Portraying mathematics teachers’ knowledge for teaching the addition of fractions through representations
Teaching the concept of fractions to students stays challenging, yet representations define an effective strategy to overcome such a challenge. Considering the impact of teachers’ knowledge on students’ achievement, this study aimed at portraying mathematics teachers’ knowledge for teaching fractions through representations, precisely, the addition process that remains a prerequisite to other operations. Hence, a purposefully selected sample of novice mathematics teachers was asked to propose a pedagogical activity through which the addition of fractions could be taught to early-age students. Later, their responses were analyzed through the study framework, which was developed by combining the five interrelated constructs of fractions with the types of activities used when teaching the addition of fractions. As a result, teachers’ knowledge was crystallized into three principal categories of utilizing the Part-whole, Measure, and Operator constructs. Furthermore, the related concepts of the unit and proportional equivalence, the fractional unit, including the iteration process, and the connection between addition and subtraction were discussed. Also, manners of representing (1) the added fractions and the result through two distinct models, (2) the added fractions and the result jointly in one model, and (3) only the added fractions emerged. These results provide a foundation for the professional development of mathematics teachers.
Abbas, N. H., ShahrilL, M., & Prahmana, R. C. I. (2020). Understanding primary school children's learning on addition of fractions. Journal of Physics: Conference Series, 1613(012046). https://doi.org/10.1088/1742-6596/1613/1/012046
Atagi, N., DeWolf, M., Stigler, J. W., & Johnson, S. P. (2016). The role of visual representations in college students' understanding of mathematical notation. Journal of Experimental Psychology Applied, 22(3), 295-304. https://doi.org/10.1037/xap0000090
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content Knowledge for Teaching: What Makes It Special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y. M. (2010). Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress. American Educational Research Journal, 47(1), 133-180. https://doi.org/10.3102/0002831209345157
Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91-126). New York: Academic Press.
Brijlall, D. (2014). Exploring practical work as a sustainable strategy in rural mathematics classrooms: A case of addition of fractions. International Journal of Educational Sciences, 7(3), 481- 490.
Bruce, C., Chang, D., Flynn, T., & Yearley, S. (2013). Foundations to learning and teaching fractions: Addition and subtraction. Curriculum and Assessment Branch: Ontario Ministry of Education. Retrieved from http://www.edugains.ca/resourcesDP/Resources/PlanningSupports/FINALFoundationstoLearningandTeachingFractions.pdf
Copur-Gencturk, Y. (2021). Teachers’ conceptual understanding of fraction operations: results from a national sample of elementary school teachers. Educ Stud Math, 107, 525-545. https://doi.org/10.1007/s10649-021-10033-4
Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School. 13(8), 490-496.
Darling-Hammond, L., & Sykes, G. (2003). Wanted: A national teacher supply policy for education: The right way to meet the “highly qualified teacher” challenge. Educational Policy Analysis Archives, 11(33). https://doi.org/10.14507/epaa.v11n33.2003
DeWolf, M., Grounds, M. A., Bassok, M., & Holyoak, K. J. (2014). Magnitude comparison with different types of rational numbers. Journal of Experimental Psychology: Human Perception and Performance, 40(1), 71-82. https://doi.org/10.1037/a0032916
Dey, K., & Dey, R. (2010). Teaching Arithmetic of Fractions Using Geometry. Journal of Mathematics Education, 3(2), 170-182.
Dhlamini, Z. B., & Kibirige, I. (2014). Grade 9 learners’ errors and misconceptions ın addition of fractions. Mediterranean Journal of Social Sciences, 5(8), 236-244. Retrieved from https://www.richtmann.org/journal/index.php/mjss/article/view/2551
Doyle, K. M., Dias, O., Kennis, J. R., Czarnocha, B., & Baker, W. (2015). The rational number sub- constructs as a foundation for problem solving. Adults Learning Mathematics: An International Journal, 11(1), 21-42.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131.
Fuchs, L. S., Malone, A. S., Schumacher, R. F., Namkung, J., Hamlett, C. L., Jordan, N. C., … Changas, P. (2016). Supported self-explaining during fraction intervention. Journal of Educational Psychology, 108(4), 493-508. https://doi.org/10.1037/edu0000073
Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4(715), 1-12. https://doi.org/10.3389/fpsyg.2013. 00715
Getenet, S., & Callingham R. (2017). Teaching fractions for understanding: addressing interrelated concepts. In A. Downton, S. Livy, & J. Hall (Eds.), 40 years on: We are still learning! Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 277-284). Melbourne: MERGA.
Godino, J. D., & Font, V. (2010). The theory of representations as viewed from the onto-semiotic approach to mathematics education. Mediterranean Journal for Research in Mathematics Education, 9(1), 189-210.
Goldin, G. (1998). Representations and the psychology of mathematics education: part II. Journal of Mathematical Behaviour, 17 (2), 135-165.
Gunawan, M. S., Putri, R. I. I., & Zulkardi. (2017). Learning Fraction through Swimming Context for Elementary School Student. 5th SEA-DR (South East Asia DEvelopment Research) International Conference 2017 (SEADRIC 2017). Makasar: Atlantis Press. https://doi.org/10.2991/seadric-17.2017.14
Gupta, D., & Wilkerson, T. L. (2015). Teaching and learning of fractions in elementary grades: Let the dialogue begin! Curriculum and Teaching Dialogue, 17(1/2), 27-44.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students. Journal for Research in Mathematics Education, 39(4), 372-400. http://www.jstor.org/stable/40539304
Janvier, C. (Ed.). (1987). Problems of representation in the teaching and learning of mathematics. Lawrence Erlbaum Associates, Inc.
Jing-Jing, H. U. (2014). A critical review of Pedagogical Content Knowledge components: nature, principle and trend. International Journal of Education and Research, 2(4), 411-424.
Kutub, A. H. W., Wijayanti, P., & Manuharawati. (2019). Relationship of Teacher’s Content Knowledge on Fraction Topic Toward Student Performance. Journal of Physics: Conf. Series, 1417 (012054). https://doi.org/10.1088/1742-6596/1417/1/012054
Lacy, S., Watson, B. R., Riffe, D., & Lovejoy, J. (2015). Issues and Best Practices in Content Analysis. Journalism & Mass Communication Quarterly, 92(4), 791-811. https://doi.org/10.1177/1077699015607338
López-Martín, M. d. M., Aguayo-Arriagada, C. G., & García López, M. d. M. (2022). Preservice Elementary Teachers’ Mathematical Knowledge on Fractions as Operator in Word Problems. Mathematics, 10(3), 423. https://doi.org/10.3390/ math10030423
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Mainali, B. (2021). Representation in teaching and learning mathematics. International Journal of Education in Mathematics, Science and Technology, 9(1), 1-21. https://doi.org/10.46328/ijemst.1111
McHugh, M. L. (2012). Interrater Reliability: The Kappa Statistic. Biochemia Medica, 22, 276-282.
Mendiburo, M., & Hasselbring, T. (2011). Technology’s impact on fraction learning: An experimental comparison of virtual and physical manipulative. SREE Conference Abstract Template.
Ministry of Education and Technical Education. (2018). Mathematics Teachers’ guide for grade 1. Retrieved from https://en.discoveryeducation.ekb.eg/curriculum/primary/#/math/
Ministry of Education and Technical Education. (2020). Mathematics Teachers’ guide for grade 3. Retrieved from https://en.discoveryeducation.ekb.eg/curriculum/primary/#/math/
Mohamed, R., Ghazali, M., & Samsudin, M. A. (2021). A systematic review on teaching fraction for understanding through representation on Web of Science database using PRISMA. LUMAT: International Journal on Math, Science and Technology Education, 9(1), 100-125. https://doi.org/10.31129/LUMAT.9.1.1449
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.
Newton, K. J. (2008). An Extensive Analysis of Preservice Elementary Teachers’ Knowledge of Fractions. American Educational Research Journal, 45(4), 1080-1110.
Olanoff, D., Lo, J. J., & Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267-310.
Pedersen, P. L., & Bjerre, M. (2021). Two conceptions of fraction equivalence. Educ Stud Math, 107, 135-157. https://doi.org/10.1007/s10649-021-10030-7
Petrou, M., Goulding, M. (2011). Conceptualising Teachers’ Mathematical Knowledge in Teaching. In T., Rowland, K., Ruthven, (Eds.) Mathematical Knowledge in Teaching. Mathematics Education Library, vol 50. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9766-8_2
Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, N. L., Rogers, A., Falle, J.,... Bennett, S. (2012). Helping children learn mathematics (1st Australian Ed.). Milton, NSW: John Wiley & Sons Australia.
Ribeiro, C. M., & Jakobsen, A. (2012). Prospective teachers' mathematical knowledge of fractions and their interpretation of the part-whole representation. In B. Maj-Tatsis & K. Tatsis (Eds.), Generalization in mathematics at all educational levels (pp. 289-298). Reszów, Poland: Wydawnictwo Uniwersytetu Rzeszowskiego.
Riccomini, P. J. (2011). Core issues of math: number sense and fraction. Kansas MTSS Symposium.
Ross, J. A., & Bruce, C. D. (2009). Student achievement effects of technology-supported remediation of understanding of fractions. International Journal of Mathematical Education in Science and Technology, 40(6), 713-727.
Samsuddin, A. F., & Retnawati, H. (2018). Mathematical representation: the roles, challenges and implication on instruction. Journal of Physics: Conf. Series, 1097 (012152). http://doi.org/10.1088/1742- 6596/1097/1/012152
Schacter, J., & Thum, Y. M. (2004). Paying for high- and low-quality teaching. Economics of Education Review, 23(4), 411-430.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296. https://doi.org/10.1016/j.cogpsych.2011.03.001
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2015). Teaching mathematics: Foundations to middle years (2nd Ed.). Melbourne: Oxford University Press.
Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503–518. https://doi.org/10.1016/j.learninstruc.2004. 06.015
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving in the classroom. New York: The Free Press.
Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: the case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.
Tobias, J. M. (2013). Prospective elementary teachers’ development of fraction language for defining the whole. Journal of Mathematics Teacher Education, 16(2), 85-103.
Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181-209. https://doi.org/10.1080/07370001003676603
Van Steenbrugge, H., Lesage, E., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students' knowledge? Journal of Curriculum Studies, 46(1), 138-161.
Watanabe, T. (2002). Representations in teaching and learning fractions. Teaching Children Mathematics, 8 (8), 457-463. Retrieved from http://academicinnovation.weebly.com/uploads/7/6/3/6/7636030/representing_fractions_nctm.pdf
Widodo, S. & Ikhwanudin, T. (2020). Students’ Understanding in Learning Fraction with Multiple Representations. Proceedings of the 7th Mathematics, Science, and Computer Science Education International Seminar, MSCEIS 2019, Bandung, West Java, Indonesia.
Wilkins, J., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International journal of STEM education, 5(1), 27. https://doi.org/10.1186/s40594-018-0119-2
Wong, M., & Evans, D. (2008). Fractions as a measure. Proceedings of the annual conference of the Mathematics Education Research Group of Australasia (pp. 597-603). Brisbane: MERGA.
Zhou, Z., Peverly, S. T., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers. Contemporary Educational Psychology, 31(4), 438-457.
Copyright (c) 2022 Samah Gamal Ahmed Elbehary, Fatima Hamada Bassiouny Aboseira
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Creative Commons License: CC BY-NC
Creative Commons Rights Expression Language (CC REL)