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THE CONNECTING QUANTITIES PROCESS TO SOLVE FRACTION MATHEMATICAL PROBLEMS OF MIDDLE SCHOOL STUDENTS
Corresponding Author(s) : Syarifuddin Syarifuddin
Humanities & Social Sciences Reviews,
Vol. 8 No. 5 (2020): September
Abstract
Purpose of the study: The concept of student fraction, in general, is to understand a small part of an intact part so that students' understanding of fractions as quantities needs to be considered in the context of quantitative reasoning. This study aimed to explore and describe the reasoning of junior high school students in Bima, Indonesia, in understanding the relationship of quantities as a fraction.
Methodology: Research was conducted with qualitative research, which was descriptive exploratory with the stages of giving the Fraction Problem Task (FPT) and task-based interviews. The subjects in this study were 4 students selected from grades 8 and grade 9, who had studied fraction material and were chosen based on the categorization shown by students on the answer sheet.
Main Findings: The results of this study described that there were two forms of student reasoning approaches, namely the variable and the non-variable approach. In the variable approach, students used multilevel variables, namely level one and level two, whereas, in a non-variable approach, students connected quantities directly, without involving variables.
Applications of this study: The process of connecting quantities involved quantitative reasoning with different quantitative operations, such as the quantity combination in the form of multiplication, addition, and concurrent combination of multiplication and addition.
Novelty/Originality of this study: In the process of linking quantities, students were connecting composite units with intact units directly, and students were connecting composite units, continuous units, and then connecting to intact units.
Keywords
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- Abdullah, A. H., Abidin, N. L. Z., & Mokhtar, M. (2017). Using Thinking Blocks to Encourage the Use of Higher Order Thinking Skills among Students When Solving Problems on Fractions. International Journal of Educational and Pedagogical Sciences.
- Atagi, N., DeWolf, M., Stigler, J. W., & Johnson, S. P. (2016). The role of visual representations in college students’ understanding of mathematical notation. Journal of Experimental Psychology: Applied. https://doi.org/10.1037/xap0000090 DOI: https://doi.org/10.1037/xap0000090
- 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 DOI: https://doi.org/10.1007/s13394-016-0171-5
- Baek, J. M., Wickstrom, M. H., Tobias, J. M., Miller, A. L., Safak, E., Wessman-Enzinger, N., & Kirwan, J. V. (2017). Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem. Journal of Mathematical Behavior, 45, 1–14. https://doi.org/10.1016/j.jmathb.2016.10.005 DOI: https://doi.org/10.1016/j.jmathb.2016.10.005
- Boyce, S., & Norton, A. (2016). Co-construction of fractions schemes and units coordinating structures. Journal of Mathematical Behavior, 41, 10–25. https://doi.org/10.1016/j.jmathb.2015.11.003 DOI: https://doi.org/10.1016/j.jmathb.2015.11.003
- Boyce, S., & Norton, A. (2017). Dylan’s units coordinating across contexts. Journal of Mathematical Behavior, 45, 121–136. https://doi.org/10.1016/j.jmathb.2016.12.009 DOI: https://doi.org/10.1016/j.jmathb.2016.12.009
- Caglayan, G. (2013). Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles. Journal of Mathematics Teacher Education, 16(5), 349–378. https://doi.org/10.1007/s10857-013-9237-4 DOI: https://doi.org/10.1007/s10857-013-9237-4
- Carraher, D. W. (1996). Learning about fractions. In P. C. B. G. L. P. Steffe, P. Nesher, G. A. Goldin (Ed.), Theories of mathematical learning (pp. 241–266). Mahwah, New Jersey: Lawrence Erlbaum Associates.
- Cetin, H., & Ertekin, E. (2011). The Relationship between Eighth Grade Primary School Students’ Proportional Reasoning Skills and Success in Solving Equations. International Journal of Instruction, 4(1), 47–62.
- Clement, J. (2000). Analysis of clinical interviews : Foundations and model viability. In A. E. Kelly & R. A. Lesh (Ed.), Handbook of research methodologies for science and mathematics education (pp. 547–589). Mahwah, NJ: Lawrence Erlbaum Associates.
- Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confrey (Ed.), The development of multiplicative reasoning in the learning of mathematics (pp. 293–332). Albany. NY: State University of New York Press.
- 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 DOI: https://doi.org/10.4135/9781452230153
- Crawford, L., Quebec Fuentes, S., Huscroft-D’Angelo, J., & Higgins, K. N. (2019). Evaluating Quantitative Reasoning Strategies for Comparing Fractions: A Tool for Teachers. Intervention in School and Clinic. https://doi.org/10.1177/1053451218782443 DOI: https://doi.org/10.1177/1053451218782443
- 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 DOI: https://doi.org/10.1017/CBO9781107415324.004
- 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 DOI: https://doi.org/10.1002/j.2333-8504.2003.tb01922.x
- 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 DOI: https://doi.org/10.1007/978-3-642-17735-4_13
- Fadhilah, N., Budiarto, M. T., & Rahaju, E. B. (2019). Mathematical Representation of Middle School Students in Solving Fractional Problems Based on Sex Difference. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1417/1/012048 DOI: https://doi.org/10.1088/1742-6596/1417/1/012048
- Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior, 28(4), 383–432. https://doi.org/10.1016/j.jmathb.2007.03.002 DOI: https://doi.org/10.1016/j.jmathb.2007.03.002
- Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction. https://doi.org/10.1080/07370008.2010.511565 DOI: https://doi.org/10.1080/07370008.2010.511565
- Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32(3), 538–563. https://doi.org/10.1016/j.jmathb.2013.06.007 DOI: https://doi.org/10.1016/j.jmathb.2013.06.007
- Hackenberg, A. J., Jones, R., Eker, A., & Creager, M. (2017). “Approximate†multiplicative relationships between quantitative unknowns. Journal of Mathematical Behavior, 48, 38–61. https://doi.org/10.1016/j.jmathb.2017.07.002 DOI: https://doi.org/10.1016/j.jmathb.2017.07.002
- Hackenberg, A. J., & Lee, M. Y. (2015). Relationships Between Students’ Fractional Knowledge and Equation Writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196 DOI: https://doi.org/10.5951/jresematheduc.46.2.0196
- Lewis, K. E. (2016). Beyond Error Patterns: A Sociocultural View of Fraction Comparison Errors in Students with Mathematical Learning Disabilities. Learning Disability Quarterly. https://doi.org/10.1177/0731948716658063 DOI: https://doi.org/10.1177/0731948716658063
- Lewis, K. E. (2017). Designing a Bridging Discourse: Re-Mediation of a Mathematical Learning Disability. Journal of the Learning Sciences. https://doi.org/10.1080/10508406.2016.1256810 DOI: https://doi.org/10.1080/10508406.2016.1256810
- Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. Journal of Mathematical Behavior, 21(1), 87–116. https://doi.org/10.1016/S0732-3123(02)00105-0 DOI: https://doi.org/10.1016/S0732-3123(02)00105-0
- Mkhatshwa, T. P., & Doerr, H. M. (2018). Undergraduate Students’ Quantitative Reasoning in Economic Contexts. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2018.1442642 DOI: https://doi.org/10.1080/10986065.2018.1442642
- Mokhtar, M. A. M., Ayub, A. F. M., Said, R. R., & Mustakim, S. S. (2019). Analysis of Year Four Pupils’ Difficulties in Solving Mathematical Problems Involving Fraction. International Journal of Academic Research in Business and Social Sciences. https://doi.org/10.6007/IJARBSS/v9-i11/6766 DOI: https://doi.org/10.6007/IJARBSS/v9-i11/6766
- 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 DOI: https://doi.org/10.5951/jresematheduc.45.1.0102
- Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. Journal of Mathematical Behavior, 28(2), 150–161. https://doi.org/10.1016/j.jmathb.2009.06.002 DOI: https://doi.org/10.1016/j.jmathb.2009.06.002
- Norton, A., & Wilkins, J. L. M. (2010). Students’ partitive reasoning. Journal of Mathematical Behavior, 29(4), 181–194. https://doi.org/10.1016/j.jmathb.2010.10.001 DOI: https://doi.org/10.1016/j.jmathb.2010.10.001
- Obersteiner, A., & Tumpek, C. (2016). Measuring fraction comparison strategies with eye-tracking. ZDM - Mathematics Education. https://doi.org/10.1007/s11858-015-0742-z DOI: https://doi.org/10.1007/s11858-015-0742-z
- Olive, J., & Çağlayan, G. (2008). Learners’ difficulties with quantitative units in algebraic word problems and the teacher’s interpretation of those difficulties. International Journal of Science and Mathematics Education, 6(2), 269–292. https://doi.org/10.1007/s10763-007-9107-6 DOI: https://doi.org/10.1007/s10763-007-9107-6
- Prayitno, L. L., Purwanto, P., Subanji, S., Susiswo, S., & As’ari, A. R. (2020). Exploring student’s representation process in solving ill-structured problems geometry. Participatory Educational Research. https://doi.org/10.17275/per.20.28.7.2 DOI: https://doi.org/10.17275/per.20.28.7.2
- Sa’dijah, C., Handayani, U. F., Sisworo, Sudirman, Susiswo, Cahyowati, E. T. D., & Sa’Diyah, M. (2019). The Profile of Junior High School Students’ Mathematical Creative Thinking Skills in Solving Problem through Contextual Teaching. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1397/1/012081 DOI: https://doi.org/10.1088/1742-6596/1397/1/012081
- Sa’diyah, M., Sa’dijah, C., Sisworo, & Handayani, U. F. (2019). How Students Build Their Mathematical Dispositions towards Solving Contextual and Abstract Mathematics Problems. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1397/1/012090 DOI: https://doi.org/10.1088/1742-6596/1397/1/012090
- 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 DOI: https://doi.org/10.1016/j.jmathb.2015.08.002
- Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309. https://doi.org/10.1016/1041-6080(92)90005-Y DOI: https://doi.org/10.1016/1041-6080(92)90005-Y
- Steffe, L. P. (2001). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307. https://doi.org/10.1016/S0732-3123(02)00075-5 DOI: https://doi.org/10.1016/S0732-3123(02)00075-5
- 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 DOI: https://doi.org/10.1007/978-1-4419-0591-8
- Swastika, G. T., Nusantara, T., Subanji, & Irawati, S. (2020). Alteration representation in the process of translation graphic to graphic. Humanities and Social Sciences Reviews, 8(1), 334–343. https://doi.org/10.18510/hssr.2020.8144 DOI: https://doi.org/10.18510/hssr.2020.8144
- 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 DOI: 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 DOI: https://doi.org/10.1088/1742-6596/1254/1/012075
- 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 DOI: https://doi.org/10.17275/per.20.35.7.3
- Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208. https://doi.org/10.1007/BF01273861 DOI: 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).
- Thompson, P. W. (1995). Notation, Convention, and Quantity in Elementary Mathematics. In J. Sowder & B. Schapelle (Ed.), Providing a Foundation for Teaching Middle School Mathematics. Albany: NY: SUNY Press.
- Thompson, P. W. (2013). In the absence of meaning…. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 57–93). New York: Springer. https://doi.org/10.1007/978-1-4614-6977-3_4 DOI: https://doi.org/10.1007/978-1-4614-6977-3_4
- Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In Research companion to the Principles and Standards for School Mathematics (pp. 95–113).
- Weber, E., Ellis, A., Kulow, T., & Ozgur, Z. (2014). Six Principles for Quantitative Reasoning and Modeling. The Mathematics Teacher, 108(1), 24–30. https://doi.org/10.5951/mathteacher.108.1.0024 DOI: https://doi.org/10.5951/mathteacher.108.1.0024
- Wijaya, A. (2017). The relationships between Indonesian fourth graders’ difficulties in fractions and the opportunity to learn fractions: A snapshot of TIMSS results. International Journal of Instruction, 10(4), 221–236. https://doi.org/10.12973/iji.2017.10413a DOI: https://doi.org/10.12973/iji.2017.10413a
References
Abdullah, A. H., Abidin, N. L. Z., & Mokhtar, M. (2017). Using Thinking Blocks to Encourage the Use of Higher Order Thinking Skills among Students When Solving Problems on Fractions. International Journal of Educational and Pedagogical Sciences.
Atagi, N., DeWolf, M., Stigler, J. W., & Johnson, S. P. (2016). The role of visual representations in college students’ understanding of mathematical notation. Journal of Experimental Psychology: Applied. https://doi.org/10.1037/xap0000090 DOI: https://doi.org/10.1037/xap0000090
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 DOI: https://doi.org/10.1007/s13394-016-0171-5
Baek, J. M., Wickstrom, M. H., Tobias, J. M., Miller, A. L., Safak, E., Wessman-Enzinger, N., & Kirwan, J. V. (2017). Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem. Journal of Mathematical Behavior, 45, 1–14. https://doi.org/10.1016/j.jmathb.2016.10.005 DOI: https://doi.org/10.1016/j.jmathb.2016.10.005
Boyce, S., & Norton, A. (2016). Co-construction of fractions schemes and units coordinating structures. Journal of Mathematical Behavior, 41, 10–25. https://doi.org/10.1016/j.jmathb.2015.11.003 DOI: https://doi.org/10.1016/j.jmathb.2015.11.003
Boyce, S., & Norton, A. (2017). Dylan’s units coordinating across contexts. Journal of Mathematical Behavior, 45, 121–136. https://doi.org/10.1016/j.jmathb.2016.12.009 DOI: https://doi.org/10.1016/j.jmathb.2016.12.009
Caglayan, G. (2013). Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles. Journal of Mathematics Teacher Education, 16(5), 349–378. https://doi.org/10.1007/s10857-013-9237-4 DOI: https://doi.org/10.1007/s10857-013-9237-4
Carraher, D. W. (1996). Learning about fractions. In P. C. B. G. L. P. Steffe, P. Nesher, G. A. Goldin (Ed.), Theories of mathematical learning (pp. 241–266). Mahwah, New Jersey: Lawrence Erlbaum Associates.
Cetin, H., & Ertekin, E. (2011). The Relationship between Eighth Grade Primary School Students’ Proportional Reasoning Skills and Success in Solving Equations. International Journal of Instruction, 4(1), 47–62.
Clement, J. (2000). Analysis of clinical interviews : Foundations and model viability. In A. E. Kelly & R. A. Lesh (Ed.), Handbook of research methodologies for science and mathematics education (pp. 547–589). Mahwah, NJ: Lawrence Erlbaum Associates.
Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confrey (Ed.), The development of multiplicative reasoning in the learning of mathematics (pp. 293–332). Albany. NY: State University of New York Press.
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 DOI: https://doi.org/10.4135/9781452230153
Crawford, L., Quebec Fuentes, S., Huscroft-D’Angelo, J., & Higgins, K. N. (2019). Evaluating Quantitative Reasoning Strategies for Comparing Fractions: A Tool for Teachers. Intervention in School and Clinic. https://doi.org/10.1177/1053451218782443 DOI: https://doi.org/10.1177/1053451218782443
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 DOI: https://doi.org/10.1017/CBO9781107415324.004
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 DOI: https://doi.org/10.1002/j.2333-8504.2003.tb01922.x
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 DOI: https://doi.org/10.1007/978-3-642-17735-4_13
Fadhilah, N., Budiarto, M. T., & Rahaju, E. B. (2019). Mathematical Representation of Middle School Students in Solving Fractional Problems Based on Sex Difference. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1417/1/012048 DOI: https://doi.org/10.1088/1742-6596/1417/1/012048
Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior, 28(4), 383–432. https://doi.org/10.1016/j.jmathb.2007.03.002 DOI: https://doi.org/10.1016/j.jmathb.2007.03.002
Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction. https://doi.org/10.1080/07370008.2010.511565 DOI: https://doi.org/10.1080/07370008.2010.511565
Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32(3), 538–563. https://doi.org/10.1016/j.jmathb.2013.06.007 DOI: https://doi.org/10.1016/j.jmathb.2013.06.007
Hackenberg, A. J., Jones, R., Eker, A., & Creager, M. (2017). “Approximate†multiplicative relationships between quantitative unknowns. Journal of Mathematical Behavior, 48, 38–61. https://doi.org/10.1016/j.jmathb.2017.07.002 DOI: https://doi.org/10.1016/j.jmathb.2017.07.002
Hackenberg, A. J., & Lee, M. Y. (2015). Relationships Between Students’ Fractional Knowledge and Equation Writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196 DOI: https://doi.org/10.5951/jresematheduc.46.2.0196
Lewis, K. E. (2016). Beyond Error Patterns: A Sociocultural View of Fraction Comparison Errors in Students with Mathematical Learning Disabilities. Learning Disability Quarterly. https://doi.org/10.1177/0731948716658063 DOI: https://doi.org/10.1177/0731948716658063
Lewis, K. E. (2017). Designing a Bridging Discourse: Re-Mediation of a Mathematical Learning Disability. Journal of the Learning Sciences. https://doi.org/10.1080/10508406.2016.1256810 DOI: https://doi.org/10.1080/10508406.2016.1256810
Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. Journal of Mathematical Behavior, 21(1), 87–116. https://doi.org/10.1016/S0732-3123(02)00105-0 DOI: https://doi.org/10.1016/S0732-3123(02)00105-0
Mkhatshwa, T. P., & Doerr, H. M. (2018). Undergraduate Students’ Quantitative Reasoning in Economic Contexts. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2018.1442642 DOI: https://doi.org/10.1080/10986065.2018.1442642
Mokhtar, M. A. M., Ayub, A. F. M., Said, R. R., & Mustakim, S. S. (2019). Analysis of Year Four Pupils’ Difficulties in Solving Mathematical Problems Involving Fraction. International Journal of Academic Research in Business and Social Sciences. https://doi.org/10.6007/IJARBSS/v9-i11/6766 DOI: https://doi.org/10.6007/IJARBSS/v9-i11/6766
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 DOI: https://doi.org/10.5951/jresematheduc.45.1.0102
Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. Journal of Mathematical Behavior, 28(2), 150–161. https://doi.org/10.1016/j.jmathb.2009.06.002 DOI: https://doi.org/10.1016/j.jmathb.2009.06.002
Norton, A., & Wilkins, J. L. M. (2010). Students’ partitive reasoning. Journal of Mathematical Behavior, 29(4), 181–194. https://doi.org/10.1016/j.jmathb.2010.10.001 DOI: https://doi.org/10.1016/j.jmathb.2010.10.001
Obersteiner, A., & Tumpek, C. (2016). Measuring fraction comparison strategies with eye-tracking. ZDM - Mathematics Education. https://doi.org/10.1007/s11858-015-0742-z DOI: https://doi.org/10.1007/s11858-015-0742-z
Olive, J., & Çağlayan, G. (2008). Learners’ difficulties with quantitative units in algebraic word problems and the teacher’s interpretation of those difficulties. International Journal of Science and Mathematics Education, 6(2), 269–292. https://doi.org/10.1007/s10763-007-9107-6 DOI: https://doi.org/10.1007/s10763-007-9107-6
Prayitno, L. L., Purwanto, P., Subanji, S., Susiswo, S., & As’ari, A. R. (2020). Exploring student’s representation process in solving ill-structured problems geometry. Participatory Educational Research. https://doi.org/10.17275/per.20.28.7.2 DOI: https://doi.org/10.17275/per.20.28.7.2
Sa’dijah, C., Handayani, U. F., Sisworo, Sudirman, Susiswo, Cahyowati, E. T. D., & Sa’Diyah, M. (2019). The Profile of Junior High School Students’ Mathematical Creative Thinking Skills in Solving Problem through Contextual Teaching. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1397/1/012081 DOI: https://doi.org/10.1088/1742-6596/1397/1/012081
Sa’diyah, M., Sa’dijah, C., Sisworo, & Handayani, U. F. (2019). How Students Build Their Mathematical Dispositions towards Solving Contextual and Abstract Mathematics Problems. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/1397/1/012090 DOI: https://doi.org/10.1088/1742-6596/1397/1/012090
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 DOI: https://doi.org/10.1016/j.jmathb.2015.08.002
Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309. https://doi.org/10.1016/1041-6080(92)90005-Y DOI: https://doi.org/10.1016/1041-6080(92)90005-Y
Steffe, L. P. (2001). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307. https://doi.org/10.1016/S0732-3123(02)00075-5 DOI: https://doi.org/10.1016/S0732-3123(02)00075-5
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 DOI: https://doi.org/10.1007/978-1-4419-0591-8
Swastika, G. T., Nusantara, T., Subanji, & Irawati, S. (2020). Alteration representation in the process of translation graphic to graphic. Humanities and Social Sciences Reviews, 8(1), 334–343. https://doi.org/10.18510/hssr.2020.8144 DOI: https://doi.org/10.18510/hssr.2020.8144
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 DOI: 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 DOI: https://doi.org/10.1088/1742-6596/1254/1/012075
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 DOI: https://doi.org/10.17275/per.20.35.7.3
Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208. https://doi.org/10.1007/BF01273861 DOI: 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).
Thompson, P. W. (1995). Notation, Convention, and Quantity in Elementary Mathematics. In J. Sowder & B. Schapelle (Ed.), Providing a Foundation for Teaching Middle School Mathematics. Albany: NY: SUNY Press.
Thompson, P. W. (2013). In the absence of meaning…. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 57–93). New York: Springer. https://doi.org/10.1007/978-1-4614-6977-3_4 DOI: https://doi.org/10.1007/978-1-4614-6977-3_4
Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In Research companion to the Principles and Standards for School Mathematics (pp. 95–113).
Weber, E., Ellis, A., Kulow, T., & Ozgur, Z. (2014). Six Principles for Quantitative Reasoning and Modeling. The Mathematics Teacher, 108(1), 24–30. https://doi.org/10.5951/mathteacher.108.1.0024 DOI: https://doi.org/10.5951/mathteacher.108.1.0024
Wijaya, A. (2017). The relationships between Indonesian fourth graders’ difficulties in fractions and the opportunity to learn fractions: A snapshot of TIMSS results. International Journal of Instruction, 10(4), 221–236. https://doi.org/10.12973/iji.2017.10413a DOI: https://doi.org/10.12973/iji.2017.10413a