# Longer bars for bigger numbers? children’s usage and understanding of graphical representations of algebraic problems

## Main Article Content

## Abstract

In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students’ understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also examined whether usage and understanding varied with grade. Secondary 2 (N = 68, M_{age} = 13.9 years) and Primary 5 students (N = 110, M_{age} = 11.1 years) were administered a production task in which they drew bar diagrams of algebraic word problems with operands of varying magnitude. In the validation task, they were presented with different bar diagrams for the same word problems and were asked to ascertain, and give explanations regarding the accuracy of the diagrams. The Küchemann algebra test was administered to the Secondary 2 students. Students from both grades drew longer bars to represent larger numbers. In contrast, findings from the validation task showed a more abstract appreciation for how the bar diagrams can be used. Primary 5 students who showed more abstract appreciations in the validation task were less likely to use the bar diagrams in a concrete fashion in the production task. Performance on the Küchemann algebra test was unrelated to performance on the production task or the validation task. The findings are discussed in terms of a production deficit, with students exhibiting a more sophisticated understanding of bar diagrams than is demonstrated by their usage.

## Article Details

*Frontline Learning Research*,

*1*(1), 81-96. https://doi.org/10.14786/flr.v1i1.49

FLR adopts the Attribution-NonCommercial-NoDerivs Creative Common License (BY-NC-ND). That is, Copyright for articles published in this journal is retained by the authors with, however, first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.

## References

Booth, J. L., & Koedinger, K. R. (2010). Facilitating low-achieving students’ diagram use in algebraic story problems. In S. Ohlsson & R. Catrambone (Eds.), Proceedings of the 32nd Annual Meeting of the Cognitive Science Society (pp. 1649-1654). Austin, TX: Cognitive Science Society.

Brown, M., Hart, K., & Kuchemann, D. (1985). Chelsea diagnostic mathematics tests and teacher's guide. Windsor: NFER-NELSON Publishing Company Ltd.

Capraro, M. M., & Joffrion, H. (2006). Algebraic Equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27(2-3), 147-164. doi: 10.1080/02702710600642467

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. Retrieved from http://ncisla.wceruw.org/publications/reports/RR-002.PDF

Carraher, D. W., Schliemann, A., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and Algebra in Early Mathematics Education. Journal for Research in Mathematics Education, 37(2), 87-115.

Dede, Y. (2004). The concept of variable and identification its learning difficulties. Educational Sciences: Theory & Practice, 4(1), 50.

Duru, A. (2011). Middle school students’ reading comprehension of mathematical texts and algebraic equations. International Journal of Mathematical Education in Science and Technology, 42(4), 447-468. doi: 10.1080/0020739x.2010.550938

Flavell, J. H. (1970). Developmental studies of mediated memory. In H. W. Reese & L. P. Lipsitt (Eds.), Advances in child development and child behavior (Vol. 5). New York: Academic Press.

Fuchs, L. S., Compton, D. L., Fuchs, D., Powell, S. R., Schumacher, R. F., Hamlett, C. L., et al. (2012). Contributions of Domain-General Cognitive Resources and Different Forms of Arithmetic Development to Pre-Algebraic Knowledge. Developmental Psychology, 48(5), 1315-1326. doi: 10.1037/a0027475

Harnishfeger, K. K., & Bjorklund, D. F. (1990). Children’s strategies: A brief history. In D. F. Bjorklund (Eds.), Children’s strategies: Contemporary views of cognitive development. Hillsdale, NJ: Lawrence Erlbaum Associates.

Hefferman, N., & Koedinger, K. R. (1997). The composition effect in symbolizing: The role of symbol production versus text comprehension. In M. G. Shafto & P. Langley (Eds.), Proceedings of the Nineteenth Annual Conference of the Cognitive Science Society (pp. 307-312). Mahwah, NJ: Lawrence Erlbaum Associates.

Hu, W. (2010). Making Math Lessons as Easy as 1, Pause, 2, Pause ... The New York Times. Retrieved from http://www.nytimes.com/2010/10/01/education/01math.html?_r=0

Hunter, J. (2007). Relational or calculational thinking: students solving open number equivalence problems. In J. Watson & K. Beswick (Eds.), Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 421-429). Adelaide: MERGA.

Khng, K. H., & Lee, K. (2009). Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving. Learning and Individual Differences, 19(2), 262-268. doi: 10.1016/j.lindif.2009.01.004

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317-326. doi: 10.1007/bf00311062

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

Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving. Cognitive Science: A Multidisciplinary Journal, 32(2), 366-397. doi: 10.1080/03640210701863933

Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129-164.

Koedinger, K. R., & Terao, A. (2002). A cognitive task analysis of using pictures to support pre-algebraic reasoning. In C.D. Schunn & W. Gray (Eds.), Proceedings of the Twenty-Fourth Annual Conference of the Cognitive Science Society (pp. 542-547). Mahwah, NJ: Lawrence Erlbaum Associates.

Küchemann, D. (1978). Children's understanding of numerical variables. Mathematics in School, 7(4), 23-26.

Kwokwc. (2011). Not able to use algebra in primary level. Retrieved from http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=27&t=18728

Lee, K., Lim, Z. Y., Yeong, S. H., Ng, S. F., Venkatraman, V., & Chee, M. W. (2007). Strategic differences in algebraic problem solving: neuroanatomical correlates. Brain Research, 1155 (June), 163-171. doi: 10.1016/j.brainres.2007.04.040

Lee, K., Ng, S. F., Bull, R., Pe, M. L., & Ho, R. H. M. (2011). Are patterns important? An investigation of the relationships between proficiencies in patterns, computation, executive functioning, and algebraic word problems. Journal of Educational Psychology, 103(2), 269-281. doi: 10.1037/a0023068

Lee, K., Yeong, S. H. M., Ng, S. F., Venkatraman, V., Graham, S., & Chee, M. W. L. (2010). Computing solutions to algebraic problems using a symbolic versus a schematic strategy. ZDM, 42(6), 591-605. doi: 10.1007/s11858-010-0265-6

Lim, B. T. (2007). Can algebra be used to solve PSLE maths problems, The Strait Times. Retrieved from http://www.moe.gov.sg/media/forum/2007/forum_letters/20070217.pdf

MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33(1), 1-19. doi: 10.1023/a:1002970913563

Mayer, R. E. (1989). Systematic thinking fostered by illustrations in scientific text. Journal of Educational Psychology, 81(2), 240.

Mayer, R. E. (2002). Multimedia learning. Psychology of Learning and Motivation, 41, 85-139.

Meter, P., & Garner, J. (2005). The Promise and Practice of Learner-Generated Drawing: Literature Review and Synthesis. Educational Psychology Review, 17(4), 285-325. doi: 10.1007/s10648-005-8136-3

Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., & Chrostowski, S. J. (2004). TIMSS 2003 International Mathematics Report: Findings From IEA's Trends in International Mathematics and Science Study at the Fourth and Eighth Grades. Chestnut Hill, MA: Boston College.

Nathan, M. J., & Koedinger, K. R. (2000). Teachers' and researchers' beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31(2), 168-190. doi: 10.2307/749750

National Council for Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: National Council for Teachers of Mathematics.

National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

Ng, S. F. (2003). How secondary two express stream students used algebra and the model method to solve problems. The Mathematics Educator, 7(1), 1-17.

Ng, S. F., & Lee, K. (2005). How primary five pupils use the model method to solve word problems. The Mathematics Educator, 9(1), 60-84.

Ng, S. F., & Lee, K. (2008). As Long As the Drawing is Logical, Size Does Not Matter. The Korean Journal of Thinking & Problem Solving, 18(1), 67-82.

Ng, S. F., & Lee, K. (2009). Model method: Singapore children's tool for representing and solving algebra word problems. Journal for Research in Mathematics Education, 40(3), 282-313.

Ng, S. F., Lee, K., Ang, S. Y., & Khng, F. (2006). Model Method: Obstacle or bridge to learning symbolic algebra. In W. Bokhorst-Heng, M. Osborne & K. Lee (Eds.), Redesigning Pedagogies (pp. 227-242). NY: Sense.

OECD (2010). PISA 2009 Results: Executive Summary. Retrieved from http://www.oecd.org/pisa/pisaproducts/46619703.pdf

Philipp, R. (1992). The many uses of algebraic variables. Mathematics Teacher, 85, 557-561.

Stacey, K., & MacGregor, M. (1999). Learning the Algebraic Method of Solving Problems. The Journal of Mathematical Behavior, 18(2), 149-167.

Steinberg, R. M., Sleeman, D. H., & Ktorza, D. (1991). Algebra students' knowledge of equivalence of equations. Journal for Research in Mathematics Education, 22(2), 112-121.

Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students' preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112.

Tolar, T. D., Lederberg, A. R., & Fletcher, J. M. (2009). A structural model of algebra achievement: computational fluency and spatial visualisation as mediators of the effect of working memory on algebra achievement. Educational Psychology: An International Journal of Experimental Educational Psychology, 29(2), 239-266.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Schulte (Eds.), The Ideas of Algebra (pp. 8-19). Reston, VA: National Council of Teachers of Mathematics.

Warren, E., & Cooper, T. (2005). Introducing functional thinking in year 2: a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6(2), 150-162.

Warren, E., & Cooper, T. J. (2009). Developing mathematics understanding and abstraction: the case of equivalence in the elementary years. Mathematics Education Research Journal, 21(2), 76-95.

Wei, W., Yuan, H. B., Chen, C. S., & Zhou, X. L. (2012). Cognitive correlates of performance in advanced mathematics. British Journal of Educational Psychology, 82(1), 157-181. doi: 10.1111/j.2044-8279.2011.02049.x