HOW EFFICIENT IS EXEMPLARY TEACHING IN LEARNING EQUATIONS AND INEQUATIONS IN PRIMARY SCHOOL?
Keywords:
exemplary teaching, efficiency, equations, inequations, taxonomy, criterion-referenced tests
Abstract
The aim of the study was to investigate the efficiency of learning equations and inequations through the application of exemplary teaching in primary school. In terms of efficiency and achievement levels, we started from the learning outcomes on equations and inequations taught in the third and fourth grade primary school mathematics curriculum and from a taxonomic model of operationalization of the goals and objectives of mathematics teaching in the cognitive domain. According to this model, the performance of the students was examined in the following 5 main taxonomy levels: recognition, reproduction, comprehension, operationalization and creative problem solving. The results of the experimental study, which was carried out on a sample of 100 students, show that the teaching method used helps students to learn equations and inequations more successfully. The use of exemplary teaching had a positive effect on students’ performance in reproduction, comprehension, operationalization, and creative problem solving, while it had no effect on students whose knowledge was at the recognition level. The results of this study could lead to changes in the presentation of teaching content, the evaluation of the efficiency of the educational process, a more efficient individualization of the learning process, etc. In addition, the methodological contribution of this work is the creation of exemplary teaching models whose use in a classroom, thanks to analogue reasoning, allows students to find their own approach to mathematics.
References
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Ableser, J. (2012). Exemplary Teaching Practices Across Educational Contexts (P-20+): Unifying Principles and an Ecological Model for Teaching for All to Learn. Journal of Teaching and Learning, 8(2), 65–75. https://doi.org/10.22329/JTL.V8I2.3133
Anđelković, S. (2022). Učenje putem otkrića na diferenciranim sadržajima algebre i njegovi efekti u početnoj nastavi matematike (Neobjavljena doktorska disertacija). Pedagoški fakultet u Užicu, Univeritet u Kragujevcu.
Anđelković, S., & Maričić, S. (2023). The Effects of Discovery-Based Learning of Differentiated Algebra Content on the Long-term Knowledge of Students in Early Mathematics Education. Facta Universitatis, Series: Teaching, Learning and Teacher Education, 7(2), 251–263. https://doi.org/10.22190/FUTLTE230611024A
Ball, L. J, Cheshire, A., & Lewis, C. (2005). Self-Explanation, Feedback and the Development of Analogical Reasoning Skills: Microgenetic Evidence for a Metacognitive Processing Account. Proceedings of the Annual Meeting of the Cognitive Science Society (p. 27). https://escholarship.org/uc/item/4mt9g0jb
Bikić, N., Maričić, S., & Pikula, M. (2016). The Effects of Differentiation of Content in Problem-solving in Learning Geometry in Secondary School. Eurasia Journal of Mathematics, Science & Technology Education, 12(11), 2783–2795. https://doi.org/10.12973/eurasia.2016.02304a
Bills, L., Mason, J., Watson, A., & Zaslavsky, O. (2006). Exemplification: The use of examples in teaching and learning mathematics. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 125–154). Prague: PME.
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446. https://psycnet.apa.org/record/2005-13920-003
Bloom, B. S., Englehart, M. D., Furst, E. J., Hill, W. H., & Krathwohl, D. R. (1956). Taxonomy of educational objectives. Handbook 1: Cognitive domain. New York: Longmans, Green.
Bogdanović, S., & Malinović-Jovanović, N. (2009). Taksonomski model i stepen ostvarenosti zadataka nastave matematike u III razredu osnovne škole. Pedagogija, 64(4), 618–632.
Brown, C. A., Carpenter, T. P., Kouba, V. L., Lindquist, M. M., Silver, E. A., and Swafford, J. O. (1988). Secondary school results for the fourth NAEP mathematics assessment: Algebra, geometry, mathematical methods, and attitudes. Mathematics Teacher, 81, 337–347.
Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference (vol. 1, pp. 155–162). Australia: The University of Melbourne.
Derbolav, J. (2003). The 'exemplary': an attempt to locate the position of learning by example. Journal of Curriculum Studies, 35(3), 319–334.
Duit, R. (1991). On the role of analogies and metaphors in learning science. Science education, 75(6), 649–672.
Dyk, C. J. V. (2006). Exemplary teaching: some possibilities for renovating and stimulating didactic practice. Journal of Curriculum Studies, 38(2), 127–134. https://doi.org/10.1080/00220270500122653
Hines, A. (2008). Global trends in culture, infrastructure and values. Futurist, 42(5), 18–23.
Holyoak, K.J., Gentner, D., & Kokinov, B.N. (2001). Introduction: The Place of Analogy in. Cognition. In D. Gentner, K. J. Holyoak, & B. N. Kokinov (Eds.), The analogical mind: Perspectives from cognitive science (pp. 1–19). Cambridge, MA: MIT Press.
Jorgensen, N. (2013). Exemplary learning in design studies: strengths and limits. In Krogh, L. & Jensen, A. A. (Eds.), Visions, challenges and strategies: PBL principles and methodologies in a Danish and global perspective, (pp. 201–218). Aalborg: Aalborg Universitets forlag.
Kalathaki, M. (2016). Reflections on a School Teaching: An Exemplary Teaching about Primary Productivity & Energy Flow in Natural Ecosystems. Journal of Education and Learning; 5(4), 291–299. https://doi.org/10.5539/jel.v5n4p291
Lew, H. С. (2004). Developing Algebraic Thinking in Early Grades: Case Study of Korean Elementary School Mathematics, The Mathematics Educator 2004, 8(1), 88–106.
Lister, R. (2006). Driving learning via criterion-referenced assessment using Bloom’s Taxonomy. UniServe Science Assessment Symposium Proceedings (pp. 80–88). Carslaw Building: UniServe Science.
Malinović-Jovanović, N. (2015). Individualizovana nastava na tri nivoa složenosti i interaktivni rad u gupama u izučavanju sadržaja o sabiranju i oduzimanju u stotini. U J. Milinković, & B. Trebješanin (Ur.), Naučni skup: Implementacija inovacija u obrazovanju i vaspitanju – izazovi i dileme, (str. 263–280). Beograd: Učiteljski fakultet.
Malinović-Jovanović, N., & Janković, S. (2014). Problemska nastava i efikasnost ostvarivanja programskih zadataka o jednačinama, Godišnjak Učiteljskog fakulteta u Vranju, 5(1), 437–451.
Malinović-Jovanović, N., & Malinović, T. (2013). Metodika osavremenjene nastave matematike. Vranje: Učiteljski fakultet.
Maričić, S., & Milinković. N. (2017). Udžbenik u stvaranju uslova za kontekstualni pristup učenju sadržaja algebre u početnoj nastavi matematike. Zbornik radova Učiteljskog fakulteta u Užicu, 19(18), 117–130.
Matejević, M. (1994). Nastavni program kao faktor opterećenosti učenika osnovne škole. Beograd: Institut za pedagogiju i andragogiju filozofskog fakulteta u Beogradu.
Milijević, S. (2003). Interaktivna nastava matematike. Banja Luka: Društvo pedagoga Republike Srpske.
Mullis, I. V. S., Martin, M. O., & Foy, P. (2005). IEA’s TIMSS 2007 International Report on Achievement in the Mathematics Cognitive Domains. Chestnut Hill, MA: Boston College.
Nayef, E. G., Rosila, N., Yaacob, N., & Ismail, H. N. (2013). Taxonomies of Educational Objective Domain. International Journal of Academic Research in Business and Social Sciences, 3(9), 165–175. http://dx.doi.org/10.6007/IJARBSS/v3-i9/199
Nikolić, M. (2015). Egzemplarna nastava o izučavanju sadržaja o jednačinama i nejednačinama (Neobjavljen master rad). Pedagoški fakultet u Vranju, Univerzitet u Nišu.
Pravilnik o programu nastave i učenja za treći razred osnovnog obrazovanja i vaspitanja. Službeni glasnik Republike Srbije – Prosvetni glasnik, br. 5/2019-6, 1/2020-1, 6/2020-1, 7/2022-1, 13/2023-457. https://pravno-informacioni-sistem.rs/slglrsViewPdf/13da23b7-5243-443a-abdd-c73e71a24ecf?fromLink=true
Pravilnik o programu nastave i učenja za četvrti razred osnovnog obrazovanja i vaspitanja. Službeni glasnik Republike Srbije – Prosvetni glasnik, br. 11/2019-1, 6/2020-20, 7/2021-671, 1/2023-1, 13/2023-458. https://pravno-informacioni sistem.rs/eli/rep/pg/ministarstva/pravilnik/2019/11/1/reg
Richland, L., Holyoak, K., & Stigler, J. (2004). Analogy Use in Eighth-Grade Mathematics Classrooms. Cognition and Instruction, 22(1), 37–60.
Author (2022).
Siegler, R. S., & Svetina, M. (2002). A microgenetic/cross-sectional study of matrix completion: comparing short-term and long-term change. Child development, 73(3), 793–809. https://doi.org/10.1111/1467-8624.00439
Somayeh, A.-M., Parvaneh, A., and Mohammad, H.B-z. (2012). Instruction of mathematical concepts through analogical reasoning skills. Indian Journal of Science and Technology, 5(6), 2916–2922. https://doi.org/10.17485/ijst/2012/v5i6.12
Teodosić, R. (Ur.). (1967). Pedagoški rečnik. Beograd: Zavod za izdavanje udžbenika Socijalističke Republike Srbije.
Van der Stoep, F., & Louw, W. J. (1979). Inleiding tot die didaktiese pedagogiek. Pretoria: Academica.
Van der Stoep, F. & Louw, W. J. (2005). Introduction to didactic pedagogics, trans. G. D. Yonge. Retrieved from http://georgeyonge.net/translations/didactic-pedagogics/introduction-didactic-pedagogics.
Vilotijević, M., & Vilotijević. N. (2008). Inovacije u nastavi. Vranje: Učiteljski fakultet.
Ableser, J. (2012). Exemplary Teaching Practices Across Educational Contexts (P-20+): Unifying Principles and an Ecological Model for Teaching for All to Learn. Journal of Teaching and Learning, 8(2), 65–75. https://doi.org/10.22329/JTL.V8I2.3133
Anđelković, S. (2022). Učenje putem otkrića na diferenciranim sadržajima algebre i njegovi efekti u početnoj nastavi matematike (Neobjavljena doktorska disertacija). Pedagoški fakultet u Užicu, Univeritet u Kragujevcu.
Anđelković, S., & Maričić, S. (2023). The Effects of Discovery-Based Learning of Differentiated Algebra Content on the Long-term Knowledge of Students in Early Mathematics Education. Facta Universitatis, Series: Teaching, Learning and Teacher Education, 7(2), 251–263. https://doi.org/10.22190/FUTLTE230611024A
Ball, L. J, Cheshire, A., & Lewis, C. (2005). Self-Explanation, Feedback and the Development of Analogical Reasoning Skills: Microgenetic Evidence for a Metacognitive Processing Account. Proceedings of the Annual Meeting of the Cognitive Science Society (p. 27). https://escholarship.org/uc/item/4mt9g0jb
Bikić, N., Maričić, S., & Pikula, M. (2016). The Effects of Differentiation of Content in Problem-solving in Learning Geometry in Secondary School. Eurasia Journal of Mathematics, Science & Technology Education, 12(11), 2783–2795. https://doi.org/10.12973/eurasia.2016.02304a
Bills, L., Mason, J., Watson, A., & Zaslavsky, O. (2006). Exemplification: The use of examples in teaching and learning mathematics. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 125–154). Prague: PME.
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446. https://psycnet.apa.org/record/2005-13920-003
Bloom, B. S., Englehart, M. D., Furst, E. J., Hill, W. H., & Krathwohl, D. R. (1956). Taxonomy of educational objectives. Handbook 1: Cognitive domain. New York: Longmans, Green.
Bogdanović, S., & Malinović-Jovanović, N. (2009). Taksonomski model i stepen ostvarenosti zadataka nastave matematike u III razredu osnovne škole. Pedagogija, 64(4), 618–632.
Brown, C. A., Carpenter, T. P., Kouba, V. L., Lindquist, M. M., Silver, E. A., and Swafford, J. O. (1988). Secondary school results for the fourth NAEP mathematics assessment: Algebra, geometry, mathematical methods, and attitudes. Mathematics Teacher, 81, 337–347.
Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference (vol. 1, pp. 155–162). Australia: The University of Melbourne.
Derbolav, J. (2003). The 'exemplary': an attempt to locate the position of learning by example. Journal of Curriculum Studies, 35(3), 319–334.
Duit, R. (1991). On the role of analogies and metaphors in learning science. Science education, 75(6), 649–672.
Dyk, C. J. V. (2006). Exemplary teaching: some possibilities for renovating and stimulating didactic practice. Journal of Curriculum Studies, 38(2), 127–134. https://doi.org/10.1080/00220270500122653
Hines, A. (2008). Global trends in culture, infrastructure and values. Futurist, 42(5), 18–23.
Holyoak, K.J., Gentner, D., & Kokinov, B.N. (2001). Introduction: The Place of Analogy in. Cognition. In D. Gentner, K. J. Holyoak, & B. N. Kokinov (Eds.), The analogical mind: Perspectives from cognitive science (pp. 1–19). Cambridge, MA: MIT Press.
Jorgensen, N. (2013). Exemplary learning in design studies: strengths and limits. In Krogh, L. & Jensen, A. A. (Eds.), Visions, challenges and strategies: PBL principles and methodologies in a Danish and global perspective, (pp. 201–218). Aalborg: Aalborg Universitets forlag.
Kalathaki, M. (2016). Reflections on a School Teaching: An Exemplary Teaching about Primary Productivity & Energy Flow in Natural Ecosystems. Journal of Education and Learning; 5(4), 291–299. https://doi.org/10.5539/jel.v5n4p291
Lew, H. С. (2004). Developing Algebraic Thinking in Early Grades: Case Study of Korean Elementary School Mathematics, The Mathematics Educator 2004, 8(1), 88–106.
Lister, R. (2006). Driving learning via criterion-referenced assessment using Bloom’s Taxonomy. UniServe Science Assessment Symposium Proceedings (pp. 80–88). Carslaw Building: UniServe Science.
Malinović-Jovanović, N. (2015). Individualizovana nastava na tri nivoa složenosti i interaktivni rad u gupama u izučavanju sadržaja o sabiranju i oduzimanju u stotini. U J. Milinković, & B. Trebješanin (Ur.), Naučni skup: Implementacija inovacija u obrazovanju i vaspitanju – izazovi i dileme, (str. 263–280). Beograd: Učiteljski fakultet.
Malinović-Jovanović, N., & Janković, S. (2014). Problemska nastava i efikasnost ostvarivanja programskih zadataka o jednačinama, Godišnjak Učiteljskog fakulteta u Vranju, 5(1), 437–451.
Malinović-Jovanović, N., & Malinović, T. (2013). Metodika osavremenjene nastave matematike. Vranje: Učiteljski fakultet.
Maričić, S., & Milinković. N. (2017). Udžbenik u stvaranju uslova za kontekstualni pristup učenju sadržaja algebre u početnoj nastavi matematike. Zbornik radova Učiteljskog fakulteta u Užicu, 19(18), 117–130.
Matejević, M. (1994). Nastavni program kao faktor opterećenosti učenika osnovne škole. Beograd: Institut za pedagogiju i andragogiju filozofskog fakulteta u Beogradu.
Milijević, S. (2003). Interaktivna nastava matematike. Banja Luka: Društvo pedagoga Republike Srpske.
Mullis, I. V. S., Martin, M. O., & Foy, P. (2005). IEA’s TIMSS 2007 International Report on Achievement in the Mathematics Cognitive Domains. Chestnut Hill, MA: Boston College.
Nayef, E. G., Rosila, N., Yaacob, N., & Ismail, H. N. (2013). Taxonomies of Educational Objective Domain. International Journal of Academic Research in Business and Social Sciences, 3(9), 165–175. http://dx.doi.org/10.6007/IJARBSS/v3-i9/199
Nikolić, M. (2015). Egzemplarna nastava o izučavanju sadržaja o jednačinama i nejednačinama (Neobjavljen master rad). Pedagoški fakultet u Vranju, Univerzitet u Nišu.
Pravilnik o programu nastave i učenja za treći razred osnovnog obrazovanja i vaspitanja. Službeni glasnik Republike Srbije – Prosvetni glasnik, br. 5/2019-6, 1/2020-1, 6/2020-1, 7/2022-1, 13/2023-457. https://pravno-informacioni-sistem.rs/slglrsViewPdf/13da23b7-5243-443a-abdd-c73e71a24ecf?fromLink=true
Pravilnik o programu nastave i učenja za četvrti razred osnovnog obrazovanja i vaspitanja. Službeni glasnik Republike Srbije – Prosvetni glasnik, br. 11/2019-1, 6/2020-20, 7/2021-671, 1/2023-1, 13/2023-458. https://pravno-informacioni sistem.rs/eli/rep/pg/ministarstva/pravilnik/2019/11/1/reg
Richland, L., Holyoak, K., & Stigler, J. (2004). Analogy Use in Eighth-Grade Mathematics Classrooms. Cognition and Instruction, 22(1), 37–60.
Author (2022).
Siegler, R. S., & Svetina, M. (2002). A microgenetic/cross-sectional study of matrix completion: comparing short-term and long-term change. Child development, 73(3), 793–809. https://doi.org/10.1111/1467-8624.00439
Somayeh, A.-M., Parvaneh, A., and Mohammad, H.B-z. (2012). Instruction of mathematical concepts through analogical reasoning skills. Indian Journal of Science and Technology, 5(6), 2916–2922. https://doi.org/10.17485/ijst/2012/v5i6.12
Teodosić, R. (Ur.). (1967). Pedagoški rečnik. Beograd: Zavod za izdavanje udžbenika Socijalističke Republike Srbije.
Van der Stoep, F., & Louw, W. J. (1979). Inleiding tot die didaktiese pedagogiek. Pretoria: Academica.
Van der Stoep, F. & Louw, W. J. (2005). Introduction to didactic pedagogics, trans. G. D. Yonge. Retrieved from http://georgeyonge.net/translations/didactic-pedagogics/introduction-didactic-pedagogics.
Vilotijević, M., & Vilotijević. N. (2008). Inovacije u nastavi. Vranje: Učiteljski fakultet.
