Álvaro Barreras
email: alvaro.barreras@unir.net

Antonio M Oller-Marcén

Generalizing a 16 th century arithmetic problem with prospective Secondary education teachers

Keywords: History of mathematics, word problems, teacher training, generalization, algebraic language

The use of history of mathematics and, in particular, of historical problems is a very useful tool for teacher training. In this work we present some partial results obtained after the design and implementation of an activity based upon an arithmetic problem from the "Conpusicion de la arte de la arismetica y juntamente de geometria" by the Spanish 16 th century Dominican friar Juan de Ortega. This activity was conducted with 48 prospective Secondary education teachers enrolled on an on-line Masters' degree. We focus on the part of the activity related to generalization and, more particularly, on the relation between generalization and use of algebraic language. Our results show that there exists a positive significant relation between generalization and the use of algebraic language. However, we also found out that many students were able to provide correct and satisfactory generalizations using just natural language.

Introduction and objectives.

The use of history of mathematics in mathematics education can be approached in many different ways and with many different motivations and objectives [START_REF] Jankvist | A categorization of the "whys" and "hows" of using history in mathematics education[END_REF]. Regarding the classroom implementation, one interesting possibility is the use of historical problems and, in particular of problems "having clever, alternative, or exemplary solutions" (Tzanakis et al., 2000, p. 224).

In the context of teacher training, [START_REF] Mosvold | How mathematical knowledge for teaching may profit from the study of history of mathematics[END_REF] show how all the different domains of teachers' Mathematical Knowledge for Teaching (MKT) might profit from the use of history of mathematics. In particular, the analysis of historical problems and their solutions can be a very interesting task for prospective teachers [START_REF] Meavilla Seguí | Problemas de Relojes. Ejemplos Históricos y Consideraciones Didácticas[END_REF].

Generalization belongs to the two main domains of the MKT model described by [START_REF] Ball | Content knowledge for teaching: What makes it special?[END_REF]: Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK). From the point of view of SMK it is clear that generalizing is one of the main components of mathematical activity [START_REF] Harel | The general, the abstract, and the generic in advanced mathematics[END_REF]. Regarding PCK, Ellis (2007, p. 221-222) points out that "one of the primary aims of educational practice is to help students to develop robust, generalizable knowledge that will support their abilities to create generalizations in the classroom". Generalizing and specializing are two key aspects of one possible approach to algebra [START_REF] Lee | An initiation into algebraic culture through generalization activities[END_REF] and the use of symbolic algebraic language is strongly related to the generalization of arithmetic properties or to the modelization of different situations [START_REF] Usiskin | Conceptions of School Algebra and Uses of Variables[END_REF].

Even in the absence of symbolic algebraic language, the search for some kind of balance between generalization and specialization can be clearly seen in many ancient mathematical texts. These texts included very long collections of problems possibly hoping that the reader would be able to infer the general rules used to solve the seemingly different problems. In the Zhou bi suan jing, a Chinese text from the 3 rd century, this idea was even explicitly stated in a dialogue between a master and a student: Illuminating knowledge of categories [is shown] when words are simple but their application is wide-ranging. When you ask about one category and are thus able to comprehend a myriad matters, I call that understanding [my] Way. (Cullen, 1996, p. 15) This underlying philosophy can still be found in Fibonacci's Liber Abaci [START_REF] Sigler | Fibonacci's Liber Abaci. A translation into modern English of Leonardo Pisano's Book of Calculation[END_REF], for example, about one thousand years later. In fact, many mathematical texts written during the 16 th and 17 th centuries contained series of problems whose solutions were just the description of the operations that had to be performed with the given data.

In this work, we analyze part of an activity designed for and implemented with prospective Secondary school teachers which is based on an arithmetic problem excerpted from a 16 th century Spanish arithmetic textbook. Our main objective is to explore if and how the participants are able to provide generalizations of the statement and the solution of the problem. More particularly, we focus on the interplay between generalization and the use of algebraic language. In other words, we focus on two ways to generalize, according to which semiotic register is used: algebraic language or natural language.

The source.

The only known biographical facts about friar Juan de Ortega is that he was born in Palencia, that he was a Dominican and that he taught mathematics in Spain and Italy both publicly and privately [START_REF] Madrid | Los Libros de Aritmética en España a lo Largo del Siglo XVI[END_REF].

As far as we know, he only wrote one book, the Conpusicion de la arte de la arismetica y juntamente de geometria [START_REF] Ortega | Conpusicion de la arte de la arismetica y juntamente de geometría[END_REF]). Ortega's book was rather popular and it was reedited several times in different countries during the 16 th century [START_REF] Rey Pastor | Los matemáticos españoles del siglo XVI[END_REF]. In fact, it was reprinted (sometimes with slightly different titles) and translated at least in Roma and Lyon (in 1515), Messina (in 1522), Sevilla (in 1534, 1537, 1542[START_REF] Aurel | Libro primero, de Arithmetica Algebratica[END_REF] and Granada (in 1563). Some authors [START_REF] Carabias | Salamanca y la medida del tiempo[END_REF] point out that the 1515 translation published in Lyon was the first text on commercial arithmetic written in French. [START_REF] Marquant | Juan de Ortega (¿1480-1568?) (OP) y la autotraducción al italiano (1515) de su Arte de la Aritmética y juntamente de Geometría[END_REF] presents an interesting discussion on the authorship of some of these translations.

Regarding its content, the book consists of 204 folios organized according to 36 chapters and it covers the usual topics in a Renaissance commercial arithmetic text: elementary operations with natural and fractional numbers, progressions, square and cubic roots, rule of three and its applications and some elements of geometry. The editions of 1534, 1537 and 1542 included a method to approximate square roots that improved the known methods at his time, on this issue we refer the reader to the classical works by Rey Pastor (1926, pp. 72-81) and [START_REF] Barinaga | Sobre los ejemplos de Fr. Juan de Ortega[END_REF]. Finally, the 1552 edition included a collection of 13 problems solved using algebraic techniques that was inserted by Gonzalo Busto, the editor of this probably posthumous edition. This is an interesting feature of the text, since the first book with a systematic introduction to algebraic language written in Spanish, the Libro Primero de Arithmetica Algebratica [START_REF] Aurel | Libro primero, de Arithmetica Algebratica[END_REF] was published that very same year [START_REF] Puig | La Arithmetica Algebratica de Marco Aurel, primer álgebra impresa escrita en español. Preliminares para su estudio[END_REF].

In addition to many classical problems [START_REF] Métin | The arithmetic of Juan de Ortega: equations without algebra[END_REF], in chapters 14 through 17, Ortega provides a collection of 34 examples of what he calls "extraordinary rules" and that he defines (Ortega, 1512, fol. 60r) as "rules outside the usual way of adding, subtracting, multiplying and dividing and that involve hidden ways to apprise those who know little". They are, in fact, a collection of arithmetic problems and their solutions presented in a merely descriptive fashion. For example, the eleventh example from chapter 14 goes as follows (our translation):

If you wanted to know, or if it was asked to you, which are those three numbers such that two fifths of the first one is the same as three sevenths of the second one and the same as four ninths of the third one, you will do the following. Put the numbers as you see here:

. Then, multiply the 5 below the 2 by the 3 above the seven to get a 15. Multiply this 15 again by the 4 above the 9 and you will get 60, which is the first number. After that, multiply the 7 below the 3 by the 4 above the 9 to get a 28, which you shall multiply by the 2 above the 5 in order to get a 56, which is the second number. After that, multiply again the 9 below the 4 by the 2 above the 5 and you will get an 18 which you shall multiply again by the 3 above the 7 to get 54. This is the third number. If you want to check that it is true, look for the two fifths of 60 and you will see that it is 24. In the same way you will see that three sevenths of 56 is 24 and that four ninths of 54 is 24, as you see in the example. (Ortega, 1512, fols. 63r-63v) Activity and participants.

The experiment was carried out in the context of the subject "Didactics of arithmetic and algebra" during the academic year 2017-2018 with 48 students of the online Master's degree in Didactic of Mathematics in Secondary education from the International University of La Rioja. The age of the participants ranged from 25 to 56 (with a mean of 37.7 and a standard deviation of 7.9) and most of them (91.7%) had at least six months of prior experience teaching mathematics at secondary level.

Even though background of the students was diverse, most of them had prior teaching experience or a degree on a STEM or education discipline and hence a good knowledge of mathematics could be assumed. The designed activity was based upon the problem transcribed above. It consisted of three exercises, each of them addressing to a different issue. Being an online Master's degree, the students had to complete this activity at their homes and they were given a two-week deadline in order to obtain as many answers as possible.

 In exercise 1, a short version of the original problem was presented using modern natural language and the students were asked to solve it using whichever method they wanted.

 In exercise 2, a short version of the original solution was presented using modern natural language and the students were asked to:  Give their opinion about this solution.

 Compare this solution with their own solution from exercise 1.  Explain which of the two solutions is "better" and in which sense.  In exercise 3, the students were asked to provide a general version of the statement of the problem from exercise 1 and a general solution based on the solution from exercise 2.

As we see, exercises 1 and 3 mostly deal with purely mathematical aspects so they are more related to SMK. Exercise 2, on the other hand, is more focused on PCK because it involves the analysis of different solutions to the same problem, their comparison and the reflection about the advantages and disadvantages of each of them. In this work, we will present the results of the analysis of exercise 3. In particular, we will focus on two variables: generalization and language. For the first one we will focus on its correctness (correct or incorrect) and on its completeness (partial or full) while for the second one we will consider the attributes natural and algebraic.

Results.

Although in this work we focus only on exercise 3, it is interesting to point out that almost all the students (46 out of 48) were able to solve the problem in exercise 1 at some extent. However, not all the answers were equally complete. example, very few students mention that the problem has an infinite number of solutions. In any case, we conclude that the students were able to understand the problem Exercise 3 was answered by 47 out of the 48 participants. Only 28 of them (about 60%) were able to provide some kind of correct generalization for the statement and the solution of the problem. In Table 2 we see that those participants that used algebraic language were more capable to provide a correct generalization of the situation. In fact, performing a χ 2 test of independence, we obtain a statistically significant relationship between both variables (significance level of about 95%).

Algebraic language Natural language Correct generalization 18 10 Incorrect generalization 7 12 Most of the wrong answers that tried to generalize the problem using only natural language simply provided incomplete statements. One student's answer, for example, was: "Do the following exercise, multiplying one denominator by the remaining numerators", which is a just simple explanation of the method rather than a generalization. The incorrect generalizations that used algebraic language mostly looked like the example in Figure 1. The statement of the problem (top of the figure) is expressed incorrectly and, at the bottom, we just find the required operations expressed symbolically.

It is noteworthy that some students seem to identify the process of generalization just with the use of symbols. Figure 2 is an interesting example of this phenomenon. There is no generalization in the student's answer, just the use of the letters X, Y and Z to denote the unknown numbers. We recall that 28 students provided a correct generalization of the original problem and its solution.

However, not all of them provided a full generalization of the situation. In Table 3 we analyze the relation between the use of algebraic language and the degree of generalization of the answer.

Algebraic language Natural language

Full generalization 4 7

Partial generalization 14 3

Table 3: Degree of generalization versus use of algebraic language

We obtain again a statistically significant relationship between both variables (significance level of about 95%) showing that the students that were able to correctly generalize the original situation using natural language provided full generalizations more often than their partners. The following transcription is a good example of full generalization that essentially uses only natural language:

If in a set of n numbers that are proportional to each other in a rational way, we multiply the denominator of one of them by the (n -1) remaining numerators, we obtain that number. If we successively proceed in this way, we will obtain the n numbers of the solution. Since it is a compatible and indeterminate system with (n -1) equations and n unknowns, the system has infinitely many solutions… Regarding the 4 answers that fully generalized the problem using algebraic language, three of them combined algebraic and natural language and only one (Figure 3) could be described as "completely algebraic". Only one of these 4 answers stated that the problem has infinitely many solutions. Now, 17 students provided correct generalizations that were only partial generalizations. In particular, they only considered a situation with three unknown numbers as in Figure 4. Even if most of these 17 answers were very similar to that in Figure 4 (only varying in the degree of use of algebraic language), two of them were particularly incomplete generalizations. For example, in Figure 5, a student considers arbitrary fractions in the statement of the problem but imposes a very particular relation between numerators and denominators inspired by the particular numbers of the original problem. The other restricted generalization imposed the additional conditions that the numerators (n i ) of the three fractions are consecutive numbers and the denominators are of the form 2n i + 1.

Discussion and final remarks.

From the purely mathematical point of view, it was somewhat surprising that about 40% of the students were not able to provide a correct generalization of the given situation. This might be caused in some cases just by some lack of SMK (Figures 1 or 2) while in other cases it might be related to a misconception about generalization that identifies it with the mere use of symbols in any way (see Figure 2). Regarding the completeness of the generalizations we see that, in terms of Ellis (2007, p. 235), many students generalize just by removing particulars (what we called partial generalization, like in Figure 4) rather than by expanding the range of applicability (what we called full generalization, like in Figure 3). It is also noteworthy that some students show a different behavior between the statement and the solution. For example, in Figure 2, the student does not generalize the statement at all (he only introduces letters for the unknown quantities) but in the solution he refers to "each fraction" instead of to particular values. This fact probably deserves more attention.

Throughout the students' responses there is a varying degree in the use of algebraic language. From those using only natural language to those using only algebraic language. Nevertheless, most of the students lie in between showing in their answers a mixture of algebraic and natural language. The results of our work suggest that the use of algebraic language facilitates the process of generalization of the considered problem, illustrating, as [START_REF] Usiskin | Conceptions of School Algebra and Uses of Variables[END_REF] points out, that one of the main uses of variables is as "pattern generalizers". However, and somewhat paradoxically, this use of algebraic language seems to constitute some kind of obstacle in order to fully generalize the considered situation. There can be several explanations for this phenomenon. One reason can be the context of the exercise, since the answer that was provided to the students only used natural language. Another reason can be related to SMK, because some students might be uncomfortable using algebraic language. Finally, the beliefs, conceptions and expectations of the students about the notion of generalization [START_REF] Strachota | Conceptualizing generalization[END_REF] might play an important role in their performance. Zazkis and Liljedahl (2002, p. 400) reached similar conclusions finding out that "there is a gap between students' ability to express generality verbally and their ability to employ algebraic notation comfortably". Nevertheless, it must be noted that they worked with preservice elementary school teachers in a context of pattern generalization, while we have worked in a purely arithmetic setting with Secondary education teachers. Either way, there exist contradictory results in the literature. For example, in [START_REF] Richardson | Prospective elementary teachers use of representation to reason algebraically[END_REF] most of the elementary prospective elementary teachers participating in the study expressed their explicit generalizations using algebraic notation.

The designed activity, an example of using history as a tool [START_REF] Jankvist | A categorization of the "whys" and "hows" of using history in mathematics education[END_REF], involved other exercises. We have already mentioned that most of the students solved the problem correctly. However, after a preliminary and incomplete analysis, we have observed different solving methods some of which involve the use of other semiotic registers besides algebraic or natural language. For example, a few students solved the problem using diagrams. It would be interesting to analyze the influence of different aspects of the students' solution to the problem over their ulterior process of generalization.

Figure 1 :

 1 Figure 1: Incorrect generalization

Figure 2 :

 2 Figure 2: Use of symbols that does not imply generalization

Figure 3 :

 3 Figure 3: Full generalization using only algebraic language

Figure 4 :

 4 Figure 4: Partial generalization using algebraic language

Figure 5 :

 5 Figure 5: A very restricted generalization

Table 1 : Some data about the sample

 1 Table 1 provides further information about the participants.

	Gender		Nationality			Degree
	Male Female Spain Colombia Ecuador Mexico STEM Education Other
	52%	48%	17%	44%	37%	2%	61%	35%	4%

Table 2 : Correctness versus use of algebraic language

 2

Acknowledgment

This work has been partially funded by the Government of Aragón and the European Social Fund via the research group S36_17D "Investigación en Educación Matemática".


Stephan Berendonk
email: stephan.berendonk@uni-due.de

On the didactical function of some items from the history of calculus

Keywords: Didactics of analysis, meaning, coherence, misconceptions

The article draws attention to some individual results from the history of calculus and explains their appearance in a course on the didactics of analysis for future high school mathematics teachers. Specifically it is shown how these results may serve as a vehicle to address and reflect on three major didactical challenges: determining the meaning of a result, concept or topic to be taught, teaching mathematics as a coherent subject and understanding the source of misconceptions.

Three didactical challengesmeaning, coherence and misconceptions

Let us consider a typical first semester university course on real analysis. With such a course we may associate the following keywords: axioms for the real numbers, convergence of sequences and series, Heine-Borel, Bolzano-Weierstrass, differentiability, monotony, continuity, integrability, intermediate value theorem, mean value theorem, fundamental theorem of calculus and epsilondelta proofs. This course has a lot to offer to its students. It introduces them to the mathematical quest for rigor, generality and structural results. It shows that the knowledge about differentiation and integration learned at school can be embedded into a beautiful large-scale deductive system, and thereby exemplifies the architectural nature of mathematics.

One might believe that this course would form a sufficient background regarding the mathematical content knowledge with respect to analysis for a future high school mathematics teacher. However, teaching analysis at school may serve quite different aims than the university course we considered. For instance one might want to teach analysis as a tool to formulate and understand some Newtonian physics or more generally speaking, on might want to teach analysis as an example showing the applicability of mathematics and its role for the other sciences. In any case, in contrast to the university course, analysis nowadays usually isn't taught architecturally at school and according to [START_REF] Freudenthal | Mathematics as an Educational Task[END_REF] and his legacy it wouldn't be a good idea to do so either. One should be acquainted with a subject, with local orderings of its results and concepts before one should work on or be confronted with a global ordering of that subject. From this situation, namely teaching analysis in a different way, a didactical challenge arises: Find the role and meaning of the concepts and results to be taught with respect to the overall approach chosen, e.g. genetic, applicational or problem-oriented. In the axiomatic approach the concepts and results receive their meaning from their place within the system. The mean value theorem for instance occupies a central position within the system. It can be seen as the linchpin of the theory. Continuity is treated as a central concept. In a different approach, however, the roles and meanings of the concepts and results might be completely different.

A central concept developed and used by the German-speaking educational sciences in order to direct the discussion of school curricula is called `Allgemeinbildung´. The concept elaborates on the functions of school and thereby it serves as a framework and a benchmark for the development and analysis of curricula. Each school subject is then requested to determine its individual contribution to the Allgemeinbildung. In Heymann's (1997, p.2) version of the concept, which is especially prominent in mathematics education, one major task of school is to teach the subjects in such a way as to endow cultural coherence, meaning not only the relationship between past, presence and future, which Heymann would rather have called `cultural continuity´, but also the interrelations of different subcultures. Analysis seems the perfect choice when trying to reveal the interrelations of mathematics and physics. However, even in mathematics itself there are subcultures or rather subdomains, whose relationship one should uncover. The didactical challenge, therefore, consists in discussing the unity of mathematics and asks the following questions: How is analysis related to geometry, how is it related to algebra? The first semester course on real analysis imagined hardly addresses this issue, since it rather focuses on presenting analysis as a self-contained subject.

Teacher students are usually keen to learn about pupils' misconceptions and the best recipes on how to treat them. It is important for them to notice, however, that the instruction itself may cause some of those misconceptions and that a different instruction, using for instance a different approach, may again cause quite different misconceptions. Thus, some misconceptions may be 'typical' in general, whereas some of them may only be 'typical' relative to a given instruction. It is an important didactical challenge to determine the misconceptions belonging to a certain instruction. Since every mathematics teacher student shares the experience of roughly the same first semester course on real analysis, it appears to be a good exercise to think about misconceptions that this course might favor.

In the preceding passages I have formulated three didactical challenges ('meaning', 'coherence' and 'misconceptions') and I have tried to indicate their importance for mathematics teacher students. These challenges form an integral part of my course on the didactics of analysis. My students, when confronted with these issues in the beginning of the course, have a hard time dealing with them. This is understandable, considering that their analytical background consists solely of a first semester course on real analysis. However, teaching a few episodes from the history of calculus, already helps them to handle the challenges in a much more appropriate and satisfying way. I will now, for each challenge separately, mention the results from the history of calculus that I teach and I will explain how they may contribute to the discussion on the respective challenge.

Meaningthe integral integrates, the calculus calculates

What is the meaning of the integral? What is the meaning of the calculus in general? What is the meaning of the fundamental theorem of calculus? A mathematics teacher needs to be able to answer these questions in a satisfying way, since 'how' he teaches a certain subject or course should of course highly depend on 'why' he regards the content as being worth to be taught in the first place. In my view, the best way to approach these questions is to look at the prehistory of the calculus. Let us consider Roberval's determination of the area (see [START_REF] Whitman | Some Historical Notes on the Cycloid[END_REF] under the cycloid (Figure 1) and his comparison of the length (see [START_REF] Pedersen | Roberval's Comparison of the Arclength of a Spiral and a Parabola[END_REF]) of an Archimedean spiral with that of a certain parabola (Figure 2). To solve the 'area' problem Roberval uses Cavalieri's principle. To solve the 'length' problem he employs his kinematic approach to tangents (see [START_REF] Pedersen | Roberval's Method of Tangents[END_REF]. Thus, the two solutions are based on two completely different methods. This is interesting, since in the world of calculus both, the length problem and the area problem, have become one and the same, in the sense that both ask you to compute some integral. Apparently the integral does what it promises. It integrates the two basic problems associated with a curve. Determine its length and its area. Of course the concept of the integral incorporates 'volumes', 'centers of gravity' and many other things as well. Wouldn't this be one possible meaning of the integral that we might wish to convey to learners of mathematics? After all Poincaré once said (see [START_REF] Verhulst | An interview with Henri Poincaré -Mathematics is the art of giving the same name to different things[END_REF]:

Mathematics is the art of giving the same name to different things. (Poincaré, 1914, p.23) Let us now think about the meaning of calculus and its fundamental theorem. To this end, let us in addition to Roberval's two results consider Archimedes' quadrature of the parabola (see Dörrie, 1965, p.239) and Huygens' determination of the area (see Maanen van, 2003) between the cissoid of Diocles and its asymptote (Figure 3). After looking at these four problems and their solutions one might get the impression that before calculus every curve was treated individually, that at that time people were extremely well acquainted with the geometrical properties of these curves and that one had to come up with a new ingenious idea for every other curve. Today, on the other hand, it needs no ingenuity to solve these questions. With the calculus at hand, these problems have become more or less straightforward routine exercises. The routine goes like this: Formulate the problem in terms of an integral and apply the fundamental theorem, that is, look for an antiderivative and plug in the limits of integration. Finding an antiderivative might be a problem, but that is where the calculus with its powerful tools, integration by parts or by substitution, sets in. We could therefore view the calculus as a facilitation of a major part of mathematics. With calculus everyone has the chance to learn how to handle areas under curves. It seems to be exactly this meaning that Leibniz himself saw in his calculus:

For what I love most in this calculus is that it gives us the same advantage over the ancients in the geometry of Archimedes as Viete and Descartes gave us in the geometry of Euclid and Apollonius; by relieving us of working with the imagination. (as cited in Blåsjö, 2016, p.17)

Coherencecalculus seen as the algebra for transcendental curves

Which of the following terms does not fit with the others? 'Similarity of triangles', 'Thales' theorem', 'Pythagoras' theorem', 'differential equation'.

Of course the last term does not fit, since the first three terms belong to geometry, whereas the last term belongs to calculus. Huygens' determination of the area of the cissoid, however, may cast some doubts about this decision. To the modern eye his problem might appear as a 'calculus problem'. His method, on the other hand, is purely geometrical. Thales, Pythagoras and similarity (compare Figure 3), these are exactly the tools that Huygens employs. Calculus and geometry, this is the morale of Huygens' result, are not as divorced from each other as the typical courses on these subjects might let one think. By looking for instance at Johann Bernoulli's lectures on the calculus [START_REF] Bernoulli | Die Differentialrechnung. Ostwald's Klassiker der exakten Wissenschaften Nr[END_REF] one realizes that differentials and their equations were just an additional tool incorporated into the old game of geometry. Thales, Pythagoras, similarity and a differential equation, in Bernoulli's determination of the tangents to a cycloid (Figure 4) we have all this at one place. We now turn to the relationship between calculus and algebra. A major challenge in teaching these two subjects coherently is to show where the algebra ends and where the calculus begins, that is, to reveal their line of demarcation. At least according to Leibniz the line runs precisely between the algebraic curves and the transcendental curves:

As far as the differential calculus is concerned, I admit that there is much in common between it and the things which were explored by both you [Wallis] and Fermat and others, indeed already by Archimedes himself. Yet now the matter is perhaps carried much further, so that now those things can be accomplished which in the past seemed closed even to the greatest geometers as Huygens himself recognized. The matter is almost the same in the analytical calculus applied to conical curves or higher: Who does not consider Apollonius and other ancients to have had theorems which present material for the equations by which Descartes later preferred to designate curves. In the meantime the matter has been reduced to calculation by the method of Descartes, so that now conveniently and without trouble that can be done which formerly required much effort of contemplation and imagination. In the same way, by our differential calculus, transcendentals too, which Descartes himself excluded in the past, are subjected to analytical operations. (as cited in Blåsjö, 2016, pp.16-17) In this picture the relationship between geometry, algebra and calculus is beautifully simple. At first we have the Euclidean geometry with compass and ruler only. Then Descartes allows some other kinds of construction tools, which on the side of the equations correspond exactly with the algebraic curves. This world can be studied by means of algebra. Leibniz, finally, goes even further and includes modes of constructions that allow also transcendental curves to enter geometry. Algebra falls short of these curves and calculus is needed. A good way to experience this situation is by starting with Descartes' double-root method for finding normals and tangents and by trying to get as far as possible with this method. This is performed in detail in (Range, 2011). One first of all shows that there is a unique line through each point of the graph which intersects the graph at that point with multiplicity at least 2. Then one generalizes this fact to rational functions and finally to algebraic functions. When it comes to transcendental functions, of course, one gets stuck. It is at that point, that one has to generalize one's definition of tangent. It is at that point, where the 'limits' come into play.

Misconceptionstangents require limits, areas require rectangles

Instead of Descartes' double-root method we might also consider the notion of tangent that Euclid employed in his third book when dealing with the circle:

A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. (Euclid, 1956, p.1) How far can we get with this notion? If we take the notion literally, it already fails when confronted with a parabola, since through each point of a parabola there are two different lines that have only one point in common with the parabola, one of them being a line parallel to the axis of symmetry of the parabola. We can, however, rule out these parallels by requiring the tangents to stay on one side of the parabola. This leads to the following modification of Euclid's definition: A line, which has a common point with a curve, is called tangent to that curve, if the curve lies on one side of the line and if it is the only line through the common point with that property. Huygens, for instance, worked with this notion, as one can see in his work on the cycloid (see Aarts, 2015, pp.3-4). For convex curves this notion works fine, but already in the case of such a simple non-convex curve as the cubical parabola (

) not one single point would have a tangent according to this notion. However, for all but one point on the cubical parabola, the problem is easily resolved by only requiring the curve to stay locally on one side of the line. The point not covered by this extended notion is the inflection point. It seems that we have to give up on Euclid's notion of tangents ultimately when it comes to inflection points. Arnold Kirsch, however, has shown in (Kirsch, 1960a) how to overcome this obstacle. After defining smoothness of convex functions by means of the modified version of Euclid's notion of tangent, he goes on to define smoothness of functions in general essentially as follows: A function is smooth in some point, if the function lies locally between two smooth convex functions that both go through that point and that share a common tangent at that point. In (Kirsch, 1960b), Arnold Kirsch shows that this notion of smoothness is equivalent to our usual notion of differentiability. This is certainly an interesting result, since it clashes with the widespread belief that one needs the concept of limit in order to define differentiability and tangents in general.

Another major misconception I come across among teacher students more often than one might imagine is the belief that one can determine areas associated with curves only by means of the Riemann integral or rather by exhausting the relevant area with rectangles. This conception seems to consolidate itself in the course of the received instruction, since children, which have not yet been 'enlightened' by a course on calculus, apparently still use all kinds of figures, triangles, trapezoids, squares or whatever suits, in order to cover the area in question. Against this background it seems important to elaborately discuss examples such as Archimedes' quadrature of the parabola (see Dörrie, 1965, p.239), which uses triangles, or Huygens' determination of the area of the cissoid (see [START_REF] Maanen Van | Precursors of Differentiation and Integration[END_REF], which uses trapezoids and triangles instead of the notorious rectangles.

Summary and discussion

In this article I have shared my approach on how to address three major challenges ('meaning', 'coherence' and 'misconceptions') in a course on the didactics of calculus for future high school mathematics teachers. In order to discuss the meaning of the calculus I provided the students with results from Roberval and Huygens, which show firstly, what can be done even without calculus and secondly, how much insight is needed for this. In this way the students may see and appreciate the calculus, just as Leibniz apparently does, as a 'mechanization' of geometry. It should be critically noted that the historical episodes chosen play within geometry and due to this we have left the meaning of calculus for other branches of mathematics and especially its role in Newtonian physics completely untouched. In order to show the coherence between calculus and geometry I offered some examples from Bernoulli's lectures on the differential calculus. To this end it is also worthwhile to consider two different rather recent geometrical approaches [START_REF] Apostol | Tangents and Subtangents Used to Calculate Areas[END_REF][START_REF] Apostol | Tangents and Subtangents Used to Calculate Areas[END_REF][START_REF] Kaenders | Flächenbestimmung bei Basisfunktionen der Schule mit Elementargeometrie[END_REF][START_REF] Kaenders | Flächenbestimmung bei Basisfunktionen der Schule mit Elementargeometrie[END_REF] to the integration of the basic functions, treated in high school. Both of them, the sweeping-tangent approach by Apostol and Mnatsakanian and the approach using geometrical transformations by Kaenders and Kirfel link calculus and geometry each in their own way. To show the coherence between calculus and algebra I focused on the difference between algebraic and transcendental curves, especially by maxing out Descartes' double-root method. Finally, regarding the misconceptions, I tried to point out that some of them might be 'instruction-generated'. For example, I showed that generalizing Euclid's notion of tangent can lead to a 'limit-free' general notion of tangent, a fact that seems to clash with the calculus related beliefs of many students. By way of the three didactical challenges I tried to show that and to indicate how courses on the didactics of mathematics may benefit from a selective and targeted recourse to the history of mathematics.

Figure 1 :

 1 Figure 1: Roberval -The area of the cycloid

Figure 3 :

 3 Figure 3: Huygens -The area of Diocles' cissoid

Figure 4 :

 4 Figure 4: Bernoulli -The tangent of the cycloid


Tanja Hamann

New Math at primary schools

come

New Math at primary schools in West Germanya theoretical framework for the description of educational reforms

Tanja Hamann Universität Hildesheim, Germany; hamann@imai.uni-hildesheim.de

To describe the implementation of New Math in West German primary schools (known as "Mengenlehre"/"set theory" up to today) in the 1970s, a framework model is developed. The model shows an educational reform as a two-dimensional process, characterized by a chronological as well as an institutional dimension wherein for the latter the educational system is divided into four layers. The model is introduced and specified when used for the description of New Math. Suitability of the framework for this suggests that it might be well transferable to the historical description of further educational reforms.

Elementary school mathematics, elementary school curriculum, history of mathematics education, New Math.

Initial questions and aim

In West Germany the New Math movement had a massive impact on mathematics education at primary schools from 1972 on, especially regarding the curricular contents and classroom methods of elementary instruction. The most remarkable innovation was the introduction of concepts from naïve set theory to young children and the use of materials such as colored blocks (the most famous being the Logic Blocks credited to Z. P. Dienes) in the classroom. However, there is still a broad consensus within a considerable part of German society that the reform failed miserably. A subsequent valuation whether the reform, its concepts and implementations were "any good" cannot be accomplished nowadays as there exists hardly any empirical data from back then. A statement whether "Mengenlehre" must be considered a failed experiment from the historical point of view, though, would require a thorough definition of the concept of "failure", whichas far as the author of this paper is awaredoes not exist in this context. An estimation of the historical significance of New Math within the development of primary mathematical education in Germany is yet to be given.

Hitherto, existing analyses of the reform in the Federal Republic are widely limited to works by contemporary witnesses, naturally lacking historical distance [START_REF] Damerow | Die Reform des Mathematikunterrichts in der Sekundarstufe I: Eine Fallstudie zum Einfluß gesellschaftlicher Rahmenbedingungen auf den Prozeß der Curriculum-Reform[END_REF][START_REF] Zumpe | Der neue Mathematikunterricht in der Grundschule: eine kritische Analyse auf dem Hintergrund bildungspolitischer Zielsetzungen Anfang der 70er Jahre[END_REF][START_REF] Keitel | Entwicklungen im Mathematikunterricht[END_REF][START_REF] Keitel | Réformes et développements de l'enseignement mathématique en R.F.A. depuis[END_REF], and short summaries as part of general overviews of the historical development of primary education (e. g. [START_REF] Padberg | Didaktik der Arithmetik: für Lehrerausbildung und Lehrerfortbildung[END_REF]. The need for a broader account that allows a historical grading of the "Mengenlehre" reform and a well-founded verdict on its historical impact and role within German mathematics education is obvious. Any judgement from a historical point of view requires a comparative integration of the event into a broader context, and this again requires a description of the historical event first. The challenges one meets when taking up the description of the introduction and implementation of New Math in West German primary schools must be considered typical, as they are inherent in any reform within a public educational system. The aim of this paper is to introduce a theoretical model, which provides a framework that has proven suitable for describing the case of "Mengenlehre" reform. It will then be shown how the model has been used for this purpose.

A framework model for the description of educational reforms

A reform is never an isolated event, but a process bound in a multitude of influences (general cultural and societal conditions as well as current developments) and outcomes, and for describing it numerous aspects and directionsor dimensions, as will be the term used in this papermust be taken into consideration and be structured in a sensible way. The obvious dimension is the chronological one as a reform does not happen in an instant but in a certain timeframe. Not only is there an initial situation as well as an aftermath, but the reform period itself takes time and in the course of events ideas, conditions and actions might be subjects to change. But especially in the case of governmental reforms there is another, non-chronological dimension, which might be called the institutional one. Numerous protagonists act on diverse institutional levels and layers and thereby take an active part in the course of reform. These institutional layers do not function independently, but they influence each other in a certain way. So altogether, the description of a public educational reform must consider multi-dimensionality of reforms. In order to allow describing the institutional dimension a model is needed, which identifies separate layers and their respective protagonists.

H. Fend (2008, pp. 30-33) has developed a model for the educational system consisting of four (more or less) separate layers that influence each other from top to bottom. In his model Fend assumes that the protagonists on each of the levels interpret the ideas, that are covered in products from the respective level above, from their very own point of view, in their own specific contexts, and that this re-interpretation always necessarily leads to substantial changes, shortenings and adjustments. He denotes these changes and adaptations as "recontextualizations" (p. 13). The four layers in Fend's model are the following:

1. the layer of culture: within each cultural sphere certain ideals of education exist and from these educational objectives are generated 2. the layer of curriculum: educational aims serve as starting point, from which subjects and contents are deduced that seem suitable to meet the aims; products on this layer are curricula as well as textbooks etc.

3. the layer of practical schooling: this is the layer where teachers plan and perform their lessons regarding the specific nature of their students 4. the layer of reception: pupils (and their parents) comprehend and assimilate contents in their specific, subjective way and make sense of it

Fend has generated his model in the context of his studies in educational sociology, it is therefore not fully applicable to the history of mathematics education. It serves as a suitable starting point but needs to be adjusted to the subject of educational reform.

As the model considers the educational system as one, the subjects play a minor role. Educational research institutions appear as part of the layer of curriculum where their function is limited to a descriptive one, whereas their normative function in designing educational concepts is not mentioned. Looking upon the educational system from a subject didactical point of view the scientific discipline as well as didactical research and design need an evident place within the model, and regarding New Math this applies even more, as on the one hand the reform has been grounded on subject science to a high degree and on the other hand mathematics educationalists were widely responsible for designing practical concepts, courses and textbooks. Hence, it is obvious that (despite some overlappings) there is a considerable difference between the protagonists responsible for writing curricula (mainly personnel in politics and administration) and those writing textbooks. Due to their importance for classroom practice mathematics didactics is given a layer of own.

Science in general is part of culture putting it on the layer of culture. However, it belongs to a global kind of culture, that is not limited by a community's borders. Undoubtedly, national culture and traditions in general are highly relevant for any educational reform within a nation's borders but they comprise such a vast number of aspects that they cannot be described in detail within the frame of a work like this and are therefore neglected. Mathematics as a theoretical discipline is different from didactical science regarding the latter's practical implementations and the former's function in providing a basis for the latter's work. That is why the two scientific branches must belong to different layers. Of course, the scientific subject is not the only discipline educationalists refer to when designing their concepts, educational sciences like pedagogy and the psychology of learning play an important role, as well.

As mentioned before, when it comes to New Math contemporary empirical data is scarce and gathering an amount of data to be able to describe the layer of reception in a satisfying way (e. g. by interviewing former pupils) is certainly a desideratum in due research, but as it cannot be part of this work either the layer is integrated in the layer of school practice in this case.

From all these specifications in consideration of an educational reform derive the following adapted layers of the educational system:

1. the layer of theoretical scientific discipline: institutions and protagonists on this layer are scientists from subject and educational science; their products are subject-specific foundations, scientific theories for example on the development of knowledge etc.

2. the layer of curriculum: institutions and protagonists are personnel from politics (e. g. from the ministry of education) and administration, sometimes educationalists as far as they contribute to decisions concerning curricula; their products are curricula, syllabi, decrees etc.

3. the layer of classroom concept: institutions and protagonists are subject didactics, textbook authors (these can be teachers, as well), editors and publishers of schoolbooks etc.; their products are textbooks, workbooks, teacher's handbooks, manipulatives, suggestions for instruction etc.

4. the layer of practical schooling: institutions and protagonists are schools, teachers in classroom, pupils, parents; their products are lessons, learning progress, knowledge as well as tests, written tasks, exercise books, portfolios, reports and so on

As for personal overlaps between the layers (i. e. someone acts on more than one level) it is assumed that the actor's function is a different one on each layer and that they must recontextualize their own ideas in another context, as well.

On each of these layers the chronological dimension additionally comes to effect. If we stick to the top-down model developed by Fend, the adapted model can be visualized as follows, the arrows meaning "recontextualized by". This framework model now allows concretization dependent on the specific reform that is about to be described, the sources that have been selected and the focus chosen. Guiding questions that are implicitly suggested by the model are:

1. What central aims and/or ideas (on content, didactical principles, methods, curriculum concept…) can be deduced from the products of each layer?

2. How can one characterize the process(es) of recontextualization between the different layers?

3. How did ideas and concepts develop and change over time?

New Math in West German primary schoolssources, focus and findings

Choice of sources and focus

We are already dealing with a model naturally underlying diverse reductions. Nevertheless, further reductions become mandatory when one chooses exemplary sources for each layer and decides on a focus.

It has been mentioned before that sources for pupils' (and parents') reception of "Mengenlehre" courses are scarce, the same is true for the whole layer of practical schooling, therefore this part of the educational system will not be the focus. From a subject didactical point of view the layers on which scientific personnel is most activethat are the layer of theoretical scientific discipline and, at least in the case of New Math, the layer of classroom conceptsseem the most relevant, that is why the emphasis will be laid on these. By this, the layer of curriculum gets assigned the role of an intermediary, which is relevant, if one wishes to describe the recontextualization that has been taking place between the other two layers, so that it must be looked at, as well.

In this case however, analysis of curricular documents leads to a specific difficulty, based on the German federal system, which involves that issue of curricular documents belongs to the federal states' field of responsibility. Hence, the number of syllabi that has been generated during the time of reform cannot be surveyed at once, and there is no work yet giving the due overview of primary school curricula in the way [START_REF] Damerow | Die Reform des Mathematikunterrichts in der Sekundarstufe I: Eine Fallstudie zum Einfluß gesellschaftlicher Rahmenbedingungen auf den Prozeß der Curriculum-Reform[END_REF] has provided of secondary school curricula. There are few nationwide documents, though, which must be taken into consideration. Explicitly concerning mathematics as a school subject, there are two directives on behalf of the Kultusministerkonferenz (the common conference of the federal states' ministers of education, referred to in figure 2 as KMK), the first one from 1968, which at the same time is seen as the starting point of reform in Germany, and the second one from 1976. In order to get an impression of at least one example of federal state curricula (Rahmenrichtlinien, referred to in figure 2 as RRL), those of Lower Saxony were chosen. The latest of these syllabi shows a massive decrease of the concept of "Mengenlehre" and therefore, its year of release, 1984, marks the terminal of the time axis.

Of course, the sources for classroom concepts are numerous, as well. Due to the facts that during the New Math reform primary school textbooks were written by mathematics educationalists and that they were widely accompanied by teachers' handbooks giving explicit insight into aims, concepts, didactical foundations and methods intended, samples from textbook series provide adequate sources for the layer of classroom concept. Here, the sample that has been chosen for thorough analysis of conceptual elements comprises 1 st grade materials from three textbook series.

The textbook series are alef by H. [START_REF] Bauersfeld | Alef 1: Wege zur Mathematik, Handbuch zum Lehrgang[END_REF], Wir lernen Mathematik ("We learn mathematics", referred to in figure 2 as N & S) by W. [START_REF] Neunzig | Wir lernen Mathematik I: Erstes Schuljahr[END_REF] and Mathematik in der Grundschule ("Mathematics in primary school", referred to in figure 2 as F & B) by A. [START_REF] Fricke | Mathematik in der Grundschule 1: Ausgabe B Lehrerband[END_REF]. Alef is a textbook series that derived from the only large-scale empirical project in the history of New Math in Germany, namely the Frankfurter Projekt, in the course of which children were educated with specially designed all new materials throughout their elementary school years. Wir lernen Mathematik was the earliest textbook containing "Mengenlehre" to be released in West Germany, even earlier than the first Kultusministerkonferenz directive, and thus it holds a special role in the process of the reform. Mathematik in der Grundschule has been chosen because it initially appeared in pre-reform times and has then been adapted to the reform standards. Of all three textbook series several editions exist, which are compared and thus contribute to the question how concepts developed and changed along the time axis.

As for the layer of theoretical scientific discipline, pedagogy has usually not been a point of reference for "Mengenlehre" concepts, so there is no need to consider the subject of pedagogy any further. Scientific mathematics, especially the Bourbaki work, is said to have had a big impact on the global New Math movement but going into the scientific subject any further would lead astray.

A major impulse to the start of reform activities within Europe is attributed to the seminar, which was organized by the OEEC (precursor of the OECD) and took place in Royaumont, France, in 1959. The seminar was attended by high school teachers and university mathematicians, who put a strong focus on how modern scientific mathematics could find its way into school, turning them into protagonists from the layer of theoretical scientific discipline. So, the report from the Royaumont seminar (OEEC, 1961) serves as source for the influence of scientific mathematics and according to the impact of the event, the date of the seminar serves as starting point of the time axis.

Another discipline the authors of textbooks have constantly referred to is the psychology of learning. Especially the landmark findings and theories by J. Piaget (e. g. [START_REF] Piaget | Die Entwicklung des Zahlbegriffs beim Kinde[END_REF] were most prominent. J. S. Bruner on the other hand has had a big influence even on modern day German curricula and his theories, which emerged in pre-"Mengenlehre" time, as well, were well-known. One of the biggest influences on primary school math certainly was Z. P. Dienes. Dienes is not easily allocated to one of the layers as he provided a vast amount of concrete classroom examples. As he never created a complete course, though, but rather provided theoretical foundations for German courses which partly arose from his own psychological research, Dienes is assigned to the scientific layer, as well.

From the above the general model can now be specified as it is filled with names of the documents that serve as sources for the description of New Math in West German primary schools. The arrow indicates the focus on the process of recontextualization from basic ideas and concepts as they can be derived from the layer of theoretical science to their concretization in textbooks on the layer of classroom concepts. Regarding the focal question for recontextualization of the scientific roots of reform it becomes obvious that the extent to which original ideas have been implemented differs. Altogether they have not been fully transferred to textbooks, but the authors of each course have selected a certain part and by this, shortened the original concept towards their own respective purpose. Bauersfeld comes closest to the ideas from the scientific layer but in this case, he was forced by decisions coming out of the curriculum layer to apply changes to his concept. The shortcomings that are caused by the incomplete transfer of the original concept altogether lead to diverse inconsistencies in the implementation of "Mengenlehre". Independently of what happened on the layer of practical schooling, this finding even allows us to state that New Math at West German primary schools has indeed been a failure, and as educational reforms never go without recontextualization processes there is a high probability for extensive innovations in the classroom to fail according to this definition of failure.

Another finding concerns the influence of the layer of practical schooling. The sources suggest that teachers' customs, beliefs and attitudes, along with an education that did not meet the demands presented by New Math, proved as an obstacle in classroom. Many parents are said to have had reservations towards reform, anyway, and these most probably increased as soon as things were not implemented according to plan. Presumably, one political reaction to this was a change in curricula, which in turn led to a change in textbooks and therefore, regarding recontextualization processes along the time axis, one finds that the process of reform cannot be described using a top-downmodel. Instead, within the framework model, mutual influences rather must be pictured the following way: In this case, even though the original model did not prove fully suitable for a description of New Math in West German primary schools, it still served as a useful device for illustrating the process of reform.

Conclusion

It was the aim of this paper to introduce a framework model for description of complex educational reforms from a historical point of view. As for the example of New Math in West German primary schools, the model proved suitable for providing a restricting frame, a helpful device for structuring the work and a starting point that could be adjusted according to the process of the reform. It therefore allowed relevant findings which may function as a ground on further research on the topic. All of this suggests that the model of recontextualization might be suitable for historical studies on other educational reforms, as well.

Figure 1 :

 1 Figure 1: Adjusted layer model

Figure 2 :

 2 Figure 2: Layer model filled with sources used for the description of "Mengenlehre" in West Germany

Figure 3 :

 3 Figure 3: Layer model showing the process of reform

This paper presents an extract from a bigger research project. Due to the focus that has been set here findings can only be displayed in a strongly abridged way. For details and a complete account of the sources see Hamann,


Michela Maschietto
email: michela.maschietto@unimore.it

Pietro Milici
email: p.milici@google.com

Dominique Tournès
email: dominique.tournes@univ-reunion.fr

Semiotic potential of a tractional machine: a first analysis

Keywords: History of mathematics, tangent, mathematical machines, tractional motion, semiotic potential

Tractional motion is an almost forgotten theory involving machines to geometrically solve inverse tangent problems. It was at the basis of Leibniz's development of calculus and interested many important mathematicians (e.g. Euler) mainly in the 17th and 18th centuries, before its disappearance because of the decrease of importance of geometric constructions. In this work, we analyse the exploration of a tractional motion machine using the framework of Theory of Semiotic Mediation. This is just a very first work in the perspective of introducing this kind of machine to students in order to mediate the meanings of transcendental curves and infinitesimal analysis.

Introduction.

The use of historic machines is a well-established practice to mediate mathematical meaning (Maschietto & Bartolini Bussi, 2011). Specifically, many studies have been published about the construction of geometric curves as traces of both ideal and practical machines, but most of such studies focus on algebraic curves. In this work, we deal with an approach to transcendental curves with machines, that is new1 in mathematics education. Such an approach, called "tractional motion," was so relevant to the conception of infinitesimal analysis (e.g. for Leibniz, who designed in 1693 a material instrument to practically solve quadrature problems). As deepened in [START_REF] Tournès | La construction tractionnelle des équations différentielles[END_REF], tractional motion mixed theoretical developments and concrete constructions of artefacts for the resolution of the inverse tangent problem. Those artefacts could be interesting for STEM education. In our approach, before proposing them to students, the analysis of cognitive processes of exploration by experts is necessary. In the paper, we aim to present the first elements of this analysis for a particular tractional machine.

Brief history of tractional motion.

As clearly exposed in [START_REF] Bos | Redefining geometrical exactness: Descartes' transformation of the early modern concept of construction[END_REF], in 17th century Descartes proposed a class of machines tracing algebraic curves as synthetic part of his method, and polynomial algebra as an analytical tool in order to simplify the problems. That determined a dualism between "geometric" and "mechanic" curves (in modern terms, algebraic and transcendental curves), being only the first one considered acceptable in the Cartesian setting. Soon after the spreading of Descartes' canon, mathematicians got habituated to polynomials, that became no longer an extremely important step for the problem solving, but directly the solution. In that period, there were only a few curves that were considered "mechanic" (e.g. the quadratrix or the cycloid), but a new general method to generate curves beyond Cartesian limits was going to emerge. The main idea underlying behind such constructions was given by a problem proposed by the architect Claude Perrault in Paris in the 1670s. The problem was easily described: if we move the end of a chain watch along a line slowly enough to avoid inertia, what curve does the watch describe? Cartesian tools appeared powerless to answer.

In the case of Perrault's watch, the motion is given by the traction of a cord: that gave the name to such constructions by "tractional motion." The described curve (called "tractrix") is constructed given its tangents: while the watch moves, the tangent to the obtained curve is given by the direction of the chain in traction. This is a new kind of tangent problem: instead of the classical tangent problem (given a curve, finding its tangent that satisfies certain properties), known at least since classical Greek period, this time the tangent properties are given, and the curve has to be determined as the solution. That's the rise of "inverse tangent problems."

But how can inverse tangent problems be concretely implemented? The reference to a heavy load having friction on a plane (as the watch) involved physical problems that should be kept away from the domain of pure geometry, e.g. the unwanted role of velocity (because of the inertia that the load acquires) or the non-perfect horizontality of the plane. It was missing a clear instrumental embodiment of the theoretical concept "driving the direction of a curve" (as a pair of compasses embeds the concept of equidistance): mathematicians as Huygens and Leibniz worked on it.

To sum up, after the habituation of mathematicians to consider polynomials as solutions, which brought to a loss of interest to machines for algebraic curves, at the end of 17th-century machines got a new importance; this time to justify the construction of transcendental curves [START_REF] Bos | Tractional motion and the legitimation of transcendental curves[END_REF].

In the 18th century, it was particularly interesting the role of Giovanni Poleni. Mathematician, physician, astronomer and also interested in engineering studies, [START_REF] Poleni | Epistolarum mathematicarum fasciculus[END_REF] proposed machines for the tractrix and the logarithmic curve, that were also built and used in cabinets for exhibitions and in teaching. In 1739, he obtained the creation, at the University of Padua, of a laboratory of experimental physics ("Teatro di filosofia sperimentale") which was unanimously praised by his contemporaries. He took the opportunity to set up one of the first university courses based on laboratory experiments (previously, he had students coming to his home in his personal scientific cabinet). For this reason, Poleni got a particular attention to the design of his tractional machines: they were no longer only theoretical machines but also practical instruments guaranteeing an adequate usability and accuracy. In this period, curves like the tractrix or the logarithmic were well known and generally accepted, thus it was no longer necessary to justify their introduction as solutions to certain problems, but it became important to give them concreteness (something similar to the possible needs in today classrooms). The main component behind such machine was the introduction of the "wheel" to drive the direction of the curve (as in the front wheel of a bike). Poleni's publication (1729) made a strong impression on the geometers of his time. Leonhard Euler (1736) read it and found in it an interesting mode of construction to address the differential equations that he could not integrate in finite form by algebraic methods. Among other results using tractional motion, Euler succeeded in integrating the famous Riccati equation, which had remained unsolved for a long time. From there, an epistolary dialogue continued between Euler and Poleni for more than four years, revealing interesting interactions between theoretical and practical concerns (Tournès 2009, p. 95-96): in particular, at Euler's request, Poleni gave the description of an instrument for drawing the tractrix of a circle.

Following on from Euler's work, Vincenzo [START_REF] Riccati | De usu motus tractorii in constructione aequationum differentialium commentarius[END_REF] published in Bologna a treatise in which he demonstrates that it is possible to exactly integrate any differential equation using tractional motion. From a theoretical point of view, Riccati's work is the culmination of the ancient tradition of problem-solving through the construction of curves. Descartes had shown the construction of algebraic curves by a simple continuous motion using linkages. Riccati, for his part, established that transcendental curves can also be constructed by a simple continuous motion from the differential equations that define them.

After the middle of the 18th century, the theory and the practice of the tractional motion got rapidly lost. The geometric paradigm was no longer dominant, and new mathematicians were interested in exploring only the analytical counterpart of infinitesimal analysis. However, in the late 19th and early 20th centuries, a new generation of mathematicians and engineers reinvented essentially the same solutions of Poleni, increasing just the complexity of these instruments (the "integraphs").

The tractional machine of our study.

One of the most interesting features of tractional motion is the deep link between its instrumental and theoretical components. Specifically, the possibility of introducing some transcendental constructions from history to classroom might be mediated by the exploration of a machine.

Out of the tractrix, the most famous construction easily given by tractional motion is for the logarithmic (or exponential) curve. As visible in Figure 1 (left and centre), for tracing this curve, a machine can implement the property of having a constant subtangent 2 . Poleni, who designed and build a machine tracing the exponential, also evinced that one tractional machine could generate both algebraic and transcendental curves. That was a fundamental idea while conceiving our machine: according to the notation of Figure 1, if we change the direction of the wheel in the machine for the exponential (i.e. we put the wheel direction perpendicular to the segment QS), the machine traces a parabola (Figure 1, right).

More specifically, according to Figure 2, the machine 3 proposed in this study is made up by a horizontal wooden table with two parallel fixed wooden guides in which a wooden frame can slide. A brass cylinder is forced to remain within the frame, free to rotate and to slide perpendicularly to the two fixed guides. Inside the cylinder, there is a wheel perpendicular to the plane. The cylinder rotation is determined by a rod that is constrained to pass through a pin fixed on the frame.

2

In a Cartesian plane, given a curve C passing through the point (x,y), the subtangent to C in (x,y) is the segment between (x,0) and the intersection of the line tangent to C in (x,y) with the x-axis. Referring to the Figure 1, the subtangent in S is the segment QP.

3 Simulations of this machine are available online: https://www.geogebra.org/m/zrMyFGdd (for the parabola) and https://www.geogebra.org/m/LXugImiH (for the exponential). This machine was introduced by Milici & Di Paola (2012) and described with more details in [START_REF] Salvi | Laboratorio di matematica in classe: due nuove macchine per problemi nel continuo e nel discreto[END_REF]. [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF] in a Vygotskian perspective. In this, a teacher uses an artefact as a tool of semiotic mediation for mediating mathematical meaning to students by managing didactical cycles (composed of small group work, collective mathematical discussion and individual work). The first proposed activity is a small group work with the artefact, because its use solicits a certain semiotic activity (where the sings can be language, gestures, drawings, ...) that is the base for the teacher's intervention. The exploration of the chosen artefact related to the educational objectives is carried out following four questions (Bartolini Bussi, Garuti, Martignone & Maschietto, 2011, p.128): "How is the machine made?", "What does the machine make?", "Why does it make it?", "What could happen if ...?". Within this framework, the choice and the educational use of an artefact is based on the analysis of its semiotic potential. In the TSM, the distinction between meanings emerging from the practice (use of artefact) and the mathematics knowledge evoked in the expert's mind is very important. [START_REF] Mariotti | Resources for the Teacher from a Semiotic Mediation Perspective[END_REF] specify the notion of the semiotic potential of an artefact to make that distinction explicit:

By semiotic potential of an artefact we mean the double semiotic link which may occur between an artefact and the personal meanings emerging from its use to accomplish a task and at the same time the mathematical meanings evoked by its use and recognizable as mathematics by an expert. (Mariotti & Maracci, 2012, p. 61) Our research questions are about if and how we could use the machine as an instrument of semiotic mediation with secondary school and/or university students. As in other research on mathematical machines, we intend to propose first a material machine rather than its simulation in a DGS, because we assume the relevance of the exploration by the hand (allowing touching, decomposing the machine, perceiving the resistance to movement or not, ...). In order to answer to those questions, the first step is the analysis of the semiotic potential of the machine, considering the description of the machine by an expert that already knows it on the one hand, and the exploration of the machine by other subjects, experts in other kinds of machines or university students in mathematics. In this way, we aim to analyse what kind of semiotic activity can emerge from the exploration and use of the machine. In particular, we focus on: which are the processes of exploration of the machine? Do people identify the role and the meaning of the wheel? How?

Methods.

We have videotaped two exploration processes of the machine. We chose three people that knew the mathematical machines and the TSM, and were experts in mathematics but without any knowledge of tractional motion. The first exploration was carried out by two secondary teachers, following the structure of four questions that they knew very well. In the exploration, the researcher had to often put the attention on the wheel and its way of use, thus, for the second case, we began proposing only some components of the machine. The second exploration was performed by a postgraduate student in mathematics that studied mathematical machines during a university course.

In this case, the machine has been disassembled and gradually reassembled by the researcher: first the cylinder with inside the rolling wheel (cf. Figure 2, (2) and ( 3)) is given free to move on the table; then the rod (cf. Figure 2,(4)) is added, in the two positions (parallel and perpendicular to the wheel direction); finally, the cylinder is put into the wood frame (cf. Figure 2, (1)).

Findings and discussion.

The analysis is based on videotapes and drawings produced by the people exploring the machine. More specifically, we analyse the part of the two explorations in which people detect the movement and constraints of the wheel and identify the link between wheel and the line tangent to the curve.

First exploration: Alberto and Bianca.

During the first moments of the exploration of the machine (corresponding to the first question, "how is it made?"), Bianca describes the machine but she does not seem to pay particular attention to the presence of the wheel, even if she considers it as a component of the machine. When the machine is then used to trace (intentionally, the ways of use are not suggested to her), this lack of attention is evident by how Bianca acts on the frame and on the wheel: sometimes the wheel rolls and sometimes it slips, tracing a curve or a straight line. The resistance to the slipping of the wheel is not noticed by Bianca, who raises doubts on how to use the machine until the researcher intervenes by asking it explicitly ("Don't you feel resistance?"), by specifying the principle of operation ("[the machine] works when the wheel turns") and by explaining the relationship between the resistance and the constraint of the machine ("you're forcing a constraint of the machine").

The machine doesn't seem to offer feedback sufficiently strong to actions that force the constraints; while forcing the wheel not to follow its direction, the resistance of the tyre is mixed with friction by other structural parts, such as the frame. This certainly is an important point to take into account for paying attention to the wheel as a new element, compared to other curve drawers (such as the ones using taut threads). To foster the identification of the role of the wheel in tracing the curve, the researcher asks to specify the physical constraints of the machine allowing that tracing; he also suggests a comparison with a tracer (which corresponds to the point of a pen) ("If, instead of the wheel, there had been a tracer, would it have been the same?"). The question corresponds to "What could happen if ...?" of our educational framework. However, it is a question that can be asked to expert people. Here are some excerpts of exchanges with Bianca, Alberto and the researcher.

Alberto Then, they turn their attention to the geometrical property of the machine. The relationship between the wheel and the tangent to the curve appears in terms of "direction."

Alberto:

[The properties] of the tangent line, isn't it? But physically ... I don't know why it is physically ... that is, the small wheel gives the direction [gesture by the hand] that is not ... any direction to the curve.

Researcher: And then, what do we know ... having the small wheel this direction? ... at the drawn point.

Alberto:

The curve must have that direction.

These excerpts show that the exploration is in discontinuity with the explorations of the curves drawers that Alberto and Bianca are familiar with. This seems to confirm the deep change quoted above from the epistemological point of view. The emergence of the link between the wheel and the tangent line to the curve is fostered by the researcher. However, some questions are not immediately understood by Alberto and Bianca, as they do not link the task and their personal meanings on curve and machine. The difference between rolling and crawling seems to emphasize such a change.

Second exploration: Corrado.

According to researchers' queries, Corrado moves the cylinder with inside the rolling wheel (cf. Figure 2, pieces 2 and 3) on the plane and defines some properties of the traced curve with consciousness ("It is not constrained to stay on a straight line or a circumference or a parabola or a branch of hyperbole, if we let it free it can go up and down in every place. ... It traces a continuous line, [...] can do edges but without jumps"). When asked to describe more precisely his interactions to make the machine move, he immediately defines the two degrees of freedom. In terms of the analysis of the semiotic potential, we could affirm that the task links the manipulation to Corrado's personal meanings of the curve.

Unlike Bianca, when Corrado is asked to explore the fully reassembled machine, he seems aware of the mechanical resistance to certain motions, and he is able to interpret it in relation with the constraints of the instrument. Specifically, when the wheel is put in the vertex of the parabola, Corrado notices that the machine "resists" when he tries to move along the direction of the axis of the parabola. To explain such behavior, he puts his attention on the idea of direction: "[the machine cannot move that way] because it cannot [with a gesture indicates to move along the axis direction], it is constrained here [indicates the pin, component 5 of Figure 2], it cannot invert, get comfortable [with a gesture of the hand: change the direction] and trace, it has to stay this way and it is unnatural to draw [gesture: in the direction of the axis], it cannot rotate this way [gesture of the rotation of the wheel perpendicularly to the direction of the wheel]. It [the wheel] draws only while rotating and in the direction of its rotation." Later on, he is asked to characterize the traced curve that he identified as a parabola. Corrado is able to find independently the role of the tangent for the wheel: "if we hypothesize that this [with the hand indicates the direction of the wheel] gives the direction of the tangent, the wheel is always perpendicular to this rod and so ... [he looks for a conjecture on the tangent perpendicular to something, but doesn't find any]."

Therefore, as a very early conclusion, we can suppose that disassembling the machine eases the understanding of the constraints of the machine. However, we cannot affirm that disassembling can also ease the characterization of the curve because Corrado tried to verify that the traced curve was a parabola focusing on the points traced on the plan and neglecting tangent properties. This corresponds to another aspect of the epistemological change that characterizes the machine.

Concluding remarks.

At our knowledge, this is the first empirical work aimed at analysing the semiotic potential of a tractional machine. As a first result, we can evince that the passage from algebraic to transcendental constructions, so groundbreaking from a historical and epistemological perspective, recalls also very different abilities in the exploration (the explorations were not trivial even for experts of mathematical machines). In particular, the relation between the wheel and the tangent did not appear immediately, but the idea of disassembling the machine in order to investigate simpler components appeared quite fruitful (even though it requires to be much more deepened). This suggests a possible task, but this is not enough sufficient for constructing worksheet for students. What done is only a very first step in this direction. We need other experimentations to clearly define the mathematical contents to be mediated (tangents, derivatives, integrals, differential equations) and the level of the exploration (high school, university).

Figure 1 :Figure 2 :

 12 Figure 1:[Left] The exponential y=a x has constant subtangent QP; [Centre] machine for exponentials; [Right] machine for parabolas

Such approach is not in use today, but it was already adopted in Italy by Giovanni Poleni in Padua (first half of the 18th century) and Ernesto Pascal in Naples (beginning of the

20th century).