Knowledge for teaching mathematical problem-solving with technology: An exploratory study of a mathematics teacher’s proficiency

Hélia Jacinto 1 * , Susana Carreira 1 2
More Detail
1 Instituto de Educação, Universidade de Lisboa, Lisbon, PORTUGAL
2 FCT, Universidade do Algarve, Faro, PORTUGAL
* Corresponding Author
EUR J SCI MATH ED, Volume 11, Issue 1, pp. 105-122. https://doi.org/10.30935/scimath/12464
Published Online: 17 September 2022, Published: 01 January 2023
OPEN ACCESS   1349 Views   1013 Downloads
Download Full Text (PDF)

ABSTRACT

This paper presents the results of an exploratory case study that examined a veteran secondary teacher’s knowledge for teaching non-routine mathematical problem-solving with digital technologies. Data was collected through the observation of a veteran mathematics teacher, in real time, solving a mathematical problem with the digital tools of his choice. The descriptive model Mathematical Problem-Solving with Technology (MPST) was used to analyse the teacher’s utterances and actions while solving a mathematical problem, with GeoGebra and a spreadsheet, and expressing his reasoning also with technology. Our findings reveal the complexity of expert problem-solving with technology, through regulation processes and several micro-cycles that mainly involve the processes integrate and explore. Mathematical problem solving with technology entails relevant mathematical knowledge as well as knowledge about the mathematical affordances of the digital tools available, and the ability to combine them to develop a conceptual model of the solution to the problem. Thus, the teacher’s techno-mathematical fluency seems crucial to successfully solve the problem and express the reasoning with technology. Based on the findings, we discuss the teachers’ knowledge for teaching mathematical problem-solving with technology as including a particular kind of proficiency, techno–mathematical fluency for solving-and-expressing problems with technology. The limitations of the study, further research topics and implications for professional development and teacher education programmes are discussed.
Keywords: mathematical problem-solving knowledge for teaching, mathematical problem-solving, mathematics teachers, techno-mathematical fluency, technology

CITATION

Jacinto, H., & Carreira, S. (2023). Knowledge for teaching mathematical problem-solving with technology: An exploratory study of a mathematics teacher’s proficiency. European Journal of Science and Mathematics Education, 11(1), 105-122. https://doi.org/10.30935/scimath/12464

REFERENCES

  • Barron, B., Martin, C., & Roberts, E. (2007). Sparking self-sustained learning: report on a design experiment to build technological fluency and bridge divides. International Journal of Technology and Design Education, 17(1), 75-105. https://doi.org/10.1007/s10798-006-9002-4
  • Bookman, J. (1993). An expert-novice study of metacognitive behavior in four types of mathematics problems. PRIMUS, 3(3), 284-314. https://doi.org/10.1080/10511979308965710
  • Borba, M., & Villarreal, M. (2005). Humans-with-media and the reorganization of mathematical thinking. Springer. https://doi.org/10.1007/b105001
  • Bray, A., & Tangney, B. (2017). Technology usage in mathematics education research–A systematic review of recent trends. Computers & Education, 114, 255-273. https://doi.org/10.1016/j.compedu.2017.07.004
  • Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent problem-solving framework. Educational Studies in Mathematics, 58(1), 45-75. https://doi.org/10.1007/s10649-005-0808-x
  • Carreira, S., & Jacinto, H. (2019). A model of mathematical problem solving with technology: The case of Marco solving-and-expressing. In P. Liljedahl & M. Santos Trigo (Eds.), Mathematical Problem solving (pp. 41-62). Springer. https://doi.org/10.1007/978-3-030-10472-6_3
  • Carreira, S., Jones, K., Amado, N., Jacinto, H., & Nobre, S. (2016). Youngsters solving mathematics problems with technology. Springer. https://doi.org/10.1007/978-3-319-24910-0
  • Carrillo, C., & Flores, M. A. (2018). Veteran teachers’ identity: What does the research literature tell us? Cambridge Journal of Education, 48(5), 639-656. https://doi.org/10.1080/0305764X.2017.1394982
  • Chapman, O. (2015). Mathematics teachers’ knowledge for teaching problem solving. LUMAT, 3(1), 19-36. https://doi.org/‌10.31129/lumat.v3i1.1049
  • Chiu, M. M., Jones, K. A., & Jones, J. L. (2013). Building on Schoenfeld’s studies of metacognitive control towards social metacognitive control. In Y. Li, & J. Moschkovich (Eds.), Mathematical proficiency and beliefs in learning and teaching–Learning from Alan Schoenfeld and Günter Toerner (pp. 69-88). Sense Publishing. https://doi.org/10.1007/978-94-6209-299-0_6
  • Clark-Wilson, A., Hoyles, C. (2019). A research-informed web-based professional development toolkit to support technology-enhanced mathematics teaching at scale. Educational Studies in Mathematics, 102, 343-359. https://doi.org/10.1007/s10649-018-9836-1
  • Drijvers, P., Tacoma, S., Besamusca, A., Doorman, M., & Boon, P. (2013). Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations. ZDM–The International Journal on Mathematics Education, 45(7), 987-1001. https://doi.org/10.1007/s11858-013-0535-1
  • Geiger, V. (2005). Master, servant, partner and extension of self: a finer grained view of this taxonomy. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), Building connections: Theory, research and practice. MERGA.
  • 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 [Mathematics education: Paths and crossroads] (pp. 83-101). Associação de Professores de Matemática [Association of Mathematics Teachers].
  • Hanin, V., & Van Nieuwenhoven, C. (2020). An exploration of the cognitive, motivational, emotional and regulatory behaviours of elementary-school novice and expert problem solvers. Canadian Journal of Science, Mathematics and Technology Education, 20(2), 312-341. https://doi.org/10.1007/s42330-020-00092-9
  • Hegedus, S. J., & Moreno-Armella, L. (2009). Introduction: The transformative nature of “dynamic” educational technology. ZDM–The International Journal on Mathematics Education, 41, 397-398. https://doi.org/10.1007/s11858-009-0201-9
  • Hernández, A., Perdomo-Díaz, J., & Camacho-Machín, M. (2020). Mathematical understanding in problem solving with GeoGebra: a case study in initial teacher education. International Journal of Mathematical Education in Science and Technology, 51(2), 208-223. https://doi.org/10.1080/0020739X.2019.1587022
  • Hervey, L. G. (2015). Between the notion and the act: Veteran teachers’ TPACK and practice in 1: 1 settings. In C. Angeli, & N. Valanides (Eds.), Technological pedagogical content knowledge (pp. 165-189). Springer. https://doi.org/10.1007/978-1-4899-8080-9_8
  • Hoyles, C., Noss, R., Kent, P., & Bakker, A. (2010). Improving mathematics at work: The need for techno-mathematical literacies. Routledge. https://doi.org/10.4324/9780203854655
  • Jacinto, H. & Carreira, S. (2017a). 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
  • Jacinto, H., & Carreira, S. (2017b). Different ways of using GeoGebra in mathematical problem-solving beyond the classroom: Evidences of techno-mathematical fluency. Bolema, 31(57), 266-288. https://doi.org/10.1590/1980-4415v31n57a13
  • Jacinto, H., & Carreira, S. (2021). Digital tools and paper-and-pencil in solving-and-expressing: How technology expands a student’s conceptual model of a covariation problem. Journal on Mathematics Education, 12(1), 113-132. http://doi.org/10.22342/.12.1.12940.113-132
  • Koehler, M. J., & Mishra, P. (2008). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9(1), 60-70.
  • Koyuncu, I., Akyuz, A., & Cakiroglu, E. (2015). Investigating plane geometry problem-solving strategies of prospective mathematics teachers in technology and paper-and-pencil environments. International Journal of Science and Mathematics Education, 13, 837-862. https://doi.org/10.1007/s10763-014-9510-8
  • Kuzle, A. (2017). Delving into the nature of problem solving processes in a dynamic geometry environment: Different technological effects on cognitive processing. Technology, Knowledge and Learning, 22(1), 37-64. https://doi.org/10.1007/s10758-016-9284-x
  • Leatham, K. (2007). Pre‐service secondary mathematics teachers’ beliefs about the nature of technology in the classroom, Canadian Journal of Science, Mathematics and Technology Education, 7(2-3), 183-207. https://doi.org/10.1080/14926150709556726
  • 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). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410607713
  • 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). Information Age Publishing and National Council of Teachers of Mathematics.
  • Leung, A. (2017). Exploring techno-pedagogic task design in the mathematics classroom. In A. Leung, & A. Baccaglini-Franck (Eds.), Digital technologies in designing mathematics education tasks (pp. 3-16). Springer. https://doi.org/10.1007/978-3-319-43423_1
  • Liljedahl, P. (2014). Approaching professional learning: What teachers want. The Mathematics Enthusiast, 11(1), 109-122. https://doi.org/10.54870/1551-3440.1294
  • Liljedahl, P., & Cai, J. (2021). Empirical research on problem solving and problem posing: A look at the state of the art. ZDM Mathematics Education, 53, 723-735. https://doi.org/10.1007/s11858-021-01291-w
  • Martin, A., & Grudziecki, J. (2006). DigEuLit: Concepts and tools for digital literacy development. Innovation in Teaching and Learning in Information and Computer Sciences, 5(4), 249-267. https://doi.org/10.11120/ital.2006.05040249
  • Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108(6), 1017-1054. https://doi.org/10.1111/j.1467-9620.2006.00684.x
  • Montague, M., & Applegate, B. (1993). Mathematical problem-solving characteristics of middle school students with learning disabilities. The Journal of Special Education, 27(2), 175-201. https://doi.org/10.1177/002246699302700203
  • Monteiro, A., Mouraz, A., & Dotta, L. T. (2020). Veteran teachers and digital technologies: Myths, beliefs and professional development. Teachers and Teaching, 26(7-8), 577-587. https://doi.org/10.1080/13540602.2021.1900809
  • Papert, S., & Resnick, M. (1995). Technological fluency and the representation of knowledge. MIT Media Laboratory.
  • Powell, A., Francisco, J., & Maher, C. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data. The Journal of Mathematical Behavior, 22(4), 405-435. https://doi.org/‌10.1016/‌j.jmathb.2003.09.002
  • Rott, B., Specht, B., & Knipping, C. (2021). A descriptive phase model of problem-solving processes. ZDM–Mathematics Education, 53, 737-752. https://doi.org/10.1007/s11858-021-01244-3
  • Saadati, F., & Felmer, P. (2021). Assessing impact of a teacher professional development program on student problem-solving performance. ZDM–Mathematics Education, 53, 799-816. https://doi.org/10.1007/s11858-020-01214-1
  • Santos-Trigo, M. (2019). Mathematical problem solving and the use of digital technologies. In P. Liljedahl, & M. Santos-Trigo (Eds.), Mathematical problem solving (pp. 63-89). Springer. https://doi.org/10.1007/978-3-030-10472-6_4
  • Santos-Trigo, M., & Camacho-Machín, M. (2013). Framing the use of computational technology in problem solving approaches. The Mathematics Enthusiast, 1-2, 279-302. https://doi.org/10.54870/1551-3440.1268
  • Santos-Trigo, M., & Reyes-Martínez, I. (2019). High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies. International Journal of Mathematical Education in Science and Technology, 50(2), 182-201. https://doi.org/10.1080/0020739X.2018.1489075
  • Santos-Trigo, M., Barrera-Mora, F., & Camacho-Machín, M. (2021). Teachers’ use of technology affordances to contextualize and dynamically enrich and extend mathematical problem-solving strategies. Mathematics, 9(8), 793. https://doi.org/10.3390/math9080793
  • Schoenfeld, A. (1985). Mathematical problem solving. Academic Press.
  • Silva, R., Barbosa, L., Borba, M., & Ferreira, A. (2021). The use of digital technology to estimate a value of pi: Teachers’ solutions on squaring the circle in a graduate course in Brazil. ZDM–Mathematics Education, 53, 605-619. https://doi.org/10.1007/s11858-021-01246-1
  • Sinclair, N. (2020). On teaching and learning mathematics–technologies. In Y. Kolikant, D. Martinovic, & M. Milner-Bolotin (Eds.), STEM teachers and teaching in the digital era (pp. 91-107). Springer. https://doi.org/10.1007/978-3-030-29396-3_6
  • Stake, R. E. (1995). The art of case study research. SAGE.
  • Yao, X., & Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. The Journal of Mathematical Behavior, 55, 1-19. https://doi.org/10.1016/j.jmathb.2019.04.002