RESEARCH PAPER
Changes of Meanings in Multiplication across Different Contexts: The Case of Amy and Beth
Kin Eng Chin 1  
,  
 
 
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1
Flinders University, Adelaide, AUSTRALIA
2
Universiti Malaysia Sabah, Sabah, MALAYSIA
Online publish date: 2019-04-11
Publish date: 2019-04-11
 
EURASIA J. Math., Sci Tech. Ed 2019;15(8):em1739
KEYWORDS
ABSTRACT
This article focuses on how two mathematics teachers (Amy and Beth – pseudonyms) cope with the changes of meanings in multiplication due to the changes of contexts. It highlights the qualitative similarities and differences between these two teachers in the sense-making process of multiplication. A potentially useful framework of supportive and problematic conceptions suggested by Chin (2013) and Chin (2014) is employed in this study. Interviews are performed with Amy and Beth in order to collect the necessary data. Findings reveal that both teachers try to make sense of multiplication by building on the conception of repeated addition across different number systems. When multiplication is operated with negative numbers then problematic aspects emerge within the conception of repeated addition. It is observed that both teachers didn’t build on the meaning of repeated addition in the multiplication of fractions.
 
REFERENCES (44)
1.
Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary students’ understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217-241. https://doi.org/10.1007/s10649....
 
2.
Bromley, D. B. (1986). The Case‐study Method in Psychology and Related Disciplines. Chicheter: John Wiley & Sons.
 
3.
Bromley, D. B. (1990). Academic contributions to psychological counselling. 1. A philosophy of science for the study of individual cases. Counselling Psychology Quarterly, 3(3), 299-308. https://doi.org/10.1080/095150....
 
4.
Chin, K. E. (2013). Making sense of mathematics: supportive and problematic conceptions with special reference to trigonometry (Unpublished doctoral thesis). University of Warwick, England. Retrieved from http://wrap.warwick.ac.uk/5840....
 
5.
Chin, K. E. (2014). Supportive and problematic aspects in mathematical thinking over the longer term. In S. Oesterle, C. Nicol, P. Liljedahl, & D. Allan (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, 6, 41. Vancouver, Canada: PME.
 
6.
Chin, K. E., & Jiew, F. F. (in press). Knowing and grasping of two university students: The case of complex numbers. The Mathematics Enthusiast.
 
7.
Chin, K. E., & Tall, D. O. (2012). Making sense of mathematics through perception, operation and reason: The case of trigonometric functions. In Tai-Yih Tso (Eds.), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 4, 264. Taipei, Taiwan: PME.
 
8.
Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1–5. Journal for Research in Mathematics Education, 27, 41–51. https://doi.org/10.2307/749196.
 
9.
Devlin, K. (2007, September). What is conceptual understanding? [Online forum]. Retrieved from Mathematical Association of America website: http://www.maa.org/external_ar....
 
10.
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.
 
11.
Forgasz, H. J. (2006). Australian year 12 mathematics enrolments: Patterns and trends – past and present. Retrieved from Australian Mathematical Science Institute website: https://amsi.org.au/wp-content....
 
12.
Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York: Macmillan.
 
13.
Izsák, A. (2004). Teaching and learning two-digit multiplication: Coordinating analyses of classroom practices and individual student learning. Mathematical Thinking and Learning, 6(1), 37-79. https://doi.org/10.1207/s15327....
 
14.
Izsák, A. (2005). You have to count the squares: Applying knowledge in pieces to learning rectangular area. Journal of the Learning Sciences, 14(3), 361–403. https://doi.org/10.1207/s15327....
 
15.
Jiew, F. F., & Chin, K. E. (in press). Supportive and problematic conceptions in making sense of multiplication: A case study. The Mathematics Enthusiast.
 
16.
Kaput, J. (1985). Multiplicative word problems and intensive quantities: An integrated software response (Technical Report). Cambridge, MA: Educational Technology Center.
 
17.
Kilhamn, C. (2011). Making sense of negative numbers (Doctoral thesis). University of Gothenburg, Sweden. Retrieved from https://gupea.ub.gu.se/bitstre....
 
18.
Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20, 147–158. https://doi.org/10.2307/749279.
 
19.
Lampert, M. (1986a). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3, 305–342. https://doi.org/10.1207/s15326....
 
20.
Larsson, K., Pettersson, K., & Andrews, P. (2017). Students’ conceptualisations of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers. The Journal of Mathematical Behavior, 48, 1-13. https://doi.org/10.1016/j.jmat....
 
21.
Lima, R. N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3-18. https://doi.org/10.1007/s10649....
 
22.
Lo, J.-J., Grant, T. J., & Flowers, J. (2008). Challenges in deepening prospective teachers’ understanding of multiplication through justification. Journal of Mathematics Teacher Education, 11(1), 5–22. https://doi.org/10.1007/s10857....
 
23.
Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267–295. https://doi.org/10.2307/749828.
 
24.
National Council of Teachers of Mathematics. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: NCTM.
 
25.
Nicholas, J., Poladin, L., Mack, J. & Wilson, R. (2015). Mathematics preparation for university: Entry, pathways and impact on performance in first-year science and mathematics subjects. International Journal of Innovation in Science and Mathematics Education, 23(1), 37-51. Retrieved from https://openjournals.library.s....
 
26.
O’Brien, T. C., & Casey, S. A. (1983a). Children learning multiplication: Part I. School Science and Mathematics, 83(1), 246-251. https://doi.org/10.1111/j.1949....
 
27.
Piaget, J. (1952). The origins of intelligence in children (Cook, M. Trans.). New York, NY: International Universities Press, Inc. https://doi.org/10.1037/11494-....
 
28.
Seah, T. K. R. (2004). An investigation of the depth and breath of students’ knowledge of multiplication as a basis for the development of multiplication thinking (Unpublished master’s thesis). Griffin University.
 
29.
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), 1–36. https://doi.org/10.1007/BF0030....
 
30.
Siemon, D. (2004). Partitioning – The Missing Link in Building Fraction Knowledge and Confidence. Retrieved from http://www.aamt.edu.au/members....
 
31.
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2015). Teaching mathematics: foundations to middle years. South Melbourne, Vic: Oxford University Press.
 
32.
Siemon, D., Breed, M., & Virgona, J. (2005). From additive to multiplicative thinking: The big challenge of the middle years. In J. Mousley, L. Bragg, & C. Campbell (Eds.), Mathematics: Celebrating achievement. Proceedings of the 42nd conference of the Mathematical Association of Victoria (pp. 278–286). Brunswick: MAV.
 
33.
Simon, M. A., & Blume, G. W. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25(5), 472-494. https://doi.org/10.2307/749486.
 
34.
Smith, A. (2018, October 15). Premier’s push to have 100 per cent of students studying HSC maths. The Sydney Morning Herald. Retrieved from https://www.smh.com.au/politic....
 
35.
Smith, S. Z., & Smith, M. E. (2006). Assessing elementary understanding of multiplication concepts. School Science and Mathematics, 106(3), 140-149. https://doi.org/10.1111/j.1949....
 
36.
Sudarshan, A., & Aye, K.M. (2008). Teacher practices in mathematics classroom with at risk students. Paper presented at the Annual Meeting of the Australian Association for Research in Education, Brisbane.
 
37.
Tall, D. O. (2013). How humans learn to think mathematically. New York, NY: Cambridge University Press. https://doi.org/10.1017/CBO978....
 
38.
Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169. https://doi.org/10.1007/BF0030....
 
39.
Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: Refining a framework. Educational Studies in Mathematics, 75(3), 241–251. https://doi.org/10.1007/s10649....
 
40.
Webel, C., & DeLeeuw, W. W. (2015). Meaning for fraction multiplication: Thematic analysis of mathematical task in three fifth grade classes. The Journal of Mathematical Behavior, 41, 123-140. https://doi.org/10.1016/j.jmat....
 
41.
Whitacre, I., & Nickerson, S. D. (2016). Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude. Journal of Mathematics Teacher Education, 19(1), 57–77. https://doi.org/10.1007/s10857....
 
42.
Wilson, R. & Mack, J. (2014). Declines in high school mathematics and science participation: Evidence of students’ and future teachers’ disengagement with maths. International Journal of Innovation in Science and Mathematics Education, 22(7), 35-48. Retrieved from https://openjournals.library.s....
 
43.
article/view/7625.
 
44.
Young-Loveridge, J. (2005). Fostering multiplication using array-based materials. Journal of Australian Mathematics Teacher, 61(3), 34-40. Retrieved from https://files.eric.ed.gov/full....
 
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