Zoom in: Exploring perceptions of the multiplication symbol (×) up close

Lia Ardiansari(1), Mohammad Sahrul Rozi(2), Shellyna Herdiyanti(3),


(1) Universitas Panca Marga Probolinggo, Probolinggo, East Java, Indonesia, 67271
(2) Universitas Panca Marga Probolinggo, Probolinggo, East Java, Indonesia, 67271
(3) Universitas Panca Marga Probolinggo, Probolinggo, East Java, Indonesia, 67271

Abstract


This study explored the perceptions of 51 prospective elementary school teachers selected using purposive sampling technique regarding the multiplication symbol (×) in arithmetic operations. Using a qualitative hermeneutic phenomenological approach, data were collected through task-based interviews administered to education students specializing in elementary education. The tasks were designed to explore candidates’ understanding and interpretation of the ‘×’ symbol, including uncovering conceptual images, understanding the ‘×’ symbol in different contexts, ability to represent the ‘×’ symbol, flexibility in relating the ‘×’ symbol among concepts, and problem-solving skills. Data analysis involved thematic coding and interpretive analysis to uncover patterns and insights into candidates’ cognitive frameworks. The findings revealed significant variation in candidates’ understanding of the ‘×’ symbol, influenced by their educational background and personal experiences with mathematics. This study highlights the need for improved mathematics instruction and curriculum design to equip future teachers with a deep and accurate understanding of arithmetic symbols, which is critical for effective mathematics teaching at the elementary level.

Keywords


Arithmetic operation; concept image; multiplication symbol; phenomenological hermeneutics; primary school teacher candidates

Full Text:

PDF

References


Amarasinghe, R., Burger, L., Nogin, M., Tuska, A., Vega, O. (2013). Exploring mathematics: Investigations for elementary school teachers (First Edition). California: Cognella.Inc.

Ardiansari, L., Suryadi, D., & Dasari, D. (2023). The misconceptions of students on equations based on their arithmetic knowledge. Jurnal Pendidikan: Teori, Penelitian, dan Pengembangan, 8(6), 387–393. http://dx.doi.org/10.17977/jptpp.v8i6.24633

Baroody, A. J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics, 13(1), 22–31. https://doi.org/10.5951/TCM.13.1.0022

Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. Jossey-Bass/Wiley.

Burns, M. (2001). Lessons for Introducing Multiplication, Grade 3 (Teaching Arithmetic). US: Math Solutions Publications.

Caron, T.A. (2007). Learning multiplication: The easy way. The Clearing House: A Journal of Educational Strategies. Issues and Ideas, 80(6), 278–282. https://doi.org/10.3200/TCHS.80.6.278-282

Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3–20. https://doi.org/10.1086/461846

Chick, H. L. (2007). Teaching and learning by example. Proceedings of the 30th annual conference of the Mathematics Education Group of Australasia 1, 3–21. https://files.eric.ed.gov/fulltext/ED503746.pdf

Chin, K. E., & Pierce, R. (2019). University students’ conceptions of mathematical symbols and expressions. EURASIA Journal of Mathematics, Science and Technology Education, 15(9), 1–12. https://doi.org/10.29333/ejmste/103736

Devlin, K. (2012). Introduction to mathematical thinking. USA: Keith Devlin.

Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Heinemann.

Gray, E. M., & Tall, D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140. https://doi.org/10.2307/749505

Haylock, D., & Manning, R. (2014). Mathematics explained for primary teachers. SAGE Publications.

Heid, M. K. (2005). Technology in mathematics education: Tapping into visions of the future. Technology-Supported Mathematics Learning Environments, 67, 345–366. http://www.fi.uu.nl/publicaties/literatuur/technology_heid_2005.pdf

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 371–404). Charlotte, NC: Information Age.

Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. https://doi.org/10.3102/00028312042002371

Hurst, C. (2007). Using task-based interviews to assess mathematical thinking of primary school students. In M. Goos, R. Brown, & K. Makar (Eds.), Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia, pp. 273–280.

Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner & C. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 33–56). Lawrence Erlbaum Associates.

Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 515–556). Macmillan.

Kline, M. (1982). Mathematics: The loss of certainty. New York: Oxford University Press.

Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In Albert A. Cuoco (Ed.), The Role of Representation in School Mathematics, 2001 Yearbook of the National Council of Teachers of Mathematics (NCTM) (pp. 146–65). Reston, VA: NCTM.

Lee H-J, Han C, Kim H-j, Herner-Patnode L. (2021). Teaching multiplication to students with mathematical learning disabilities (MLD): Analysis of preservice teachers’ lesson design. Sustainability, 13(21), 11813. https://doi.org/10.3390/su132111813

Lesh, R.A., & Doerr, H.M. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (1st ed.). Routledge. https://doi.org/10.4324/9781410607713

Leung, A., Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In Watson, A., Ohtani, M. (Eds.), Task Design in Mathematics Education. New ICMI Study Series (pp.191–225). Springer, Cham. https://doi.org/10.1007/978-3-319-09629-2_6

Leung, K.T., & Cheung, P.H. (1988). Fundamental concept of mathematics. Hong Kong University Press.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563

Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38(1), 103–125. https://doi.org/10.1023/A:1003622804002

McMullen, J., & Hoof, J. (2020). The role of rational number density knowledge in mathematical development. Learning and Instruction, 65, 101228. https://doi.org/10.1016/j.learninstruc.2019.101228

Mejía-Ramos, J.P., Weber, K. (2020). Using task-based interviews to generate hypotheses about mathematical practice: mathematics education research on mathematicians’ use of examples in proof-related activities. ZDM Mathematics Education, 52, 1099–1112. https://doi.org/10.1007/s11858-020-01170-w

Moschkovich, J. (1999). Understanding the needs of Latino students in reform-oriented mathematics classrooms. In Ortiz-Franco, L., Hernandez, N., & Cruz, Y.D.L. (Eds), Changing the Faces of Mathematics (Vol. 4): Perspectives on Latinos (pp.5–12). NCTM. https://www.researchgate.net/publication/242599058

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

NCTM. (2010). Why is teaching with problem solving important to student learning?. Reston, VA: NCTM.

NIST. (2020). Multiplication: A fundamental concept in mathematics. Retrieved from [NIST](https://www.nist.gov/publications/multiplication-fundamental-concept-mathematics).

Nunes, T., Dorneles, B.V., Lin, P-J., & Rathgeb-Schnierer, E. (2016). Teaching and Learning About Whole Numbers in Primary School. Springer Open. https://doi.org/10.1007/978-3-319-45113-8_1

Núñez, R. E., Edwards, L. D., & Matos, J. F. (1999). Embodied cognition as a framework for understanding mathematical ideas. Educational Studies in Mathematics, 39(1), 45–65. https://doi.org/10.1023/A:1003759711966

Önal, H. (2023). Primary school students' understanding of four operation symbols (+, ?, x, ÷, =) and using them in arithmetic operations and word problems. Mathematics Teaching Research Journal, 5(1), 152–173. https://files.eric.ed.gov/fulltext/EJ1391475.pdf

Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge: Does one lead to the other?. Journal of Educational Psychology, 91(1), 175–189. https://doi.org/10.1037/0022-0663.91.1.175

Ryan, R., Santesso, N., Lowe, D., Hill, S., Grimshaw, J., Prictor, M., Kaufman, C., Cowie, G., & Taylor, M. (2014). Interventions to improve safe and effective medicines use by consumers: an overview of systematic reviews. The Cochrane database of systematic reviews, 2014(4), CD007768. https://doi.org/10.1002/14651858.CD007768.pub3

Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. Routledge

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36. https://doi.org/10.1007/BF00302715

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.3102/0013189X015002004

Skemp, R.R. (1987). The psychology of learning mathematics: Expanded american edition (1st ed.). Routledge. https://doi.org/10.4324/9780203396391

Smith III, J. P., diSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163. https://doi.org/10.1207/s15327809jls0302_1

Steinle, V., & Stacey, K. (2004). Persistence of Decimal Misconceptions and Readiness to Move to Expertise. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004, 4. 225–232. https://files.eric.ed.gov/fulltext/ED489646.pdf

Taber, K. S., & Akpan, B. (2017). Science education: An international course companion. Netherlands: Sense Publishers.

Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. https://doi.org/10.1007/BF03217474

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169. https://doi.org/10.1007/BF00305619

Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts ofrate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning ofmathematics (pp. 181-234). Albany, NY: SUNY Press. https://www.researchgate.net/publication/272177940

Van de Walle, J. A, Karp, K. S., & Bay-Williams, J. M., Wray, J., Brown. E.T. (2019). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Upper Saddle River, NJ: Pearson.

Van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24(2), 244–276. https://doi.org/10.1016/j.tate.2006.11.005

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305. https://doi.org/10.1080/0020739830140305

Wu, H. (2011). Understanding numbers in elementary school mathematics. Providence, RI: American Mathematical Society.




DOI: https://doi.org/10.34007/jdm.v5i2.2281

Refbacks

  • There are currently no refbacks.


Journal of Didactic Mathematics

Mahesa Research Center