# Relational thinking: The bridge between arithmetic and algebra

## Main Article Content

## Abstract

The purpose of this study is to investigate the development of relational thinking skill, which is an important component of the transition from arithmetic to algebra, of 5th grade students. In the study, the qualitative research method of teaching experiment was used. The research data were collected from six secondary school 5th grade students by means of clinical interviews and teaching episodes. For observing the development of relational thinking, pre and post clinical interviews were also conducted before and after the eight-session teaching experiment. Qualitative analysis of the research data revealed that the relational thinking skills of all the students developed. It was also found that there was an interaction between the development of fundamental arithmetic concepts and relational thinking; that the students developed concepts related to arithmetical operations such as addend and sum; minuend, subtrahend and difference; multiplicator and product; and dividend, divisor and quotient. Moreover, students were able to use these concepts effectively although they failed to provide formal explanations about the relations between them. In addition, the students perceived the equal sign not only finding a result but also as a symbol used to establish a relation between operations and expressions.

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## Article Details

**International Electronic Journal of Elementary Education**, [S.l.], v. 10, n. 1, p. 131-145, oct. 2017. ISSN 1307-9298. Available at: <https://iejee.com/index.php/IEJEE/article/view/305>. Date accessed: 23 oct. 2017.

## References

Baroody, A. J. & Ginsburg, H. P. (1983). The effects of instruction on children's understanding of the "equals" sign. The Elementary School Journal, 84 (2), 198-212.

Behr, M., Erlwanger, S., & Nichols E. (1980). How children view the equal sign. Mathematics Teaching, 92, 13-15.

Blanton, M.L. (2008). Algebra and the elementary classroom: transforming thinking, transforming practice. Portsmouth: Heinemann.

Boulton G. M., Lewis G., Cooper T. J., Atweh B., Pillay H. & Wills L. (2000). Readiness of algebra. In T. Nakahara & M. Koyama (Eds.), Proceeding of the 24th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 89-105). Hiroshima, Japan: International Group for the Psychology of Mathematics Education.

Cai, J. & Moyer, P. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In C. E. Greenes (Ed.), Algebra and Algebraic thinking in School Mathematics (pp.169-179). Reston, VA: NCTM.

Carpenter, T. P., & Franke, M. L. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (Vol. 1, pp. 155-162). Melbourne, Australia.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann.

Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. (Res. Rep. 00-2). Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science.

Carpenter,T. P., Levi, L., Franke, M. L., & Zeringue, J. K. (2005). Algebra in elementary school: Developing relational thinking, ZDM, 37(1), 53-59.

Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 547-589). London: Lawrence Erlbaum Associates, Publishers.

Dede Y. & Argün Z. (2003). Cebir, öğrencilere niçin zor gelmektedir? [Why do students have difficulty with algebra?] Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24, 180-185.

Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-236.

Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78.

Hunter, J. (2007). Relational or calculational thinking: Students solving open number equivalence problems. In. J. Watson & K. Beswick (Eds.), Proceeding of the 30th Annual Conference of the Mathematics Education Research Group of Australasia-Mathematics: Essential Research, Essential Practice vol. 1 (pp. 421-429). MERGA.

Kieran C. (1981) Concepts associated wıth the equality symbol. Educational Studies in Mathematics, 12, 317-326.

Kieran C. (2004). The equation/inequality connection in constructing meaning for inequality situations. Psychology of Mathematics Education, 1, 143-147.

Kieran,C. (2007). What do students struggle with when first introduced to algebra symbols? Algebra Research Brief, retrieved from NCTM: http://www.nctm.org/news/content.aspx?id=12332

Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students’ understanding of core algebraic concepts: Equivalence & Variable. ZDM, 37(1), 68-76.

Knuth, E., Stephens, A., McNeil, N., & Alibali, M. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297-312.

Knuth, E., Stephens, A., Blanton, M. & Gardiner,A. (2016). Build an early foundation for algebra success. Phi Delta Kappan, 97(6), 65-68. DOI: 10.1177/0031721716636877

Koehler, J.L. (2004). Learning to think relationally: Thinking relationally to learn (University of Wisconsin-Madison) Retrieved from ProQuest Dissertations and Theses. (AAT 3143187).

Köse, N., & Tanışlı, D. (2011). İlköğretim matematik ders kitaplarında eşit işareti ve ilişkisel düşünme [Equal sign and relational thinking in elementary mathematics textbooks]. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi (EFMED), 2(5), 251-277.

Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to Algebra: Perspectives for Research and Teaching (pp.87-106). Dordrecht, The Netherlands: Kluwer.

Li, X., Ding, M., Capraro, M. M. & Capraro, R. M. (2008). Sources of differences in children’s understanding of mathematical equality: Comparative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26, 195-217. DOI: 10.1080/07370000801980845

Matthews, P., Rittle-Johnson, B., McEldoon, K. & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43 (3), 316-350.

McNeil, N., & Alibali, M. (2005). Knowledge change as a function of mathematics experience: All contexts are not created equal. Journal of Cognition and Development, 6(2), 285-306.

McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. L. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24 (3), 367-385.

Molina, M., & Ambrose, R. C. (2006). Fostering relational thinking while negotiating the meaning of the equals sign. Teaching Children Mathematics, 13(2), 111-117.

Molina, M. & Ambrose, R. (2008). From an operational to a relational conception of the equal sign. Thirds graders’ developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1), 61-80.

Molina, M., Castro, E., & Ambrose, R. (2005). Enriching arithmetic learning by promoting relational thinking. International Journal of Learning, 12(5), 265-270.

Molina, M., Castro, E., & Mason, J. (2008). Elementary school students’ approaches to solving true/false number sentences. PNA, 2(2), 75-86.

Molina, M. & Mason, J. (2009). Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic. Canadian Journal of Science, Mathematics and Techology Education, 9 (4), 224-242. DOI: 10.1080/14926150903191885

Napaphun, V. (2012). Relational thinking: Learning arithmetic in order to promote algebraic thinking. Journal of Science and Mathematics Education in Southeast Asia, 35 (2), 84-101.

Rittle-Johnson, B. , Matthews, P. G., Taylor, R. S. & McEldoon, K. L. (2011). Assessing knowledge of mathematical eqivalence: A construct modeling approach. Journal of Educational Psychology, 103 (1), 85-104. DOI: 10.1037/a0021334

Saenz-Ludlow, A. & Walgamuth, C. (1998). Third graders’ interpretations of equality and the equal symbol. Educational Studies in Mathematics, 35(2), 153-87.

Seo, K.-H., & Ginsburg, H. P. (2003). ‘‘You’ve got to carefully read the math sentence...”: Classroom context and children’s interpretations of the equals sign. In A. J. Baroody & A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills. Mahwah, NJ: Lawrence Erlbaum.

Simon, M. A. (2000). Research on the development of mathematics teachers: The teacher development experiment. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp.335-359). London: Lawrence Erlbaum Associates Publishers

Stephens, M. (2006). Describing and exploring the power of relational thinking. Retrieved from http://www.merga.net.au/documents/RP552006.pdf

Stephens, M. & Ribeiro, A. (2012). Working towards algebra: The importance of relational thinking, Revista Latinoamericana de Investigación en Mathemática Educativa, 15 (3), 373-402.

Usiskin, Z. (1997). Doing algebra in grades K-4. In B. Moses (Ed.), Algebraic thinking, Grades K-12 (pp.5-7). Reston, VA: NCTM.

Warren, E. (2004). Generalising arithmetic: Suporting the process in the early years, In M. J. Hoines & A. B. Fuglestad (Eds.), Proceeding of the 28th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 417-424), Bergen, Norway: International Group for the Psychology of Mathematics Education.

Warren, E. (2006). Comparative mathematical language in the elementary school: A longitudinal study. Educational Studies in Mathematics, 62, 169-189. DOI: 10.1007/s10649-006-4627-5

Warren, E. (2009). Early childhood teachers' professional learning in early algebraic thinking: A model that supports new knowledge and pedagogy. Mathematics Teacher Education and Development, 10, 30-45.

Watanabe, T. (2008). Algebra in elementary school: A Japanese perspective. In C. Greenes (Ed.), Algebra and Algebraic Thinking in School Math: 70th Yearbook (pp. 183–194). Reston, VA:NCTM.

Yaman H., Toluk Z., & Olkun S. (2003). İlköğretim Öğrencileri Eşit İşaretini Nasıl Algılamaktadırlar? [How the elementary school students would perceive equal sign?] Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24, 142-151.