Proceedings of MACAS – 2015 Mathematics and its

Proceedings of MACAS – 2015 International Symposium of

Mathematics and its Connections to the Arts and Sciences

10th Anniversary of the MACAS- Symposia

University of Education Schwäbisch Gmünd, Germany Edited by

Astrid Beckmann, Viktor Freiman, Claus Michelsen

Proceedings of the International Symposium MACAS – 2015

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Index

Introduction

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Plenaries

Uffe Thomas Jankvist: Primary historical sources in the teaching and learning of mathematics – short and long term effects

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George Gadanidis: Aesthetic attention & young mathematicians

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Jean-Luc Dorier: Vectors and translations in mathematics and physics

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Gesche Pospiech: Interplay of mathematics and physics in physics education

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Section 1. Mathematics and Science

Simon Zell: Doing math with motion sensors

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Claus Michelsen: Mathematical modeling – the didactical link between mathematics and biology

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Section 2. Mathematics and Language

Shuzhu Gao & Weiwei Chen & Ang Li: Historical Meanings of Vocabularies in Chinese Mathematical Curriculum

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Silke Ladel & Julia Knopf: Communication via Text Messages – The Network Between Mathematics and Language

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Section 3. Mathematics and Arts

Xavier Robichaud & Viktor Freiman: Exploring Deeper Connections Between Mathematics, Music and ICT: hat Avenues for Research in Education?

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Hans Peter Nutzinger: The Connection of Mathematics and Music as an Opportunity to change Beliefs

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Dietmar Guderian: Mathematics in Contemporary Art

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Section 4. Mathematics and Technology

Reinhard Oldenburg: Functions in computer science and in mathematics – ideas to bridge the gap

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Viktor Freiman & Caitlin Furlong & Manon LeBlanc & Xavier Robichaud: Digital Skills needed for Mathematics Students and Teachers: What does research say?

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Section 5. Search for new ground in pedagogy

Dominic Manuel & Anniee Savard & David Reid: Observing Teachers: The Mathematics Pedagogy of Quebec Francophone and Anglophone Teachers

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Hans Walser: Puzzles and Dissections

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Elena Klimova & Sabine Prinz: A Mathematical Excusion

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Introduction

The symposium series MACAS have been founded in 2005 by an international group of re-searchers at the University of Education Schwäbsich Gmünd, Germany. In 2007, the 2nd Symposium was hosted by the University of Southern Denmark. In 2009, the 3rd Symposium has moved to the North America, to Uniersité de Moncton, New Brunswick, Canada. In 2015, MACAS is celebrating its tenth anniversary returning to its roots in Schwäbsich Gmünd for this special occasion.

The MACAS-vision is based on the goal to achieve a humanistic way of education that is combining various disciplines in one curriculum – an approach, which has been suggested by Renaissance philosophers. According to this philosophical notion, the aim is to educate stu-dents, in the way to enable them to pursue diverse fields of study, while getting insight into the aesthetic and scientific connection between arts and science. In view of the challenges of the 21st century, a modern education with a focus on inter- and multi-disciplinarity has gained a new and larger importance. In this perspective, the field of mathematics is taking over a key role through its connections to all other disciplines and can serve as a bridge between them. This holistic interdisciplinary and transdisciplinary approach is the heart of the MACAS (Mathematics and its Connections to the Arts and Sciences) philosophy.

The MACAS-2015 was targeting in particular scientists from mathematics, science, arts, hu-manities, philosophy, educational sciences and other disciplines that are scientifically con-nected to mathematics. The main idea of the MACAS symposia was to bring at one table sci-entists who are interested in the connection between arts and science in educational curricu-lum, while emphasizing on, as well as researching about, the role of mathematics. This role can be considered from different viewpoints, as previous MACAS activities have shown. Thus, at MACAS-2015 these different approaches and viewpoints between mathematics, arts and science, we pooled together, so that possible synergies and paths for future collaborations could be discovered.

This implied the following focus areas:

• Theoretical investigation of the relation between mathematics, arts and science • Curricular approach to integrate mathematics and science • Importance of the mathematical modelling and the inter-disciplinarity for the learning

and studying of mathematics • Meaning of arts and humanities for the understanding of the connection between arts,

humanities and mathematics in ordinary daily situations • Intercultural dimension of studying mathematics.

The MACAS-2015 also succeeded to bring together researchers, including emerging scholars, whose interests are focused on or connected to these fields of study. The conference thus promoted sharing scientific expertise, initiated new cooperations and enabled the reflection on commonalities and differences between different viewpoints.

The 3-days scientific program (http://www.macas.ph-gmuend.de/?page_id=8) featured 5 key-note presentations by the scholars from Canada, Denmark, Germany, and Switzerland, as well as 11 oral communications by researchers from Canada, China, Denmark, Germany and Swit-zerland. The MACAS proceedings present 16 peer-reviewed papers grouped in six sections.


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In the plenaries section, the paper by Uffe Thomas Jankvist (Denmark) discusses teaching modules on aspects of the History, Application, and Philosophy of mathematics which al-lowed upper-secondary mathematics students to work with historical primary sources. George Gadandis (Canada) shared his work with elementary schools, collaborating with teachers to develop an aesthetic dimension for engaging young children with big math ideas, while look-ing at parallels between "story" and "mathematics". By looking into connections between mathematics and physics, Jean-Luc Dorier (Switzerland) describes students’ learning about vectors and translation in mathematis and physics, while inverstigating if they make the con-nection between these concepts introduced in different disciplines. Gesche Pospiech (Germa-ny) analyses the interplay mathematics - physics while looking at possible sources of the often complained deficiencies of students in applying mathematical elements in physics.

In the first section presenting oral talks, connections between mathematics and science were investigated by Simon Zell (Germany) who studied the use a motion sensor to explore com-mon aspects in mathematics and physics as possible way to enhance learning in upper sec-ondary level. The paper by Claus Michelsen (Denmark) focused on how to strengthen the ed-ucational relations between mathematics and biology by interdisciplinary teaching centred on modeling activities.

The second section deals with connections between mathematics and language. Namely, Shu-zhu Gao et al. (China) discussed vocabularies or terms in Chinese Mathematical Curriculum referring to the characters, words and phrases that indicate the mathematical objects and could be the sources of obstacles relevant to the understanding of mathematical concepts. Silke Ladel and Julia Knopf (Germany) showed basic principles for interdisciplinary lessons of Mathematics and language using the example of text message-communication.

The third section is devoted to papers exploring connections between mathematics and arts. Xavier Robichaud and Viktor Freiman (Canada) analyze connections between music and mathematics, while discussing some of the possible gains that could be obtained in making more explicit connections between the two disciplines in our classrooms by using technology. Hans Peter Nutzinger (Germany) challenges views that music is a subject, which you can learn and perform with delight, whereas mathematics often provokes a rather anxious feeling in learners as well as in many teachers and suggests some possible ways in changing these beliefs. Dietmar Guderian (Germany) presents modern examples of some mathematics appli-cations in art like: mirroring, patterns and numbers, combinatorial analysis, hazard, paral-lels in mathematics/science and art, computer science, “prescientific mathematics” and con-crete art, aesthetics of information, falsification.

The fourth section looks into connections between mathematics and technology. The paper by Reinhard Oldenburg (Germany) analyses functions a a key concept both of mathematics and of computer science while introducing several contexts in which the power of functions in computer science becomes more visible. Viktor Freiman et al. (Canada) reviewed several ini-tiatives implemented in New Brunswick, Canada by connecting technology and other disci-plines by using robotics, online problem-soving, as well as Wiki collaborative environements. The attention needs to be shifted to the development of transdisciplinary skills, often referred as 21st century skills, while looking specifically at the context of mathematics education and digital competences.

The fifth section concludes the proceedings by investigating new paths in pedagogy. Dominic Manuel et al. (Canada) describe regional differences in mathematics teaching and underlying


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pedagogies in Canada, and relate these differences to student achievement in mathematics. Hans Walser (Switzerland) brings different aspects of equivalence by dissection: variations on the theorem of Pythagoras, differences between methods and creativity, symmetry, optimiz-ing, rational and irrational rectangles, color and esthetics. Elena Klimova and Sabine Prinz, from Germany, analyse mathematical excursion as a ‘window’ to look for various opportuni-ties given by nature just outside the window. Their paper (1) illustrates a concept for the pro-motion of mathematically gifted and interested pupils in reference to popular theories about this issue and (2) investigates the positive effects of a mathematical excursion on the devel-opment of mathematically gifted and interested pupils.

The overall success of the Symposium as result of a very productive scientific work magnifi-cently supported by the great enthousiasm, devotion and hospitality of the local organising team lead by Professor, Dr. Astrid Beckmann, President of the University promots for contin-uation of the MACAS symposia in the coming years. The 5th one is planned in 2017 in Co-penhagen, Denmark.

Astrid Beckmann, Claus Michelsen and Viktor Freiman, co-chairs of the symposium

Uffe Thomas Jankvist Proceedings of the International Symposium MACAS – 2015

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Plenaries

PRIMARY HISTORICAL SOURCES IN THE TEACHING AND LEARNING OF MATHEMATICS

– SHORT AND LONG TERM EFFECTS Uffe Thomas Jankvist, Danish School of Education, Aarhus University

Author’s email: [email protected]

Abstract: The study of primary historical sources is often described as a rewarding pursuit worth the effort, despite being extremely demanding for both teachers and students. In the present talk, focus shall be on recent empirical research findings from a Danish study and on the short and long term effects for students of having been exposed to readings of historical primary sources. The Danish study revolved around two specially designed, so-called, HAPh-modules, which are teaching modules on aspects of the History, Application, and Philosophy of mathematics. One of these modules concerned the early history of graph theory and its later application to shortest path algorithms, and the other concerned the history of Boolean algebra and its later application to electric circuit design. Upper secondary mathematics stu-dents, exposed to readings of primary source material as part of these modules in 2010-11, illustrate the short term effects; while undergraduate mathematics students exposed to the same material in 2012, and interviewed in 2015, illustrate the long term effects.

Introduction Among the various possible activities by which historical aspects might be inte-grated into the teaching of mathematics, the study of an original source is the most demanding and the most time consuming. In many cases a source requires a de-tailed and deep understanding of the time when it was written and of the general context of ideas; language becomes important in ways which are completely new compared with usual practices of mathematics teaching. Thus, reading a source is an especially ambitious enterprise, but [...] rewarding and substantially deepening the mathematical understanding. (Jahnke et al., 2000, p. 291)

The above quote is taken from the ICMI-Study on History in Mathematics Education from 15 years ago, but is still as true today as back then. What Jahnke and colleagues point out is that although reading and working with primary historical – or original - sources in mathematics education is not an easy and straightforward task to undertake, it is certainly one which is worth the effort. As is also implicitly addressed in the quote is that the inclusion of original sources may serve more than just one educational purpose. A now somewhat common distinc-tion is that of history of mathematics, including the studying of original sources, serving mainly as a ‘tool’ for the teaching and learning of mathematics, or as a ‘goal’ (e.g. Jankvist, 2009; 2014a).

History as a tool concerns the learning of so-called mathematical in-issues, e.g. that the study of original sources can teach students mathematical concepts, mathematical ideas and notions, theorems, proofs, etc. and that history in this respect can deepen their “mathematical under-standing” (cf. quote above). Other arguments include that original sources offer a truer mode


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of presentation when compared to that of textbooks, which usually goes: definitions, theo-rems, proofs, (constructed) examples of application, while the actual historical development often is the reverse – both on the scale of details, but also on a scale of mathematical topics (e.g. Jankvist, 2014). Furthermore, with original sources the introduction of abstract notions is motivated mathematically, and may therefore be easier for students to comprehend (Barnett et al., 2014).

On the other hand, if it is considered a goal in itself to develop students’ images of mathemat-ics as a (scientific) discipline and their mathematical ‘awareness’, then original sources have equally much to offer in terms of addressing the metaperspective issues – or meta-issues – of mathematics. For example, original sources can illustrate that mathematics actually comes from somewhere, and is brought to life by human beings, that it interacts with culture, society, and other disciplines (and a bunch of other meta-issues, e.g. see Jankvist, 2009). Of course, to develop ‘awareness’ of such matters, a given original source must be placed in its historical setting, etc., as also pointed to in the quote from Jahnke and colleagues. Finally, from a meta-issue as well as an educational point of view, Fried (2001) points out that an original source is not ‘pre-digested’, meaning that it is open to students’ own interpretations, which a secondary source often is not.

Of course, the above distinction between history as a tool and history as a goal should not be understood as this being an either/or. As we shall see, the main purpose can certainly be one of history as a goal where the meta-issues aimed for cannot be reached without learning about some mathematical in-issues along the way. (And the other way around is certainly also pos-sible.)

Design principles of the HAPh-modules The modules to be described are placed in a Danish setting of being concerned with students’ development of mathematical competencies and so-called ‘overview and judgment’ regarding the subject of mathematics (Niss & Højgaard, 2011). While mathematical competencies (e.g. mathematical thinking, reasoning, problem handling, etc.) comprise a “well-informed readi-ness to act appropriately in situations involving a certain type of mathematical challenge”, the three types of overview and judgment are “‘active insights’ into the nature and role of math-ematics in the world” and Niss and Højgaard state that “these insights enable the person mas-tering them to have a set of views allowing him or her overview and judgement of the rela-tions between mathematics and in conditions and chances in nature, society and culture” (Niss & Højgaard, 2011, pp. 49, 73, italics in original). The three types of overview and judgment (OJ) are:

o OJ1: the actual application of mathematics in other subject and practice areas – in par-ticular the actual application of mathematics to extra-mathematical purposes within areas of everyday;

o OJ2: the historical development of mathematics, both internally and from a social point of view – in particular the fact that mathematics has developed in time and space, in culture and society;

o OJ3: the nature of mathematics as a subject – in particular that mathematics as a sub-ject areas has its own characteristics and that some of these characteristics it has in common with other subject areas, but also that some of them are unique.

The HAPh-modules were based on an idea of having aspects of these three dimensions in one and same teaching module for upper secondary school. That is to say, to design a teaching module involving History of mathematics, Applications of mathematics, and Philosophy of


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mathematics – hence, the name HAPh – since these topics related to the three types of over-view and judgment (see Jankvist, 2013). For the actual design of a module, three main ideas – or design principles – were identified:

o To have one primary historical source for each of the three dimensions; o To follow an approach of guided readings (Barnett et al., 2014); o To have the students do essay-assignments (Jankvist, 2011).

The guided reading approach offers a sensible way of dealing with the occasional inaccessi-bility of original sources. The idea is to supply or interrupt a student’s reading of the text by explanatory comments and illustrative tasks along the way, while not in any way altering the original text itself. While such interrupting tasks often deal with the mathematical in-issues, at the end of each module the students were asked to prepare a set of essay-assignments dealing explicitly with meta-issues. Here, the students were to relate the three original texts of the module to each other, discuss various aspects of history, application, and philosophy (see Jankvist, 2013), as well as discuss which text they preferred and why (see Jankvist, 2014b). More precisely, two modules were prepared. One in which the students were to read Danish translations of:

o Leonhard Euler (1736). Solutio problematis ad geometriam situs pertinentis o Edsger W. Dijkstra (1959). A Note on Two Problems in Connexion with Graphs o David Hilbert (1900). Mathematische Probleme: Vortrag, gehalten auf dem inter-

nationalen Mathematiker-Kongreß zu Paris 1900 (the introduction)

And another module in which the students read and worked with Danish translations of:

o George Boole (1854). An Investigation of the Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilities (chapters II and III)

o Claude E. Shannon (1938). A Symbolic Analysis of Relay and Switching Circuits (first parts)

o Richard W. Hamming (1980). The Unreasonable Effectiveness of Mathematics

Implementations The implementation of these two modules in Danish upper secondary school consisted of groups of students working with the texts, while the teacher circled the classroom answering questions. Each module ran over approximately ten 90-minutes lessons. No actual blackboard teaching took place. The module on early graph theory and Dijkstra’s shortest path algorithm was implemented in the fall of 2010, while that on Boolean algebra and electric circuit design was implemented in the fall of 2011. Students were provided with questionnaires before, in-between, and after the implementation of the HAPh-modules, and a selection of students were interviewed about their questionnaire answers as well as their hand-in essay-assignments in 2010, 2011, and 2012 (cf. Jankvist, 2015a). These implementations provide the basis for ad-dressing the question of short term effects of using primary historical sources. (Further expla-nation may be found in Jankvist, 2014b; 2015a). In the spring of 2012 both HAPh-modules were implemented as ‘projects’ in an undergradu-ate course on “Discrete Mathematics and Its Applications” at Roskilde University. Three years after the completion of this course, I was able to track down three students who had been exposed to the HAPh-modules, and who were now about to complete their master’s de-grees in mathematics. These students were interviewed in the spring of 2015 (see also Jankvist, 2014c).


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HAPh-module on Euler’s early graph theory and Dijkstra’s shortest path algorithm The overall theme for this module was ‘mathematical problems’, which is what Hilbert ad-dressed in general terms in the introduction of his lecture from 1900. Besides posting a series of criteria for a good mathematical problem (for example that it must be explainable to lay-men and that it must be challenging but not inaccessible, etc.), he also discussed the origin of good mathematical problems and the relationship between mathematics and the real world through these:

Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. [...] But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. [...] In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. (Hilbert, 1902, quoted from the 2000-reprint, p. 409)

In a certain sense, parts of the history of graph theory illustrate Hilbert’s observations. Usual-ly, Euler’s paper from 1736 on the Königsberg bridge problem – how to take a stroll through Königsberg crossing each of its seven bridges once and only once – is considered the begin-ning of mathematical graph theory (Fleischner, 1990). Euler’s approach to the problem was first to model the real world situation into a more mathematical one represented by his figure 1 (left) and then generalize the problem to: “whatever the shape of the river and its distribu-tion of arms may be, and whatever the number of bridges crossing them may be, to discover whether one can cross the individual bridges once and only once.” (Euler, 1736, p. 129 in Fleischner, 1990, p. II.12). By considering the number of bridges leading to a given region of land and whether this number is even or uneven (I shall not go into all the mathematical de-tails here, for a discussion of those see e.g. Biggs et al., 1976), Euler concludes:1

Therefore, in every possible case, one can immediately and very easily decide, with the help of the following rule, whether or not a walk without repetition can be taken across all bridges:

- If there are more than two regions with an odd number of bridges leading to them, it can be declared with certainty that such a walk is impossible.

- If, however, there are only two regions with an odd number of bridges leading to them, a walk is possible provided the walk starts in one of these two regions.

- If, finally, there is no region at all with an odd number of bridges leading to it, a walk in the desired manner is possible and can begin in any region.

Consequently by this given rule the posed problem can be completely solved. (Eu-ler, 1736, p. 139 in Fleischner, 1990, p. II.19)

As seen from the above, already from first instance Euler began generalizing the problem by bringing it into mathematics and giving it a life of its own which resulted in a much more general solution than the original problem actually required – the Königsberg bridge problem

1 The proof for the third part of Euler’s result is ascribed to Carl Hierholzer (published posthumous in 1873).


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fell in Euler’s first category, since every region (or node) had more than two bridges (or edg-es) leading to it (see figure 1, left and right), meaning that the requested stroll is not possible.

Figure 1. Left: Euler’s 1736 simplification of Königsberg’s bridges. Right: A modern graph representation. Two centuries later, with the dawn of the computer era, graph theory in its fine-tuned mathe-matical form (and discrete mathematics in general) found new applications. Dijkstra’s algo-rithm from 1959 solved the problem of finding shortest path in a connected and weighted graph (Dijkstra, 1959), and as previously mentioned, today Dijkstra’s algorithm finds its use in almost every Internet application that has to do with shortest distance, fastest distance or lowest cost. Furthermore, Dijkstra also discussed and provided a solution for the problem for finding minimum spanning trees:

We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Construct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) [...] Problem 2. Find the path of minimum total length between two given nodes P and Q. (Dijkstra, 1959, pp. 269-270)

The problem of finding minimum spanning trees was considered in order to minimize the amount of expensive copper wire to be used in the building of computers at the time,2 this illustrating Hilbert’s point of the outer world forcing new questions upon mathematics which in this case are solved by means of a well-developed graph theory – as evident also from the terminology used by Dijkstra above.

HAPh-module on Boolean algebra and Shannon circuits The philosophical theme for this module was Hamming’s (1980) comment to a paper by the physicist Eugene Wigner from 1960, where he discussed the “unreasonable effectiveness of mathematics in the natural sciences (Wigner, 1960). While Wigner’s examples stem from the physical sciences, Hamming set out to illustrate this unreasonable effectiveness of mathemat-ics drawing on his own experiences from engineering – and aspects of what we today would consider to be computer science:

During my thirty years of practicing mathematics in industry, I often worried about the predictions I made. From the mathematics that I did in my office I con-

2 The problem of finding minimum spanning trees had on several occasions been solved before though: by Prim in 1957; by Kruskal in 1956; by Jarník in 1930; and by Borůkva in 1926.


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fidently (at least to others) predicted some future events – if you do so and so, you see such and such – and it usually turned out that I was right. How could the phe-nomena know what I had predicted (based on human-made mathematics) so that it could support my predictions? It is ridiculous to think that is the way things go. No, it is that mathematics provides, somehow, a reliable model for much of what happens in the universe. And since I am able to do only comparatively simple mathematics, how can it be that simple mathematics suffices to predict so much? (Hamming, 1980, p. 83)

Hamming did point out that even though the standards of rigor in mathematics may change over time, and with that definitions and proofs, the mathematical results often stay intact. Af-ter having discussed the effectiveness of mathematics and what mathematics is, Hamming (1980, pp. 88-89) goes on to provide some tentative (and partial) explanations for the unrea-sonable effectiveness of mathematics arranged under four headings: (1) “We see what we look for”; (2) “We select the kind of mathematics to use”; (3) “Science in fact answers compara-tively few problems”; (4) “The evolution of man provided the model”. Hamming concluded that indeed mathematics is unreasonably effective, but that his explanation for this still was far from satisfactory in its account: “The logical side of the nature of the universe requires further exploration” (Hamming, 1980, p. 90). Boole’s The Laws of Thought... from 1854 is an example of Hamming’s explanation 2, that we select what mathematics fits the situation, since Boole selected the elements from standard (arithmetic) algebra that applied to his logic system, the purpose of which he described as follows:

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to estab-lish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the na-ture and constitution of the human mind. (Boole, 1854, p. 1)

In chapters II and III of his treatise, Boole considered the role of language in relation to the above and introduces a number of signs and laws to do so. More precisely, he introduced lit-eral symbols x, y, etc. representing classes, and signs of operation +, -, x (times) and the sign of identity = to be used on these classes. For example, if x stands for ‘white things’ and y for ‘sheep’, then the class xy stands for ‘white sheep’, similarly if z stands for ‘horned things’, then zyx stands for ‘horned white things’. After associating the sign + with the words ‘and’ and ‘or’, Boole deduced a number of laws which have their equivalent counterparts in stand-ard arithmetic, for example the commutative law x+y = y+x, the distributive law z(x+y) = zx+zy, and the associative law, and he deduced laws for the operation x (times) as well. The more interesting thing, however, is Boole’s observation that in the context of his investigation we have that xx = x (or x2 = x). If for example x stands for ‘good’, then saying ‘good, good men’ is the same as saying ‘good men’. Boole then drew the consequence of comparing this to standard algebra:

Now, of the symbols of Number there are but two, viz. 0 and 1, which are subject to the same formal law. We know that 02 = 0, and that 12 = 1; and the equation x2 = x, considered as algebraic, has no other roots than 0 and 1. Hence, instead of de-termining the measure of formal agreement of the symbols of Logic with those of Number generally, it is more immediately suggested to us to compare them with


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symbols of quantity admitting only of the values 0 and 1. Let us conceive, then, of an Algebra in which the symbols x, y, z, etc. admit indifferently of the values 0 and 1, and of these values alone. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone di-vide them. (Boole, 1854, pp. 26-27)

Boole’s ideas went on to be adapted within mathematical logic and set theory, and the notion Boolean algebra was conceived. Some eighty years later, however, the ideas showed valuable in a very different setting than that of language and thought, namely design of electric cir-cuits. Shannon was a student at MIT when he got the idea for describing electric circuits by use of logic. With a set of postulates from Boolean algebra (0·0 = 0; 1+1 = 1; 1+0 = 0+1 = 1; 0·1 = 1·0 = 0; 0+0 = 0; and 1·1 = 1) and their interpretations in terms of circuits (0·0 = 0 meaning that a closed circuit in parallel with a closed circuit is a closed circuit; 1+1 = 1 meaning that an open circuit in series with an open circuit is an open circuit), he was able to deduce a num-ber of theorems which could be used to simplify electric circuits (see below). In an interview from 1987 in the magazine Omni, Shannon explained his use of Boolean algebra:

It’s not so much that a thing is ‘open’ or ‘closed,’ the ‘yes’ or ‘no’ that you men-tioned. The real point is that two things in series are described by the word ‘and’ in logic, so you would say this ‘and’ this, while two things in parallel are de-scribed by the word ‘or.’ The word ‘not’ connects with the back contact of a relay rather than the front contact. There are contacts which close when you operate the relay, and there are other contacts which open, so the word ‘not’ is related to that aspect of relays. All of these things together form a more complex connection be-tween Boolean algebra, if you like, or symbolic logic, and relay circuits. The people who had worked with relay circuits were, of course, aware of how to make these things. But they didn’t have the mathematical apparatus of the Boole-an algebra to work with them, and to do them efficiently. [...] They all knew the simple fact that if you had two contacts in series both had to be closed to make a connection through. Or if they are in parallel, if either one is closed the connection is made. They knew it in that sense, but they didn’t write down equations with plus and times, where plus is like a parallel connection and times is like a series connection. (Shannon, 1987 in Sloane & Wyner; 1993, p. xxxvi)

For a given electric circuit a-b, Shannon defined the hindrance function Xab to be 1 if a-b is open and 0 if closed. For example, figure 2 (left) has the hindrance function Xab = W +W’(X+Y)+(X+Z)·(S+W’+Z)·(Z’+Y+S’V), where + indicates series, · parallel and W’ is the negation of W. Now, by means of manipulations according to his theorems of the expression for Xab, Shannon was able to reduce this to Xab = W+X+Y+ZS’V, the circuit of which is illus-trated on figure 2 (right). (For exact reductions and theorems used, see Shannon, 1938b, p. 715.)

Figure 2. Left: The circuit to be simplified. Right: The simplified circuit after reductions on the hindrance function (Shannon, 1938a).


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Examples of short term effects: four upper secondary school students When asked about what they thought of the two HAPh-modules, some of the more immediate reactions of the upper secondary students often concerned whether or not they found it inter-esting. After the completion of the second HAPh-module, a student, Sophia, said:

Well, it’s been dry getting through it, it has, but it’s also been very... Well, it has provided insight, I think, on how mathematics has been used before and how it has come into being, quite precisely. That was really cool, I think. Even though it wasn’t mega exciting and even though it was enormously difficult to interpret, it also gave something in a sense. You got a lot of information about how mathe-matics was applied or about how some clever fellow formulated it back then and so. That was exciting, I think. Okay, maybe not necessarily exciting, but I think it was really cool to see that, how it worked. (Sophia, November, 2011)

In the above quote, Sophia does touch upon aspects of both OJ1 and OJ2, i.e. actual applica-tions of mathematics and the historical development, respectively. But in the following quote from the student Nikita, the second HAPh-module’s relation to OJ1 and OJ2 is more explicit-ly formulated:

For me, I personally think that I get much more interested, when I see it all, than if I’m only told that now we are studying vectors and we must learn how to dot these vectors and then we must be able to calculate the length, right. That’s all very good, but what am I to use it for? Whereas, when you know about the back-ground, the development up till today, that I think was exciting. Because when we began with the first text [Boole’s text], it was kind of like, yeah, that’s alright, he can figure out this thing here, and this equals that, I can follow that, and ‘white sheep’ and so... That was good for starters. Then more is built on top, and all of a sudden we see: Why, it’s a [electric] circuit we are doing! You could begin to re-late it to your own reality; that is, something you knew already. So, the thing about starting from scratch, which I kind of felt I did, and suddenly seeing it form a whole, what it was used for today, and be able to relate it to something. Some-thing you knew about. That, I think, was way cooler. (Nikita, November, 2011)

A third student, Katharine, touches upon the modules’ – and the original texts’ – dealing with both mathematical in-issues and the meta-issues. For her the in-issues were clearly easier to relate to than the meta-issues (we shall later something similar for the undergraduate student, Ada):

Hamming was a little like being on the moon for me. Well, I understood it, but I had to read it twice before I could do the connection. [...] of course he [Hamming] taught me something, but for me it was on this high strange level, because I’m like; I just want the math, and then calculate and stuff, right. So, it was a bit high floating, I preferred the other two better [Boole and Shannon]. But it did provide an incredibly good connection between the parts, that there were the three dimen-sions. (Katharine, November, 2011)

Approximately five months after the completion of the second HAPh-module, some of the upper secondary students were interviewed again (for details, see Jankvist, 2015a). At this time, Sophia was able to deepen some of her previous considerations regarding the modules, and reading original sources, and aspects of the historical development of mathematics (OJ2):

Regarding the modules, even though it has been a little dry from time to time, I do think that it has been nice to get the historical [dimension], to read the original texts, and do it the way they did, the people who developed things. [...] ... in order


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to get it at a slower pace... to try and figure out ‘what the f*ck is going on here?’ That is, instead of just sitting and doing exercises, which you do in school. To try something completely different, something which might be more similar to what they [the originators] actually did. (Sophia, March, 2012)

A fourth student, Christopher, at this time related to aspects of OJ3, the nature of mathemat-ics, and OJ1, its actual applications in other practice areas:

Well, you can say that what gave me some [insight] was all this philosophy, which lies behind, but also the way in which it has evolved... that it has evolved in order to describe a certain thing; for example that Boole used it to describe one thing, and then Shannon saw, okay, I apply it for this other thing and then develop it according to that. This connection; that it is two completely different things they are working with and they then can use the same [mathematics]... that this math-ematics can be applied in so many different contexts. (Christopher, March, 2012)

Examples of long term effects: undergraduate students Ada and Moby Having been exposed to the HAPh-module on Boolean algebra and Shannon circuits three years ago as a ‘project’ as part of their discrete mathematics course in the spring of 2012, it took Ada and Moby a few moments to recall its design and content:

Ada: What I remember the clearest was 𝑥2=𝑥 and this Boolean algebra – because I didn’t like it. I just didn’t. So I remember that quite well. And that he used ‘lan-guage’ to say something about mathematics – that was kind of strange, I thought. But quite fun to read…

Moby: What I remember most clearly was that with the Shannon system...

Ada: Yes, that was clever!

Moby: … the electric circuits. Partly I recall this because, I saw it as again as late as this semester. […] Logic is also part of my study at philosophy – often actually. And I of course also encounter logic here at mathematics – and every time you see logic, propositional logic, truth tables, well yeah, then you might as well view it as electric circuits. (Ada & Moby, April, 2015)

Ada, who was now a graduate student of mathematics and physics, eventually went on:

Mathematically speaking, I don’t think I benefitted that much really. It was more of a new angle on something which I may have known already, I think. […] It was more the perspective which was in play. And I think it would have been the same, if I had been presented to this in upper secondary school.

The difficult thing would have been to see math in this perspective. Because I’m good at things that are ‘just math’, you have rules, etc., etc. But the thing with see-ing ‘what is mathematics’ really; this I have always found challenging. I mean, when you have to see it in, what you call, a ‘meta-perspective’. I’ve spent a lot of time at Roskilde University on such perspectives… and it has been fun to chal-lenge myself in this way. But I’ve never found it easy. It is not ‘logical’ to me, not in the way math is. So, something like this would certainly also have challenged me while in upper secondary school. (Ada, April, 2015)

Ada, whom I recall as a particularly skilled student of mathematics, seems to find the meta-issues of mathematics challenging, while the mathematical in-issues for her are more straight-forward (as we saw for Katharine above). It appeared easier for Moby, now a graduate student


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of mathematics and philosophy, to relate to the meta-issue discussions. In the following quote he touches upon aspects of OJ2, the historical development of mathematics, both internally and from a social point of view, and in particular he lingers at the fact that mathematics de-veloped in one context may be used in another:

This nuance exactly; that you find out that one thing can be used for something else. This is not something that I have come across that often. I have written quite a few student projects at Roskilde University… […]. And I think that I’ve used what we learned in this module in the student projects I made subsequently.

How?

The entire idea with having a meta-perspective on math, and try to figure out – contrary to what we learned in upper secondary school – that mathematics isn’t something which is just there; ‘bang, this is it’ in your ready-made book, where it says ‘here is the math!’. Math is something which has been built up gradually; you tried out something, tossed things back and forth, found some things, and ‘perfect, let’s go with this’. Such things, this line of thought, I don’t think I’ve worked much with that prior to this module. But afterwards, I have done it quite a bit. (Moby, April, 2015)

Examples of long term effects: undergraduate student Julius Unlike Ada and Moby, Julius did not have to recall the content and design of the HAPh-module on early graph theory and shortest-path algorithms. Julius, now a graduate student of mathematics and history, immediately exclaimed:

I’ve used the material later on. I’ve actually lend it out to others. I remember this; that we read these original texts, that we did these tasks. In a way, I actually think this is one of the things that I remember the clearest from my undergraduate mathematics courses. (Julius, February, 2015)

In following quote, Julius touches upon aspects of both OJ1 and OJ2, i.e. the application of mathematics to extra-mathematical purposes within areas of everyday, and the development of mathematics:

… Actually, it is a good example of exactly that, i.e. what began as bridges in a town ended up being used for something with telegraph cables. And then you real-ized – because you had the general theory – that telegraph cables and cobber wire connecting components in computer hardware, things which at first sight are not related in any way, actually is the same problem, only in different forms, because you realize that it can be treated as the same mathematical problem… Yes, the minimum spanning tree. Yes! Exactly, right.

And you remember this after 3 years? Yes, yes. I remember we talked about this. I thought it was funny because of these problems, which one wouldn’t immediately think to be related […] It has to do with how time changes. At one point in time you need telegraph cables, later you develop the computer, and you realize it involves a similar problem, only in a new form. […] It has to do with the external world, how it develops, and how mathe-matics is brought into play. It is fun to see that it begins with an external question, which is dragged into mathematics, and then develops within the field of mathe-


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matics – internally, right – […] and then it plays back at a problem that is not re-lated to the initial problem. (Julius, February, 2015)

From the last sentences in the above quote, it is also clear that Julius has taken to heart some of the points of Hilbert’s text on mathematical problems. In the following quote we see an example of Julius’ insights in terms of OJ3, i.e. the nature and characteristics of mathematics as a discipline and its relation to other disciplines, based on the historical cases of the HAPh-module:

If you are to compare how Euler worked to some of the other disciplines, then it is obvious that it is something different. If a physicist is to solve a problem, he would work within physics; he would use some theories and tools from physics. But he would always return to the outset, return to the original problem. But in the Euler text there is almost no connection facing backwards – only forward. He generalizes, generalizes, generalizes, and fairly soon he couldn’t care less about the Königsberg problem. He quickly realizes that he can answer this. But he doesn’t answer it by actually answering the Königsberg problem. He answers it by being capable of answering a long line of similar questions; his answer is a ‘theo-ry’. He won’t accept to solve just the one problem. He could have done that, of course. I mean spent three pages on solving only this one problem. But if he had only solved the Königsberg problem, then we would hardly ever have heard about it afterwards. And this I think is typical for mathematics. You generalize and take things further and further away from the outset. And this you get an impression of – or you get an image of how mathematics is as a discipline. (Julius, February, 2015)

Concluding remarks In relation to ‘short term’ effects of the HAPh-modules and the exposure to historical primary sources as an integral part of these, it appears clear that the upper secondary students get a glimpse of mathematics being used for something (OJ1); how it has come into being (OJ2); the students experience that mathematics is ‘created’ by human beings (OJ2) and sometimes with a specific extra-mathematical purpose (OJ2); and they get a glimpse of philosophy also being a relevant aspect of the discipline of mathematics (OJ3). As for the ‘long term’ effects, as illustrated by the three undergraduate – now graduate – mathematics students, we also ob-serve several aspects which may be related to the three types of overview and judgment. Not surprisingly, perhaps, some of these appear somewhat deeper and are clearly connected to the meta-issues of mathematics as a discipline (cf. what Ada says about this). Both Moby and Julius talk about the fact that mathematics developed in one context (and for one purpose) surprisingly find its use in another context later. Julius talks about how mathematics is differ-ent from other disciplines and is able to ‘deduce’ more general aspects of the nature of math-ematics as a discipline from the case of early graph theory and Euler’s work.

It seems fair to say that for the upper secondary students as well as the undergraduate students their work with the original sources have provided them a different perspective both on what mathematics teaching and learning can be, and on what mathematics is – Julius says: “you get an image of how mathematics is as a discipline.” For the undergraduate students as well as for the upper secondary students, it is clear that they are able to provide evidence for their views (and beliefs) by means of examples from the HAPh-modules and the original texts in these. Hence, it seems fair to argue that the original sources of the HAPh-modules came to serve as evidence – or knowledge – for the students in their development of overview and judgment. Of course, we cannot expect students’ to change or alter their views (or beliefs) and conse-


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quently their overview and judgment are they not provided with concrete evidence to ‘meas-ure’ these against. If the reader will forgive me for quoting myself:

Not until students have access to evidence – or counter-evidence – are they likely to criticize rationally, reason about, and reflect upon their images of mathematics as a discipline, and possibly accommodate and change them. (Jankvist, 2015b, p. 55)

As ought also to be pointed out in the context of this MACAS symposium, the potential of using primary historical sources does not end here. A use of original historical sources may potentially contribute to some of the more general problems which the mathematical sciences are facing, as for example: the problem of recruiting students to study mathematics and the mathematical sciences at university level (see e.g. Jankvist, 2014d); the problem transition between educational levels, e.g. that between upper secondary school and undergraduate level (see e.g. Jankvist, 2014a); the problem of retaining students in the mathematics (and science) programs once they have entered (Pengelley at HPM2012, as described in Jankvist, 2014a); and last but not least, the increasing demand for interdisciplinary teaching at both upper sec-ondary level and undergraduate level (Jankvist, 2014a).

References Barnett, J. H., Lodder, J. & Pengelley, D. (2014). The pedagogy of primary historical sources

in mathematics: classroom practice meets theoretical frameworks. Science & Educa-tion, 23(1), 7-27. Special issue on history, philosophy and mathematics education, edited by V. J. Katz, U. T. Jankvist, M. N. Fried & S. Rowlands.

Biggs, N. L., Lloyd, E. K. & Wilson, R. J. (1976). Graph Theory 1736-1936. Oxford: Claren-don Press.

Boole, G. (1854). An Investigation of the Laws of Thought on Which are Founded the Mathe-matical Theories of Logic and Probablities. London: Walton and Maberly.

Dijkstra, E. W. (1959). A note on two problems in connexion with graphs, Numerische Math-ematik 1: 269–271.

Euler, L. (1736). Solutio prolematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae 8 (1736), 1741, pp. 128-140.

Fleischner, H. (1990). Eulerian Graphs and Related Topics. Amsterdam: Elsevier Science Publishers B.V.

Fried, M. (2001). Can history of mathematics and mathematics education coexist? Science & Education, 10(4), 391-408.

Hamming, R. W. (1980). The unreasonable effectiveness of mathematics, The American Mathematical Monthly 87(2), 81–90.

Hilbert, D. (1900). Mathematische Probleme – Vortrag, gehalten auf dem internationalen Ma-thematiker-Kongreß zu Paris 1900, Göttinger Nachrichten, 253–297.

Hilbert, D. (1902). Mathematical problems, Bulletin of the American Mathematical Society, 8: 437–479. Reprinted in: Bulletin (New Series) of the American Mathematical Society, 37(4), 407-436, Article electronically published on June 26, 2000.

Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F. El Idrissi, A., da Silva, C. M. S. & Weeks, C. (2000). The use of original sources in the mathematics classroom.


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In: J. Fauvel and J. van Maanen (eds.): History in Mathematics Education, The ICMI Study, pp. 291-328. Dordrecht: Kluwer Academic Publishers.

Jankvist, U. T. (2009). A categorization of the ‘whys’ and ‘hows’ of using history in mathe-matics education. Educational Studies in Mathematics, 71(3), 235–261.

Jankvist, U. T. (2011). Anchoring students’ meta-perspective discussions of history in math-ematics. Journal of Research in Mathematics Education, 42(4), 346-385.

Jankvist, U. T. (2013). History, Applications, and Philosophy in mathematics education: HAPh - a use of primary sources. Science & Education, 22(3), 635-656.

Jankvist, U. T. (2014a). On the use of primary sources in the teaching and learning of mathe-matics. In M. R. Matthews (ed.) International Handbook of Research in History, Phi-losophy and Science Teaching, pp. 873-908. Vol. 2 Dordrecht: Springer.

Jankvist, U. T. (2014b). A historical teaching module on ‘the unreasonable effectiveness of mathematics’: Boolean algebra and Shannon circuits. BSHM Bulletin, 29(2), 120–133.

Jankvist, U. T. (2014c). Long term effects of exposure to primary historical sources in under-graduate studies – the case of Julius. Education Sciences, Special Issue, 2014, 152-171. Guest edited by M. Kourkoulos & C. Tzanakis.

Jankvist, U. T. (2014d). The use of original sources and its potential relation to the recruit-ment problem. For the Learning of Mathematics, 34(3), 8-13.

Jankvist, U. T. (2015a). History, application, and philosophy of mathematics in mathematics education: accessing and assessing students’ overview and judgment. In S.J. Cho (ed.), Selected Regular Lectures from the 12th International Congress on Mathemat-ical Education, pp. 383-404. Switzerland: Springer International Publishing.

Jankvist, U. T. (2015b). Changing students’ images of “mathematics as a discipline”. Journal of Mathematical Behavior, 38, 41–56.

Niss, M., & Højgaard, T. (eds.) (2011). Competencies and mathematical learning - ideas and inspiration for the development of mathematics teaching and learning in Denmark. English Edition, October 2011. IMFUFA tekst no. 485. Roskilde: Roskilde Universi-ty. (Published in Danish in 2002).

Shannon, C. E. (1938a). A Symbolic Analysis of Relay and Switching Circuits, Master’s the-sis. Cambridge: Massachusetts Institute of Technology.

Shannon, C. E. (1938b). A symbolic analysis of relay and switching circuits, American Insti-tute of Electrical Engineers Transactions, 57, 713–723.

Sloane, N. J. A. & Wyner, A. D. (eds.) (1993). Claude Elwood Shannon: Collected Papers. New York: IEEE Press.

Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences, Communications in Pure and Applied Mathematics, 13(1), 1–14.

George Gadanidis Proceedings of the International Symposium MACAS – 2015

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AESTHETIC ATTENTION & YOUNG MATHEMATICIANS George Gadanidis, Faculty of Education, Western University, Canada


Abstract: In my work in elementary school classrooms, collaborating with teachers to devel-op an aesthetic dimension for engaging young children with big math ideas, we have looked at parallels between "story" and "mathematics". Paraphrasing Boorstin’s (1990) criteria of what makes movies work, we have strived to develop math experiences that afford us three distinct pleasures: (1) the pleasure of experiencing the new, the wonderful and the surprising in mathematics; (2) the pleasure of experiencing emotional mathematical moments (either our own, or vicariously those of others); and, (3) the visceral pleasure of sensing mathematical beauty. In this paper I discuss this work with examples from project classrooms.

Introduction The poet Kathleen Norris (1996) writes,

… in exasperation at some muddle I’d made with a math problem on the blackboard, an experience that always terrified me, (my teacher) grabbed my chalk, solved the problem, and said, in a sarcastic voice, “You see, it’s simple, as simple as two plus two is always four. And, without thinking, I said, “That can’t be.” Suddenly, I was sure that two plus two could not possibly always be four. And, of course, it isn't. In Boolean algebra, two plus two can be zero ... I had stumbled onto ... a truth about numbers that I had no language for. ... I staggered away from my epiphany and went back to my seat, feeling certain of the truth I'd seen but also terribly confused. Briefly, numbers had seemed much more exciting than I had been led to believe. But if two plus two was always four, then numbers were too literal, too boring, to be worth much attention. I wrote math off right then and there, and, of course, ended up with a classic case of math anxiety. (p. 38)

Sullivan (2000, p.211) asks: "What exactly are teachers asking for when they say, 'Pay atten-tion'?" My experience tells me that typically few of our students are actively attending to the mathematics ideas in play. I think the majority of students have learned to be passive observ-ers, waiting to be "explained-to". This is not their natural state. "Children begin their lives as eager and competent learners. They have to learn to have trouble with learning in general and mathematics in particular" (Papert, 1980, p. 40). In many cases, a teacher's call to “pay atten-tion” is simply an authoritative statement rather than an invitation to an experience worthy of mathematical attention. In this paper I discuss how we might solve the puzzle of creating mathematics experiences that are worthy of student attention by considering the aesthetic quality of these experiences.

Aesthetics and mathematics? Sinclair (2001) notes:


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Many would agree that we need to make mathematics more relevant and inter-esting to students, yet most recommendations for increased relevance have ig-nored the aesthetic dimension of student interest and cognition. (p. 25)

In my work in elementary school classrooms (Gadanidis, 2012; Gadanidis & Hughes, 2011; Gadanidis & Borba, 2008; Gadanidis, Hughes & Borba, 2008), collaborating with teachers to develop an aesthetic dimension for engaging young children with big math ideas, we have looked at parallels between "story" and "mathematics". Paraphrasing Boorstin’s (1990) crite-ria of what makes movies work, we have strived to develop math experiences that afford us three distinct pleasures: (1) the pleasure of experiencing the new, the wonderful and the sur-prising in mathematics; (2) the pleasure of experiencing emotional mathematical moments (either our own, or vicariously those of others); and, (3) the visceral pleasure of sensing math-ematical beauty. The connection between mathematics and narrative is suggested by Devlin (2000, p. 12), who states that ‘reasoning about mathematical relationships between mathematical (abstract) ob-jects is no different from reasoning about … human relationships between people’. Devlin also claims that this relationship applies to higher level mathematical skills, such as abstrac-tion and relational reasoning. O’Neill, Pearce & Pick (2004) designed a study to test Devlin's conjecture. They tested 3-4 year olds on their general narrative ability and two years later tested the same children on their mathematical ability. They found no correlation between lower level abilities such as vocabulary or computation. They did find a correlation between higher level narrative skills--such as the ability to see a narrative from different perspectives and the ability to identify common plots--and higher level mathematical patterning and rela-tional skills. Wilson (2001, p. xiv) suggests that “Science, like the rest of culture, is based on the manufac-ture of narrative. […] We all live by narrative, every day and every minute of our lives. […] By narrative we take the best stock we can of the world and our predicament in it.” Our world, unlike the traditional mathematics classroom, is not only about paradigmatic thinking but also about narrative thinking (Bruner, 1986). And some would suggest that our world is “a performance-based, dramaturgical culture” (Denzin, 2003, p. x). Story is a biological necessity, an evolutionary adaptation that "train(s) us to explore possibil-ity as well as actuality, effortlessly and even playfully, and that capacity makes all the differ-ence" (Boyd, 2009, p. 188). Boyd (2001) notes that good storytelling involves solving artistic puzzles of how to create situations where the audience experiences the pleasure of surprise and insight. Watson and Mason (2007, p. 4) "see mathematics as an endless source of sur-prise, which excites us and motivates us. ... The challenge is to create conditions for learners so that they too will experience a surprise." Movshovitz-Hadar (1994) and Watson and Mason (2007) see mathematics as full of surprises and Burton (1999) sees mathematicians' "world of knowing" as "a world of uncertainties and explorations," with "feelings of excitement, frustration and satisfaction" (p. 138). Sinclair and Watson (2001) highlight the importance of mathematical surprise and moments of insight that "reveal our logics of feeling, as Gattegno (1974) put it, those intuitive and aesthetic modes of thinking that allow us to formulate conjectures and ideas" (p. 40). In the design of mathemat-ics tasks, Zaslavsky (2005) highlights the important role of "cognitive conflict" and "perplexi-ty, confusion and doubt" (p. 299).


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Artistic puzzles in mathematics pedagogy Gadanidis, Hughes and Cordy (2011), discussing students artistic communication of mathe-matics ideas, note that:

We do not believe that artistic mathematical expression is possible in a mathe-matics program which focuses on procedural rather than conceptual knowledge. Students can add artwork to “decorate” procedural knowledge, thus adding a layer of “sugar-coating” to otherwise dry mathematical ideas, but mathematical art, like art in general, requires a deeper engagement and under-standing. Thus, for us, challenging mathematics is a co-requisite for artistic mathematical expression. (pp. 423-424)

Movshovitz-Hadar (1994) notes the need for non-trivial mathematical relationships in elicit-ing mathematical surprise. In my collaborative work with teachers we have taken up the chal-lenge of solving artistic puzzles of how to create situations where young mathematicians ex-perience the pleasure of mathematical surprise and insight. To do this, we had to put aside theories of what children cannot do, such as Piaget's stages of development. Piaget (1972/2008) himself raises some caution about how generally his stages of development might apply, noting that

we used rather specific types of experimental situations ... it is possible to question whether these situations are, fundamentally, very general and therefore applicable to any school or professional environment. (p. 46)

Like Egan (1997), Fernandez-Armesto (1997), Papert (1980), and Schmittau (2005) we seek to challenge Piaget's notion that young children are not capable of abstract thinking, which Egan identifies as integral to language development. We distinguish between content--what teachers are mandated to teach--and context--the con-nections we make to other mathematical ideas to enrich the experience. Purposely, to chal-lenge common conceptions of what mathematics young children can engage with, we have used contexts from the secondary school curriculum. For example, we have used the Calculus ideas of infinity and limit as a context for grade 3 students to study the content of area repre-sentations of fractions and discover that "I can hold infinity in my hand!" (see Figure 1) We have also used ideas from the topic of sequences and series as a context for grade 2 students to study growing patterns and discover that "Odd numbers hide in squares!" (see Figure 2) And we have used ideas from non-Euclidean geometry as a context for grade 2 students to study longitude lines on a flat map and on a sphere and discover that "Parallel lines can meet!" (see Figure 3) Documentaries of these and other examples are available at our project website: www.researchideas.ca.

Figure 1. Infinity. Figure 2. Odd numbers. Figure 3. Parallel lines.

http://www.researchideas.ca/


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Math worth talking about When mathematics is worthy of student attention, it is also worth talking about. We offer stu-dents opportunities to share their learning with peers, with family and friends, and with the wider community. We involve students in creating skits or comics or lyrics to songs that an-wer the question, "What did you do in math today?" We also use research data (from student thinking and parental feedback) to author data-based songs that share classroom experiences. Students perform these songs for their school, and they are posted on the Web for public ac-cess. We also have a band, funded by the Fields Institute for Research in Mathematical Sci-ences, which perfoms these songs in math concerts for schools across Ontario. These songs are available at our project website: www.researchideas.ca. We also involve mathematicians, by engaging them with the same activities done by young mathematicians, and asking them to comment from their perspective. For example, Figure 4 shows mathematician Lindi Wahl, from Western University, discussing an activity done by grades 1-2 students, where they explored what changes and what stays the same in concrete and graphical representations of linear functions.

Figure 4. Representations of linear functions.

The interviews with mathematician are available at our project website: www.researchideas.ca

Aesthetic attention Humans are storytelling beings. We think in terms of stories, we understand the world in terms of stories that we have already understood, we learn by living and accommodating new stories, and we define ourselves through the stories we tell ourselves (Schank 1990; Bruner 1990, 1996; Wilson, 2001). When we frame our pedagogical goals in terms of engaging stu-dents with mathematics that is worth talking about, mathematics that can be a good story to share, it is natural to turn to the arts for guidance. Boorstin's (1990) model has focused our pedagogy on creating opportunities for students to attend aesthetically, as well as cognitively, through experiences of mathematical surprise and insight, emotional and vicarious connec-tions, and visceral sensations of mathematical beauty.

http://www.researchideas.ca/


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Root-Bernstein (1996) notes that although it is common today to generalize that the sciences are "objective, analytical, and rational" and the arts are "subjective, emotional, and based on intuition", not long ago the sciences and the arts were "considered to be very similar, certainly complimentary, and sometimes even overlapping ways of understanding the world" (p. 49). He describes the "common aesthetic" (p. 50) shared by the sciences and the arts:

The integration of thought and emotion, analysis and feeling is as typical in science as in poetry, music, or painting, and many scientists have been explicit-ly clear on this matter. (p. 55) The details of the personal aesthetics are always unique, but the nature of an aesthetic experience seems to be universal. (p. 62)

Although we have used the arts for framing our pedagogical model, the core elements of that model are not limited to the arts. The aethetic dimension of mathematics--which can engage students' aesthetic attention--is natural human quality. This aesthetic that is common to math-ematics, the arts, and other disciplines is the aesthetic that makes the experience of these dis-ciplines human.

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197-214. Boyd, B. (2009). On the origin of stories: evolution, cognition, and fiction. Cambridge, MA:

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ture. Thousand Oaks, California: Sage Publications. Devlin, K. (2000). The math gene: How mathematical thinking evolved and how numbers are

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Jean-Luc Dorier Proceedings of the International Symposium MACAS – 2015

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VECTORS AND TRANSLATIONS IN MATHEMATICS AND

PHYSICS Jean-Luc Dorier, Université de Genève, Switzerland


Abstract: In mathematics, students learn about vectors and translation, in physics they model forces, speed, acceleration, etc. with vectors and study movements of translation. Do they make the connection between these concepts introduced in different disciplines or do they put things in separate boxes? In this paper we will start with some partial considerations on the history of vectors and we will give some references. Then we will show some examples of na-ïve illustrations of vectors from physics in mathematics textbooks. We will then present a non-conventional example and the difficulties it created for both mathematics and physics teach-ers. Finally we will develop an example of a possible collaboration between teachers of both disciplines in relation to movement of translation.

Introduction Our purpose in this paper is to see, with the example of vectors, how mathematics can be ac-tually connected to physics and to give propositions to make this connection more efficient for the benefit of both mathematics and physics. This talk is based on the supervision of Ba’s doctorate (Ba 2007) and some common publications (Ba & Dorier 2006, 2007, 2010, 2011) and our own work on vectors and linear algebra (Dorier 1995, 1996, 1998, 2000). Mathematics is often seen as a very specific subject, either by students, parents, media or even mathematicians themselves. In many contexts, mathematics is feared and seen as a subject for selection, disconnected from interesting applications in real life. Moreover, the structure of teaching institutions, in many cases, makes the collaboration between teachers from different disciplines very difficult. At the same time, mathematics is more and more invisible in every-day life, since high technology tends to hide the mathematics necessary for its creation in so-phisticated black boxes. As a result, it is quite a challenge to give an adequate answer to those who, legitimately, wonder what mathematics is useful for. Our study reveals that, most often, teachers know very little about other subjects, even in rela-tion to their own subject. Mathematics teachers do not want to get involved in too specialised applications while physics teachers send their students back to their mathematics teacher for explanations on the use of mathematics in their field. As a result, students are used to seeing mathematics and physics as disconnected. This is reinforced by cultural differences, especial-ly visible in the use of vocabulary or recipes that create artificial gaps between different disci-plines. In this paper we will start with some partial considerations on the history of vectors and we will give some references. Then we will show some examples of naïve illustrations of vectors from physics in mathematics textbooks. We will then present a non-conventional example and the difficulties it created for both mathematics and physics teachers. Finally we will develop an example of a possible collaboration between teachers of both disciplines in relation with movement of translation.


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History of vectors: some comments Like we argued in our history of linear algebra, the links between vectors and traditional ge-ometry on the one side and the emergence of modern linear algebra on the other are not as obvious as one may think:

One of the myths (supported by traditional teaching) about the natural link between geometry and linear algebra comes from the extensive use of common vocabulary in the two fields. For instance, the fact that the linear structure is called a ‘vector space’ automatically certifies the geometrical origins of the theory. Another reason for assuming a natural link comes from the use of geometrical representation: e.g. the sum of two vectors can be represented as the diag-onal of a parallelogram (as in the parallelogram of velocities and forces, a very ancient type of representation used in physics to symbolize the combined action of two velocities or forces applied at a same point). However, there is a big gap between the traditional use of this rep-resentation and the modern interpretation of the algebraic sum of vectors (see Crowe 1967, p.2) (Dorier 2000, pp.12-13). The real starting point in the history of vectors lies in the invention of analytical geometry, independently by René Descartes (1596–1650) and Pierre de Fermat (160?–1665) and the criticism made by Gottfried Wilhelm Leibniz (1646–1716) as expressed in a letter to Chris-tian Huyghens, dated 8th September 1679:

I am still not satisfied with Algebra, because it does not give the shortest methods or the most beautiful constructions in Geometry. That is why I believe that we need still another analysis which is distinctly geometrical or linear (see footnote above), and which will express situation directly, as Algebra expresses magnitude directly. And I believe that I have found the way and that we could represent figures and even machines and movements by characters, as algebra represents numbers or magnitudes. (Translation by Crowe (1967, p. 3) from the original in French published in (Leibniz 1850, vol.1, p.382)). Leibniz tried to invent a geometrical calculus on the basis of his criticisms towards analytic geometry but never managed to take into account the idea of direction of a line. A specificity of vectors lies in their double nature as geometrical entities with algebraic properties. In this sense, Leibniz failed to introduce the idea of negative in geometry. Moreover, negative quan-tities remained problematic for many mathematicians until the 19th century. For instance, in his Algebra published in 1673, John Wallis (1616–1703) claims: “But it is also impossible that any Quantity can be Negative. Since it is not possible that any Magnitude can be Less than Nothing, or any Number Fewer than None.” (vol. II, p. 264). To this vision of a quantity less than nothing, Wallis opposes the idea of a magnitude in an opposite direction: “Yet when it comes to Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense.” (Ibid., p. 265).

To illustrate this idea, he imagines a person who goes 5 yards forward from A to C and then 8 yards backward from C to D.

How much he is Advanced when at D, or how much Forwarder than when he was at A: I say –3 Yards. (Because +5 – 8 = – 3.) That is to say, he is advanced 3 Yards less than nothing. Which in property of speech, cannot be, (since there cannot be less than nothing.) And there-fore as to the Line AB Forward, the case is Impossible. But if (contrary to the Supposition) the Line from A, be continued Backward, we shall find D, 3 Yards Behind A. (Which was presumed to be Before it). (Ibid., p.265)


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The difficulty here is due to the dominant model of the negative quantities being less than nothing, under zero (thermometer, lift, underwater levels…). The obvious, yet revolutionary idea is to see the zero not as a pushing point but as an articulation point, where one can turn back and face the other direction. Wallis’ text is one of the first, in which the two lines AB and BA are seen as opposite to each other and the negative is associated with a change of sense on a right line (a half turn). This represents the first conceptualisation of one-dimensional vectors. This idea of directed line segments will be developed further when rep-resenting imaginary quantities geometrically in the 2-dimensional plane. One of the pioneer works in this sense was published in 1806 by Jean-Robert Argand (1768-1822) an amateur mathematician from Geneva. In his treatise, like Wallis, he explains that the negative quanti-ties are on a symmetrical line of the positive from the origin 0. He then interprets the quantity (–1) as the product (+1).(–1) and the square root √(-1) as the geometrical means of this prod-uct, therefore a quantity with magnitude 1 and with direction perpendicular to the line of real numbers, either in one sense or the other.

We can see here that the point K which is the origin, the zero, is an articulation point and that with the half turn from positive to negative, the line opens on the whole plane (or even space) (see Châtelet 1993, p.128). In this sense the representation of imaginary quantities is a crucial step towards the creation of vectors. Yet, it would take another 50 years to reach some deci-sive discovery.

Hermann Grassmman (1809–1877) published in 1844 his first version of Die lineale Ausdehnungslehre, a revolutionary treatise in which he invented, in a total disconnection to mathematics of his time, not only vectors but linear and multi-linear algebra. This book was ignored and never understood in its time (see Schubring 1996). In his introduction, Grass-mann points out very clearly the inspiration he got from the introduction of negatives in ge-ometry.

The initial incentive was provided by the consideration of negatives in geometry; I was used to regarding the displacements AB and BA as opposite magnitudes. From this, it follows that: if A,B,C are points of a straight line, then AB+BC=AC is always true, whether AB and BC are directed similarly or oppositely; that is, even if C lies between A and B. In the latter case, AB and BC are not interpreted merely as lengths, but rather their directions are simultane-ously retained as well, according to which they are precisely oppositely oriented. Thus the distinction can be drawn between the sum of lengths and the sum of such displacements, in which the directions were taken into account. From this there followed the demand to estab-lish this latter concept of a sum, not only for the case that the displacements were similarly or oppositely directed, but also for all other cases. This can most easily be accomplished if the law AB+BC=AC is imposed even when A, B, C do not lie on a single straight line. While I was pursuing the concept of the product in geometry as it had been established by my father, I concluded that not only rectangles, but also parallelograms in general, may be re-garded as products of an adjacent pair of their sides, provided one again interprets the prod-uct, not as the product of their lengths, but as that of the two displacements with their direc-tions taken into account. (Grassmann 1844, translation from Kannenberg 1995, p. 9)


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I cannot give more information in this text about the history of vectors. One can see (Crowe 1967) or Dorier (1995, 1996, 1998 and 2000, part I) for more details. However, I would like to stress two points for what is going to follow:

1. Vectors are, by nature, hybrid objects with a dual geometrical/algebraic feature, which is an essential component in their learning.

2. The history of vectors is largely independent from the history of linear algebra (which is linked with infinite dimensional function spaces). Vectors are linked with the histo-ry of physics (Maxwell equations).

In Ba & Dorier (2006, 2010 and 2011) we have shown by analysing the history of the teach-ing of vectors in France that in the early XXe century a didactical object “the geometrical vec-tor” has been artificially created and cut from its roots in physics. Then modern mathematics made vectors the prototype of linear algebra for ideological reasons. Since the 80s vectors are struggling to find a correct place at the junction between physics and mathematics. That is what we are going to analyse briefly now.

Naïve situations from physics in mathematics teaching of vectors It is well known that when situations from extra-mathematical contexts are used in mathemat-ics it is often very naïve. In his work, Ba (2007) analysed several mathematics textbooks3 and showed that the situa-tions from physics contexts used in the teaching of vectors are not very numerous and usually very naïve. I will give only two examples.

First naïve example An Indian in a canoe wants to cross a river whose sides are parallel. The canoe is subject to two forces: the stream force represented by vec-tor and the force exerted by the rower repre-sented by vector . One considers that the canoe starts at point D and moves in the direction of vector defined by the equality In the whole exer-cise, it is assumed that the length of is the dou-ble of the length of . 1. Reproduce the schema above and represent vector and the trajectory of the canoe. One calls A the arrival point of the canoe and as-sumes that the river is 35m wide 2. Calculate the angle and the length DA 3. Make a drawing at scale 1/1000 and verify the results on the drawing.

Here it is quite clear that the context is purely a pretext to develop some basic geometrical skill; there is no motivation to make use of the sum of vectors, since the vector equation is given. Moreover physicists can argue that the use of the term “force” to designate the effects

3 He analysed textbooks of last year of lower secondary and first year of upper secondary (age 15-16 years old)

which is when vectors are introduced in the mathematics curriculum in France., while forces and velocity are introduced only in the two first years of upper secondary school.


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of both the stream of the river and the rowing could be subject to a debate… which is clearly ignored here. However, at least, the directions of the forces are discussed, which not always the case. Indeed, various studies have shown that, although a force is characterised by a mag-nitude and a direction, tasks given in physics focus on the magnitude only (Genin et al. 1987 and Lounis 1989).

Second naïve example In a rugby training, one player J1 is chal-lenged by two other players J2 and J3 at-tached with two ropes. In each case, is player J1 going to move forward or backward?

In the three cases the sum has exactly the opposite direction of (with magnitude respectively bigger, equal and smaller!!!): too nice to be true… one can imagine the type of discussion if the directions had been totally different, while in this simplified cases, the main argument is one dimensional, therefore insufficiently representative of the use of vectors and suspicious regarding the credibility of the context.

In order to make the use of vectors more substantial in a physics context, we created our own exercise… and faced other difficulties…

A non conventional example This specific situation concerns a variation from a very classical problem in dynamics: the pendulum. We designed this version in order to make the determination of the direction of a force essential for its solution.

Here is the text of the problem:

An iron small ball (comparable to a point M) with mass m is hung to the ceiling by a thread (whose mass will be neglected). A magnet attracts the ball, the direction of the force makes an angle θ under the horizontal line (see drawing) and its magnitude is F. When in equilibrium, the thread makes an angle α with the vertical (see drawing) The only forces are: the weight of the ball, the attraction of the magnet and the tension of the thread.

Data: m=200g, θ=30° , F=2N, take g=10N/g. 1. Write the equilibrium equation. 2. Represent with the scale (1cm=1N) the forces in action. 3. What are the characteristics of the tension of the thread? We now give the answers:


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1. The equilibrium equation is given by the first fundamental law of dynamics:

2.

3. With use of relations in an isosceles triangle, it is easy to see that makes an angle of 30° with the vertical and has a magnitude of 2√2 N.

The interesting point in this problem is that in question 2, one has to draw and first in order to draw (since this the opposite of their sum). Then the direction of gives the direc-tion of the thread. Therefore in order to draw the thread, one has to use the sum of two vec-tors, which is the essential key to the problem. However, this task is problematic in the context of physics. Indeed, the construction required in question 2 has to take place in a mathematical model, which is not reality. Moreover, in this model the point where the thread is attached to the ceiling can only be determined at the end of the process. Once this theoretical construction is made, one can come back to the drawing representing the reality and use the results of question 3 to represent the situation starting with the fact that the thread makes an angle of 30° with the vertical.

In his work, Ba (2007) submitted this problem both to students and teachers in Première S (second scientific class of upper secondary school, Lycée, in France, students aged 16) both in mathematics and in physics lessons. The students, tested during physics lessons, did not have any problem with question 1. But they met real difficulties in question 2. They could not transfer the problem into the mathematical model. As a matter of fact, they did not see that there were two levels in the representation of the situation (reality and model) and were blocked because of the missing thread. On the other hand, different studies show that students at this level have acquired sufficient knowledge about vectors to be able to draw the sum of two vectors and to answer questions like question 3, when given in a purely geometrical set-ting. This shows that students have sufficient knowledge of mathematics but are not able to mobilise it, when required, in physics. Moreover, they do not identify the mathematics at stake in a physics problem. The difficulty here is typical of modelling situations.

Furthermore we interviewed physics teachers and asked them if they would give such a prob-lem to their students, and if so, what difficulty they think would appear. Massively, they ad-mitted that this problem was close to a typical situation of dynamics, but at the same time they felt uncomfortable with the formulation. They did not believe that their students would handle

Ceil-

Horizon-

M

30

30

30

T

P

F

P+


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the geometrical construction. For the solving of question 3, they also massively prefer a solu-tion using projections on two orthogonal axes, a technique widely used in physics.

Mathematics teachers, on the other hand, would not be ready to give such a problem to their students because they do not consider it as part of mathematics. Moreover, the physics notions at stake are only taught one or two years after the sum of vectors is studied in mathematics, so there is a problem of priority.

This problem appears to be typical of the difficulty in building a bridge between mathematics and physics even when two notions are naturally related like vectors and forces. Teachers of both disciplines do not want to take the burden of linking the two, and students cannot trans-fer their knowledge from one to the other. Only a joint effort from teachers of both disciplines can solve the problem. Moreover a real change in culture is necessary to fight against the compartmentalization of disciplines

In order to show some possible collaboration between mathematics and physics teachers, we are now going to explore the connection between translation in mathematics and movement of translation in physics.

Movement of translation and translation: an impossible dialogue between mathematics and physics? The question now is quite different from the previous case of vectors and forces. Indeed, here, the relation between physics and mathematics seems more obvious, since the same terms are used but, on the other hand, it is more mysterious. Indeed, it is well known that geometrical transformations are cognitively attached to dynamical representations. A mathematical trans-formation only takes into account an initial and a final state (i.e. an element and its image), but one, often implicitly, attaches an idea of movement between those two states. In this sense, the effect of a rotation on a geometrical object can be seen as a movement of rotation of the object. This representation of a geometrical rotation is coherent with the concept of movement of rotation in physics. However, it is quite different with translation, since the dy-namical representation of the translation of a geometrical object is attached to the special case of rectilinear movement of translation only and does not take into account all the other types of movement of translation studied in physics.

Indeed, in physics an object is said to have a movement of translation when any segment attached to the solid remains parallel to itself during the movement (def.1).

Therefore, the trajectory of the object can be non-rectilinear, but follow any type of curve:

Figure 1. The general case of a movement of

translation

Figure 2. Ferris Wheel: the prototypical example of a circular movement of trans-lation, often confused with a movement of rotation


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Experiments have been made involving mathematics and physics teachers about their repre-sentation of a movement of translation, and it shows that most mathematics teachers only think of rectilinear movement of translation and are totally puzzled when physics teachers try to explain what is a movement of translation by showing a movement with their hand follow-ing a non-rectilinear trajectory, yet with the hand remaining parallel to itself (Gasser 1996).

Another puzzling question is that most French physics textbooks (at the level of Première S), in the chapter introducing the definition of a movement of translation as given above, also give illustrations with objects on which vectors are drawn (like on the figure above), although the definition only mentions segments. Indeed, the objects are always supposed not to change their shape during the movement, therefore a segment [M,N] on the solid cannot change its length. In a movement of translation any segment remains parallel to itself. So vector can have only two opposite directions, and cannot change length. Thus, according to a basic con-tinuity principle, it is clear that vector cannot change its direction (because it would have to go from one direction to the opposite without being able to have any intermediary positions in between). Having the same direction and the same length it therefore remains identical.

In other words, a movement of translation can be characterised by the fact that every vector on the solid remains identical (def.2).

One can wonder why such a formulation is never used in physics, while vectors appear in practically all drawings. Certainly, the fear of being too abstract is the main reason. This is characteristic of the distance separating physics and mathematics.

Let us now see what the connection between movement of translations and mathematical translation can be and why this is neither explicit in physics nor in mathematics teaching. Let us introduce the time in the notation, what physics teachers usually do not do at this level in order to avoid abstraction and formal notations. For each value t of [0,T] (the duration of the movement) and any point M of the solid S, one calls M(t) the position of the point M at time t. Then the definition of a movement of translation becomes:

S has a movement of translation if, for any t, t’ of [0,T] and M, N of S: . (def3)

In terms of translation, the condition can be expressed by:

S has a movement of translation if, for any M, N of S, there is a translation τMN (independent of the time) such that for any t of [0,T] : τMN (M(t)) = N(t). (def.4)

Of course τMN is the translation of vector . This is a first characterisation of a movement of translation using the mathematical notion of translation. Moreover, if one applies what is sometimes known as the parallelogram rule:

is equivalent to : , one gets another characterisation of a movement of translation using the mathematical notion of translation:

S has a movement of translation if, for any t, t’ of [0,T] there exists a translation τtt’ (inde-pendent of the point) such that for any M of S : τtt’ (M(t)) = M(t’). (def.5)

The difficulty here is that this translation does not give any information about the trajectory followed by the solid S between t and t’. Finally, if, for distinct t and t’, one divides the preceding equality by (t’ – t), one gets:


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Which becomes, when t’ tends to t: == , which means that at any time during the movement all points have the same velocity. Reciprocally, by integrating between t and t’ the equality of velocity, one gets that:

. This gives another characterisation of a movement of translation that students see in physics without any proof:

S has a movement of translation if, at any time, all points have the same velocity. (def.6)

In the teaching of physics in France, only definitions 1 and 6 are given to the students without any proof of the fact that they are equivalent and no attempt to connect them to mathematical translations, either in books or by teachers (according to a questionnaire sent to a large num-ber of teachers).

Moreover, physics teachers do not care about this connection or simply believe that transla-tion and movement of translation are the same thing, while most mathematics teachers reduce movement of translation to the rectilinear case.

Most students are used to not trying to make bridges between physics and mathematics and therefore use the same word in two different disciplines without trying to find a connection. However, they have difficulties with movement of translation. They often get confused, for instance, between circular movement of translation and movement of rotation (like for the Ferry wheel). They also have difficulties in non-“classical” examples in making their defini-tion operational when trying to prove that a given movement is of translation, while they have the mathematical skills at hand (Ba, 2007). This situation is not satisfactory. Especially since students have all the necessary knowledge at hand to be able to understand, with a minimum of time and work, the different connections we have briefly established above. Again, the question is to know who, among the mathemat-ics teachers and the physics teachers, should be in charge of making the connection explicit. Moreover such clarification would benefit to physics teaching, of course, since it enriches the notion of movement of translation, but also to mathematics teaching, since it provides a use of vectors and translations in a rich context, with a challenging use of notations. For these rea-sons, it seems that this should be a joint effort of both teachers, either in parallel, in the math-ematics class and the physics class, or even better, in a common session with both teachers. In his work Ba (2007) experimented the second solution in Senegal with a joint teaching involv-ing the mathematics and physics teachers on the basis of the previous analysis. This was a relatively positive experiment, although it would need to be repeated in different contexts, in order to evaluate the real impact.

Conclusion Nowadays, the teaching of mathematics is subject to a social pressure that requires more ap-plications and raises issues about modelling. The outside world forces mathematics to come out of its ivory tower. This is true for all levels of education in any context. However, it is even more essential for students whose major interest lies outside mathematics. It is not pos-sible anymore for mathematicians to remain isolated, away from applications, in a position of superiority. This is the best thing that could have happened to mathematics, which needs to make itself more visible.


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Our belief is that mathematics will not sell its soul by getting more interested in other disci-plines. We hope to have shown with the example of vector and translation that by connecting itself to outside contexts, mathematics can be taught in a richer way, without reducing the value of its concepts. As we have shown here but also in other works in relation with econom-ics (Dorier 2005) the connection with other disciplines is also a way of making the formal aspect of mathematics accessible. Making the connection with another discipline is not only a question of psychological motivation, but also an epistemological challenge. Indeed, using an example from another discipline is not only a (fashionable) way to motivate students, but it is also a way to present a richer context where issues on the meaning of mathematics will auto-matically be addressed and questioned. This is not just an abdication of supremacy, but also a humble recognition of the power of mathematics as a provider of models to other disciplines, which has always been an essential part of its history. Literature Argand, J.-R. (1806). Essai sur une manière de représenter les quantités imaginaires, dans les

constructions géométriques. Annales de mathématiques IV, 134–147.

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Ba, C. & Dorier, J.-L. (2011). Die Entwicklung der Vektorrechnung im französischen Ma-thematikunterricht seit Ende des 19. Jahrhunderts. Mathematik in der Lehre 58, 215–232.

Ba, C. & Dorier, J.-L. (2010), The teaching of vectors in mathematics and physics in France during the 20th century. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Azarello (Eds.) Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education – CERME6 – January 28th-February 1st 2009 Lyon (France) (pp. 2682 – 2691). Lyon: Editions INRP - http://www.inrp.fr/editions/editions-electroniques/cerme6

Ba, C. & Dorier, J.-L. (2007). Liens entre mouvement de translation et translation mathéma-tique : une proposition pour un cours intégrant physique et mathématiques. Repères IREM 69, 81-93.

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Châtelet, G. (1993). Les enjeux du mobile. Paris: Seuil. Crowe, M.J. (1967). A history of vector analysis : the evolution of the idea of a vectorial sys-

tem. Notre Dame : University Press. Reed., New-York : Dover, 1985.

Dorier J.-L. (2005). An introduction to mathematical modelling – An experiment with stu-dents in economics. In M. Bosch (Ed.) e-Proceedings of CERME4 – 17-21 Feb 2005 – Sant Feliu de Guixols, http://ermeweb.free.fr/CERME4/ pp.1634-1644.

Dorier J.-L (Ed.) (2000). On the teaching of linear algebra, Dordrecht: Kluwer Academic Pub-lisher.

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Dorier J.-L. (1996). Basis and Dimension, from Grassmann to van der Waerden. In G. Schu-bring (Ed.). Hermann Günther Grassmann (1809-1877): Visionnary Mathematician, Scientist and Neohumanist Scholar -Papers from a Sesquicentinnial Conference. Dordrecht: Kluwer Academic Publishers (Boston Studies in the Philosophy of Sci-ence). 175-196.

Dorier J.-L. (1995). A general outline of the genesis of vector space theory, Historia Mathe-matica, 22(3). 227-261.

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Genin, C., Michaud-Bonnet J. & Pellet A. (1987). Représentation des élèves en mathéma-tiques et en physique sur les vecteurs et les grandeurs vectorielles lors de transition collège-lycée. Petit x, 14/15. 39–63.

Grassmann, H. (1844). Die lineale Ausdehnungslehre, Leipzig: Otto Wigand. English transla-tions are taken from [Hermann Grassmann, A new branch of mathematics. The Ausdehnungslehre of 1844 and other works, trans. Lloyd C. Kannenberg, 1995, Chi-cago, La Salle: Open court].

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Lounis A. (1989). L’introduction aux modèles vectoriels en physique et en mathématiques : conceptions et difficultés des élèves, essai et remédiation, Thèse de l’université de Provence Aix-Marseille I.

Schubring, G. (Ed.) (1996). Hermann Günther Grassmann (1809-1877): Visionnary Mathe-matician, Scientist and Neohumanist Scholar -Papers from a Sesquicentinnial Con-ference. Dordrecht: Kluwer Academic Publishers (Boston Studies in the Philosophy of Science).

Wallis, J. (1673). Algebra. London. (see extracts pp. 46–54, in Smith, D. E. (1959). A source book in mathematics. New-York : Dover).

Gesche Pospiech Proceedings of the International Symposium MACAS – 2015

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INTERPLAY OF MATHEMATICS AND PHYSICS IN PHYSICS EDUCATION

Gesche Pospiech, TU Dresden, Germany Author’s email: [email protected]

Abstract: Mathematics and physics are strongly interrelated in a fruitful relationship. Since centuries the use of mathematics is an important part of the methodology of physics. There-fore this interplay should be taught at school in order to give students an adequate insight into physics and the nature of physics. The key difficulty students mostly face is establishing the connection between the concrete physical phenomena and the abstract mathematical world. In order to analyse the necessary cognitive activities and the possible difficulties of students a physical-mathematical model, displaying the interplay between mathematics and physics with the focus on the structural role of mathematics in physics, is used. An additional important aspect is the corresponding communication about physics with the help of mathe-matical representations, mostly graphs or formula. After describing central aspects of the interplay mathematics - physics we analyse possible sources of the often complained deficiencies of students in applying mathematical elements in physics. Empirical research shows that the attitudes of students have to be considered and often are even more positive towards mathematical elements than is often said. Then we look in detail at the strategies and problems students in lower secondary school might show. Herewith we address mostly the representational forms of graphs and formula. We derive hints for important aspects in teaching the interplay mathematics and physics. Introduction Science education as an important part of general education aims at scientific literacy. There-fore also physics education should contribute to the development of the students' ability to take an own well founded standpoint on questions of everyday life or societal relevance relat-ed to physics. Because among other aspects mathematical modelling in developing scientific knowledge gains increasing importance students should learn about it. As the use of mathe-matical elements and structures is most pronounced in physics this is a suitable place to learn about mathematization. Famous is the saying of Wigner about the "unbelievable effective-ness" of mathematics in physics (Wigner, 1960). Even this often quoted citation only presents a spotlight onto the complicated and rich interrelationship between physics and mathematics. This richness should influence the teaching of physics and its methodology. That mathematics in physics education plays an eminent but also problematic role was among others stated by the German physics educator Martin Wagenschein along the lines of Wigner: „Dass die Ma-thematisierbarkeit des durch die Physik herausgehobenen Naturzusammenhangs ein Faktum ist, über das wir nur staunen können, dies die Kinder erfahren zu lassen, ist eines der kost-barsten Ziele des physikalischen Unterrichts. Dass er oft das Gegenteil bewirkt, muss uns zu denken geben.“ Accordingly the task of teaching the "art of mathematization" proves to be difficult and is accompanied by many quick statements concerning the interest and abilities of students: The most often stated assumptions are that mathematical applications in physics seem to be demo-tivating and far too difficult and that students just do not master the mathematical techniques. These statements deserve a deeper analysis. In order to do so we will very briefly describe the


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role of mathematics in physics. Among other points we will discuss the communicative role of mathematical representations, also analysed from a historical perspective. Then we will present some results on students' views, difficulties and strategies in the use of algebraic rep-resentations (formula) with the help of a model of mathematization. Some possible difficulties with graphical representations will be described afterwards with an example. In the end we draw some conclusions for further research.

Mathematics in Physics and in Physics Education The central goal of physics is to describe nature with the help of physical concepts such as force, energy, heat etc. and to derive physical laws, models and theories. These then contrib-ute to the explanation of physical problems or everyday phenomena. The related physical method rests on two pillars: performing experiments with their evaluation as well as the use of mathematical tools and structures both in a deductive and inductive way. Both pillars are strongly interrelated with each other. E.g. the prediction, the evaluation and interpretation of experiments cannot be thought of without use of mathematics. The onset of mathematization in physics is often dated back to the 17th century to the times of Newton and Huygens. How-ever, from then until today there still was a long way until the mathematical structures used in physics were fully developed and reached its modern undoubted relevance. Alone this long development suggests that teaching mathematization will not be an easy task. However, it is necessary because knowledge of physical laws as well as basic insight into the physical meth-od lie at the core of scientific literacy and are part of understanding physics and the nature of physics. This requires insight into the structure of physics and its outside relations, especially to mathematics.

Mathematics in Physics

If analysed in detail, the role of mathematics in physics has multiple aspects: it serves as a tool (pragmatic perspective), it acts as a language (communicative function) and it provides a logical and structural framework (Krey 2012).

• Mathematics as a technical tool The role as a technical tool is the most prominent role in the perception of students as well as many researchers. Mathematics provides many structures e.g. functions or equations, differen-tial equations and algorithms. These contribute to the precise formulation of physical laws and therefore allow calculating numerical results and hence quantitative predictions.

• Structural role of mathematics In its role as a structural means mathematics builds the skeleton of physical theories and gives valuable general theorems allowing to proceed into the unknown as e.g. in the case of the No-ether Theorems. Pietrocola (2008) identifies three main aspects of the structural role of math-ematics: 1. Physics inherits the formal operations and definitions of mathematical objects if these are used (use of vectors, derivatives,..). 2. Mathematics orders the physical phenomena according to underlying patterns (analogies). 3. Mathematics orders physical thought by the physical (concrete) meanings of its operations (limiting cases, functions,..) Generally it can be said that the structuring role of mathematics becomes more and more important the more ad-vanced the physical theory is, as is obvious e.g. in the theories of the 20th century. In these theories often mathematics is needed as a guidance even for conceptual explanations or rea-soning.


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• Communicative Role of Mathematics Mathematics provides possibilities that help in representing physical relations in a symbolic and at the same time precise way. These representational means - numbers, graphs, algebraic forms and geometrical objects - contribute to its communicative role. The power of these symbolic representations can easily be recognized in reading a physical paper in an unknown language. Nevertheless representations have to be interpreted in the framework of the physical knowledge which requires a fixed syntax and semantics with some general rules. These different roles have to be considered also in teaching physics and its mathematization.

Mathematics in Physics Education

The perception of the importance and relevance of mathematics in physics education strongly depends on the age of the students. Whereas in doing physics at university level mathematics is regarded an integral part of teaching and receives high attention this high esteem is already greatly reduced at high school level. Here only some aspects of mathematics are considered suitable and important. Mostly only its formal use as a technical tool is required, not so much the structural role (Schoppmeier 2013). The lesser regard of mathematics is even more pro-nounced in physics education at lower secondary school. Here often a more qualitative ap-proach in teaching physics is followed as is seen in many curricula (see e.g. KMK 2004).

However, mathematics as “structuring physical thought” (Pietrocola 2008) should play a cen-tral role in order that students get used to the specific interplay and acquire an insight into the nature of physics. As the full appreciation of the interplay requires long experience its teach-ing should start as early as possible. This requires careful analysis of the essence of mathe-matical structures for physics in order to find elementary and important elements and meth-ods. Concerning mathematical operations one pathway is to analyse the so called "Grundvor-stellungen" from mathematics education. The inner-mathematical meaning of the mathemati-cal operations changes when used in a physics context because then the symbols are laden with physics notions and ideas (Sherin 2001). Teachers should be aware of this shift in mean-ing of e.g. multiplication or division in order to recognize corresponding difficulties.

Concerning the technical role even the syntax may change e.g. in writing functions, deriva-tives or just using different letters and symbols. So it cannot be expected that students simply transfer the mathematical structures into physics but an explicit connection has to be made between usage of structures in mathematics and in physics.

The communication role is connected to these shifts in denoting and in meaning: working with formula and graphs always needs the physical framework and interpretation.

Mathematical Representations and the Interplay Mathematics - Physics

Abstract mathematical objects have to be represented in order that they can be used in phys-ics. The first and most simple representation which already shows a significant difference to mathematics education is the use of numbers. In physics numbers are generally coupled with units belonging to the respective physical quantity. Also the magnitude of numbers can be important in order to decide on the relevance of an effect. If a number is the result of a calcu-lation corresponding to a physical problem it has to be interpreted with respect to the original situation. If the numbers are data from an experiment they are often ordered in a table and then transferred into a graph. On this way some information (perhaps the precision) is lost but once e.g. a regression curve is drawn the relation between the data becomes visible, can be


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interpreted as relation between the corresponding physical quantities and be memorized more easily. Therefore graphs have a strong advantage before numbers concerning visualization. Perhaps the most prominent mathematical representations are the algebraic forms (by which we mean formulas resp. equations). They serve for calculating concrete values and making precise quantitative predictions to be evaluated and validated. On a more advanced level for-mula can be used to derive new laws. If the formula are thought of as functions it becomes important to distinguish the dependent and independent variables and to identify constant pa-rameters. Once the functional dependencies are identified the mathematical techniques as e.g. differentiation or integration can be used for analysing and exploring the scope of the func-tion. These theoretical implications make the algebraic representation a powerful tool of phys-ics, even its trademark, and show how intimately mathematical structures and techniques are intertwined with physics.

Representations from a historical perspective

The relevance of a suitable representation will now be shown with the example of the ideal gas law. Today the ideal gas law seems to be a fairly easy law of physics which can be derived with ease from experiments or with the assumption of the particle model. The historical development however shows that this is by no means the case but that its derivation took more than a century. One of the first physicists studying the behaviour of gases was Robert Boyle in 1662 ("Boyles Law") who conducted the experiments and published the results (Fig. 1). As representations he choose the data themselves, i.e. numbers, and a verbal description:

Figure.1. Original table of Boyle (left, after De Berg 1995) and a graphical representation as it would be used today in a publication “For this being considered, it (the data) will appear to agree rarely-well with the hypothesis, that as according to it the air in that degree of density and correspondent measure of re-sistance, to which the weight of the incumbent atmosphere had brought it, was able to coun-ter-balance and resist the pressure of a mercurial cylinder of about 29 inches, as we are taught by the Torricellian experiment; so here the same air being brought to a degree of den-sity about twice as great as that it had before obtains a spring twice as strong as formerly.“

The choice of the representation and corresponding level of abstractness has a big impact on the communication, on teaching and the possible conclusions to be drawn: The table and the description need far more time and effort to extract the conclusion, the pressure-volume rela-


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tion of gases, than the graph. Nevertheless, some qualitative predictions could e.g. be drawn from the extrapolation of the numbers in the table: "historical data ... shows us that the alge-braic format of a law is not necessary for making predictions." (De Berg 1995). Today we would use graphs for representing the data and formula for describing the relations. These representations are more abstract but also more concise. Especially the algebraic formulation enables deeper insights and derivations of relations including abstract ideas: “.. algebraic expressions are primarily useful for theoretical development which leads not only to new data but new concepts. This development takes place through the laws of mathematics." (De Berg 1995) In thermodynamics the derivation of the state function, its total differential and conclu-sions thereof would be conclusions from the algebraic formulation. In addition it allows for connecting the microscopic particle model with the macroscopic phenomena by establishing the functional dependence between temperature and average velocity of the particles or be-tween pressure and number of collisions and momentum transfer between the particles and the walls.

The teaching goal of the mathematization therefore determines which representation is most suitable. The historical development gives hints which representations are more demanding for students or which seem more natural. Herewith it has to be considered which competen-cies are required and which difficulties and advantages for students are connected with each representation. We chose formula and graphs for a deeper consideration.

Students' Views and Competences in Physics

Besides general complaints by educators about the physical-mathematical abilities of students there are only few well founded research results concerning the views and the competences of students concerning the use of formula and graphs.

That the students' underlying epistemic views are important for achievement is underlined by a study with first year university students (Ataide& Greca 2013). They identified three groups of students. Some view mathematics as a tool: it is used by the physicist to facilitate numeri-cal calculations. Others view it as translator: mathematics is a translator of physical thought, a mere manifestation of physics, with the task of representing it in an understandable way. Some hold the view of mathematics as giving structure: it helps to structure physical thought and leads to new concepts. The problem solving techniques used by college students were studied by Tuminaro&Redish (2007). They found a broad variety fo so-called "epistemic games" from "Plug'n chug" to more refined strategies.

For school students it is often asserted that formulae and calculations are not interesting (Hoffmann et al 1998). On the other hand Krey (2012) has found that students see formula as characteristic elements of physics which serve as a tool and as a means of precise and direct communication. In line with these results it was observed that formula are not seen as far too difficult (Strahl 2010). An analysis of the required technical-mathematical competencies in the higher secondary schools showed that they are mainly on the level of the middle grades which indicates that technical problems cannot be the sole source of difficulties but that the structural aspect needs special attention (Schoppmeier 2013, Trump&Borowski 2014).

In a questionnaire study (192 students, grade 8, 14 y old) it was found that a group of students can be identified with a significant positive attitude towards formula, and a group with a sig-nificant positive attitude towards graphs. Already in this age students show a quite well dif-ferentiated view on the role of formula in physics and even have a glimpse of the structural role (Pospiech&Oese 2014). This was confirmed in interviews (20 students, grade 8, 14 y old). More than half of the statements concerning formula had quantification (technical role)


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as their main aspect, but one quarter of the statements alluded to the structural role and gain of insight and another quarter mentioned reduction of cognitive load (communicative role) (Naumann 2015). More than half of the statements concerning graphs held that they help to reduce cognitive load, mostly because of the visualization function; one third stressed the structural role and gain of insight. So students perceive formula and graphs as each having a well defined role for representing physical relations and for learning and understanding (Naumann 2015). The higher their self esteem the more positive they think about the structur-al role of graphs or formula. In spite of this more positive than expected view on formula there are undoubted difficulties.

Concrete Difficulties with Formula

In order to analyse the problems of students concerning the structural role a model is needed. This was developed by Uhden et al. (2012) (see Fig.2) and was used for categorising the spe-cific difficulties of students in grades 9 and 10 in applying formula (Uhden 2015).

Figure 2. In order to stress the interplay between mathematics and physics this is located in the "red box" which mirrors the structural role of mathematics. It contains mathematization as well as the interpretation (Uhden et al 2012, modified by Geyer 2015 (unpublished)).

It focusses on the interplay of mathematics and physics and stresses the connection between qualitative reasoning and mathematical formulation. In the "red box" two aspects are located: strategies of mathematization and ways of interpretation. Concerning the mathematization Uhden (2015) could identify unfavourable strategies such as „Plug'n Chug“ or superficial translation, i.e. just remembering a formula where no connection between physical process and formula was explicitly made, similar to the "epistemic games" observed by Tuminaro& Redish (2007). Especially in idealization the use and meaning of e.g. "0" sometimes was dom-inated by intuitive physical notions and not by physical-mathematical reasoning. Students focussed either on the mathematics or on the physics ideas without setting both sides into re-lation. But with some students it turned out that the connection between physics and mathe-matics was helpful for them in identifying errors. Concerning the interpretation Uhden (2015) observed a deficient understanding of formula where e.g. the conditions of usability or the analysis of limiting cases were not taken into account. The explanation of the physical mean-ing of functions sometimes showed misunderstandings when e.g. a t was read as a(t). These results strongly show that teachers should stress the structuring role of mathematics: understanding of functions, the concept of variables or parameters or exploring limiting cases.


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Concrete Difficulties with Graphs

Function graphs in mathematics depict a relation in the form {(x, f(x))| x in D}. In physics we call graphs (i.e. line graphs or curves) the representation of a functional relation between two (or more) physical quantities. If a graph stems from a formula it is normally depicted by cal-culating some values and sketching the corresponding curve. In this case the graph visualizes the abstract relation and might serve as a memory aid. The derivation of a graph from an ex-periment is connected with severe fundamental and principal difficulties. First the data from the experiment mostly are put into the coordinate system. Then the points can be simply con-nected by lines or approximated by a smooth regression curve, if a functional relation is con-jectured or known. In textbooks from lower secondary school this complex procedure is not always carefully considered. Often students are forbidden to just connect points because the teacher knows the (theoretical) dependence between the quantities. Hence the students often are not aware that they have to consider several aspects at the same time: a) there are measur-ing uncertainties, the data are not "true"; b) there is an additional interpolation between the measured values; c) a theory is underlying the required smooth curve (e.g. a straight line or a parabola), which does not follow uniquely alone from the data.

How young students (grade 6) try to make sense of this procedure is indicated by an example. After an experiment of heating water a student said: “Temperature should increase steadily, but we make errors and that's why they do not lie on a straight line. The points in a diagram ought to lie on a straight line.” This attempt to make meaning from the graphical representa-tion shows the complexity of the process. Also systematic errors cause deviations from the idealized law: “The temperature rises always, but the higher the water is heated up the slower the temperature rises. Temperature → rises steadily.” The verbal description of the tempera-ture increase does not uniquely define the proportionality: Mostly the grade 6 students speak about “The temperature of ice/ water increases steadily.“. Here the additional difficulty arises of aligning verbal explanation and mathematical terms (Pospiech et al 2012). In grade 8 there is already more awareness of mathematical terms: “Supplied energy and temperature in-crease are proportional if no change of state takes place.” or “The heat Q supplied to an ob-ject is proportional to the temperature increase." These statements indicate that managing and interpreting graphs is a demanding task and requires experience from the side of the stu-dents.

Conclusion

In modern society, in social contexts or in the discussion of environmental problems mathematical models play an increasingly important role. Empirical data show that in principle many students have a certain awareness of the scope and role of mathematization in physics. But it is also seen, that its competent application requires a careful preparation along the school career also with respect to use of representation and interpretation. The described students' difficulties indicate that the consciousness of teachers concerning the structural role of mathematics with focus on the differences between the use of mathematical structures in mathematics and in physics has to be promoted in order that students be enabled to reconstruct the pathway from physics to mathematics and backwards. The teachers have to be especially clear on the intricacy of the interplay. Therefore the views and knowledge of teachers has to come into focus of future research.


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