The Teaching of Initial Multiplication Concepts and Skills in Croatian Textbooks
Abstract
The goal of this paper is to describe the teaching of initial multiplication concepts and skills, up to the multiplication table, in the Croatian educational system. As Stiegler and Hiebert (1999) concluded, teaching is a complex system rooted in a cultural script of a given society. To describe it without ignoring certain features of it that appear to be self-evident to an insider, it is necessary to step out of this cultural frame. For that reason, we study the teaching of initial multiplication in Croatia by comparing Croatian mathematical textbooks with textbooks from Singapore, Japan, and England. For the textbook analysis, we adapt the framework of Charalambous, Delaney, Hsu, and Mesa that examines a textbook as an environment for the construction of knowledge of a single mathematical concept. The analysis provides evidence that practice and automation are at the centre of the initial learning of multiplication in Croatia. The meaning of multiplication usually is not clear, and pupils are not provided additional tools for developing understanding, nor they are encouraged to use different calculation strategies in a flexible manner. The study also indicates that Croatian textbooks present mathematics as a practice that is closed and pre-given, restricted to the one and the only right way through it.
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References
Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 20(4), 367–385. https://doi.org/10.1007/BF00315607
Baković, T., TrupÄević, G., & Valent, A. (2019). Treatment of initial multiplication in textbooks from Croatia and Singapore. In Z. Kolar-Begović, R. Kolar-Å uper, & Lj. Jukić Matić (Eds.), Towards new perspectives on mathematics education (pp. 216–228). Element.
Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and context. Educational Studies in Mathematics, 15(2), 129–147. https://doi.org/10.1007/BF00305893
Boaler, J. (2016). Mathematical mindsets: unleashing students potential through creative math, inspiring messages. and innovative teaching. Jossey-Bass.
Boonlerts, S., & Inprasitha, M. (2013). The Textbook Analysis on Multiplication: The Case of Japan, Singapore and Thailand. Creative Education, 4(4), 259–262. https://doi.org/10.4236/ce.2013.44038
Charalambous, C.Y., Delaney, S., Hsu, H.-Y., & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12(2), 117–151. https://doi.org/10.1080/10986060903460070
Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27(1), 41–51. https://doi.org/10.2307/749196
Department for Education [DfE]. (2013). Mathematics programmes of study: Key stages 1 and 2. National curriculum in England. DfE.
Fan, L., & Kaeley, G. S. (2000). The influence of textbooks on teaching strategies: An empirical study. Mid-Western Educational Researcher, 13(4), 2–9.
Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: Development status and directions. ZDM: The International Journal on Mathematics Education, 45(5), 633–646. https://doi.org/10.1007/s11858-013-0539-x
Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17. https://doi.org/10.2307/748969
Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work. Constructing multiplication and division. Heinemann.
Foong, P. Y., Chang S.H., Lim L. G. P., & Wong O.H. (2014). Primary mathematics 1B. Shinglee.
Foong, P. Y., Chang S.H., Lim L. G. P., & Wong O.H. (2015). Primary mathematics 3A. Shinglee.
Foong, P. Y., Lim L. G. P., & Wong O. H. (2014). Primary mathematics 2A. Shinglee.
Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. Aldine.
Glasnović Gracin, D. & Domović, V. (2009). Upotreba matematiÄkih udžbenika u nastavi viÅ¡ih razreda osnovne Å¡kole [The use of mathematics textbooks in teaching of upper primary grades]. Odgojne znanosti, 18(2), 45–65. https://hrcak.srce.hr/48441
Glasnović Gracin, D., & Jukić Matić, Lj. (2016). The role of mathematics textbooks in lower secondary education in Croatia: An empirical study. The Mathematics Educator 16(2), 31–58.
Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan.
Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German classrooms: Who gets an opportunity to learn what? British Educational Research Journal, 28(4), 567–590. https://doi.org/10.1080/0141192022000005832
Hong, D. S., & Choi, K. M. (2014). A comparison of Korean and American secondary school textbooks: the case of quadratic equations. Educational Studies in Mathematics, 85(2), 241–263. https://doi.org/10.1007/s10649-013-9512-4
Isoda, M. (Ed.) (2010). Elementary school teaching guide for the Japanese course of study mathematics (Grade 1-6). CRICED, University of Tsukuba. https://www.criced.tsukuba.ac.jp/math/apec/ICME12/Lesson_Study_set/Elementary_School_Teaching_Guide-Mathematics-EN.pdf
Jones, K., & Fujita, T. (2013). Interpretations of national curricula the case of geometry in textbooks from England and Japan. ZDM Mathematics Education, 45(5), 671–683. https://doi.org/10.1007/s11858-013-0515-5
Jukić Matić, Lj. (2019). The teacher as a lesson designer. Center for Educational Policy Studies Journal, 9(2), 139–160. Ttps://doi.org/10.26529/cepsj.722
Kheong, F. H., Ramakrishnan, C., & Choo, M. (2014). My pals are here maths 2A. MC Education.
Kheong, F. H., Ramakrishnan, C., & Choo, M. (2015). My pals are here maths 3A. MC Education
Kheong, F. H., Ramakrishnan, C., & Wah, B. L. P. (2013). My pals are here maths 1B. MC Education.
Kolar, V. M., 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. https://doi.org/10.26529/cepsj.342
Kouba, V. L. (1989). Children's solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20(2), 147–158. https://doi.org/10.2307/749279
Markovac, J. (2015). Matematika 2 [Mathematics 2]. Alfa.
Miklec, D., Prtajin, G., & Jakovljević Rogić, S. (2015). Moj sretni broj 2 [My lucky number 2]. Školska knjiga.
Ministarstvo znanosti obrazovanja i Å¡porta [MZOÅ ]. (2006). Nastavni plan i program za osnovnu Å¡kolu. [Course of study for primary school]. MZOÅ .
Ministry of Education Singapore [MOE]. (2012). Mathematics syllabus: Primary one to Six. MOE. https://www.moe.gov.sg/-/media/files/primary/mathematics_syllabus_primary_1_to_6.pdf
Mulligan, J. T., & Mitchelmore, M. C. (1997). Young children's intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28(3), 309. https://doi.org/10.2307/749783
Ministarstvo znanosti i obrazovanja [MZO]. (2019). Odluka o donošenju kurikuluma za nastavni predmet Matematike za osnovne škole i gimnazije u Republici Hrvatskoj [Decision of declaration of the curriculum of Mathematics for primary and secondary school]. Narodne novine 7/2019. https://narodne-novine.nn.hr/clanci/sluzbeni/2019_01_7_146.html
Ministarstvo znanosti obrazovanja i sporta [MZOS]. (2014). KonaÄne liste odabranih udžbenika i pripadajućih dopunskih nastavnih sredstava prema odabiru struÄnih aktiva u osnovnim i srednjim Å¡kolama [Final lists of selected textbooks and related supplementary teaching aids according to the selection of professional boards in primary and secondary schools]. https://mzo.gov.hr/vijesti/konacne-liste-odabranih-udzbenika-i-pripadajucih-dopunskih-nastavnih-sredstava-prema-odabiru-strucnih-aktiva-u-osnovnim-i-srednjim-skolama/1031
Nunes, T., & Bryant, P. (1996). Children doing mathematics. Blackwell.
Nunes, T., Bryant, P., Evans, D., & Bell, D. (2010). The scheme of correspondence and its role in children's mathematics. British Journal of Educational Psychology, 2(7), 83–99. https://doi.org/10.1348/97818543370009X12583699332537
Park, J., & Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development, 16(3), 1–11. https://doi.org/10.1016/S0885-2014(01)00058-2
Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: A way to understand teaching and learning cultures. ZDM: The International Journal on Mathematics Education, 33(5), 158–175. https://doi.org/10.1007/BF02656616
Piaget, J. (1965). The child’s conception of number. Norton.
Polak, S., Cindrić, D., & Duvnjak, S. (2014). Nove matematiÄke price 2 [New mathematical tales 2]. Profil.
Sansu Math 2. (2015). Koyo Publishing.
Sansu Math 3. (2015). Koyo Publishing.
Staneff, T. (Ed.) (2017). Power maths 1C. Pearson.
Staneff, T. (Ed.) (2017). Power maths 2A. Pearson.
Staneff, T. (Ed.) (2018). Power maths 3A. Pearson.
Staneff, T. (Ed.) (2018). Power maths 3B. Pearson.
Staneff, T. (Ed.) (2018). Power maths 4A. Pearson.
Steffe, L. (1994). Children's multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–40). SUNY Press.
Stiegler, J., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. The Free Press.
StruÄna radna skupina [SRS]. (2016). Nacionalni kurikulum nastavnog predmeta Matematika, Prijedlog [The national curriculum of Mathematics, A proposal]. http://www.kurikulum.hr/wp-content/uploads/2016/03/Matematika.pdf
StruÄna radna skupina [SRS]. (2016). Odgovori na pristigle priloge struÄnoj raspravi Nacionalni kurikulum nastavnoga predmeta Matematika [Answers to the received contributions to the expert discussion of the national curriculum of Mathematics]. http://www.kurikulum.hr/wp-content/uploads/2016/06/komentari-struÄna-rasprava-matematika.pdf
Tirosh, D., Tsamir, P., Barkai, R., & Levenson, E. (2018). Engaging young children with mathematical activities involving different representations: Triangles, patterns, and counting objects. Center for Educational Policy Studies Journal, 8(2), 9–30. https://doi.org/10.26529/cepsj.271
Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Kluwer.
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in middle grades (pp.141–161). NCTM.
Yang, D. C., & Sianturi, I. A. (2017). An analysis of Singaporean versus Indonesian textbooks based on trigonometry content. EURASIA Journal of Mathematics Science and Technology Education, 13(7), 3829–3848. https://doi.org/10.12973/eurasia.2017.00760a
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