The Korea Society Of Educational Studies In Mathematics
[ Article ]
Journal of Educational Research in Mathematics - Vol. 30, No. 2, pp.353-374
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 May 2020
Received 10 Apr 2020 Revised 15 May 2020 Accepted 15 May 2020

Alternative Method of Irrational Numbers Using Approximate Fractions and Slopes of Straight Lines in a Spreadsheet Environment

Yoo, Jae-Geun* ; Yi, Jung-A**, ; Park, Moon Hwan*** ; Chang, Hyewon****
*Teacher, Hongcheon middle school, South Korea
**Teacher, Pungduck high school, South Korea
***Professor, Chuncheon National University of Education, South Korea
****Professor, Seoul National University of Education, South Korea

Correspondence to: Teacher, Pungduck high school, South Korea,


The purpose of this study was to explore the possibility of interconnection between non-fractional representation and non-circular decimal representation in irrational teaching and learning of school mathematics and to examine the possibility of field application. First, through the analysis of previous studies, it was found that the core of understanding irrational concepts is the intuitive perception of convergence and the inference of acyclicity. And through mathematical analysis, we confirmed that finite decimal sequences and approximate fractional sequences should be learning contents. In this regard, we applied engineering tools in teaching and learning activity of irrational numbers and analyzed the process by which engineering tools become learners’ mathematical tools and then learning occurs, that is, instrument genesis. As a result, in the activity of finding finite fractions of square roots, students gained a meaningful understanding by focusing on the meaning of square root symbols and square roots as objects from inputting functions and operations and using drag. And in the activity to solve the task ‘Can a graph cross a grid point?’, Students could infer the acyclicity of the decimal representation of irrational numbers by observing the approximate fraction as the slope. Thus, this study is expected to be a basic reference of teaching and learning of irrational numbers to construct a mental image of real numbers completeness axiom.


irrational number, fractional representation, instrumental genesis, teaching experiment, slopes of straight lines, a spreadsheet environment


  • Byun, H. H. (2005). A Didactical Analysis of the Decimal Fraction Concept. Doctoral dissertation, Seoul National University Graduate School.
  • Byun, H. H. & Park, S. Y. (2002). Teaching and Learning Irrational Number with Its Conceptual Aspects Stressed : Consideration of Irrational Number through the Conception of ‘Incommensurability’ School Mathematics, 4(4), 643-655.
  • Chang, K. Y., et al. (2020). Middle School Mathematics 3. Seoul: Jihaksa.
  • Choi, E. A. & Kang, H. I. (2016). Pre-Service Teachers' Understanding of the Concept and Representations of Irrational Numbers. School Mathematics, 18(3), 647-666.
  • Drijvers, P. & Gravemeijer, K. (2005). Computer algebra as an instrument: examples of algebraic schemes. In Guin, D., Ruthven, K., & Trouche, L. (eds.), The Didactical Challenge of Symbolic Calculators. Boston, MA: Springer.
  • González-Martín, A. S., Giraldo, V., & Souto, A. M. (2013). The introduction of real numbers in secondary education: an institutional analysis of textbooks. Research in mathematics education, 15(3), 230-248. []
  • Han, S. H. & Chang, K. Y. (2009). Instrumental Genesis of Computer Algebra System(CAS) in Mathematical Problem Solving among High School Students. School Mathematics, 11(3), 527-546.
  • Kang, H. I. & Choi, E. A. (2017). Teacher Knowledge Necessary to Analyze Student's Errors and Difficulties about the Concept of Irrational Numbers. School Mathematics, 19(2), 319-343.
  • Kang, J. G. (2016). Difficulties and Alternative Ways to learn Irrational Number Concept in terms of Notation. Journal of the Korean School Mathematics Society, 19(1), 63-82.
  • Kim, B. Y. & Chung, Y. W. (2008). Inducing Irrational Numbers in Junior High School. The Korean Journal for History of Mathematics, 21(1), 139-156.
  • Kim, B. Y. & Lee, J. S. (2008). Technology as Instruments and the Change of Paradigm in Mathematics Learning. Journal of the Korean School Mathematics Society, 47(3), 261-271.
  • Kim, S. R., et al. (2013). Middle School Mathematics 3. Seoul: Chunjae.
  • Klein, F. (1924). Elementary Mathematics from an advanced standpoint- Arithmetic⋅Algebra·Analysis. New York: Dover Publications.
  • Lee, J. H. (2014). The Infinite Decimal Representation: Its Opaqueness and Transparency. Journal of Educational Research in Mathematics, 24(4), 595-605.
  • Lee, J. H. (2015). Beyond the Union of Rational and Irrational Numbers: How Pre-Service Teachers Can Break the Illusion of Transparency about Real Numbers?. Journal of Educational Research in Mathematics, 25(3), 263-279.
  • Lee, S. B. (2013). Preservice secondary matheamtics teachers' understanding of irrational numbers. Journal of the Korean School Mathematics Society, 16(3), 499-518.
  • Lee, Y. R. & Lee, K. H. (2006). A Case Study on the Introducing Method of Irrational Numbers Based on the Freudenthal's Mathematising Instruction Theory. Journal of Educational Research in Mathematics, 16(4), 297-312.
  • Loborde, C. & StraBer, R. (2010). Place and use of new technology in the teaching of mathematics: ICMI activities in the past 25 years. ZDM, 42, 121-133. []
  • Oh, K. H., Park, J. S. & Kwon, O. N. (2017). A textbook analysis of irrational numbers unit: focus on the view of process and object. The Mathematical Education, 56(2), 131-145.
  • Park, S. Y. (2016). Defining the Infinite Decimal without Using the 'Limit to a Real Number'. Journal of Educational Research in Mathematics, 26(2), 159-172.
  • Park, Y. H., Park, D. W. & Jung I. C. (2004). Study on learneer's understanding of the concept of irrational number in middle school. Journal of the Korean School Mathematics Society, 7(2), 99-116.
  • Trouche, L. (2005). An instrumental approach to mathematics learning in symbolic calculator environments. In Guin, D., Ruthven, K., & Trouche, L. (eds.), The Didactical Challenge of Symbolic Calculators: turning a computational device into a mathematical instrument (pp. 137-162). N.Y.: Springer. []
  • Trouche, L. (2018). Understanding the work of teachers through their interaction with their teaching resources -a history of trajectories. Educación Matemática, Sociedad Mexicana de Investigación y Divulgación de la Educación Matemática A. C., 30(3), 9-40. available from:
  • Woo, J. H. (2017). School math educational foundation (revised) A. Seoul: Seoul National University Publishing Council.