pp. 749-767 | Article Number: iejme.2017.044
Published Online: November 15, 2017
Article Views: 79 | Article Download: 86
This paper deals with precursory (propaedeutic) learning of the concept of number in the elementary mathematical education. The authors’ objective is to suggest a method allowing for the increase of the effectiveness of interactive expansion of the concept of number by using a grade-appropriate learning framework for elementary mathematical education content. A theoretical background for the creation of this method is based on the description of various characteristics of precursory learning and interactive teaching of mathematics as well and the flexible differentiation approach. The paper especially emphasizes the possibilities of propaedeutic understanding of the concept of fraction and examine the effects of such approach in terms of student achievement in elementary mathematics education, on the basis of a methodological approach. Results obtained during the experimental research suggest that under the influence of the methodological approach of introducing fractions through propaedeutic learning, students achieve significantly better results in learning compared to students who have not used this method.
Keywords: Fractions ∙ Propaedeutic learning ∙ Methodological approach ∙ Flexible differentiation ∙ Empirical evaluation
Abramovich, S., Easton, J. & Hayes, V.O. (2012). Parallel structures of computer – assisted signature pedagogy: the case of integrated spreadsheets, Computers in the Schools, 29(1-2), 174-190.
Amato, S. A. (2005). Developing students‘understanding of the concept of fractions as numbers. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th PME International Conference, 2, 49–56.
Behr, M., Lesh, R., Post, T. & Silver, E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press.
Behr, M., Harel, G., Post, T. & Lesh, R. (1993). Rational Numbers: Toward a Semantic Analysis-Emphasis on the Operator Construct. In T. P. Carpenter, E. Fennema, & T.A. Romberg, (Eds.), Rational Numbers: An Integration of Research (pp. 13-47). NJ: Lawrence Erlbaum.
Berlin, D. F. & White, A. L. (1995). ,,Connecting School Science and Mathematics”. In: Connecting Mathematics across the Curriculum, Ed. House, P. A. & Coxford, A. F., National Council of Teachers of Mathematics, 1995. Yearbook, Reston, Virginia.
Charalambous, C.Y. & Pitta-Pantazi, D. (2005), Revisiting a theoretical model on fractions: Implications for teaching and researching, In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 233-240), Melbourne: PME.
Charalambous, C.Y. & Pitta-Pantazi, D. (2007), Drawing on a theoretical model to study students’ understanding of fractions, Educational Studies in Mathematics, 64(3): 293–316.
Fosnot, C. (2007). Field Trips and Fund Raisers: Introducing Fractions. Portsmouth: Heineman.
Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., Herpen, E. & Keijzer, R. (2008). Fractions, percentages, decimals and proportions: A learning-teaching trajectory for grade 4, 5 and 6. Rotterdam: Sense.
Gleizer, D. G. (1997). Geometry in the school: problems and judgments. Norma, 3 (1-2), 9-20.
Hackenberg, A.J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383-342.
Hasegawa, J. (2000). Classroom discussion on the representation of quantity by frаctions: Stability of misconcepuon and implications to practice. In T. Nakahara & M. Koyama (Eds.). Proceedings of the 24th PME International Conference, 3, 41–48.
Kieren, T.E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and Measurement: Papers from a Research Workshop (pp. 101-144). Columbus, OH: ERIC/SMEAC.
Kieren, T.E. (1995), Creating Spaces for Learning Fractions. In: J. T. Sowder & B. P. Schappelle (Eds.), Providing a Foundation for Teaching Mathematics in the Middle Grades, (pp. 31-66). Albany: State University of New York Press.
Klippert, H. (2001). How to successfully teach the team. Zagreb: Eduka.
Lamon, S.J. (2012). Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers. New York, NY and London, UK. Routledge.
Lazić, B. (2015). Propaedeutic Introduction of Fractions in Arithmetics for Lower Grades of Primary School, Ph.D. Thesis, University of Belgrade, Teaching Training Faculty.
Lazić, B. & Maričić, S. (2015). Propaedeutic formation of the concept of fraction in elementary mathematics education; In: Novotná, J. & Moraová, H. (Eds.), Proceedings if Developing Mathematical Language and Reasoning, (pp. 212-221), Charles University, Faculty of Education, Prague.
Lazić, B., Milinković, J. & Petojević, A. (2012). Connecting mathematics in propaedeutic exploration of the concept of fraction in elementary grades, In: Brankovic, N. (Ed.), Theory and Practice of Connecting and Integrating in Teaching and Learning Process (pp. 123–137). Sombor: Faculty of Education in Sombor.
Mamede, E., Nunes, T. & Bryant, P. (2005). The equivalence and ordering of fractions in part-whole and quotient situations. In H. L. Chick & J. L. Vincent (Eds.). Proceedings of the 29th PME International Conference, 3, 281–288.
Marshall, S.P. (1993). Assessment of rational number understanding: A schema-based approach, in T.P. Carpenter, E. Fennema & T.A. Romberg (Eds.), Rational Numbers: An Integration of Research, (pp. 261–288) Lawrence Erlbaum Associates, New Jersey.
Milinković, J. (2007). Methodological aspects of the introduction to probability and statistics. Belgrade: Faculty of Pedagogy.
Mrdja, M., Crvenković, S. & Milovanovic, J. (2015). The increase in efficiency of interactive learning of mathematics through the implementation of mini exemplary teaching, IMVI Open Mathematical Education Notes, 5(2): 87–99.
Petrovic, N., Mrdja, M. & Lazic, B. (2011). Models of differentiated interactive classroom teaching of mathematics. Norma, 15(2): 211-228.
Pitta-Pantazi, D., Gray, E. M. & Christou, C. (2004). Elementary school students’ mental representations of fractions. In M. J. Hoines & A. D. Fuglestad (Eds.). Proceedings of the 28th PME International Conference, 4, 41–48.
Prediger, S. (2013). Focussing structural relations in the bar board – a design research study for fostering all students’ conceptual understanding of fractions. In B. Ubuz, C. Haser & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in mathematics Education, Antalya, 343–352.
Rasmussen, P. (2004). Towards flexible differentiation in higher education?: recent changes in Danish higher education, In Fägerlind, I. and Strömqvist, G. (Eds.), Reforming higher education in the Nordic countries –studies of change in Denmark, Finland, Iceland, Norway and Sweden, International Institute for Educational Planning, Paris.
Santos, D.A. (2008). Andragogic propaedeutic mathematics, Available online at: http://www.freemathtexts.org/Santos/PDF/Arithmetic(2008).pdf..
Schon, D.A. (1963). Invention and the evolution of ideas. London: Social science paperbacks.
Shulman, L. S. (2005). Signature pedagogy in the professions. Daedalus, 134(3): 52-59.
Siegler, R.S., Thompson, C.A. & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4): 273–296.
Siegler, R.S., Fazio, L.K., Bailey, D.H. & Zhou, X. (2013). Fractions: the new frontier for theories of numerical development, Trends in Cognitive Sciences, 17(1): 13–19.
Small, M. (2009). Teaching to the Big Ideas K-3, Mathfocus K-3, Nelson.
Smith, J. P. (2002). The development of students’ knowledge of fractions and ratios. In Litwiller, B. ve Bright, G. (Eds). Making Sense of Fractions, Ratios, and Proportions: Yearbook. P. 1-2. NCTM: Reston, VA.
Steffe, L.P. & Olive, J. (2009). Children’s Fractional Knowledge. New York: Springer.
Strang, T. (1990). The fraction-concept in comprehensive school at grade-levels 3-6 in Finland. In G. Booker, P. Cobb & T. N. Mendicuti (Eds.), Proceedings of the 14th PME International Conference, 3, 75–80.
Tennyson, R. D. & Park, O. (1980). The teaching concept: A review of instructional design research literature. Review of Educational Research, 50(1): 55–70.
Torbeyns, J., Schneider, M., Xin, Z. & Siegler, R.S. (2014). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37(1): 5-13.
Vasilyev, N.S. & Gromyko, V.I. (2015). Propaedeutic Mathematical courses in the context of continuous learning, [Н.С. Васильев, В.И. Громыко, Пропедевтические курсы математики в условиях непрерывного образования. Гуманитарный вестник, МГТУ им. Н.Э. Баумана, 2(28):1-17].
Vygotsky, L.S. (1987). Thinking and Speech. In R.W. Rieber and A.S. Carton (Eds.), The collected works of L.S. Vygotsky (vol. 1, pp.39-285). New York: Plenum Press.
Watanabe, T. (2006). Teaching and learning of fractions: A Japanese perceptive, Teaching Children Mathematics, 12(7), 368–372.
Watanabe, T. (2012). Thinking about learning and teaching sequences for the addition and subtraction of fractions. In C. Bruce (Chair), Think Tank on the Addition and Subtraction of Fractions. Think Tank conducted in Barrie, Ontario.
Wittmann, G. (2013). The consistency of students’ error patterns in solving computationalproblems with fractions. In B. Ubuz, C. Haser & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in mathematics Education, Antalya, 393–402.
Zech, F. (1999). Grundkurs Mathematikdidaktik - Theoretische und praktische Anleitungen für das Lehren und Lernen von Mathematik, Beltz Verlag - Weinheim und Basel.