Research Article
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Students' Thinking Processes Connecting Quantities in Solving Covariation Mathematical Problems in High School Students of Indonesia

Year 2020, Volume: 7 Issue: 3, 59 - 78, 01.12.2020
https://doi.org/10.17275/per.20.35.7.3

Abstract

The purpose of this study was to describe students' thinking processes in relating quantities to the problem of covariation in the process of solving mathematical problems. This study used a descriptive exploratory approach within the scope of qualitative research involving 87 students as prospective subjects from three different high schools. The schools were chosen from three different areas; where two schools were on Sumbawa island namely in Bima district and Bima town. The subject described was chosen by giving assignments related to the covariation problem. Data obtained from the answer sheets from the subjects while doing think aloud, the results of task-based interviews, and field notes. Supporting equipment for video and audio recording was used to take data of think aloud and interview. Data analysis was conducted retrospectively by combining answer sheets, think aloud, interview results and field notes with reference to APOS theory. The results obtained from students' thinking processes in connecting the quantities of covariation were two, namely Connecting Quantity by direct analytic and equation analytic. In general, it was found that students connected quantities in covariation problem solving by linking changing-quantity along with the changes in other quantities internally or externally through mathematical analysis to form new quantities based on quantitative operations.

Supporting Institution

LPDP Indonesia, Universitas Negeri Malang, STKIP Bima

Thanks

The researchers would like to express their gratitude to the Ministry of the Research, Technology, and Higher Education of Republic of Indonesia, LPDP Indonesia, Universitas Negeri Malang, and STKIP Bima.

References

  • Ayalon, M., Watson, A., & Lerman, S. (2015). Functions represented as linear sequential data: relationships between presentation and student responses. Educational Studies in Mathematics, 90(3), 321–339. https://doi.org/10.1007/s10649-015-9628-9
  • Ayalon, M., Watson, A., & Lerman, S. (2016). Reasoning about variables in 11 to 18 year olds: informal, schooled and formal expression in learning about functions. Mathematics Education Research Journal, 28(3), 379–404. https://doi.org/10.1007/s13394-016-0171-5
  • Brown, C. (2007). Cognitive psychology. In Cognitive Psychology. SAGE Publications India Pvt Ltd. https://doi.org/10.4135/9781446212967
  • Burns-Childers, A., & Vidakovic, D. (2018). Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative. International Journal of Mathematical Education in Science and Technology, 49(5), 660–679. https://doi.org/10.1080/0020739X.2017.1409367
  • Bybee, R. W. (1982). Piaget for Educators. Charles E Merri Publising. co Colombus Ohio.
  • Carlson, M. (1998). A cross-sectional investigation of the development of the function concept (pp. 114–162). Research in collegiate mathematics education. III. CBMS issues in mathematics education. https://doi.org/10.1090/cbmath/007/04
  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. https://doi.org/10.2307/4149958
  • Carroll, J. B. (1993). Human cognitive abilities: A survey of factor-analytic studies. England: Cambridge University Press.
  • Clement, J. (1989). The concept of variation and misconceptions in cartesian graphing. Focus on Learning Problems in Mathematics, 11(1–2), 77–87.
  • Confrey, J., & Smith, E. (1995). Splitting, Covariation, and Their Role in the Development of Exponential Functions. Journal for Research in Mathematics Education, 26, 66–86. https://doi.org/10.2307/749228
  • Copi, I. M. (1978). Introduction to Logic. New York: Macmillan.
  • Corbin, J., & Strauss, A. (2012). Basics of Qualitative Research (3rd ed.): Techniques and Procedures for Developing Grounded Theory. In Basics of Qualitative Research (3rd ed.): Techniques and Procedures for Developing Grounded Theory. London: Sage Publications. https://doi.org/10.4135/9781452230153
  • Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. In Educational Research. Thousand Oaks, CA: Sage. https://doi.org/10.1017/CBO9781107415324.004
  • Dubinsky, E., & Mcdonald, M. A. (2011). APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research. In D. Holton et al. (Ed.), The Teaching and Learning of Mathematics at University Level (pp. 273–280). An ICMI Study. Kluwer Academic Publisher. https://doi.org/10.1007/0-306-47231-7_25
  • Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58(3), 335–359. https://doi.org/10.1007/s10649-005-2531-z
  • Dwyer, C. A., Gallagher, A., Levin, J., & Morley, M. E. (2003). WHAT IS QUANTITATIVE REASONING? DEFINING THE CONSTRUCT FOR ASSESSMENT PURPOSES. ETS Research Report Series. https://doi.org/10.1002/j.2333-8504.2003.tb01922.x
  • Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
  • Ellis, A. B. (2011). Algebra in the Middle School: Developing Functional Relationships Through Quantitative Reasoning. In Cai Jinfa & E. and Knuth (Ed.), Early Algebraization: A Global Dialogue from Multiple Perspectives (pp. 215–238). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_13
  • Johnson, Heather L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change. Journal of Mathematical Behavior, 31(3), 313–330. https://doi.org/10.1016/j.jmathb.2012.01.001
  • Johnson, Heather Lynn. (2015). Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2015.981946
  • Kabael, T., & Akin, A. (2018). Prospective Middle-School Mathematics Teachers’ Quantitative Reasoning and Their Support for Students’ Quantitative Reasoning. International Journal of Research in Education and Science, 4(1), 178–197. https://doi.org/10.21890/ijres.383126
  • Kaput, J. (1995). A research base for algebra reform: Does one exist. In & G. M. M. D. Owens, M. Reed (Ed.), Proceedings of the 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 71–94). Columbus, OH: The ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Köklü, Ö. (2007). An investigation of college students’ covariational reasoning. In Doctoral dissertation. Florida State University.
  • Krulik, S., Rudnick, J. A., & Milou, E. (2003). Teaching mathematics in middle school: A practical guide (Allyn and Bacon (ed.)). Boston.
  • Moore, K. C. (2014). Quantitative reasoning and the sine function: The case of Zac. Journal for Research in Mathematics Education, 45(1), 102–138. https://doi.org/10.5951/jresematheduc.45.1.0102
  • Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. Journal of Mathematical Behavior, 31(1), 48–59. https://doi.org/10.1016/j.jmathb.2011.09.001
  • NCTM. (2000). Principle and Standards for Schools Mathematics. Resto, VA.
  • Ramful, A. (2009). Reversible Reasoning in Multiplicative Situations: Conceptual Analysis, Affordances and Constraints. Doctoral dissertation, University of Georgia.
  • Rosdiana, Budayasa, I. K., & Lukito, A. (2019). Pre-Service Primary School Teachers’ Mathematical Reasoning Skills from Gender Perspectives: A Case Study. Journal for the Education of Gifted Young Scientists, 7(4), 1107–1122. https://doi.org/10.17478/jegys.620234
  • Saldanha, L. A., & Thompson, P. W. (1998). Re-thinking Covariation from a Quantitative Perspective: Simultaneous Continuous Variation. In K. N. & L. S. S. B. Berensah, K. R. Dawkings, M. Blanton, W. N. Coulombe, J. Kolb (Ed.), Proceedings of the Annual Meeting of the Psychology of Mathematics Education - North America (pp. 298–303). Columbus, OH: ERIC.
  • Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259–281. https://doi.org/10.1023/A:1002937032215
  • Smith, J., & Thompson, P. (2007). Quantitative Reasoning and the Development of Algebraic Reasoning. In & M. L. B. J. J. Kaput, D. W. Carraher (Ed.), Algebra in the early grades (pp. 95–132). New York: Lawrence Erlbaum.
  • Stalvey, H. E., & Vidakovic, D. (2015). Students’ reasoning about relationships between variables in a real-world problem. Journal of Mathematical Behavior, 40, 192–210. https://doi.org/10.1016/j.jmathb.2015.08.002
  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. USA: Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-0591-8
  • Sümen, Ö. Ö., & Çalışıcı, H. (2016). The Relationships Between Preservice Teachers’ Mathematical Literacy Self Efficacy Beliefs, Metacognitive Awareness And Problem Solving Skills. Participatory Educational Research, 3(5), 11–19. https://doi.org/10.17275/per.16.spi.2.2
  • Syarifuddin, Nusantara, T., Qohar, A., & Muksar, M. (2019a). Quantitative reasoning process in mathematics problem solving: A case on covariation problems reviewed from Apos theory. Universal Journal of Educational Research, 7(10), 2133–2142. https://doi.org/10.13189/ujer.2019.071011
  • Syarifuddin, Nusantara, T., Qohar, A., & Muksar, M. (2019b). The Identification Difficulty of Quantitative Reasoning Process toward the Calculus Students’ Covariation Problem. Journal of Physics: Conference Series, 1254(1). https://doi.org/10.1088/1742-6596/1254/1/012075
  • Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208. https://doi.org/10.1007/BF01273861
  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In The development of multiplicative reasoning in the learning of mathematics (pp. 179–234).
  • Weber Eric, Amy Ellis, Torrey Kulow, & Zekiye Ozgur. (2014). Six Principles for Quantitative Reasoning and Modeling. The Mathematics Teacher, 108(1), 24–30. https://doi.org/10.5951/mathteacher.108.1.0024
  • Yemen-Karpuzcu, S., Ulusoy, F., & Işıksal-Bostan, M. (2017). Prospective Middle School Mathematics Teachers’ Covariational Reasoning for Interpreting Dynamic Events During Peer Interactions. International Journal of Science and Mathematics Education, 15(1), 89–108. https://doi.org/10.1007/s10763-015-9668-8
Year 2020, Volume: 7 Issue: 3, 59 - 78, 01.12.2020
https://doi.org/10.17275/per.20.35.7.3

Abstract

References

  • Ayalon, M., Watson, A., & Lerman, S. (2015). Functions represented as linear sequential data: relationships between presentation and student responses. Educational Studies in Mathematics, 90(3), 321–339. https://doi.org/10.1007/s10649-015-9628-9
  • Ayalon, M., Watson, A., & Lerman, S. (2016). Reasoning about variables in 11 to 18 year olds: informal, schooled and formal expression in learning about functions. Mathematics Education Research Journal, 28(3), 379–404. https://doi.org/10.1007/s13394-016-0171-5
  • Brown, C. (2007). Cognitive psychology. In Cognitive Psychology. SAGE Publications India Pvt Ltd. https://doi.org/10.4135/9781446212967
  • Burns-Childers, A., & Vidakovic, D. (2018). Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative. International Journal of Mathematical Education in Science and Technology, 49(5), 660–679. https://doi.org/10.1080/0020739X.2017.1409367
  • Bybee, R. W. (1982). Piaget for Educators. Charles E Merri Publising. co Colombus Ohio.
  • Carlson, M. (1998). A cross-sectional investigation of the development of the function concept (pp. 114–162). Research in collegiate mathematics education. III. CBMS issues in mathematics education. https://doi.org/10.1090/cbmath/007/04
  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. https://doi.org/10.2307/4149958
  • Carroll, J. B. (1993). Human cognitive abilities: A survey of factor-analytic studies. England: Cambridge University Press.
  • Clement, J. (1989). The concept of variation and misconceptions in cartesian graphing. Focus on Learning Problems in Mathematics, 11(1–2), 77–87.
  • Confrey, J., & Smith, E. (1995). Splitting, Covariation, and Their Role in the Development of Exponential Functions. Journal for Research in Mathematics Education, 26, 66–86. https://doi.org/10.2307/749228
  • Copi, I. M. (1978). Introduction to Logic. New York: Macmillan.
  • Corbin, J., & Strauss, A. (2012). Basics of Qualitative Research (3rd ed.): Techniques and Procedures for Developing Grounded Theory. In Basics of Qualitative Research (3rd ed.): Techniques and Procedures for Developing Grounded Theory. London: Sage Publications. https://doi.org/10.4135/9781452230153
  • Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. In Educational Research. Thousand Oaks, CA: Sage. https://doi.org/10.1017/CBO9781107415324.004
  • Dubinsky, E., & Mcdonald, M. A. (2011). APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research. In D. Holton et al. (Ed.), The Teaching and Learning of Mathematics at University Level (pp. 273–280). An ICMI Study. Kluwer Academic Publisher. https://doi.org/10.1007/0-306-47231-7_25
  • Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58(3), 335–359. https://doi.org/10.1007/s10649-005-2531-z
  • Dwyer, C. A., Gallagher, A., Levin, J., & Morley, M. E. (2003). WHAT IS QUANTITATIVE REASONING? DEFINING THE CONSTRUCT FOR ASSESSMENT PURPOSES. ETS Research Report Series. https://doi.org/10.1002/j.2333-8504.2003.tb01922.x
  • Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
  • Ellis, A. B. (2011). Algebra in the Middle School: Developing Functional Relationships Through Quantitative Reasoning. In Cai Jinfa & E. and Knuth (Ed.), Early Algebraization: A Global Dialogue from Multiple Perspectives (pp. 215–238). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_13
  • Johnson, Heather L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change. Journal of Mathematical Behavior, 31(3), 313–330. https://doi.org/10.1016/j.jmathb.2012.01.001
  • Johnson, Heather Lynn. (2015). Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2015.981946
  • Kabael, T., & Akin, A. (2018). Prospective Middle-School Mathematics Teachers’ Quantitative Reasoning and Their Support for Students’ Quantitative Reasoning. International Journal of Research in Education and Science, 4(1), 178–197. https://doi.org/10.21890/ijres.383126
  • Kaput, J. (1995). A research base for algebra reform: Does one exist. In & G. M. M. D. Owens, M. Reed (Ed.), Proceedings of the 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 71–94). Columbus, OH: The ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Köklü, Ö. (2007). An investigation of college students’ covariational reasoning. In Doctoral dissertation. Florida State University.
  • Krulik, S., Rudnick, J. A., & Milou, E. (2003). Teaching mathematics in middle school: A practical guide (Allyn and Bacon (ed.)). Boston.
  • Moore, K. C. (2014). Quantitative reasoning and the sine function: The case of Zac. Journal for Research in Mathematics Education, 45(1), 102–138. https://doi.org/10.5951/jresematheduc.45.1.0102
  • Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. Journal of Mathematical Behavior, 31(1), 48–59. https://doi.org/10.1016/j.jmathb.2011.09.001
  • NCTM. (2000). Principle and Standards for Schools Mathematics. Resto, VA.
  • Ramful, A. (2009). Reversible Reasoning in Multiplicative Situations: Conceptual Analysis, Affordances and Constraints. Doctoral dissertation, University of Georgia.
  • Rosdiana, Budayasa, I. K., & Lukito, A. (2019). Pre-Service Primary School Teachers’ Mathematical Reasoning Skills from Gender Perspectives: A Case Study. Journal for the Education of Gifted Young Scientists, 7(4), 1107–1122. https://doi.org/10.17478/jegys.620234
  • Saldanha, L. A., & Thompson, P. W. (1998). Re-thinking Covariation from a Quantitative Perspective: Simultaneous Continuous Variation. In K. N. & L. S. S. B. Berensah, K. R. Dawkings, M. Blanton, W. N. Coulombe, J. Kolb (Ed.), Proceedings of the Annual Meeting of the Psychology of Mathematics Education - North America (pp. 298–303). Columbus, OH: ERIC.
  • Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259–281. https://doi.org/10.1023/A:1002937032215
  • Smith, J., & Thompson, P. (2007). Quantitative Reasoning and the Development of Algebraic Reasoning. In & M. L. B. J. J. Kaput, D. W. Carraher (Ed.), Algebra in the early grades (pp. 95–132). New York: Lawrence Erlbaum.
  • Stalvey, H. E., & Vidakovic, D. (2015). Students’ reasoning about relationships between variables in a real-world problem. Journal of Mathematical Behavior, 40, 192–210. https://doi.org/10.1016/j.jmathb.2015.08.002
  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. USA: Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-0591-8
  • Sümen, Ö. Ö., & Çalışıcı, H. (2016). The Relationships Between Preservice Teachers’ Mathematical Literacy Self Efficacy Beliefs, Metacognitive Awareness And Problem Solving Skills. Participatory Educational Research, 3(5), 11–19. https://doi.org/10.17275/per.16.spi.2.2
  • Syarifuddin, Nusantara, T., Qohar, A., & Muksar, M. (2019a). Quantitative reasoning process in mathematics problem solving: A case on covariation problems reviewed from Apos theory. Universal Journal of Educational Research, 7(10), 2133–2142. https://doi.org/10.13189/ujer.2019.071011
  • Syarifuddin, Nusantara, T., Qohar, A., & Muksar, M. (2019b). The Identification Difficulty of Quantitative Reasoning Process toward the Calculus Students’ Covariation Problem. Journal of Physics: Conference Series, 1254(1). https://doi.org/10.1088/1742-6596/1254/1/012075
  • Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208. https://doi.org/10.1007/BF01273861
  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In The development of multiplicative reasoning in the learning of mathematics (pp. 179–234).
  • Weber Eric, Amy Ellis, Torrey Kulow, & Zekiye Ozgur. (2014). Six Principles for Quantitative Reasoning and Modeling. The Mathematics Teacher, 108(1), 24–30. https://doi.org/10.5951/mathteacher.108.1.0024
  • Yemen-Karpuzcu, S., Ulusoy, F., & Işıksal-Bostan, M. (2017). Prospective Middle School Mathematics Teachers’ Covariational Reasoning for Interpreting Dynamic Events During Peer Interactions. International Journal of Science and Mathematics Education, 15(1), 89–108. https://doi.org/10.1007/s10763-015-9668-8
There are 41 citations in total.

Details

Primary Language English
Subjects Other Fields of Education
Journal Section Research Articles
Authors

Syarifuddin Syarifuddin 0000-0003-3352-0910

Toto Nusantara 0000-0003-1116-9023

Abd. Qohar 0000-0001-8532-102X

Makbul Muksar 0000-0002-5829-8650

Publication Date December 1, 2020
Acceptance Date May 28, 2020
Published in Issue Year 2020 Volume: 7 Issue: 3

Cite

APA Syarifuddin, S., Nusantara, T., Qohar, A., Muksar, M. (2020). Students’ Thinking Processes Connecting Quantities in Solving Covariation Mathematical Problems in High School Students of Indonesia. Participatory Educational Research, 7(3), 59-78. https://doi.org/10.17275/per.20.35.7.3