Skip to main navigation menu Skip to main content Skip to site footer
Logo Revista Latinoamericana de Investigación en Matemática Educativa

Artículos

Vol. 15 No. 2 (2012): Julio

HIGH SCHOOL STUDENTS' STRUCTURAL SENSE IN THE CONTEXT OF SIMPLIFICATION OF ALGEBRAIC FRACTIONS THAT INVOLVE NOTABLE EQUATIONS

Submitted
July 14, 2023
Published
2012-07-01

Abstract

This article analyzes the structural sense that high school students (16 to 18 years old) demonstrate when working with simple and complex algebraic expressions, in the context of simplification of algebraic fractions that involve notable equations: square of the sum, square of the difference, difference of squares and distributive property or common factor. A written test prepared for this purpose detects a wide spectrum of levels of structural sense. These levels reveal the relevance of students' knowledge of notable equations. The analysis carried out allows us to make progress in understanding the construct structural sense by identifying some of its component skills and its role in the proposed tasks.

References

  1. Arcavi, A. (1994). Symbol sense; informal sense-making in formal mathematics. For the Learning of Mathematics 14 (3), 24-35.
  2. Booth, L. R (1982). Ordering your operations. Mathematics in School II 0),5-6 Caraça, B. J. (1998). Conceitos Fundamentais da Matemática, Lisboa, Portugal Gradiva.
  3. Carpenter, T. P. & Franke, M. L. (2001). Developing algebraic reasoning in the elementary school generalization and proof. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.). Proceedings of the 12 ICMI Study Conference. The Future of the Teaching and Learning of Algebra (pp.155-162). Melbourne, Australia: University of Melbourne.
  4. Cerdán, F. (2010), Las igualdades incorrectas producidas en el proceso de traducción algebraico un catálogo de errores. PNA 4 (3), 99-110.
  5. Chang, C. K. & Tsai, Y. L. (2005). An alternative Approach for the Learning of (a+b)-a-2ab+b Electronic Proceedings the 3 East Asia Regional Conference in Mathematics Education (pp. 7-12), 7-12 august 2005, Shanghai, China. Avaliable http://www.sciedu.ncue.edu.twteacher-in.php?ts=3
  6. Demby, A. (1997). Algebraic procedures used by 13-to-15-years-olds. Educational Studies in Mathematics 33 (1), 45-70. DOI: 10.1023/A:1002963922234
  7. Esty, W. W. (1992). Language concepts of mathematics. Focus on Learning Problems in Mathematics 14 (4), 31-54.
  8. Gray, E. M. & Tall, D. O. (1994). Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic. Journal for Research in Mathematics Education 26 (2), 115-141.
  9. Herscovics, N. & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics 27 (1), 59-78. DOI: 10.1007/BF01284528
  10. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates.
  11. Hoch, M. (2003). Structure sense. In M. A. Mariotti (Ed.), Proceedings of the 3 Conference for European Research in Mathematics Education (CD). Bellaria, Italy: ERME
  12. Hoch, M. & Dreyfus, T. (2004), Structure sense in high school algebra: The effect of brackets. In M. J. Hoines & A. B. Fuglestad (Eds.). Proceedings of the 28 Conference of the International Group for the Psychology of Mathematics Education (Vol.3, pp. 49-56). Bergen, Norway: Bergen University College.
  13. Hoch, M. & Dreyfus, T. (2005). Students' difficulties with applying a familiar formula in an unfamiliar context. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29 Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 145-152). Melbourne, Australia: University of Melbourne.
  14. Hoch, M. & Dreyfus, T. (2006). Structure sense versus manipulation skills: an unexpected result. In J. Novotná, H. Moraová, M. Krátká & N. Stehlíková (Eds.). Proceedings of the 30 conference of the International Group for the Psychology of Mathematics Education (Vol.3.pp. 305-312). Prague, Czech Republic: Faculty of Education, Charles University in Prague.
  15. Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wanger & C. Kieran (Eds.). Research Issues in the Learning and Teaching of Algebra (Vol. 4, pp. 33-59). Reston, VA.: Lawrence Erlbaum Associates y NCTM.
  16. Kieran, C. (1991). A procedural structural perspective on Algebra Research. In F. Furinghetti (Ed), Proceedings of the 15 International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 245-253), Assisi, Italy: PME Program Committee.
  17. Kieran, C. (1992). The learning and teaching of school algebra In A Grouws, Handbook of Research on Mathematics Teaching and Learning (A project of the NCTMO (pp. 390-419) New York, USA. Macmillan
  18. Kietan, C. (2006), Research on the learning and teaching of algebra. A broadening of sources of meaning. In A. Gutiérrez & P. Boero (Eds.), Handbook of Research on the Prychology of Mathematics Education Past, Prevent and Future (pp. 11-49) Rotterdam, Netherlands Sense Publishers.
  19. Kieran, C (2007), Learning and teaching algebra at the middle school through college levels building meaning for symbols and their manipulation. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 707-762). Charlotte, USA: National Council of Teachers of Mathematics.
  20. Kilpatrick, J. Swafford, J. O. & Findell, B. (Eds.). (2001). Adding it up. Helping children learn mathematics. Washington, DC: National Academy Press.
  21. Kirshner, P. (1989). The Visual Sintax of Algebra. Journal for Research in Mathematics Education 20 (3), 274-287.
  22. Kirshner, D. & Awtry. T. (2004). The visual salience of algebraic transformations. Journal for Research in Mathematics Education 35 (4), 224-257.
  23. Liebenberg, R., Sasman, M. & Olivier, A. (1999). From Numerical Equivalence To Algebraic Equivalence. Mathematics Learning and Teaching Initiative (MALATI), Proceedings of the Annual Congress of the Association for Mathematics Education of South Africa (Vol. 2, pp. 173-183), Port Elizabeth, South Africa: Port Elizabeth Technikon.
  24. Linchevski, L. & Herscovies, D. (1994). Cognitive obstacles in pre-algebra: In J. P. Ponte & 1.F. Martos (Eds), Proceedings of the 18 International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 176-183). Lisboa, Portugal: Universidad de Lisboa.
  25. Linchevski, L. & Livneh, D. (1999) Structure sense: the relationship between algebraic and numerical contexts. Educational Studies in Mathematics 40 (2), 173-196. DOI: 10.1023/A: 1003606308064
  26. Malaespina, U. (2007). Intuición, rigor y resolución de problemas de optimización. Revista Latinoamericana de Investigación en Matemática Educativa 10 (3), 365-399.
  27. MacGregor, M. (1996). Aspectos curriculares en las materias aritmética y álgebra. UNO Revista deDidáctica de las Matemáticas 9, 65-69.
  28. Mason, J., Graham, A. & Wilder, J. S. (2005). Developing thinking in algebra. Londres, Reino Unido: The Open University.
  29. Ministerio de Educación & Ciencia (2006). Real Decreto 1631/2006 de 29 de diciembre, por el que se establecen las enseñanzas minimas correspondientes a la Educación Secundaria Obligatoria. Boletin Oficial del Estado (BOE) 5, 677-773.
  30. Molina, M. (2006). Desarrollo de pensamiento relacional y comprensión del signo igual por alumnos de tercero de Primaria. Tesis doctoral, Universidad de Granada, Granada, España Disponible en http://funes.uniandes.edu.co/544/
  31. Molina, M. (2009). Una Propuesta de Cambio Curricular: Integración del Pensamiento Algebraico en Educación Primaria, PNA 3 (3), 135-156.
  32. Molina, M. (2010). Una visión estructural del trabajo con expresiones aritméticas y algebraicas Suma 65,7-15.
  33. Molina, M., Ambrose, R., Castro, E. & Castro, B. (2009). Breaking the addition addiction: creating the conditions for knowing-to act in early algebra. In S. Lerman & B. Davis (Eds.), Mathematical Action & Structures Of Noticing Studies inspired by John Mason (pp. 121 134). Rotterdam, The Netherlands: Sense Publisher.
  34. Molina, M. & Mason, J. (2009), Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic Canadian Journal of Science. Mathematics and Technology Education 9 (4), 224-242
  35. NCTM (2000), Principios y Estándares para la Educación Matemática, Sevilla, España: Sociedad Andaluza de Educación Matemática Thales.
  36. Nortes, A. y Nortes, R. (2010). Resolución de Problemas de Matemáticas en las pruebas de acceso a la Universidad. Errores significativos. Educatio Siglo XXI 28 (1), 317-342.
  37. Novotná, J. & Hoch, M. (2008). How Structure sense for algebraic expression or equations is related to structure sense for abstract algebra. Mathematics Education Research Journal 20 (2), 93-104.
  38. Peltier, M. (2003). Problemas aritméticos. Articulación, significados y procedimiento de resolución. Educación Matemática 15 (3), 29-55.
  39. Piric, S. & Martin, L. (1997). The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations. Educational Studies in Mathematics 34 (2). 159-181. DOI: 10.1023/A:1003051829991
  40. Rico, L., Castro, E., Castro, E., Coriat, M., Marin, A., Puig, L. et al. (1997). La educación matemática en la enseñanza secundaria, Barcelona, España: Editorial Horsori e Institut de Ciències de l'Educació.
  41. Ruano, R. M.. Socas, M. M. y Palarea, M. M. (2008). Análisis y clasificación de errores cometidos por alumnos de secundaria en los procesos de sustitución formal, generalización y modelización en álgebra. PNA 2 (2), 61-74.
  42. Schifter, D. (1999). Reasoning about operations. Early algebraic thinking in grades K-6. In L. V. Stiff & F. R. Curcio (Eds.), Developing mathematical reasoning in grades K-12. NCTM Yearbook (pp. 62-81). Reston, VA: NCTM.
  43. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections and objects as different sides of the same coin. Educational Studies in Mathematics 22 (1), 1-36. DOI:10.1007/BF00302715
  44. Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra Educational Studies in Mathematics 26 (2-3), 191-228. DOI: 10.1007/BF01273663
  45. Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher 26 (3), 9-15.
  46. Slavit, D. (1998). The role of operation sense in transitions from arithmetic to algebraic thought Educational Studies in Mathematics, 37 (3), 251-274. DOI: 10.1023/A:1003602322232
  47. Sowder, J. (1992). Estimation and Number Sense. En D. A. Grouws (Ed.), Handbook of research anmathematics teaching and learning (pp. 371-389). New York, U.S.A.: Macmillan Publishing Company y NCTM.
  48. Steinberg, R. M., Sleeman, D. H. & Ktorza, D. (1990). Algebra students' knowledge of equivalence of equations. Journal for Research in Mathematics Education 22 (2), 112-121.
  49. Tall, D., Thomas, M., Davis, G., Gray E. & Simpson, A. (2000). What is the object of the encapsulation of a process? Journal of Mathematical Behavior 18 (2), 223-241
  50. Trujillo, P. A., Castro, E. y Molina, M. (2009). Un estudio de casos sobre el proceso de generalización. En M. J. González, M. T. González y J. Murillo (Eds.), Investigación en Educación Matemática XIII (pp. 511-521). Santander: Sociedad Española de Investigación en Educación Matemática.
  51. Wagner, S., Rachlin, S. L. & Jensen, R.J. (1984). Algebra learning project: Final report. Athens, Greece: University of Georgia.
  52. Warren, E. (2001). Algebraic understanding and the importance of operation sense. In M.Heuvel-Penhuizen (Ed.), Proceedings of the 25 International Group for the Psychology of Mathematics Education (Vol. 4, pp. 399-406). Utrecht, The Netherlands: Freudenthal Institute.
  53. Warren, E. (2004). Generalizing arithmetic: supporting the process in the early years. In M. Johnsen y A. Berit (Eds.), Proceedings of the 28 International Group for the Psychology of Mathematics Education (Vol. 4 pp. 417-424). Bergen, Norway: Bergen University College.

Downloads

Download data is not yet available.

Similar Articles

<< < 1 2 3 4 5 6 7 8 > >> 

You may also start an advanced similarity search for this article.