## Abstract

Teacher education is central to the development of the professional knowledge of pre-service teachers. The main goal of this paper is to reflect on the development that the analysis (done by a group of pre-service secondary teachers) of a set of tasks, based on elements related to domains of KTMT—*Knowledge for Teaching Mathematics with Technology*—can bring to the knowledge of pre-service teachers of mathematics. Specifically, the goal was to investigate the following questions: (1) What are the factors that guide the pre-service teachers’ task discussion? (2) Which KTMT domains are emphasized by pre-service teachers during task discussion? The elements taken into account are the characteristics of the tasks (focus on cognitive level, structuring level and technology role), the use of representations (focus on balance and articulation of representations), and the equilibrium between experimentation (focus on digital technology affordances) and justification (focus on argumentation and proof). The methodology of this case study involves a qualitative approach. The main conclusions suggest that influences in the pre-service teachers’ discussion of tasks fell into the following categories: the potentialities of technology, the type of tasks, and the prospective teachers’ experience with a set of tasks, and analysis of some real students’ reports. With regard to KTMT, although it was possible to identify some global development, *Teaching and Learning and Technology Knowledge* was the domain in which stronger development took place.

This is a preview of subscription content, access via your institution.

## References

Biza, I., Nardi, E., & Zachariades, T. (2010). Teachers’ views on the role of visualization anddidactical intentions regarding proof. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.),

*Proceedings of CERME 6*(pp. 261–270). Paris: INRP.Burril, G. (2017). Designing interactive dynamic technology activities to support the development of conceptual understanding. In A. Leung & A. Baccaglini-Frank (Eds.),

*Digital technologies in designing mathematics education tasks—Potential and pitfalls*(pp. 303–328). Cham: Springer.Clark-Wilson, A., & Timotheus, J. (2013). Designing tasks within a multi-representational technological environment: An emerging rubric. In C. Margolinas (Ed.),

*Task design in Mathematics Education—Proceedings of ICMI Study 22*(pp. 45–52). Oxford: ICMI.De Villiers, M. (1990). The role and function of proof in mathematics.

*Pythagoras,**24,*17–24.Dreher, A., Kuntze, S., & Lerman, S. (2016). Why use multiple representations in the mathematics classroom? Views of English and German preservice teachers.

*International Journal of Science and Mathematics Education,**14,*S363–S382.Drijvers, P., Tacoma, S., Besamusca, A., Doorman, M., & Boon, P. (2013). Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations.

*ZDM,**45*(7), 987–1001.Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.

*Educational Studies in Mathematics,**61*(1–2), 103–131.Goos, M., & Bennison, A. (2008). Surveying the technology landscape: Teachers’ use of technology in secondary mathematics classrooms.

*Mathematics Education Research Journal,**20*(3), 102–130.Hanna, G. (2001). Proof, explanation and exploration: An overview.

*Educational Studies in Mathematics,**44,*5–23.Haspekian, M. (2011). The co-construction of a mathematical and a didactical instrument. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.),

*Proceedings of the 7*^{th}*CERME*(pp. 2298–2307). Rzeszow: ERME, University of Rzeszow.Hegedus, S., Laborde, C., Bradey, C., Dalton, S., Siller, H.-S., Tabach, M., et al. (2017).

*Uses of technology in upper secondary mathematics education*. Hamburg: Springer.Heid, M., & Blume, G. (2008). Technology and the development of algebraic understanding. In M. Heid & G. Blume (Eds.),

*Research on technology and the teaching and learning of Mathematics*(Vol. 2, pp. 55–108). Charlotte: NCTM, IAP.Hsieh, F., Horng, W., & Shy, H. (2012). From exploration to proof production. In G. Hanna & M. de Villiers (Eds.),

*Proof and proving in mathematics education*(pp. 279–304). Dordrecht: Springer.Joubert, M. (2017). Revisiting theory for the design of tasks: Special considerations for digital environments. In A. Leung & A. Baccaglini-Frank (Eds.),

*Digital technologies in designing mathematics education tasks—Potential and pitfalls*(pp. 17–40). Cham: Springer.Kaput, J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.),

*Research issues in the learning and teaching of algebra*(pp. 167–194). Reston, VA: NCTM.Kaur, B. (2017). Impact of the course teaching and learning of mathematics on preservice grades 7 and 8 mathematics teachers in Singapore.

*ZDM,**49*(2), 265–272.Kendal, M., & Stacey, K. (2001). Influences on and factors changing technology privileging. In M. Heuvel-Panhuizen (Ed.),

*Proceedings of the 25*^{th}*PME, vol. 4*(pp. 217–224). Utrecht: PME.Laborde, C. (2001). Integration of technology in design of geometry tasks with Cabri-geometry.

*International Journal of Computers for Mathematical Learning,**6,*283–317.Lepak, J., Wernet, J., & Ayieko, R. (2018). Capturing and characterizing students’ strategic algebraic reasoning through cognitively demanding tasks with focus on representations.

*Journal of Mathematical Behavior,**50,*57–73.Lesseig, K. (2016). Investigating mathematical knowledge for teaching proof in professional development.

*International Journal of Research in Education and Science,**2,*253–270.Leung, A. (2011). An epistemic model of task design in dynamic geometry environment.

*ZDM,**43,*325–336.Lopes, S. (2016). Funções no 10.º ano (unpublished work). Lisbon: FCT-UNL.

Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge.

*Teachers College Record,**108,*1017–1054.Molenje, L., & Doerr, H. (2006). High school mathematics teachers’ use of multiple representations when teaching functions in graphing calculator environments. In S. Alatorre, J. Cortina, M. Sáiz, & A. Méndez (Eds.),

*Proceedings of the 28**th**NA-PME*. Mérida: Universidad Pedagógica Nacional.Monaghan, J., Trouche, L., & Borwein, J. (2016).

*Tools and mathematics: Instruments for learning*. Cham: Springer.Picciatto, H. (1996). Make these designs.

*Mathematics Teacher,**89*(5), 424–427.Ponte, J. (2005). Gestão curricular em Matemática. In GTI (Eds.),

*O professor e o desenvolvimento curricular*(pp. 11–34). Lisbon: APM.Rabardel, P. (1995).

*Les hommes et les technologies, approche cognitive des instruments contemporains*. Paris: Armand Colin.Rocha, H. (2013). Knowledge for Teaching Mathematics with Technology—A new framework of teacher knowledge. In A. Lindmeier & A. Heinze (Eds.),

*Proceedings of the 37**th**PME, vol. 4*(pp. 105–112). Kiel: PME.Rocha, H. (2016). Teacher’s representational fluency in a context of technology use.

*Teaching Mathematics and its Applications,**35*(2), 53–64.Shulman, L. (1986). Those who understand: Knowledge growth in teaching.

*Educational Researcher,**15*(2), 4–14.Tabach, M. (2011). A Mathematics teacher’s practice in a technological environment: A case study analysis using two complementary theories.

*Technology, Knowledge and Learning,**16*(3), 247–265.Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y. (2012). Cognitivedevelopment of proof. In G. Hanna & M. de Villiers (Eds.),

*Proof and proving inMathematics Education*(pp. 13–49). Dordrecht: Springer.Thomas, M., & Hong, Y. (2013). Teacher integration of technology into mathematics learning.

*International Journal of Technology in Mathematics Education,**20,*69–84.Thomas, M., & Lin, C. (2013). Designing tasks for use with digital technology. In C. Margolinas (Ed.),

*Task design in mathematics education—Proceedings of ICMI Study 22*(pp. 109–118). Oxford: ICMI.Trgalova, J., Clark-Wilson, A., & Weigand, H.-G. (2018). Technology and resources in mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.),

*Developing research in mathematics education: Twenty years of communication, cooperation and collaboration in Europe*(pp. 142–161). Cham: Springer.Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations.

*International Journal of Computers for Mathematical Learning,**9,*281–307.Yeo, J. (2017). Development of a framework to characterize the openness of mathematical tasks.

*International Journal of Science and Mathematics Education,**15,*175–191.Yin, R. (2003).

*Case study research—Design and methods*. Thousand Oaks: Sage Publications.Zbiek, R., Heid, M., Blume, G., & Dick, T. (2007). Research on technology in mathematics education. In F. Lester (Ed.),

*2*nd*Handbook of Research on Mathematics Teaching and Learning Handbook of Research on Mathematics Teaching and Learning*(pp. 1169–1207). Charlotte: NCTM.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Rocha, H. Using tasks to develop pre-service teachers’ knowledge for teaching mathematics with digital technology.
*ZDM Mathematics Education* **52**, 1381–1396 (2020). https://doi.org/10.1007/s11858-020-01195-1

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11858-020-01195-1

### Keywords

- Teacher’s knowledge
- KTMT
- Technology
- Mathematics
- Functions