Article Sidebar

Download
PDF
Statistic
Read Counter : 678 Download : 352

Downloads

Download data is not yet available.

Main Article Content

Abstract

Proportional reasoning has been greatly influencing the development of students’ mathematical abilities. Along with the area conservation ability, it helps elementary students comprehend area measurement. This exploratory study aimed to produce qualitative-descriptive data on elementary students’ proportional reasoning in solving the conservation of plane figures. The study used guided-unguided area conservation problems using a proportional reasoning level as the analysis framework. Data were collected from 4 primary school students in Sidoarjo, Indonesia, who were in fifth-grade class. The students' strategies used were identified to analyze the students' proportional reasoning in solving area conservation. Results show that the level of proportional reasoning varies from zero to two. Regarding the students' proportional reasoning levels, most of the students' strategies use visual clues and cute paste strategies. Only one student can reach the level of quantitative reasoning by using a formula to compare both area measurements. Interestingly, the problem of the conservation of plane figures failed to reveal students' formal proportional reasoning due to their insufficient knowledge of fractions, division, multiplication, and decimals. Some implications regarding students' proportional reasoning and interventions in the area conservation problem are discussed.

Keywords

Area Conservation Guided-Unguided Problems Part-Whole Relation Proportional Reasoning

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite
Sari, Y. M., Fiangga, S., Milla, Y. I. E., & Puspaningtyas, N. D. (2023). Exploring students’ proportional reasoning in solving guided-unguided area conservation problem: A case of Indonesian students. Journal on Mathematics Education, 14(2), 375–394. https://doi.org/10.22342/jme.v14i2.pp375-394
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Abreu-Mendoza, R. A., Coulanges, L., Ali, K., Powell, A. B., & Rosenberg-Lee, M. (2021). From Non-Symbolic to Symbolic Proportions and Back: A Cuisenaire Rod Proportional Reasoning Intervention Enhances Continuous Proportional Reasoning Skills. Frontiers in Psychology, 12, 633077. https://doi.org/10.3389/fpsyg.2021.633077
  2. ACARA. (2019). Australian Curriculum: Mathematics. Australian Curriculum, Assessment and Reporting Authority (ACARA). https://www.australiancurriculum.edu.au/
  3. Aktaș, M. C. (2022). A Comparison of Solution Strategies for Proportional and Non-Proportional Problems of Students at Different Education Levels: A Cross-Sectional Study. International Journal of Education Technology and Scientific Researches, 7(18), 1064–1082. https://doi.org/10.35826/ijetsar.484
  4. Ayan, R., & Isiksal-Bostan, M. (2018). Middle School Students’ Proportional Reasoning in Real Life Contexts in the Domain of Geometry and Measurement. International Journal of Mathematical Education in Science and Technology, 50(1), 65–81. https://doi.org/10.1080/0020739X.2018.1468042
  5. Beckmann, S., & Izsák, A. (2015). Two Perspectives on Proportional Relationships: Extending Complementary Origins of Multiplication in Terms of Quantities. Journal for Research in Mathematics Education, 46(1), 17–38. https://doi.org/10.5951/jresematheduc.46.1.0017
  6. Ben-Chaim, D., Keret, Y., & Ilany, B.-S. (2012). Ratio and Proportion: Research and Teaching in Mathematics Teachers’ Education (Pre- and In-Service Mathematics Teachers of Elementary and Middle School Classes). Sense Publishers.
  7. Beswick, K. (2011). Putting Context In Context: An Examination of the Evidence for the Benefits of “Contextualised” Tasks. International Journal of Science and Mathematics Education, 9, 367–390. http://dx.doi.org/10.1007/s10763-011-9323-y
  8. Boyer, T. W., & Levine, S. C. (2012). Child Proportional Scaling: Is 1/3 = 2/6 = 3/9 = 4/12? Journal of Experimental Child Psychology, 111(3), 516–533. https://doi.org/10.1016/j.jecp.2011.11.001
  9. Boyer, T. W., & Levine, S. C. (2015). Prompting Children to Reason Proportionally: Processing Discrete Units as Continuous Amounts. Developmental Psychology, 51(5), 615–620. https://doi.org/10.1037/a0039010
  10. Christodoulou, T. (2022). Why are the Students Unable to Distinguish the Proportional from the Non-Proportional Mathematical Situations? A Review of the Relative Research on the Illusion of Proportionality. Mediterranean Journal for Research in Mathematics Education, 19, 37–54.
  11. Clements, D. H., & Stephan, M. (2004). Measurement in Pre-K to Grade 2 Mathematics. In D. H. Clements & J. Sarama (Eds.), Engaging Young Children in Mathematics (pp. 299–317). Lawrence Erlbaum Associates.
  12. Cohen, L., Manion, L., & Morrison, K. (2007). Research Methods in Education (6th ed.). Routledge.
  13. Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. T. Owens (Ed.), Research Ideas for the Classroom: Middle Grades Mathematics (pp. 159–178). Macmillan.
  14. Cramer, K., & Post, T. (1993). Making Connections: A Case For Proportionality. The Arithmetic Teacher, 40(6), 342–346. https://doi.org/10.5951/AT.40.6.0342
  15. Creswell, J. W. (2013). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches (4th ed.). Sage Publications Inc.
  16. Ekawati, R., Kohar, A. W., Imah, E. M., Amin, S. M., & Fiangga, S. (2019). Students’ Cognitive Processes in Solving Problem Related to the Concept of Area Conservation. Journal on Mathematics Education, 10(1), 21–36. https://doi.org/10.22342/jme.10.1.6339.21-36
  17. Ergül, A., & Artan, İ. (2017). Children Explain Their Reasoning about Measurement: “Because It’s Big and Heavy!” Journal of Education and Practice, 8(22), 47–57.
  18. Fernández-Verdú, C., Llinares, S., Modestou, M., & Gagatsis, A. (2010). Proportional Reasoning: How Task Variables Influence the Development of Students’ Strategies from Primary to Secondary School. Acta Didactica Universitatis Comenianae (ADUC) - Mathematics, 10, 1–18.
  19. Fiangga, S. (2013a). Designing Tangram Game Activity as an Introduction to the Concept of Area Conservation in the Topic of Area Measurement [Master Thesis]. State University of Surabaya.
  20. Fiangga, S. (2013b). First Cycle on Designing the Tangram Game Activities as an Introduction to the Concept of Area Conservation: Game Activity for 3rd Grade (9-10 Years Old). In Zulkardi (Ed.), The First South East Asia Design/Development Research (SEA-DR) International Conference (pp. 409–415).
  21. Fiangga, S. (2014). Tangram Game Activities, Helping the Students Difficulty in Understanding the Concept of Area Conservation Paper Title. Proceedings of International Conference on Research, Implementation, and Education of Mathematics and Sciences, 453–460.
  22. Françoise, T. (1986). Proportions in Elementary School. Educational Studies in Mathematics, 17, 401–412.
  23. Funny, R. A. (2013). Developing Students’ Understanding of the Concept of Conservation of Area as a Preparatory for Learning Area Measurement [Dissertation]. State University of Surabaya.
  24. Funny, R. A. (2014). Students’ Initial Understanding of the Concept of Conservation of Area. Journal on Mathematics Education, 5(1), 57–65. https://doi.org/10.22342/jme.5.1.1449.57-65
  25. Gouet, C., Carvajal, S., Halberda, J., & Peña, M. (2020). Training Nonsymbolic Proportional Reasoning in Children and Its Effects on Their Symbolic Math Abilities. Cognition, 197, 104154. https://doi.org/10.1016/j.cognition.2019.104154
  26. He, W., Yang, Y., & Gao, D. (2018). Proportional Reasoning in 5- to 6-Year-Olds. Journal of Cognition and Development, 19(4), 389–412. https://doi.org/10.1080/15248372.2018.1495218
  27. Hsu, L. M., & Field, R. (2003). Interrater Agreement Measures: Comments on Kappa n, Cohen’s Kappa, Scott’s π, and Aickin’s α. Understanding Statistics, 2(3), 205–219. https://dx.doi.org/10.1207/s15328031us0203_03
  28. Huang, H.-M. E., & Witz, K. G. (2013). Children’s Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems. Journal of Curriculum and Teaching, 2(1), 10–26. http://dx.doi.org/10.5430/jct.v2n1p10
  29. Jacob, S. N., Vallentin, D., & Nieder, A. (2012). Relating Magnitudes: The Brain’s Code for Proportions. Trends in Cognitive Sciences, 16(3), 157–166. https://doi.org/10.1016/j.tics.2012.02.002
  30. Johar, R., Yusniarti, S., & Saminan. (2018). The Analysis of Proportional Reasoning Problem in the Indonesian Mathematics Textbook for the Junior High School. Journal on Mathematics Education, 9(1), 55–68. https://doi.org/10.22342/jme.9.1.4145.55-68
  31. Kaitera, S., & Harmoinen, S. (2022). Developing Mathematical Problem-Solving Skills in Primary School by Using Visual Representations on Heuristics. Lumat Special Issue: Mathematical Thinking and Understanding in Learning of Mathematics, 10(2), 111–146. https://doi.org/10.31129/LUMAT.10.2.1696
  32. Karplus, R., Pulos, S., & Stage, E. K. (1983). Early Adolescents’ Proportional Reasoning on ‘Rate’ Problems. Educational Studies in Mathematics, 14, 219–233. https://doi.org/10.1007/BF00410539
  33. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press.
  34. Kordaki, M., & Balomenou, A. (2006). Challenging Students to View the Concept of Area in Triangles in a Broad Context: Exploiting the Features of Cabri-II. International Journal of Computers for Mathematical Learning, 11(1), 99–135. https://doi.org/10.1007/s10758-005-5380-z
  35. Kordaki, M., & Potari, D. (2002). The Effect of Area Measurement Tools on Student Strategies: The Role of a Computer Microworld. International Journal of Computers for Mathematical Learning, 7(1), 65–100. https://doi.org/10.1023/A:1016051411284
  36. Kospentaris, G., Spyrou, P., & Lappas, D. (2011). Exploring Students’ Strategies in Area Conservation Geometrical Tasks. Educational Studies in Mathematics, 77(1), 105–127. https://doi.org/10.1007/s10649-011-9303-8
  37. Langrall, C. W., & Swafford, J. (2000). Three Balloons for Two Dollars: Developing Proportional Reasoning. Mathematics Teaching in the Middle School, 6(4), 254–261.
  38. Lesh, R., Post, T. R., & Behr, M. (1988). Proportional Reasoning. In Number Concepts and Operations in the Middle Grades (pp. 93–118). Lawrence Erlbaum & National Council of Teachers of Mathematics.
  39. Lim, K. H. (2009). Burning the Candle at just One End: Using Non Proportional Examples Helps Students Determine when Proportional Strategies Apply. Mathematics Teaching in the Middle School, 14(8), 492–500.
  40. Lo, J.-J., & Watanabe, T. (1997). Developing Ratio and Proportion Schemes: A Story of a Fifth. Journal for Research in Mathematics Education, 28(2), 216–236. https://doi.org/10.2307/749762
  41. Lourenco, S. F., Bonny, J. W., Fernandez, E. P., & Rao, S. (2012). Nonsymbolic Number and Cumulative Area Representations Contribute Shared and Unique Variance to Symbolic Math Competence. Proceedings of the National Academy of Sciences of the United States of America, 109(46), 18737–18742. https://www.pnas.org/cgi/doi/10.1073/pnas.1207212109
  42. Maschietto, M., & Bartolini Bussi, M. G. (2009). Working with Artefacts: Gestures, Drawings and Speech in the Construction of the Mathematical Meaning of the Visual Pyramid. Educational Studies in Mathematics, 70(2), 143–157. https://doi.org/10.1007/s10649-008-9162-0
  43. Miles, M. B., & Huberman, A. M. (1994). An Expanded Sourcebook: Qualitative Data Analysis (2nd ed.). Sage Publications.
  44. Möhring, W., Newcombe, N. S., & Frick, A. (2015). The Relation between Spatial Thinking and Proportional Reasoning in Preschoolers. Journal of Experimental Child Psychology, 132, 213–220. https://doi.org/10.1016/j.jecp.2015.01.005
  45. Möhring, W., Newcombe, N. S., Levine, S. C., & Frick, A. (2016). Spatial Proportional Reasoning is Associated with Formal Knowledge about Fractions. Journal of Cognition and Development, 17(1), 67–84. https://doi.org/10.1080/15248372.2014.996289
  46. Nasution, A. A., & Lukito, A. (2015). Developing Student’s Proportional Reasoning Through Informal Way. Journal of Science and Mathematics Education in Southeast Asia, 38(1), 77–101.
  47. NCTM. (2000). Principles and Standards for School Mathematics. The National Council of Teachers of Mathematics, Inc.
  48. Nelson, G., Hunt, J. H., Martin, K., Patterson, B., & Khounmeuang, A. (2022). Current Knowledge and Future Directions: Proportional Reasoning Interventions for Students with Learning Disabilities and Mathematics Difficulties. Learning Disability Quarterly, 45(3), 159–171. https://doi.org/10.1177/0731948720932850
  49. Nugraha, Y., Sujadi, I., & Pangadi, P. (2016). Penalaran Proporsional Siswa Kelas VII [Proportional Reasoning of Class VII Students]. Beta Jurnal Tadris Matematika, 9(1), 34. https://doi.org/10.20414/betajtm.v9i1.2
  50. Piaget, J., Inhelder, B., & Szeminska, A. (1960). Child’s Conception of Geometry. Routledge and Kegan Paul.
  51. Saleh, M., Prahmana, R. C. I., Isa, M., & Murni. (2018). Improving the Reasoning Ability of Elementary School Students through the Indonesian Realistic Mathematics Education. Journal on Mathematics Education, 9(1), 41–54. https://doi.org/10.22342/jme.9.1.5049.41-54
  52. Sari, P., & Fong, N. S. (2022). Exploring Quantitative Relationship through Area Conservation Activity. Journal on Mathematics Education, 13(1), 31–50. https://doi.org/10.22342/jme.v13i1.pp31-50
  53. Sari, Y. M., Widyaningrum, R., & Fiangga, S. (2017). Identification of Student’s Concept on Area Conservation in Solving Proof Task Based on Witkin’s Cognitive Styles: A Case of Indonesian Primary Student. In C. Kusumawardani, D. Darmawan, & E. Yulianti (Eds.), Proceedings of the 4th International Conference on Research, Implementation and Education of Mathematics and Science (pp. 232–236). The Faculty of Mathematics and Natural Sciences.
  54. Sawatzki, C., Downton, A., & Cheeseman, J. (2019). Stimulating Proportional Reasoning through Questions of Finance and Fairness. Mathematics Education Research Journal, 31, 465–484. https://doi.org/10.1007/s13394-019-00262-5
  55. Schoevers, E. M., Leseman, P. P. M., & Kroesbergen, E. H. (2020). Enriching Mathematics Education with Visual Arts: Effects on Elementary School Students’ Ability in Geometry and Visual Arts. International Journal of Science and Mathematics Education, 18(8), 1613–1634. https://doi.org/10.1007/s10763-019-10018-z
  56. Seah, R. T. K., & Horne, M. (2020). The Influence of Spatial Reasoning on Analysing about Measurement Situations. Mathematics Education Research Journal, 32, 365–386. https://doi.org/10.1007/s13394-020-00327-w
  57. Smith, J. P., van den Heuvel-Panhuizen, M., & Teppo, A. R. (2011). Learning, Teaching, and Using Measurement: Introduction to the Issue. ZDM Mathematics Education, 43(5), 617–620. https://doi.org/10.1007/s11858-011-0369-7
  58. Sophian, C. (2000). Perceptions of Proportionality in Young Children: Matching Spatial Ratios. Cognition, 75, 145–170. https://doi.org/10.1016/S0010-0277(00)00062-7
  59. Stylianides, A. J. (2007). Proof and Proving in School Mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
  60. Toluk-Ucar, Z., & Bozkus, F. (2018). Elementary School Students’ and Prospective Teachers’ Proportional Reasoning Skills. International Journal for Mathematics Teaching and Learning, 19(2), 205–222.
  61. van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2021). Elementary and Middle School Mathematics: Teaching Developmentally (10th Global Editions). Pearson.
  62. Vanluydt, E., Degrande, T., Verschaffel, L., & Van Dooren, W. (2020). Early Stages of Proportional Reasoning: A Cross-Sectional Study with 5- to 9-Year-Olds. European Journal of Psychology of Education, 35(3), 529–547. https://doi.org/10.1007/s10212-019-00434-8
  63. Vanluydt, E., Verschaffel, L., & van Dooren, W. (2022a). The Role of Relational Preference in Early Proportional Reasoning. Learning and Individual Differences, 93, 102108. https://doi.org/10.1016/j.lindif.2021.102108
  64. Vanluydt, E., Verschaffel, L., & van Dooren, W. (2022b). The Early Development of Proportional Reasoning: A Longitudinal Study of 5-to 8-year-olds. Journal of Educational Psychology, 114(6), 1343–1358. https://doi.org/10.1037/edu0000734
  65. Yuliani, R., Nurhayati, & Alfin, E. (2021). Analisis kemampuan penalaran proporsional siswa [Analysis of students' proportional reasoning abilities]. Jurnal Bayesian: Jurnal Ilmiah Statistika dan Ekonometrika, 1(1), 24–39. https://doi.org/10.46306/bay.v1i1.3
  66. Zacharos, K. (2006). Prevailing Educational Practices for Area Measurement and Students’ Failure in Measuring Areas. Journal of Mathematical Behavior, 25(3), 224–239. https://doi.org/10.1016/j.jmathb.2006.09.003