References
Allain, A. (2000). Development of an instrument to measure proportional reasoning among fast-track middle school students.Unpublished master’s thesis, North Carolina State University, Raleigh.
Artut, P. D., & Pelen, M. S. (2015). 6th grade students’ solution strategies on proportional reasoning problems. Procedia-Social and Behavioral Sciences, 197(11), 113-119.
Ben-Chaim, D., Keret, Y., & Ilany, B. (2012). Ratio and proportion: Research and teaching in mathematics teachers’ education. pre-and in-service mathematics teachers of elementary and middle school classes. Rotterdam, the Netherlands: Sense. https://www.researchgate.net/publication/268007155
Bjorg Steinthorsdottir, O. (2006). Proportional reasoning: variable influencing the problems difficulty level and one's use of problem solving strategies. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, pp. 169-176. Prague: PME.
Broekman, H. & van der Valk, T. (1999) The lesson preparation method: A way to investigate preservice teachers’ pedagogical content knowledge, European Journal of Teacher Education, 21( 2), 11-22.
Burgos, M., & Chaverri, J. (2024). Knowledge and Competencies of Prospective Teachers for the Creation of Proportionality Problems. Journal of Acta Scientiae, 24(6), 270–306.
https://doi.org/10.17648/acta.scientiae.7061
Confrey, J., Maloney, A., Nguyan, K., Mojica, G., & Myers, M. (2009). Equipartition/splitting as a foundation of rational number reasoning using learning trajectories. Paper presented at the 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece.
Cramer, K., & Post, T. (1993). Connecting Research to Teaching Proportional Reasoning. Journal of Mathematics Teacher, 86(5), 404–407.
Fernández, C. Llinares, S. Van Dooren, W. De Bock, D. and Verschaffel, L. (2011) Effect of number structure and nature of quantities on secondary school students’ proportional reasoning. Journal of studia psychologica. 53 (1). 69-81
Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. p. 178-209
Howe, C., T. Nunes, and P. Bryant. (2011). Rational number and proportional reasoning: Using intensive quantities to promote achievement in mathematics and science. International Journal of Science and Mathematics Education, 9(2), 391-417.
Glassmeyer, D., Brakoniecki, A, & Amador, J. M. (2021). Identifying and supporting teachers’ robust understanding of proportional reasoning. Journal of Mathematical Behavior, 62, 100873.
Glassmeyer, D., Brakoniecki, A, & Amador, J. M. (2023). Comparing Elementary and Secondary Teachers’ Robust Understanding of Proportional Reasoning. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-023-10437-z
Karplus, R., Pulos, S., & Stage, E. K. (1983b). Early adolescents’ proportional reasoning on rate problems. Journal of Educational Studies in Mathematics, 14(3), 219–233. https://doi.org/10.1007/BF00410539
Kaput, J. West, M, M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (235-287). Albany, NY: SUNY press.
Lamon, S. J. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24(1), 41-61.
Lamon, S. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics, (pp. 89-121). New York Press.
Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essentiol content 83 and instructional strategies for teachers. https://doi.org/10.4324/9781003008057
Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed), Second Handbook of Research on Mathematics Teaching and Learning (pp. 629-667). Reston, VA: NCTM
Lawton, C.A. (1993) Contextual factors affecting errors in proportional reasoning. Journal for Research in
Mathematics Education, 24, 5, pp. 460-466.
Lesh, R. Post, T., Behr, M. (1988). Proportional Reasoning. In J. Hiebert & M. Behr (Eds.) Number Concepts and .Operations in the Middle Grades (pp.93-118) Reston; VA: Lawrence Erlbaum.
Lin, D. Jia, H & Leung, F. K.SH. (2014). Relations between Subject Matter Knowledge and Pedagogical Content Knowledge: A Study of Chinese Pre-Service Teachers on the Topic of Three-Term Ratio. Journal of The Mathematics Educator, 15 (2), 50-76
Misailidou, C. Williams, J. (2002). Ratio: raising teachers’ awareness of children’s thinking. the Proceedings of the 2nd International Conference on the Teaching of Mathematics (at the undergraduate level), ICTM2, http://www.math.uoc.gr/~ictm2/Proceedings/pap143.pdf
Mitsue, A. (2020). Elementary school teacher’s problems in the process of pedagogical reasoning in proportion from the perspective of curriculum maker. Hiroshima journal of mathematics education, 13. 1-10
NCTM, (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
NCTM. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2013). Teaching ratio and proportion in the Middle grades. Retrieved from https://www. nctm. org/Research-and-Advocacy/Research-Brief-and-Clips/Ratio-_-Proportion. https:// pdfs. semanticscholar. org/7587/385ec6bad6ece0d182d9217d23b7d0970d47. pdf.
Ohtani, M. (2007). Designing umit for teaching proportion based on cultural-historical activity theory: process of symbolizing through collective discourse. Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 33-40. Seoul: PME
Orrill, C. H., Izsák, A, Lobato, J., Cohen, A., Templin, J., de Araujo, Z., Bradshaw, L., Brown, R., Caglayan, G., Druken, B., Jacobson, E., Lee, S. Stanton, S. S., & Wang, A. (2010). Preliminary observations of teachers’ multiplicative reasoning: Insights from Does it Work and Diagnosing teachers’ multiplicative reasoning projects. Fairhaven, MA: Kaput Center for Design and Innovation in STEM Education & University of Massachusetts Dartmouth.
Orrill, C. H., & Millett, J. E. (2021). Teachers’ abilities to make sense of variable parts reasoning.
Mathematical Thinking and Learning, 23(3), 254–270
Ortiz, Martinez. (2015). Examining Students' Proportional Reasoning Strategy Levels as Evidence of the Impact of an Integrated LEGO Robotics and Mathematics Learning Experience. Journal of Technology Education, 26(2) 1-16. doi: 10.21061/jte.v26i2.a.3.
Parish, L. (2010). Facilitating the Development of Proportional Reasoning through Teaching Ratio. Mathematics Education Research Group of Australasia, 469-476.
Ruiz, E. F., & Lupiáñez Gómez, J. L. (2009). Detecting psychological obstacles to teaching and learning the topics of ratio and proportion in sixth grade primary pupils. Electronic Journal of research in Educatinal Psychology, 7(1) 396-424.
Ruiz Ledesma, E. F. (2011). Primary and secondary teachers' knowledge, interpretation, and approaches to students errors about ratio and proportion topics. Creative Education, 2(03), 264-269.
Ruiz Ledesma, E. F. R. (2013). Activities to learn the proportion concept using technology. Journal of International Review of Social Sciences and Humanities, 5(1), 175-184.
Rupley, W. H.: 1981, ‘The effects of numerical characteristics on the difficulty of proportional problems’, Doctoral dissertation, University of California, Berkeley.
Steinthorsdottir, O, B. (2006). Proportional reasoning: variable influencing the problems difficulty level and one’s use of problem solving strategies. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, pp. 169-176. Prague: PME.
Silvestre, A. I., & da Ponte J. P. (2012). Missing value and comparison problems: What pupils
know before the teaching of proportion. PNA, 6(3), 73-83.
HANDLE: http://hdl.handle.net/10481/19500.
Son, J. W. (2013). How preservice teachers interpret and respond to student errors: ratio and proportion in similar rectangles. Educational Studies in Mathematics. 84(1): 49-70.
Sowder, J., Armstrong, B., Lamon, S., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education,. 1, 127-155. https://doi.org/10.1023/A:1009980419975
Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational studies in Mathematics, 16(2), 181-204.
Van Dooren, W. De Bock, D. Evers, M and Verschaffe, L. (2008). Students’ Overuse of Proportionality on Missing-Value Problems:How Numbers May Change Solutions. Journal for Research in Mathematics Education. 40(2), 1–25
Van Dooren, W. De Bock, D. Vleugels, K. Verschaffel, L. (2010), Just Answering … or Thinking? Contrasting Pupils' Solutions and Classifications of Missing-Value Word Problems, Mathematical Thinking and Learning, 12:1, 20-35.
Weiland, T., Orrill, C. H., Nagar, G. G., Brown, R. E., & Burke, J. (2021). Framing a robust understanding
of proportional reasoning for teachers. Journal of Mathematics Teacher Education, 24(2),179–202.