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http://dx.doi.org/10.7468/mathedu.2021.60.1.21

Teachers' understanding of the definition of rational exponent  

Shin, Bomi (Chonnam National University)
Publication Information
The Mathematical Education / v.60, no.1, 2021 , pp. 21-39 More about this Journal
Abstract
The aim of this study was to deduce implications of the growth of mathematics teachers' specialty for effective instruction about the formulae of exponentiation with rational exponents by analyzing teachers' understanding of the definition of rational exponent. In order to accomplish the aim, this study ascertained the nature of the definition of rational exponent through examining previous literature and established specific research questions with reference to the results of the examination. A questionnaire regarding the nature of the definition was developed in order to solve the questions and was taken to 50 in-service high school teachers. By analysing the data from the written responses by the teachers, this study delineated four characteristics of the teachers' understanding with regard to the definition of rational exponent. Firstly, the teachers did not explicitly use the condition of the bases with rational exponents while proving f'(x)=rxr-1. Secondly, few teachers explained the reason why the bases with rational exponents must be positive. Thirdly, there were some teachers who misunderstood the formulae of exponentiation with rational exponents. Lastly, the majority of teachers thought that $(-8)^{\frac{1}{3}}$ equals to -2. Additionally, several issues were discussed related to teacher education for enhancing teachers' knowledge about the definition, features of effective instruction on the formulae of exponentiation and improvement points to explanation methods about the definition and formulae on the current high school textbooks.
Keywords
Definition of rational exponent; Formulae of exponentiation with rational exponents; Teacher knowledge;
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1 Fan, L. (2013). Textbook research as scientific research: Towards a common ground on issues and methods of research on mathematics textbooks. ZDM: the international journal on mathematics education, 45, 765-777. DOI: 10.1007/s11858-013-0530-6   DOI
2 Fischbein, E. (1993). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. Scholz, R. Straber, & B. Winkelmann (Eds.), Didactics of Mathematics as a Scientific Discipline (pp.231-245). Dordrecht: Kluwer.
3 Foerster, A. P. (2003). Precalculus with Trigonometry. CA: Key Curriculum Press.
4 Guberman, R. & Gorev, D. (2015). Knowledge concerning the mathematical horizon: A close view. Mathematics Education Research Journal, 27, 165-182. DOI:10.1007/s13394-014-0136-5   DOI
5 Adler, J. & Davis, Z. (2006). Opening another black box: Researching mathematics for teaching in mathematics teacher education, Journal for Research in Mathematics Education, 37(4), 270-296. DOI: 10.2307/30034851   DOI
6 Bernardo, G. & Carmen, B. (2010). The ambiguity of the sign √. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 509-518). France: CERME.
7 Kim, I. S., Byun, C. H., & Ahn, S. H. (2012). Calculus. Seoul: Kyungmoonsa.
8 Kim, W. K., Jo. M. S., Bang, G. S., Yoon, J. G., Shin, J. H., Yim, S. H. ..., Jeong, J. H. (2017). Mathematics I. Seoul: Visang.
9 Ko, S. E., Lee, J. H., Lee, S. W., Choi, S. G., Kim, Y. H., Oh, T. G., & Jo, S. C. (2017). Mathematics I. Seoul: Sinsago.
10 Even, R. & Tirosh, D. (1995). Subject matter knowledge and knowledge about students as sources of teacher presentations of the subject matter. Educational Studies in Mathematics, 29(1), 1-20. DOI:10.1007/BF01273897   DOI
11 Lavy, I. & Shriki, A. (2010). Engaging in problem posing activities in a dynamic geometry setting and the development of prospective teachers' mathematical knowledge. Journal of Mathematical Behavior, 29, 11-24. DOI: 10.1016/j.jmathb.2009.12.002   DOI
12 Kwak, D. Y., Kim, D. S., Seo, D. Y., Lee, S. Y., & Jin, G. T. (2001). Calculus. Seoul: Kyungmoonsa.
13 Landau, S. I. (2001). Dictionaries: The Art and Craft of Lexicography. Cambridge: Cambridge University Press.
14 Lang, S. (2001). Short Calculus. New York: Springer.
15 Leikin, R. & Zazkis, R. (2010). On the content-dependence of prospective teachers' knowledge: A case of exemplifying definitions. International Journal of Mathematical Education in Science and Technology, 41(4), 451-466. DOI: 10.1080/00207391003605189   DOI
16 Levenson, E. (2012). Teachers' knowledge of the nature of definitions: The case of the zero exponent. Journal of Mathematical Behavior, 31, 209-219. DOI: 10.1016/j.jmathb.2011.12.006   DOI
17 Lewin, J. (2003). An Interactive Introduction to Mathematical Analysis. New York: Cambridge University Press.
18 Ministry of Education (2015). Mathematics curriculum. Seoul: Ministry of Education.
19 Magiera, T. M., van den Kieboom, A. L., & Moyer, C. J. (2011). Relationships among features of pre-service teachers' algebraic thinking. In B. Ubuz (Ed.), Proceedings 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 169-176). Turkey: PME.
20 Matic, L. J. & Grancin, D. G. (2016). The use of the textbook as an artefact in the classroom: A case study in the light of a socio-didactical Tetrahedron. Journal fur Mathematik-Didaktik, 37(2), 349-374. DOI: 10.1007/s13138-016-0091-7   DOI
21 Robin, M. J., Fuller, E., & Harel, G. (2013). Double negative: The necessity principle, commognitive conflict, and negative number operations. Journal of Mathematical Behavior, 32, 649-659. DOI: 10.1016/j.jmathb.2013.08.001   DOI
22 Movshovitz-Hadar, N. (2011). Bridging between mathematics and education courses: Strategy games as generators of problem solving and proving tasks. In O. Zaslavsky & P. Sullivan (Eds.), Constructing Knowledge for Teaching Secondary Mathematics (pp. 117-140). New York: Springer.
23 National Council of Teachers of Mathematics(2000). Principles and Standards for School Mathematics. Reston: NCTM.
24 National Council of Teachers of Mathematics(2015). Principles to Actions: Ensuring Mathematics Success for All. Reston: NCTM.
25 Ohkamoto, K. (2017). Mathematics II. Tokyo: Gikkosubang.
26 Remillard, J. T., Harris, B., & Agodini, R. (2014). The influence of curriculum material design on opportunities for student learning. ZDM: the international journal on mathematics education, 46(5), 735-749. DOI: 10.1007/s11858-014-0585-z   DOI
27 Ryu, H. C., Sunwoo, H. S., Shin, B. M., Jo, J. M., Lee, B. M., Kim, Y. S., ..., Jeong. S. Y. (2017). Mathematics I. Seoul: Chunjae.
28 Sangwin, J. C. (2019). Textbook accounts of the rules of indices with rational exponents. International Journal of Mathematical Education in Science and Technology, 50(8). 1191-1209. DOI: 10.1080/0020739X.2019.1597935   DOI
29 Seaman, C. & Szydlik, J. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10(3), 167-182. DOI: 10.1007/s10857-007-9033-0   DOI
30 Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. DOI: 10.3102/0013189X015002004   DOI
31 Verberg, D., Purcell, J. E., & Ridgon. E. S. (2000). Calculus. New York: Prentice Hall.
32 Tall, D. (2013). How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics. New York: Cambridge University Press.
33 Telecommunication Technology Association (2021). Wolfram Alpha. Retrieved Jan. 21, 2021, from http://terms.tta.or.kr/dictionary/searchList.do
34 Thomas, B. G., Finney, L. R., & Weir, D. M. (2003). Calculus. New York: Addison Wesley.
35 Tirosh, D. & Even, R. (1997). To define or not to define: The case of $(-8)^{\frac{1}{3}}$. Educational Studies in Mathematics, 33, 321-330. DOI: 10.1023/A:1002916606955   DOI
36 Turner, F. & Rowland, T. (2011). The knowledge Quartet as an organizing framework for developing and deepening teachers' mathematics knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical Knowledge in Teaching (pp. 195-212). London: Springer.
37 Watson, J., Beswick, K., & Brown, N. (2006). Teachers' knowledge of their students as learners and how to intervene. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, Cultures and Learning Spaces: Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 551-558). Adelaide: MERGA.
38 Woo, J. H. & Cho, Y. M. (2001). A study on the definitions presented in school mathematics. The Journal of Education Research in Mathematics, 11(2), 363-384.
39 Woo, J. H. & Yim, J. H. (2008). Revisiting 0.999... and $(-8)^{\frac{1}{3}}$ in school mathematics from the perspective of the algebraic permanence principle. For the Learning of Mathematics, 28(2), 11-16.
40 Yang, S. A. & Lee, S. J. (2019). Secondary teachers' advanced knowledge for teaching algebra. School Mathematics, 21(2), 419-439. DOI: 10.29275/sm.2019.06.21.2.419   DOI
41 Edwards, B. & Ward, M. (2004). Surprises from mathematics education research: Student (mis)use of mathematical definitions. American Mathematical Monthly, 111(5), 411-424. DOI:10.1080/00029890.2004.11920092   DOI
42 Chung, Y. J. & Lee, K. H. (2018). Using what-if-not strategy for teaching definitions: Focusing on the exterior angle of polygon. School Mathematics, 20(3), 379-394. DOI : 10.29275/sm.2018.09.20.3.379   DOI
43 Choi, G. S. (2014). Teaching geometry through Geogerba 5. Proceedings of the KSME 2014 Spring Conference on Mathematics Education (pp. 433-437). Seoul: KSME.
44 Do, J. H. & Park, Y. B. (2011). Comments on the definition of the rational exponent $a{\frac{m}{n}}$ in contemporary Korean highschool mathematics textbooks. The Mathematical Education, 50(1), 61-67. DOI:10.7468/mathedu.2011.50.1.061   DOI