text:
Regina D Möller
email: regina.moeller@uni-erfurt.de

Peter Collignon
email: peter.collignon@uni-erfurt.de

The concept of infinitydifferent meanings through the centuries

Keywords: Infinity, countable set, denumerable set, very small/large numbers

The concept of infinity and its use is one with different meanings through the centuries within various contexts reflecting mathematical historical development. This development is scarcely clear to pupils during school time and rarely stressed in teacher education although it offers a lot of potential to understand mathematics. The field of arithmetic is one example in which to study infinity within a range that student teachers are able to understand and that is useful for and in their future teaching. The paper focusses on the potential of this concept within the arithmetic field using an original article of Cantor and on examples, also from Hilbert, that stress different counting methods and various illustrations of infinity.

Introduction

Like many other mathematical concepts ideas of infinity developed and diversified through the centuries. Well known is its appearance within the Elements of Euclid within the ninth chapter: There are more prime numbers than any given number of prime numbers, cf. Euclid (1997, IX, §20). This statement answers the question, if there is an end of prime numbers. The proof uses the fact that the product of arbitrary prime numbers added by 1 has a new prime number in its prime factorization. That is, to conceptualize the concept of infinity of prime numbers Euclid uses the idea that there is always another one apart from the given ones which has a somewhat operative aspect. Also within geometric contexts there are ideas about infinity regarding the extensions of space, planes and lines as well as the number of points on lines and figures. There are considerations of the behavior of parallel lines regarding infinity, like for example the parallel axiom states.

Already these examples show that there exist different meanings of infinity depending on the various mathematical objects. Also, the meanings are verbalized without formulization. Doubtless does the use of very large numbers within everyday experiences come close to a sense of infinity. For example, the addition of two numbers, one of which is very much larger than the other, form a sum that does not differ very much from the large number in terms of the relative error concept.

Infinity and the use of it deserves a closer look because terms like "infinite", "endless" and "unlimited" have a colloquial meaning that sometimes provides a reasonable starting point for the understanding of abstract patterns and sometimes not. In some instances infinity incorporates the imagination of a very large extension, endlessness is for many just a word for a very large but still finite set of objects or a very large extension like the ocean.

First-year students often show a vague notion of the term "infinity", an observation that was done also by [START_REF] Woerner | Zum Verständnis des Unendlichkeitsbegriff im Mathematikunterricht -Eine empirische Untersuchung[END_REF]. She even points out that a thorough understanding of infinity is neither a goal nor is it a step towards understanding mathematics. [START_REF] Dötschel | Zum Verständnis der Unendlichkeitsbegriffs im Mathematikunterricht[END_REF] even finds that the understanding of infinity does not vary much between teacher students and pupils of secondary level. In school they learned the lemniscate symbol "" and used it associated with the limit of sequences, series andat the bestthe differential quotient. At least in principle they know that it is not allowed to use this symbol like a number or a variable in all respects (like it was used partly in the 18th century), but one can observe a high degree of uncertainty. For instance, what does it mean when the differential quotient is interpreted in terms like "zero divided by zero"? And what is the outcome? Standard lectures like the ones for Analysis and Linear Algebra usually do not change the perspective and are continued with refined conceptions of limits and the definition of concepts like that of an infinitely dimensional space. Concerning the cardinality of sets there remains often the sketchy notation A =  or the like.

Precisely because counting, the determination of a number, is a fundamental concept at all school levels, it is surprising that the issue of cardinality seems to be somewhat neglected. Nevertheless, it is not too difficult to provide a base of knowledge due to Georg Cantor. In many cases original mathematical treatises are definitely unsuitable within teacher education, but there are notable exceptions. One of these is the article of [START_REF] Cantor | Beiträge zur Begründung der transfiniten Mengenlehre[END_REF], which is understandable in large parts. The reason for this is that students are able to acquire understanding with supporting examples on the side of the teacher which will be shown later in the paragraph.

Our paper describes the used examples during the seminar held in the summer term 2016 together with some details around the concept of infinity.

Various ways towards an understanding of infinity

The mathematical education of future elementary math teachers at the University of Erfurt includes among mathematical survey lectures a seminar on the basic principles of arithmetic and algebra. One of the goals of this seminar is to improve the student's understanding of (natural) numbers and their properties since this field will constitute one of the bases of their future teaching. During the seminar students are often encountering the question in how far they have an understanding of the set of natural numbers being infinite. The upcoming discussions circle around the question how one could find out "what is true". And the discussion ends with the question why. The phenomenon of infinite many natural numbers very often brought about an astonished attitude on the part of the students who pondered about the reasoning of Aristotle: There will be always one number that can be added.

In our seminar there was a focus on aspects of numbers, especially of the natural numbers. Apart from the Peano axioms, the aspect of ways of counting provides a reliable foundation, especially since the cardinalities of sets may be included. This approach promotes a formalization due to the concept of bijective mappings, which is of value in its own. The natural numbers can be regarded as cardinal numbers of finite sets. But what do students know about the cardinality of the set of the natural numbers? Being asked, students claim "infinite" and denote the lemniscate symbol which they know from limits of sequences or functions. In interviews they show an obvious uncertainty about the arithmetical behavior of the object "infinite". In case of doubt they often suggest treating it like a "usual" number.

We feel responsible for giving a brief insight into the cultural heritage of the different approaches dealing with "infinity". In particular we seek to show with what kind of caution Euclid, and Cantor too, got closer to proper descriptions, depending on the actual contexts. Especially we are interested in giving insight into the richness of mathematical thoughts and ideas: "It is possible to regard the history of the foundations mathematics as a progressive enlarging of the mathematical universe to include more and more infinities" (Rucker, 1982, p. 2). With regard to Cantor, we know that "... soon obtained a number of interesting results about actually infinite sets, most notably the result that the set of points on the real line constitutes a higher infinity than the set of all natural numbers. That is, Cantor was able to show that infinity is not an all or nothing concept: there are degrees of infinity." (Rucker, 1982, p. 9) There are a lot of ways how to understand mathematical statements. Some point out their proof, some stress their genetic development, some point out their formal argument. Our actual understanding of infinity allows us to give statements like: the set of natural numbers or the set of natural numbers between 0 and 100. The first is an infinite set, the second is a finite one. Stressing the idea of a potential infinity which we could not grasp as a solid concept, we help ourselves by a stepwise approach knowing that we will never succeed. This very constructive standpoint or procedure permits a very simple activity and that is adding one, again and again: |, | |, | | |, … In this manner one can distinguish finite and infinite sets. For the first set the procedure ends with a certain number, for the second one there is no certain last number and it becomes clear that the procedure never ends. In both cases the counting is mathematically a 1-1-correspondence.

The different meanings of the term infinity show the richness of mathematics and its historical development. Needless to say that mathematical history does not develop in a regular and uniform way (Dieudonné, 1985, p. 16). Some epoch does not show any development in a field, in some there is a continuous change because of new developments. The fact that we use the word infinity the way we do with numbers goes back to [START_REF] Cantor | Beiträge zur Begründung der transfiniten Mengenlehre[END_REF]. It was the upcoming of new ideas, e.g. the idea of a set that changed the understanding of infinity.

How very much different this meaning is in contrast to the "old" Greek meaning shows when student teachers learn about it in their first mathematical lectures: infinity is hard to grasp and the use of it shows that school mathematics does not at all build a proper foundation. Because of its lack it is even more important to build a solid understanding during mathematical studies especially for student teachers as there are many potential links to basic notions of counting in their future teaching.

German mathematical education often refers to three basic experiences ("Grunderfahrungen"), by [START_REF] Winter | Mathematikunterricht und Allgemeinbildung[END_REF]:

• perceiving phenomena of nature, society and culture;

• knowing (and appreciating) mathematical issues, represented by language, symbols, images and formulas;

• acquiring heuristic competencies.

We like to refer to [START_REF] Winter | Mathematikunterricht und Allgemeinbildung[END_REF] because he stresses a connection between everyday life experiences, heuristics and beginning formalization. In order to get aware of basic experiences and deepen the understanding there is a strategy necessary that gets students involved. [START_REF] Kattou | Teachers' perceptions about infinity: A process or an object?[END_REF] points out:

In particular, academic programs offered to teachers should include mathematical knowledge regarding to infinity in combination with instructional approaches related to the concept. A proposed teaching approach could include the following steps: presentation with several typical tasks aimed at uncovering teachers' intuitions about the concept, discussion about infinity's applications in real life, introduction of the formal definition of infinity and the two aspectspotential and actual-and attempt to distinguish them in examples. (Kattou et al.2009).

Within this this context we pinpoint the following aspects:

1. The notion of infinity changed its meaning through the centuries. In the late 19 th century the notion of aleph 0, aleph 1 and so forth came up.

2. The way infinity appears in mathematical textbooks follows the idea of Freudenthal's antididactical inversion. It is common to introduce infinity by using the lemniscate symbol, mostly just informing about it. The mathematical developments are neglected.

3. Some examples of infinite sets can be solved with simple steps used with finite sets. This presents an approach with a low barrier to student teachers and enhances their understanding of infinity.

Our didactical approach is influenced by [START_REF] Vollrath | Begriffsbildung als schöpferisches Tun im Mathematikunterricht[END_REF] who proposed a phase model showing the process of understanding mathematical concepts:

intuitive and content-related  formal / integrated  critical

We therefore stress the finding of variations of standard examples and of counting strategies on the side of the students. The integration of Cantor's text provides some formalism and fostered discussions about the historical circumstances which were not controversial, that is the conflict between Cantor and Kronecker e.g.

The following paragraph presents examples that proved useful within elementary school teacher education.

Methods of counting

In many cases it turns out difficult to provide an original text to students with the expectation of an adequate comprehension. But just mathematical topics that lead to very fundamental issues may prove appropriate in order to their connection with intuition and imagination. Cantor (1895) develops a concept of elementary set theory, which includes transfinite cardinalities and their arithmetic properties. During teaching it became evident, that important parts of this text are quite understandable and can be an opportunity to discuss an historical treatise and express own reflections.

Before we start investigating into various ways of counting infinite sets we observe that there is no uniform definition of the concept infinity. The word occurs as an adjective to characterize sets especially. We concentrate therefore on the arithmetic field.

The following sections present a couple of examples that may foster the understanding of infinite sets.

Counting as one-to-one correspondence

The concept of a set, as introduced from Cantor (1895), surely fits into these frameworks. To him we owe the so-called "naïve" definition of a set.

By an "aggregate" we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought. These objects are called the "elements" of M.

A counting or numerating of a finite set M with exactly n elements means, that every number 1, 2, 3, …, n is assigned to exactly one element of the set. This is linked to the concepts of maps and functions and more over bijectivity.

The set ℚ is countably infinite

This follows out of a scheme in which every positive rationale number shows one time and is arranged like this: The way the scheme is counted goes back to Cantor and is called the "diagonal method".

1/1 1/2 1/3 1/4 1/5 1/6 … 2/1 2/2 2/3 2/4 2/5 2/6 … 3/1 3/2 3/3 3/4 3/
Following the presentation of the scheme students were invited to vary it and write down their proposals. Are there other suitable paths? What do they have in common? Furthermore, how could repetitions of numbers be avoided? In the scheme above every positive rational number is repeated infinitely. Does this cause problems? What options do we have to be represented by a reduced table of fractions? Is this already an indicative of the countability of even "larger" sets? After all most students could design various methods for counting even all rational numbers, for instance by designing spiral paths or the like.

The set ℝ is uncountable

Suppose that there is an enumeration ..... , , , ,

4 3 2 1     of the interval [0, 1[,
          [0, 1[ such that       1 , 7 1 , 1 ii ii i if if   
for all positive integers i. Obviously, the representation of  possesses at least one decimal digit that differs from i, namely, the i-th digit. Therefore  cannot occur in the enumeration above, which is inevitably incomplete. Now, if the given interval is already uncountable, then all the more the real numbers are. This scheme originates from Cantor, too, and is called the second diagonal method.

To promote an adequate understanding, students did vary this scheme in a written form, also regarding other b-adic representations. At this point, the fundamental significance of place value systems in general is to be clarified. During teaching lessons students were encouraged to replace the digits by other symbols such as letters or notes from sheet music etc., and it has become clear, that the relating interpretations ("the entity of 'texts' is uncountable") can foster an adequate understanding in the sense that students are able to make a transfer.

Hilbert's Hotel

The cardinality of the set of the natural numbers is denoted by ℵ0. In set theory several properties of this first transfinite cardinality are elaborated, as there are 1 + ℵ0 = ℵ0, n + ℵ0 = ℵ0 for any n  ℕ, as well as 2ℵ0 = ℵ0 and nℵ0 = ℵ0, again for any natural number n. To illustrate this, the thought experiment of Hilbert's hotel is helpful: Suppose that there is a hotel with an unlimited number of single rooms, which are numbered according to the natural numbers. The hotel is fully occupied and one other person is knocking on the door. Will the hotel be able to accommodate this person, too? In the classical version each present guest moves up to the room that is numbered one greater as yet. In this way the first room (numbered by 0 or 1) becomes available and no one has to leave the hotel. The situation is very similar if two or a finite number of new guests ask to come in: The present guests move up in the rooms that are numbered n greater than now.

A bit more challenging is the arrival of a "Hilbertian bus" with an infinite number of passengers, named or numbered due to the natural numbers. In this case a constant moving up will not be successful. But the past guests could double their initial room number, and every passenger gets an oddly numbered room. This is not the only option available, students should contribute alternatives. More general, if there are two or a finite number n of "Hilbertian busses", one can multiply every original room number with n + 1 and assign the passengers of the first bus those rooms, which numbers are congruent n + 1 modulo 1. The occupants of the second bus move into the rooms that are numbered by natural numbers congruent n + 1 modulo 2 and so on. Students are expected to formulate a proper mapping rule and to come up with their own ideas relating alternatives.

Where is the border line? Even a "Hilbertian bus-fleet" of infinite number of "Hilbertian busses" numbered according the natural numbers, is still not able to overstrain the hotel. For example, one can assign a double index to every passenger due to his bus number and his seat number within this bus. Now Cantor does the work by applying his first diagonal method to this matrix structure. Also here students could consider a formula or a formal description of an algorithm.

Students varied the above solutions in several and diverse ways. For example, prime numbers were used and alternating methods of simultaneous counting. Of course, the most important task is the clarification of the impossibility of lodging an uncountable amount of recent arrivals.

The above considerations go along with the equation

ℵ0 + ℵ0 + ℵ0 + … = ℵ0,
where the number of the summands on the left side is countable.

The given examples above, which were part of the studies of our futures teachers, have certain potential to support understanding.

Conclusions with respect to understanding the concept infinity

We referred to Cantor especially when we stressed the 1-1-coresspondance (or mapping) and some insights of arithmetic rules including infinity. Since all examples are rather basic but initially unknown to most of the students they gained competencies with counting and the notion of bijectivity. It is important to realize that the arithmetic rules known from the basic arithmetic operations may vary, depending on the context. Another example, but in a different relationship is the "double distributivity" in case of unions and intersections of sets. The phenomenon "infinity" holds in itself ambiguities which contradict common sense at first glance. It is of great educational value to become acquainted with some of them, namely in two respects: in terms of general education, which should be a concern of mathematics education and for the purpose of educating "good" teachers. The well-educated primary teacher is then in the position to react properly when children ask smart questions or questions that show insight but do not use proper wording. Pupils occasionally may achieve even philosophical significanceso long as the teacher recognizes its meaning.

Table 1 : Cantor's first diagonal method

 1

	5	3/6	…

  which is a subset of ℝ, and the numbers are represented in the decimal system, i.e.

	1	, 0	11	12	13	14	15	.....

Proceedings of CERME10


domain: math shs

text:
Rainer Kaenders
email: r.kaenders@uni-bonn.de

Ysette Weiss
email: weisspid@uni-mainz.de

Historical methods for drawing anaglyphs in geometry teaching

Keywords: Anaglyph, binocular geometry, historical methods of visualization, 3D-representations

The historical anaglyphic method was in use for more than hundred years to create spatial illusions of mathematical objects and for technical constructions. While algebraisation is predominant at school, students lack experience in understanding the causalities of technical tools. Modern technical devices rarely allow for direct investigations on underlying technical principles. Here we use the historic anaglyphic method to enable the students to produce high-standard 3D illusions just by using coloured pencils, a ruler and glasses with colour filters. We developed an approach to the anaglyphic method that uses nothing but similarity and especially surpasses projective geometry. The presented approach relates plane and spatial geometry, and can be grasped by all students that have some understanding of the similarity of triangles.

Introduction

Concept development in high school mathematicsin particular A-level subjects in many countries is characterized by predominant algebraisation. This seems to match successful and effective heuristics, attitudes and problem-solving strategies in modern everyday lived experience. Out of school it fosters the learning of pattern recognition, algorithmic procedures and trial and error. The progressive digitalisation of most fields of experience and actions of contemporary students seems to make the search for causalities and functional principals superfluous and unnecessary. Modern technical devices rarely for allow direct investigations on underlying technical principles. However, approaches and attitudes aiming only for the use and not the understanding of technical devices involve the risk of simple manipulation of the user and restrict creative developments of the tools by their users. In order to help our students to become autonomous, mature individuals, we need in mathematics (and also as a foundation for technical sciences) teaching designs where students start to wonder: How does it work? How can I accomplish it on my own?

In the following we present materials for discovery learning, supporting the described educational goals. The design of teaching and learning materials allows for the implementation of the history of engineering and technical drawing in different ways and with different aims. A historical investigation using original sources can be conducted as an introductory part of a workshop on 3D presentations. Historical investigations can also be undertaken as a part of an individual student's presentations after the workshop on anaglyphs and binocular geometry. We start with a short historical introduction into anaglyphs as well as into the literature and original sources, which are applicable by the students at school or by student teachers in teacher education.

We then give a short summery of how binocular geometry can be related to the mathematics curriculum and discuss a textbook presentation of the topic. The following section deals in detail with the geometrical properties of central projections, which enable the drawing of anaglyphs without using three-dimensional analytic geometry.

We tried out the developed set-up and the material in four workshops at an international Kangaroo Camp, each with 15-20 high school students between the ages of 15-18, and in a day course at the Hausdorff Center with local high school students in Bonn, with a lecture and problem sessions with about 60 high school students.

Historical background

Sir Charles Wheatstone invented the earliest type of stereoscope in 1838 [START_REF] Wheatstone | Contributions to the physiology of vision. Part the first. On some remarkable, and hitherto unobserved phenomena of binocular vision[END_REF]. However, as David Brewster writes in (Brewster, 1856, p.27) a certain Mr Elliot was led to the study of binocular vision … as early as 1823. Wheatstone and Elliot used mirrors, whereas Brewster invented lenses. The mathematician Wilhelm Rollmann (1853) invented the anaglyph stereographic method. The Greek word Anaglyph is derived from 𝛼 ´𝜈𝛼 ´(aná), meaning "on" or "on each other" and 𝛾𝜆𝜐 ´𝜑𝜔(glýphō) "to carve", "engrave", or "represent". In his method, two pictures in the mutually different colours blue and yellow are superposed onto each other. The observer in Rollmann's method separates these pictures by using glasses with colour filters, i.e. a red glass for the left eye and a blue one for the right eye.1 All cited sources are digitalised and available online. The descriptions of the methods are mainly verbal. The few calculations can be performed and understood using elementary middle school mathematics. This makes the cited literature a good and suitable source for historical investigations in the classroom.

Rather than being a goal as such, the anaglyph method was already used by Rollmann to illustrate mathematical facts and insights in a three-dimensional fashion. Another impressive example of this kind of illustration is the Imre Pal's beautiful book (1961). Although the books of Rollmann and Pale describe the three-dimensional model of spatial seeing, nevertheless they do not instruct how to draw two-dimensional anaglyph stereographic pictures. For the latter we designed a special workshop, which can be related to schoolbook exercises in analytic geometry (Körner et al., 2010, p. 92) or can serve independently for project work.

Anaglyphs in teaching

Binocular geometry is not a canonical topic for mathematics lessons. So we were rather surprised to find a set of exercises related to the anaglyph method as well as well as materials for teacher training courses. The following excerpts in Figure 1 are taken from a German school textbook for grade 11 (one year before A-levels). The book is one of the five most widely used textbooks in high schools in Germany. The excerpts stem from a chapter in analytic geometry dealing which the calculation of intersections of lines and planes given in Cartesian coordinates. In addition to perspective drawings of a house and bowling pins, some general properties of perspective drawings are given (left). In preparation for the work with perspective mappings, the textbook authors posed the exercise to compute the image of the shadow of a cube illuminated by a lamp given in Cartesian coordinates. The shadow has to be calculated in a given plane using the formulas for intersection points of lines with planes. The green image (right) is the cube shadow in the given plane. The red image is the result for the same calculation but with a slightly shifted illuminant. The students are asked to make redgreen glasses themselves and to look at the picture through these glasses. Among the textbook materials for computer-based learning, there are 3D dynamic geometry applets with the green and the red calculated images corresponding to different dynamic light sources. Even though the exercises deal with a geometrical contextperspective drawingsthe proposed approach is purely algebraic. The pictures are visualizations of the results of algebraic calculations. In [START_REF] Färber | Erzeugung von 3D-effekten bei Bildern[END_REF] the author proposes designing the excursion on the anaglyph stereographic method as discovery learning in groups. Nevertheless, the lesson planning only involves algebraic manipulations. As we already discussed in [START_REF] Kaenders | Algebra without context is empty, visualizations without concepts are blind[END_REF], the high degree of abstraction and technical complexity of algebraic symbolic language gives students few opportunities to question the underlying rules, to introduce their own situated notations and notions reflecting their individual understanding of a problem and its context or to develop their own mathematical questions. To deal with this problem we develop a geometrical context for discovery learning of the anaglyph stereographic method.

A simple approach: Lifting plane figures

When we considered using the historical anaglyph method for teaching, we expected a couple of difficulties. First, we were convinced that we would have to give a quick introduction to projective geometry and then apply it to the anaglyph method. We were then surprised to discover an approach to the fabrication of such binocular illusions that does not use projective geometry at all. It is the technique of lifting a point. Well understood, it allows not only for the lifting of points but also the lifting of any figure from the plane as long as it is supposed to become an illusion of a congruent figure in a plane parallel to the initial plane. We give a description of the course of action in the workshops.

Practical preparation

Before the students are given the task of creating their own anaglyphs, we show them some exemplars. By looking through the red-green-glasses they begin to get a feeling for what such a painting could look like and which type of effects are generally possible. It turned out to be a helpful practical hint to let the student put their fingertip on the spatial spot where they expect the figure to be. A few students do not succeed in recognizing the intended illusion. The reason for this might be problems like shifted eyesight, where one eye has a stronger visual faculty than the other, or a red green deficiency. Nevertheless, these students could successfully participate in the workshop.

Erecting a little stick

The first exercise is to erect virtually a little stick of thumb size to an illusion that appears to be an orthogonal stick on the paper. In a second step, we can also let the stick appear slightly levitated. In order to find such a representation of the desired illusion, the students can turn the question around: the stick is given and we seek the red-green drawing on our paper. If we put a stick (like a pencil) orthogonally on the paper, we can conceive the red and the green drawing as the shadow we obtain when we imagine our eyes to be light sources. Almost all students draw a short red and a short green line segment that produce two lines that intersect at the point that is ought to be the orthogonal projection of the stick on the paper. By discovering this, two questions arise:

 What is the angle between the red and the green line segment? The students relate it to the position of the eyes and some conjecture that the lines prolong to the pedals of the eye points.  How far do we have to draw the line segments for a perfect illusion? After having explored the situation, the students give two conjectures: The ratio of the red and green line segment is the same as the ratio of line segments between the pedal points of the eyes and the point where the stick touches the paper. They indicate that because of similarity, both assertions are equivalent.

Analysis of the exercise

To analyse the situation, we assume the position of the eyes 𝐴 and 𝐵 on fixed height ℎ over the table.

We assume the eyes 𝐴 and 𝐵 to be parallel to the table plane and to have a distance of 7 cm, which is about the average eye distance for adult human beings. When we project the eyes 𝐴 and 𝐵 orthogonally onto the table, we obtain two pedal points 𝐴′ and 𝐵′. We consider an arbitrary point 𝑃′ in the plane. Now we want to find points 𝑃 𝐴 and 𝑃 𝐵 in the plane with the effect that they create the illusion of a point 𝑃 in space, that lifts our point 𝑃′ to some height ℎ.

In Figure 3 we see that this illusion arises when the lines 𝐴𝑃 𝐴 and 𝐵𝑃 𝐵 cross in the point 𝑃. Then 𝐴𝐵𝑃 forms a plane. Then the three planes 𝐴 ′ 𝐵 ′ 𝐵𝐴, 𝐴 ′ 𝐵 ′ 𝑃 and 𝐴𝐵𝑃 have three intersection lines, two of which are parallel. Then the third one is parallel as well. The reason is what one can call the Theorem of the Tent: Given three planes 𝐸 1 ,𝐸 2 and 𝐸 3 that intersect in three lines 𝑔 12 = 𝐸 1 ∩ 𝐸 2 and𝑔 23 = 𝐸 2 ∩ 𝐸 3 as well as 𝑔 13 = 𝐸 1 ∩ 𝐸 3 . When two of these lines are parallel to each other, the third is parallel to both as well. The proof of this proposition is an occasion to show the efficiency of the set theoretic language: Assume that the lines 𝑔 12 = 𝐸 1 ∩ 𝐸 2 and 𝑔 23 = 𝐸 2 ∩ 𝐸 3 are not parallel; they then intersect in a point 𝑃, since both lie in a plane. Then {𝑃} = 𝑔 12 ∩ 𝑔 23 = 𝐸 1 ∩ 𝐸 2 ∩ 𝐸 3 and 𝑔 23 = 𝐸 2 ∩ 𝐸 3 intersect in 𝑃 as well. Hence, we can lift the point 𝑃′ to an illusionist point 𝑃, when we draw the lines 𝐴′𝑃′ and 𝐵′𝑃′ and end up at points 𝑃 𝐴 and 𝑃 𝐵 , such that 𝑃 𝐴 𝑃 𝐵 is parallel to 𝐴′𝐵′. In order to find out how far the point will be lifted, we use the point of view of similarity as the students uttered it. We consider one of the two triangles 𝐴 ′ 𝑃 𝐴 𝐴 or likewise 𝐵 ′ 𝑃 𝐵 𝐵 (see Figure 4). We especially want to understand the relation between the height ℎ and the distance 𝑑 ≔ 𝑃 𝐴 𝑃 𝐵 . In Figure 4 we read off the following ratios: .

Note the remarkable fact that the distance 𝑑 does not depend on the special position of 𝑃′. Hence, we have one method to lift not just one point but also a whole figure to a certain fixed height ℎ. For instance, we can lift a square by lifting its vertices on the same height and then connect the corresponding red and green points. If we do that twice, we can construct the illusion of spatial box.

Similarity as key concept

Finally, we want to understand how we can lift a figure that does not consist of line segments, e.g. a circle. For this, we need to understand how to lift an arbitrary figure to a fixed height ℎ. For this fixed ℎ, we consider the map of the plane to itself, that maps 𝑃′ to 𝑃 𝐴 and likewise the map that maps 𝑃′ to 𝑃 𝐵 . We know already the answer, since These considerations on similarity can be used to construct tasks for instructional scaffolding as well as materials for explorative learning.

Observations during the workshops and development of research questions

In our workshops, we developed most of the tasks and drawings together with the students on the blackboard. This gave us the possibility of choosing between small-step guiding tasks and rather open activity-oriented tasks corresponding to the work of the students. The groups in the four workshops at the international Kangaroo Camp were very inhomogeneous regarding their English language skills as well as their mathematical preparation. These groups had students interested in mathematics but without any training for competitions or mathematical extracurricular experiences and participants of the International Mathematical Olympics. None of the participants had ever dealt with binocular geometry. In spite of their age (15-18 years old) there was no problem to get the students to draw pictures with crayons. It became also evident that the problem 'How to draw an anagram' is extremely suitable for inhomogeneous groups. The students worked in small groups and were quickly fascinated by their own experiments and pictures. Our objective that was reached in all four workshops was to get the students to search for the underlying principles of constructions and to produce their own drawings by using the principles. As soon as the students are acquainted with the technique of point lifting, there are many possible projects to tackle.

The day course in Bonn was organized differently. The workshops were hold by mathematics teacher students. We gave introductions into anaglyphs first for the tutors than for the students, attended the different workshops and moderated the presentations of the results. For the day course in Bonn, we prepared a script for the tutors with problems they were supposed to solve. Before the day course there were several meetings were the tutors could asked questions and discuss the concept. There was a noticeable difference between tutors who tried to grasp on their own the concept of drawing anaglyphs by lifting points and curves using the script more to look up some of the details and tutors who first read the script and tried to solve the problems by using the methods described there. The first type of tutors led their workshops in a more explorative experimental way; some of the second type tutors had inserted into their workshops small lectures on the basic of the script. The students asked questions from different perspectives: from a phenomenological perspective (What are conditions for two points to create the illusion of a floating point?), from the perspective of geometrical invariants (Which properties do not depend on the position of the centre of the projection?), from the perspective of geometric transformations (How to place the figure to support its three-dimensional illusion?). Nevertheless, the students of all groups of the day course were very motivated to understand how to draw a 3D picture, experimented with pencils, lifted points and produced their own 3D images (Fig. 6).

During our reconsiderations of the first repeated workshops, we tried to describe the atmosphere when the students started to construct central projections and calculate distances. At the beginning -may be because of our technical explorations into the past -we called it the mind of engineers and inventors. During the next workshops non the less we realized that a substantial part of workshops the students were engaged with thought experiments: projections through transparent tables, cutting up spheres, building houses out of cubes and tetrahedrons, rotating trefoil knots. This led us to think about the role of thought experiments in physic and mathematic lessons. May be it was so easy to inspire so different students for experiments because there were not real experiments? Students meet nowadays in their everyday life not often people who both are said to be cool and decompose technical devises in order to understand basic functional principles. Choosing advanced courses in science for A-levels does not imply one has grown up with a soldering iron or a chemistry box.

One could think that our students good performance and experiences in virtual worlds could give the historical extremely important thought experiment a new place in physics lessons in order to develop interest in functional principles even if there are only very limited prior technical experiences. But at least in German physic lessons, it seems to be rather the other way around: Many computer based visualisations and experiments transform the very nature of the thought experiment and replace it by a confirmation experiment of a virtual programmed reality (as it is also done in the earlier discussed mathematics textbook task using the applet to compute the anaglyphs).

May be it became rather a task for mathematics educators to integrate thought experiments and virtual engineering into mathematics teaching to support the perspective: How does it work?

Conclusion

The fact that the participants in our workshops were able to pose independent research questions in geometrical terms with geometrical meaning is for us an indication of their development of a geometrical language and concept. The students gave their proofs by construction and in different geometrical notions using invariants and similarity mappings. We were especially impressed by the ability to switch between plane and spatial perspectives on the stereographic pictures developed by the participants during the workshops.

Figure 1 :

 1 Figure 1: Erecting a stick of thumb size by lifting a point.

Figure 2 :

 2 Figure 2: Erecting a stick of thumb size by lifting a point.

Figure 3 :

 3 Figure 3: The basic principle of point lifting.

Figure 4 :

 4 Figure 4: One of the triangles of the basic figure.

.

  Combining this with the ratio between the triangles 𝐴𝐵𝑃 and 𝑃𝑃 𝐴 𝑃 𝐵 , we conclude

  with centre 𝐴′, and the other with centre 𝐵′.

Figure 5 :

 5 Figure 5: Central dilation with factor 𝒂 𝒉 .

Figure 6 :

 6 Figure 6: Examples of student products: a box with concave top, a house, a tetrahedron.

Figure 7 :

 7 Figure 7: Borromean rings made of three golden rectangles.

Proceedings of CERME10

In our case, we use a red glass for the left eye and a blue or better turquoise one for the right eye.Thematic Working Group 12Proceedings of CERME10

Acknowledgment

The authors thank Alexander Unger from the Humboldt University Berlin and his Kangaroo team as well as the Hausdorff Center for Mathematics in Bonn for the many occasions they gave us to develop material and to conduct workshops on different topics.


text:
Katalin Gosztonyi
email: katalin.gosztonyi@gmail.com

Understanding didactical conceptions through their history: A comparison of Brousseau's and Varga's experimentations

Keywords: History of mathematics education, didactical theories, New Math, probability teaching, abstraction

In this article, I compare two experimental projects led in the 1960s and 1970s by Tamás Varga in Hungary and by Guy Brousseau in France, concerning the teaching of combinatorics and probabilities. I attempt to show that this comparison can contribute to a better understanding of their theoretical reflection on the teaching of mathematics and the dependence of their didactical conceptions on the particular historical context in which these reflections were realized.

Introduction

In the Hungarian mathematics education community, the "complex mathematics education" reform led by Tamás Varga during the 1960s and 1970s is considered as one of the most important milestones of the history of mathematics education in Hungary. Varga's conception is viewed as a representative of the so called "discovery based" tradition of mathematics education. However, this conception was never developed in a theoretical level: it can be understood from the documents of the reform (as the curriculum, the textbooks, teacher's handbooks etc.) and from some articles of Varga, but most of the time, these texts present his approach on very concrete examples and give only limited, indirect access to the conceptual basis of his didactical conception.

The research presented in this article is part of a more complex work, aiming to analyse Varga's conception with a combined, historical and didactical approach, and in comparison with the French reforms and mathematics educational movements of the same period (Gosztonyi, 2015a). This analysis contains three levels: the study of the historical context, of the epistemological background (the conceptions about the nature of the mathematics in the background of both reforms) and a didactical analysis of the reforms' documents. 1 For this third part, French didactical theories were used, Brousseau's Theory of Didactical Situations (TDS) [START_REF] Brousseau | La théorie des situations didactiques[END_REF] among others. The TDS proved to be an expedient theory to reveal characteristics of Varga's conception; however, some difficulties of the analysis let to think there were also some important differences between Varga's and Brousseau's didactical approach.

These theoretical questions led to consider Brousseau's work not only as a theoretical framework for my research, but also as a research object: actually, Brousseau himself was one of the actors of the French math education scene of the 1960s and 1970s. He led experimentations during this period, and he developed his didactical theory in relationship with these experimentations, in the very particular historical context of the French reform movement. So, a historical analysis of Brousseau's projects 1 [START_REF] Gosztonyi | The 'New Math' reform and pedagogical flows in Hungarian and French mathematics education[END_REF] gives a short summary of this work, showing how the difference of the epistemological background, represented by influent groups of mathematicians in both of these countries, can explain, at least partly, the differences of the two reforms.

Thematic Working Group 12

Proceedings of CERME10 of the period could contribute to the comparison of the two countries' reform movements and to the understanding of Varga's project.

One more reason explains my interest for Brousseau's experimentations. One of the main domains of Varga's activity concerns the teaching of combinatorics and probability: so I was interested to do comparative analysis partly on these domains. At the same time, these domains are missing from the French obligatory education of the period which makes the comparison obviously difficult between the two countries. However, during the 1970s, these subjects appear in France in certain experimental projects, and one of the most developed examples is Brousseau's experiment on teaching probabilities, described in a retrospective article [START_REF] Brousseau | An experiment on the teaching of statistics and probability[END_REF].

In the following, I will resume the comparison of Brousseau's experiment with Varga's teaching projects and experiments, and I attempt to show how this comparison helps to understand the authors' didactical conceptions.

Some elements of the historical context

The international discourse on mathematics education during the 1960s and 1970s is dominated by the New Math movement and the debates following the introduction of New Math related reforms in several countries. France is one of the leading countries of the international reform movement, with the "mathématiques modernes" reform introduced in 1969/70. Varga declares also being influenced by the international movement: he starts his experimentations in 1963, inspired by Z. P. Dienes' lectures in Hungary in 1960 and a UNESCO conference organized in Budapest in 1962, and follows the reforms of several countries during the following decades. His experimentations concerning the primary and later also the middle-school are progressively spread in the country, and give the basis of a new official curriculum in 1978.

Although Brousseau is not among the main actors of the French curricular reform movement, his early experiments can also be interpreted in the context of the reform. In France, various experimentations on primary and secondary school mathematics education were encouraged and supported during the 1960s and 1970s. For example the system of the IREMs2 was progressively created from 1968, with the mission to form in-service teachers to the implementation of the reform, but also to continue experimentations. During the 1970s, the IREMs became also centers of critical debates on the reform, and principal sources of the emergence of didactical research in France. Brousseau's early work fits clearly in this context: he led experiences since the 1960s, and his projects led in the Jules Michelet experimental primary school from the beginning of the 1970s became the basis of the development of his later theory. The experimentations on the teaching of probability described in his retrospective article were also led in 1973 and 1974 in this school.3

Brousseau's and Varga's teaching projects and experimentations

The reasons to teach combinatorics and probability

In France and Hungary, combinatorics and probability are not present in primary and middle-school curricula before the New Math period. So, the experiments concerning these domains attempt to introduce new elements of curricula or introduce them much earlier than in precedent cases. Varga explains in several papers why he found important to introduce these domains very early during mathematics education. According to him, these domains contribute to the diversity of mathematical subjects treated in school and help to implement dialectic relationships between mathematical domains -which is one of the main principles of his reform curriculum's structure. He says combinatorics and probabilities can be studied on the basis of very concrete, material experiences and so can give a good example of the process of mathematical abstraction very early, using a diversity of representational tools, and without the need of complex theoretical background. These domains can also give occasion to many playful activities. More particularly the teaching of probabilities represents a "different kind" of mathematics: the mathematics of uncertain things, which is especially important to describe real world phenomena. Furthermore, in the frame of tasks related to probabilities, and especially through estimations, student's autonomous thinking and the expression of students' various opinion can be easily encouraged. The quotation below shows also that this last question is closely related to more general pedagogical considerations concerning the teacher's and student's role in the learning process or students' education to democracy.

My own view is that estimating, guessing, predicting, mentally representing, the future and expressing our opinion about it is a human ability which should play a greater role in education than it does now. All these activities [...] get kids personally involved in learning. […] Reasons must be strong, maybe not unrelated to sentences from "a child should not have an opinion" to "a child should have not will." If this is a correct conjecture, then the issue is a more general one about education, not necessarily school education or school math in particular. (Varga, 1982, p. 30) All the arguments resumed above are important characteristics of Varga's conception on mathematics and its teaching (Gosztonyi, 2016a, b). Thus, one can understand that the emphasis made on the teaching of combinatorics and probabilities is closely related to the general educational goals of Varga's reform, and these domains' curricula and tasks can serve as efficient examples to understand the realization of these goals in Varga's reform. In Brousseau's examined experiment, there is no question of a general curricular reform: probabilities are not included in the official curriculum of primary school, and Brousseau's experimentation is realized in the frame of general activities ("activités d'éveil") and not during the mathematics lessons. However, Brousseau underlines also some general arguments to explain the interest of teaching probabilities. The idea of "another kind of mathematics" appears also in his work (Brousseau, Brousseau & Warfield, 2002, p. 397). Beyond that, students' autonomous work and the repartition of responsibilities between the teacher and the students is in the center of his theoretical thinking, and as he admit in the quoted paper (e.g. p. 384, p. 411), the experiments on teaching probabilities played a crucial role in the development of his didactical theory.

Some tasks of Varga on probabilities

In his articles, Varga describes several probabilistic tasks developed in the frame of his experimentations (see e.g. [START_REF] Varga | Probability through games. A sample of three games[END_REF][START_REF] Varga | New topics for the elementary school math curriculum[END_REF]. In the case of the subtraction game, Each kid draws boxes for the digits [of two two-digit numbers] the way they are used in writing subtractions. The goal is to make the difference as great as possible. They can fill the boxes in any order, but only with random numbers produced by rolling dice. (Varga, 1982, p. 28) In another game, the "game with three disks" (Varga, 1970, p. 424), three discs are in a box, one of them has a cross in each side, one has a cross on one side only and one is blank in both sides. The teacher draws a disk at random and shows one of its faces at random to the students. They have to guess the other face.

In each of these situations, there is a competition between students, which motivates the development of a strategy. The comparison of these strategies, their test during further experiments contributes to the development of probabilistic thinking. According to Varga's description, frequentist approach, based on repeated experiments and the observation of relative frequencies is alternated with a classical approach to probabilities, based on logical arguments concerning models with equally probable events. However, the frequentist approach is rarely developed in deep details: it seems more to serve as an experimental basis to develop classical models which are then confirmed by logical and combinatorial arguments.

In the situations described by Varga, teacher and students are in permanent dialogue: Varga gives several examples how to guide these dialogues in order to help the development of mathematical notions, but to give also important autonomy to the students in this process.

Beyond the description of tasks and problem situations, Varga describes also the conception of longterm teaching processes. Even if some texts help to understand the long-term conception of probability-teaching, the clearest descriptions on the construction of long-term teaching processes concern the domain of combinatorics. We study this in the next section.

Varga's series of problems to teach combinatorics

In Varga's conception, ordered series of problems play a crucial role in the construction of long-term teaching processes [START_REF] Gosztonyi | Séries de problèmes dans une tradition d'enseignement des mathématiques en Hongrie au 20e siècle[END_REF]). A particularly interesting example concerning combinatorics is described in the first grade teacher's handbook associated to Varga's reform (C. Neményi et al., 1978, pp. 243-258). It is interesting not only because it is particularly clearly structured (compared to other parts of the handbook), but also because it gives quite explicit explanation about the principles of ordering problems. This ordering is not completely given in advance: teachers have to conceive (and reconceive) it depending on the particularities of the class, and on the reactions of the students.

The series contains activities with different materials: the students build towers with coloured cubes; thread beads, draw flags or build houses with three parts of different colour; write 'words' (letter series) with a given number of characters or play music with a given number of notes. One organizing principle is the variety of experiences, apparently fare from each-other, and stimulating a diversity of senses. But there is also certain progressiveness in their order, namely in the level of abstraction: starting from the manipulation of physical objects, through drawing and until the manipulation of symbols.

The activities with one material follow also a progressiveness, which is explained in detail on the example of building towers: after a free game with the material (elements with, for example three different colours), students are asked to build towers with a given height; then different towers with the same height; and then come the question of the number of possibilities. The handbook suggests variations of the mathematically important variables as the number of levels or colours, but also the mathematically neutral elements as the type of the material or the colours of the elements. It suggest also some additional restrictions (as for example two neighbouring levels cannot have the same colour; or a given colour can have on the top level, etc.). A similar process is described for different materials. The handbook indicates the analogies between the corresponding phases of these different activities, and explains also their differences which make them problems of different nature in the eyes of students: for example, in the case of building houses, the order of the elements is not immediately given, contrarily to the building of towers or the colouring of (three-stripe) flags. Students have to recognize progressively the links and the analogies between these different problems: that is what will lead to a progressive generalization of methods and solutions.

The process is planned for several years: the object at the first grade is mostly the collection of experiences in structuring possible cases and in looking for the number of all possibilities, in concrete situations. The systematic variation of conditions and the formulation of rules for calculating the number of cases come some years later. In a later article, [START_REF] Varga | New topics for the elementary school math curriculum[END_REF] describes in detail a process on the long-term, based on the example of building coloured towers: there he also explains how the progressive construction of representation tools (as tables and trees) leads in his conception to creating proofs and general formulas in combinatorics.

Brousseau's experimentation

In his experimentation on the teaching of probabilities, Brousseau also conceives a long and coherent teaching process, although the structure of this process is quite different from those of Varga, as we will see below. The whole process emerges from one situation: the teacher fills three opaque sacks with black and white balls, five in each sacks but the proportion of blacks and withes is unknown. The students have to find out the exact composition of each sack by drawing balls one by one from the sacks. The process goes through several phases that the authors describe as follows:

i.

"An introduction to hypothesis testing" (5 sessions) ii.

"Modelling and experimenting" (3 sessions) iii.

"Graphic representation of long series" (8 sessions) iv.

"Convergence and statistical decision" (4 sessions) v.

"Decision intervals" (5 sessions) vi.

"Events and their probabilities" (7 sessions)

After a number of drawings and hypotheses made on the compositions, they decide to model the sacks with transparent bottles where they put five balls in different compositions, in order to compare the outcomes of these bottles with those of the sacks. Students work almost autonomously during the first two phases, only with some "regulative" interventions of the teacher: that's what Brousseau calls later an adidactic situation. The teacher intervenes more directly during the third phase, in order to stabilise the method of students' experimentation. A difficulty emerge in this phase, as even after long series of drawings, students are still not able to prove a decision between the possible compositions and they start to loose motivation. The teacher refuses to open the sacks but suggest proving the compositions by elaborating a methodology which allows students to do predictions. That is what happens during the fourth and fifth phases using computers for simulations and different tools and methods of representation (as tables and graphics).

The described process represents a typically frequentist type of approach to probabilities, with the use of complex statistical methods and tools, and with a long process of repeated experimentations. The first five phases emerge from one initial problem. The classical approach as well as other problem situations appear in the sixth phase but, considered more conventional as the precedent ones, this phase is not developed in the article (p. 405)

A comparison of Brousseau's and Varga's projects

Comparing Brousseau's and Varga's teaching projects briefly described above, one can observe some common points but also several interesting differences.

Although both frequentist and classical approach appears in both of the authors' projects, Brousseau puts more emphasis on the first one while Varga emphasize more the second one. This difference can be explained by different things, as by the coherence with other parts of the curricula (links with statistics on one hand, with logic and combinatorics, privileged domains of Varga on the other; the use of decimal numbers, privileged by the French curriculum of the period, or the use of fractions, emphasized in Varga's curriculum), but also by some pedagogical questions.

One of the common motivations to introduce probabilities into the primary school mathematics education is, in both of the cases, the opportunity offered by this domain to work with students guessing and estimations, and develop mathematical thinking with an important responsibility provided to students during the learning process. However, the repartition of responsibilities between teacher and students does not happen in the same way by the two authors. For Brousseau, this repartition has to be provided by the alternation of adidactic phases, where students work fundamentally autonomously and teacher does not intervene on the level of mathematical knowledge, and phases of institutionalization, where teacher intervenes essentially in order to transform students' context-dependent discoveries into stable, decontextualized and institutionally accepted knowledge. As Brousseau underlines, the experiments on teaching probabilities contributed essentially to develop these key-notions of his later theory. In Varga's case, although students' responsibility in the learning process plays also a key role, the teaching situations are more based on a permanent dialogue between the teacher and the class: the teacher acts as an experimented guide to develop progressively collective knowledge.

In both cases, we can see sophistically constructed long-term teaching processes. But the structure of these processes is quite different. Brousseau's experiment is developed starting from one problem situation. This is something that he calls later fundamental situation: a problem situation which is rich enough to lead to the emergence of a whole theme. The process goes through the alternation of adidactic situations and institutional phases. According to him, the transmission of knowledge constructed by students in the context of a particular situation is not possible without the decontextualisation realized during the institutionalization (Brousseau, Brousseau, & Warfield, 2002, p. 407). Indeed, in the examined experiment, other problem situations appear only in the last sixth phase. Varga's approach is quite different to the construction of long-term teaching processes. He suggests starting from a diversity of problem-situations, with a big variety of contexts. The construction of series of problems offer occasions to recognize similitudes, analogies between these different problems, and leads to a progressive generalization of the solutions.

Behind this difference of the two authors' conception on long-term teaching processes we can recognize a fundamental difference concerning their conception on the mathematical abstraction. As for Brousseau, abstraction is decontextualisation, for Varga, the abstraction process does not mean the elimination of the context, more a progressive generalization on the basis of a diversity of contexts.

Conclusion and discussion

The comparison of Brousseau's and Varga's experimentations led in the 1970s showed several common points and also some important differences. This comparison can help to understand better the conceptions of the two authors on mathematics and its teaching, and also the relationship of these conceptions to their special context. Both of the two authors' projects inscribe clearly in the context of the international New Math movement, with the reforms of the content of the curricula, the debates on the nature of the mathematics and on the psychology of mathematics education. For both of them, Piaget's constructivist theory represents an important reference, but they are both critical with it: the alternation of the adidactical situations with institutionalization in Brousseau's case, and the dialogic relationship between the teacher and the student in Varga's case can be interpreted as two different answers to the limits of constructivism considered by these authors. Moreover, Brousseau's notion of adidactical situation seems to be related to the discourses about the French reform, more precisely to the notion of situation and its role in pedagogical practices and students' learning processes [START_REF] Artigue | Problem solving in France: Research and curricular perspectives[END_REF]). Varga's choices seem also to respond to several elements of local discourses in mathematics education as well as in pedagogy and psychology (Gosztonyi 2015a).

An interesting relationship can also be observed with the dominant epistemological background of each countries reform movement, concerning the author's conception on mathematical abstraction: Brousseau's decontextualisation approach seems to be related to the "bourbakian" conception on abstract mathematical notions, as Varga's ideas on the progressive generalization can be find in the writings of several Hungarian mathematicians supporting his reform (Gosztonyi, 2015a, part II).

This case study reveals how a didactical theory, like Brousseau's one, may depend on the particular context in which it emerges. Thus, further studies on the history of didactical theories are susceptible to contribute to their understanding, and enrich also projects connecting and comparing didactical theories as the "Networking Theories" project (Bikner-Ahsbahs & Prediger, 2014).

The comparative analysis presented above contributed importantly to the original objectives of my research, namely to a reconstitution of Varga's missing didactical theory. But it can also contribute to recent didactical discourse in several further ways, as to the research on the teaching of probabilities or to the recent reflections on the notion of mathematical abstraction.

Insittut de Recherche sur l'Enseignement des Mathématiques

For more details and references on the historical background of the two projects, see(Gosztonyi, 2015a, part I.) Thematic Working Group 12 Proceedings of CERME10

Proceedings of CERME10

Acknowledgment I would like to acknowledge for the financial support of the MTA-ELTE Complex Mathematics Education Research Group, working in the frame of the Content Pedagogy Research Program of the Hungarian Academy of Sciences (ID number: 471028).


text:
Kathleen Clark
email: kclark@fsu.edu

Daniel Otero
email: otero@xavier.edu

Nicholas A Scoville
email: nscoville@ursinus.edu

Primary source projects in an undergraduate mathematics classroom: A pilot case in a topology course

Keywords: Instructional materials, primary historical sources, topology, worldviews

TRIUMPHS) is a five-year, seven-institution collaborative project funded by the US National Science Foundation to design, develop, test and evaluate curricular materials for teaching standard topics in the undergraduate mathematics curriculum in the United States via the use of primary historical sources. Three short

Introduction

There are numerous motivations and benefits for incorporating the use of primary sources into undergraduate mathematics teaching. Primary among the cited motivations is that providing students experience with reading texts in which the genetic development of a topic is presented gives them an opportunity to expand their mathematical education in such a way that includes both traditional and modern methods of the discipline [START_REF] Fried | History of mathematics and mathematics education[END_REF][START_REF] Laubenbacher | Recovering motivation in mathematics: Teaching with original sources[END_REF]. Another motivation is that using original sources in the teaching of mathematics makes it possible to contextualize the mathematics in ways that many textbook treatments do not afford. That is, original sources place particular mathematical ideas in the context and setting of the investigations in which the author was engaged at the time. As a result, the problems with which the author was struggling, and the motivations for solving them, are often more clearly and naturally described, and more compelling than traditional textbook expositions. Exposing the original motivations behind the development of esoteric mathematical concepts may be especially critical for placing the subject "within the larger mathematical world," in the hope of making it more accessible to students [START_REF] Scoville | Georg Cantor at the dawn of point-set topology[END_REF]. Furthermore, primary texts seldom contain the specialized vocabulary that comes with later formalism, promoting access to the ideas by students with a wider range of backgrounds than is achieved with more standard presentations.

Many mathematics instructors interested in bringing the history of mathematics to the classroom question the use of primary historical sources in light of the increased availability of high quality secondary historical sources (e.g., [START_REF] Katz | A history of mathematics: An introduction[END_REF]. Such resources may suffice to help students reap some of the benefits of original works; however, they carry their own difficulties. For example, there is a risk of placing too much emphasis on the history of mathematics per se, as opposed to using that history to support the learning of mathematics. Other key differences between using primary and secondary sources are described by [START_REF] Jankvist | A categorization of the 'whys' and 'hows' of using history in mathematics education[END_REF]:

When using secondary sources, the students are exposed to a given historian's presentation and, possibly, interpretation of history, and they must make their choices based on this (Furinghetti, 2007, p. 136) In this final statement, regarding "what students may gain from just 'sniffing' at" selections from an original source, Jankvist points to the primary focus of the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project.

The TRIUMPHS project

In 2015, a seven-institution collaborative project to design, test, and evaluate curricular materials for teaching standard topics in the university mathematics curriculum in the United States via the use of primary historical sources was funded by the National Science Foundation. The TRIUMPHS project seeks to help students learn and develop a deeper interest in, and appreciation and understanding of, these mathematical concepts by crafting educational materials in the form of Primary Source Projects (PSPs) based on original historical sources written by mathematicians involved in the discovery and development of the topics being studied. These PSPs contain excerpts from one or several historical sources, a discussion of the mathematical significance of each selection, and student exercises designed to illuminate the mathematical concepts that form the focus of the sources. PSPs are meant to guide students in their explorations of these original texts in order to promote their own understanding of those ideas. TRIUMPHS plans to work with mathematics faculty and graduate students from over forty institutions in the United States who will participate in the development and testing process of these PSPs. As part of this five-year project, impacts of the materials and approaches to implementing them will be investigated in terms of teaching, student learning, and departmental and institutional change.

Organization of the paper

The remainder of the paper is organized into four sections. First, we describe the three "mini-PSPs" implemented by the third author during his topology course in spring of 2016. Next, we present a broad overview of one of the components of research we are conducting during the five-year project, and then discuss a small subset of the data collected from five student participants. The paper concludes with a discussion of the implications from this small data set from the pilot year of the project, and a review of next directions planned for the research.

Primary source projects in topology: The case of three mini-projects

The third author developed and implemented three "mini-PSPs" during the Spring 2016 semester at Ursinus College, a small liberal arts school outside Philadelphia, Pennsylvania. His experience using primary source materials in the classroom began in 2012 when he introduced them into a discrete mathematics course. His current interest in using primary sources in the classroom involves teaching topology. Three "mini-PSPs," each taking up two 50-minute class periods, were written for use in an Introduction to Point-Set Topology course, an upper-division elective course for mathematics majors and a designated "capstone" course intended to provide an experience for mathematics students to test and apply previously acquired mathematical knowledge and skills. The author taught this course twice before in a standard lecture-based style. A short description of each of the three projects is provided below.

The course was introduced with a mini-PSP titled "Topology from Analysis" that investigates a paper by Georg [START_REF] Cantor | Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen[END_REF] in which he considered a problem in Fourier series, namely, if a function has a Fourier series expansion, when is such an expansion uniquely determined? Cantor proved in a previous paper that two convergent trigonometric series with equal sums have the same coefficients (uniqueness theorem), even iffor a finite number of values of the variablethe series either fail to converge, or converge toward different sums. Could this theorem be extended to certain infinite sets of points? After reading Cantor's statement of the problem, the students explore simple examples of infinite sets where such an extension is possible. They investigate what properties these examples have that allow for such an extension. The desirable properties ultimately prompted Cantor to define concepts like limit points, derived sets, point sets, and iterated derived sets. The students were then able to use these concepts to prove a more general theorem. Even though these concepts were used to prove a result in analysis concerning Fourier series, they are naturally topological concepts. Hence the project helped to connect analysis and topology, thereby motivating new definitions through the need to clarify concepts, rather than introducing them as standard jargon.

The second mini-PSP focused on the topological concept of connectedness. It again considers a work of [START_REF] Cantor | Über unendliche, lineare Punktmannigfaltigkeiten, 5[END_REF] and his study of the continuum. Students follow Cantor's musings concerning the best way to define such a concept. After he defined a perfect point set based on derived sets, he investigated whether the property of perfectness is sufficient to characterize a continuum. The students wrestle with this question, and eventually observe that such a definition will not suffice. Cantor then proposed an additional property, connectedness, which he defined using a metric. After reading Cantor's definition of connectedness, students examine a work of Jordan (1883, pp. 24-27) which viewed connectedness in terms of separation of point sets into components. This introduces the students to a new conceptual perspective for the same notion considered by Cantor. Next, students read from a paper by Schoenflies (1904, pp. 209) in which it was proven that connectedness is a topological invariant. To do this, Schoenflies required a definition of connectedness that does not appeal to a metric, and is therefore purely topological. Finally, students read from a work of Lennes (1911, pp. 287, 303) in which he attempted to give a proof of the Jordan Curve Theorem and gave yet another definition of connectedness. Students are then asked to show that Lennes' definition is equivalent to the definition used today.

The final mini-PSP used in the course studies excerpts from a paper of John Henry Smith (1874) on discontinuous but integrable functions. Smith intended to provide a counterexample to a "theorem" of Henkel. He constructed an integrable function that is "very badly" discontinuous. Students are led through Smith's work involving the definition of the concept of nowhere dense set in order to construct such functions. As in the first mini-PSP, students examine a problem that motivates the need for a new mathematical concept by abstracting the essence of particular examples in order to capture the essential properties that a set must satisfy to prove a result. Ultimately, this project culminates in Smith's construction of a generalized Cantor set. A function that is continuous except on a generalized Cantor set is then seen to provide a counterexample to Henkel's claim.

A major benefit of these mini-PSPs is that they naturally induce sophisticated discussions about the mathematics by students. During classroom implementation, students were observed to be carefully and thoughtfully working to understand concepts, answer questions, and pose their own questions and conjectures. One such instance occurred when students began to question Schoenflies' definition of connectedness, without prompting from the instructor. The question was raised as to what Schoenflies meant by the phrase "... can be decomposed ..." One student suggested that he meant a partition, but it was soon realized that such an interpretation would be too general. Others then began to modify the partition idea to make it work. This sort of high-level engagement had not occurred in any previous topology course taught by the instructor.

Description of the research

The TRIUMPHS project includes an evaluation-with-research (EwR) study, designed to provide both formative and summative evaluation of the key project activities and defined goals for each. In the original EwR study design, we designated three project components for which we would conduct extensive research and evaluation, and designated these as "student change," "faculty expertise," and "development cycle." Here we only describe a small piece of the "student change" component of the study.

Rationale and research questions

For decades much of the research literature on the impact of the history of mathematics on students, particularly at the secondary level or post-secondary level, was focused on students' attitudes (e.g., [START_REF] Marshall | Using history of mathematics to improve secondary students' attitudes toward mathematics[END_REF][START_REF] Mcbride | The effects of history of mathematics on attitudes towards mathematics of college algebra students[END_REF]. There was scant focus on the use of primary sources as a classroom tool in the early work in the field of history in mathematics education. However, more recent work on the use of primary sources has been done in countries such as Denmark (e.g., [START_REF] Kjeldsen | Beyond motivation: History as a method for learning metadiscursive rules in mathematics[END_REF] and Brazil (e.g., [START_REF] Bernardes | History of matrices: Promoting commognitive conflicts and encouraging reflection on meta-discursive rules in prospective teachers[END_REF], while such research has not yet been conducted with student populations in the United States. Thus, we are committed to investigating the ways in which mathematics students respond to concepts within the undergraduate curriculum that are taught via primary sources. To this end, we developed several research questions, of which we provide a subset here:

1. As a result of students' work with or study of a PSP, what changes do students report in their attitudes and beliefs about learning mathematics? 2. As a result of engaging with PSPs, what do students report as challenges and benefits of learning from primary sources? 3. What is the dominant mathematical worldview reported by students on pre-and post-course surveys? And, does academic major (or gender, or race or other attribute) make a difference in the reported worldviews?

Pilot study: The case of a topology course

This work took place in the first year (pilot year) of the project, which was focused on developing instruments and refining the questions that were formulated in the original grant proposal. The only data sources available for analysis were student pre-and post-course surveys, student work samples (from the three mini-PSPs), and instructor surveys and post-implementation reports. Data were collected from four undergraduate mathematics courses during the pilot year of the TRIUMPHS project: two courses in Fall 2015 (geometry; analysis) and two in Spring 2016 (abstract algebra; topology). In the topology course, the three mini-PSPs described above were implemented and tested for the first time by the third author. In this paper we report data that inform the first, second, and fourth research questions listed above for the five of eight students enrolled in that course who consented to participate in the research and for whom we obtained a complete set of data. For this small group of students, pre-and post-course responses were mostly stable for this pair of questions. This could be a function of the fact that all five students were in the final two years of their undergraduate program. However, we discuss two interesting pre-/post-course survey pairs of responses below.

First, in response to the second prompt, Student 2 stated on the pre-course survey that she was "not fond of professors/texts that state a definition/theorem/idea without giving any hint to how that conclusion was derived, either historically or through proof/explanation." By the end of the course, however, her response no longer referred to lacking historical grounding or a thorough proof or explanation. Instead, her response focused on the fact that "it is very easy to go through math classes and fall behind. If you don't understand one topic, the class often moves on without you…" A similar change in identifying what he enjoyed least about studying mathematics occurred with Student 3. On the pre-course survey, Student 3 stated that he least enjoyed "how everything is given to you but never comes with an explanation of where the math is coming from." However, in response to the same item on the post-course survey, Student 3 only commented that he disliked having to remember definitions and equations. It is possible that for Students 2 and 3 that the historical sources related directly to the formal content of the course satisfied their initial "sore spot" with regard to what they previously enjoyed least about studying mathematics. Since there was no effort in the pilot year to ask for student clarification (e.g., via follow up interviews), we cannot link the change in the sample responses presented here as resulting from students' engagement with the PSPs. However, the responses signal potential interesting outcomes and we have modified our pre-and post-course surveys and have added post-PSP surveys to further investigate this potential influence.

We also asked students to describe their experience using the mini-PSPs in the topology course, as a way to explore the benefits and obstacles they identified. The student responses were encouraging, and in important ways their responses point to the underlying effect of engaging with materials that provide an opportunity to understand the evolution of a mathematical concept. Here, we provide a sample of three of the students' descriptions: Student 1 (Mathematics major): "We used these sources to learn about topics such as connectedness as we thought about origins of the idea and read about how the definitions changed the longer it was studied."

Student 2 (Computer Science major): "Each PSP was an interesting addition to class, and it was unique to be able to see the process of mathematical ideals through the eyes of the various mathematicians."

Student 5 (Physics & Mathematics major): "I enjoyed the document on definitions of connectedness, because it showed how mathematics is really done and how it takes time to accurately articulate an idea. I also liked the first document because it helped motivate topological ideas and illustrate the connection between topology and analysis."

Exploring student responses: Research question 3

To address the third research question, we included a subset of 20 items from [START_REF] Törner | Views of German mathematics teachers on mathematics[END_REF] on the pre-and post-course surveys. Törner and his colleagues surveyed 310 German secondary mathematics teachers in order to identify attitudes about mathematics that captured "the essence of mathematics," where they defined "mathematics as a field and not as a subject taught in school" (p. 119). In doing so, they identified attitudes relating to four aspects of a mathematical worldview: scheme, formalism, process, and application. Tables 1 and2 report the pre-and post-survey means of students' responses for these items. Item responses ranged from 1 to 5, where higher values indicate stronger association with that particular worldview. The predominant worldview (in bold in tables 1 and 2) of the Spring 2016 topology students on both the pre-and post-course survey was the process view. 1 Törner (1998) described each of the four aspects. For example, the process aspect characterizes mathematics "as a process and as an activity in thinking about problems and gaining knowledge" (p. 122); whereas the schema aspect represents a view of mathematics as "a "tool-box and bundle of formulas" and an idea oriented with algorithm and schemes" (p. 123).

2 In the college and university system in the United States, typical undergraduate degrees are four years in duration. Törner (1998) reported on the relations among these four aspects, stating, "the formalism and schem[a] scale represent both aspects of the static view of mathematics as a system and intercorrelate highly" (p. 125). However, these two aspects of a static paradigm "correlate with the process scale in a significantly negative way" (p. 125), which is a finding that appears to hold for several students in our sample. For example, where there are lower means on the scheme and formal aspects (e.g., Student 2 pre/post; Student 3 post; Student 4 post), higher mean values occurred on the process aspect.

Discussion

This paper highlights the promise of robust investigations and implications that may result from the EwR efforts of the TRIUMPHS project. The project's pilot year enabled us to trial student and instructor survey instruments and data collection procedures. We chose the topology course as a case because of the particular nature of the PSPs ("mini" as opposed to full-length), the students' mathematical maturity (juniors and seniors), and the expectation that many of the courses participating in TRIUMPHS will also likely have small enrollments. As a result of the pilot year, we have significantly modified our survey instruments and research questions, and our aggregate student population has now increased. As we move forward, our research plans include conducting multiple analyses to identify trends in students' mathematical worldviews (within courses, across courses, and disaggregated by other demographic characteristics). We have also developed and incorporated post-PSP surveys and are currently developing and piloting multiple interview protocols.

  . When reading original sources, the students must, on the other hand, perform their own interpretation of what actually took place, why a certain mathematician developed a theory in one way or another, whether or not what is written is true, what internal and/or external forces drove the development of the work, etc. … The extent to which original sources are being used does, of course, have an impact on what the students learn: what students may gain from just "sniffing" at a few picks from an original source and what they might learn from being exposed to systematic readings of original sources are immensely different things. (p. 250)

  Our initial pre-and post-course surveys (to which students responded before instruction on any PSP occurred and again at the end of the course) only contained one pair of open-ended questions asking students to identify what they enjoyed most and least about studying mathematics: What do you enjoy most about studying mathematics. (Explain briefly.) What do you enjoy least about studying mathematics. (Explain briefly.)

Table 1 : Mathematical worldview of Topology students, Spring 2016 sample (pre-survey)

 1

	1
	Mathematical	Student 1	Student 2	Student 3	Student 4	Student 5
	worldview	(female; 4 th	(female; 4 th	(male; 3 rd	(male; 3 rd	(male; 4 th
		year 2 )	year)	year)	year)	year)
	Schema	3.4	3.2	3.4	4.2	1.6
	Formal	3.8	3.4	3.8	4	4.4
	Process	4	4.6	5	3.6	4.8
	Application	3.6	3.4	3.6	3.4	3.4
	Mathematical	Student 1	Student 2	Student 3	Student 4	Student 5
	worldview
	Schema	3.4	2.6	4.2	3.6	1.4
	Formal	4.8	3.4	4	3.8	4
	Process	4.2	4.4	5	4	4.4
	Application	3.8	3.4	4.4	3.8	3.4

Table 2 : Mathematical worldview of Topology students, Spring 2016 sample (post-survey)

 2

	Thematic Working Group 12

Proceedings of CERME10

Acknowledgments

This material is based upon work supported in part by the National Science Foundation under grants numbers 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

The authors wish to thank Janet H. Barnett for her helpful comments and suggestions on earlier drafts of this paper.