Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study

Verónica Díaz; Maria Aravena; George Flores

doi:10.29333/ejmste/8547

Full Text (PDF)

Verónica Díaz ¹ ^* , Maria Aravena ² ^* , George Flores ³ ^*

More Detail

¹ Universidad de Los Lagos, Departamento de Ciencias Exactas, Osorno, CHILE² Universidad Católica del Maule, Facultad de Ciencias Básicas, Talca, CHILE³ Universidad de Los Lagos, Dirección de Acceso, Equidad y Permanencia, Osorno, CHILE^* Corresponding Author

Abstract

The article aims to determine the academic performance and errors in the resolution of types of problems of application of the quadratic function, of high school students from the Los Lagos Region and Los Rios Region in Chile. The approach is qualitative and descriptive with case studies. A math test with open response problems and an opinion questionnaire were developed and applied. Through the results, the highest academic performance is evidenced in the routine problems of purely mathematical context and fantasist context, but with difficulty in the resolution of non-routine problems. In addition, errors originating in affective and emotional attitudes associated with blockages at the time of initiating the resolution, forgetfulness at the time of posing the quadratic function, prevail over cognitive errors originating in an obstacle and errors originating in the absence of meaning.

Keywords

errors
problem solving
quadratic function
secondary education

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 16, Issue 11, November 2020, Article No: em1896

https://doi.org/10.29333/ejmste/8547

Publication date: 26 Sep 2020

Article Views: 3442

Article Downloads: 3066

Open Access References How to cite this article

References

Adams, D. M., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., & Van Velsen, M. (2014). Using erroneous examples to improve mathematics learning with a web-based tutoring system. Computer in Human Behavior, 36, 401-411. https://doi.org/10.1016/j.chb.2014.03.0530747-5632/Ó2014
Agustyaningrum, N., Abadi, A., & Mahmudi, A. (2018). An analysis of students’ error in solving abstract algebra tasks. Journal of Physics: Conference Series. 1097. 012118. https://doi.org/10.1088/1742-6596/1097/1/012118
Akgün, L., Işık, C., Tatar, E., İşleyen, T., & Soylu, Y. (2012). Transfer of mathematical knowledge: series. Australian Journal of Teacher Education, 37(3), 83-89. https://doi.org/10.14221/ajte.2012v37n3.2
Akyüz, G. (2020). Non-routine problem solving performances of mathematics teacher candidates. Educational Research Review, 15(5), 214-224. https://doi.org/10.5897/ERR2020.3907
Bajaña, C. (2019). Didactic proposal for teaching quadratic equations and functions, through the use of “Desmos Graphing Calculator” [Propuesta didáctica para la enseñanza de funciones y ecuaciones cuadráticas, a través del uso de: “Desmos Graphing Calculator”]. (Bachelor), Universidad de Cuenca, Ecuador.
Beilock, S. L., Schaeffer, M. W., & Rozek, C. S. (2017). Understanding and addressing performance anxiety. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), Handbook of competence and motivation: Theory and application (p. 155-172). New York, NY: The Guilford Press.
Benning, I., & Agyei, D. D. (2016). Effect of using spread sheet in teaching quadratic functions on the performance of senior high school students. International Journal of Education, Learning and Development, 4(1), 11-29.
Berger, N., Mackenzie, E., & Holmes, K. (2020). Positive attitudes towards mathematics and science are mutually beneficial for student achievement: a latent profile analysis of TIMSS 2015. The Australian Educational Researcher, 47, 409-444. https://doi.org/10.1007/s13384-020-00379-8
Blanco, L., Guerrero, E., & Caballero, A. (2013). Cognition and affect in mathematics problem solving with prospective teachers. The Mathematics Enthusiast, 10(1-2), 335-364.
Boesen, J., Lithner, J., & Palm, T. (2010). The relation between types of assessment tasks and the mathematical reasoning students use. Educational Studies in Mathematics, 75(1), 89-105. https://doi.org/10.1007/s10649-010-9242-9
Budd, C., & Sangwin, C. (2004). 101 uses of a quadratic equation: Part II. Magazine Living Mathematics, 30 [on-line]. Retrieved on 3 April 2020 from http://plus.maths.org/content/os/issue30/features/quadratics/index
Çelik, Ö., & Güzel, B. (2019). An instructional sequence triggering students’ quantitative reasoning during learning of quadratic functions. Turkish Journal of Computer and Mathematics Education, 10(1), 157-194. https://doi.org/10.16949/turkbilmat.446403
Chapman, O. (2013). Mathematical-task knowledge for teaching. Journal of Mathematics Teacher Education, 16, 1-6. https://doi.org/10.1007/s10857-013-9234-7
Davis, J. D., Smith, D. O., Roy, A. R., & Bilgic, Y. K. (2014). Reasoning-and-proving in algebra: The case of two reform-oriented U.S. Textbooks. International Journal of Educational Research, 64, 92-106. https://doi.org/10.1016/j.ijer.2013.06.012
Díaz V. (2020). Difficulties and performance in mathematics competences: solving problems with derivatives. International Journal of Engineering Pedagogy, 10(4), 35-53. https://doi.org/10.3991/ijep.v10i4.12473
Díaz, V. (2019). Limits Problem Solving in Engineering Careers: Competences and Errors. International Journal of Business, Human and Social Sciences, 13(7), 951-957. https://doi.org/10.5281/zenodo.3299965
Díaz, V., & Poblete, A. (2001). Categorizing types of problems in algebra [Categorizando tipos de problemas en álgebra]. UNO. Revista de Didáctica de las Matemáticas, 27, 93-103.
Díaz, V., & Poblete, A. (2018). Use of didactic models of the teachers of mathematics in the teaching of logarithmic, quadratic and exponential functions [Uso de modelos didácticos de los docentes de matemáticas en la enseñanza de funciones logarítmicas, cuadráticas y exponenciales]. Paradigma, 39(1), 353-372.
Díaz, V., Belmar, H., & Poblete, A. (2018). Emotional manifestation and modeling of a mathematical function [Manifestación emocional y modelamiento de una función matemática]. Bolema Boletim de Educação Matemática, 32(62), 1198-1218. https://doi.org/10.1590/1980-4415v32n62a22
Didis, M. G., Bas, S., & Erbas, A. (2011). Students’ reasoning in quadratic equations with one unknown. Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education. University of Rzeszów, Poland. Retrieved on 18 April 2020 from http://www.ezproxy.uvm.edu/cerme7.univ.Rzeszowpl/WG/3/CERME7_WG3_Gozde.pdf
Dündar, S. (2015). The investigation of teacher candidates’ skills of solving exercises and non-routine problems related to the topic of series. Kastamonu Education Journal, 23(3),1293-1310.
Durkin, K., & Rittle-Johnson, B. (2012). The efectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206-214. https://doi.org/10.1016/j.learninstruc.2011.11.001
Ellis, A. B., & Grinstead, P. (2008). Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. The Journal of Mathematical Behavior, 27(4), 277-296. https://doi.org/10.1016/j.jmathb.2008.11.002
Eraslan, A. (2008). The notion of reducing abstraction in quadratic functions. International Journal of Mathematical Education in Science and Technology, 39(8), 1051-1060. https://doi.org/10.1080/00207390802136594
Esquer, M. P., Robles, A., Cosmes, S., & Ansaldo, J. (2017). Didactic proposal with quadratic functions of problems in context at a higher level [Propuesta didáctica con funciones cuadráticas de problemas en contexto a nivel superior]. In F.Rodríguez & R. Rodríguez (Eds.), Memoria de la XVII Escuela de Invierno en Matemática Educativa (pp. 376-381). Oaxaca: Red Cimates.
Farez, M. (2018). Solving quadratic functions through the use of learning objects by second year high school students, in the feedback phase [Resolución de funciones cuadráticas mediante el uso de objetos de aprendizaje por parte de estudiantes de segundo año de bachillerato, en la fase de retroalimentación]. (Master Thesis), Universidad de Cuenca, Ecuador.
Fuadi, I., Minarni, A., & Banjarnahor, H. (2017). Analysis of students’ mathematical problem solving ability in ix grade at junior high school ar-rahman percut. International Journal of Novel Research in Education and Learning, 4(2), 153-159.
Fulgar, A. (2020). Comparative analysis of mathematics proficiency and attitudes toward mathematics of senior high school student 2020. International Journal of Scientific and Research Publications, 10(5), 211-222. https://doi.org/10.29322/IJSRP.10.05.2020.p10125
Gómez-Blancarte, A., Guirette, R., & Morales-Colorado, F. (2017). Proposal for treating global interpretation of the quadratic function using the software GeoGebra [Propuesta para el tratamiento de interpretación global de la función cuadrática mediante el uso del software GeoGebra]. Educación Matemática, 29(3), 189-224. https://doi.org/10.24844/em2903.07
Good. T. L., & Lavigne, A.L. (2018). Looking in classrooms (11th ed.). New York, NY: Routledge. https://doi.org/10.4324/9781315627519
Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: can this foster learning outcomes? Learning and Instruction, 17(6), 612-634. https://doi.org/10.1016/j.learninstruc.2007.09.008
Haciomeroglu, G. (2013). The field experiences of student teachers and effective mathematics teaching in Turkey. Australian Journal of Teacher Education, 38(2), 131-142. https://doi.org/10.14221/ajte.2013v38n2.5
Hall, N., & Goetz, T. (2013). Emotion, motivation, and self-regulation: A handbook for teachers. UK: Esmerald Group Publishing Limited.
Heinze, A., & Reiss, K. (2007). Mistake-handling activities in the mathematics classroom: Effects of an in-service teacher training on students’ performance in geometry. In J.Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education. Seoul: PME.
Henderson, S. (2012). Why the journey to mathematical excellence may be long in Scotland’s primary schools. Scottich Education Review, 44(1), 46-56.
Hernández, J., Castañeda, A., & González, R. (2019). The solution of an unconventional mathematical problem by university students [La solución de un problema matemático no convencional por estudiantes universitarios]. Revista Científica, 35(2). https://doi.org/10.14483/23448350.14863
Hernández, R., Fernández, C., & Baptista, P. (2014). Investigation methodology [Metodología de la Investigación]. México: McGraw-Hill.
Jäder, J., Lithner, J., & Sidenvall, J. (2019). Mathematical problem solving in textbooks from twelve countries. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2019.1656826
Kaya, S., & Kablan, Z. (2018). The Analysis of the Studies on Non-Routine Problems. Necatibey Faculty of Education. Electronic Journal of Science and Mathematics Education, 12(1), 44-25. https://doi.org/10.17522/balikesirnef.437652
Kilic, H. (2011). Preservice secondary mathematics teachers’ knowledge of students. Turkish Online Journal of Qualitative Inquiry, 2(2), 17-35.
Kotsopoulos, D. (2007). Unravelling student challenges with quadratics. Australian Mathematics Teacher, 63(2), 19-24.
Langat, A. C. (2015). Students’ attitudes and their effects on learning and achievement in Mathematics: A Case study of public secondary schools in Kiambu County, Kenya. (Master Thesis), Kenyatta University, Kenya.
Loibl, K., & Rummel, N. (2014). Knowing what you don’t know makes failure productive. Learning and Instruction, 34, 74-85. https://doi.org/10.1016/j.learninstruc.2014.08.004
Martinez-Planell, R., Torres, J., & Hernandez, I. (2015). Students’ understanding of quadratic equations. International Journal of Mathematical Education in Science and Technology, https://doi.org/10.1080/0020739X.2015.1119895
McLaren, B. M., Adams, D. M., & Mayer, R. E. (2015). Delayed learning effects with erroneous examples: a study of learning decimals with a web-based tutor. International Journal Artificial Intelligence in Education, 25, 520-542. https://doi.org/10.1007/s40593-015-0064-x
McLaren, B. M., Adams, D., Durkin, K., Goguadze, G., Mayer, R. E., Rittle-Johnson, B., & Van Velsen, M. (2012). To err is human, to explain and correct is divine: a study of interactive erroneous examples with middle school math students. Proceedings in the 21st Century learning for 21st Century skills (pp. 222-235). Berlin: Springer. https://doi.org/10.1007/978-3-642-33263-0_18
Mensah, J. K., Okyere, M., & Kuranchie, A. (2013). Student attitude towards mathematics and performance: Does the teacher attitude matter. Journal of Education and Practice, 4(3), 132-139.
Metcalf, R. C. (2007). The nature of students’ understanding of quadratic functions. (Doctoral Thesis), University of New York at Buffalo, USA.
Ministerio de Educación de Chile MINEDUC (2019). Fundamentals curricular bases 3rd and 4th middle [Fundamentos bases curriculares 3° y 4° medio]. Santiago: Ministerio de Educación.
Mutambara, L.H.N., Tendere, J., & Chagwiza, C.J. (2020). Exploring the conceptual understanding of the quadratic function concept in teachers’ colleges in Zimbabwe. EURASIA Journal of Mathematics, Science and Technology Education, 16(2), em1817. https://doi.org/10.29333/ejmste/112617
National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: ensuring mathematical success for all. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). (2018). Catalyzing Change in High School Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). (2020). Standards for the Preparation of Secondary Mathematics Teachers. Reston, VA: NCTM.
Nielsen, L. E. J. (2015). Understanding quadratic functions and solving quadratic equations: An analysis of student thinking and reasoning. EURASIA Journal of Mathematics, Science and Technology Education, 3(4), 351-361.
OECD. (2019). OECD skills strategy 2019. Skills to shape a better future. Paris, France: OECD Publishing.
Oser, F., & Spychiger, M. (2005). Learning is painful - on the theory of negative knowledge and a practice and error culture [Lernen ist schmerzhaft. zur theorie des negativen wissens und zur praxis der fehlerkultur]. Weinheim (Germany): Beltz.
Özaltun, C. & Bukova, G. (2019). An instructional sequence triggering students’ quantitative reasoning during learning of quadratic functions. Turkish Journal of Computer & Mathematics Education, 10(1), 157-194. https://doi.org/10.16949/turkbilmat.446403
Parent, J. S. S. (2015). Students’ understanding of quadratic functions: Learning from students’ voices. (Doctoral Thesis), University of Vermont, USA.
Pekrun, R., & Linnenbrink-Garcia, L. (2014). Educational psychology handbook series. International handbook of emotions in education. Washington: Routledge/Taylor & Francis Group.
Peralta-García, J., Encinas-Pablos, F., & Cuevas-Salazar, O. (2019). Diagnosis of previous knowledge about the parábola in university students [Diagnóstico de conocimientos previos sobre la parábola en estudiantes universitarios]. Revista de Educación Superior, 3(8), 1-11. https://doi.org/10.35429/JHS.2019.8.3.1.11
Peranginangin, S. (2017). An analysis of students’ mathematics problem solving ability in vii grade at smp negeri 4 pancurbatu. International Journal of Sciences: Basic and Applied Research, 33(2), 57-67.
Peteros, E., Columna, D., Etcuban, J. O., Almerino J. P., & Almerino, J. G. (2019). Attitude and Academic Achievement of High School Students in Mathematics Under the Conditional Cash Transfer Program. International Electronic Journal of Mathematics Education, 14(3), 583-597. https://doi.org/10.29333/iejme/5770
Radatz, H. (1980). Students’errors in the mathematical learning process. For the Learning of Mathematics, 1(1), 16-20.
Rico, L. (1995). Errors in learning mathematics [Errores en el aprendizaje de las Matemáticas]. In J. Kilpatrick, P. Gómez & L. Rico (Eds.), Educación Matemática (pp.69-108). México: Grupo Editorial Iberoamérica.
Ruli, R. M., Priatna, N., Sufyani, P., & Endang, M. (2018). Analysis learning obstacle on quadratic function topic. International Journal of Information and Education Technology, 8(9), 681-684. https://doi.org/10.18178/ijiet.2018.8.9.1122
Rushton, S.J. (2018). Teaching and learning mathematics through error analysis. Fields Mathematics Education Journal, 3(4). https://doi.org/10.1186/s40928-018-0009-y
Schoenfeld, A. H. (2012). Problematizing the didactic triangle. ZDM, 44(5), 587-599. https://doi.org/10.1007/s11858-012-0395-0
Sisman, G. T., & Aksu, M. (2015). A study on sixth grade students’ misconceptions and errors in spatial measurement: length, area, and volume. International Journal of Science and Mathematics Education, 14(7), 1293-1319. https://doi.org/10.1007/s10763-015-9642-5
Skjong, R., & Wentworth, B. (2001). Expert judgement and risk perception. Proceedings of Eleventh the International Offshore and Polar Engineering Conference. Stavanger: Norway.
Socas, M. (1997). Difficulties, obstacles and errors in the learning of Mathematics in Secondary Education [Dificultades, obstáculos y errores en el aprendizaje de las Matemáticas en la Educación Secundaria]. In L. Rico et al., (Eds.), La Educación Matemática en la enseñanza secundaria (pp. 125-154). Barcelona: Horsori.
Strickland, T. K. (2011). The effects of blended instruction and visual representations on area problems involving quadratic expressions for secondary students with mathematics learning difficulties. (Doctoral Thesis), University of Maryland, College Park, USA.
Tsovaltzi, D., Melis, E., McLaren, B. M., Meyer, A. K., Dietrich, M., & Goguadze, G. (2010). Learning from erroneous examples: when and how do students beneft from them? Sustaining TEL: from innovation to learning and practice (pp. 357-373). Berlin: Springer. https://doi.org/10.1007/978-3-642-16020-2_24

How to cite this article

APA

Díaz, V., Aravena, M., & Flores, G. (2020). Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study. Eurasia Journal of Mathematics, Science and Technology Education, 16(11), em1896. https://doi.org/10.29333/ejmste/8547

Vancouver

Díaz V, Aravena M, Flores G. Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study. EURASIA J Math Sci Tech Ed. 2020;16(11):em1896. https://doi.org/10.29333/ejmste/8547

AMA

Díaz V, Aravena M, Flores G. Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study. EURASIA J Math Sci Tech Ed. 2020;16(11), em1896. https://doi.org/10.29333/ejmste/8547

Chicago

Díaz, Verónica, Maria Aravena, and George Flores. "Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study". Eurasia Journal of Mathematics, Science and Technology Education 2020 16 no. 11 (2020): em1896. https://doi.org/10.29333/ejmste/8547

Harvard

Díaz, V., Aravena, M., and Flores, G. (2020). Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study. Eurasia Journal of Mathematics, Science and Technology Education, 16(11), em1896. https://doi.org/10.29333/ejmste/8547

MLA

Díaz, Verónica et al. "Solving Problem Types Contextualized to the Quadratic Function and Error Analysis: A Case Study". Eurasia Journal of Mathematics, Science and Technology Education, vol. 16, no. 11, 2020, em1896. https://doi.org/10.29333/ejmste/8547