Puspita Sari, Mimi Nur Hajizah, Swida Purwanto


The number line and the neutralization model have been used very extensively in teaching integer additions and subtractions for decades. Despite their advantages, issues concerning subtractions on these models are still debatable. Therefore, the neutralization on an empty number line (NNL) model is proposed in this research to help students better understand the meaning of integer subtractions as well as additions. This report is a part of a design research study conducted in a classroom of 28 elementary school students at the fifth grade. Data were gathered by collecting students’ written work, conducting interviews and observations during the teaching experiment. This paper focuses on students’ perceptions of the NNL model and what factors that might contribute to students’ achievements in understanding integer additions and subtractions. The analysis revealed that most students overemphasized on the use of the NNL model as a procedural method instead of as a model for thinking. Moreover, students’ mathematical beliefs and conceptions on the use of the column strategy and the absence of a discussion on the need of using the model are found to be some factors that could cause students’ misunderstandings. However, with a thorough planning, the NNL model has a potential to help students developing a meaning of integer additions and subtractions.


Addition; Subtraction; Negative; Neutralization on an Empty Number Line (NNL) Model

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Aris, R.M., Putri, R.I.I., & Susanti, E. (2017). Design study: Integer subtraction operation teaching learning using multimedia in primary school. Journal on Mathematics Education, 8(1), 95-102.

Bakker, A. (2004), Design Research in Statistic Education on Symbolizing and Computer Tools. Utrecht: Cd-ß Press.

Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194-245.

Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.

Cobb, P., Jackson, K., & Dunlap, C. (2016). Design research: An Analysis and Critique. Handbook of International Research in Mathematics Education Third Edition (pp. 481-503). New York: Routledge.

Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practice. The Journal of the Learning Science, 10(1&2), 113-163.

Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrecht: D. Reidel Publishing.

Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Dordrecht: Kluwer Academic Publisher.

Fuson, K.C., Wearne, D., Hiebert, J.C., Murray, H.G., Human, P.G., Olivier, A.I., Carpenter, T.P., & Fennema, E. (1997). Children's conceptual structures for multi digit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130-162.

Gravemeijer, K., & Cobb, P. (2006). Design Research from a Learning Design Perspective, Educational Design Research. London and New York: Routledge.

Gravemeijer, K., & Terwel, J. (2000). Hans Freudenthal: A mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777-796.

Küchemann, D. (1981). Positive and negative numbers. In K.M. Hart (Ed.), Children's Understanding of Mathematics: 11-16 (pp. 82–87). London: John Murray.

Lesh, R., & Doerr, H.M. (2000). Symbolizing, communicating, and mathematizing: Key components of models and modeling. In Cobb, P., Yackel, E., & Mc Clain, K. (Eds), Symbolizing and Communicating in Mathematics Classroom, Perspectives on Discourses, Tools, and Instructional Design (pp. 361-385). New Jersey: LEA Publishers.

Liebeck, P. (1990). Scores and forfeits—An intuitive model for integer arithmetic. Educational Studies in Mathematics, 21(3), 221-239.

McKenney S., & Reeves T.C. (2014). Educational design research. In: Spector J., Merrill M., Elen J., Bishop M. (eds). Handbook of Research on Educational Communications and Technology, (pp. 131-140). New York: Springer.

Muslimin, Putri, R.I.I., & Somakim. (2012). An instructional design on subtraction of integers by traditional game ‘congklak’ based on realistic mathematics education in Indonesia at the 4th grade elementary school [in Bahasa]. Jurnal Kreano, 3(2), 100-112.

NCTM. (2000). Principles and Standards for School Mathematics. United States of America: The National Council of Teachers of Mathematics, Inc.

Prahmana, R.C.I. (2017). The hypothetical learning trajectory on addition in mathematics GASING. Southeast Asian Mathematics Education Journal, 5(1), 49-61.

Prahmana, R.C.I., Zulkardi, & Hartono, Y. (2012). Learning multiplication using Indonesian traditional game in third grade. Journal on Mathematics Education, 3(2), 115-132.

Sahat, N., Tengah, K.A., & Prahmana, R.C.I. (2018). The teaching and learning of addition and subtraction of integers through manipulative in Brunei Darussalam. Journal of Physics: Conference Series, 1088(1), 012024.

Sari, P., Purwanto, S., & Hajizah, M.N. (2019). The ‘Neutralization on a Number Line’ (NNL) model for integer addition and subtraction. In Y. Rahmawati & P. Taylor (Eds.), Empowering Science and Mathematics for Global Competitiveness (pp. 495–504). London: CRC Press.

Shanty, N.O. (2016). Investigating students’ development of learning integer concept and integer addition. Journal on Mathematics Education, 7(2), 57-72.

Shutler, P.M.E. (2017). A symbolical approach to negative numbers. The Mathematics Enthusiast, 14(1), 207-240.

Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.

Simon, M.A. (2018). An emerging methodology for studying mathematics concept learning and instructional design. The Journal of Mathematical Behavior, 52(December), 113-121.

Steiner, C.J. (2009). A study of pre-service elementary teachers’ conceptual understanding of integers. Electronic Dissertation. Ohio: Kent University.!etd.send_file?accession=kent1248466399&disposition=inline.

Stephan, M. & Cobb, P. (2013). Teachers engaging in mathematics design research. In T.Plomp, & N.Nieveen (Eds.), Educational Design Research – Part B: Illustrative Cases (pp. 277-298). Enschede: SLO.

Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464.

Teppo, A., van den Heuvel-Panhuizen. M. (2014). Visual representations as objects of analysis: The number line as an example. ZDM: The International Journal on Mathematics Education, 46(1), 45-58.

Treffers, A. (1987). Three Dimensions (A Model of Goal and Theory Description in Mathematics Instruction – The Wiskobas Project). Dordrecht, Boston, Lancaster, Tokyo: D. Reidel Publishing Company.

Van de Walle, J.A. (2004). Elementary and Middle School Mathematics: Teaching Developmentally Fifth Edition. Boston: Pearson.

Whitacre, I., Bishop, J.P., Lamb, L.L.C., Philipp, R.A., Schappelle, B.P., & Lewis, M.L. (2012). Happy and sad thoughts: An exploration of children’s integer reasoning. Journal of Mathematical Behavior, 31(3), 356-365.



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Journal on Mathematics Education
Doctoral Program on Mathematics Education
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