Mathematics and living together: Why, how, what?

The theme of the CIEAEM70 conference "Mathematics and living together" is in perfect harmony with the orientations of the commission since its creation.

The CIEAEM was created in 1950 just after the Second World War by mathematicians, psychologists, teachers from many devastated and war-ravaged countries in Europe, all faced with major difficulties in the teaching of mathematics, Reconstruction of society and even entire countries: lack of teachers, materials, and many refugees, displaced persons and orphans. Personalities such as Piaget, Choquet, Gategno, Dieudonné and then Krygowska and Castelnuovo gathered to rethink the teaching of mathematics in light of these events.

From the beginning the founders of the CIEAEM tried to restore living together by working with German teachers and mathematicians and in 1953 the meeting of the Commission takes place in Germany in Calw.

The current situation in the world has several points in common with that of the post-war period, and Europe is confronted with problems quite similar to those of the creation of the CIEAEM. Living together has become a necessity. It is a challenge for mathematical education.

How can we rethink mathematical education to better live together ?

What is living together ?

Problematic :

To live together is to do together: how can we rethink mathematical education for a “better living together”?

Can mathematics courses not be an opportunity to confront mathematical thinking and / or to learn by exchanging between pupils, pupils and professors, between teachers, both mathematics and also the art of living together? to evolve together through speech and reasoning?

On the other hand, what positive impact for living together can mathematics have in human society and not just in the classroom? This can be achieved by the students, actors of today and citizens of tomorrow, by the influence they can have on their family by modifying their behavior, by mathematical actions outside the walls of schools and universities.

From its history, CIEAEM is a european creation. However, the particular scheme of CIEAEM has increasingly attracted participants from other parts of the world. Their views and concerns occupy an ever-growing part of CIEAEM activities and open up exciting – and also worrying – perspectives on mathematics education as a global enterprise. CIEAEM faces the challenge of exchanging and sharing views, of offering aid and of co-operating in solutions without imposing Euro-centric views, and without fostering cultural alienation. The experience of CIEAEM is that mutual understanding, human and professional esteem, and an honest and attentive discourse can overcome these risks. The seriousness of the work, the "family"-character of the meetings, and the continuity of contacts have all proved to be assets of CIEAEM. (CIEAEM Manifesto[1])

Apart from changing labour market demands for new, extended or upgraded qualifications, education institutions are scrutinised more often than in previous decades in terms of their contribution to local, regional, national and even global social and economic needs. Public service, technology transfer, wealth distribution, solutions to various problems, the production of a highly qualified workforce, reduction of inequality ... to name but a few of the multitude of expectations and demands. (CIEAEM Manifesto)

Today: The 17 United Nations Sustainable Development Goals[2] are part of the curricula of many countries. Of these, the following five objectives are more particularly related to education.

• Goal 4. Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all.
• Goal 5. Achieve gender equality and empower all women and girls.
• Goal 10. Reduce inequality within and among countries.
• Goal 16. Promote peaceful and inclusive societies for sustainable development, provide access to justice for all and build effective, accountable and inclusive institutions at all levels.
• Goal 17. Strengthen the means of implementation and revitalize the Global Partnership for Sustainable Development.

To live together is to do together. What could be the contribution of mathematics?

Some questions about Mathematics and Living Together from the CIEAEM Manifesto

• How to create appropriate social conditions to establish a teaching and learning practice guided by principles of social justice and equity ?
• Who benefits from and whose interests are served by mathematics education today - is there a major change compared to 50 years ago ?
• How to make the society aware that mathematics education could promote accountability and give full scope to a democratic vision to establish new forms of social contracts, communications, and discourse ?
• Given that mathematics education is a powerful tool in basic democratic virtues: How to empower people to think critically and to adopt critical attitudes ?
• How could mathematics education emphasise more the development of judgement and wisdom than of particular skills ?
• Given the ubiquity of poverty and violence in a major part of the world, can co-operation in the field of mathematics education contribute anything to alleviate this situation ?
• How can communities with different political, cultural, and social conditions find ways to learn from each other more productively ?

Subtheme 1: Mathematics and Sustainable Education

Mathematics and Sustainable Development Goals of the United Nations. Mathematics and gender. Mathematics and Cooperative Pedagogies.

This sub-theme will focus on the United Nations Sustainable Development Goals, as set out in the introduction, which includes in particular gender issues in mathematics education and also collaboration issues within a class or between classes. The use of cooperative games in mathematics can be a tool for socialization and living together.

Mathematics and Sustainable Development Goals [SDGs]of the United Nations.

Mathematics and sustainable development can be understood in two different ways. First, it means using sustainable development concepts to solve problems such as carbon equilibrium using mathematics, as the example is explained in the Sesamath review [Revue Sésamath] (Hache, 2011). It is also to use in mathematics classes examples that make people aware of specific problems related to the ecological management of the world.

A second way to understand this concept is for educators to consider mathematics so that their teaching belongs to the field of sustainable education and is part of the basic culture necessary for the future of humanity with the goal of promoting sustainable development. This is an objective in relation to the United Nations SDGs.

"Sustainable Mathematical Education is the project of reorienting mathematical education towards conscious environmental thinking and sustainable practices" (our translation, Renert, 2011).

In Europe, sustainable development is part of the curricula (Baudrit, 2005, Pêtre, 2010, Adomssent & al., 2014, O'Byrne & al., 2015) and mathematics can play an important role in this teaching and can be used as a subject of study for problems related to sustainable development (Ruhlmann, Mathey, 2010). This is also the case in many other parts of the world.

Mathematics and Gender.

The research concerning "mathematics and gender" is plentiful, and relies on theoretical frameworks and methodologies which are well synthesized in the article by Leyva (2017):

"The purpose of this journal is to present a critical analysis highlighting the contributions and limitations of the methodological approaches of different studies for future research on gender in mathematical education" (p. 398).

Even if the principles of equality and cooperation are put forward and promoted in institutions, the inequalities persist and challenge educators and more generally the ways of to apprehending mathematics education. Several studies on gender stereotypes in textbooks have been done (Cromer, & Hassani-Idrissi, 2011).

Although the principle of gender equality has been consolidated in Europe over the centuries, based on international texts and the acquis communautaire, gender inequalities persist, including in education systems where schooling girls and boys are parity” (see in particular Baudino, 2007, Jarlegan, A Tazouti, Y (2010), Eurydice network 2010). A dual finding is unanimously established in OECD countries (Organization for Economic Co-operation and Development) :

"A supremacy of girls doubled everywhere a more favorable orientation of boys who reach the best levels" (Baudelot, Establet, 2007, p. 91).

How to explain the paradoxical success of boys, monopolizing prestige studies and being offered more opportunities while experiencing more academic difficulties, and the paradoxical failure of girls, doing their job better but having restricted choice of orientation? The significative stereotypes of sex through practices, speeches, representations, which are also relayed by the school, constitute a partial answer (Brugeilles & Cromer, 2011).

Mathematics and Cooperative Pedagogies

Cooperative pedagogies are part of this movement of living together. In particular, cooperative mathematical games, set up in kindergarten, modify human relations within the classroom. Without going into the diversity of definitions of cooperation and collaboration, the work highlights the social importance and beneficial educational consequences of such an approach :

"To make pedagogy of co-operation a basic principle of the social and educational life of its class is to adhere to a conception of man and society, of values, of a relationship between people and to a movement of ideas that go beyond the school" (Legall, 1999).

"Cooperative pedagogy is an interactive approach to work organization (...) where students of different abilities and strengths (...) each have a specific task and work together to achieve a common goal" (Howden & Martin, 1997, p. 7) cited by (Lavoie, Drouin, & Héroux, 2012).

Whether it is cooperative pedagogies such as reciprocal teaching, help, mutual aid, tutoring or others (Barkley, Cross, & Major, 2005), this type of pedagogy facilitates learning and seems to be particularly useful in the case of tutoring to "guardian"(Slavin, 1996). In addition, problem solving becomes more effective (Vidakovic & Martin, 2004) through the use of such an approach. In mathematics, the collaborative resolution of problems initiated by researchers (Gowers & Nielsen, 2009) in the Polymath project, for example, are parallel to collaborative research projects between students (ResCo, 2014, Aldon & Garreau, 2017).

Some questions

• What activities should be put in place to enable the teaching of mathematics to be sustainable teaching ?
• What elements of sustainable development can be used as a starting point for teaching mathematics ? Can multicultural classes be an asset for this ?
• Concepts such as "mathematical skill" or "gifted student" are part of the collective ideology, and are based on convictions and prejudices. In addition, the "math gift" a priori is easily associated with other hereditary characteristics, such as gender or ethnicity: how can we react against this ? (CIEAEM Manifesto)

Références

• Adomßent, M., Fischer, D., Godemann, J., Herzig, C., Otte, I., Rieckmann, M., & Timm, J. (2014). Emerging areas in research on higher education for sustainable development–management education, sustainable consumption and perspectives from Central and Eastern Europe. Journal of cleaner production, 62, 1-7
• Aldon, G., Garreau, O. (2017). Un dispositif de recherche de problèmes en cycle 3, Repères IREM, 108
• Barkley, E.F., Cross, K.P., & Major, C.H. (2005). Collaborative Learning Techniques: A Handbook for College Faculty. San Francisco, ed Jossey Bass and Willey Brand
• Baudelot, Establet, 2007, Quoi de neuf chez les filles ? Entre stéréotypes et libertés. Nathan
• Baudino, C. (2007) Revue de la littérature récente sur les inégalités de genre dans les méthodes d’enseignement et la gestion des relations entre pairs dans l’aire francophone. UNESCO document de référence préparé pour le rapport mondial de suivi sur l’éducation pour tous. http://unesdoc.unesco.org/images/0015/001555/155506f.pdf
• Baudrit, A (2005). Apprentissage coopératif et entraide à l'école,  Revue française de pédagogie, 121-149  Retrieved from http://ife.ens-lyon.fr/publications/edition-electronique/revue-francaise-de-pedagogie/INRP_RF153_10.pdf
• Brugeilles, C., Cromer, S. (2011).  Genre et mathématiques dans les images des manuels scolaires en France,  revue  Trema Valeurs et stéréotypes dans les manuels scolaires de la Méditerranée  de Montpellier ,  142-154, N° 35 - 36 https://trema.revues.org/2599
• Cromer, S. & Hassani-Idrissi, M. (2011) Echanges humains et culturels en Méditerranée dans les manuels scolaires un thème d'actualité  revue  Trema Valeurs et stéréotypes dans les manuels scolaires de la Méditerranée  Université de Montpellier , 1 - 7, N° 35 - 36  https://trema.revues.org/2596
• Réseau Eurydice (2010) Gender Differences in Educational Outcomes: Study on the Measures Taken and the Current Situation in Europe http://eacea.ec.europa.eu/Education/eurydice/documents/thematic_reports/120EN.pdf
• Gowers, T., & Nielsen, M. (2009). Massively collaborative mathematics. Nature, 461(7266), 879-881.
• Hache, S. (2011). Mathématiques et développement durable : une première approche de l’équilibre carbone, Revue Sésamath, 23, http://revue.sesamath.net/spip.php?rubrique78
• Jarlegan, A Tazouti, Y ( 2010) La perception des relations enseignant(e)-élèves : regards croisés des filles et des garçons. Actes du congrès de l’Actualité de la recherche en éducation et en formation (AREF), Université de Genève, septembre
• Lavoie, A.,Drouin, M.Héroux,S. (2012)  Pédagogie coopérative une approche à redécouvrir Pédagogie Collégiale Vol. 25 n 3 Printemps  2012   https://cdc.qc.ca/ped_coll/v25/Lavoie-Drouin-Heroux-25-3-2012.pdf   Le Gal, J. (1999). Coopérer pour développer la citoyenneté, Paris, Hatier.
• Leyva, L. A. (2017). Unpacking the male superiority myth and masculinization of mathematics at the intersections: A review of research on gender in mathematics education. Journal for Research in Mathematics Education, 48(4), 397-433.
• O’Byrne, D., Dripps, W., & Nicholas, K. A. (2015). Teaching and learning sustainability: An assessment of the curriculum content and structure of sustainability degree programs in higher education. Sustainability Science, 10(1), 43-59.
• Pêtre, C. (2010).  Mathématiques et developpement durable, IREM de Clermont Ferrand
• Renert, M. (2011). Mathematics for life: sustainable mathematics education. For the Learning of Mathematics, 31(1), 20-26. Retrieved from http://www.jstor.org/stable/41319547
• ResCo, G. (2014). La résolution collaborative de problèmes comme modalité de la démarche d’investigation. Repères IREM, 96, 73-96.
• Ruhlmann, A., Mathey, A. (2010). Mathématiques et développement durable, Repères Irem, 78, 5 - 12

• Slavin, R. E. (1996).  Cooperative learning and intergroup relations .  J. A. Banks & C. A. McGee Banks (éd.), Handbook of research on multicultural education. New York : Prentice Hall ; London : Macmillan, p. 628-634.
• Vidakovic, D., & Martin, W. O. (2004). Small group researches for mathematics proofs and individual reconstructions of mathematics concepts. Journal Mathematics Behaviour, 23, 465-492.http://dx.doi.org/10.1016/j.jmathb.2004.09.006

Subtheme 2 : Cross-border Mathematics

Globalization and the Internet: information technology, EAD, MOOC, cooperation of classes and schools across borders. Teaching in multicultural classes. Training of teachers to teach in a multicultural environment.

By cross-border mathematics we mean all the devices, tools, ways of thinking that transcend boundaries, including geographic, political, professional or cultural in the digital age. Thus, new forms of emerging teaching and learning (EAD, MOOC) are part of this theme, as is the work on multicultural classes but also on collaborative research and the necessary interactions between different communities.

Globalization and the Internet: information technology, EAD, MOOC, cooperation of classes and schools across borders.

Several universities have set up cross-border cooperation, such as the University of Pau (France) and the University of Zaragoza (Spain). This cross-border cooperation in mathematics proposes bilateral cooperation actions in mathematics between the two universities since the end of the 1980s. They concern either teaching and training activities, or research collaborations concretized, if necessary, by publications in international journals and participations in thesis juries.

(See http://lma-umr5142.univ-pau.fr/fr/collaborations/cooperations-transfrontalieres.html)

The UniGR “University of the Grande Région” which brings together the six founding universities : the University of Kaiserslautern, the University of Liège, the University of Lorraine, the University of Luxembourg, the University of Saarland, and the University of Trier was created in 2008. It works towards training and research "without borders", improving and facilitating the conditions of mobility of students, PhD students, teachers and researchers from partner universities; supporting cross-border collaborations through the network and expertise developed, and supporting the visibility of cross-border initiatives in institutions. Thanks to the impulses of the organs and numerous networks of the group, a "cross-border culture" must emerge within the partner universities. The UniGR must thus exceed its status as a recognized player in the Greater Region to become an engine of its development. The priorities set for excellence in research and teaching within the grouping are as much oriented towards visibility, and therefore international competitiveness, as towards the development strategy of the large regional area. The group also promotes multilingualism in the cross-border region and prepares students and PhD students for the Greater Region's labor market."

"Cross-border and intercultural research in partnership is an ongoing process, which aims to create solid knowledge, while generating mutual trust and learning and a shared sense of ownership." Excerpt from "a guide for cross-border research partnerships’ 11 principles 7 questions Swiss Commission for Science Partnership with Developing Countries (KFPE) 2011 (See https://sciencesnaturelles.ch/uuid/e70572c4-64ca-58d8-86a9-2b7af19fd2ef?r=20170706115333_1499301166_adeb9728-2ce0-54b4-aec1- 4cbbb3bf4ef3)

The African Virtual University and its resource model, the conditions for the viability of distance learning programs for mathematics teachers, is also an example of collaboration between teachers and researchers from a training perspective (Sokhna & Sarr, 2009).

These examples show cooperation at the university level, but also with educational institutions, and presuppose the reorganization of teaching at the international level. The MOOCs, moreover, were born at the beginning of the 2000s initiated by prestigious American universities (MIT, Harvard, Stanford, Duke University, ...) seeking to attract students from all over the world. It should be noted that what is the specificity of MOOCs as compared to a distance course is their massive side: with 10 000 participants, it is impossible for a trainer to follow each student. It is therefore necessary to develop peer-to-peer exchanges in the whole process of learning. A new form of collaboration between learners emerges from the very context of teaching in the development of communities of practice (Wenger, 1998, Shah, 2015, Panero & al., 2016).

"International school correspondence makes it possible to work on education for sustainable development, the social and solidarity economy, and cooperation. Martine Trinel (Millac school) and Patrick Galland (Moussac) report on a partnership experience with Ricotte in Senegal, which led to the electrification of a school and a village. (ICEM 2009)

The resources provided by the Internet could be more:

• Allow work teams to contact experts from the same region. (teamwork)

• Allow teachers and students around the world 24 hours a day to find where and how to get in touch with a work group.

• Facilitate the creation of an effective transnational work system.

This can only be done if teachers and researchers are trained in the use of these new technologies (Niess, 2005) and if we take into account the students in this training (Druin, 2002).

Teaching in Multicultural Classes, Teacher Training in a Multicultural Context

"Cultural diversity is an internationally recognized fact today. The idea of an international convention on the subject, launched by France at the World Summit on Sustainable Development held in Johannesburg in 2002, culminated in October 2005 with the Convention on the Protection and Promotion of the Diversity of Cultural Expressions adopted by Unesco.

With this text, States have elevated cultural diversity to the rank of common heritage of humanity and affirmed that its protection, promotion, enrichment and maintenance are essential conditions for sustainable development.

Living together and promoting cultural diversity are major axes of a sustainable development policy. It is therefore necessary to establish a space of citizen dialogue with the inhabitants and associations of multiple horizons. Association des médiations interculturelles.

As Avruch and Black, or DuPraw and Axner remind us:

"Our culture is the lens through which we see the world, the logic with which we order it, and the grammar with which we decipher its meaning (Avruch & Black, in Berg et al., 2012).

We share common experiences that shape our understanding of the world. These include the group in which we are born and those to which we join (DuPraw & Axner, in Berg & al., 2012).

Favilli, Oliveras, & César, (2003) studied the activity of mathematics teachers in multicultural classes in Southern Europe.

These realities of teaching in multi cultural classrooms can be approached from the point of view of the different social and ethnic cultures of students and from a professional point of view (Akkari, 2006). Indeed, the resolution of certain technical practical problems may require the establishment of different methods compared to theoretical knowledge. The gap between the performance of students who come from the dominant culture and those who are not, continues to widen. In classes where the teacher uses methods rooted in the dominant culture, students who are not of this culture will be left behind. In these classes, students who are already marginalized experience more difficulties and often fail, which only reinforces their feeling of being "null" (Osborne, 2001, Steinback, 2012).

Several researchers have stressed the importance of taking into account cultures, the socio-cultural background of students, culture-related thought systems to teach mathematics effectively. This is the case of Spagnolo & Di Paola, (2010), Barton, & Frank, (2001), Radford, (1997).

Some questions

• Which strategies, both from research and practice, could make progress in bringing the essential and the most appropriate to teaching and learning, and ensure access to all levels of schooling, both primary and secondary school, as well as adults who have not been to university ?
• Does perception of excellence or mathematical achievement vary by culture and community? Does it depend on class, gender or ethnicity ?
• How can international cooperation foster dialogue and fairness in debates rather than a one-way relationship ?
• Regarding the consideration of cultural diversity, how much of the responsibility for education is attributable to politicians, and more specifically to mathematics education? What actions can be taken to make the teaching of mathematics more sustainable in an increasingly diverse world?
• Is multiculturalism a force for teaching and learning? How can each culture enrich itself with diversity and multiculturalism?
• What will become of the cultural and social diversities in globalization? Will the internationalization of mathematics education and globalization equally respect the equity and autonomy of partners? What is the impact of competition between and within educational institutions ?
• How can new technologies serve the teaching of mathematics especially in relation to multiculturalism ?
• How can taking into account the cultural diversity of the mathematics class be an aid for the learning of all students ?
• How can the teacher use these differences within the classroom to enrich students' mathematical learning?
• Why are social networks like Facebook, Twitter, Facebook, LinkedIn, Tumblr, Pinterest, ... not widely implemented in mathematics in the classroom? Could this not help to improve living together in the group

Références

• Akkari A.-J. (2006). Les approches multiculturelles dans la formation des enseignants: entre recherche et pédagogie critique. Formation et pratiques d’enseignement en questions, 4, 233-258.http://revuedeshep.ch/site-fpeq-n/Site_FPEQ/4_files/2006-4-Akkari.pdf
• Association des médiations interculturelles  http://www.ami-oimc.org/multiculturalisme-interculturalite/
• Barton, B., & Frank, R. (2001). Mathematical ideas and indigenous languages. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), Sociocultural research on mathematics education: An international perspective (pp. 135–149). Mahwah, NJ: Lawrence Erlbaum
• Berg, A., Dhillon, S., Kershaw, J., & Maheu B. (2012). Teachers’ Association. Ici tout le monde est le bienvenu. Alberta University, ISBN 978-1-927074-04-6
• Convention sur la protection et la promotion de la diversité des expressions culturelles 2005 http://portal.unesco.org/fr/ev.php-URL_ID=31038&URL_DO=DO_TOPIC&URL_SECTION=201.html
• Druin A.(2002) The role of children in the design of new technology. University of Mariland. Behavior&Inforation Technology.
• Favilli, F., Oliveras, M. L., & César, M. (2003). Maths teachers in multicultural classes: findings from a Southern European project. In Proceedings of the Third Congress of the European Society for Research in Mathematics Education (pp. 1-10).
• ICEM August 2009 https://www.icem-pedagogie-freinet.org/node/418
• https://www.teachers.ab.ca/SiteCollectionDocuments/ATA/Publications/Human-Rights-Issues/MON-3F%20%20Ici,%20tout%20le%20monde%20est%20le%20bienvenu.pdf
• Niess N. L. (2005) Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge. «Teaching and Teacher Education » Volume 21, Pages 509-523.
• Osborne, A. B. (2001). Teaching, diversity and democracy. Melbourne: Common Ground.
• Panero, M., Aldon, G., Trgalova, J., & Trouche, L. (2016). Analysing MOOCs in terms of teacher collaboration potential and issues : The French experience. In CERME proceedings, Dublin, 1 – 5 February, 2017.
• Radford, L. (1997). On psychology, historical epistemology and the teaching of mathematics: Towards a socio-cultural history of mathematics. For the Learning of Mathematics, 17(1), 26-30.
• Shah, D. (2015). Less experimentation, more iteration: A Review of MOOC Stats and Trends in 2015. Online Class Central, www.class-central.com/report/moocs-stats-and-trends-2015.
• Sokhna, M., Sarr, J. (2009). L'Université Virtuelle Africaine: Passage d'une formation d'enseignants aux mathématiques à une formation d'enseignants de mathématiques. In Actes de EMF 2009, Dakar 6-10 avril, 2009.
• Sommet mondial pour le développement durable http://www.un.org/french/events/wssd/coverage/summaries/envdev33.htm
• Spagnolo, F., & Di Paola, B. (2010). European and Chinese cognitive styles and their impact on teaching mathematics.Studies in Computational Intelligence, 277, 1-267. http://doi.org/10.1007/978-3-642-11680-3_1
• Steinbach, M. (2012).  Élargir les perspectives interculturelles des futurs enseignants, Université de Sherbrooke Revue des sciences de l’éducation de McGill, 47(2). http://mje.mcgill.ca/index.php/MJE/article/viewFile/8824/6812
• Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: A design research study. Journal of Mathematics TeacherEducation, 10(4-6), 217-237.
• Wang, F., & Hannafin, M. J. (2005). Design-based research and technology-enhanced learning environments. Educational technology research and development, 53(4), 5-23.
• Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge University Press.

Subtheme 3 : Mathematics and Dialogue with Other Disciplines and Between Teachers and Researchers. 

Transdisciplinary, interdisciplinary and multidisciplinary activities. Collaborative research between teachers and researchers.

Living together means also to be able to change our point of view and to work together from a perspective of a better understanding the phenomena that a single discipline can not catch completely.

Transdisciplinary, interdisciplinary and multidisciplinary activities.

In literature, we find the terms trans-, inter- and multi disciplinarity. We will rely on the definition of Bourguignon (1997):

"In a first approach, it could be said that in multidisciplinarity several disciplines combine to study a common object, none of which can observe all aspects with the only techniques at its disposal, whereas in interdisciplinarity there is a need to establish cooperation between autonomous disciplines to broaden understanding of a particular area or to achieve a common goal."

Transdisciplinarity is then a superior stage of collaboration between disciplines in which connections are built within a system where disciplines meet and intersect (Piaget, 1972). Bourguignon continues: "Transdisciplinarity offers a new vision of Nature, opening up the disciplines to what passes through them and goes beyond them. It goes beyond the realm of the exact sciences that it must reconcile with the sciences of man. Transdisciplinarity places Man in the Universe. It postulates that the economy must be at the service of humanities. It dialogues with all humanist and non-totalitarian ideologies” (1997)

In the 1970s and the 1980s, CIEAEM took a new direction under the influence of Anna Sofia Krygowska, a Polish teacher and mathematician, Emma Castelnuovo, an Italian pedagogue, Claude Gaulin, a Canadian professor, and Hans Freudenthal, a Dutch mathematician. These people tried to break the "noble isolation" of mathematics and their orientation towards pure mathematics, to bring them closer to other sciences, social reality and the social practice of mathematics. It is thanks to their initiative that the themes of the CIEAEM meetings were formulated and perceived more and more in terms of transdisciplinarity and interdisciplinarity.

"From the beginning, mathematics is the language of physics, indispensable in the very formulation of the fundamental laws of nature. Mathematical modeling plays an important role in the social sciences, especially in economics, and an ever-increasing role in the life sciences. Conversely, as we all too often ignore, mathematics has always been fed with concepts and problems from other scientific disciplines. In their development through the centuries” (CREM, 2005).

Interdisciplinarity often makes it possible to approach a problem from a different angle than the one usually used by mathematicians. Currently, interdisciplinarity in the sciences, technology, engineering and mathematics (STEM) offers teachers and students a global approach to the apprehension of scientific phenomena (Becker & Park, 2011), CREM (2005), (Arnoux, 2006).

This sub-theme studies models in which mathematics is partially or largely involved, and through which, social, economic, ecological processes, etc. can be described, predicted and prescribed. (Alshwaikh & Straehler-Pohl, 2017).

Cross-disciplinary research in mathematics education linking the sciences of education, didactics and mathematics themselves is not plentiful but has proved fruitful (Will & Ryden, 2015). The questions of obstacles encountered and ways to overcome them can be studied in this sub-theme.

However, a precise reference is the text written by Piaget in 1970 on the occasion of a colloquium on interdisciplinarity : "Finally, at the stage of interdisciplinary relations, we can hope to see succeed a higher stage that would be ‘trans-disciplinary,’ which would not be satisfied with reaching interactions or reciprocities between specialized researches, but would situate these links within a total system without stable boundaries between disciplines."

Gunter Pauli, father of the Blue Economy (Pauli, 2010) has set up multidisciplinary teams of researchers. In the field of education, the Zero Emissions Research and Initiatives (ZERI) Foundation has developed ecological fables[3] based on real scientific facts that “living together", ecology and “know being” and that can be exploited in a trans, inter or pluridisciplinary activities. Mathematics are fully able to find their place. (Pauli, 2015). These fables are used in many classes in the People's Republic of China.

Collaborative Research Between Teachers and Researchers

In research work in education, researchers and teachers as well as engineers or computer scientists work on common topics and objects with different objectives related to their institutional positions. The aim is to foster collaboration between several communities for the construction and mobilization of scientific and professional knowledge in a pooling perspective for the benefit of all. Discussions on educational research paradigms (Wang & Hanafin, 2005, Swan, 2007, Sanchez & Monod-Ansaldi, 2015, Battaglia, Di Paola, & Fazio C. 2017). will help advance both the theoretical frameworks for explaining the interactions between different communities and the professional competences of all actors.

Some questions

• In what way does the dialogue between disciplines make possible the enlightening of a coherence of learning for students? How does this dialogue strengthen the teaching and learning of mathematics?
• How does interdisciplinarity contribute to the creation of concepts specific to the mathematics discipline?
• Any discipline is characterized by some markers or preliminary elements: history, literature, technical language, symbolic place of installation, recognition from outside, shared values and scientific tools. How do these markers promote cross-disciplinary dialogue and how can teachers take hold of these reflections from the perspective of improving mathematics education?
• How is it possible to establish relationships between investigation and hypotheses on learning?
• How is it possible to identify the different modalities arising from the convergence of disciplines and why?
• How to overcome institutional obstacles to build effective interdisciplinary teaching?
• What concepts are needed to describe and analyze collaborative research processes? How do they work?

Références

• Alshwaikh, J., & Straehler-Pohl, H. (2017). Interrupting passivity: Attempts to interrogate politcal agency in Palestinian school mathematics. In H. Straehler-Pohl, N. Bohlmann, & A.Pais (Eds.), The disorder of mathematics education: Challenging the sociopolitical dimensions of research (pp.191–208). Cham : Springer.
• Arnoux, P. (2006). Etude de rapports récents sur l'enseignement des mathématiques et des sciences, mai 2006. http://educmath.ens-lyon.fr/Educmath/ressources/etudes/pierre-arnoux/mad2.pdf
• Battaglia, O.R., Di Paola B, & Fazio C. (2017). A quantitative analysis of Educational Data through the Comparison between Hierarchical and Not-Hierarchical Clustering. Eurasia journal of mathematics, science and technologie education, 13(8), 4491-4512.
• Becker, K., & Park, K. (2011). Effects of integrative approaches among science, technology, engineering, and mathematics (STEM) subjects on students' learning: A preliminary meta-analysis. Journal of STEM Education: Innovations and Research, 12(5/6), 23-.
• Bourguignon, A. (1997). De la pluridisciplinarité à la transdisciplinarité. Congrès de Locarno, 30 avril-2 mai 1997. Annexe au document de synthèse UNESCO http://ciret-transdisciplinarity.org/locarno/loca5c1.php
• Pauli G. (2010) The blue Economy. Report to the club of Rome
• Pauli G. (2015) Les fables de Gunter, Gunter’s Fables « pour ne jamais cesser de rêver » édition bilingue Paris les éditions blu

• Piaget, J.(1972). L’épistémologie des relations interdisciplinaires. In OCDE, L’interdisciplinarité, problème d’enseignement et de recherche (pp. 131-144).
• CREM (2005). L’enseignement des mathématiques en relation avec les autres disciplines. Bulletin de l'APMEP, 458, 354-374, http://numerisation.irem.univ-mrs.fr/AAA/AAA05036/AAA05036.pdf
• Sanchez, É., & Monod-Ansaldi, R. (2015). Recherche collaborative orientée par la conception. Education & Didactique, 9(2), 73-94.
• Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: A design research study. Journal of Mathematics Teacher Education, 10(4-6), 217-237.
• Wang, F., & Hannafin, M. J. (2005). Design-based research and technology-enhanced learning environments. Educational Technology Research and Development, 53(4), 5-23.
• Will M., Rydén L. (2015). Trans-disciplinarity in sustainability science and education. In W. Leal Filho, , L. Brandli, O. Kuznetsova, & A. Paço (Eds)Integrative approaches to sustainable development at university level. World Sustainability Series. Cham: Springer.
• video Mathématiques et Musique Karim Zayana et Nicolas Ngo
• http://videos.education.fr/MENESR/eduscol.education.fr/2016/Ressources2016/EPI/EPI_la_portee_des_maths.mp4
• Mathématiques et autres disciplines http://culturemath.ens.fr/math%C3%A9matiques-et-autres-disciplines-281

Sub-theme 4: Rethinking history of mathematics.

Forgotten Mathematics and Forgotten mathematicians. Classroom activities and history of mathematics. Ethnomathematics.

Forgotten Mathematics and Forgotten Mathematicians

It is essential, if we want to give a common culture to students, to revisit the History of Mathematics by highlighting the contribution of different cultures to the evolution of Mathematics) for example Djebbar (2001, 2005) describes the contributions of the Arab Muslim culture. For Thomas (2015), in his note concerning Rashed (2011), it is about "breaking the chronological boundaries inherited from political history (ancient, medieval, classical, modern mathematics), and to think about the place of the History of Sciences, between epistemology and social sciences."

The Arab Muslim world has become an issue of political and scholarly controversy. We must therefore give a scientific answer concerning the History of Mathematics. Several researchers (Büttgen, and all, 2009) Rashed (2011), have examined the question and have appreciated the true value of what the terms "Arabic," "Latin," "Greek," and "Jewish" meant in the Middle Ages and the Renaissance, thus allowing the restoration of the place of each in the transmission of sciences - among others - through the history of scholarly transmissions. Let us mention the many works of Aissani concerning the history of mathematics in Algeria and exchanges between the two shores of the Mediterranean. (Aissani & al 2014, Aïssani D. et al. 2014), (Verdier N., Romera-Lebret P. and Aïssani D., May 2016). Similarly, the update in Algeria of unknown ancient mathematical manuscripts can also help to change the way scientists view the role played by Arab mathematicians in the evolution of this discipline (Mechehed DE, Bekli MR and Aïssani D., 2013 ).

All these works has undoubtedly made it possible to bring down European-centric representations concerning the transmission of the History of Science and Mathematics in particular. This approach is essential if we want to restore living together because it allows to take into account the contribution of each.

In the same way, we are rediscovering today the role of black mathematicians in the United States in the great successes of the American aerospace program of the 1960s. They exploded prejudices and social barriers at the time of racial segregation. And who knows today that the only woman in the world to be awarded the Fields Medal is Iranian? It is of utmost importance to rediscover these forgotten mathematicians if we want to enable students from different cultures to learn about the contribution of different civilizations to the advancement of mathematics and to many discoveries (Büttgen et al., 2009).

Classroom activities using the history of mathematics.

Many researchers, for example in the French IREM (Research Institute for Mathematics Education), have shown that using activities related to the History of Mathematics in the classroom helps to promote the mathematical activity of students, and also to create a common culture integrating the contributions of different civilizations to mathematical discoveries (Collectif IREM, 1993,1997; Cerquetti-Aberkane, Johan, & Rodriguez, 1997; Cerquetti-Aberkane, & Rodriguez, 2002; & Cerquetti-Aberkane, 2000, 2004). This is fundamental, especially in multicultural classes.

In France, the group M.A.T.H. (Mathematics: Historical Literacy Approach) has been working since the early 1980s on the introduction of a historical perspective in mathematics education with the main objective of designing and experimenting with teaching scenarios incorporating the work on mathematics historical texts (Barbin & Maltret, 2015).

Ethnomathematics

Introduced in the 1970s by the Brazilian, Ubiratan D'Ambrosio (D'Ambrosio 1985), Ethnomathematics allows us to better understand human cultural diversity (Rosa, and All 2016).

According to socio-cultural groups there are, for example, different representations of geometric figures or bodily representations of numbers. The study of the use of traditional African games made it possible to highlight the setting up of sophisticated logical reasoning and the resolution of graph problems, among others, to play for example the game of Shongo. Claudia Zaslavsky has shown that the 24 different types of patterns enumerated by the mathematician Coxeter (1969) are present in the African art of many ethnic groups (Zaslavsky, 1979,1999; Dhombres, 1992). Moreover, we find these 24 types of motifs in Islamic art present at the Alhambra in Granada. Several researchers have worked on this subject.

Vellard, 1982) have shown, by a choice of very precise examples concerning the mathematical activities of calculation and logic put in place in several African cultures, that "contrary to a very widespread belief, French, (no more than any European language) is the language of logic and reason without which it would be impossible to work scientifically ... .The argument of "greater logical rigor" of French (or English) in relation to languages Africans to develop a scientific thought is a false argument, not devoid of understatement (...)”(p. 274).

Vellard (1982) has shown the importance of introducing training in mathematical activities in traditional African cultures into higher education. She introduced African Mathematics into her Mathematics History course for mathematics students at the University of Bukavu in the late 1970s and early 1980s, and she said, "I was able to to notice with pleasure, the astonishment first, then the interest that they (the students) brought to this unusual course for them. For the first time, Africans, students of mathematics, heard both the words ‘mathematics’ and ‘African,’ the juxtaposition of these two words having always appeared to them, until then, as an absolute nonsense" (pp.275-276).

During this same period, Zavslavky (1979) and Cambell (1976) also set up in the United States similar teaching practices with the same success.

Poirier (2005 and 2006) highlighted the traditional Inuit mathematical practices that need to be known and taken into account when teaching mathematics to Inuit students.

The ethnomathematical perspective can inform our practices and interventions in the schools, while ethnomathematical programs can be enacted in broader community projects by youth who lead with the processes of dignity, recognition, and reconciliation. One might also say that ethnomathematics can give the floor to critical voices, to other minority voices, and to different voices in mathematics education (Gilligan 1982; François & Van Bendegem, 2007).

The ethnomathematical perspective can inform our practices and interventions in the schools, while ethnomathematical programs can be enacted in broader community projects by youth who lead with the processes of dignity (Butler and Al 2016), recognition, and reconciliation. One might also say that ethnomathematics can give the floor to critical voices, to other minority voices, and to different voices in mathematics education (Gilligan 1982; François & Van Bendegem, 2007).

Stathopoulou, & Appelbaum, (2016) describe how such an approach might inform mathematics education in classrooms and in community organizations locally and via global collaborations. In light of “Living Together”, they describe how Ethnomathematics prompts mathematics educators to consider the pursuit of dignity, recognition and reconciliation as forms of forgiveness – not in the common sense notion of acceptance, but rather in Hannah Arendt’s framework of maintaining awareness of violations and dispossession while refusing to allow these violations and dispossession to affect our present or future (Arendt, 1963/2013; Biesta, 2013; Knott, 2011). embracing dignity and recognition through forms of refusal that enact forgiveness.

Currently this type of education is not implemented systematically in the university mathematics curriculum. Could not the introduction of such a teaching in mathematics contribute to a better life together?

Some questions

• How and why to update a common culture concerning the history of mathematics? How can this promote the Living Together ?
• What activities to put in place concerning the history of mathematics to promote Living Together ?
• History of mathematics and multicultural classes: how to show the links between different cultures thanks to the history of mathematics ?
• How to introduce into the university curriculum teaching about African mathematics among others ?
• How can Ethnomathematics contribute to the pursuit of dignity, recognition an reconciliation as forms of forgivness for better present and future of Living Together in classrooms, at the school level, or across multiple school collaborations ?

• How can Ethnomathematics report being useful in cross-border collective projects for teaching and learning mathematics ?

References

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