Knowledge of preservice elementary teachers on fractions

Keywords: Didactic analysis, Pedagogical content knowledge, Didactic content, Preservice teacher training, Fractions, Mathematical knowledge, Mathematics education

Abstract

In this paper, we studied the knowledge on didactic contents reflected by future elementary teachers when answering questions related to teaching and learning fractions. Following a qualitative methodology, specifically a case study, 9 senior pre-service elementary teachers were interviewed using a structured approach. The topic was presented to the subjects using a narrative they had previously written on how to initiate the concept of fractions with school children. After reading it, they were asked questions regarding task design, learning objectives, and mistakes and difficulties. Results identified two trends in the participants’ knowledge: a procedural or technical trend in which the stated knowledge emphasizes procedures, processes, or action modes, and a conceptual or cognitive trend in which the stated knowledge emphasizes the functional understanding of fractions and their relationships. As a conclusion, it is essential that initial teacher training emphasizes mathematical as well as didactic contents.

References

Adu-Gyamfi, K., Schwartz, C. S., Sinicrope, R. and Bossé, M. (2019). Making sense of fraction division: domain and representation knowledge of preservice elementary teachers on a fraction division task. Mathematics Education Research Journal, 31, 507-528. https://doi.org/10.1007/s13394-019-00265-2
Behr, M. J., Lesh, R. Post, T. R. and Silver, E. A. (1983). Rational number concept. In R. Lesh and M. Landau (Editors), Acquisitions of Mathematics Concepts and Processes (p. 91-126). Academy Press.
Behr, M. J., Wachsmuth, I., Post, T. R. and Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323–341. https://doi.org/10.2307/748423
Castro-Rodríguez, E., Pitta-Pantazi, D., Rico, L. and Gómez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129-146. https://doi.org/10.1007/s10649-015-9673-4
Charalambous, C. Y. (2016). Investigating the Knowledge Needed for Teaching Mathematics: An Exploratory Validation Study Focusing on Teaching Practices. Journal of Teacher Education, 67(3) 220-237. https://doi.org/10.1177/0022487116634168
Charalambous, C. Y., Hill, H. C. and Ball, D. L. (2011). Prospective teachers’ learning to provide instructional explanations: How does it look and what might it take? Journal of Mathematics Teacher Education, 14(6), 441-463. https://doi.org/10.1007/s10857-011-9182-z
Cramer, K. A., Post, T. R., and del Mas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: a comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111-144. https://doi.org/10.2307/749646
D’Ambrosio, B. S. and Mendonça-Campos, T. N. (1992). Pre-service teachers’ representations of children’s understanding of mathematical concepts: Conflicts and conflict resolution. Educational Studies in Mathematics, 23(3), 213-230. https://doi.org/10.1007/BF02309530
Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssen, R., Kelchtermans, G., and Van Dooren, W. (2015). Teachers’ content and pedagogical content knowledge on rational numbers: A comparison of prospective elementary and lower secondary school teachers. Teaching and teacher education, 47, 82-92. https://doi.org/10.1016/j.tate.2014.12.009
Domoney, B. (2001). Student teachers’ understanding of rational numbers. In J. Winter (Ed.), Proceedings of the British Society for Research into Learning Mathematics, (Vol. 21, num. 3, pp. 13-18). BSRLM.
Isiksal, M. and Cakiroglu, E. (2011). The nature of prospective mathematics teachers’ pedagogical content knowledge: The case of multiplication of fractions. Journal of Mathematics Teacher Education, 14(3), 213-230. https://doi.org/10.1007/s10857-010-9160-x
Jacobson, E. and Izsák, 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. https://doi.org/10.1007/s10857-015-9320-0
Jacobson, E., Lobato, J. and Orrill, C. H. (2018). Middle school teachers’ use of mathematics to make sense of student solutions to proportional reasoning problems. International Journal of Science and Mathematics Education, 16, 1541-1559. https://doi.org/10.1007/s10763-017-9845-z
Kang, R. and Liu, D. (2018). The importance of multiple representations of mathematical problems: Evidence from Chinese preservice elementary teachers’ analysis of a learning goal. International Journal of Science and Mathematics Education, 16, 125-143. https://doi.org/10.1007/s10763-016-9760-8
Kieren, T. E. (1993). The Learning of Fractions: Maturing in a Fraction. Paper presented at the Conference Fraction Learning and Instruction, GA, Athens.
Klemer, A., Rapoport, S. and Lev-Zamir, H. (2019). The missing link in teachers’ knowledge about common fractions division. International Journal of mathematical education in science and technology, 50(8), 1256-1272. https://doi.org/10.1080/0020739X.2018.1522677
Krippendorff, K. (1990). Metodología de análisis de contenido. Teoría y práctica. Paidós. In Spanish.
Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410617132
Lee, M. Y. and Lee, J. E. (2019). Pre-service Teachers’ Perceptions of the Use of Representations and Suggestions for Students’ Incorrect Use. Eurasia Journal of Mathematics, Science and Technology Education, 15(9), em1745. https://doi.org/10.29333/ejmste/103055
Lee, S. J., Brown, R. E. and Orrill, C. H. (2011). Mathematics teachers’ reasoning about fractions and decimals using drawn representations. Mathematical Thinking and Learning, 13(3), 198-220. https://doi.org/10.1080/10986065.2011.564993
Li, Y. and Kulm, G. (2008). Knowledge and confidence of pre-service mathematics teachers: the case of fraction division. ZDM, 40, 833-843. https://doi.org/10.1007/s11858-008-0148-2
Lo, J. and Grant, T. (2012). Preservice elementary teachers’ conceptions of fractional units. In T. Y. Tso (Ed.), Proceedings of the 36th PME (Vol. 3, p. 169-176). PME.
Lupiáñez, J. L. (2009). Expectativas de aprendizaje y planificación curricular en un programa de formación inicial de profesores de matemáticas de secundaria [Doctoral Dissertation]. Universidad de Granada, España. https://fqm193.ugr.es/produccion-cientifica/tesis/ver_detalles/5489/. In Spanish.
Lupiáñez, J. L. (2013). Análisis didáctico: La planificación del aprendizaje desde una perspectiva curricular. In L. Rico, J. L. Lupiáñez and M. Molina (Editors), Análisis didáctico en educación matemática (p. 81-102). Comares. In Spanish.
Mack, N. K. (1990). Learning fractions with understanding: building on informal knowledge. Journal for Research in Mathematics Education, 21(1), 16-32. https://doi.org/10.5951/jresematheduc.21.1.0016
Marks, R. (1990). Pedagogical Content Knowledge: From a Mathematical Case to a Modified Conception. Journal of Teacher Education, 41(3), 3-11. https://doi.org/10.1177/002248719004100302
Morris, A. K., Hiebert, J. and Spitzer, S. M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can preservice teacher learn? Journal for Research in Mathematics Education, 40(5), 491-529. https://doi.org/10.5951/jresematheduc.40.5.0491
Movshovitz-Hadar N., Zaslavsky O. and Inbar S. (1987). An empirical classification model for errors in high school mathematics. Journal for Research in Mathematics Education, 18, 3-14. https://doi.org/10.2307/749532
Newton, K. J. (2008). An Extensive Analysis of Preservice Elementary Teachers’ Knowledge of Fractions. American Educational Research Journal, 45(4), 1080-1110. https://doi.org/10.3102/0002831208320851
Rico, L. (2016). Matemática y análisis didáctico. In L. Rico and A. Moreno (Editors), Elementos de didáctica de la matemática para el profesor de secundaria (p. 85-100). Pirámide. In Spanish.
Rojas, N., Flores, P. and Carrillo, J. (2015). Conocimiento especializado de un profesor de matemáticas de Educación Primaria al enseñar los números racionales. Boletim de Educação Matemática, 29(51), 143-166. https://doi.org/10.1590/1980-4415v29n51a08. In Spanish.
Rosli, R., Goldsby, D., Onwuegbuzie, A. J., Capraro, M. M., Capraro, R. M. and González, E. G. Y. (2020). Elementary preservice teachers’ knowledge, perceptions, and attitudes towards fractions: A mixed-analysis. Journal on Mathematics Education, 11(1), 59-76. http://doi.org/10.22342/jme.11.1.9482.59-76.
Şahin, O., Gökkurt, B. and Soylu, Y. (2016). Examining prospective mathematics teachers’ pedagogical content knowledge on fractions in terms of students’ mistakes. International Journal of Mathematical Education in Science and Technology, 47(4), 531-551. https://doi.org/10.1080/0020739X.2015.1092178
Sánchez, M. (2011). A review of research trends in mathematics teacher education. PNA, 5(4), 129-145.
Socas, M. M. (1997). Dificultades, obstáculos y errores en el aprendizaje de las matemáticas en la educación secundaria. In L. Rico (Coord.), E. Castro, E. Castro, M. Coriat, A. Marín, L. Puig, et al., La educación matemática en la enseñanza secundaria (p. 125-154). Horsori. In Spanish.
Stake, R. E. (2010). Investigación cualitativa: El estudio de cómo funcionan las cosas. The Guilford Press. In Spanish.
Steffe, L. P. and Olive, J. (1990). Constructing fractions in computer microworlds. In G. Booker, P. Cobb and T. N. de Mendicuti (Editors), Proceedings of the 14th PME (p. 59-66). CONACYT.
Streefland, L. (1991). Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research. Kluwer Academic Publishers. https://doi.org/10.1007/978-94-011-3168-1
Tirosh, D. (1999). Forms of mathematical knowledge: learning and teaching with understanding. In D. Tirosh (Ed.), Forms of mathematical knowledge (p. 1-9). Springer. https://doi.org/10.1007/978-94-017-1584-3_1
Tröbst, S., Kleickmann, T., Depaepe, F., Heinze, A. and Kunter, M. (2019). Effects of instruction on pedagogical content knowledge about fractions in sixth-grade mathematics on content knowledge and pedagogical knowledge. Unterrichtswissenschaft, 47(1), 79-97. https://doi.org/10.1007/s42010-019-00041-y
Tröbst, S., Kleickmann, T., Heinze, A., Bernholt, A., Rink, R. and Kunter, M. (2018). Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic. Journal of Educational Psychology, 110(8), 1049-1065. https://doi.org/10.1037/edu0000260
Whitacre, I., Atabaş, Ş. and Findley, K. (2019). Exploring unfamiliar paths through familiar mathematical territory: Constraints and affordances in a preservice teacher’s reasoning about fraction comparisons. The Journal of Mathematical Behavior, 53, 148-163. https://doi.org/10.1016/j.jmathb.2018.06.006
Wu, H. H. (2018) The content knowledge mathematics teachers need. In Y. Li, W. Lewis, J. Madden (Editors), Mathematics matters in education. Advances in STEM education (p. 43-91). Springer. https://doi.org/10.1007/978-3-319-61434-2_4
Published
2021-07-31
How to Cite
Castro-Rodríguez, E., & Rico, L. (2021). Knowledge of preservice elementary teachers on fractions. Uniciencia, 35(2), 1-20. https://doi.org/10.15359/ru.35-2.10
Section
Original scientific papers (evaluated by academic peers)

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