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Analysis of Misconceptions and Errors Regarding Exponential and Radical Expressions Through the Theory of Reducing Abstraction

Year 2024, Volume: 8 Issue: 2, 281 - 295
https://doi.org/10.54535/rep.1520588

Abstract

Investigating the reasons for misconceptions and errors is essential and important for making improvements in mathematics teaching. The idea of reducing abstraction, utilized as the framework for this research, can essentially be based on students' tendency to work with a lower abstraction level than the concepts they encounter in the course or what experts (mathematicians, teachers, etc.) expect from them. Since the process of reducing abstraction often occurs unconsciously, it can lead to misconceptions and errors. Exponential and radical expressions, which students first encounter in secondary school, are significant topics in mathematics, offering ease of representation and various calculations in many fields of basic sciences and engineering. Research on exponential and radical expressions, perceived by secondary and high school students as a collection of unnecessary formulas unrelated to daily life and difficult to understand, has revealed various misconceptions and errors. In this study, through the theory of reducing abstraction, the possible reasons for the misconceptions and errors revealed by research will be interpreted from an alternative viewpoint. Thus, a new perspective on student approaches regarding these subjects will be provided to teachers and mathematics educators.

References

  • Avcu, R. (2010). Eight graders’ capabilities in Exponents: Making mental comparisons. Practice and Theory in Systems of Education, 5(1), 39-48.
  • Aydın, U., & Özdemir, E. (2010). Analysis of students' misconceptions concerning exponent and root expressions. Educational Sciences: Theory & Practice, 10(3), 1775-1800.
  • Bingölbali, E. & Özmantar, M. F. (2012). Matematiksel kavram yanılgıları: Sebepleri ve çözüm arayışları [Mathematical misconceptions: Causes and solutions]. In E. Bingölbali & M. F. Özmantar (Eds.), İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri [Mathematical difficulties encountered in primary education and solution suggestions] (pp. 1 – 30). Ankara, Turkiye: Pegem Akademi.
  • Corbetta, P. (2003). Social research: Theory, methods and techniques. Thousand Oaks: Sage.
  • Corbin, J. & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks: Sage.
  • D. Ancheta, C. M. & Subia, S. (2020). Error analysis of engineering students' misconceptions in algebra. International Journal of Engineering Trends and Technology, 68(12), 66-71.
  • Denbel, D. G. (2019). Students’ difficulties of understanding exponents and exponential expressions. Engineering and Technology Research, 1(3), 083-088.
  • Dreyfus T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Boston, MA: Kluwer.
  • Elstak, I.R. (2007). College student’s understanding of rational exponents: A teaching experiment (Doctoral Dissertation). Retrieved from https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=osu 1186505864&disposition=inline
  • Eraslan, A. (2008): The notion of reducing abstraction in quadratic functions. International Journal of Mathematical Education in Science and Technology, 39(8), 1051-1060. http://dx.doi.org/10.1080/00207390802136594
  • Erlandson, K. M. (2013). A study of college students’ misconceptions of radical expressions (Master’s Thesis). Retrieved from https://soar.suny.edu/handle/20.500.12648/211
  • Ferrari, P. L. (2003). Abstraction in mathematics. Philosophical Transactions: Biological Sciences, 358(1435), 1225-1230.
  • Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts. Educational Studies in Mathematics, 40, 71–90.
  • Hazzan, O. (2001). Reducing abstraction: The case of constructing an operation table for a group. Journal of Mathematical Behaviour, 20(2), 163-172.
  • Hazzan, O. (2003a). How students attempt to reduce abstraction in the learning of mathematics and in the learning of computer science. Computer Science Education, 13(2), 95-122.
  • Hazzan, O. (2003b). Reducing abstraction when learning computability theory. Journal of Computers in Mathematics and Science Teaching, 22(2), 95-117.
  • Hazzan, O. and Zaskis, R. (2005). Reducing abstraction: The case of school mathematics. Educational Studies in Mathematics, 58, 101–119.
  • Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
  • Nesher, P. (1987). Towards an instructional theory: The role of student's misconceptions. For the Learning of Mathematics, 7(3), 33-39.
  • Oliver, A. (1989). Handling Pupils’ Misconceptions. Presidential address delivered at the Thirteenth National Convention on Mathematics, Physical Science and Biology Education, Pretoria, 3-7 July 1989. Retrieved from http://academic.sun.ac.za/mathed /Malati/Misconceptions.htm
  • Raychaudhuri, D. (2014). Adaptation and extension of the framework of reducing abstraction in the case of differential equations. International Journal of Mathematical Education in Science and Technology, 45(1), 35-57, http://dx.doi.org/10.1080/0020739X.2013.790503
  • Rushton, N. (2014). Common errors in Mathematics. Research Matters: A Cambridge Assessment publication, 17(1), 8-17.
  • 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.
  • Sikora-Press, C. M. (2016). Writing Radical Wrongs: A Study of Students' Misconceptions with Radicals and Rational Exponents (Master’s Thesis). Retrieved from https://soar.suny.edu/handle/20.500.12648/403
  • Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115-163.
  • Stacey, K. (2006). Addressing mathematical misconceptions: Using research to guide teaching. Australian Primary Mathematics Classroom, 11(2), 14-18.
  • Şenay, Ş. C. (2002). Üslü ve köklü sayıların öğretiminde öğrencilerin yaptıkları hatalar ve yanılgıları üzerine bir araştırma [A study on students' mistakes and misconceptions in teaching exponential and radical numbers] (Master’s thesis, Selçuk University, Konya, Turkiye). Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/
  • Şenay, Ş. C. ve Özdemir, A. Ş. (2014). Matematik öğretmen adaylarının lineer kongrüanslar ile ilgili soyutlamayı indirgeme eğilimleri [Pre-Service Mathematics Teachers’ Tendencies of Reducing Abstraction about Linear Congruence]. Eğitim ve İnsani Bilimler Dergisi: Teori ve Uygulama [Journal of Education and Humanities: Theory and Practice], 5(10), 59 – 72.
  • Şenay, Ş. C. and Özdemir, A. Ş. (2019). Analysis of Errors and Misconceptions about Number Theory Through the Notion of Reducing Abstraction. In Abstract Book of International Conference on Mathematics and Mathematics Education (ICMME 2019), p. 359 http://www.icmme2019.selcuk.edu.tr/ICMME2019_Abstract_Book.pdf
  • Tall, D. (2004). Building concepts in calculus: The role of intuitive concept images and formal concept definitions. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121-128).
  • Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematical education. In I. Harel & S. Papert (eds.), Constructionism (pp. 193-203). Norwood, NJ: Ablex Publishing Corporation.
  • Yenilmez, K. ve Yaşa, E. (2008). İlköğretim Öğrencilerinin Geometrideki Kavram Yanılgıları [Primary School Students' Misconceptions in Geometry]. Uludağ Üniversitesi Eğitim Fakültesi Dergisi [Journal of Uludag University Faculty of Education], XXI (2), 461-483.
  • Zazkis, R., & Liljedahl, P. (2004). Understanding the structure of powers and roots in secondary school mathematics. Journal of Mathematical Behavior, 23(2), 179-202.
Year 2024, Volume: 8 Issue: 2, 281 - 295
https://doi.org/10.54535/rep.1520588

Abstract

References

  • Avcu, R. (2010). Eight graders’ capabilities in Exponents: Making mental comparisons. Practice and Theory in Systems of Education, 5(1), 39-48.
  • Aydın, U., & Özdemir, E. (2010). Analysis of students' misconceptions concerning exponent and root expressions. Educational Sciences: Theory & Practice, 10(3), 1775-1800.
  • Bingölbali, E. & Özmantar, M. F. (2012). Matematiksel kavram yanılgıları: Sebepleri ve çözüm arayışları [Mathematical misconceptions: Causes and solutions]. In E. Bingölbali & M. F. Özmantar (Eds.), İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri [Mathematical difficulties encountered in primary education and solution suggestions] (pp. 1 – 30). Ankara, Turkiye: Pegem Akademi.
  • Corbetta, P. (2003). Social research: Theory, methods and techniques. Thousand Oaks: Sage.
  • Corbin, J. & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks: Sage.
  • D. Ancheta, C. M. & Subia, S. (2020). Error analysis of engineering students' misconceptions in algebra. International Journal of Engineering Trends and Technology, 68(12), 66-71.
  • Denbel, D. G. (2019). Students’ difficulties of understanding exponents and exponential expressions. Engineering and Technology Research, 1(3), 083-088.
  • Dreyfus T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Boston, MA: Kluwer.
  • Elstak, I.R. (2007). College student’s understanding of rational exponents: A teaching experiment (Doctoral Dissertation). Retrieved from https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=osu 1186505864&disposition=inline
  • Eraslan, A. (2008): The notion of reducing abstraction in quadratic functions. International Journal of Mathematical Education in Science and Technology, 39(8), 1051-1060. http://dx.doi.org/10.1080/00207390802136594
  • Erlandson, K. M. (2013). A study of college students’ misconceptions of radical expressions (Master’s Thesis). Retrieved from https://soar.suny.edu/handle/20.500.12648/211
  • Ferrari, P. L. (2003). Abstraction in mathematics. Philosophical Transactions: Biological Sciences, 358(1435), 1225-1230.
  • Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts. Educational Studies in Mathematics, 40, 71–90.
  • Hazzan, O. (2001). Reducing abstraction: The case of constructing an operation table for a group. Journal of Mathematical Behaviour, 20(2), 163-172.
  • Hazzan, O. (2003a). How students attempt to reduce abstraction in the learning of mathematics and in the learning of computer science. Computer Science Education, 13(2), 95-122.
  • Hazzan, O. (2003b). Reducing abstraction when learning computability theory. Journal of Computers in Mathematics and Science Teaching, 22(2), 95-117.
  • Hazzan, O. and Zaskis, R. (2005). Reducing abstraction: The case of school mathematics. Educational Studies in Mathematics, 58, 101–119.
  • Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
  • Nesher, P. (1987). Towards an instructional theory: The role of student's misconceptions. For the Learning of Mathematics, 7(3), 33-39.
  • Oliver, A. (1989). Handling Pupils’ Misconceptions. Presidential address delivered at the Thirteenth National Convention on Mathematics, Physical Science and Biology Education, Pretoria, 3-7 July 1989. Retrieved from http://academic.sun.ac.za/mathed /Malati/Misconceptions.htm
  • Raychaudhuri, D. (2014). Adaptation and extension of the framework of reducing abstraction in the case of differential equations. International Journal of Mathematical Education in Science and Technology, 45(1), 35-57, http://dx.doi.org/10.1080/0020739X.2013.790503
  • Rushton, N. (2014). Common errors in Mathematics. Research Matters: A Cambridge Assessment publication, 17(1), 8-17.
  • 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.
  • Sikora-Press, C. M. (2016). Writing Radical Wrongs: A Study of Students' Misconceptions with Radicals and Rational Exponents (Master’s Thesis). Retrieved from https://soar.suny.edu/handle/20.500.12648/403
  • Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115-163.
  • Stacey, K. (2006). Addressing mathematical misconceptions: Using research to guide teaching. Australian Primary Mathematics Classroom, 11(2), 14-18.
  • Şenay, Ş. C. (2002). Üslü ve köklü sayıların öğretiminde öğrencilerin yaptıkları hatalar ve yanılgıları üzerine bir araştırma [A study on students' mistakes and misconceptions in teaching exponential and radical numbers] (Master’s thesis, Selçuk University, Konya, Turkiye). Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/
  • Şenay, Ş. C. ve Özdemir, A. Ş. (2014). Matematik öğretmen adaylarının lineer kongrüanslar ile ilgili soyutlamayı indirgeme eğilimleri [Pre-Service Mathematics Teachers’ Tendencies of Reducing Abstraction about Linear Congruence]. Eğitim ve İnsani Bilimler Dergisi: Teori ve Uygulama [Journal of Education and Humanities: Theory and Practice], 5(10), 59 – 72.
  • Şenay, Ş. C. and Özdemir, A. Ş. (2019). Analysis of Errors and Misconceptions about Number Theory Through the Notion of Reducing Abstraction. In Abstract Book of International Conference on Mathematics and Mathematics Education (ICMME 2019), p. 359 http://www.icmme2019.selcuk.edu.tr/ICMME2019_Abstract_Book.pdf
  • Tall, D. (2004). Building concepts in calculus: The role of intuitive concept images and formal concept definitions. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121-128).
  • Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematical education. In I. Harel & S. Papert (eds.), Constructionism (pp. 193-203). Norwood, NJ: Ablex Publishing Corporation.
  • Yenilmez, K. ve Yaşa, E. (2008). İlköğretim Öğrencilerinin Geometrideki Kavram Yanılgıları [Primary School Students' Misconceptions in Geometry]. Uludağ Üniversitesi Eğitim Fakültesi Dergisi [Journal of Uludag University Faculty of Education], XXI (2), 461-483.
  • Zazkis, R., & Liljedahl, P. (2004). Understanding the structure of powers and roots in secondary school mathematics. Journal of Mathematical Behavior, 23(2), 179-202.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematics Education
Journal Section Articles
Authors

Ş. Can Şenay 0000-0001-8437-180X

Early Pub Date December 18, 2024
Publication Date
Submission Date July 23, 2024
Acceptance Date October 4, 2024
Published in Issue Year 2024 Volume: 8 Issue: 2

Cite

APA Şenay, Ş. C. (2024). Analysis of Misconceptions and Errors Regarding Exponential and Radical Expressions Through the Theory of Reducing Abstraction. Research on Education and Psychology, 8(2), 281-295. https://doi.org/10.54535/rep.1520588

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