Volume 65 | Issue 2 | Year 2019 | Article Id. IJMTT-V65I2P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I2P511

Relying on the fact that fractions are of a long time for researchers a challenging cognitive area for research, as students of all levels of education face particular difficulties on their understanding, this paper presents a review of contemporary literature on the subject of the representations of fractions. In particular, the international literature was investigated in order to study and record the results of researches which have been published in this time on the representations of fractions. In other words, this paper aiming to answers to what representations have been emerged by this research as the most appropriate or inappropriate for understanding the concept of fractions from the students. In this way, this study is expected to give to teachers of mathematics education and researchers who are examining the area of fractions a useful guide.

[1] R. Adjiage, and F. Pluvinage, ―An experiment in teaching ratio and proportion‖, Educational Studies in Mathematics vol. 20, pp.149–175, 2007.

[2] E. Avgerinos, R. Vlachou, and K. Kantas, Comparing different age student abilities on the concept and manipulation of fractures, in E. Avgerinos and A. Gagatsis (Eds), Research on Mathematical Education and Mathematics Applications, pp. 159-168, Rhodes: University of the Aegean, 2012.

[3] R. Vlachou, and E. Avgerinos, ―Multiple representatıons and development of students' self-confidence on rational number‖, Experiences of Teaching with Mathematics, Sciences and Technology, vol. 4, pp.567-586, 2018.

[4] E. Avgerinos, and R. Vlachou, ―The perceptions of the prospective teacher on the concepts of the number liny, the equal parts of the unit and the improper fractions‖, Proceedings of the 15th Pancyprian Congress of Mathematical Education and Science, Cyprus: Cypriot Mathematical Society (in Greek), 2013.

[5] G. Brousseau, N. Brousseau, and Warfield V., ―Rationals and decimals as required in the school curriculum Part 2: From rationals to decimals‖, The Journal of Mathematical Behavior, vol. 26, pp. 281-300, 2007.

[6] J. Cai, and Wang, T., ―U.S. and Chinese teachers‘ conceptions and constructions of representations: a case of teaching ratio concept‖, International Journal of Science and Mathematics Education, vol. 4, pp. 145-186, 2006.

[7] X. Chen, and Y. Li, ―Instructional coherence in Chinese mathematics classroom—a case study of lessons on fraction division‖, International Journal of Science and Mathematics Education, vol. 8, pp. 711-735, 2009.

[8] S. J. Courey, and Siker, J. R., ―Academic music: music instruction to engage third-grade students in learning basic fraction concepts‖, Educational Studies in Mathematics, vol. 81, pp. 251-278, 2012.

[9] K. Cramer, and T. Wyberg, ―Efficacy of Different Concrete Models for Teaching the Part-Whole Construct for Fractions‖, Mathematical Thinking and Learning, vol. 11, pp. 226-257, 2009.

[10] S. B., Empson, D.Junk, H. Dominguez, and, E. Turner, ―Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children‘s thinking‖, Educational Studies in Mathematics, vol. 63, pp. 1–28, 2006.

[11] L. S. Fuchs, D. Fuchs, C. L. Hamlet, and S. R. Powell, ―The effects of computer-assisted instruction on number combination skill in at-risk first graders‖, Journal of Learning Disabilities, vol. 39, pp. 467-475, 2006.

[12] A., J. Hackenberg, ―Units coordination and the construction of improper fractions: A revision of the splitting hypothesis‖, The Journal of Mathematical Behavior, vol. 26, pp. 27-47, 2007.

[13] A., J. Hackenberg, and Ε., S. Tillema, ―Students‘ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes‖, The Journal of Mathematical Behavior, vol. 28, pp. 1-18, 2009.

[14] J. Hodgen, D. Küchemann, M. Brown, and Coe, R., ―Lower secondary school students' knowledge of fractions‖, Research in Mathematics Education, vol. 12, pp. 75-76, 2010.

[15] A. Izsák, ―Mathematical Knowledge for Teaching Fraction Multiplication‖, Cognition and Instruction, vol. 26, pp. 95-143, 2008.

[16] C. Janvier, Translation Processes in Mathematics Education. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics, pp. 27-32. Hillsdale, NJ: Lawrence Erlbaum, 1987.

[17] C. Jiang, and B. L. Chua, ―Strategies for Solving Three Fraction-Related Word Problems on Speed: a Comparative Study Between Chinese and Singaporean Students‖, International Journal of Science and Mathematics Education, vol. 8, pp. 73-96, 2010.

[18] S. Lamon, ―The development of unitizing: Its role in children‘s portioning strategies‖, Journal for Research in Mathematics Education, vol. 27, pp. 170-193, 1996.

[19] H. S. Lee, and P. Sztajn, ―Focusing on Units to Support Prospective Elementary Teachers‘ Understanding of Division in Fractional Contexts‖, School Science and Mathematics, vol. 108, pp. 20-27, 2008.

[20] C., Y. Lin, ―Web-Based Instruction on Preservice Teachers‘ Knowledge of Fraction Operations: School Science and Mathematics, vol. 110, pp. 59-70, 2010.

[21] J-J. Lo, ―Conceptual Bases of young Children‘s Solution Strategies of Missing value Proportional Tasks‖, Psychology of Mathematics Education, Proceedings of Seventeenth PME International Conference, pp. 162-177, 1993.

[22] L. Ma, Knowing and teaching elementary mathematics: Teachers‘ knowledge of fundamental mathematics in China and the United States, Mahwah, NJ: Lawrence Erlbaum, (1999).

[23] B. J. Moseley, Y. Okamoto, and Ishida J., ―Comparing us and Japanese elementary school teachers‘ facility for linking rational number representations‖, International Journal of Science and Mathematics Education, vol. 5, pp. 165-185, 2006.

[24] B. Moseley, and Y. Okamoto, ―Identifying Fourth Graders‘ Understanding of Rational Number Representations: A Mixed Methods Approach‖, School Science and Mathematics, vol. 108, pp. 238-250, 2008.

[25] T., M. Muzheve, and R., M., Capraro, ―An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals and percents‖, The Journal of Mathematical Behavior, vol.31 pp. 1-14, 2012.

[26] M. I. Fandiño Pinilla, ―Fractions: conceptual and didactic aspects‖, Acta Didactica Universitatis Comenianae, vol. 7, pp. 2345. 2007.

[27] Y. Ni, ―Semantic Domains of Rational Number and the Acquisition of Number Equivalence‖, Contemporary Educational Psychology, vol. 26, pp. 400-417, 2001.

[28] A. Norton, and L.M. J. Wilkinsb, ―A quantitative analysis of children‘s splitting operations and fraction schemes‖, Journal of Mathematical Behavior‖, vol. 28, pp. 150-161, 2009.

[29] J. Olive, and E. Vomvoridi, ―Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim‖, Journal of Mathematical Behavior, vol. 25, pp. 18–45, 2006.

[30] D. Pagni, ―Fractions and decimals‖, Australian Mathematics Teacher, vol. 60, pp. 28–30, 2004.

[31] D.M. Peck, and S.M. Jencks, ―Conceptual Issues in the Teaching and Learning of Fractions‖, Journal for Research in Mathematics Education, vol. 12, pp., 339-348, 1981.

[32] T. Post, K. Cramer, M. Behr, R. Lesh, and G. Harel, Curriculum implications of research on the learning, teaching and assessing of rational number concepts. In T. Carpenter, E. Fennema & T. Romberg (Eds.), 1993.

[33] A. Hansen, M. Mavrikis, and E. Geraniou, ―Supporting teachers‘ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative‖, Journal of Mathematics Teacher Education, vol. 19, pp. 205-226, 2016.

[34] G. Psycharis, and C. Kynigos, ―Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion‖, Research in Mathematics Education, vol. 11, pp. 149-166, 2009.

[35] A. E. Ryken, ―Multiple representations as sites for teacher reflection about mathematics learning‖, Journal of Mathematics Teacher Education, vol. 12, pp. 347-364, 2009.

[36] T. Satwicz and R. Stevens, ―Playing with Representations: How Do Kids Make Use of Quantitative Representations in Video Games?‖ International Journal of Computers for Mathematical Learning, vol. 13, pp. 179-206, 2008.

[37] R. Y. Schorr, and G. A. Goldin, ―Students‘ expression of affect in an inner-city SimCalc classroom‖, Educational Studies in Mathematics, vol. 68, pp. 131-148, 2008.

[38] K. Sedig, and M. Sumner, ―Characterizing interaction with visual mathematical representations‖, International Journal of Computers for Mathematical Learning, vol. 11, pp. 1–55, 2006.

[39] A. Sfard, ―On the Dual Nature of Mathematical Conceptions: Reflections on processes and objects as different sides of the same coin‖, Educational Studies in Mathematics, vol. 22, pp. 1-36, 1991.

[40] N. Sinclair, P. Liljedahl and A. Zazkis, ―Coloured window on pre-service teachers‘ conceptions of rational numbers‖, International Journal of Computers for Mathematical Learning, vol. 11, pp. 177–203, 2006.

[41] L. Steffe, Children‘s construction of number sequences and multiplying schemes, in J. Hiebert & M. Behr (Eds.), Number concepts and operations in middle grades, pp. 119–140, Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum Associates, Inc, 1988.

[42] L. Steffe, Children‘s multiplying schemes, in G. Harel & J. Confrey (Eds.), the development of multiplicative reasoning in the learning of mathematics, pp. 3–39, Albany: State University of New York Press, 1994.

[43] L. Streefland, Fractions in Realistic Mathematics Education: A paradigm of developmental research, Dordrecht, Τhe Netherlands: Kluwer, 1991.

[44] E., S. Sweeney, and R., J. Quinn, ―Concentration: Connecting fractions, decimals & percents‖, Mathematics Teaching in the Middle School, vol. 5, pp. 324–328, 2000.

[45] W. Widjaja, K. Stacey, and V. Steinle, ―Locating negative decimals on the number line: Insights into the thinking of pre-service primary teachers‖, Journal of Mathematical Behavior, vol. 30, pp. 80–91, 2011.

[46] Vamvakoussi, X., & Vosniadou, S., Understanding the structure of the set of rational numbers: a conceptual change approach. Learning and Instruction, 14, 453-467, (2004).

[47] X. Vamvakoussi, and S. Vosniadou, ―Bridging the Gap Between the Dense and the Discrete: The Number Line and the ―Rubber Line‖ Bridging Analogy‖, Mathematical Thinking and Learning, vol. 14, pp. 265-284, 2012.

[48] D., C. Yang, R., E. Reys, and L., L. Wu, ―Comparing the Development of Fractions in the Fifth- and Sixth-Graders‘ Textbooks of Singapore, Taiwan, and the USA‖, School Science and Mathematics, vol. 110, 2010.

[49] H. Yoshida, & K. Sawano, ―Overcoming cognitive obstacles in learning fractions: Equal-partitioning and equal-whole‖, Japanese Psychological Research‖, vol. 44, pp. 183-195, 2002.

[50] A. Dreher, S. Kuntze, ―Teachers‘ professional knowledge and noticing: The case of multiple representations in the mathematics classroom‖, Educational Studies in Mathematics, vol. 88, pp. 89-114, 2015.

[51] A. Dreher, S. Kuntze, S. Lerman, ―Why use multiple representations in the mathematics classroom? Views of English and German Preservice teachers‖, International Journal of Science and Mathematics Education, vol. 14, pp.363-382, 2016.

[52] L. P. Steffe, J. Olive, Children ‘ s Fractional Knowledge, Springer, New York, 2010.

[53] E. Avgerinos, and R. Vlachou, ―Towards reducing the difficulties of students in fractions: Instructional practices and theoretical context‖, International Journal of Latest Research in Humanities and Social Science, vol. 1, pp. 9-20, 2018.

[54] E. Avgerinos, R. Vlachou, ―The consistency between the concepts of equal parts of the unit, improper fractions and problem solving at candidate teachers of education departments‖, Proceedings of the 30th Hellenic Conference on Mathematical Education 2013, pp.135-147, Greece, Hellenic Mathematical Society (in Greek), 2013.

[55] F. Rønning, ―Making sense of fractions in different contexts‖, Research in Mathematics Education, vol. 15, pp.201-202, 2013.

[56] C. Howe, S. Luthman, K. Ruthven, N. Mercer, R. Hofmann, S. Ilie, P. Guardia, ―Rational number and proportional reasoning in early secondary school: towards principled improvement in mathematics‖, Research in Mathematics Education, vol. 17, pp.38-56, 2015.

[57] A. J. Shahbari, I. Peled, ―Resolving cognitive conflict in a realistic situation with modeling characteristics: coping with a changing reference in fractions‖, International Journal of Science and Mathematics Education vol. 13, pp.891-907, 2015.

[58] R. Vlachou, and E. Avgerinos, ―Visualization and understanding in mathematics education: The case of fractions‖. The Journal of the ISIS, (accepted), 2016.

[59] E. Deliyianni, A. Gagatsis, I. Elia, A. Panaoura, ―Representational flexibility and problem-solving ability in fraction and decimal number addition: A structural model‖, International Journal of Science and Mathematics Education, vol. 14, pp. 397-417, 2016.

[60] E. Dubinsky, I. Arnon, K. Weller, ―Preservice teachers‘ understanding of the relation between a fraction or integer and its decimal expansion: The case of 0.9 and 1‖, Canadian Journal of Science, Mathematics and Technology Education, vol. 13, pp. 232-258, 2013.

[61] M. J. Tobias, ―Prospective elementary teachers‘ development of fraction language for defining the whole‖, Journal of Mathematics Teacher Education, vol. 16, pp.85-103, 2013.

[62] O. Şahin, B. Gökkurt, Y. Soylu, ―Examining prospective mathematics teachers‘ pedagogical content knowledge on fractions in terms of students‘ mistakes‖, International Journal of Mathematical Education in Science and Technology, vol. 47, pp.531-551, 2016.

[63] E. Thanheiser, D.Olanoff, A. Hillen, Z. Feldman, M.J. Tobias, M.R. Welder, ―Reflective analysis as a tool for task redesign: The case of prospective elementary teachers solving and posing fraction comparison problems‖, Journal of Mathematics Teacher Education, vol. 19, pp.123-148, 2016.

[64] I. Whitacre, D.S. Nickerson, ―Investigating the improvement of prospective elementary teachers‘ number sense in reasoning about fraction magnitude‖, Journal of Mathematics Teacher Education, vol. 19, pp.57-77, 2016.

[65] E. Castro-Rodríguez, D. Pitta-Pantazi, L. Rico, and P. Gómez, ―Prospective teachers ‘ understanding of the multiplicative part-whole relationship of fraction‖, Educational Studies in Mathematics, vol. 92, pp. 129-146, 2016.

[66] S. Card, J. MacKinlay, and B. Shneiderman, Readings in information visualization: Using vision to think, San Francisco: Morgan Kaufmann Publishers, 1999.

[67] A. A. Cuoco, and F.R. Curcio, The roles of representation in school mathematics: 2001 Yearbook, Reston, VA: National Council of Teachers of Mathematics, 2001.

[68] P. Cheng, ―Electrifying diagrams for learning: Principles for complex representational systems‖, Cognitive Science vol. 26, pp. 685–736, 2002.

[69] National Council of Teachers of Mathematics, Principles and standards for school mathematics, Reston, VA: National Council of Teachers of Mathematics, 2000.

[70] E. Jacobson and A. Izsa´k, ―Knowledge and motivation as mediators in mathematics teaching practice: the case of drawn models for fraction arithmetic‖, Journal of Mathematics Teacher Education, vol. 18, pp. 467-488, 2015.

[71] A. Hansen, M. Mavrikis, and E. Geraniou, ―Supporting teachers‘ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative‖, Journal of Mathematics Teacher Education, vol. 19, pp. 205-226, 2016.

[72] R. Duval, ―Registers de représentations sémiotique et fonctionnement cognitif de la pensée. Annales de didactique et de sciences cognitives‖ ULP, IREM Strasbourg. vol. 5, pp. 37-65, 1993.

[73] M. P. Kshetree, ―Nature of Mathematical Content as a Contributing Factor for Students Mathematical Errors‖, International Journal of Mathematics Trends and Technology ( IJMTT ), vol.58, pp. 309-317, 2018.

Roza Vlachou, Evgenios Avgerinos, "Current Trend and Studies on Representations in Mathematics: The Case of Fractions," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 65, no. 2, pp. 54-72, 2019. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V65I2P511