# Different Levels of Sophistication in Solving and Expressing Mathematical Problems with Digital Tools

## Abstract

All over the world, several organizations have nurtured the development of students’ problem-solving abilities by organizing competitions and tournaments of different kinds. This is the case of the Mathematical Competitions SUB12 and SUB14, promoted by the University of Algarve, addressing students from grades 5 to 8 (10–14 year olds) in the south of Portugal. To each proposed problem, participants are required to explain their problem-solving process and find ways to express their thinking. They may use any of the digital tools they have available and they find useful for solving a given problem. Our research has uncovered the aptitudes of young competitors in taking advantage of everyday digital tools and its representational expressiveness to give form and substance to their reasoning and strategies. Another emerging aspect is the apparent existence of different degrees of robustness of the solutions submitted, mainly in terms of the strategies that competitors develop, with a particular technological tool, to solve the problems. In this chapter, we are taking a selection of solutions submitted to two problems, in which competitors resort to GeoGebra, in one case, and to Excel, in the other. We offer a proposal for identifying levels of sophistication and robustness of technology-based solutions to the problems, according to the characteristics of the tool use and its connection to the conceptual models underlying students’ thinking on the problems.

## Keywords

Affordances Co-action Conceptual model Excel GeoGebra Humans-with-media## References

- Barbeau, E., & Taylor, P. (Eds.). (2009).
*Challenging mathematics in and beyond the classroom. The 16th ICMI study*. New York, NY: Springer.Google Scholar - Borba, M., & Villarreal, M. (1998). Graphing calculators and the reorganization of thinking: The transition from functions to derivative. In A. Olivier & K. Newstead (Eds.),
*Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 136–143). Stellenbosch, South Africa: PME.Google Scholar - Borba, M., & Villarreal, M. (2005).
*Humans-with-media and the reorganization of mathematical thinking*. New York, NY: Springer.Google Scholar - Carreira, S. (2015). Mathematical problem solving beyond school: Digital tools and students’ mathematical representations. In S. Cho (Ed.),
*Selected regular lectures from the 12th international congress on mathematical education*(pp. 93–113). Cham, Switzerland: Springer.CrossRefGoogle Scholar - Carreira, S., Jones, K., Amado, N., Jacinto, H., & Nobre, S. (2016).
*Youngsters solving mathematical problems with technology*. New York, NY: Springer.CrossRefGoogle Scholar - Chemero, A. (2003). An outline of a theory of affordances.
*Ecological Psychology, 15*(2), 181–195.CrossRefGoogle Scholar - Gibson, J. (1979).
*The ecological approach to visual perception*. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Gravemeijer, K. (1997). Solving word problems: A case of modelling?
*Learning and Instruction, 7*(4), 389–397.CrossRefGoogle Scholar - Gravemeijer, K. (2005). What makes mathematics so difficult, and what can we do about it? In L. Santos, A. P. Canavarro, & J. Brocardo (Eds.),
*Educação matemática: Caminhos e encruzilhadas*(pp. 83–101). Lisboa, Portugal: APM.Google Scholar - Greeno, J. (1994). Gibson’s affordances.
*Psychological Review, 101*(2), 336–342.CrossRefGoogle Scholar - Hegedus, S., & Moreno-Armella, L. (2009a). Intersecting representation and communication infrastructures.
*ZDM Mathematics Education, 41*, 399–412.CrossRefGoogle Scholar - Hegedus, S., & Moreno-Armella, L. (2009b). Introduction: The transformative nature of “dynamic” educational technology.
*ZDM Mathematics Education, 41*, 397–398.CrossRefGoogle Scholar - Hegedus, S., & Moreno-Armella, L. (2010). Accommodating the instrumental genesis framework within dynamic technological environments.
*For the Learning of Mathematics, 30*(19), 26–31.Google Scholar - Hegedus, S., & Moreno-Armella, L. (2011). The emergence of mathematical structures.
*Educational Studies in Mathematics, 77*, 369–388.CrossRefGoogle Scholar - Jacinto, H. (2017). A atividade de resolução de problemas de matemática com tecnologias e a fluência tecno-matemática de jovens do século XXI (Unpublished doctoral dissertation). Instituto de Educação, Universidade de Lisboa, Lisboa, Portugal.Google Scholar
- Jacinto, H., & Carreira, S. (2013). Beyond-school mathematical problem solving: A case of students-with-media. In A. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th conference of the IGPME*(Vol. 3, pp. 105–112). Kiel, Germany: PME.Google Scholar - Jacinto, H., & Carreira, S. (2017). Mathematical problem solving with technology: The techno-mathematical fluency of a student-with-GeoGebra.
*International Journal of Science and Mathematics Education*,*15*(6), 1115–1136. https://doi.org/10.1007/s10763-016-9728-8 - Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations.
*Educational Studies in Mathematics, 44*, 55–85.CrossRefGoogle Scholar - Lesh, R., & Doerr, H. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. Doerr (Eds.),
*Beyond constructivism – Models and modeling perspectives on mathematics problem solving, learning, and teaching*(pp. 3–33). Mahwah, NJ: Lawrence Erlbaum Associates.CrossRefGoogle Scholar - Lesh, R., & Zawojewski, J. (2007). Problem Solving and Modeling. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 763–804). Charlotte, NC: Information Age Publishing and National Council of Teachers of Mathematics.Google Scholar - Lester, F., & Kehle, P. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. Doerr (Eds.),
*Beyond constructivism – Models and modeling perspectives on mathematics problem solving, learning, and teaching*(pp. 501–517). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Moreno-Armella, L., & Hegedus, S. (2009). Co-action with digital technologies.
*ZDM Mathematics Education, 41*, 505–519.CrossRefGoogle Scholar - Moreno-Armella, L., Hegedus, S., & Kaput, J. (2008). From static to dynamic mathematics: Historical and representational perspectives.
*Educational Studies in Mathematics, 68*, 99–111.CrossRefGoogle Scholar - Nobre, S., & Amado, N. (2013). Combining the spreadsheet with paper and pencil: a mixed environment for learning algebraic methods. In E. Faggiano & A. Montone (Eds.),
*11th International Conference on Technology in Mathematics Teaching – Conference Proceedings*(pp. 226–231). Bari, Italy: Università degli Studi di Bari Aldo Moro.Google Scholar - Nobre, S., Amado, N., & Carreira, S. (2012). Solving a contextual problem with the spreadsheet as an environment for algebraic thinking development.
*Teaching Mathematics and Its Applications, 31*, 11–19. https://doi.org/10.1093/teamat/hrr026CrossRefGoogle Scholar - Quivy, R., & Campenhoudt, L. (2008).
*Manual de Investigação em Ciências Sociais*. Lisboa, Portugal: Gradiva.Google Scholar - Villarreal, M., & Borba, M. (2010). Collectives of humans-with-media in mathematics education: Notebooks, blackboards, calculators, computers and… notebooks throughout 100 years of ICMI.
*ZDM Mathematics Education, 42*(1), 49–62.CrossRefGoogle Scholar - Wijers, M., Jonker, V., & Drijvers, P. (2010). MobileMath: Exploring mathematics outside the classroom.
*ZDM Mathematics Education, 42*(7), 789–799.CrossRefGoogle Scholar - Wilson, K. (2006). Naming a column on a spreadsheet.
*Research in Mathematics Education, 8*, 117–132.CrossRefGoogle Scholar - Wilson, K., Ainley, J., & Bills, L. (2005). Spreadsheets, pedagogic strategies and the evolution of meaning for variable. In H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 321–328). Melbourne, Australia: PME.Google Scholar