# Innovations Through Institutionalized Infrastructures: The Case of Dimitris, His Students and Constructionist Mathematics

## Abstract

The paper discusses the case of Dimitris, a secondary mathematics teacher, who selected three micro-experiments from an institutionalized portal, re-mixed them and then gave his version to his students who in turn made their own changes and constructions. The case is discussed in the frame of the potential for institutionalized portals and digital infrastructures to afford pedagogical innovation which in this particular instance was about designing and re-mixing digital artefacts as an activity for educators, designers, teachers and students alike. Innovation is considered simultaneously at diverse levels, the representational affordances of digital artefacts, the potential for experiential mathematics for students, the potential for teacher-designer expressivity and the potential for economy-of-scale interventions. Dimitris’ changes were about the level of abstraction of the available linked representations in a simulation, about restructuration by bringing up front the notion of equivalence in solving equations, about encouraging the use of the negation of a property in a geometrical justification and about laying the ground for students to discover the usefulness of linear functions in working with geometrical properties. The students employed equivalence in a situated context, created an auxiliary point and segment to think around a geometrical property and embedded a linear relationship between segment lengths to create a rectangle which can never be a square. The paper discusses the potential for accredited large-scale institutionalized infrastructures to become the starting point for the generation of personalized living digital artifacts for both teachers and students rather than a showcase of exemplary interactive artifacts.

## Keywords

Institutionalized portals Constructionist re-mixes Half-baked Micro-experiments## Notes

### Acknowledgements

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 610467—project “M C Squared”, http://mc2-project.eu. The c-book technology is based on the widely used Freudenthal Institute’s DME portal and is being developed by a consortium of nine partner organisations, led by CTI&Press “Diophantus”.

This work has been supported by the Greek National project “Digital School Platform, Interactive Books, and Learning Object Repository” (Contract Νο 296441/2010-2015) that is co-financed by the European Union (ESF) and National funds in the context of Operational Programme “Education and Lifelong Learning” of the Greek National Strategic Reference Framework (NSRF), and is being implemented by CTI “Diophantus”.

## References

- Abelson, H., & DiSessa, A. (1981).
*Turtle geometry: The computer as a medium for exploring mathematics*. Cambridge, MA: MIT Press.Google Scholar - Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work.
*International Journal of Computers for Mathematical Learning,**7*(3), 245–274.CrossRefGoogle Scholar - Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments.
*Zentralblatt für Didaktik der Mathematik,**34*(3), 66–72.CrossRefGoogle Scholar - Bussi, M. G. B., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. G. B. Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.),
*Handbook of international research in mathematics education,*second revised edition (pp. 746–805). Mahwah: Lawrence Erlbaum.Google Scholar - Benton, L., Hoyles, C., Kalas, I., & Noss, R. (2016) Building mathematical knowledge with programming: insights from the ScratchMaths project. In
*Proceedings of the Fourth Constructionism Conference*, Bangok, Thailand.Google Scholar - Chevallard, Y. (2012). Teaching mathematics in tomorrow’s society: A case for an oncoming counterparadigm. In
*Plenary lecture at 12th International Congress on Mathematical Education*, Seoul, Korea.Google Scholar - Clinton, G., & Hokanson, B. (2012). Creativity in the training and practice of instructional designers: the Design/Creativity Loops model.
*Educational Technology Research and Development,**60,*111–130. doi: 10.1007/s11423-011-9216-3 CrossRefGoogle Scholar - DiSessa, A. A. (2001).
*Changing minds: Computers, learning, and literacy*. MIT Press.Google Scholar - Fischer, G. (2012). Meta-design: Empowering all stakeholders as Co-designers. In R. Luckin, P. Goodyear, B. Grabowski, S. Puntambeker, J. Underwood, & N. Winters (Eds.),
*Handbook on Design in Educational Research*. Routledge.Google Scholar - Gueudet, G., & Trouche, L. (2012). Communities, documents and professional geneses: Interrelated stories. In G. Gueudet, B. Pepin & L. Trouche (Eds.),
*Mathematics curriculum material and teacher documentation: From textbooks to lived resources*(pp. 305–322). New York: Springer.Google Scholar - Hoyles, C., & Noss, R. (1987). Seeing what matters: Developing an understanding of the concept of parallelogram through a logo microworld. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.),
*Proceedings of the 11th Conference of the International Group for the Psychology of Mathematics Education*, Montreal, Canada (Vol. 2, pp. 17–24).Google Scholar - Kafai, Y., & Resnick, M. (Eds.). (1996).
*Constructionism in practice. Designing, thinking and learning in a digital world*. Wahwah, NJ: Lawrence Earlbaum Associates.Google Scholar - Kaput, J., Noss, R., & Hoyles, C. (2002). Developing new notations for a learnable mathematics in the computational era. In L. English (Ed.),
*Handbook of international research in mathematics education*(pp. 51–75). Hillsdale, NJ, USA: Lawrence Erlbaum.Google Scholar - Kynigos C. (2002). Generating cultures for mathematical microworld development in a multi-organisational context.
*Journal of Educational Computing Research*, Baywood Publishing Co. Inc. (1 and 2), 183–209.Google Scholar - Kynigos, C. (2004). Α black and white box approach to user empowerment with component computing,
*Interactive Learning Environments*, Carfax Pubs, Taylor and Francis Group,*12*(1–2), 27–71.Google Scholar - Kynigos, C. (2007) Half-baked microworlds in use in challenging teacher educators’ knowing.
*International Journal of Computers for Mathematical Learning. 12*(2), 87–111 (Kluwer Academic Publishers, Netherlands).Google Scholar - Kynigos, C. (2012). Niches for Constructionism: Forging connections for practice and theory. In C. Kynigos, J. Clayson, & N. Yiannoutsou (Eds.),
*Proceedings of the “Constructionism 2012” International Conference*, Athens, (pp. 40–51).Google Scholar - Kynigos, C., & Diamantidis, D. (2014). The design principles of micro-experiments in the Interactive Digital Student Books’ Portal and their role as a basis for developing new artefacts. In C. Skoumpourdi & M. Skoumios (Eds.),
*Proceedings of the 1st Conference: The Development of Educational Material for Mathematics and Science*(pp. 823–842). Greece: Rhodes.Google Scholar - Kynigos, C. (2015). Designing constructionist e-books: New mediations for creative mathematical thinking?
*Constructivist Foundations,**10*(3), 305–313.Google Scholar - Morgan, C., & Kynigos. C. (2014) Digital artefacts as representations: forging connections between a constructionist and a social semiotic perspective. Special Issue in digital representations in mathematics education: Conceptualizing the role of context and networking theories. In J. B. Lagrange & C. Kynigos (Eds.),
*Educational Studies in Mathematics,*Springer Science + Business Media, Dordrecht (Vol. 85, No. 3, pp. 357–379).Google Scholar - Papert, S. (1972). Teaching children to be mathematicians versus teaching about mathematics.
*Journal of Mathematics in Science and Technology, 31,*249–262.Google Scholar - Pepin, B., Gueudet, G., & Trouche, L. (Eds.). (2013). Re-sourcing teacher work and interaction: New perspectives on resource design, use and teacher collaboration. Special issue of ZDM,
*The International Journal on Mathematics Education, 45*(7), 929–944.Google Scholar - Ruthven, K. (2009). Towards a naturalistic conceptualisation of technology integration in classroom practice: The example of school mathematics.
*Education & Didactique,**3*(1), 131–149.CrossRefGoogle Scholar - Stahl, G., Çakir, M. P., Weimar, S., Weusijana, B. K., & Ou, J. X. (2010). Enhancing mathematical communication for virtual math teams.
*Acta Didactica Napocensia. 3*(2), 101–114. Web: http://GerryStahl.net/pub/adn2010.pdf - Vergnaud, G. (2009). The theory of conceptual fields.
*Human Development,**52,*83–94. doi: 10.1159/000202727 CrossRefGoogle Scholar - Wilensky, U., & Papert, S. (2010). Restructurations: Reformulations of knowledge disciplines through new representational forms. In
*Proceedings of the Constructionism 2010 Conference*, Paris, France, 2010, (p. 97).Google Scholar - Zantzos I., & Kynigos C. (2012) Differential approximation of a cylindrical helix by secondary school students. In
*Proceedings of the “Constructionism 2012” International Conference*. Kynigos, Clayson, & Yiannoutsou (Eds.), Athens, 136–145.Google Scholar