# Difficulties in initial algebra learning in Indonesia

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## Abstract

Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students’ achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students’ difficulties in algebra. In order to do so, a literature study was carried out on students’ difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.

## Keywords

Algebra Difficulties Indonesian students Linear equations Linear inequalities## Notes

### Acknowledgments

This study was funded by the Indonesia Ministry of Education project BERMUTU IDA CREDIT NO.4349-IND, LOAN NO.7476-IND DAN HIBAH TF090794. We would like to thank Jan van Maanen for his valuable and constructive comments and suggestions. We also thank the teachers and students for their participation, as well as the external assistant and the second coder for their contributions.

## References

- Adinawan, M. C., & Sugijono, S. (2007).
*Matematika untuk SMP kelas VII semester 1*. [Mathematics for junior secondary school grade VII semester 1.] Jakarta: Erlangga.Google Scholar - Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics.
*For the Learning of Mathematics, 25*(2), 24–35.Google Scholar - Arcavi, A. (2005). Developing and using symbol sense in mathematics.
*For the Learning of Mathematics, 14*(3), 42–47.Google Scholar - Bokhove, C., & Drijvers, P. (2010). Symbol sense behavior in digital activities.
*For the Learning of Mathematics, 30*(3), 43–49.Google Scholar - Bokhove, C. (2011).
*Use of ICT for acquiring, practicing, and assessing algebraic expertise. Dissertation*. Utrecht: Utrecht University.Google Scholar - Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A.F. Coxford (Ed.),
*The ideas of algebra, K–12(1988 Yearbook)*(pp. 20–32). Reston, VA: National Council of Teachers of MathematicsGoogle Scholar - Budhi, W. S. (2007).
*Matematika untuk SMP kelas VII semester 1.*[Mathematics for junior secondary school grade VII semester 1.] Jakarta: Erlangga.Google Scholar - De Lange, J. (1987).
*Mathematics, insight, and meaning*. Utrecht: OW & OC, Rijkuniversiteit Utrecht.Google Scholar - DEPDIKNAS (2006).
*Kurikulum tingkat satuan pendidikan sekolah menengah pertama [Curriculum of unit of education for junior secondary school.]*. Jakarta: Department of National Education.Google Scholar - Drijvers, P. H. M. (2003).
*Learning algebra in a computer algebra environment: Design research on the understanding of the concept of parameter. Dissertation*. Utrecht: CD-B Press.Google Scholar - Drijvers, P. (Ed.) (2010).
*Secondary algebra education. Revisiting topics and themes and exploring the unknown.*Rotterdam: Sense.Google Scholar - Filloy, E., & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra.
*For the Learning of Mathematics, 9*(2), 19–25.Google Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education: China lectures*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., & Brenwald, S. (2008).
*Highlights From TIMSS 2007: Mathematics and Science Achievement of U.S. Fourth-and Eighth-Grade Students in an International Context*(NCES 2009–001 Revised). Washington, DC: National Center for Education Statistics, Institute of Education Sciences, US Department of Education.Google Scholar - Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra.
*Educational Studies in Mathematics, 27*(1), 59–78.CrossRefGoogle Scholar - IEA (2003).
*TIMSS 2003 mathematics items: Released set eight grade.*Boston: International Association for the Evaluation of Educational Achievement (IEA).Google Scholar - Johar, R. (2010). PMRI in Aceh. In R. Sembiring, K. Hoogland, & M. Dolk (Eds.),
*A decade of PMRI in Indonesia*(pp. 115–122). Bandung, Utrecht: APS International.Google Scholar - Jones, I., & Pratt, D. (2012). A substituting meaning for the equal sign in arithmetic notating tasks.
*Journal for Research in Mathematics Education, 43*(1), 2–33.CrossRefGoogle Scholar - Katz, V. J. (Ed.) (2007).
*Algebra: Gateway to a technological future.*The Mathematical Association of AmericaGoogle Scholar - Kendal, M., & Stacey, K. (2004). Algebra: a world of difference. In K. Stacey, H. Chick, & M. Kendal (Eds.),
*The future of the teaching and learning of algebra: The 12th ICMI Study*(pp. 329–346). Dordrecht: Kluwer Academic Publishers.Google Scholar - Ketterlin-Geller, L. R., Jungjohann, K., & Chard, D. J. (2007). From arithmetic to algebra.
*Educational Leadership, 65*(3), 66–71.Google Scholar - Kieran, C. (1981). Concepts associated with the equality symbol.
*Educational Studies in Mathematics, 12*(3), 317–326.CrossRefGoogle Scholar - Kolovou, A. (2011).
*Mathematical problem solving in primary school. Dissertation*. Utrecht: Utrecht University.Google Scholar - Kolovou, A., Van den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks-A needle in a haystack.
*Mediterranean Journal for Research in Mathematics Education, 8*(2), 29–66.Google Scholar - Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data.
*Biometrics, 33*(1), 159–174.CrossRefGoogle Scholar - Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: a definition of pre-algebra.
*Journal of Mathematical Behavior, 14*(1), 113–120.CrossRefGoogle Scholar - Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: operating on the unknown in the context of equations.
*Educational Studies in Mathematics, 30*(1), 39–65.CrossRefGoogle Scholar - MacGregor, M., & Stacey, K. (1998). Cognitive models underlying algebraic and non-algebraic solutions to unequal partition problems.
*Mathematics Education Research Journal, 10*(2), 46–60.CrossRefGoogle Scholar - Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., Gregory, K. D., Garden, R. A., O’Connor, K. M., Chrostowski, S. J., & Smith, T. A. (2000).
*TIMSS 1999 international mathematics report: Findings from IEA’s repeat of the third international mathematics and science study at the eighth grade*. Boston: International Study Center Lynch School of Education.Google Scholar - Mullis, I. V. S., Martin, M. O., Foy, P., in collaboration with Olson, J. F., Preuschoff, C., Erberber, E., Arora, A., & Galia, J. (2008).
*TIMSS 2007 international mathematics report: Findings from IEA’s trends in international mathematics and science study at the fourth and eighth grades*. Boston: TIMSS & PIRLS International Study CenterGoogle Scholar - Nuharini, D., & Wahyuni, T. (2008).
*Matematika konsep dan aplikasinya untuk kelas VII SMP dan MTs [Mathematics concepts and its applications for grade VII junior secondary school]*. Jakarta: Book Publishing Center, Department of National Education.Google Scholar - OECD (2006).
*PISA released items-mathematics*. Paris: OECD Publishing.Google Scholar - OECD (2010). PISA 2009 results: What students know and can do—Student performance in reading, mathematics and science (Volume 1). Retrieved March 9, 2011, from http://dx.doi.org/10.1787/9789264091450-en
- Pillay, H., Wilss, L., & Boulton-Lewis, G. (1998). Sequential development of algebra knowledge: a cognitive analysis.
*Mathematics Education Research Journal, 10*(2), 87–102.CrossRefGoogle Scholar - Polya, G. (1973).
*How to solve it: a new aspect of mathematical method*(2nd ed.). Princeton: Princeton University Press.Google Scholar - Rosnick, P. (1981). Some misconceptions concerning the concept of variable.
*Are you careful defining your variables? Mathematics Teacher, 74*(6), 418–420.Google Scholar - Saenz-Ludlow, A., & Walgamuth, C. (1998). Third graders’ interpretations of equality and the equal symbol.
*Educational Studies in Mathematics, 35*(2), 153–187.CrossRefGoogle Scholar - Schmidt, W. H., McKnight, C. C., Valverde, G. A., Houang, R. T., & Wiley, D. E. (1997).
*Many visions, many aims: A cross-national investigation of curricular intentions in school mathematics (Vol. 1)*. Dordrecht: Kluwer.Google Scholar - Sembiring, R. K., Hadi, S., & Dolk, M. (2008). Reforming mathematics learning in Indonesian classrooms through RME.
*ZDM, The International Journal on Mathematics Education, 40*(6), 927–939.CrossRefGoogle Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*(1), 1–36.CrossRefGoogle Scholar - Stacey, K., Chick, H., & Kendal, M. (2004).
*The future of the teaching and learning of algebra: the 12th ICMI Study*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Thomas, M., & Tall, D. (1991). Encouraging versatile thinking in algebra using the computer.
*Educational Studies in Mathematics, 22*(2), 125–147.CrossRefGoogle Scholar - Treffers, A. (1987).
*Three dimensions. A model of goal and theory description in mathematics instruction-The Wiskobas project*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford (Ed.),
*The ideas of algebra, K-12*(pp. 8–19). Reston: National Council of Teachers of Mathematics.Google Scholar - Van Amerom, B. A. (2002).
*Reinvention of early algebra: developmental research on the transition from arithmetic to algebra. Dissertation*. Utrecht: CD-B Press.Google Scholar - Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra.
*Educational Studies in Mathematics, 54*(1), 63–75.CrossRefGoogle Scholar - Van den Heuvel-Panhuizen, M. (2000).
*Mathematics education in the Netherlands: a guided tour. Freudenthal Institute Cd-rom for ICME9*. Utrecht: Utrecht University.Google Scholar - Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: an example from a longitudinal trajectory on percentage.
*Educational Studies in Mathematics, 54*(1), 9–35.CrossRefGoogle Scholar - Wagiyo, A., Surati, F., & Supradiarini, I. (2008).
*Pegangan belajar matematika untuk SMP/MTs kelas VII. [Handbook of learning mathematics for junior secondary school grade VII.]*. Jakarta: Book Publishing Center, Department of National Education.Google Scholar - Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra.
*Mathematics Education Research Journal, 15*(2), 122–137.CrossRefGoogle Scholar - Watson, A. (2009).
*Algebraic reasoning. Key understanding in mathematics learning.*University of Oxford: Nuffield Foundation.Google Scholar - Zulkardi (2002).
*Developing a learning environment on realistic mathematics education for Indonesian student teachers. Dissertation*. Enschede: University of Twente.Google Scholar