Jurnal Riset Pendidikan Matematika


addition and subtraction of integers, design research, metacognitive teaching-learning culture

Document Type



For the goal of raising the mathematical performance of students, the quality of teaching has to be improved in the direction that critical thinking and a step-by-step controllable argumentation have to be established. Metacognitive activities of teachers and students can support this. The goal of the design research presented in this paper is to show how metacognitive activities in classroom discussions can be fostered and the mechanisms can be uncovered. For this purpose, on the one hand, a new learning environment - designed for learning addition and subtraction of integers in grade four - has been created. On the other hand, the teacher and the students have been consequently educated to practice new behavior during the lessons. This pilot study is carried out in Sumba, in a regular class, grade four, consisting of 28 students. The lessons had been video graphed; selected pieces of the public discussions have been transcribed. Teacher's and learners' utterances and their interplay have been classified using a particular coding system developed for those purposes. The pattern of classifications has been the basis for interpreting the metacognitive-discursive learning culture. One result of the study is that students even in grade four accept such a learning environment; they practice inventing and analyzing strategies in this field of algebra as it would be a game with step-by-step controllable argumentations. After that, an astonishingly short time of educating students to practice metacognitive activities, both in oral discussions as well as in written answers to relevant formulated questions.

Page Range

73 - 82





Digital Object Identifier (DOI)





Bofferding, L. (2010). Addition and Subtraction with negatives: Acknowledging the multiple meanings of the minus sign. https://social.education.purdue.edu/bofferding/wp-content/uploads/2011/08/2010-Bofferding-Addition-and-subtraction.pdf

Brainerd, C. (1979). The origins of the number concept. Praeger.

Cohors-Fresenborg, E. (2013). Das hüpfen auf der zahlenlinie als evidenzbasis für vertragsgemässes rechnen. Beiträge zum Mathematikunterricht 2013, 240-243.

Cohors-Fresenborg, E., & Kaune, C. (2007). Modeling classroom discussions and categorising discursive and metacognitive activities. In D Pitta-Pantazi & G Philippou. Proceedings of CERME 5, 1180-1189. University of Cyprus.

Cohors-Fresenborg, E., Kaune, C., & Zülsdorf-Kersting, M. (2014). Klassifikation von metakognitiven und diskursiven aktivitäten im mathematik- und geschichtsunterricht mit einem gemeinsamen kategoriensystem. Forschungsinstitut für Mathematikdidaktik.

Cohors-Fresenborg, E., & Nowińska, E. (2021). Pengantar kategorisasi kegiatan metakognitif-diskursif para siswa dan guru dalam pelajaran matematika. Lembaga Matematika Kognitif - STKIP Weetebula.

Fosnot, C. T., & Dolk, M. (2001). Young mathematics at work, constructing number sense, addition, and subtraction. Heinemann

Gravemeijer, K., & Cobb, P. (2006). Design research from a learning perspective. In J. Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 45-85). Taylor and Francis.

Gray, E., & Tall, D. (1994). Duality, ambiguity, and flexibility: A "proceptual" view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116-140. https://doi.org/10.2307/749505

Kaune, C., & Cohors-Fresenborg, E. (2017). Kontrak untuk perhitungan - Buku Kerja untuk Siswa Kelas 7. Lembaga Matematika Kognitif - STKIP Weetebula.

Muslimin, M., Putri, R. I. I. P., & Somakim. (2012). Desain pembelajaran pengurangan bilangan bulat melalui permainan tradisional congklak berbasis pendidikan matematika realistik Indonesia di kelas IV sekolah dasar. Jurnal Matematika Kreatif-Inovatif (Kreano), 3(2), 100-112. https://journal.unnes.ac.id/nju/index.php/kreano/article/view/2642/2710

Musser, G. L., Burger, W. F., & Peterson, B. E. (2005). Mathematics for elementary teacher: A approach (7th ed.). John Wiley & Sons.

NowiÅ„ska, E. (2015). The benefit of the metaphor "contract" for understanding of concept formation in mathematics. Didactica Mathematicae, 1(36), 58 - 103.

Pintér, K. (2010). Creating game from mathematical problems. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21(1), 73-90. https://doi.org/10.1080/10511970902889919

Prahmana, R. C. I., Zulkardi, Z., & Hartono, Y. (2012). Learning multiplication using Indonesian traditional game in third grade. Journal on Mathematics Education, 3(2), 115-132. https://doi.org/10.22342/jme.3.2.1931.115-132

Putri, R. I. I. (2011). Implementasi alat peraga operasi bilangan bulat bagi guru sekolah dasar (SD) se-Kecamatan Ilir Barat I Palembang. Jurnal Pendidikan Matematika, 5(1), 73-79. https://doi.org/10.22342/jpm.5.1.823

Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM - The International Journal on Mathematics Education, 42(2), 149-161. https://doi.org/10.1007/s11858-010-0240-2

Schwank, I. (1995). The role of microworlds for constructing mathematical concepts. In M. Behara, R. Fritsch, & R Lintz (Eds.), Symposia Gaussiana, Conference: A mathematics and theoretical physics. Proceedings of the 2nd Gauss Symposium (pp. 101-120). W. de Gruyter.

Schwank, I. (2005). Die Schwierigkeit des Dazu-Denkens. In M. von Aster & J. -H. Lorenz (Eds.), Rechenstörungen bei Kindern - Neurowissenschaft, Psychologie, Pädagogik (pp. 93-133). Vandenhoeck & Ruprecht.

Schwank, I. (2010). Erlebniswelt zahlen - spielereien mit der rechenwendeltreppe für vorschulkinder. Forschungsinstitut für Mathematikdidaktik.

Schwank, I. (2011). Mathematisches grundverständnis: Denken will erlernt werden. In H. Keller (Ed.), Handbuch der kleinkindforschung: 4 korrigierte, überarbeitete und erweiterte Auflage (pp. 1154-1176). Huber.

Schwank, I., & Schwank, E. (2015). Development of mathematical concepts during early childhood across cultures. In J.D. Wright (Ed.), The international encyclopedia of the social and behavioral sciences (2nd ed., pp. 772-784). https://doi.org/10.1016/B978-0-08-097086-8.23068-7

Sembiring, R. K., Hadi, S. & Dolk, M. (2008). Reforming mathematics learning in Indonesia classrooms through RME. ZDM-The International Journal on Mathematics Education, 40(6), 927-939. https://doi.org/10.1007/s11858-008-0125-9

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2008). Elementary and middle school mathematics: Teaching developmentally (8th ed). Pearson.

Veenman, M., Van Hout-Wolters, B., & Afflerbach, P. (2006). Metacognition and learning: Conceptual and methodological considerations. Metacognition and Learning, 1(1), 3-14. https://doi.org/10.1007/s11409-006-6893-0

Zulkardi, Z. (2002). Developing a learning environment on realistic mathematics education for Indonesian student teachers [Doctoral dissertation, University of Twente]. Sriwijaya University Institutional Repository. http://doc.utwente.nl/58718/1/thesis_Zulkardi.pdf