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:

"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."

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.

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:

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.

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

"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.

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.

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.

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.