Developmental Process of Quadratic Equations from Past to Present and Reflections On Teaching-Learning
Year 2016,
Volume: 13 Issue: 3, 149 - 163, 29.12.2016
Pınar Güner
,
Tuğba Uygun
Abstract
The
mathematical concept of quadratic equations is one of the important topics in
algebra and has deep developmental process. It is also an inseparable component
of history of mathematics and mathematics curriculum. In this study, it was
aimed to present historical development of quadratic equations through periodic
examples with reference to using history of mathematics that may
help students to pay attention to the subject and improve meaningful
understanding. Besides, the purpose of the current study was to examine the
reflections of the developmental process on education and to produce
implications for quadratic equations training. Data were presented in time
sequence. The analysis showed that making sense of quadratic equations was
difficult in terms of the students and they had various misconceptions.
Therefore, this topic should be focused on it and some implications such as
using historical examples and explaining developmental process may facilitate
learning and teaching quadratic equations.
References
- Abramovich S. & Norton, A. (2006). Equations with parameters: A locus approach. Journal of Computers in Mathematics and Science Teaching, 25(1), 5-28.
- Allaire, P. R., & Bradley, R. E. (2001). Geometric approaches to quadratic equations from other times and places. Mathematics Teacher, 94(4), 308-319.
- Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics: Motivation for more techniques. Teaching Mathematics and its Applications: An International Journal of the IMA, 24(4), 143-153.
- Boyer, C. B. & Merzbach, U. C. (2011). A history of mathematics. New York: John Wiley & Sons.
- Cooley, J. W. (1993). Decartes analitic method and the art of geometric imagineering in negotiation and mediation. Valparaiso University Law Review, 28(1), 83-166.
- Gallardo. A. & Rojano, T. (1994). School algebra: 4 syntactic difficulties in the operativity with negative numbers. Proceedings of the XVI International Group for the Psychology of Mathematics Education, North American Chapter. Louisiana State Univesity, USA, Vol. I, pp. 159-165.
- Gallardo, A. (2000). Historical-Epistemological analysis in mathematics education: Two works in didactics of algebra. Perspective on School Algebra, 121-139.
- Gandz, S. (1937). The origin and development of the quadratic equations in Babylonian, Greek, and Early Arabic algebra. History of Science Society, 3, 405-557.
- Gandz, S. (1940). Studies in Babylonian mathematics III: Isoperimetric problems and the origin of the quadratic equations. Isis, 3(1), 103-115.
- Harding, A. & Engelbrecht, J. (2007). Sibling curves and complex roots 1: Looking back. International Journal of Mathematical Education in Science and Technology, 38(7), 963–973
- Hoffman, N. (1976). Factorisation of quadratics. Mathematics Teaching, 76, 54-55.
- Intanku, S., S. (2003). Diagnosis jenis kesilapan pelajar dalam pembelajaran perbezaan. [Diagnosis for the type of error in differentiation]. Master of Education Research Project. Universiti Kebangsaan Malaysia.
- Jensen P.L. (2005) Integer factorization. Master Thesis, Department of Computer science, University of Copenhagen, Denmark.
- Katz, V. J. (1997), Algebra and its teaching: An historical survey. Journal of Mathematical Behavior, 16(l), 25-36.
- Katz, V., J. (1998). A history of mathematics (2nd edition). Harlow, England: Addison Wesley Longman Inc.
- Katz, J. V. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66, 185–201.
- Kemp, A. (2010). Factorizing quadratics. Mathematics in School: for secondary and college teachers of mathematics, 39(4), 44-45.
- Kendal, M., & Stacey, K. (2004). Algebra: A world of difference. In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra (pp. 390-419). Dordrecht: Kluwer Academic.
- Kennedy, P. A., Warshauer, M. L. & Curtin, E. (1991). Factoring by grouping: Making the connection. Mathematics and Computer Education, 25(2), 118-123.
- Kotsopouslos, D. (2007). Unravelling students’ challenges with quadratics: a cognitive approach. Australian Mathematics Teacher, 63(2), 19-24.
- Leong, Y. H., Yap, S. F., Yvoone, T. M., Mohd Zaini, I. K. B., Chiew, Q. E., Tan, K. L. K., & Subramaniam, T. (2010). Concretising factorisation of quadratic expressions. The Australian Mathematics Teacher, 66(3), 19-24.
- Liew, S.T. & Wan Muhamad Saridan Wan Hasan. (1991). Ke arah memahami dan mengurangkan kesukaran dalam pembelajaran matematik. [Understanding and minimizing difficulty in learning mathematics] Berita Matematik, 38, 22-29.
- López, J. Robles, I. & Martínez-Planell, R. (2016). Students' understanding of quadratic equations, International Journal of Mathematical Education in Science and Technology, 47(4), 552-572.
- MacDonald, T. H. (1986). Problems in presenting quadratics as a unifying topic. Australian Mathematics Teacher, 42(3), 20.
- Nataraj, M. S., & Thomas, M. O. J. (2006). Expansion of binomials and factorisation of quadratic expressions: Exploring a vedic method. Australian Senior Mathematics Journal, 20(2), 8-17.
- National Council of Teachers of Mathematics [NCTM]. (2000). Curiculum and evaluation etandards for school mathematics. Reston. VA: National Council of Teachers of Mathematics.
- Norasiah, A. (2002). Diagnosis jenis kesilapan dalam hierarki Pembelajaran Serentak. [Error type diagnosis in learning simultaneous equation]. Master of Education Research Project, Universiti Kebangsaan Malaysia.
- Obermeyer, D. D. (1982). Another look at the quadratic formula. Mathematics Teacher, 75(2), 146-152.
- Olteanu, C. (2007). "Vad skulle x kunna vara": Andragradsekvation och andragradsfunktion som objekt för lärande. Kristianstad: Institutionen för beteendevetenskap, Högskolan i Kristianstad.
- Ontario Ministry of Education [OME] (2005). The Ontario curriculum grades 9 and 10 mathematics - Revised. Toronto: Queen’s Printer for Ontario.
- Parish, C.R. & Ludwig, H.J. (1994). Language, intellectual structures and common mathematical errors: a call for research. School Science and Mathematics. 94(5), 235-239.
- Rahim, Mohd Nor (1997). Kemahiran penyelesaian masalah matematik dikalangan pelajar menengah rendah. [Problem solving skills among lower secondary school students]. Master of Education Research Project. Universiti Kebangsaan Malaysia.
- Rauff, J. V. (1994). Constructivism, factoring, and beliefs. School Science and Mathematics, 94(8), 421.
- Rosen, Frederic (Ed. and Trans). (1831). The algebra of Mohumed Ben Muss. London: Oriental Translation Fund; reprinted Hildesheim: Olms, 1986, and Fuat Sezgin, Ed., Islamic Mathematics and Astronomy, Vol. 1. Frankfurt am Main: Institute for the History of Arabic-Islamic Scicence 1997.
- Roslina, R. (1997). Keupayaan algebra asas pelajar tingkatan empat sekolah menengah kerajaan Daerah Hulu Langat. [The ability of form four students in basic algebra]. Master of Education Research Project. Universiti Kebangsaan Malaysia.
- Sangwing, C. J. (2007). Assessing elementary algebra with STACK. International Journal of Mathematical Education in Science and Technology, 38(8), 987–1002.
- Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Research, 15(2), 4-14.
- Smith, D. (1951). History of mathematics, Vol. 1. New York: Dover.
- Smith, D. (1953). History of mathematics, Vol. 2. New York: Dover.
- Stols, H. G. (2004). Sketching the general quadratic equation using dynamic geometry software. International Journal of Mathematical Education in Science and Technology, 36(5), 483–488.
- Stover, D. W. (1978). Teaching quadratic problem solving. Mathematics Teacher, 71(1), 13-16.
- Sönnerhed, W. W. (2011). Mathematics textbooks for teaching. Retrieved from http://gupea.ub.gu.se/handle/2077/27935.
- Usiskin, Z. (1995). “Why is algebra important to learn?” American Educator 19, 30-37.
- Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students' understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47.
- Vinogradova, N. (2007). Solving quadratic equations by completing squares. Mathematics Teaching in the Middle School, 12(7), 403-405.
- Yakes, C. & Star, J. R. (2011), Using comparison to develop flexibility for teaching Algebra. Journal of Mathematics Teacher Eduction, 14, 175–191.
- Yong, L. L. (1970). The geometrical basis of the ancient Chinese square-root method. The History of Science Society, 61(1), 92-102.
- Zakaria, E. & Matt, S. M. (2010). Analysis of students’ error in learning of quadratic equations. International Education Studies, 3(3), 105-110.
- Zhu, X. & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition and Instruction, 4(3), 137-16.
DEVELOPMENTAL PROCESS OF QUADRATIC EQUATIONS FROM PAST TO PRESENT AND REFLECTIONS ON TEACHING-LEARNING
Year 2016,
Volume: 13 Issue: 3, 149 - 163, 29.12.2016
Pınar Güner
,
Tuğba Uygun
Abstract
The
mathematical concept of quadratic equations is one of the important topics in
algebra and has deep developmental process. It is also an inseparable component
of history of mathematics and mathematics curriculum. In this study, it was
aimed to present historical development of quadratic equations through periodic
examples with reference to using history of mathematics that may
help students to pay attention to the subject and improve meaningful
understanding. Besides, the purpose of the current study was to examine the
reflections of the developmental process on education and to produce
implications for quadratic equations training. Data were presented in time
sequence. The analysis showed that making sense of quadratic equations was
difficult in terms of the students and they had various misconceptions.
Therefore, this topic should be focused on it and some implications such as
using historical examples and explaining developmental process may facilitate
learning and teaching quadratic equations.
References
- Abramovich S. & Norton, A. (2006). Equations with parameters: A locus approach. Journal of Computers in Mathematics and Science Teaching, 25(1), 5-28.
- Allaire, P. R., & Bradley, R. E. (2001). Geometric approaches to quadratic equations from other times and places. Mathematics Teacher, 94(4), 308-319.
- Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics: Motivation for more techniques. Teaching Mathematics and its Applications: An International Journal of the IMA, 24(4), 143-153.
- Boyer, C. B. & Merzbach, U. C. (2011). A history of mathematics. New York: John Wiley & Sons.
- Cooley, J. W. (1993). Decartes analitic method and the art of geometric imagineering in negotiation and mediation. Valparaiso University Law Review, 28(1), 83-166.
- Gallardo. A. & Rojano, T. (1994). School algebra: 4 syntactic difficulties in the operativity with negative numbers. Proceedings of the XVI International Group for the Psychology of Mathematics Education, North American Chapter. Louisiana State Univesity, USA, Vol. I, pp. 159-165.
- Gallardo, A. (2000). Historical-Epistemological analysis in mathematics education: Two works in didactics of algebra. Perspective on School Algebra, 121-139.
- Gandz, S. (1937). The origin and development of the quadratic equations in Babylonian, Greek, and Early Arabic algebra. History of Science Society, 3, 405-557.
- Gandz, S. (1940). Studies in Babylonian mathematics III: Isoperimetric problems and the origin of the quadratic equations. Isis, 3(1), 103-115.
- Harding, A. & Engelbrecht, J. (2007). Sibling curves and complex roots 1: Looking back. International Journal of Mathematical Education in Science and Technology, 38(7), 963–973
- Hoffman, N. (1976). Factorisation of quadratics. Mathematics Teaching, 76, 54-55.
- Intanku, S., S. (2003). Diagnosis jenis kesilapan pelajar dalam pembelajaran perbezaan. [Diagnosis for the type of error in differentiation]. Master of Education Research Project. Universiti Kebangsaan Malaysia.
- Jensen P.L. (2005) Integer factorization. Master Thesis, Department of Computer science, University of Copenhagen, Denmark.
- Katz, V. J. (1997), Algebra and its teaching: An historical survey. Journal of Mathematical Behavior, 16(l), 25-36.
- Katz, V., J. (1998). A history of mathematics (2nd edition). Harlow, England: Addison Wesley Longman Inc.
- Katz, J. V. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66, 185–201.
- Kemp, A. (2010). Factorizing quadratics. Mathematics in School: for secondary and college teachers of mathematics, 39(4), 44-45.
- Kendal, M., & Stacey, K. (2004). Algebra: A world of difference. In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra (pp. 390-419). Dordrecht: Kluwer Academic.
- Kennedy, P. A., Warshauer, M. L. & Curtin, E. (1991). Factoring by grouping: Making the connection. Mathematics and Computer Education, 25(2), 118-123.
- Kotsopouslos, D. (2007). Unravelling students’ challenges with quadratics: a cognitive approach. Australian Mathematics Teacher, 63(2), 19-24.
- Leong, Y. H., Yap, S. F., Yvoone, T. M., Mohd Zaini, I. K. B., Chiew, Q. E., Tan, K. L. K., & Subramaniam, T. (2010). Concretising factorisation of quadratic expressions. The Australian Mathematics Teacher, 66(3), 19-24.
- Liew, S.T. & Wan Muhamad Saridan Wan Hasan. (1991). Ke arah memahami dan mengurangkan kesukaran dalam pembelajaran matematik. [Understanding and minimizing difficulty in learning mathematics] Berita Matematik, 38, 22-29.
- López, J. Robles, I. & Martínez-Planell, R. (2016). Students' understanding of quadratic equations, International Journal of Mathematical Education in Science and Technology, 47(4), 552-572.
- MacDonald, T. H. (1986). Problems in presenting quadratics as a unifying topic. Australian Mathematics Teacher, 42(3), 20.
- Nataraj, M. S., & Thomas, M. O. J. (2006). Expansion of binomials and factorisation of quadratic expressions: Exploring a vedic method. Australian Senior Mathematics Journal, 20(2), 8-17.
- National Council of Teachers of Mathematics [NCTM]. (2000). Curiculum and evaluation etandards for school mathematics. Reston. VA: National Council of Teachers of Mathematics.
- Norasiah, A. (2002). Diagnosis jenis kesilapan dalam hierarki Pembelajaran Serentak. [Error type diagnosis in learning simultaneous equation]. Master of Education Research Project, Universiti Kebangsaan Malaysia.
- Obermeyer, D. D. (1982). Another look at the quadratic formula. Mathematics Teacher, 75(2), 146-152.
- Olteanu, C. (2007). "Vad skulle x kunna vara": Andragradsekvation och andragradsfunktion som objekt för lärande. Kristianstad: Institutionen för beteendevetenskap, Högskolan i Kristianstad.
- Ontario Ministry of Education [OME] (2005). The Ontario curriculum grades 9 and 10 mathematics - Revised. Toronto: Queen’s Printer for Ontario.
- Parish, C.R. & Ludwig, H.J. (1994). Language, intellectual structures and common mathematical errors: a call for research. School Science and Mathematics. 94(5), 235-239.
- Rahim, Mohd Nor (1997). Kemahiran penyelesaian masalah matematik dikalangan pelajar menengah rendah. [Problem solving skills among lower secondary school students]. Master of Education Research Project. Universiti Kebangsaan Malaysia.
- Rauff, J. V. (1994). Constructivism, factoring, and beliefs. School Science and Mathematics, 94(8), 421.
- Rosen, Frederic (Ed. and Trans). (1831). The algebra of Mohumed Ben Muss. London: Oriental Translation Fund; reprinted Hildesheim: Olms, 1986, and Fuat Sezgin, Ed., Islamic Mathematics and Astronomy, Vol. 1. Frankfurt am Main: Institute for the History of Arabic-Islamic Scicence 1997.
- Roslina, R. (1997). Keupayaan algebra asas pelajar tingkatan empat sekolah menengah kerajaan Daerah Hulu Langat. [The ability of form four students in basic algebra]. Master of Education Research Project. Universiti Kebangsaan Malaysia.
- Sangwing, C. J. (2007). Assessing elementary algebra with STACK. International Journal of Mathematical Education in Science and Technology, 38(8), 987–1002.
- Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Research, 15(2), 4-14.
- Smith, D. (1951). History of mathematics, Vol. 1. New York: Dover.
- Smith, D. (1953). History of mathematics, Vol. 2. New York: Dover.
- Stols, H. G. (2004). Sketching the general quadratic equation using dynamic geometry software. International Journal of Mathematical Education in Science and Technology, 36(5), 483–488.
- Stover, D. W. (1978). Teaching quadratic problem solving. Mathematics Teacher, 71(1), 13-16.
- Sönnerhed, W. W. (2011). Mathematics textbooks for teaching. Retrieved from http://gupea.ub.gu.se/handle/2077/27935.
- Usiskin, Z. (1995). “Why is algebra important to learn?” American Educator 19, 30-37.
- Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students' understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47.
- Vinogradova, N. (2007). Solving quadratic equations by completing squares. Mathematics Teaching in the Middle School, 12(7), 403-405.
- Yakes, C. & Star, J. R. (2011), Using comparison to develop flexibility for teaching Algebra. Journal of Mathematics Teacher Eduction, 14, 175–191.
- Yong, L. L. (1970). The geometrical basis of the ancient Chinese square-root method. The History of Science Society, 61(1), 92-102.
- Zakaria, E. & Matt, S. M. (2010). Analysis of students’ error in learning of quadratic equations. International Education Studies, 3(3), 105-110.
- Zhu, X. & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition and Instruction, 4(3), 137-16.