# Μultıple representatıons and development of students' self-confidence on rational number

### Abstract

- In spite of the fact that analytical programs change and mathematical school texts adapt to new education needs, students, internationally, continue to have difficulties when handling fractions. This paper presents the results of a research conducted on students of the 5th and 6th grade of elementary school and its purpose was to investigate at what extent the use of multiple representations is possible to help students cope with difficulties on mathematics and thus to boost their self -confidence in mathematics. For that purpose, the concepts of fractions was chosen are classification of fractions as a representation on the number line, as well as the concepts of the unit’s division in equal parts and the concept of the improper fractions. Thus, we present education practices were applied by our research team. These teaching practices take into account the results as stated by international bibliographies as well as years of research of our team on rational numbers. They emphasize on multiple representations, use of experiential activities and activities carried out on electronic platforms. In additional, the present research deepens with semi-structured interviews of the participants. The results of the research indicate that students after instructive interventions with the use of multiple representations performed better on fractions and increased their self -esteem in mathematics.

### References

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

Boyce S., Norton A., (2016). Co-construction of fractions schemes and units coordinating structures. The Journal of Mathematical Behavior, 41, 10-25.

Brousseau G., Brousseau N., Warfield V., (2007). Rationals and decimals as required in the school curriculum Part 2: From rationals to decimals. The Journal of Mathematical Behavior, 26(4), 281-300.

Card S., MacKinlay J., Shneiderman B., (1999). Readings in information visualization: Using vision to think. San Francisco: Morgan Kaufmann Publishers.

Castro-Rodríguez E., Pitta-Pantazi D., Rico L., Gómez P., (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129-146.

Cheng P., (2002). Electrifying diagrams for learning: Principles for complex representational systems. Cognitive Science 26(6), 685–736.

Chen X., Li Y., (2009). Instructional coherence in Chinese mathematics classroom a case study of lessons on fraction division. International Journal of Science and Mathematics Education, 8(4), 711-735.

Chen X., Li Y., (2009). Instructional coherence in Chinese mathematics classroom a case study of lessons on fraction division. International Journal of Science and Mathematics Education, 8(4), 711-735.

Cohen L., Manion L., Morrison K., (2011). Research methods in education. UK: Routledge.

Cuoco A. A., Curcio F.R., (2001). The roles of representation in school mathematics: 2001 Yearbook. Reston, VA: National Council of Teachers of Mathematics.

Deliyianni E., Gagatsis A., Elia I., Panaoura A., (2016). Representational flexibility and problem-solving ability in fraction and decimal number addition: A structural model. International Journal of Science and Mathematics Education, 14(2), 397-417.

Dreher A., Kuntze S., (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89-114.

Dreher A., Kuntze S., Lerman, S., (2016). Why use multiple representations in the mathematics classroom?

Views of English and German preservice teachers. International Journal of Science and Mathematics Education, 14(2), 363-382.

Duval R. (1993). Registres de représentations sémiotique et fonctionnement cognitif de la pensée. Annales de didactique et de sciences cognitives, ULP, IREM Strasbourg. 5, 37-65.

Empson S. B., Levi L., Carpenter T. P., (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai, & E. J. Knuth (Eds.), Early algebraization (pp. 409 – 428). Berlin, Germany: Springer.

Fandiño Pinilla, M. I. (2007). Fractions: conceptual and didactic aspects. Acta Didactica Universitatis Comenianae. 7, 23-45.

Gras R., Peter P., Briand H., Philippe J., (1997). Implicative statistical analysis. In C. Hayashi, N. Ohsumi, N. Yajima, Y. Tanaka, H. Bock, & Y. Baba (Eds.), Proceedings of the 5th Conference of the International Federation of Classification Societies (pp. 412-419). Tokyo, Berlin, Heidelberg, New York: Springer-Verlag.

Hackenberg A. J., (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. The Journal of Mathematical Behavior, 26(1), 27-47.

Hackenberg J. A., (2013). The fractional knowledge and algebraic reasoning of students with the ﬁrst multiplicative concept. The Journal of Mathematical Behavior, 32(4), 538-563.

Hansen A., Mavrikis M., Geraniou, E., (2016). Supporting teachers’ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative. Journal of Mathematics Teacher Education, 19(2-3), 205-226.

Hannula M. S., Maijala H., Pehkonen E., Soro R. (2002). Taking a step to infinity: Student’s confidence with infinity. Tasks in School Mathematics. In S. Lehti & K. Merenluoto (Eds.) Third European Symposium on Conceptual Change – A Process Approach to Conceptual Change (pp. 195–200). University of Turku: Dept Teacher Education in Turku.

House J. (2000). Student self-beliefs and science achievement in Ireland: Findings from the third international mathematics and science study (TIMMS). International Journal of Instructional Media 27(1), 107-115.

Howe C., Luthman S., Ruthven K., Mercer N., Hofmann R., Ilie S., Guardia P., (2015). Rational number and proportional reasoning in early secondary school: towards principled improvement in mathematics. Research in Mathematics Education, 17(1), 38-56.

Jacobson E., Izsa´k A., (2015). Knowledge and motivation as mediators in mathematics teaching practice:

the case of drawn models for fraction arithmetic. Journal of Mathematics Teacher Education, 18(5), 467-488.

Janvier C., (1987). 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.

Jordan N. C., Hansen N., Fuchs L. S., Siegler R. S., Gersten R., Micklos D., (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116(1), 45 – 58.

Kieren T. E., (1992). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. In R. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 323 – 371). Hillsdale, NJ: Lawrence Erlbaum Associates.

Lester F. K., Garofalo J., Kroll, D. L. (1989). Self-confidence, interest, beliefs, and metacognition: Key influences on problem-solving behavior. In D. B. McLeod & V. M. Adams (Eds.), Affect and Mathematical Problem Solving (pp. 75-88). New York: Springer-Verlag.

Lee J. S., Shin J., (2015). Distributive partitioning operation in mathematical situations involving fractional quantities. International Journal of Science and Mathematics Education, 13(2), 329-355.

Lee M. Y, Hackenberg A. J., (2014). Relationships between fractional knowledge and algebraic reasoning:

The case of Willa. International Journal of Science and Mathematics Education, 12(4), 975-1000.

Lo J-J., (1993). Conceptual bases of young children’s solution strategies of missing value proportional tasks. Proc. of the Seventeenth International Conference of Psychology of Mathematics Education (pp. 162-177). Tsukuba, Japan: University of Tsukuba.

Mack N. K., (2001). Building on informal knowledge through instruction in a complex content domain:

Partitioning, units and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267 – 295.

National Council of Teachers of Mathematics, (2000). Principles and standards for school mathematics.

Reston, VA: National Council of Teachers of Mathematics.

Olive J., Vomvoridi E., (2006). Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim. Journal of Mathematical Behavior, 25(1), 18–45.

Petakos, K., (2016). Comparing fractions at the age of 11 through the use of the zone of proximal development. Experiences of Teaching with Mathematics, Sciences and Technology, 2(2), 369-375.

Rønning F., (2013). Making sense of fractions in different contexts. Research in Mathematics Education, 15(2), 201-202.

Ryken A., (2009). Multiple representations as sites for teacher reﬂection about mathematics learning.

Journal of Mathematics Teacher Education, 19(2-3), 205-226.

Schoenfeld A. (1992). Learning to think mathematically: problem solving, metacognition and sense making in mathematics. In A. D. Grows (Ed.), Handbook of research on mathematics learning and teaching, (pp.334-370).

Sedig K., Sumner M., (2006). Characterizing interaction with visual mathematical representations.

International Journal of Computers for Mathematical Learning, 11(2), 1–55.

Sfard A., (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.

Shahbari A. J, Peled I., (2015). Resolving cognitive conflict in a realistic situation with modeling characteristics: coping with a changing reference in fractions. International Journal of Science and Mathematics Education, 13(4), 891-907.

Steffe L. P., Olive J., (2010). Children ’ s fractional knowledge. New York: Springer.

Streefland L., (1991). Fractions in realistic mathematics education: A paradigm of developmental research.

Dordrecht, Τhe Netherlands: Kluwer.

Tobias M. J., (2013). Prospective elementary teachers’ development of fraction language for deﬁning the whole. Journal of Mathematics Teacher Education, 16(2), 85-103.

*EDiMaST: Experiences of Teaching With Mathematics, Sciences and Technology*,

*4*, 567-586. Retrieved from https://www.edimast.it/index.php/edimast/article/view/59

Copyright (c) 2018 Roza Vlachou, Evgenios Avgerinos

This work is licensed under a Creative Commons Attribution 4.0 International License.