2.1 – 4 European Museums of non-formal Mathematics
2.1.1 – In Germany MathematiKu
Mathematikum is a mathematics museum, located in Gießen, in Germany. It was founded in 2002 on the initiative of Albrecht Beutelspacher who, organising a seminar for future teachers, gave them the task of creating a geometric model and explaining its mathematical functioning. Their commitment and enthusiasm made the professor of geometry realise the importance of founding an innovative museum in the fields of mathematics. After several temporary exhibitions, Abrecht Beutelspacher was awarded a major prize by the Stifterverband für die Deutsche Wissenschaft, an association that seeks to identify and meet challenges in higher education, science and research. The idea of a “German mathematics museum” was finally recognised by the nation and the project was launched in 2002.
As soon as it opened, the attendance figures exceeded all their expectations, they expected 60,000 visitors, and more than double that number visited the museum in the first year. In 2010, the million – visitor mark was even passed, showing that public interest has not changed over time.
The Mathematikum museum is based on three pillars:
- Mathematics for all (all ages, all categories, with or without a passion for mathematics);
- Cooperative mathematics (where people can help each other);
- And, finally, practical and fun mathematics (where people experiment and manipulate).
This museum offers multiple interactive experiments to better understand mathematical concepts (from geometry to functions and probabilities). These experiments are designed to be experienced by all ages and to stimulate all of our senses: we are offered to lock ourselves in a giant soap bubble, to measure our size with the binary system, or to see ourselves an infinite number of times. More than 170 new experiences are spread over 1200m² of floor space.
The success of the museum is explained by the fact that visitors are really taken into account and are a significant part of their visits: they can move freely and leave at any time. Furthermore, the mathematics presented are taken seriously and the material conditions (buildings and facilities) are subject to extreme vigilance (they are regularly maintained and restored).
At the same time, the Mathematikum is developing its traveling exhibitions in Germany, France, and even more widely in Europe. There are three of them:
- The first, “Hands – on mathematics”, presents a selection of the most popular experiments;
- The second, “Mini – Mathematikum”, follows the general concept of the Mathematikum and is accessible from the age of 3;
- And finally, the last, “What a coincidence! “offers experiments on the theme of coincidence.
2.1.2 – In Spain, MMACA
The “Museu de Matemàtiques de Catalunya” (MMACA) was created in 2006 in Cornellà de Llobregat, Catalonia. This museum is the only mathematics museum in Spain and therefore represents a great opportunity to unite and bring together its citizens around mathematics, bringing them together through interactive experiences and manipulative activities.
When it was created in 2006, the museum only had travelling exhibitions, i.e. , it did not receive the public onsite but made its activities travel around Spain. Today, there are more than ten travelling exhibitions, with different levels of difficulty and on different themes such as intuition, geometry or the tangram.
Alongside this activity, MMACA has opened its doors on the upper floor of the Mercader de Palau Museum in Cornellà since 2014. It covers an area of more than 300m² , where you can find its permanent exhibition ” Mathematical Experiments”. This exhibition is made up of games, construction games and a series of elements related to geometry, calculations and statistics or strategy. One of the museum’s mottos is: ” Forbidden NOT to touch “, because the important thing f or this museum is that people can manipulate and experiment with the elements they are offered and perhaps change their minds about mathematics.
With the aim of reaching out to all audiences, the Mathematics Museum of Catalonia seeks to cover both the playful and the rigorous aspects of mathematics so that the Catalan mathematics scene continues to support it.
In addition, MMACA is increasing their European and international projects in order to support the European mathematical culture and to participate in the development of frameworks that will subsequently favour the learning and understanding of mathematics by students.
The Navet Science Centre is one of four municipal associations in Borås Sjuhärad. This “informal region” was established in 1999 when a new county – Västra Götaland – was created with the aim of safeguarding the interests of the municipalities in the Borås sub – region. The region’s aim in establishing the four municipal associations was to create a spirit of 19 initiative, ingenuity, creativity and entrepreneurship among its 280,000 inhabitants. The Navet Science Centre was built precisely to participate in and support the development of the skills of the school staff by designing teaching methods and materials.
This science centre covers more than 3500m² and raises awareness on a wide range of subjects such as mathematics, astronomy, creative technology, space creation and the environment (from climate to food and agriculture). It seeks to reach out to all age groups by expanding its offer and allowing anyone to manipulate, in order to offer continuous training of the different subjects. For instance, offer training on the environment for companies seeking to learn more about the subject, but also offer teachers training which can be further explained to their students. Students of all ages (from kindergarten to high school) can also come to the centre to do activities involving mathematics and other scientific fields. In addition to its permanent exhibition, the Navet Science Centre also offers several travelling exhibitions for rental, particularly on the subjects of geometry, algebra, and probability. This allows the knowledge acquired during the permanent visit to be deepened and internalized. Teachers particularly appreciate this offer and many have now fully integrated it into their teaching: it allows them to discover a new way of presenting science and to explain it mo re clearly to their pupils.
The Science Centre is one of the 19 Swedish science centres grouped together by the association Svenska Science Centres with the aim of stimulating scientific curiosity. According to the 2011 Science Centre World Congress statement, these centres and their scientific commitments are based on three pillars: scientific knowledge, hands-on interaction and co-creation of experiments with scientists and the public. Each year, the National Agency for Education checks the quality of the science centres offer before offering them a national grant or not. In 2016, 2017, 2018 and 2019, the Navet Science Centre scored the first place, with a total of 35 points out of a possible 40 (in 2019).
2.1.4 – In France, Fermat Science
Fermat Science was founded in 1996 in the town of Beaumont-de-Lomagne, near Toulouse, following the excitement and enthusiasm due to Andrew Wiles’ demonstration of “Fermat’s great theorem” in 1995. From its inception, Fermat Science has sought to promote mathematical culture in a fun way to all audiences while linking this field to other cultural or heritage knowledge. Indeed, the association relies on the history of its town and surroundings as well as on the history of Pierre de Fermat to enrich and diversify its offer.
Fermat Science offers numerous workshops in the Maison de Fermat, the place where Pierre de Fermat was born and lived occasionally. These workshops always involve one or more mathematical concepts and can concern both the public school (from kindergarten to high school) and the general public. To name a few, the workshops may focus on maths and art like optical illusions, origami, stained glass and mosaics, or on maths and cryptology or maths and puzzles, such as tangrams, pavements, labyrinths, etc. Fermat Science also travels and offers these same workshops in schools. Some of these workshops are also available for rental and travel to schools, colleges and high schools all around France or other foreign countries. For instance, the exhibitions Voyage en Mathematique or Curiosités Mathématiques, or kits like Divertimaths, Festimaths etc. In addition, the association presents an annual exhibition on a theme related to mathematics such as coincidence, geometry or weather and climate.
The association organises events such as La fête des Maths, Femmes en Sciences, Matermaths or bâtiTMaths. These events aim to give a different image of mathematics on specific themes for different audiences. Moreover, Fermat Science, anchored in the local heritage of its surroundings, also seeks to make its contribution to the dissemination of mathematical culture on a European scale. To this end, the association participates in European projects such as STEAMbuilders and Math Reality.
2.2 – A Museum’s Aspiration
Places of scientific heritage and culture are constantly redefining themselves and rethinking their role in and for society as well as in their relationship to the public. If the notion of “inclusive museum” originated in the early 1980s in an Anglo-Saxon professional community, this posture is now widespread in the French-speaking and European space. Beyond the accessibility of certain spaces or the adaptation of systems, the museum space is open to everyone, inside and outside the walls, in urban, peri-urban or rural areas.
Museums of all categories are constantly redefining themselves socially. The notion of inclusion is particularly questioned by museums at the level of their institutional and public policy, but also of their cultural and scientific communication project, their managerial policy and their economic model. Over the last twenty years or so, there have been numerous initiatives to include and make room for different audiences in museums, thus creating new avenues for reflection on professional practices and the evolution of professions. As an actor for and in society, the museum is, for example, developing participative mediation, working on accessibility, questioning the representations of professionals and the public to be “included”. Partnerships with cultural, health and economic structures and associations are being created.
Access to “culture for everyone” is one of the objectives of democratisation. Making the place accessible and open to all people regardless of their social origin, level of knowledge, or health becomes an objective, in the name of the general interest. It concerns all cultural and museum structures that are part of the public or associative service. Individuals and citizens are equal and as such share common interests.
The care taken in welcoming the public is reflected as much in the design process as in the first moments of the meeting. Indeed, from the very beginning, it is a question of creating confidence in order to put visitors at ease. This consideration of the public continues afterwards by adapting the discourse and a capacity to listen and be open.
The social and inclusive museum is a place where sharing and exchange are privileged between the institution and the individual whether they are local, disabled, socially distant or a tourist. A place for reflection and debate, in search of dialogue, highlights the identity of territories, their rich heritage, cultures, people and their knowledge.
For students, the museum can help to motivate the study of mathematics, from the point of view of the emotional approach. But it can also provide insights and tools to enrich those experiences that are the basis of the formation of the mathematical idea.
If we start from the consideration that a session in the museum must be special, exciting and motivating and essentially different from a school activity, it is clear that the exhibits of an exhibition must be “special” by size, design, materials and interaction. In this case, the main goal must be to keep in touch and/or inspire class activities and contents.
In this case, the main goal must be to keep in touch and/or inspire class activities and contents.
- Materials must be simple, easy to create.
- If possible, easy to be built by students themselves.
- New technologies, as 3D printers or the FabLabs network, can deeply and quickly increase the possibilities to self-produce pull-up exhibitions.
However, an exhibit must immediately motivate an action (hands-on or virtual) and communicate the basic concept that inspires it, requiring the minimum instruction and intervention by the educator.
On the contrary, an exhibit has to stir discussion, exchange and collaboration among the users, to develop into the basic elements of making a visit to the exhibition a real moment of learning.
Additional information (historical placement, uses, concepts involved …) can be present as elements of the exhibition (panels, short videos, still or moving images, simulations on a computer …) or entrusted to a virtual visit through the museum’s web page or a follow-up period at school or at home, either individually or in groups.
Finally, a reflection is needed about the concept of proximity: with materials, activities, relationship with the public and the territory, but also, if not above all, with the concepts and the methods of mathematics. Most of the society considers maths abstract, dry and distant. It is then fundamental that, without trivializing or spectacularising it, we can highlight aspects that bring mathematics to each one’s experience and stimulate the pleasurable aspects of discovery, resolution, and improvement of self-performance.
2.3 – Conclusions deriving from the project partners’ discussion with math museums’ experts
This report is intended to provide a contributive, comprehensive, and thorough preview of the scope, collection, and educational opportunities of Math Museums. The consortium of the project has connected with the 4 prementioned Math Museums from 4 distinct European countries – Fermat Science in France, MMACA in Spain, Mathematikum in Germany, and Navet in Sweden – to contribute and to render assistance to the provision of necessary information for the recreaMATHS project. The Math Museums’ experts were provided with a questionnaire of 5 sections: Museum Details, Museum Scope-Aims-Visions, The museum’s collection and exhibits, Museum Education and Training, and Mathematics at the museum. Despite all of them being museums with an identical area of expertise – being mathematics – the responses of all four museums are inspirational and significant. The creation of a report based on Math Museums aims to increase educators’ understanding of the museums’ aspirations and to popularize the educational methodologies and pedagogies developed in a museum.
The following results were obtained by a google form questionnaire:
|Museum Name||Country||Year of establishment||Area of Expertise||How many (on average) preschool children visit the museum|
|Fermat Science||France||1995||Mathematics||5400 students (2019)|
|MMACA||Spain||2014||Mathematics / Hands-on material||10000|
|Navet Science Center||Sweden||1996||Mathematics and Science||2000|
How about we have a broader perspective of the museums’ responses and focus individually on each question, starting with section 2 and the museum’s scope. Fermat Science’s scope is to introduce school children to a non-formal approach for learning mathematics through the use of several mathematical activities, such as games, visits, exhibitions, and workshops. The museum has an aim to create an interactive mathematical space in the birthplace of Pierre Fermat, to popularize mathematics with non-formal tools, to promote and disseminate the topic of mathematics and create educational tools. Similarly, MMACA in Spain has a scope to promote the topic of mathematics and offer visitors a deep and exceptional mathematical experience. They aim to promote mathematics, show the kind and powerful aspect of mathematics, offer a joyful math experience as well as provide a good vision of mathematics and support for their teachers. Moving on to Germany, the Mathematikum museum has set its scope to give the opportunity to as many people as possible – especially younger people – to participate in mathematics. Last, Navet Science Center’s scope is a 3500square meter exhibition space, focusing on topics such as mathematics, water, astronomy, creative technology, maker space, sustainability Hub – among other things; digital hybrid arena, 3 labs, facilities for in-service teacher training. Navet’s aims include inspiring the curiosity and the will of all visitors to learn, being an inspiration for mathematics, science, and technology, helping teachers and schools develop new methods for teaching in mathematics, science and technology, and encouraging entrepreneurial learning and action.
Besides the museums’ aims, the visions that each museum has is as important. Fermat Science museum’s vision includes, the popularization of mathematics, to show their objects and how they work in both historical and recent discoveries, presenting mathematics in a fun way that is accessible to as many people as possible, and emphasizing the links of mathematics with other fields of knowledge and culture. The visions of MMACA and Mathematikum respectively, are to develop a pleasant reputation and material which are highly rated, and for people (families, friends, etc.) to come in their leisure time to “do math” and leave the museum happier. Finally, Navet Science’s vision is to be a noticeable component and collaboration partner for the realization of the sustainable development goals.
Proceeding to section 3, the museums were asked about their main collection of exhibits. Fermat Science responded that at the moment they have the Szilassi and Pierre Fermat sculptures, exhibits of Fermat called ‘enfant de la Lomagne’, ‘Voyage en mathématique’ and ‘Curiosités Mathématiques’. The most acknowledged exhibits of the museum for kindergarten children are mostly based on experiencing hands-on because at such an age touching and handling are very important. Exhibits include a space for games and manipulations, a mathematical manipulation tool in the form of challenges, a mathematical workshop with big geometric shapes. MMACA’s main collection of exhibits consists of some of Pythagoras’ visual proofs, joyful puzzles and enigmas, polyhedral mirrors, dissection of geometric shapes, and optical illusions. Kindergarten children’s favorite exhibit is to build free structures or polyhedrons using big geometrical pieces. Moving on to the main collection of exhibits of Mathematikum. The museum’s exhibits range from puzzles – such as Conway’s cube – over the investigation of an object y – for instance light and shadow or mirrors – to ‘big’ exhibits, such as a turntable, giant soap film, as well as very few computer-based exhibits with a nice and simple surface. Children have different favorite exhibits, with some of them playing “building a city” and some playing the gear-wheels or even lying in the mirror house. Lastly, Navet Science Center’s main collection of exhibits includes Berta the Dragon’s chemistry lab, the sustainability Hub, Bagdad and the infinity, Crime lab, Maker space, and Astronoma (an astronomy exhibition). The most acknowledged exhibit of the museum is Berta the Dragon’s chemistry lab because there are not many possibilities for kindergarten children to experience real chemistry.
Further, we should also consider how the museums are connected to European cultural heritage. Fermat Science is linked with the European cultural heritage through the internationally recognized famous mathematician Pierre Fermat who communicated and worked with European scientists of his century. Furthermore, Mathematikum is linked to European cultural heritage by preserving and disseminating the incredible cultural heritage of mathematics of the last 5000 years, particularly Greek – but not exclusively – Mesopotamian, Chinese, Mayan, all of whom have developed breathtaking mathematics. Navet museum works with the following components of European cultural heritage: tourism and heritage in promoting sustainable cultural tourism, engaging the younger generation, heritage-related skills in enhancing education and training for the traditional and new professions, fostering participation and social innovation as well as using research, innovation, science and technology for the better conservation and presentation of heritage.
Moving on, the partners were asked to give 2 examples of exhibits that can be easily modelled and printed by kindergarten educators, and additionally if the museum’s collection supports inclusive education. In other words, to recognize all children’s entitlement to a learning experience that respects diversity enables participation, removes barriers and considers a variety of learning needs and preferences. Starting with Fermat Science, the museum suggested two exhibits that could be 3D printed, La chasse aux forms (The hunt for shapes) and Les apprentis géomètres (Apprentice Surveyors) which is an educational tool created with the help of national education specialists for inclusive education. MMACA museum suggested an exhibit for matching shapes and colors, Tangram and “Patrons”. All of the museum’s exhibits are designed for inclusive education; the exhibits are free of barriers and some exhibits are even suitable for blind people. Similarly, Mathematikum in Germany embraces a collection of exhibits supporting inclusive education as well as Navet in Sweden. The Swedish museum is constantly working with the problems of inclusion and lately, they have even worked with groups of mentally disabled individuals through the use of creative technology, immigrant families in topics such as sustainability and art, and the museum is working long term with groups of children who would have limited access to Navet museum thru their families.
Hence, the museums were asked to describe the educational opportunities they offer, and if they have a rich array of training offers for kindergartens. According to Fermat Science, the museum offers school activities such as workshops and games where students manipulate several mathematical concepts learned in class. Fermat offers the following training workshops for kindergartens: shapes, colors, numbers, algorithms and labyrinths, spatial geometry, shape hunting. Furthermore, they propose a game of recognizing shapes in Beaumont de Lomagne city as well as construction games. MMACA in Spain offers an opportunity to experience a new and different understanding in schools. The museum has some didactic suitcases with hands-on material to bring to school and can use them for pop-up exhibitions or even for creating some activities for the classroom.
Moving on to Mathematikum; several times a year the museum invites kindergarten educators for a course and offers approximately 30 courses that connect “Haus der kleinen Forscher” (House of Little Researchers) to STEM topics. Finally, Navet Science Center offers in-service and pre-service training for teachers, public events in local communities, kindergartens and schools, students of all ages, outdoor facilities and activities for all, city events, sustainability training for companies and other organizations, etc. The museum offers many programs for kindergarten teachers, for instance, programming and digital tools, chemistry, physics, mathematics and so on. Navet considers that working with younger preschool children on mathematics is a better way to pursue such concepts “in-house”.
The museums cover a number of preschool mathematical concepts. Concepts such as shapes and numbers are covered by all four museums, yet through different exhibitions and workshops. The following table portrays the distinct concepts covered by each Math Museum:
|Museum||Mathematical Concepts covered in museum’s exhibits|
|Fermat Science||geometry shapes, counting, colors, and find one’s way in space|
|MMACA||shapes and forms, number concepts, early operations, reasoning, solving problems, and creativity|
|Mathematikum||numbers, shapes, functions (movement), as well as metacontexts such as communicating, arguing, describing, etc.|
|Navet Science Center||games, puzzles, measurements, digital math and counting|
The last section of the questionnaire is about the mathematics at the museum, how effective is the teaching of non-formal mathematics to preschool children and if the children leave with a different perspective towards mathematics after visiting the museum. As stated by Fermat, when teaching non-formal mathematics, it is best to manipulate appropriate mathematical concepts since this approach always succeeds. According to MMACA, non-formal mathematics offers a really good and pleasant mathematical experience which is difficult to achieve at school and this is, therefore, very important for the future development of mathematical skills. Mathematikum believes that the teaching of non-formal mathematics to kindergarten children is a very effective method and is definitely much more powerful than teaching mathematical syntax. Nonetheless, Navet museum thinks that non-formal mathematics is more effective when carried out on a children’s every-day basis at kindergarten or in outdoor facilities. In the long run, the best outcome comes from training the teachers that work in kindergartens and support the children’s work. After visiting the museum, children have a different view towards mathematics, and this can be confirmed by all museums. The children have experienced a new view of the world, they now know: ‘Mathematics is there’ and ‘I can understand the world using math’. All children have tried new challenges, worked with mathematics in new ways and even experienced the joy of doing mathematics in groups/teams.
The purpose of this questionnaire was to identify the effectiveness of teaching non-formal mathematics to younger students through the work – exhibits, workshops, etc. – of math museums around Europe. Based on the analysis conveyed from these four Math Museums (Fermat Science, MMACA, Mathematikum, and Navet Science Center), it can be concluded that the work of the museums is of high importance since it influences, inspires, and encourages the young generation of students to grow up having a great devotion towards mathematics. The playful and engaging exhibits of math museums has emboldened the recreaMATHS project in developing tailor-made e-books as well as 3D printed exhibits inspired by the ones of math museums. By sharing the Math Museums’ valuable input, we hope educators understand the greatness of teaching mathematics from an early age as well as provide them with various ideas of how to incorporate mathematics in their day-to-day kindergarten basis.
The European Mathematical Society was established in 1991. Its main objective was to take advantage of the more general establishment of a European community, politically, economically and culturally, to federate initiatives and projects capable of promoting mathematics on the European territory and beyond. But, as in other fields, arguments were based not on the opportunities and needs of the present (or the future) but instead on a recourse to the history of mathematics which sometimes resurfaced to justify this gathering. According to these arguments, mathematics is basically a creation particularly European. This view is supported by a commonplace account of the development of mathematics which, in its full form, can be summarised as follows: Born in Greek antiquity and forgotten in the dark obscurantism of the Middle Ages, mathematics was created a second time in Western Europe in the 18th century by Galileo, Descartes, Newton, and Leibniz; gradually showing its effectiveness thanks to the development of all the possible applications, and would therefore spread spontaneously over the whole planet.
For most contemporary scholars of the history of science, this account is considered misleading because it neglects the crucial contributions and autonomous thinking of cultures that developed outside the geographical territory of Europe: for example, China, Islamic countries. The mathematicians of these civilizations are presented as mere transmitters of European knowledge. Descartes’ generation drew on a vast heritage that was not limited to the sources of classical antiquity. The relationship between mathematics and economic life was important, and multiple, long before aerodynamic modelling or contemporary medical statistics: commerce, architecture, demography, fortifications, thus justified or promoted mathematical development before the 18th century.
The history of mathematics occupies an important place in the dissemination of mathematics or more generally in reflection on this field, whether philosophical, economic or educational. History by force is simplified, just as the mathematics that is popularised. The nature of the simplifications, the rhythm of the overall story presented, the emphasis on one aspect or another, all reflect and simplify the choices made by the community that elaborates and orders the basic elements. In this case, it is a certain European identity that is forged or activated.
Mediators use this European heritage to develop and promote mathematics. European museums are investing in it, staging it, updating it, sharing it through exhibitions and workshops.
The MMACA presents Leonardo da Vinci (Italian painter, visionary inventor, architect and theoretician) through his work “Vitruvian Man” which shows us the ideal proportions of Man. The body, inscribed in a circle with its navel as the centre, the square with the genitals as the centre. Leonardo da Vinci is referring here to the measurements of length used by the architects of the time. These measures follow a progression linked to the Fibonacci sequence (Italian mathematician); the ratio of one term of this sequence to the previous one approaches the golden ratio as one advance in this sequence.
Another activity at MMACA is the collaborative construction of a dome 1 metre high and 4 to 5 metres in diameter, where the pieces are assembled and form particular geometric patterns to reproduce Leonardo da Vinci’s dome.
The European cultural heritage is the medium for the transmission of current mathematics.
Another example is their Eratosthenes room (Greek astronomer, geographer, philosopher and mathematician) with a pole explaining his method for calculating the size of the earth. Manipulation is ubiquitous and a globe and coin are available to reproduce its method.
The Mathematikum takes us on a trip with the graphs of Euler (Swiss mathematician and physicist), children and adults manipulate and try to find a path that passes through all the edges only once, between brainteasers and maths, there is something for everyone. We also find Leonardo da Vinci with his bridge built with small wooden pieces.
Pythagoras (Greek philosopher, mathematician, and scientist) with his monochord invented in the 6th century B.C. where visitors can discover fractions and deepen their musical knowledge with the Pythagorean scale.
Source: Euler’s Trails Our European heritage of sculptors and painters is also not to be neglected, many museums are inspired by them for workshops. Geometric shapes and optical illusions are all examples of media that can be animated using mathematical techniques.
Fermat Science also uses this heritage for its workshops. For example, in its “Maths and Arts” section. Vasarely (a Hungarian artist who became a French citizen) shows us with curved and straight lines how to make objects appear without drawing their contours. Pierre de Fermat (French mathematician) invites us to take mathematical challenges combining maths and heritage in his home town, Beaumont de Lomagne. Alan Turing (British mathematician and cryptologist) introduces us to cryptography in a workshop where secret codes will no longer hold any secrets for you.
The richness of Europe’s cultural heritage is a real gold mine for mathematics museums. They can use the work of a mathematician, a monument, a movement, or even a city, as they wish, and transcribe it into a workshop or an exhibition. European museums have understood this and all of them bring it to life, use it, and of course transmit it.