[논문]유리수 지수 정의에 대한 교사 이해 분석

신보미

doi:10.7468/mathedu.2021.60.1.21

[국내논문] 유리수 지수 정의에 대한 교사 이해 분석
Teachers' understanding of the definition of rational exponent 원문보기

Journal of the Korean Society of Mathematical Education. Series A. The Mathematical Education, v.60 no.1, 2021년, pp.21 - 39

신보미 (전남대학교)

초록
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본 연구는 유리수 지수 정의에 대한 교사의 이해 특징을 분석하여 교사 교육에의 시사점을 구체화하고, 지수의 확장을 지도하는 수업에서 정의의 본질 및 그 이면의 사고를 다루기 위해 고려할 필요가 있는 교수학적 쟁점을 밝히는 데 목적을 두었다. 이를 위해 지필검사 도구를 개발하여 현직 고등학교 교사 50명의 답변을 분석하였으며, 이를 토대로 유리수 지수 정의에 대한 교사의 이해 특징이 교사 교육에 주는 시사점 및 교수학적 쟁점을 기술하였다. 또한 이러한 시사점 및 교수학적 쟁점을 국내 교과서 전개 방식에 비추어 해석하여 수업을 통해 지수의 확장과 관련된 정의의 본질을 의미있게 다루기 위해 교사 및 교과서가 좀 더 주목할 필요가 있는 측면을 제언하였다.

Abstract ▼ AI-Helper

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.

Keyword

표/그림 (22)

그림 [Fig. 1] The definition of irrational exponent (Ryu et al., 2017, p. 26)
그림 [Fig. 2] The proof of f'(x) = rx^r-1 (Lang, 2001, p. 127)
표 [Table 1] Summary of the teachers' responses to Q1
표 [Table 2] Summary of the teachers' responses to Q2
그림 [Fig. 3] A case of teachers' responses in Category 2-2
그림 [Fig. 4] A case of teachers' responses with explaining x >0 in Category 2-1
그림 [Fig. 5] A case of teachers' responses in Category AB-AB1
그림 [Fig. 6] The other case of teachers' responses in Category AB-AB1
그림 [Fig. 7] A case of teachers' responses in Category A-A1
그림 [Fig. 8] A case of teachers' responses in Category B-B1
그림 [Fig. 9] A case of teachers' responses in Category B-B3
그림 [Fig. 10] The case of teacher's response in Category B-B4
그림 [Fig. 11] The case of teacher's response in Category A-B1
그림 [Fig. 12] A case of teachers' responses in Category B-A1
그림 [Fig. 13] A case of teachers' responses in Category B-B2
그림 [Fig. 14] A case of teachers' responses in Category B-A2
$[Fig. 15] The graph of y = <TEX>$x^{\frac{1}{3}}$</TEX> from Geogebra$ 그림 [Fig. 15] The graph of y = $x^{\frac{1}{3}}$ from Geogebra
$[Fig. 16] The graph of y = <TEX>$x^{\frac{1}{3}}$</TEX> from Wolfram Alpha$ 그림 [Fig. 16] The graph of y = $x^{\frac{1}{3}}$ from Wolfram Alpha
그림 [Fig. 17] A problem related to the definition of rational exponent on Mathematics I (Ko et al., 2017, p. 18)
그림 [Fig. 18] A problem related to the restriction of bases in exponential function on an American textbook (Foerster, 2003, p. 277)
그림 [Fig. 19] A table to draw the graph of y = 2^x on Mathematics I (Kim et al., 2017, p. 39)
그림 [Fig. 20] A table to draw the graph of y = 2^x on a Japanese textbook (Ohkamoto, 2017, p. 149)

참고문헌 (44)

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

상세보기
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.
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

상세보기
Choi, G. S. (2014). Teaching geometry through Geogerba 5. Proceedings of the KSME 2014 Spring Conference on Mathematics Education (pp. 433-437). Seoul: KSME.
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

원문보기 상세보기
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

상세보기
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

상세보기
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

상세보기
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.
Foerster, A. P. (2003). Precalculus with Trigonometry. CA: Key Curriculum Press.
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

상세보기
Kim, I. S., Byun, C. H., & Ahn, S. H. (2012). Calculus. Seoul: Kyungmoonsa.
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.
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.
Kwak, D. Y., Kim, D. S., Seo, D. Y., Lee, S. Y., & Jin, G. T. (2001). Calculus. Seoul: Kyungmoonsa.
Landau, S. I. (2001). Dictionaries: The Art and Craft of Lexicography. Cambridge: Cambridge University Press.
Lang, S. (2001). Short Calculus. New York: Springer.
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

상세보기
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

상세보기
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

상세보기
Lewin, J. (2003). An Interactive Introduction to Mathematical Analysis. New York: Cambridge University Press.
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.
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

상세보기
Ministry of Education (2015). Mathematics curriculum. Seoul: Ministry of Education.
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.
National Council of Teachers of Mathematics(2000). Principles and Standards for School Mathematics. Reston: NCTM.
National Council of Teachers of Mathematics(2015). Principles to Actions: Ensuring Mathematics Success for All. Reston: NCTM.
Ohkamoto, K. (2017). Mathematics II. Tokyo: Gikkosubang.
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

상세보기
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

상세보기
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.
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

상세보기
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

상세보기
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. DOI: 10.3102/0013189X015002004

상세보기
Tall, D. (2013). How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics. New York: Cambridge University Press.
Telecommunication Technology Association (2021). Wolfram Alpha. Retrieved Jan. 21, 2021, from http://terms.tta.or.kr/dictionary/searchList.do
Thomas, B. G., Finney, L. R., & Weir, D. M. (2003). Calculus. New York: Addison Wesley.
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

상세보기
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.
Verberg, D., Purcell, J. E., & Ridgon. E. S. (2000). Calculus. New York: Prentice Hall.
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.
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.
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.
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

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