[1] R. Lesh, T. R. Post and M. Behr, Number Concepts and Operations in the Middle Grades, National Council of Teachers of Mathematics, Lawrence Erlbaum Associates, 1988 93–118.
[2] Y. Copur-Gencturk, C. Baek and T. Doleck, A closer look at teachers’ proportional reasoning, International Journal of Science and Mathematics Education, 21 (2023) 113–129.
[3] R. Lesh, M. Behr and T. Post, Rational number relations and proportions, In C. Janvier (Ed.), Problems of representations in the teaching and learning of mathematics, Hillsdale, NJ: Lawrence Erlbaum Associates, 1987 41–58.
[4] S. J. Lamon, Rational numbers and proportional reasoning: Toward a theoretical framework for research, Second handbook of research on mathematics teaching and learning, 1 (2007) 629–667.
[5] J. Lobato, C. Orrill, B. Druken and E. Jacobson, Middle school teachers’ knowledge of proportional reasoning for teaching, In Conference: Annual Meeting of the American Educational Research Association (AERA), New Orleans, (2011).
[6] M. Arican, The development and application of an interview structure on determining preservice mathematics teachers’ competence in proportional reasoning, Mathematics Education Research Journal, 35 (2023) 55–79.
[7] S. J. Lamon, Teaching Fractions and Ratios for Understanding, Essential Content 83 and Instructional Strategies for Teachers, Routledge, New York, 2020.
[8] M. M. Petit, R. E. Laird, M. F. Wyneken, F. R. Huntoon, M. D. Abele-Austin and J. D. Sequeira, A focus on ratios and proportions, Bringing mathematics education research to the classroom, Routledge, New York, 2020.
[9] K. Hino and H. Kato, Teaching whole-number multiplication to promote children’s proportional reasoning: A practice-based perspective from Japan, ZDM, 51 (2018) 125–137.
[10] K. A. Cramer, T. Post and S. Currier, Learning and teaching ratio and proportion: Research implications: Middle grades mathematics, In Conference:Research ideas for the classroom: Middle grades mathematics, Macmillan Publishing Company, New York, NY, (1993) 159–178.
[11] A. S. Supply, E. Vanluydt, W. Van Dooren, and P. Onghena, Out of proportion or out of context? Comparing 8-to 9- year-olds’ proportional reasoning abilities across fair-sharing, mixtures, and probability contexts, Educ. Stud. Math., 113 (2023) 371–388.
[12] D. Ben-Chaim, J. T. Fey, W. M. Fitzgerald, C. Benedetto and J. Miller, Proportional reasoning among 7th grade students with different curricular experiences, Educ. Stud. Math., 36 (1998) 247–273.
[13] J. M. Amador, A. Brakoniecki and D. Glassmeyer, Secondary teachers’ analytic stance of noticing based on video of proportional reasoning, International Journal of Mathematical Education in Science and Technology, (2022) pp. 21.
[14] S. L. D. Benson, The influence of studying students’ proportional reasoning on middle school mathematics teachers’ content and pedagogical content knowledge, University of Houston, 2011.
[15] K. N. Begolli, T. Dai, K. M. McGinn and J. L. Booth, Could probability be out of proportion? Selfexplanation and example-based practice help students with lower proportional reasoning skills learn probability, Instructional Science, 49 (2021) 441–473.
[16] A. Kahaki E. Reyhani E. Bahrami, Assessment of Understanding and Understanding of Eighth Grade Students of Probability, Andishe-ye Amari, 24 (2019) 57–80.
[17] S. Dole, D. Clarke, T. Wright and G. Hilton, Students’ proportional reasoning in mathematics and science, In Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, (2012) 195–202.
[18] H. Amiri, et al, Mathematics for Year 6 in elementary school, Organization for Educational Research and Planning, Ministry of Education, Tehran (Iran), Printing and Publishing Company of Iran Textbooks, 2021a.
[19] H. Amiri, et al, Mathematics for Year 8 in high school, Organization for Educational Research and Planning, Ministry of Education, Tehran (Iran), Printing and Publishing Company of Iran Textbooks, 2021b.
[20] H. Amiri, et al, Mathematics for Year 9 in high school, Organization for Educational Research and Planning, Ministry of Education, Tehran (Iran), Printing and Publishing Company of Iran Textbooks, 2021c.
[21] E. Reyhani, H. Barkhordari and S. Haghjoo, Investigating the performance of students in solving proportional problems, Journal of Educational and Scholastic Studies, (2023).
[22] E. Vanluydt, T. Degrande, L. Verschaffel and W. Van Dooren, Early stages of proportional reasoning: A cross-sectional study with 5-to 9-year olds, European Journal of Psychology of Education, 35 (2019) 529–547.
[23] S. Haghjoo and E. Reyhani, A qualitative meta-analysis of assessment frameworks of mathematical problemposing skills, Research in School and Virtual Learning, 9 (2022) 9–28.
[24] L. Baumanns and B. Rott, The process of problem posing: development of a descriptive phase model of problem posing, Educational Studies in Mathematics, 110 (2022) 251–269.
[25] E. A. Silver, On mathematical problem posing, For the Learning of Mathematics, 14 (1994) 19–28.
[26] NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.
[27] S. J. Lamon, More: In-depth discussion of the reasoning activities in Teaching fractions and ratios for understanding, Routledge, 2005.
[28] M. J. Behr, R. Lesh, T. Post and E. A. Silver, Rational number concepts, Acquisition of mathematics concepts and processes, New York: Academic Press, 1983 91–125.
[29] J. A. Van de Walle, K. S. Karp and J. M. Bay-Williams, Elementary and middle school mathematics, London: Pearson Education UK, 2016.
[30] L. Fisher, Strategies used by secondary mathematics teachers to solve proportion problems, Journal for Research in Mathematics Education, 19 (2) (1988) 157–168.
[31] E. Reyhani, S. Bakhshalizadeh and M. Dosti, Grade 6th students understanding of fraction, Journal of Curriculum Studies, 9 (34) (2014) 133–164.
[32] T. Degrande, L. Verschaffel and W. V. Dooren, Proportional word problem solving through a modeling lens: A half-empty or half-full glass?, In Posing and solving mathematical problems, Springer, Cham, (2016) 209–229.
[33] W. Van Dooren, X. Vamvakoussi and L. Verschaffel, Proportional reasoning, (Educational Practices Series, 30). International Academy of Education (IAE), (2018).
[34] T. K. Ulger, I. Bozkurt and M. Altun, Analyzing in-service teachers’ process of mathematical literacy problem posing, International Electronic Journal of Mathematics Education, 17 (3) (2022) pp. 19.
[35] E. Stoyanova and N. F. Ellerton, A framework for research into students’ problem posing, In P. Clarkson (Ed.), Technology in Mathematics Education, Melbourne: Mathematics Education Research Group of Australasia, (1996) 518–525.
[36] M. Eskandari and E. Reyhani, Investigating the process of problem posing, Journal of Theory & Practice in Curriculum, 2 (3) (2014) 117–140.
[37] E. Reyhani and S. Haghjoo, Mathematical problem solving : from theory to practice: A perspective for mathematics teacher education, Shahid Rajaee Teacher Training University, Tehran, Iran, 2020.
[38] M. Izadi and E. Reyhani, Using an unusual Task to investigate elementary school teachers’ mathematical -task knowledge and common content knowledge of fraction concept in tehran province, Research in School and Virtual Learning 7 (4) (2020) 55–70.
[39] E. Stoyanova, N. F. Ellerton, A framework for research into students’ problem posing, In P. Clarkson (Ed.), Technology in Mathematics Education, Melbourne: Mathematics Education Research Group of Australasia, (1996) 518–525.
[40] L. D. English, The development of fifth-grade children’s problem-posing abilities, Educational Studies in Mathematics, 34 (3) (1997) 183–217.
[41] S. H. Im and A. K. Jitendra, Analysis of proportional reasoning and misconceptions among students with mathematical learning disabilities, The Journal of Mathematical Behavior, 57 (2020).
[42] M. Nedaei, F. Radmehr and M. Drake, Exploring undergraduate engineering students’ mathematical problem-posing: the case of integral-area relationships in integral calculus, Mathematical Thinking and Learning, 24 (2) (2022) 149–175.
[43] S. S. Leung, Teachers implementing mathematical problem posing in the classroom: challenges and strategies, Educational studies in mathematics, 83 (1) (2013) 103–116.
[44] J. E. A. Cai, An investigation of U. S. and Chinese students’ mathematical problem posing and problem solving, Mathematics Education Research Journal, 10 (1) (1998) 37–50.
[45] M. Carney, K. Paulding and J. Champion, Efficient assessment of students’ proportional reasoning, Applied Measurement in Education, 35 (1) (2022) 46–62.
[46] C. K. Bilir, Pre-service Teachers’ Use of Proportional Reasoning Skills to Solve the Area Measurement Problems of the Rectangles, In international online conference on mathematics education, 26-29 Istanbul/turkey (2021) p. 63.
[47] E. Reyhani, F. Hamidi and F. Kolahdouz, A study on algebraic proof conception of high school second graders, Procedia-Social and Behavioral Sciences, 31 (2012) 236–241.
[48] A. Peng, M. Li, L. Lin, L. Cao and J. Cai, Problem posing and its relationship with teaching experience of elementary school mathematics teachers from ethnic minority area in southwest china, EURASIA Journal of Mathematics, Science and Technology Education, 18 (2) (2022) pp. 14.
[49] C. Vistro-Yu, Using innovation techniques to generate ‘new’ Problems, In book: Mathematical problem solving, Singapore: World Scientific Publishing Co. Pte. Ltd., 2009 185–207.
[50] E. Vysotskaya, A. Lobanova, I. Rekhtman and M. Yanishevskaya, The challenge of proportion: does it require rethinking of the measurement paradigm?, Educational Studies in Mathematics, 106 (3) (2021) 429–446.
[51] A. Rafipour and L. Goya, Why was the mathematical performance of Iranian students in TIMMS unique?, Journal of Mathematical Education Development, (75) (2004) 15–22.
[52] G. Howson, Looking back-and looking forward, The Mathematical Gazette, 80 (487) (1996) 129–136.
[53] J. M. Eckert, Trends in mathematics and science study (TIMSS): international accountability and implications for science instruction, Research in Comparative and International Education, 3 (2) (2008) 202–210.
[54] A. Rafi pour and L. Jokār, The role of gender and grade level in students’ math performance to solve a none-routine problem, Journal of Educational Innovation, 12 (4) (2014) 27–44.
[55] T. Weiland, C. H. Orrill, G. G. Nagar, R. E. Brown and J. Burke, Framing a robust understanding of proportional reasoning for teachers, J. Math. Teach. Educ., 24 (2) (2021) 179–202.
[56] A. M. Gallagher and R. De Lisi, Gender differences in scholastic aptitude test: Mathematics problem solving among high-ability students, Journal of Educational Psychology, 86 (2) (1994) 204–211.
[57] M. Gorian, Different learning of girls and boys, Payk Bahar, Tehran, Iran, 2004.
[58] E. G. Fierros, Examining gender differences in mathematics achievement on the Third International Mathematics and Science Study (TIMSS), Boston College, 1999.
[59] H. P. Sharifi and M. Ghodrati, The Investigation of gifted girl’s creativity in talented-students, non-profit and public achools, Jornal of Analytical-Cognitive Psychology, 1 (3) (2010) 31–40.
[60] A. Rafi pour, The study of the role of type of schools in proportional reasoning of students, Educational Innovations, 11 (2014) 24–33.
[61] M. J. Behr, G. Harel, T. Post and R. Lesh, Rational numbers: Toward a semantic analysis-emphasis on the operator construct, New Jersey, 1993 13–47.
[62] G. Harel and M. Behr, Structure and hierarchy of missing value proportion problems and their representations, J. Math. Behav., 8 (1989) 77–119.