# Problem solving and the use of digital technologies within the Mathematical Working Space framework

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## Abstract

The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological and cognitive actions in problem solving approaches. Participants were encouraged to construct dynamic representations of tasks and always to look for different ways to identify and support mathematical relations. To this end, three interrelated geneses (instrumental, semiotic, discursive) proposed in the Mathematical Working Space frame are used to document ways in which the participants constructed dynamic configuration of mathematical tasks as a means initially to explore invariants and eventually formulate conjectures based on empirical arguments. Technology affordances that involve dragging objects, quantifying parameters, graphing loci and using sliders were important for the participants to articulate semiotic and discursive geneses. Thus, the participants’ approaches that appeared individual, small group and plenary discussions contributed to identifying initial conjectures that later were analyzed via empirical, geometric or algebraic arguments. In addition, the participants recognized that a working space to understand mathematical ideas and to solve problems should foster mathematical discussions beyond formal settings. The use of digital technologies plays an important role in both representing and exploring mathematical tasks and in following the discussion outside the formal class. As a result, the participants outlined a possible route to implement a route for learners to integrate activities to account for instrumental, semiotic, and discursive genesis.

## Keywords

Problem solving Empirical and formal reasoning Mathematical Working Space Digital technologies## Notes

### Acknowledgments

The authors would like to acknowledge the support received from projects Conacyt-168543 and Plan Nacional I + D + I del MCIN, Reference EDU2011-29328 during the development of this study.

## References

- Blume, W. (2013). Introduction: Content related research on PISA. In M. Prenzel, M. Kobarg, K. Schöps & S. Rönnebeck (Eds.),
*Research on PISA. Research outcomes of the PISA research conference 2009*(pp. 2–5). New York: Springer.Google Scholar - Duval, R. (2006). A cognitive analysis of problem of comprehension in a learning of mathematics.
*Educational Studies in Mathematics,**61*, 103–131.CrossRefGoogle Scholar - Ellis, W., Bauldry, W. C., Fiedler, J. R., Giordano, F. R., Judson, P. T., Lodi, E., et al. (1999).
*Calculus, mathematics and modeling*. New York: Addison-Wesley.Google Scholar - Geeraerts, L., Venant, F., & Tanguay, D. (2014). Subterranean structures of technological tools and teaching issues in geometry.
*Proceedings of EDULEARN14 conference*(pp. 257–264). Spain.Google Scholar - Hegedus, S. J., & Moreno-Armella, L. (2009). Introduction: the transformative nature of ‘dynamic’ educational technology.
*ZDM - The International Journal on Mathematics Education,**41*(4), 397–398.CrossRefGoogle Scholar - Hegedus, S. J., & Tall, D. O. (2016). Foundations for the future: The potential of multimodal technologies for learning mathematics. In L. English & D. Kirshner (Eds.),
*Handbook of international research in mathematics education*(3rd ed., pp. 543–562). New York: Taylor & Francis.Google Scholar - Kuzniak, A., Parzysz, B., & Vivier, L. (2013). Trajectory of a problem: a study in teacher training.
*The Mathematics Enthusiast,**10*(1&2), 407–440.Google Scholar - Kuzniak, A., & Richard, P.R. (2014). Spaces for mathematical work: Viewpoints and perspectives.
*Revista Latinoamericana de Investigación en Matemática Educative, RELIME*,*17*(4-I), 17–26.Google Scholar - Leung, A. (2011). An epistemic model of task design in dynamic geometry environment.
*ZDM - The International Journal on Mathematics Education,**43*(3), 325–336.CrossRefGoogle Scholar - Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A. Watson & M. Ohtani (Eds.),
*Task design in mathematics education*(pp. 191–225). New ICMI Study Series. New York: Springer.CrossRefGoogle Scholar - Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge.
*Teacher College Record,**108*(6), 1017–1024.CrossRefGoogle Scholar - Moreno-Armella, L., & Santos-Trigo, M. (2016). The use of digital technologies in mathematical practices: Reconciling traditional and emerging approaches. In L. English & D. Kirshner (Eds.),
*Handbook of international research in mathematics education*(3rd ed., pp. 595–616). New York: Taylor & Francis.Google Scholar - NCTM. (2009).
*Focus in high school mathematics reasoning and sense making*. Reston, VA: NCTM.Google Scholar - Rabardel, P. (1995).
*Les Hommes et les Technologies*. Paris: Armand Colin.Google Scholar - Santos-Trigo, M. (2014). Problem solving in mathematics education. In S. Lerman (Ed.),
*Encyclopedia of mathematics education*(pp. 496–501). New York: Springer.Google Scholar - Santos-Trigo, L.M., & Camacho, M. (2009). Towards the construction of a framework to deal with routine problems to foster mathematical inquiry.
*Problems, Resources, and Issues in Mathematics Undergraduate Studies (PRIMUS)*,*19*(3), 260–279.Google Scholar - Santos-Trigo, M., & Camacho-Machín, M. (2013). A problem solving framework that enhances the use of computational technology.
*The Mathematics Enthusiast Journal,**10*(1 & 2), 279–302.Google Scholar - Santos-Trigo, M., & Ortega-Moreno, F. (2013). Digital technology, dynamic representations, and mathematical reasoning: Extending problem solving frameworks.
*International Journal Learning Technology,**8*(2), 186–200.CrossRefGoogle Scholar - Santos-Trigo, M., & Reyes-Rodríguez, A. (2015). The use of digital technology in finding multiple paths to solve and extend an equilateral triangle task.
*International Journal of Mathematical Education in Science and Technology,*. doi: 10.1080/0020739X.2015.1049228.Google Scholar - Sinclair, N., & Baccaglini-Frank, A. (2016). Digital technologies in the early primary school classroom. In L. English & D. Kirshner (Eds.),
*Handbook of international research in mathematics education*(3rd ed., pp. 662–685). New York: Taylor & Francis.Google Scholar - Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations.
*International Journal of Computers for Mathematical Learning,**9*(3), 281–307.CrossRefGoogle Scholar