interesting answers were given to the questions. For instance, it did turn out that some students had the same basic ideas as Newton and Bolzano mentioned above.

The authors argue that there are parallelisms that could be exploited to

Reason enough to include history of mathematics in teacher education, in my opinion...

This article combines historical study with classroom study, to examine the "relation between historical evolution of mathematical concepts and the process of their teaching and learning". They wanted to look at both whether there is a "parallelism" here and what such a parallel "between a creative mathematician of the past and a student learning mathematics in a modern classromm" might consist of.

The mathematical topic considered is the order relation on the number line - that is (for instance); is -4 greater than or smaller than -2? In section 2 of the article, the authors trace the development of mathematicians' thought on this. For instance, Newton wrote that "the greatest negative [root]" was the one "most remote [from zero]", that is -4 > -2. Also, Bolzano used the notation e<±1 for which we would write -1< e< 1.

The study of students posed three questions, of which the first one was:

"What are the solutions of the inequality x² > 9 when x ∈ ℜ?"

Lots of interesting answers were given to the questions. For instance, it did turn out that some students had the same basic ideas as Newton and Bolzano mentioned above.

The authors argue that there are parallelisms that could be exploited to

"forsee possible persistent difficulties of the students; and to make teachers more tolerant towards their students' errors, by increasing their awareness that these errors and difficulties do not simply mean that "the student has not studied enough" but may have deeper epistemological roots which should be explored and understood thoroughly."but also

"become more tolerant towards non-conventional, but essentially correct, views of their students, even though they may be wrong or insufficient by modern standards of logical rigor and clarity; [...] encourage these and other students to express their possibly idiosyncratic views on specific mathematical problems and in this way to implicitly guide them towards conceiving mathematics as an exciting human activity".

Reason enough to include history of mathematics in teacher education, in my opinion...

## Thursday, July 30, 2009

## Wednesday, July 29, 2009

### Article: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking

Issue number 2 of Educational studies in mathematics 2007 was a special issue on history of mathematics. For some reason, I haven't read it in full before, but now is as good a time as any. In the following days I will blog about the articles in this special issue.

Luis Radford and Luis Puig: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking, Educational studies in mathematics (2007) 66: 145-164.

There are a few main concepts in this article, and by pointing to these, I will give an idea of what the article is about:

The Embedment Principle: "our cognitive mechanisms (e.g. perceiving, abstracting, symbolizing) are related, in a crucial manner, to a historical conceptual dimension ineluctably embedded in our social practices and in the signs and artifacts that mediate them."

Zones of proximal development of the culture: "new mathematical ideas are answers worked out in the historically situated zones of proximal development of their cultures".

The authors argue that "phylogenesis cannot recapitulate ontogenesis". "To learn algebra is not to construct the objects of knowledge (for they have already been constructed) but to make sense of them."

The article goes on to look at how this can be explored in an 8th grade class' work on fractional equations. The authors look at a word problem, in which students successfully writes the algebraic equation. In this form, the students recognize every part of the equation as a representation of a corresponding part of the word problem. However, when trying to solve the equation, the correspondence with the word problem is lost, and the operations take over as the "main personages".

This is an interesting point which I will try to keep in mind next time I teach equations.

The point about learners not "constructing" the objects of knowledge ("for they have already been constructed") is not entirely convincing to me, however. Why not "construct anew"? I agree that when students learn algebra, they do not on their own construct the objects of knowledge from nothing, but the word "construct" still makes sense to me. Or maybe I would prefer to think of it as a combination - partly (re)constructing and partly making sense?

(Obviously, I need to think more about this...)

Luis Radford and Luis Puig: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking, Educational studies in mathematics (2007) 66: 145-164.

There are a few main concepts in this article, and by pointing to these, I will give an idea of what the article is about:

The Embedment Principle: "our cognitive mechanisms (e.g. perceiving, abstracting, symbolizing) are related, in a crucial manner, to a historical conceptual dimension ineluctably embedded in our social practices and in the signs and artifacts that mediate them."

Zones of proximal development of the culture: "new mathematical ideas are answers worked out in the historically situated zones of proximal development of their cultures".

The authors argue that "phylogenesis cannot recapitulate ontogenesis". "To learn algebra is not to construct the objects of knowledge (for they have already been constructed) but to make sense of them."

The article goes on to look at how this can be explored in an 8th grade class' work on fractional equations. The authors look at a word problem, in which students successfully writes the algebraic equation. In this form, the students recognize every part of the equation as a representation of a corresponding part of the word problem. However, when trying to solve the equation, the correspondence with the word problem is lost, and the operations take over as the "main personages".

This is an interesting point which I will try to keep in mind next time I teach equations.

The point about learners not "constructing" the objects of knowledge ("for they have already been constructed") is not entirely convincing to me, however. Why not "construct anew"? I agree that when students learn algebra, they do not on their own construct the objects of knowledge from nothing, but the word "construct" still makes sense to me. Or maybe I would prefer to think of it as a combination - partly (re)constructing and partly making sense?

(Obviously, I need to think more about this...)

## Tuesday, July 14, 2009

### Teacher pay

Freakonomics writes about an idea for teacher pay: "What if teachers were paid based on the future income their students make."

It's interesting to see this idea. I wrote about the same idea in a Dagbladet article in 1999".

And I continued: "This would have the interesting consequence that a teacher that encourages his pupils to become stock brokers, can get a better salary than a teacher that gets his pupils to become nurses. And that is good - the society does award stock brokers better than nurses because stock brokers are more productive, doesn't it? But if the teacher himself becomes a role model for his pupils, so that they also become teachers, he would probably not get a raise - again exactly what society wants, isn't it?"

I do hope the irony shines through...

It's interesting to see this idea. I wrote about the same idea in a Dagbladet article in 1999".

And I continued: "This would have the interesting consequence that a teacher that encourages his pupils to become stock brokers, can get a better salary than a teacher that gets his pupils to become nurses. And that is good - the society does award stock brokers better than nurses because stock brokers are more productive, doesn't it? But if the teacher himself becomes a role model for his pupils, so that they also become teachers, he would probably not get a raise - again exactly what society wants, isn't it?"

I do hope the irony shines through...

## Sunday, July 12, 2009

### Ken ken

This summer, I tried Ken Ken for the first time. This is a puzzle related to Sudoku, but with even more connection to mathematics. Depending on the difficulty level, addition, subtraction, multiplication and division can be involved.

I think this could be a welcome diversion when there is time to kill – if a few pupils get hooked on Ken Ken, their mathematics skill may improve considerably. Have a look at KenKen.com.

I think this could be a welcome diversion when there is time to kill – if a few pupils get hooked on Ken Ken, their mathematics skill may improve considerably. Have a look at KenKen.com.

## Thursday, July 9, 2009

### A Mathematician Reads the Newspaper

I came across “A Mathematician Reads the Newspaper” by John Allen Paulos in a bookstore the other day, and had to pick it up, partly because of the blurbs on the back, for instance: “A wise and thoughtful book, which skewers much of what everyone knows to be true.” (Los Angeles Times)

I do understand that I’m not in the target group for the book. For me, quite a lot of what the book says is things that I already knew (which may not be surprising, as I have worked in mathematics education for 12 years, following my mathematics studies). The book is still interesting to read, however, as there are many interesting examples included. A bit more worrisome is that Paulos obviously struggled to fill a whole book, and at times the link between the newspaper and the mathematics he wants to discuss is a bit strained (as in the incidence matrices on page 189-).

However, many of the topics he covers are important and well worth revisiting: voting, chaos, coincidences, Ponzi schemes, statistical tests and meaningless precision, to name just a few.

(To exemplify the two last ones from the list: He is wary of journalists writing about polls without mentioning the margin of errors and of recipes stating the number of calories as 761, for instance.)

To conclude: it was an interesting book, but also a bit disappointing, given the hype on the cover.

I do understand that I’m not in the target group for the book. For me, quite a lot of what the book says is things that I already knew (which may not be surprising, as I have worked in mathematics education for 12 years, following my mathematics studies). The book is still interesting to read, however, as there are many interesting examples included. A bit more worrisome is that Paulos obviously struggled to fill a whole book, and at times the link between the newspaper and the mathematics he wants to discuss is a bit strained (as in the incidence matrices on page 189-).

However, many of the topics he covers are important and well worth revisiting: voting, chaos, coincidences, Ponzi schemes, statistical tests and meaningless precision, to name just a few.

(To exemplify the two last ones from the list: He is wary of journalists writing about polls without mentioning the margin of errors and of recipes stating the number of calories as 761, for instance.)

To conclude: it was an interesting book, but also a bit disappointing, given the hype on the cover.

## Friday, July 3, 2009

### InSITE: Didactics of ICT

Said Hadjerrouit of University of Agder had a talk on Didactics of ICT in Secondary Education. Hadjerrouit first described a lack of research on the didactics of ICT, and that he believed that there would be much to learn from the didactics of mathematics. I agree. In mathematics, there has been much research done on separate areas of mathematics (algebra, numbers, geometry, probability etc), and this has meant that the field of mathematics education has never become a purely theoretical subject, but has remained close to the actual teaching tasks. It seems to me that ICT does not have such a clear division into subfields, and that therefore, didactics of ICT may risk becoming too general and not too useful.

Hadjerrouit has taken part in small-scale projects with teachers, students and pupils in secondary school to learn ICT (not just use it), but from the presentation I got no clear sense of what the outcome of these was. Surely, the paper could be consulted for more on this.

Hadjerrouit has taken part in small-scale projects with teachers, students and pupils in secondary school to learn ICT (not just use it), but from the presentation I got no clear sense of what the outcome of these was. Surely, the paper could be consulted for more on this.

## Wednesday, July 1, 2009

### InSITE: Ground Rules in Team Projects

Janice Whatley’s talk titled Ground Rules in Team Projects was based on her PhD. She has created a computer system to automate parts of the process of deciding on “ground rules” when students are to start group work. The students answer a poll on which ground rules they find useful for their group, and based on the answers of the students in the group, a set of rules is suggested. The students concluded that it was useful to have a discussion on group rules early on, but it is a bit unclear for me whether the use of the computer system was a critical factor.

## Tuesday, June 30, 2009

### InSITE: ICTs and Network Relations

Inge Hermanrud of Hedmark University College in Norway had a talk on ICTs and Network Relations. He has been working with people from two different “distributed” organizations – the workplace security organization and part of the tax authorities. What they have in common is that there often comes up cases which look like nothing they have seen before, but which may be similar to something from another part of the country. It is, of course, a goal that similar cases are treated similarly across the country, so a certain amount of sharing of experiences is needed.

It turned out that people who had frequent relations used several communication channels, for instance both phone and email simultaneously. Those having infrequent relations, on the other hand, had problems with “multitasking” (doing other things while supposedly taking part in a discussion online) and individual drop-out.

As a lot of the Norwegian government is based on this kind of organizations, it would be important to improve the cooperation between different geographical regions. Therefore, Inge’s research is important.

It turned out that people who had frequent relations used several communication channels, for instance both phone and email simultaneously. Those having infrequent relations, on the other hand, had problems with “multitasking” (doing other things while supposedly taking part in a discussion online) and individual drop-out.

As a lot of the Norwegian government is based on this kind of organizations, it would be important to improve the cooperation between different geographical regions. Therefore, Inge’s research is important.

## Monday, June 29, 2009

### AT&T Mathematics Error

I'm currently following donttrythis on twitter - he apparently was quoted a price of .015 cents per kb when he signed up to a service, but was charged 1.5 cents per kb. When he complained, they said that "data is charged at .015 cents, or a penny and a half, per kb".

Apparently, the difference between .015 cents and 1.5 cents is lost on AT&T. (On the other hand, his more than 50,000 Twitter followers (at the moment: 63,856) seem to help him get them to accept his claim - even though I'm not sure they still understand the mathematics.)

This case is of course similar to a maths problem Verizon wireless had some time ago.

Apparently, the difference between .015 cents and 1.5 cents is lost on AT&T. (On the other hand, his more than 50,000 Twitter followers (at the moment: 63,856) seem to help him get them to accept his claim - even though I'm not sure they still understand the mathematics.)

This case is of course similar to a maths problem Verizon wireless had some time ago.

## Sunday, June 28, 2009

### InSITE: Meaningful Learning in Discussion Forums

Raafat Saade and Qiong Huang’s paper was on Meaningful Learning in Discussion Forums. In online courses with maybe 1300 students and where use of online discussion forums (ODF) is mandatory, there is obviously needed some smart strategy to evaluate the participation.

One could look at what the students do in various levels of detail. At the macro level, one could look at numbers showing each student’s 1) participation and 2) interaction. At the micro level, one could analyze 3) the interaction or 4) cognition. One of the claims in the talk was that, from a learning point of view, cognition is the most interesting of these, while participation is obviously the easiest one to measure.

However, in my context I think that I can find a measure that is both interesting and (quite) easy to measure. I will come back to that later if it proves useful.

I will surely have to read this article closely, as the topic is so close to one of my projects for next semester.

One could look at what the students do in various levels of detail. At the macro level, one could look at numbers showing each student’s 1) participation and 2) interaction. At the micro level, one could analyze 3) the interaction or 4) cognition. One of the claims in the talk was that, from a learning point of view, cognition is the most interesting of these, while participation is obviously the easiest one to measure.

However, in my context I think that I can find a measure that is both interesting and (quite) easy to measure. I will come back to that later if it proves useful.

I will surely have to read this article closely, as the topic is so close to one of my projects for next semester.

## Friday, June 26, 2009

### InSITE: Departmental Collaboration for the Community

Joseph Chao gave a talk about Cross-Departmental Collaboration for the Community. The talk centered on a project where programmers and technical writers work together in projects with real-world customers to solve real-world problems. One major problem detected was – unsurprisingly – that the programmers was so busy programming to meet the deadline that they didn’t have time to discuss documentation with the technical writers. Thereby, the technical writers were left to themselves trying to figure out how the program worked.

It’s an interesting project, although not very relevant to my setting.

It’s an interesting project, although not very relevant to my setting.

### HPM Newsletter 71

The new issue of the HPM Newsletter (no. 71) was published today. It is available from the HPM website.

In this issue:

- a note on Giorgio T. Bagni, an Italian colleague who tragically died earlier this month

- information on new books: Geir Botten's book on the first Norwegian textbook in mathematics and three books related to Paulus Gerdes' work

- the first announcement for the ESU 6, which will be in Vienna, Austria 19th-23rd of July 2010

- information on how Narges Assarzadegan uses history of mathematics in her work

- a call for possible collaboration on history of mathematics for primary school students.

In addition, there are the usual stuff (information on new articles in the field, interesting links and announcements of events).

In this issue:

- a note on Giorgio T. Bagni, an Italian colleague who tragically died earlier this month

- information on new books: Geir Botten's book on the first Norwegian textbook in mathematics and three books related to Paulus Gerdes' work

- the first announcement for the ESU 6, which will be in Vienna, Austria 19th-23rd of July 2010

- information on how Narges Assarzadegan uses history of mathematics in her work

- a call for possible collaboration on history of mathematics for primary school students.

In addition, there are the usual stuff (information on new articles in the field, interesting links and announcements of events).

## Wednesday, June 24, 2009

### InSITE: Design Alternatives for a MediaWiki

Sumonta Kasemvilas had a talk about Design Alternatives for a MediaWiki. Earlier attempts at using Wikis in higher education have made apparent that there are some challenges. For instance, only using watchlists to keep track of changes has turned out not to be sufficient, students feel that the wiki is not easy to navigate and an overall “summary table” of what is going on is missing. Her efforts to enhance MediaWiki to make it better suited for higher education seems promising.

## Monday, June 22, 2009

### InSITE: Learning objects vs informing objects

A panel discussion with participants Robert Skoriba, Alex Koohang, Fred Kohun and Richard Will discussed the novel concept of “informing object”. As far as I understood, the main problem with the concept “learning object” was considered to be that one should not call something a “learning object” before one has established that learning has taken place. I never got round to ask this question, but I suppose that the same goes for “informing object” – that some sort of informing has to take place before that name is warranted.

To me, who is not familiar with all the preexisting concepts, the discussion soon turned confusing. Someone asked how “informing object” would relate to “information object” and “knowledge object”. And all the time, what the “object” would be, was a bit unclear.

I was, however, made aware of the depository of learning objects called Merlot, which I should look at.

To me, who is not familiar with all the preexisting concepts, the discussion soon turned confusing. Someone asked how “informing object” would relate to “information object” and “knowledge object”. And all the time, what the “object” would be, was a bit unclear.

I was, however, made aware of the depository of learning objects called Merlot, which I should look at.

## Saturday, June 20, 2009

### InSITE: Learning Objects and OOP

Namdar Mogharreban’s talk Regaining the ‘Object’ of Learning Objects, tried to link the concept of “learning objects” back to the idea of object oriented programming (OOP). He envisaged a model for creating learning objects based on a Learning Pod, which is, in the language of OOP, be a class. A learning object should be an instance of a Learning Pod with certain properties, values and behaviours (for instance the age of the learner, bandwidth, learning style, learning history and bandwidth).

I am a sceptical person, and often my first thought when confronted with a theoretical framework is “does it make sense in real life?” That is also my question here. The complexity of the concepts one wants the learner to learn, means, in my mind, that the specifications needed to get a useable learning object will be very extensive. For instance: if a pupil works on a multiplication algorithm for two-digit numbers, which learning object that will make sense to him and help him will depend of his understanding of the numeral system, on whether he has worked on the area model of multiplication, on whether his parents have shown him an algorithm already, on whether he knows the basic multiplication facts and so on. Thus, answers to all of these questions will need to be parameters for the Learning Pod when making a new learning object.

A counterargument might be that I compare the learning object to a perfect teacher who knows all his pupils’ backgrounds and has the time to individualize everything. Perhaps that is unfair, and the learning objects created in this way should rather be compared to the imperfect teaching many pupils get today, because they are taught by undereducated and overworked teachers. But still, I want to see the Learning Pod in action before I hoist the flag.

I am a sceptical person, and often my first thought when confronted with a theoretical framework is “does it make sense in real life?” That is also my question here. The complexity of the concepts one wants the learner to learn, means, in my mind, that the specifications needed to get a useable learning object will be very extensive. For instance: if a pupil works on a multiplication algorithm for two-digit numbers, which learning object that will make sense to him and help him will depend of his understanding of the numeral system, on whether he has worked on the area model of multiplication, on whether his parents have shown him an algorithm already, on whether he knows the basic multiplication facts and so on. Thus, answers to all of these questions will need to be parameters for the Learning Pod when making a new learning object.

A counterargument might be that I compare the learning object to a perfect teacher who knows all his pupils’ backgrounds and has the time to individualize everything. Perhaps that is unfair, and the learning objects created in this way should rather be compared to the imperfect teaching many pupils get today, because they are taught by undereducated and overworked teachers. But still, I want to see the Learning Pod in action before I hoist the flag.

## Thursday, June 18, 2009

### InSITE: Taxonomy as a Vehicle for Learning

Not surprisingly, I was present at Brodahl and Smestad’s talk on A Taxonomy as a Vehicle for Learning.

In this talk, we described the development of a taxonomy (classification system) of VaniMaps (a subgroup of learning objects that we have defined ourselves – see the link above for details) and also described a teaching sequence in which this taxonomy was used to foster discussion in a class of (mainly) teacher students. The teaching sequence used polling (electronic questionnaire) as a tool for collecting the opinions of the students. When students were confronted with their different opinions, they got curious about what the reasons might be for the variations – thereby leading to discussions.

It is terribly hard to describe all of this in just 20 minutes, especially since both the context of teacher education, the basics from mathematics education research as well as the theory of learning objects must be assumed to be unknown for some of the listeners. I think it went quite well, however.

In this talk, we described the development of a taxonomy (classification system) of VaniMaps (a subgroup of learning objects that we have defined ourselves – see the link above for details) and also described a teaching sequence in which this taxonomy was used to foster discussion in a class of (mainly) teacher students. The teaching sequence used polling (electronic questionnaire) as a tool for collecting the opinions of the students. When students were confronted with their different opinions, they got curious about what the reasons might be for the variations – thereby leading to discussions.

It is terribly hard to describe all of this in just 20 minutes, especially since both the context of teacher education, the basics from mathematics education research as well as the theory of learning objects must be assumed to be unknown for some of the listeners. I think it went quite well, however.

## Tuesday, June 16, 2009

### InSITE-Connect: A Model for Dynamic Interdisciplinary Interaction

The keynote speaker of the InSITE 2009 conference is Gerry McKiernan. He gave a talk on social networking and Web 2.0 technology and how that could be used to improve the interaction within such a network as Informing Science.

One major problem with research networks today is that they are "paper-based", in the sense that the main output of the researchers are papers published in journals and presented at conferences. This is a very slow discussion, in which you will not get an "answer" until someone cites you and discusses your work. Of course, there are also lots of oral discussions in conferences, as well as lots of email discussions and so on. However, these are not archived, so in a sense they are here today, gone tomorrow.

Gerry McKiernan discussed several different networks already existed to (as I understood him) motivate Informing Science to make a move to some form of Web 2.0 technology. He mentioned several niche online social networks. One of them is ResearchGATE, which is a "professional network for scientists". SciSpace.net is a "collaborative network for scientists", which you need an invitation to join. Social Science Research Network is created to facilitate early distribution of research results.

Then there is Ning, of course. There are lots of communities on Ning, including the Norwegian Del og bruk, a sharing network for teachers which already has almost 2000 members.

One particular important sentence which could have been a motto for his whole talk, is this: "It is not about publication, it's about ideas." With all the silly index systems and systems for publication points, it is easy to forget this. In today's academia, it's better to have one moderately good idea that you can write five sufficiently different articles about, than to have one excellent idea that you only write one article about before moving on to other areas. The systems ask us to ask ourselves first if we could write more articles on the same old thing rather than doing the hard work of thinking about something new. The social networks, on the other hand, are not about publishing in the publication point way of thinking, but it is about distributing ideas to get our ideas further. Therefore, I wish I could ignore the publication points and just be Web 2.0-oriented...

One major problem with research networks today is that they are "paper-based", in the sense that the main output of the researchers are papers published in journals and presented at conferences. This is a very slow discussion, in which you will not get an "answer" until someone cites you and discusses your work. Of course, there are also lots of oral discussions in conferences, as well as lots of email discussions and so on. However, these are not archived, so in a sense they are here today, gone tomorrow.

Gerry McKiernan discussed several different networks already existed to (as I understood him) motivate Informing Science to make a move to some form of Web 2.0 technology. He mentioned several niche online social networks. One of them is ResearchGATE, which is a "professional network for scientists". SciSpace.net is a "collaborative network for scientists", which you need an invitation to join. Social Science Research Network is created to facilitate early distribution of research results.

Then there is Ning, of course. There are lots of communities on Ning, including the Norwegian Del og bruk, a sharing network for teachers which already has almost 2000 members.

One particular important sentence which could have been a motto for his whole talk, is this: "It is not about publication, it's about ideas." With all the silly index systems and systems for publication points, it is easy to forget this. In today's academia, it's better to have one moderately good idea that you can write five sufficiently different articles about, than to have one excellent idea that you only write one article about before moving on to other areas. The systems ask us to ask ourselves first if we could write more articles on the same old thing rather than doing the hard work of thinking about something new. The social networks, on the other hand, are not about publishing in the publication point way of thinking, but it is about distributing ideas to get our ideas further. Therefore, I wish I could ignore the publication points and just be Web 2.0-oriented...

## Monday, June 15, 2009

### Wikis in teacher education

In one project, I will try to develop a wiki for students in teacher education, which should function as a “Wikipedia” for teacher education. By that, I mean that the articles should be within the context of teacher education. For instance, an article on multiplication would not only say what multiplication is or how it is done, but also include examples of pupils’ misconceptions, ideas of teaching materials and so on.

The wiki ("eleviki"), is currently on a very early stage, but I hope to start using it with students in August.

A lot more could be said on this and my other main interests, but this should suffice for now.

The wiki ("eleviki"), is currently on a very early stage, but I hope to start using it with students in August.

A lot more could be said on this and my other main interests, but this should suffice for now.

## Sunday, June 14, 2009

### Workshops at InSITE 2009

The InSITE 2009 conference started off on Friday with some workshops (the day before the official opening of the conference).

Tim Ellis and Bill Hafner gave a talk on “Security and the Professor”. They looked at security issues for the professor in the modern age. Only some years ago, most professors would only use the PC at work, thereby leaving all security concerns to the institution. Nowadays, however, work is brought to the home PC, to a laptop or mobile device using wireless networks and to internet cafes everywhere. That makes a whole range of new security problems which we should be aware of, particularly if we are working on confidential information, such as student grades. Ellis and Hafner analyzed the security risks from three perspectives: the environment, the network and the computer. It was an interesting reminder, although personally I tend not to work on confidential materials, luckily.

William H. “Bill” Burkett talked about “Academic Uses of Second Life and other emerging/converging technologies in your classroom”, with rather more weight on the “other” part. Personally, I’m not too convinced about the use of Second Life – it needs to be used quite a lot to be worth the fuss of making all students install the program (on every computer they use) and to register and then design their avatar. However, he also gave the link to a blog with some links to interesting resources.

Then there was a workshop presenting the Informing Science Institute’s journals and books. There are no less than eight different journals connected to the institute, which all publish all articles online free of charge (as well as on paper for a fee). Moreover, the institute publishes books that are also available online (but also on paper). I do like this policy a lot, as it means that the ideas will be more easily available to others.

Thereafter, there was a panel on the topic “What is Informing Science?” One major difference of opinion in the panel was whether it would be better to stick to the original idea of Eli Cohen, which pointed out three main components: a client, a delivery system and an informing environment. Actually, Eli Cohen himself disagreed with this, thinking that the field had evolved so that we could no longer simply stick to the original, founding ideas. Obviously, for a newcomer like me, it is far too early to have opinions on this.

Finally, Linda Knight and Terry Steinbach held a very useful workshop titled “Creating Research Manuscripts for Publication”. Including editor and reviewer experience, they gave helpful advice on how to work on a research article. For instance, they had a checklist of ten points – if you follow these, you will be sure to get your paper published. They also showed an interesting diagram of an article, showing which links should be there. For instance, obviously any problem you mention in the introduction should be revisited in the end. While much of what they said sound like common sense, it is still easy to miss one or more of them when you are busily writing an article. So I think it will be nice to look back to this workshop in future.

Tim Ellis and Bill Hafner gave a talk on “Security and the Professor”. They looked at security issues for the professor in the modern age. Only some years ago, most professors would only use the PC at work, thereby leaving all security concerns to the institution. Nowadays, however, work is brought to the home PC, to a laptop or mobile device using wireless networks and to internet cafes everywhere. That makes a whole range of new security problems which we should be aware of, particularly if we are working on confidential information, such as student grades. Ellis and Hafner analyzed the security risks from three perspectives: the environment, the network and the computer. It was an interesting reminder, although personally I tend not to work on confidential materials, luckily.

William H. “Bill” Burkett talked about “Academic Uses of Second Life and other emerging/converging technologies in your classroom”, with rather more weight on the “other” part. Personally, I’m not too convinced about the use of Second Life – it needs to be used quite a lot to be worth the fuss of making all students install the program (on every computer they use) and to register and then design their avatar. However, he also gave the link to a blog with some links to interesting resources.

Then there was a workshop presenting the Informing Science Institute’s journals and books. There are no less than eight different journals connected to the institute, which all publish all articles online free of charge (as well as on paper for a fee). Moreover, the institute publishes books that are also available online (but also on paper). I do like this policy a lot, as it means that the ideas will be more easily available to others.

Thereafter, there was a panel on the topic “What is Informing Science?” One major difference of opinion in the panel was whether it would be better to stick to the original idea of Eli Cohen, which pointed out three main components: a client, a delivery system and an informing environment. Actually, Eli Cohen himself disagreed with this, thinking that the field had evolved so that we could no longer simply stick to the original, founding ideas. Obviously, for a newcomer like me, it is far too early to have opinions on this.

Finally, Linda Knight and Terry Steinbach held a very useful workshop titled “Creating Research Manuscripts for Publication”. Including editor and reviewer experience, they gave helpful advice on how to work on a research article. For instance, they had a checklist of ten points – if you follow these, you will be sure to get your paper published. They also showed an interesting diagram of an article, showing which links should be there. For instance, obviously any problem you mention in the introduction should be revisited in the end. While much of what they said sound like common sense, it is still easy to miss one or more of them when you are busily writing an article. So I think it will be nice to look back to this workshop in future.

## Friday, June 12, 2009

### ICT in mathematics education

I have been interested in ICT in mathematics education for many years.

The most important result so far is the paper A Taxonomy as a Vehicle for Learning (written with Cornelia Brodahl of University of Agder) which will be presented at a conference this week.

The most important result so far is the paper A Taxonomy as a Vehicle for Learning (written with Cornelia Brodahl of University of Agder) which will be presented at a conference this week.

## Wednesday, June 10, 2009

### Gay issues in teacher education

Last year, the Norwegian newspaper Blikk did a little research on what Norwegian teacher education included on gay issues. The conclusion was depressing: "Ingen undervisning om homofili” (”No teaching about homosexuality"), claiming that the only teaching about homosexuality is the three-hour lecture that I give every year.

I don’t believe that the situation is quite this terrible, but I do believe that there is little in the way of systematic teaching in this area. Luckily there are some efforts. There are textbooks to come, and some information is provided by the authorities. Moreover, I am editing a “resource” for lecturers who want to include the topic in their teaching (on behalf of the group FHiOHL, and with money from the rector of the university). The second edition will be available this autumn.

Just a few weeks ago, a new study showed that almost half of Norwegian gay, male 10th graders are bullied in school. There is plenty of work to do.

I don’t believe that the situation is quite this terrible, but I do believe that there is little in the way of systematic teaching in this area. Luckily there are some efforts. There are textbooks to come, and some information is provided by the authorities. Moreover, I am editing a “resource” for lecturers who want to include the topic in their teaching (on behalf of the group FHiOHL, and with money from the rector of the university). The second edition will be available this autumn.

Just a few weeks ago, a new study showed that almost half of Norwegian gay, male 10th graders are bullied in school. There is plenty of work to do.

## Tuesday, June 9, 2009

### Equations in WolframAlpha

Tor Espen writes about how WolframAlpha can be used while working on equations in high school. WolframAlpha not only solves the equations, but can give a stepwise explanation.

Of course, there's been software around for a long while which can do this, but to have it in the same search engine that you use for every other task, is very helpful. (And if it's in WolframAlpha today, it will be in Google tomorrow.)

Of course, there's been software around for a long while which can do this, but to have it in the same search engine that you use for every other task, is very helpful. (And if it's in WolframAlpha today, it will be in Google tomorrow.)

## Sunday, June 7, 2009

### History of mathematics in mathematics education

One of my main fields of interest is how history of mathematics may enrich mathematics teaching. It’s only natural that I have this interest, as I have a Masters degree on history of mathematics.

My main work in the area has been a study of Norwegian textbooks, a study of the historical content in the TIMSS 1999 Video Study, and an interview study on teachers’ conceptions of history of mathematics. I have also been a co-editor of the HPM Newsletter since 2004.

I believe that the history of mathematics may enrich mathematics teaching in important ways, but that it’s currently not easy for the individual teacher to start including history of mathematics, and my main work so far has been to analyse the situation to try to see what can be done to improve the situation in future.

My main work in the area has been a study of Norwegian textbooks, a study of the historical content in the TIMSS 1999 Video Study, and an interview study on teachers’ conceptions of history of mathematics. I have also been a co-editor of the HPM Newsletter since 2004.

I believe that the history of mathematics may enrich mathematics teaching in important ways, but that it’s currently not easy for the individual teacher to start including history of mathematics, and my main work so far has been to analyse the situation to try to see what can be done to improve the situation in future.

### Four main areas

In this blog, I will write about anything connected to my work, but it will tend to include more about my main research interests, which are:

1) History of mathematics in mathematics education

2) Gay issues in teacher education

3) ICT in mathematics education

4) Wikis in teacher education

To get this blog started, I will, in the days to come, write short posts on these four topics to introduce them.

1) History of mathematics in mathematics education

2) Gay issues in teacher education

3) ICT in mathematics education

4) Wikis in teacher education

To get this blog started, I will, in the days to come, write short posts on these four topics to introduce them.

## Sunday, May 31, 2009

### InSITE 2009

Currently, I'm preparing for the InSITE 2009 conference, which will this year take place in Macon, Georgia, US from June 12th to June 15th. I'm giving a talk called A Taxonomy as a Vehicle for Learning with my colleague Cornelia Brodahl from the University of Agder.

This is my first InSITE conference. It seems to cover a wide area, judging from the titles of the talks. (But one should NEVER judge from the titles of the talks...)

I will probably be blogging during the conference on the most interesting talks.

This is my first InSITE conference. It seems to cover a wide area, judging from the titles of the talks. (But one should NEVER judge from the titles of the talks...)

I will probably be blogging during the conference on the most interesting talks.

## Monday, May 25, 2009

### ESU 6

I have received notice that ESU 6 (European Summer University on the history and epistemology of mathematics education) will be in Vienna (Austria) from 19 to 23 July 2010. (More information will be available in the next HPM Newsletter.)

Personally, I have enjoyed taking part in the ESU 4 in Uppsala and the ESU 5 in Prague, and am looking forward to going to Vienna for this conference.

Personally, I have enjoyed taking part in the ESU 4 in Uppsala and the ESU 5 in Prague, and am looking forward to going to Vienna for this conference.

## Friday, May 22, 2009

### Triangles in triangles

In an in-service course for teachers at “mellomtrinnet” (pupils age 9-13), I gave the participants the following exercise in GeoGebra: draw a triangle. Find the midpoint of each side of the triangle. Use these three midpoints as corners for a new triangle.

The somewhat interesting fact is that the new triangle will have the same shape as the original one. This can of course be easily proved, but at this level, the pupils could also just explore the triangles in GeoGebra and see that it seems right.

I then asked the teachers what else they could use this quite simple figure for. Are other mathematical concepts involved? Indeed, there are lots of things to be found:

You can find all the angles of the original triangle repeated several times inside this figure. In particular, you can find them repeated next to each other, so that you see that the three angles taken together is 180 degrees.

You can observe that the original triangle has now been divided into four congruent triangles. Thus, dividing each of the original sides into two, has lead to dividing the area into four. This could be a useful reminder for the pupils. If you also discuss the (omkrets) of each of the new triangles, you will find that they are half of the original (omkrets). Again: a useful reminder of the difference in the behaviour of area and (omkrets).

Of course, there are also parallel lines involved.

If you repeat the operation (that is, find the midpoints of the sides of the new triangle and create yet another triangle using these midpoints as corners) several times, you will get in touch with the idea of fractals. Moreover, you can keep discussing the lengths and areas of each triangle. (Repeating this operation many, many times is very simple in GeoGebra if you create a new tool for this job.)

Thus, by means of a very simple drawing, many important concepts from the curriculum can be discussed. Of course, formal proofs can also be introduced if the teacher wishes.

Do you see other interesting aspects of this drawing – or want to draw my attention to similar examples – please leave a comment below.

## Monday, May 18, 2009

### New blog!

Welcome to this new blog!

Previously, I have had a blog called Bjørn's maths blog where I have blogged about mathematics. This turned out to be a bit too narrow, as I would like to blog also on other topics related to my job, such as teacher education in general. Therefore I've started this blog, which will include both mathematics-related and other subjects.

Enjoy!

Previously, I have had a blog called Bjørn's maths blog where I have blogged about mathematics. This turned out to be a bit too narrow, as I would like to blog also on other topics related to my job, such as teacher education in general. Therefore I've started this blog, which will include both mathematics-related and other subjects.

Enjoy!

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