A Learning Trajectory of Indonesian 12-years Old Students Understanding of Division of Fractions

Achmad Nizar, Siti Maghfirotun Amin, Agung Lukito


The purpose of this study was to describe mathematical hands-on activities that can support students to gain better understanding of dividing fractions. This preliminary research phase was started by testing, analyzing, and refining the initial hypothetical learning trajectory (HLT), then in the pilot experimental phase the revised HLT was implemented, and ended with the teaching experimental phase by developing a learning trajectory for 12-year old students in understanding division of fractions. In developing the trajectory, a design research methodology was employed by using four contextual-based learning series (sharing biscuit, sharing remaining chocolate bar, arranging bedroom mats, and running around school yard), including providing some concrete materials or pictorial models as manipulative tools. Seven mathematics experts and twenty five 12-year old students were involved during the research. The four designed learning goals were determining the quotient of division of integer by proper fraction, proper fraction by integer, two proper fractions, and two fractions. Students succeeded in demonstrating their understanding and stated that 16:1/2 = 32; 25/36:5 = 5/36; 24/64:1/2 = 6/8; and 15/2:3/4 = 10 respectively at the end of each designed activity. The interesting results of this study are not a proof, so that a much larger study is needed to determine if the results are due to this approach or due to the teachers’ enthusiasm or what is known as the Hawthorne Effect.


division of fractions; learning trajectory; understanding; contextual learning; design research

Full Text:



Armanto, D. (2002). Teaching multiplication and division realistically in Indonesia primary schools: A prototype of local instructional theory (Doctoral’s dissertation). University of Twente, Endschede.

Cramer, K., Monson, D., Whitney, S., Leavitt, S., & Wyberg, T. (2010). Dividing fractions and problem solving. Mathematics Teaching in the Middle School, 15(6), 338-346.

Gravemeijer, K. (2004a). Creating opportunities for students to reinvent mathematics. Paper presented at the 10th International Congress on Mathematical Education (ICME), Copenhagen, Denmark.

Gravemeijer, K. (2004b). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105-128. https://doi.org/10.1207/s15327833mtl0602_3

Gregg, J., & Gregg, D. U. (2007). Measurement and fair-sharing models for dividing fractions. Mathematics Teaching in the Middle School, 12(9), 490-496.

Holisin, I. (2002). Pembelajaran pembagian pecahan di SD dengan menggunakan pendekatan konkrit dan semikonkrit (Master’s thesis). Universitas Negeri Surabaya, Surabaya.

Meng, W. L., & Sivasubramaniam, P. A. P. (2009). The conceptual understanding of fractions as part of a set among year 5 students. Paper presented at the 3rd International Conference on Science and Mathematics Education (CoSMEd), Penang, Malaysia.

Rifa’at, M., Parwati, N. N., & Tambelu, J. V. A. (1996). Pembelajaran pengertian pecahan bersama siswa kelas satu SMP IKIP Malang. Forum Penelitian Kependidikan, 8, 71-82.

Saondi, O. (2011). Psikologi belajar matematika [blog post]. Retrieved from https://mathedu08.files.wordpress.com/2010/05/psikologi-belajar-matematikadiktat.doc

Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks on conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91-104. https://doi.org/10.1207/ s15327833mtl0602_2

Skemp, R. R. (1987). The psychology of learning mathematics. New York: Routledge.

Valanides, N. (1998). Formal operational performance and achievement of lower secondary school students. Studies in Educational Evaluation, 24(1), 1-23. https://doi.org/10.1016/s0191-491x(98)00001-7

Winarti, D. W. (2011). Learning the concepts of area and perimeter by exploring their relation (Master’s thesis). Universitas Negeri Surabaya, Surabaya.

Worthington, M. (2005). Reflecting on creativity and cognitive challenge: visual representations and mathematics in early childhood - some evidence from research. Retrieved from http://www.tactyc.org.uk/pdfs/Reflection-worthing ton.pdf

Zulkardi. (2000). Realistic mathematics education theory meets web technology. Paper presented at the National Conference on Mathematics, Bandung, Indonesia.

DOI: https://doi.org/10.46517/seamej.v7i2.52


  • There are currently no refbacks.

Indexed by:


Southeast Asian Mathematics Education Journal
SEAMEO Regional Centre for QITEP in Mathematics
Jl. Kaliurang Km 6, Sambisari, Condongcatur, Depok, Sleman
Yogyakarta, Indonesia
Telp. +62 274 889955
Email: seamej@qitepinmath.org

p-ISSN: 2089-4716 | e-ISSN: 2721-8546

Southeast Asian Mathematics Education Journal is licensed under a Creative Commons Attribution 4.0 International License

View My Stats

Supported by: