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Mathematical Problem Solving and the Use of Digital Technologies

  • Manuel Santos-TrigoEmail author
Chapter
Part of the ICME-13 Monographs book series (ICME13Mo)

Abstract

The goal in this chapter is to analyze and document ways in which the use of digital technologies provides affordances for teachers/students to develop knowledge and solve mathematical problems. What types of representations and problem explorations appear and characterize a technology enhanced problem-solving approach? What does the systematic and coordinated use of digital technologies bring to the students’ process of delving into problem statements, to the ways of representing and exploring tasks, and to the development of their problem-solving competencies? Four groups of problems are examined to illustrate what strategies and ways of reasoning emerge and are enhanced in a technological problem-solving approach. Focusing on the reconstruction of figures that often appear in problem statements, the transformation of a routine problem into an investigation task, the representation and exploration of a variation phenomenon task, and the construction of a dynamic configuration to pose and pursue questions shed lights on ways to frame a problem-solving approach that fosters and values the use of digital technologies. The discussion of these types of problems might provide students an opportunity to pay attention to and develop ways of reasoning that include the construction of dynamic models, the controlled movement of some model elements, the search and exploration of loci of points or lines to analyze some variation phenomena, the use of sliders to control parameters to delve into concepts, and to examine object attributes to formulate, validate, and support conjectures.

Keywords

Mathematical Problem solving Digital technologies Mathematical tasks Mathematical reasoning Dynamic geometry systems (DGS) 

Notes

Acknowledgements

This chapter is part of a project that deals with teachers and students’ use of digital technologies in extending both mathematics and didactic knowledge. I acknowledge the support received from research projects with references Conacyt-168543 and EDU2017-84276-R.

References

  1. Barbeau, E. (2009). Introduction. In E. J. Barbeau, P. J. Taylor (Eds.), Challenging mathematics in and beyond the classroom (p. 97). New ICMI Study Series 12.  https://doi.org/10.1007/978-0-387-09603-2. New York: Springer.Google Scholar
  2. Blaschke, L. M., & Hase, S. (2016). A holistic framework for creating twenty-first-Century self-determined learners. In B. Gros et al. (Eds.), The future of ubiquitous learning. Learning design for emerging pedagogies (pp. 25–40). New York: Springer.Google Scholar
  3. Connected Geometry. (2000). Developed by Educational Development Center, Inc., Chicago Illinois: Everyday Learning.Google Scholar
  4. Freiman, V., Kadijevich, D., Kuntz, G., Pozdnyakov, S., & Stedoy, I. (2009). Technological environments beyond the classroom. In E. J. Barbeau & P. J. Taylor (Eds.), Challenging mathematics in and beyond the classroom (p. 97). New ICMI Study Series 12.  https://doi.org/10.1007/978-0-387-09603-2_4. New York: Springer.Google Scholar
  5. Gros, B. (2016). The dialogue between emerging pedagogies and emerging technologies. In. B. Gros, et al. (Eds.). The future of ubiquitous learning. Learning design for emerging pedagogies (pp. 3–23). New York: Springer.Google Scholar
  6. Gros, B., Kinshuk, & Maina, M. (2016). Preface. In B. Gros, Kinshuk, & M. Maina (Eds.), The future of ubiquitous learning. Learning design for emerging pedagogies (pp. v–x). New York: Springer.Google Scholar
  7. Hokanson, B., & Gibbons, A. (Eds.). (2014). Design in educational technology. Design thinking, design process, and the design studio. London: Springer.Google Scholar
  8. Kinshuk, R. H., & Spector, J. M. (Eds.). (2013). Reshaping learning. Frontiers of learning technologies in a global context. New York: Springer.Google Scholar
  9. Leikin, R., Koichu, B., Berman, A., & Dinur, S. (2017). How are questions that students ask in high level mathematics classes linked to general giftedness? ZDM Mathematics Education, 49, 65–80.CrossRefGoogle Scholar
  10. Lester, F. K., & Cai, J. (2016). Can mathematical problem solving be taught? Preliminary answers from 30 years of research. In P. Felmer, et al. (Eds.), Posing and solving mathematical problems (pp. 117–135). Research in Mathematics Education. Switzerland: Springer.  https://doi.org/10.1007/978-3-319-28023-3_8.CrossRefGoogle Scholar
  11. Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13, 135–157.CrossRefGoogle Scholar
  12. Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM Mathematics Education, 43, 325–336.CrossRefGoogle Scholar
  13. Leung, A. (2017). Exploring techno-pedagogic task design in the mathematics classroom. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing
mathematics education tasks (pp. 3–16). Switzerland: Springer.  https://doi.org/10.1007/978-3-319-43423-0_1.Google Scholar
  14. Leung, A., & Baccaglini-Frank, A. (2017a). Digital technologies in designing mathematics educational tasks. Potential and pitfalls. Berlin: Springer.  https://doi.org/10.1007/978-3-319-43423-0.Google Scholar
  15. Leung, A., & Baccaglini-Frank, A. (2017b). Introduction. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing mathematic education tasks (pp. vii–xvi). Switzerland: Springer.  https://doi.org/10.1007/978-3-319-43423-0.Google Scholar
  16. 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). Switzerland: Springer.  https://doi.org/10.1007/978-3-319-09629-2_6.CrossRefGoogle Scholar
  17. Liljedahl, P., Santos-Trigo, M., Malaspina, U., & Bruder, R. (2016). Problem solving in mathematics education. ICME-13 Topical Surveys.  https://doi.org/10.1007/978-3-319-40730-2_1.Google Scholar
  18. Margolinas, C. (2013). Task design in mathematics Education. In Proceedings of ICMI Study 22 (hal-00834054v2).Google Scholar
  19. Mason, J. (2016). When is a problem…? “When” is actually the problem! In P. Felmer, et al. (Eds.), Posing and solving mathematical problems (pp. 263–283). Research in Mathematics Education. Switzerland: Springer.  https://doi.org/10.1007/978-3-319-28023-3_8.CrossRefGoogle Scholar
  20. Mason, J., Burton, L., & Stacy, K. (2010). Thinking mathematically (2nd ed.). New York: Pearson.Google Scholar
  21. Melzak, Z. A. (1983). Invitation to geometry. NY: Dover.Google Scholar
  22. Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108(6), 1017–1054.CrossRefGoogle Scholar
  23. 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
  24. Polya, G. (1945). How to solve it. Princeton: Princeton University Press.Google Scholar
  25. Reyes-Martínez, I. (2016). The design and results of implementing a learning environment that incorporates a mathematical problem-solving approach and the coordinated use of digital technologies (Unpublished doctoral dissertation). Mathematics Education Department, Cinvestav-IPN, Mexico.Google Scholar
  26. Santos-Trigo, M. (2007). Mathematical problem solving: An evolving research and practice domain. ZDMThe International Journal on Mathematics Education, 39(5, 6), 523–536.CrossRefGoogle Scholar
  27. 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
  28. Santos-Trigo, M., & Camacho-Machín, M. (2016). Digital technologies and mathematical problem solving: Redesigning resources, materials, and extending learning environments. In K. Newton (Ed.), Problem-solving: Strategies, challenges and outcomes (pp. 31–49). New York: Nova Science Publishers.Google Scholar
  29. Santos-Trigo, M., & Moreno-Armella, L. (2016). The use of digital technologies to frame and foster learners’ problem-solving experiences. In P. Felmer, et al. (Eds.), Posing and solving mathematical problems (pp. 189–207). Research in Mathematics Education. Switzerland: Springer.  https://doi.org/10.1007/978-3-319-28023-3_8.CrossRefGoogle Scholar
  30. Santos-Trigo, M., Moreno-Armella, L., & Camacho-Machín, M. (2016a). Problem solving and the use of digital technologies within the Mathematical Working Space framework. ZDM Mathematics Education, 48, 827–842.CrossRefGoogle Scholar
  31. Santos-Trigo, M., & Reyes-Martínez, I. (2018). High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies. International Journal of Mathematical Education in Science and Technology.  https://doi.org/10.1080/0020739x.2018.1489075.CrossRefGoogle Scholar
  32. Santos-Trigo, M., Reyes-Martínez, I., & Aguilar-Magallón, D. (2016). Digital technologies and a modeling approach to learn mathematics and develop problem solving competencies. In L. Uden, D. Liberona, & B. Feldmann (Eds.), Learning technology for education in cloud (pp. 193–206). Switzerland: Springer.  https://doi.org/10.1007/978-3-31942147-6_18.
  33. Santos-Trigo, M., & Reyes-Rodríguez, A. (2016). 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, 47(1), 58–81.  https://doi.org/10.1080/0020739X.2015.1049228.CrossRefGoogle Scholar
  34. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press.Google Scholar
  35. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grows (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.Google Scholar
  36. Selden, A., Selden, J., Hauk, S., & Mason, A. (2000). Why can’t calculus students access their knowledge to solve non-routine problems? CBMS Issues in Mathematics Education, 8, 128–153.CrossRefGoogle Scholar
  37. Silver, E. A. (1990). Contribution of research to practice: Applying findings, methods, and perspectives. In T. Cooney & C.R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s. 1990 yearbook (pp. 1–11). Reston VA: The Council.Google Scholar
  38. Silver, E. A. (2016). Mathematical problem solving and teacher professional learning: The case of a modified PISA mathematics task. In P. Felmer, et al. (Eds.), Posing and solving mathematical problems (pp. 345–360). Research in Mathematics Education. Switzerland: Springer.  https://doi.org/10.1007/978-3-319-28023-3_8.CrossRefGoogle Scholar
  39. Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing. From research to effective practice. London: Springer.Google Scholar
  40. Stanic, G., & Kilpatrick, J. (1988). Historial perspectivas on problem solving in the mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 1–22). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  41. Törner, G., Schoenfeld, A. H., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. ZDM Mathematics Education, 39(5–6), 353.CrossRefGoogle Scholar
  42. Walling, D. R. (2014). Designing learning for tablet classrooms. Innovations in instruction. London: Springer.CrossRefGoogle Scholar
  43. Weiser, M. (1991). The computer for the 21st century. Scientific American, 265(3), 94–104.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Center for Research and Advanced StudiesCinvestav-IPNMexico CityMexico

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