Providing Online Learning Situations for In-Service Mathematics Teachers' External Transposition Knowledge During COVID-19 Pandemic: Case of Indonesia




online learning situation, teachers' professional development, external didactic transposition, didactical design research, Moodle, GeoGebra Classroom


Updating mastery of mathematics scholarly knowledge is vital for mathematics teachers' professional development in Indonesia. External transposition refers to the transposition of mathematics scholarly knowledge to the school curriculum. During the COVID-19 pandemic, online learning becomes the primary strategy for strengthening the external transposition knowledge of mathematics teachers in Indonesia. This study aims to describe the process of providing online learning situations in lifting the in-service mathematics teachers' external transposition knowledge during the COVID-19 pandemic. The online learning involved 57 in-service high school mathematics teachers in Indonesia as participants. The scholarly knowledge to be delivered was the visualization of matrix multiplication. This qualitative study is a didactical design research that includes a prospective learning situation analysis, a meta-technopedadidactic analysis, and a retrospective analysis. In the prospective learning situation, the solutions for re-situating face-to-face learning to online learning were disclosed. Moodle was used as the Learning Management System, and GeoGebra Classroom was used as the Task Response System. A meta-technopedadidactic is an educator's ability to comprehensively analyze pedagogic interaction, didactic interaction, and didactic pedagogy anticipation in using technology in a learning situation. The finding in the meta-technopedadidactic analysis showed the role of technology as a visualization tool in online learning, evidence of participants' external didactic transposition, constraints, and collaboration in learning. Finally, the retrospective analysis suggested some future improvements for better online learning situations. This study contributes to the comprehensive organization of e-learning courses in Indonesia for mathematics teachers' professional development based on pedagogic interaction, didactic interaction, and didactic pedagogy anticipation in the use of technology. Despite some of the study's limitations, the findings have the potential to be an advantage for the long-term sustainability of e-learning as a new norm for teachers' professional development programs, rather than just a foreshadowing of the pandemic.


Adelabu, F.M., Makgato, M., and Ramaligela, M.S., 2019. The importance of dynamic geometry computer software on learners’ performance in geometry. Electronic Journal of e-Learning, 17(1), pp. 52–63.

Allyson Hallman-Thrasher, Erin T. Litchfield, and Kevin E. Dael, 2016. Original recipes for matrix multiplication. The Mathematics Teacher, 110(3), pp. 182–190.

Arcavi, A., 2003. The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), pp. 215–241.

Astriawati, N. and Pratama, H.A., 2021. Cadets ’ effectivity and perception on Moodle online learning in economy mathematics course. In: Advances in Social Science, Education and Humanities Research. Atlantis Press SARL, pp. 34–39.

Aytekin, C. and Kiymaz, Y., 2019. Teaching linear algebra supported by GeoGebra visualization environment. Acta Didactica Napocensia, 12(2), pp. 75–96.

Azhari, B. and Fajri, I., 2021. Distance learning during the COVID-19 pandemic: School closure in Indonesia. International Journal of Mathematical Education in Science and Technology.

Barahmand, A., 2020. On the definition of matrix multiplication. International Journal of Mathematical Education in Science and Technology, 51(7), pp. 1137–1145.

Basitere, M. and Ivala, E., 2015. Mitigating the mathematical knowledge gap between high school and first year university chemical engineering mathematics course. Electronic Journal of e-Learning, 13(2), pp. 68–83.

Bosch, M., Hausberger, T., Hochmuth, R., Kondratieva, M., and Winsløw, C., 2021. External didactic transposition in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 7(1), pp. 140–162.

Brousseau, G., 2002. Theory of Didactical Situations in Mathematics. Edited and translated by N. Balacheff et al. Kluwer Academic Publishers.

Brousseau, G. and Warfield, V., 2020. Didactical Situations in Mathematics Education. In: S. Lerman, ed. Encyclopedia of Mathematics Education. 2nd edition. Springer, pp. 206–213.

Cao, Y., Zhang, S., Chan, M.C.E., and Kang, Y., 2021. Post-pandemic reflections: Lessons from Chinese mathematics teachers about online mathematics instruction. Asia Pacific Education Review, 22, pp. 157–168.

Chaka, C., Nkhobo, T., and Lephalala, M., 2020. Leveraging MoyaMA, Whatsapp and online discussion forum to support students at an open and distance e-learning university. Electronic Journal of e-Learning, 18(6), pp. 494–515.

Chevallard, Y., 2019. Introducing the anthropological theory of the didactic: An attempt at a principled approach. Hiroshima Journal of Mathematics Education, 12, pp. 71–114.

Chevallard, Y. and Bosch, M., 2020. Didactic transposition in mathematics education. In: Stephen Lerman, ed. Encyclopedia of Mathematics Education: 2nd edition. London: Springer, pp. 214–218.

Cook, J.P., 2018. An investigation of an undergraduate student’s reasoning with zero-divisors and the zero-product property. Journal of Mathematical Behavior, 49, pp. 95–115.

Cook, J.P., Zazkis, D., and Estrup, A., 2018. Rationale for matrix multiplication in linear algebra textbooks. In: S. Stewart, C. Andrews-Larson, A. Berman and M. Zandieh, ed. Challenges and Strategies in Teaching Linear Algebra (ICME-13 Monographs). Switzerland: Springer International Publishing, pp. 103–125.

Corfman, T. and Beck, D., 2019. Case study of creativity in asynchronous online discussions. International Journal of Educational Technology in Higher Education, 16(22).

Creswell, J.W. and Creswell, J.D., 2018. Research Design: Qualitative, Quantitative and Mixed Methods Approaches. 5th edition. SAGE.

Denzin, N.K., 2017. The Research Act: A Theoretical Introduction to Sociological Methods. New York: Routledge.

Dreher, A., Lindmeier, A., Heinze, A., and Niemand, C., 2018. What kind of content knowledge do secondary mathematics teachers need?: A conceptualization taking into account academic and school mathematics. Journal fur Mathematik-Didaktik, 39(2), pp. 319–341.

Fauzi, A., Wandira, R., Sepri, D., and Hafid, A., 2021. Exploring students’ acceptance of Google Classroom during the Covid-19 pandemic by using the technology acceptance model in West Sumatera Universities. The Electronic Journal of e-Learning, 19(4), pp. 233–240.

Ferdiánová, V., 2017. GeoGebra materials for LMS Moodle focused Monge on projection. Electronic Journal of e-Learning, 15(3), pp. 259–268.

Fuadiah, N.F., Suryadi, D., and Turmudi, 2019. Teaching and learning activities in classroom and their impact on student misunderstanding: A case study on negative integers. International Journal of Instruction, 12(1), pp. 407–424.

Galikyan, I. and Admiraal, W., 2019. Students’ engagement in asynchronous online discussion: The relationship between cognitive presence, learner prominence, and academic performance. The Internet and Higher Education, 43.

Garad, A., Al-Ansi, A.M., and Qamari, I.N., 2021. The role of e-learning infrastructure and cognitive competence in distance learning effectiveness during the COVID-19 pandemic. Cakrawala Pendidikan, 40(1), pp. 81–91.

Gómez-Blancarte, A., Romo-Vázquez, A., Miranda, I., Aguilar, M.S., Castañeda, A., and Lezama, J., 2019. An online learning community for the professional development of mathematics teachers in Mexico. Interciencia, 44(4), pp. 247–252. Available at:

Guba, E.G. and Lincoln, Y.S., 2005. Paradigmatic controversies, contradictions, and emerging confluences. In: N.K. Denzin and Y.S. Lincoln, ed. The SAGE Handbook of Qualitative Research. 3rd edition. SAGE, pp. 191–216.

Harel, G., 2017. The learning and teaching of linear algebra: Observations and generalizations. Journal of Mathematical Behavior, 46, pp. 69–95.

Harel, G., 2019. Varieties in the use of geometry in the teaching of linear algebra. ZDM Mathematics Education, 51(7), pp. 1031–1042.

Hennink, M.M., 2014. Focus Group Discussions. Oxford University Press.

Herbst, P. and Chazan, D., 2020. Mathematics teaching has its own imperatives: Mathematical practice and the work of mathematics instruction. ZDM Mathematics Education, 52(6), pp. 1149–1162.

Hohenwarter, M. and Fuchs, K., 2005. Combination of dynamic geometry , algebra and calculus in the software system GeoGebra. In: Computer Algebra Systems and Dynamic Geometry Systems in Mathematics Teaching Conference 2004.

Joubert, J., Callaghan, R., and Engelbrecht, J., 2020. Lesson study in a blended approach to support isolated teachers in teaching with technology. ZDM Mathematics Education, 52, pp. 907–925.

Joubert, M., 2017. Revisiting theory for the design of tasks: Special considerations for digital environments. In: A. Leung and A. Baccaglini-Frank, ed. Digital Technologies in Designing Mathematics Education Tasks: Potential and Pitfalls. Springer International Publishing Switzerland, pp. 17–40.

Juandi, D., Kusumah, Y.S., Tamur, M., Perbowo, K.S., and Wijaya, T.T., 2021. A meta-analysis of Geogebra software decade of assisted mathematics learning: what to learn and where to go? Heliyon, 7(5), p. e06953.

Kansanen, P., 2003. Studying - The realistic bridge between instruction and learning. An attempt to a conceptual whole of the teaching-studying-learning process. Educational Studies, 29(2–3), pp. 221–232.

Kilpatrick, J., 2019. A double discontinuity and a triple approach: Felix Klein’s perspective on mathematics teacher education. In: H.-G. Weigand, W. McCallum, M. Menghini, M. Neubrand and G. Schubring, ed. The Legacy of Felix Klein. ICME-13 Monographs.

Koutsabasis, P., Stavrakis, M., Spyrou, T., and Darzentas, J., 2011. Perceived impact of asynchronous E-learning after long-term use: Implications for design and development. International Journal of Human-Computer Interaction, 27(2), pp. 191–213.

Kusumah, Y.S. and Nurhasanah, F., 2017. The endless long-term program of mathematics teacher professional development in Indonesia. In: B. Kaur, O.N. Kwon and Y.H. Leong, ed. Professional Development of Mathematics Teachers: An Asian Perspective. Springer, pp. 33–45.

Kynigos, C. and Kalogeria, E., 2012. Boundary crossing through in-service online mathematics teacher education: The case of scenarios and half-baked microworlds. ZDM Mathematics Education, 44(6), pp. 733–745.

Lamanauskas, V. and Makarskaitė-Petkevičienė, R., 2021. Distance lectures in university studies: Advantages, disadvantages, improvement. Contemporary Educational Technology, 13(3).

Mailizar, M., Almanthari, A., Maulina, S., and Bruce, S., 2020. Secondary school mathematics teachers ’ views on e-learning implementation barriers during the Covid-19 pandemic: The case of Indonesia. EURASIA Journal of Mathematics, Science and Technology Education, 16(7).

Marfuah, M., 2017. Profiling self-regulated learning in online mathematics teacher training: A case study of a GeoGebra course. Southeast Asian Mathematics Education Journal, 7(1), pp. 45–55.

Martínez, S., Guíñez, F., Zamora, R., Bustos, S., and Rodríguez, B., 2020. On the instructional model of a blended learning program for developing mathematical knowledge for teaching. ZDM Mathematics Education, 52, pp. 877–891.

Martinovic, D. and Manizade, A.G., 2020. Teachers using GeoGebra to visualize and verify conjectures about trapezoids. Canadian Journal of Science, Mathematics and Technology Education, 20(3), pp. 485–503.

Miles, M.B., Huberman, A.M., and Saldana, J., 2014. Qualitative Data Analysis: A Methods Sourcebook. 3rd edition. Sage Publications.

Ministry of Education and Culture, 2018. Kompetensi Inti dan Kompetensi Dasar Pelajaran pada Kurikulum 2013. Available at: Nomor 37 Tahun 2018.pdf [Accessed: 30 March 2020].

Mlotshwa, N., Tunjera, N., and Chigona, A., 2020. Integration of MOODLE into the classroom for better conceptual understanding of functions in Mathematics. South African Journal of Education, 40(3), pp. 1–14.

Moodle, 2021. Statistics. Available at: [Accessed: 18 June 2021].

Mulyanti, B., Purnama, W., and Pawinanto, R.E., 2020. Distance learning in vocational high schools during the COVID-19 pandemic in West Java Province, Indonesia. Indonesian Journal of Science and Technology, 5(2), pp. 271–282.

Pölzl-Stefanec, E., 2021. Challenges and barriers to Austrian early childhood educators’ participation in online professional development programmes. British Journal of Educational Technology, pp. 1–17.

Price, F. and Kadi-Hanifi, K., 2011. E-motivation! the role of popular technology in student motivation and retention. Research in Post-Compulsory Education, 16(2), pp. 173–187.

Reijers, W. and Coeckelbergh, M., 2020. Narrative and Technology Ethics. Springer Nature Switzerland AG.

Ricoeur, P., 2016. Hermeneutics and The Human Sciences. 4th edition. Edited and translated by J.B. Thompson. Cambridge University Press.

Rudi, R., Suryadi, D., and Rosjanuardi, R., 2020. Teachers ’ perception as a crucial component in the design of didactical design research-based teacher professional learning community in Indonesia. European Online Journal of Natural and Social Sciences, 9(3), pp. 642–654.

Ruthven, K., 2012. The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics. ZDM Mathematics Education, 44, pp. 627–640.

Schoenfeld, A.H., 2012. Problematizing the didactic triangle. ZDM Mathematics Education, 44, pp. 587–599.

Stevenson, M., Hedberg, J., Highfield, K., and Diao, M., 2015. Visualizing solutions: Apps as cognitive stepping-stones in the learning process. Electronic Journal of e-Learning, 13(5), pp. 366–379. Available at:

Sumaryanta, Mardapi, D., Sugiman, and Herawan, T., 2018. Assessing teacher competence and its follow-up to support professional development sustainability. Journal of Teacher Education for Sustainability, 20(1), pp. 106–123.

Suryadi, D., 2013. Didactical Design Research (DDR) dalam pengembangan pembelajaran matematika. Prosiding Seminar Nasional Matematika dan Pendidikan Matematika, 1, pp. 3–12. Available at:

Suryadi, D., 2019. Landasan Filosofis Penelitian Desain Didaktis (DDR). Bandung: Pusat Pengembangan DDR Indonesia.

Turmudi, T., Kusumah, Y.S., Juandi, D., and Mulyana, E., 2014. Development of didactical design of mathematics pedagogy through professional program of mathematics teacher education. Jurnal Pendidikan dan Pembelajaran, 21(1), pp. 10–23. Available at: [Accessed: 26 February 2021].

UNESCO, 2020. Education in a post covid world: Nine ideas for public action international commission on the futures of education. Available at: [Accessed: 30 June 2021].

Wasserman, N.H., 2018. Exploring advanced mathematics courses and content for secondary mathematics teachers. In: N.H. Wasserman, ed. Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers. New York: Springer, pp. 1–15.

Zainuddin, N., Idrus, R., and Jamal, A.F.M., 2016. Moodle as an ODL teaching tool: A perspective of students and academics. Electronic Journal of e-Learning, 14(4), pp. 282–290.

Zöchbauer, J. and Hohenwarter, M., 2020. Developing a collaboration tool to give every student a voice in a classroom discussion. In: Seventh ERME Topic Conference on Language in the Mathematics Classroom, pp. 195–202.



1 Feb 2022