Learning how to axiomatise through paperfolding

Transcription

1 Learning how to axiomatise through paperfolding D. Nedrenco Abstract: Mathematical paperfolding can be seen as a helpful tool to start a course with some axiomatisation issues of a mathematical theory. In an iterative series of one semester courses for prospective secondary level mathematics teachers, 1-fold-origami has been (re)axiomatised by and with the students. In the paper we describe how this axiomatisation can be accomplished in class, how it motivates discussions about an axiomatisation of the Euclidean plane, and why this could be beneficial in students education. We then describe how we analyse the students understanding of axioms before and after the course and show some examples of the results. 1 Introduction At the beginning of our PhD project, one of the questions was how to integrate paperfolding into the education of pre-service secondary level mathematics teachers on the university level. It was obvious that paperfolding in a classroom setting embodies a very promising tool for teaching mathematics in general and geometry in particular [Golan and Jackson 09], [Arici and Aslan-Tutak 15]. But it was not clear which origami-related mathematics to teach at the university and why. For there are lots of wonderful constructions, which by mere folding of paper enable students to observe or discover beautiful mathematics [Hull 13]. Some of these instructions go back as far as to Sundara Row [Row 93] more than a hundred years ago, some are more modern such as [Olson 75], some contain very recent developments of paperfolding such as the ingenious book [Hull 13]. These books are united by the fascinating idea of illustrating and producing mathamatics by paperfolding. It is hard to find any topics in mathematics which cannot be accessed through paperfolding: from polygons to infinite series to quadratic equations to Euclidean constructions to polyhedra to field theory. There is one subtelty, though. In the most cases mathematical paperfolding is used to demonstrate, to make visible, even touchable, some piece of mathematics. Think of folding polyhedra (modular or not), the pythagorian theorem, folding of unit fractions, folding of symmetries etc. But only recently we think of 1980 s and 1990 s people began to exploit mathematics of paperfolding more systematically. Two prominent examples are flat foldability, cf. [Hull 13], [Hull 94] and 1-fold-origami, cf. [Cox 04, 10.3] or [Martin 98b]. These two theories, which we later describe in some detail, ask a very mathematical question: What is the underlying mathematics of the foldings? But describing flat

2 NEDRENCO foldability and 1-fold-origami require some mathematical preparation and knowledge from the lecturer (in the latter case even some axiomatic thinking). So one can ask why not better do in class more standard origami mathematics like folding pyramids or unit fractions? In our experience there are some misunderstandings about the nature of mathematical paperfolding. As more and more books and instructions of how to fold fascinating objects and visualise important mathematical topics appear, more and more teachers are able to integrate paperfolding in their (math) classes. But despite the fact that folding of cubes, pentagons, square roots or tall boxes obviously entails mathematical objects, from our point of view it lacks an argument for why it should be considered a mathematical activity. For instance, consider a more or less typical situation: In class a teacher suggests to fold a pyramid from a square and gives visual and verbal instructions ( look what I am doing, fold this on top of that etc.). After a finite number of steps students are very excited to hold a pyramid in their hands, maybe to observe its surface, to count edges and vertices or to calculate its volume. This motivating hands-on-activity allows students both to construct and to analyse the pyramid. So one tends to say that mathematical paperfolding was done. But there is a pitfall. We believe from our experience and analysis (but without any solid data!) that students do not perceive the folding and the following calculations of some properties of the pyramid as one continuous process, but rather as two separate processes. We believe that they first fold probably without major mathematical thinking and then, after the pyramid is folded, they start calculating. So, apart from the important psychological effect of having created an object by oneself, we believe that the students interest and benefit of computing the volume of the pyramid would be nearly the same as when they, for instance, cut the net of the pyramid from a piece of paper and glued some parts together. From this perspective we propose a rather broad definition of mathematical paperfolding, in order to reduce the misuse of this expression and to give a frame for a discussion on paperfolding in math classes. Definition. By mathematical paperfolding we understand a branch of paperfolding, where paper is being folded by some declared, and in a mathematical sense, describable rules, with the goal to analyse the result mathematically. It is obvious, for reasons not given here, that, for instance, 1-fold-origami, flat foldability, rigid foldability and many other origami branches can be seen as mathematical paperfolding. Folding a crane or a cube may well be mathematical paperfolding, too. But folding a cube just for having folded a cube is not. But of course one is still allowed and welcomed to fold cranes and cubes in math classes, one just has to be careful about its mathematical impact. This definition suggests finding and mathematically describing rules of folding. This would be most certainly a mathematical activity and will lead us to looking for basic folds, which will automatically bring us to axiomatisation issues!

3 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING 2 Main ideas In this section we describe connections between paperfolding and axiomatics as suggested in the title. First, we give an overview of the mathematics of paperfolding relevant to us. Second, we describe why axiomatic thinking and axiomatisation of a mathematical theory can be of interest in students education and, third, how we can combine these two rather differents topics origami and axioms. 2.1 Why paperfolding Paperfolding is widely accepted as a highly inspiring activity for both teachers and students with a great number of educational benefits and an enormous mathematical potential, cf. Review of Literature in [Arslan 12, p. 16ff]. There is a number of studies about the benefits of paperfolding at all school types from elementary schools e.g. [Golan and Jackson 09] to middle schools e.g. [Boakes 09] to even universities e.g. [Boakes 11], [Arslan 12]. These studies cover a great number of different topics such as spatial abilities [Boakes 09], [Arici and Aslan-Tutak 15], reasoning, geometric knowledge and beliefs [Arslan 12] etc. In [Arici and Aslan-Tutak 13, p. 2] the authors state that most research is concentrated on elementary and middle schools and that more research on benefits of paperfolding in high schools is needed. We try to fill this gap. Mathematical paperfolding is nowadays understood as a rich mathematical area, which considers and answers such diverse questions as: 1. How can a desired object be folded, e.g. a polyhedron or a swan, from a piece of paper? 2. Can we decide whether a given crease pattern produces a 2D-model, a 3D-model or whether it is not even foldable? 3. Can we construct as many points in the Euclidean plane with just paperfolding as with a ruler and compass? All these questions were well studied in the last few decades and a lot of answers (but new question, too) were found. Let us concentrate on the last question. If we allow folding a piece of paper in such a way that from some already folded lines and points only one single fold line is produced after one folding step (this constitutes 1-fold-origami), then one can show, that every point constructible by ruler and compass is constructible by 1-fold-origami, too. As a matter of fact, 1-fold-origami can even solve classical problems of geometry, such as the doubling the cube or the trisection of an arbitrary angle (not solvable by Euclidean constructions). Even a solution of a generic cubic equation is 1-foldable, cf. [Hull 11,Martin 98b,Cox 04]. One can go even further and study 2-fold-origami (or n-fold-origami) cf. [Alperin and Lang 09] and solve general septic equations with it [König and Nedrenco 16], but this turns out to be rather technical and not suitable for teaching. All in all, we see that there are lots of possibilities to study origami on a higher level. We chose 1-fold-origami to represent mathematical paperfolding in the course, because it both covers essential geometric constructions rather elegantly as well as formal math.

4 NEDRENCO 2.2 Why axiomatisation It is quite obvious that math students should understand the basic principle of mathematics: if you make a statement, then you should try to give a sound argument for it, based on already proved statements. Explaining to students the need for a proof can be challenging and one will expect that at least the school teachers do see this need. But who explains it to the teacher? And moreover: the teacher must understand how and where this process of decomposition of statements into more atomic statements will end. The teacher needs to understand the process of axiomatisation. There is a consensus that university students should be confronted with axiomatisation of a mathematical theory during their studies, cf. [Freudenthal 73, p. 133], for they should see how a mathematical theory evolves. In addition, we think that prospective secondary school teachers should see or learn some basics of the axiomatics of the Euclidean plane at some point in their studies, because Euclidean geometry constitutes a huge part of the curriculum. Hence, a sound background in this topic is needed and the Standards for the education of mathematics teachers support this point of view, cf. [DMV, GDM, MNU. Standards für die Lehrerbildung im Fach Mathematik 08, p. 5]. Since, according to Freudenthal [Freudenthal 73, p. 451], there is no place for prefabricated mathematics in students education, axiomatisation rather than axiomatics should be chosen for this goal. It is worth mentioning that by axiomatics we mean the building of a theory by starting with some given set of axioms. By axiomatisation we mean, to some extent, a reverse way of finding a set of axioms in a given or known theory, which fully describes this particular theory. Unfortunately, there seems to be no simple path to axiomatisation of the Euclidean geometry, although there are some very elaborate suggestions as in [Martin 98a], [Schnabel 81]. Apart from the difficulty of teaching Euclidean geometry rigorously, it appears difficult (but possible) to make plausible to the students why axiomatisation is needed (in [Yannotta 13] the author describes students problems with axiomatisation and possible successful interventions). Above all, students need to have or gain an ability to reach a certain level of abstraction in order to understand the axiomatisation process. From personal experiences supported by recent studies cf. [Arslan 12] and the references therein, we believe that mathematical paperfolding offers such a way. 2.3 Why axiomatisation of paperfolding There is a rather intuitive way of axiomatisation of 1-fold-origami, cf. [Alperin and Lang 09]. By that we mean to find all the basic folds, which produce exactly one fold line, working in the Euclidean plane. So the main idea is to let students guided by the lecturer analyse 1-fold-origami, such that they succeed in finding all of the seven basic folds and, thus, in axiomatising the theory. Combining this idea with observations made in the above two sections, we find it very natural to propose 1-fold-origami as a tool in the following sense: students learn an intuitive and manageable theory which they can axiomatise and furthermore use this experience to understand the more complicated axiomatics of the Euclidean plane. While the students advance to dealing with global ordering instead of local

5 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING ordering, their understanding of the essence of the abstraction process increases. They are thus able to treat axiomatics of the Euclidean plane more rigorously. (At least, this is what we hope that they do.) Note that we are not suggesting to axiomatise the Euclidean plane, as it is still too tedious and time consuming. We would be happy to convey an understanding of the modern point of view about axioms and axiom systems to the students, its meaning and basic knowledge of the topic, cf. precise goals in Course description With the ideas from section 2 in mind, we designed a course on paperfolding and axiomatisation for prospective secondary level mathematics teachers at the University of Wuerzburg, Germany. The course was held three times, was optimised and changed according to previous experiences. In this section we describe the design and structure of the courses. 3.1 Organisation and some data All three courses were taught once a week over the period of one semester each, in total weeks. Students were not preselected for the course, every pre-service secondary level math teacher could enroll on a voluntary basis. The number of participants was limited to a maximum of fifteen. This number seemed reasonable for group discussions. Our experience showed that nine to eleven participants are optimal for our purposes. The actual number of participants varied over the courses from nine to fourteen mainly due to the fact that the courses offered only few credit points and were not included in the main curriculum. As no restrictions were imposed on the level or age or mathematical skills of the students, the level of preparation and experience of participants also varied greatly. 3.2 Premisses The course deals with mathematical paperfolding. Students learn some major developments in mathematical paperfolding. As we know, mathematical paperfolding has been more and more growing since approximately Interesting theorems, assertions and constructions arose from natural questions motivated by folding of paper. These constructions, although well known in the origami community, are still not established at universities. Therefore, we had to teach them from scratch. Due to time limits and research interests, in the courses we only dealt with flat foldability and 1-fold-origami. Students axiomatise 1-fold-origami. Students learn the mathematical structure of 1-fold-origami. When folding paper in the sense of 1-fold-origami, we observe typical patterns, which occur over and over again. For instance, one puts one segment (we imagine this segment extended to both sides infinitely) onto another, one folds one point on another etc. With more and advanced foldings students distill other basic folds, too: one can fold perpendiculars, connecting lines, but also seemingly strange constructions like one point onto a line, such that the fold line

6 NEDRENCO passes a second point. They try to describe these strange folding instructions mathematically, analyse what can be constructed using them, and try to find a definition of 1-fold-origami. If we consider a piece of paper as a small region of the unbounded plane and mark two points, then we can define the following basic folds to obtain further points and lines. Moreover we can prove that each folding which produces one new line from already folded points and lines must be one of these basic steps. segment bisector connecting line angle bisector perpendicular tangent to a parabola another tangent to a parabola simultaneous tangent to 2 parabolas Figure 1: Basic 1-fold-origami folds (Huzita-Justin axioms). Step by step students find through experiment and analysis of different foldings all 1 of those seven basic steps and describe them mathematically. We call 2 this procedure axiomatisation of 1-fold-origami. Later on we discuss whether all those folds are really needed, going towards the metamathematical analysis of the axiom system in the Hilbertian sense. Students discuss difficulties with a mathematisation and formalisation of the Euclidean geometry and grasp historical developments of this branch. Despite the fact, that deductive methods in geometry classes have become less important in the last decades, we believe, that teachers should know the modern point of view (compared to one of Euclid and contemporaries) of their discipline. In the beginning of the course we agree to work in the drawing plane. Therefore, after the axiomatisation of 1-fold-origami (which lives in the plane) we realised that the plane itself has not been formally described yet. Relatively few students instantly know that the plane can be understood as the vector space R 2 together with the Euclidean inner product. In this case we confront them with Euclidean axioms, which they often vaguely remember from school, and ask what these two representations of the term Euclidean plane have in common. Oth- 1 This axiomatisation of 1-fold-origami was settled in late 1990 s and early 21th century, but significant results were found back in 1930 s by Mrs. Beloch, cf. [Hull 11]. 2 It is not an axiomatisation in a pure sense. Students would need to know a lot of theorems and constructions of 1-fold-origami and then distill the axioms from them by first local and then global ordering. This we can not expect. This is why we proceed by a mixture of axiomatics and axiomatisation.

7 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING erwise some students think that Euclidean plane has something to do with the parallel postulate. Hereby a more or less seamless transition from origami to the Euclidean plane takes place. Summarising, we note that a full discussion of the developments and ideas of the past 2500 years regarding the Euclidean plane is utopian within the scope of our course. Therefore, we discuss only the milestones, e.g. Euclid s postulates and motivation as well as some issues with these by his successors. Due to limits of the course, we do not do any mathematics of the Euclidean plane, but remain instead on a meta level. 3.3 Selected topics We begin the course with flat foldability, based on [Hull 13, Act ] and explained in some detail in [Nedrenco and Beck 16]. In our opinion, flat foldability is a great start into mathematical paperfolding as such, but it is also a first impression of local ordering of statements. This four-to-five hour sequence serves as a preparation for an axiomatisation of a mathematical theory, i.e. global ordering in the sense of [Freudenthal 73]. Then we switch to 1-fold-origami Construction We consider construction as a key notion for a mathematical treatment of origami. By way of discussions we want to determine which foldings are to be considered 3 : What should be allowed to fold? How do we mathematically define this activity? Instantly we dismiss sheer crumbling of paper from our agenda. The first proposal is: Folding of paper should be controllable and repeatable. This coincides with basic requirements of an scientific experiment. Students further propose to only consider uniquely defined foldings. By that they mean, that a construction can be repeated by anyone without further explanations or demonstrations. As a counterexample we fold a corner of a square to a side of square, cf. Fig. 2. This folding is not repeatable in exact same way: Figure 2: This fold is not unique resp. not repeatable in exact the same way. Thus it is not a construction. There are infinitely many possibilities to create the fold line. Hence we dismiss such foldings in valid constructions. Then the question arises: What do we want to consider a valid construction? This is not an easy but multifaceted question, which let us dangle for a while Euclidean constructions It is obviously possible to fold an already folded line onto another one. Students do not hesitate to recognise that the fold line can be described as an angle bisector 3 In the course we concentrate on 2D-foldings and fold 3D objects only as a warm up or relaxing activity, but do not study them mathematically, since their origami mathematics seem complicated to us.

8 NEDRENCO of the two lines. But here a problem occurs: There are two angle bisectors of two nonparallel lines, cf. Fig. 3 on the left. Thus folding one line onto another is not an unique folding in the above sense and should be dismissed. But everybody thinks that such a natural fold should be allowed in our theory. Therefore, we weaken our too strict uniqueness requirement. We agree to allow 4 constructions in which the number of possible fold lines is finite (if necessary one wound specify which exactly angle bisector is meant.) D m A l f B C Figure 3: Two ways to fold a line onto another (left); different ways to fold a single line (right): angle resp. segment bisector of AD and AC resp. of DC; the connecting line of A and B. By this and other examples students link and compare paperfolding with ruler & compass and bridge mathematical paperfolding with school geometry Number folding After we gathered some experiences with folding, we start experimenting with folding numbers, by which we first denote ratios of lengths of line segments. Which number are origami i.e. 1-foldable numbers? Students immediately say: 1 2, fold the middle of a side of a square and explain that they folded a segment bisector, which is 1-foldable. Thus we obtain that all the powers of 2 1 are 1-foldable. Students further propose 1 3 by letter-folding cf. Fig. 4. Figure 4: First impressions of unit fractions: 1 2 on the left, 1 3 on the right. Studens are now requested to decide whether this folding can be considered 1-foldorigami. It is overwhelming how rich and controverse the arising discussions based on this rather primitive fold are! After a vivid discussion the insight wins, that there is no way to separate this folding into two 1-fold steps. Some students provide the argument that this zigzag-fold is only an approximative 5 trisection anyway, for how should one describe it different than fold until it looks equal? Moreover it is very enriching to show students the Fujimoto quintisection [Hull 13, Act. 3] and to think 4 It is worth noticing that students play around here with different definitions and possibly understand that definitions are made and not given. 5 This folding is accepted as a construction in 2-fold-origami, though, cf. [Alperin and Lang 09].

9 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING about why we do not consider it 1-foldable. These examples stimulate discussions about a sound definition of 1-fold-origami and constructions in general, they lift our dialog to a more profound level. Subsequently we learn proceedures and methods (like Haga s theorems or diagonal constructions) for folding all the unit fractions and then all rational numbers Axiomatising 1-fold-origami After having seen several examples and counterexamples for 1-fold-origami, we gather and sort our experiences and basic folds so far. Among them are: Fold line segment and angle bisectors, connecting lines, perpendiculars. In an unit about folding of triangles (for more details cf. [Hull 13, Act. 1] and [Nedrenco 17]) students use a new fold without further concerns: Put one point upon a line such that the fold line passes a second point, cf. Fig. 5. This fold is essentially a construction of tangents to a certain parabola. It widens our horizon and enables us to fold all solutions of quadric equations as well as to fold many regular polygons [Geretschläger 08]. Students analyse this basic, but new fold, rediscover the connection between the synthetic and analytic definition of a parabola and strengthen once again the bond between origami and plane geometry. From the axiomatic point of view this connection is obvious, but students find it very eyeopening. They express the insight to not anymore think of ruler and compass as the natural instruments for constructions. They even ask why these have yet not been replaced by origami. Figure 5: Folding regular triangles leads to tangents of parabolas. After a while someone will eventually come up with the next basic fold, proposing to fold two points onto two lines. This insight can be stimulated, if needed, by the analysis of the image of a fixed point being reflected at variing tangents to a parabola, which is defined by a point and a line cf. Fig. 6. There is still another possibility to introduce this surprising fold. On can first informally construct Peter Messer s solution of the Delian problem and then let students analyse it (but we considered this approach to some extent dishonest and try not to introduce the so called HJ6 this way). From here there is no great trouble to describe this fold as a simultaneous tangent to two parabolas. Students do usually not conceive this insight as remarkable, but rather difficult to fold and to analytically describe. Obviously this part of 1-fold-origami is the most challenging to motivate and to analyse, but it is very inspiring thereafter, because students now hold a proof in their hands, that paperfolding is stronger as an instrument than the Euclidean instruments we can fold

10 NEDRENCO solutions of irreducible cubic equations 6. To do this we utilise the cubic curve from Fig.6, which is build by moving points, as well as Beloch-Hull origamising of Lill s method [Hull 11]. Q P f Q P l l Q f (a) Folding P to l produces f. Reflecting Q at f produces Q f. (b) As the image of P on l varies, so does f and Q f (represented by some dashed red lines). The points Q f build a cubic curve (green). Figure 6: The origami cubic curve, here with Q = ( 2,0), P = (0,1), l = 1 and a parametrisation t 1 t 2 +4 (t3 + 2t 2 8, 2t 2 8t). Up to this point of the course we tried to give a formal definition of 1-foldorigami. After all, one needs to exactly know what we are talking about when working mathematically with origami. Students admit that fold in such a way that only one fold line emerges does not qualify anymore as a formal definition. They come up with rather accurate definitions. But there is always a gap or uncertainty left. So we decide to narrow down the definiton by introducing half-reflexions (omitted here). From it we derive all the six axioms of 1-fold-origami and try to prove that those are all there are. Surprisingly we spot a gap in the proof and discover a new, seventh, axiom (Hatori-Justin s seventh) and finish our proof. We find this method very inspiring and typical in mathematics: Students learn that mathematics is not always straight forward, from time to time one has to go back and forth to finally close all gaps. Moreover, this kind of proof shows them why a sound formalisation is needed. In the course, we do not consider it nessessary to analyse the axioms of 1-foldorigami as it is done for instance in [Ghourabi et al. 13]. But we do go in some depth on dependencies of those basic folds and realise that in algebraic resp. field terms there is only one axiom we actually need Further topics In the course we explicitly deal with following school relevant topics: folding of unit fractions, folding of solutions of linear, quadratic and, to some extent, cubic 6 This is not possible by ruler and compass.

11 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING equations (in particular we solve the Delian problem with 1-fold-origami), folding of some regular polygons and polyhedra. We give a formal definition of 1-foldorigami. The (in)dependence of the axioms of 1-fold-origami is discussed and transferred to the axioms of the Euclidean plane. As we already mentioned, we conceive the Euclidean plane from a meta level. We discuss typical problems with Euclid s axioms from modern point of view and learn possible solutions for them. Moreover we learn one axiomatisation of the Euclidean plane cf. [Martin 98a], [Schnabel 81]. During the course we also learn typical difficulties of the didactics of origami and find solutions for them. We often talk about how to apply our knowledge in schools. Students are asked to prepare and present some foldings to others. 4 Analysis In this section we shortly describe the methods for analysing the collected data and for answering the research questions as well as some results one can already observe. 4.1 Research questions As mentioned above, one goal of our research was to create a course on paperfolding and axiomatisation 7. Our second goal is to answer the following research questions: 1. From mathematical point of view, which problems and troubles do students have with axioms and understanding of the axiomatic thinking? 2. Which conceptions regarding axioms, axiomatisation and axiomatic thinking do students have? 3. How does our course on paperfolding change the understanding of axioms, axiomatisation, axiomatic thinking? 4.2 Methods of the analysis The third research question above uses the word understanding, which should be explained here. First, we tried to apply the van Hiele model (inspired by [Golan 11]) to observe whether there is a change in the van Hiele levels (vhl) of the participants before and after the course. As the highest vhl is often neglected, there are few examples of what it exact means to arrive at this level and how to test it, cf. [Gutierrez and Jaime 98]. We tried to modify the test from [Gutierrez and Jaime 98] in such a way that it focuses more on the axiomatic thinking (assuming that students already mastered the basic vhl). We used some questions from e.g. [Burger and Shaughnessy 86] for this purpose. Unfortunately, after reviewing some data, we weren t able to verify that the modified test reliably describes students levels of thinking. That is why we have not pursued any assessments of vhl levels in order to observe 7 Note that [Hull 13] resp. the wonderful lectures on paperfolding by Eric Demaine have a somewhat different focus.

12 NEDRENCO effects of the course. Instead, we proceeded, simply put, as follows: We want to observe any change in contentfulness of the students answers before and after the course. To do so we code the answers (by means of grounded theory) and then evaluate the content of each category we found in the data. By contentful we understand here mathematical quality of the categories and therefore the answers given by the students. It is crucial to state that it was not our intent to teach students how to give a better or a correct definition of a specific notion like axiom or axiomatisation. We were and still are concerned about them understanding the concept of the axiomatic method and the need for it. To give an example of this method we show from a pretest two answers 8 to the question: How would you explain to someone the word axiom?. 9 8th semester A simplest possible statement which is valid resp. predetermined and need not to be proved. With the help of an axiom many conclusions can be drawn. Therefore, they are something like the cornerstones in a particular domain. 5th semester The properties typical for particular things. 3th semester An axiom is a fundamental law predetermined by a panel, for instance in physics or mathematics there are certain rules, (axioms) upon which physics or mathematics are built. One can immediately observe some differences in the precision, concreteness and choice of words of the answers. One easily observes some interesting misconceptions like the deciding panel. At first it seems quite wrong, but is actually not very far away from reality, cf. [Thurston 94]. 4.3 Data collection and analysis Following our research questions we wanted to openly talk to students about axioms and axiomatisation 10. So on the one hand we decided to interview them. On the other hand, our intention was to observe how the students way of thinking changes about this topics. That is why we needed a pretest-posttest design. But due to organisational difficulties there was no possibility to assume that students will attend the course regulary or will actively participate in it. So in order to not scare them away we decided not to interview students before the course. Therefore, we decided to let students complete a paper-and-pencil test at the first meeting. Unfortunately, it is difficult to compare answers from written tests with those from interviews. The analysis is still ongoing. For the interviews students were asked to not prepare (as they would for an exam) in order to be able to speak to them more freely. Despite the fact that the 8 The test and answer were in German. Translation by the author. 9 Note that we are not asking to define the notion axiom for we cannot expect students to know one. We are rather interested in what they tell us about it. 10 In the course and in the interviews axioms were never studied for their own sake, but just as a necessary companion of formalising of paperfolding resp. Euclidean plane.

13 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING atmosphere during the courses was always very friendly and collaborative, we expected students to be, to some extent, tense on the record opposite the interviewer. Therefore, we split the participants into groups of two for the interviews, because we believed they would speak more freely in the presence of others and there would be an opportunity to confront one of them with the ideas and thoughts of the other. The partitions were rather random, but we tried to pair students of similar age or experience. In total, we conducted 16 pair interviews with participants of the courses. Moreover we conducted 2 single interviews with PhD-students, who did not attend the course, in order to observe whether students and mathematicians manage the questions differently. As the data has not been fully analysed yet, we only state this: it is obvious from the data, that PhD-students understand more about axioms and can better deal with them, although one can not assume that they dealt with axioms more often than our students. Still their answers are not as precise and correct as one wished them to be Conclusions In this paper, we explained how to use paperfolding and especially 1-fold-origami as an example for an axiomatisation of a mathematical theory. This method seems to be very inspiring for students and covers and combines a large number of topics from school mathematics. With our courses we tried to demonstrate the possibility to teach mathematical paperfolding not only throughout the school curriculum but in a systematic way at universities. We hope that mathematical paperfolding will gain acceptance in the education of pre-service mathematics teachers and will become an inherent part of the geometric curriculum. As for our study we expect in the near future to present results of the analysis and to show how the axiomatisation of paperfolding affects students perception and understanding of axioms. There are still lots of open questions about the impact of mathematical paperfolding. Despite some results about slightly improved spacial ability, it is important to understand more deeply how folding of paper impacts spacial thinking of children and adults. It should be considered important to analyse the learners perception of folding and its impact on the understanding of the mathematical background thereof. References [Alperin and Lang 09] Roger C. Alperin and Robert J. Lang. One-, two-, and multi-fold origami axioms. In Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education, edited by Robert J. Lang, pp CRC Press, [Arici and Aslan-Tutak 13] Sevil Arici and Fatma Aslan-Tutak. Using Origami To Enhance Geometric Reasoning And Achievement. In CERME8 (European Research in 11 Interestingly enough, even PhD-students did not always answer basic (van Hiele motivated) questions about rhombi and squares correctly.

14 NEDRENCO Mathematics Education), Available online ( tr/wgpapers/wg4/wg4_arici.pdf). [Arici and Aslan-Tutak 15] Sevil Arici and Fatma Aslan-Tutak. The Effect of Origami- Based Instruction on Spatial Visualization, Geometry Achievement, and Geometric Reasoning. International Journal of Science and Mathematics Education 13 (2015), [Arslan 12] Okan Arslan. Investigation of Relationship between Sources of Self-Efficacy Beliefs of Secondary School Students and Some Variables. Ph.D. thesis, Middle East Technical University, [Boakes 09] Norma Boakes. Origami Instruction in the Middle School Mathematics Classroom: Its Impact on Spatial Visualization and Geometry Knowledge of Students. RMLE Online 32:7 (2009), [Boakes 11] Norma Boakes. Origami and Spatial Thinking of College-Age Students. In Origami 5: Fifth international meeting of origami science, mathematics, and education (5OSME), edited by Patsy Wang-Iverson, Robert J. Lang, and Mark YIM, pp CRC Press, [Burger and Shaughnessy 86] William Burger and Michael Shaughnessy. Characterizing the van Hiele levels of development in geometry. Journal for research in mathematics education, pp [Cox 04] David A. Cox. Galois Theory. Wiley & Sons, [De Villiers 86] Michael De Villiers. The role of axiomatization in mathematics and mathematics teaching. University of Stellenbosch Stellenbosch, [DMV, GDM, MNU. Standards für die Lehrerbildung im Fach Mathematik 08] DMV, GDM, MNU. Standards für die Lehrerbildung im Fach Mathematik. Empfehlungen von DMV, GDM und MNU. Mitteilungen der DMV 16 (2008), [Freudenthal 73] Hans Freudenthal. Mathematics as an Educational Task. Reidel, Dordrecht, [Geretschläger 08] Robert Geretschläger. Geometric Origami. Arbelos, [Ghourabi et al. 13] Fadoua Ghourabi, Asem Kasem, and Cezary Kaliszyk. Algebraic Analysis of Huzita s Origami Operations and Their Extensions. In Automated Deduction in Geometry, edited by Tetsuo Ida and Jacques Fleuriot, pp Springer, [Golan and Jackson 09] Miri Golan and Paul Jackson. Origametria: A program to teach geometry and to develop learning skills using the art of origami. In Origami 4: Forth International Meeting of Origami Science, Mathematics, and Education, edited by Robert J. Lang, pp CRC Press, [Golan 11] Miri Golan. Origametria and the van Hiele Theory of Teaching Geometry. In Origami 5: Fifth international meeting of origami science, mathematics and education (5OSME), edited by Patsy Wang-Iverson, Robert J. Lang, and Mark YIM, pp CRC Press, 2011.

15 LEARNING HOW TO AXIOMATISE THROUGH PAPERFOLDING [Gutierrez and Jaime 98] Angel Gutierrez and Adela Jaime. On the Assessment of the Van Hiele Levels of Reasoning. Focus on Learning Problems in Mathematics 20 (1998), [Hull 94] Thomas Hull. On the mathematics of flat origamis. Congressus Numerantium 100 (1994), [Hull 11] Thomas Hull. Solving Cubics With Creases: The Work of Beloch and Lill. The American Mathematical Monthly 118:4 (2011), [Hull 13] Thomas Hull. Project origami: activities for exploring mathematics. CRC Press, [König and Nedrenco 16] Joachim König and Dmitri Nedrenco. Septic Equations are Solvable by 2-fold Origami. Forum Geometricorum :16 (2016), [Martin 98a] George E. Martin. The foundations of geometry and the non-euclidean plane. Springer, [Martin 98b] George E. Martin. Geometric constructions. New York, Springer, [Nedrenco and Beck 16] Dmitri Nedrenco and Johannes Beck. Flachfaltbarkeit: Mathematik mit eigenen Händen schaffen. frontdoor/index/index/docid/13364, [Nedrenco 17] Dmitri Nedrenco. Gestaltung und Durchführung eines Kurses Axiomatisieren lernen mit Papierfalten für das gymnasiale Lehramt. In Papierfalten im Mathematikunterricht, edited by Michael Schmitz, pp Jenaer Schriften zur Mathematik und Informatik, [Olson 75] Alton T. Olson. Mathematics Through Paper Folding. National Council of Teachers of Mathematics, [Row 93] Sundara T. Row. Geometrical exercises in paper folding. Addison Co, Madras, [Schnabel 81] Rudolf Schnabel. Euklidische Geometrie. Habilitation dissertation, Kiel, [Thurston 94] William P. Thurston. On proof and progress in mathematics. Bulletin of the AMS 30:2 (1994), [Yannotta 13] Mark Yannotta. Students axiomatizing in a classroom setting. In Proceedings of the 16th Annual Conference on Research in Undergraduate Mathematics Education, 2, pp , Dmitri Nedrenco Institute of Mathematics, University of Wuerzburg, Germany, dmitri.nedrenco@mathematik.uni-wuerzburg.de