Fraction Sense: An Analysis of Preservice Mathematics Teachers’ Cognitive Obstacles
Abstract
Research on cognitive obstacles related to fraction sense in preservice mathematics teachers is significant, because their success depends on their skills. The acquisition of fraction sense is a complicated problem requiring a strategy to solve it. This study presents cognitive obstacles with fraction sense tests in preservice who will teach in secondary schools. It focuses on the following categories of cognitive obstacles: epistemological (language representation, tendency to generalise and rely on intuition) and didactic (less meaningful learning, and strategy). This paper takes a qualitative descriptive approach to examine 20 preservice mathematics teachers. The preservice teachers who encountered cognitive obstacles related to fraction sense testing were then grouped based on the similarity of their answers, and seven of them were selected to be interviewed. The research findings showed that five preservice teachers had overlapping obstacles: language representation and tendency to generalise; tendency to generalise and less meaningful learning; language representation, tendency to rely on intuition and trial and error strategy in; language representation and trial and error; and language representation and tendency to rely on intuition.
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Akkaya, R. (2016). An investigation into the number sense performance of secondary school students in Turkey. Journal of Education and Training Studies, 4(2), 113–123.
Alenazi, A. A. (2016). Development of an instrument to measure fraction division operation sense. IJAEDU- International E-Journal of Advances in Education, 2(4), 32–39.
Ali, P. (2014). Assessing developmental students’ number sense: A case study. National Association for Developmental Education Digest, 8(1), 2–9.
Bezpalko, O. V., Klishevych, N. A., Liakh, T. L., & Pavliuk, R. O. (2016). Criteria and indicators of university education quality: The results of expert interview. New Educational Review, 46(4), 61–71.
Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2014). Obstacles and affordances for integer reasoning: An analysis of children’ s thinking and the history of mathematics. Journal for Research in Mathematics Education, 45(1), 19–61.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer Academic Publishers.
Charalambous, C. Y., & Pitta-Pantazi, D. (2005). Revisiting a theoretical model on fractions: Implications for teaching and research. In H. L. Chick & J. L. Vincent (Eds.), Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 233–240). PME.
Chattopadhyay, K. N., Sarkar, K. C., & Koner, S. (2017). Number sense of high school students: An assessment. International Research Journal of Interdisciplinary & Multidisciplinary Studies (IRJIMS), III(V), 212–218.
Clarke, D., Roche, A., & Mitchell, A. (2011). Fractions: teaching for understanding. The Australian Association of Mathematics Teachers Inc.
Cornu. (1991). Limits. In H. Bauersfeld, J. Kilpatrick, G. Leder, S. Tumau, & G. Vergnaud (Eds.), Advanced mathematical thinking. (pp. 153–166). Kluwer Academic Publishers.
Cortina, J. L., Visnovska, J., & Zúñiga, C. (2014). Equipartition as a didactical obstacle in fraction instruction. Acta Didactica Universitatis Comenianae Mathematics, 14(1), 1–18.
Courtney-Clarke, M., & Wessels, H. (2014). Number sense of final year preservice primary school teachers. Pythagoras, 35(1), 1–9.
Fatqurhohman, Sa’dijah, C., Irawan, E. B., & Sulandra, I. M. (2017). Representation of secondary school students in solving fractions. International Journal of Innovation in Science and Mathematics, 5(6), 172–176.
Fennell, F. S., & Karp, K. (2017). Fraction sense: Foundational understandings. Journal of Learning Disabilities, 50(6), 648–650.
Iuculano, T., & Butterworth, B. (2011). Rapid communication: Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology, 64(11), 2088–2098.
Lamberg, T., & Wiest, L. R. (2014). Dividing fractions using an area model: A look at in-service teachers’ understanding. Mathematics Teacher Education and Development, 17(1), 30–43.
Lemonidis, C., Tsakiridou, H., & Meliopoulou, I. (2018). In-service teachers’ content and pedagogical content knowledge in mental calculations with rational numbers. International Journal of Science and Mathematics Education, 15(83), 1–19.
Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38(4), 201–221.
Magajna, Z. (2013). Overcoming the obstacle of poor knowledge in proving geometry tasks. Center for Educational Policy Studies Journal, 3(4), 99–116.
Manfreda Kolar, V., Hodnik Čadež, T., & Vula, E. (2018). Primary teacher students’ understanding of fraction representational knowledge in Slovenia and Kosovo. Center for Educational Policy Studies Journal, 8(2), 71–96.
Milles, M.B. & Huberman, A.M. 2009. Analisis data kualitatif [Qualitative data analysis]. UI-Press.
Murniasih, T. R., Sa’dijah, C., Muksar, M., & Susiswo, S. (2018). Errors in representation translation in solving problems related to number sense of preservice math teachers. Annual Conference on Social Sciences and Humanities (ANCOSH) (pp. 393–399). SCITEPRESS – Science and Technology Publications, Lda. A.
Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110.
Nyikahadzoyi, M. R., Mapuwei, T., & Chinyoka, M. (2013). Some cognitive obstacles faced by “a†level mathematics students in understanding inequalities: A case study of bindura urban high schools. International Journal of Academic Research in Progressive Education and Development, 2(2), 2226–6348.
Olanoff, D., Feldman, Z., Welder, R., Tobias, J., Thanheiser, E., & Hillen, A. (2016). Greater number of larger pieces: A strategy to promote prospective teachers’ fraction sense development. In M. B. Wood, M. E. E. Turner, M. Civil, & J. A. Eli (Eds.), Annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 799–805). The University of Arizona.
Ontario Ministry of Education. (2016). A guide to effective instruction in mathematics, grades 1 to 3 – number sense and numeration. Queen’s Printer for Ontario.
Ormond, C. (2012). Developing “algebraic thinkingâ€: Two key ways to establish some early algebraic ideas in primary classrooms. Australian Primary Mathematics Classroom, 17(4), 13–22.
Osana, H. P., & Royea, D. A. (2011). Obstacles and challenges in preservice teachers’ explorations with fractions: A view from a small-scale intervention study. Journal of Mathematical Behavior, 30(4), 333–352.
Pinilla, M. I. F. (2007). Fractions: Conceptual and didactic aspects. Acta Didactica Universitatis Comenianae, 7(1), 23–45.
Prayitno, L. L., Purwanto, P., Subanji, S., & Susiswo, S. (2018). Identification errors of problem posed by prospective teachers about fraction based meaning structure. International Journal of Insights for Mathematics Teaching, 1(1), 76–84.
Prediger, S. (2008). Discontinuities for mental models - A source for difficulties with the multiplication of fractions. In D. D. Bock, B. D. Sondergaard, B A. Gomez, & C. C. L. Cheng. ICME – Research and Development of Number Systems and Arithmetic (pp. 29–36). Monterrey.
Prediger, S. (2008). The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions. Learning and Instruction, 18(1), 3–17.
Prediger, S., & Wessel, L. (2010). Relating registers for fraction – Multilingual students on their way to conceptual understanding. ICMI Study 21 - Mathematics and Language Diversity (pp. 324–333). Sao Paulo.
Purnomo, Y. W., Kowiyah, Alyani, F., & Assiti, S. S. (2014). Assessing number sense performance of Indonesian elementary school students. International Education Studies, 7(8), 74–84.
Rodrigues, J., Dyson, N. I., Hansen, N., & Jordan, N. C. (2017). Preparing for algebra by building fraction sense. Teaching Exceptional Children, 49(2), 134–141.
Sa’dijah, C. (2013). Kepekaan bilangan siswa SMP melalui pembelajaran matematika kontekstual dan mengintegrasikan ketrampilan berpikir kreatif [Number sense of junior high school students through contextual mathematics learning integrates creative thinking skills]. Jurnal Pendidikan Dan Pembelajaran, 20(2), 222–227.
Sbaragli, S., Arrigo, G., D ’amore, B., Isabel, M., Pinilla, F., Frapolli, A., Frigerio, D., & Villa, O. (2011). Epistemological and didactic obstacles: The influence of teachers’ beliefs on the conceptual education of students. Mediterranean Journal for Research in Mathematics Education, 10(1), 1–2.
Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359–371.
Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the learning through activity research program. Journal of Mathematical Behavior, 52(4), 122–133.
Şengül, S. (2013). Identification of number sense strategies used by preservice elementary teachers. Educational Sciences: Theory & Practice, 13(3), 1965–1974.
Sengul, S., & Gulbagci, H. (2012). An investigation of 5th grade Turkish students’ performance in number sense on the topic of decimal numbers. In G. A. Baskan, F. Ozdamli, Sezer Kanbul, & D. Özcan (Eds). Procedia - Social and Behavioral Sciences (pp. 2289–2293). Elsevier Ltd.
Son, J., & Lee, J. (2016). Preservice teachers’ understanding of fraction multiplication, representational knowledge, and computational skills. Mathematics Teacher Education and Development. 18(2), 5–28.
Subanji. (2016). Peningkatan pedagogical content knowledge guru matematika dan praktiknya dalam pembelajaran melalui model pelatihan TEQIP [Improved pedagogical content knowledge teacher mathematics and practices in learning through TEQIP training model]. Jurnal Ilmu Pendidikan, 21(1), 71–79.
Way, J. (2011). Developing fraction sense using digital learning objects. The Australian Association of Mathematics Teachers Inc.
Whitacre, I., & Nickerson, S. D. (2016). Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude. Journal of Mathematics Teacher Education, 19(1), 57–77.
Woodward, T. L. (1998). An exploration of grade 8 students’ fraction sense. Simon Fraser.
Yaman, H. (2015). The mathematics education I and II courses’ effect on teacher candidates’ development of number sense. Educational Sciences: Theory & Practice, 15(4), 1119–1135.
Yang, D. C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by preservice teachers in Taiwan. International Journal of Science and Mathematics Education, 7(2), 383–403.
Yoshida, H., & Sawano, K. (2002). Overcoming cognitive obstacles in learning fractions : Equal-partitioning and equal-whole. Japanese Psychological Research, 44(4), 183–195.
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