Skip to main content
SpringerLink
Log in

Meanings given to algebraic symbolism in problem-posing

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

Some errors in the learning of algebra suggest that students might have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing to analyze the students’ capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process, depending on the characteristics of the algebraic statements given. We designed a written questionnaire composed of eight closed algebraic statements expressed symbolically, which was administered to 55 students who had finished their compulsory education and who had some previous experience in problem-posing. In our analysis of the data, we examine both syntactic and semantic structures of the problem posed. We note that in most cases students posed problems with syntactic structures different to those given. They did not include computations within variables, and changed the kinds of relationships connecting variables. Students easily posed problems for statements with additive structures. Other differences in the type of problems posed depend on the characteristics of the given statements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. We use the term algebraic statement to refer to general propositions of relations among quantities, some of them unknown, that can be expressed using algebraic symbolism (Rodríguez-Domingo et al., 2015).

  2. The course is aimed at helping students deepen their understanding of elementary mathematics, focusing on the multiple meanings and representations of the mathematics concepts included in the primary education curriculum. It is normally attended by students studying the primary education teacher’s degree.

  3. Expressions not containing an equal sign.

  4. The students presented the problems in Spanish; thus, all the examples in this paper have been translated into English by the authors.

References

  • Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 1(3), 24–35.

    Google Scholar 

  • Arcavi, A. (2006). El desarrollo y el uso del sentido de los símbolos. In I. Vale, T. Pimentel, A. Barbosa, L. Fonseca, L. Santos, & P. Canavarro (Eds.), Números e álgebra na aprendizagem da Matemática e na formaçâo de profesores (pp. 29–47). Caminha: Sociedade Portugesa de Ciências de Eduacaçâo.

    Google Scholar 

  • Bonotto, C., & Dal Santo, L. (2015). On the relationship between problem posing, problem solving, and creativity in the primary school. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 103–123). New York: Springer.

    Chapter  Google Scholar 

  • Booth, L. R. (1982). Ordering your operations. Mathematics in School, 11(3), 5–6.

    Google Scholar 

  • Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011a). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal of Mathematics Education, 6(3), 113–133.

    Google Scholar 

  • Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011b). Translations among mathematical representations: Teacher beliefs and practices. International Journal of Mathematics Teaching and Learning, June. Retrieved from http://www.cimt.plymouth.ac.uk/journal/bosse4.pdf

  • Brown, S., & Walter, M. (2005). The art of problem posing (3rd ed.). New York: Routledge.

  • Cai, J. (1998). An investigation of U.S. and Chinese students’ mathematical problem posing and problem solving. Mathematics Education Research Journal, 10, 37–50.

    Article  Google Scholar 

  • Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21(4), 401–421.

    Article  Google Scholar 

  • Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 3–39). New York: Springer.

    Chapter  Google Scholar 

  • Cañadas, M. C., & Figueiras, L. (2011). Uso de representaciones y generalización de la regla del producto. Infancia y Aprendizaje, 34(4), 409–425.

    Article  Google Scholar 

  • Castro, E. (2011). La invención de problemas y sus ámbitos de investigación. In J. L. Lupiáñez, M. C. Cañadas, M. Molina, M. Palarea, & A. Maz (Eds.), Investigaciones en Pensamiento Numérico y Algebraico e Historia de la Matemática y Educación Matemática – 2011 (pp. 1–16). Granada: Dpto. Didáctica de la Matemática, Universidad de Granada. Retrieved from http://funes.uniandes.edu.co/2015/

  • Castro, E. (2012). Dificultades en el aprendizaje del álgebra escolar. In A. Estepa, A. Contreras, J. Deulofeu, M. C. Penalva, F. J. García, & L. Ordóñez (Eds.), Investigación en Educación Matemática XVI (pp. 75–94). Baeza: SEIEM.

    Google Scholar 

  • Castro, E., Rico, L., & Gil, F. (1992). Enfoques de investigación en problemas verbales aritméticos aditivos. Enseñanza de las Ciencias, 10(3), 243–253.

    Google Scholar 

  • Cerdán, F. (2010). Las igualdades incorrectas producidas en el proceso de traducción algebraico: Un catálogo de errores. PNA, 4(3), 99–110.

    Google Scholar 

  • Dede, Y. (2005). Interpretation of the first-degree equations: A study on freshmen students in education faculty. Cumhuriyet University Social Sciences Journal, 29(2), 197–205.

    Google Scholar 

  • Fernández-Millán, E., & Molina, M. (2016). Indagación en el conocimiento conceptual del simbolismo algebraico de estudiantes de secundaria mediante la invención de problemas. Enseñanza de las Ciencias, 34(1), 53–71.

    Google Scholar 

  • Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.

    Article  Google Scholar 

  • Goldin, G. A., & Mcclintock, C. E. (Eds.). (1980). Task variables in mathematical problem solving. Pennsylvania: The Franklin Institute Press.

    Google Scholar 

  • Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 276–295). New York: Macmillan.

    Google Scholar 

  • Heller, J. I., & Greeno, J. G. (1979). Information processing analysis of mathematical problem solving. In R. W. Tyler & S. H. White (Eds.), Testing, teaching, and learning: Report of a conference on research on testing. Washington: National Institute of Education.

    Google Scholar 

  • Isik, C., & Kar, T. (2012). The analysis of the problems the pre-service teachers experience in posing problems about equations. Australian Journal of Teacher Education, 37(9), 6. Retrieved from http://ro.ecu.edu.au/ajte/vol37/iss9/6

  • Kaput, J. (1987). Representation and mathematics. In C. Janvier (Ed.), Problems of representation in the learning of mathematics (pp. 19–26). Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kieran, C. (2006). Research on the learning and teaching of algebra. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 11–50). Rotterdam: Sense.

    Google Scholar 

  • Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning. Nueva York: NCTM.

    Google Scholar 

  • Kilpatrick, J. (1978). Variables and methodologies in research on problem solving. In L. L. Hatfield & D. A. Bradbard (Eds.), Mathematical problem solving: Papers from a research workshop (pp. 7–20). Columbus: ERICISMEAC.

    Google Scholar 

  • Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20, 274–289.

    Article  Google Scholar 

  • Koichu, B., & Kontorovich, I. (2012). Dissecting success stories on mathematical problem posing: A case of the billiard task. Educational Studies in Mathematics, 83(1), 71–86.

    Article  Google Scholar 

  • Leikin, R. (2015). Problem posing for and through investigations in a dynamic geometry environment. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Problem posing: From research to effective practice (pp. 373–391). Dordrecht: Springer.

    Chapter  Google Scholar 

  • MacGregor, M., & Stacey, K. (1993). Cognitive models underlying students’ formulation of simple linear equations. Journal for Research in Mathematics Education, 24(3), 217–232.

    Article  Google Scholar 

  • Marshall, S. P. (1995). Schemas in problem solving. New York: Cambridge University Press.

    Book  Google Scholar 

  • Ministerio de Educación y Ciencia. (2015). Real Decreto 1105/2014, de 26 de diciembre, por el que se establece el currículo básico de la Educación Secundaria Obligatoria y del Bachillerato. BOE, 3, 169–546.

    Google Scholar 

  • Molina, M. (2009). Una propuesta de cambio curricular: Integración del pensamiento algebraico en educación primaria. PNA, 3(3), 135–156.

    Google Scholar 

  • Molina, M., Rodríguez-Domingo, S., Cañadas, M. C., & Castro, E. (2017). Secondary school students’ errors in the translation of algebraic statements. International Journal of Science and Mathematics Education, 15(6), 1137–1156.https://doi.org/10.1007/s10763-016-9739-5

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: Author.

    Google Scholar 

  • Ngah, N., Ismail, Z., Tasir, Z., Said, M., & Haruzuan, M. N. (2016). Students’ ability in free, semi-structured and structured problem posing situations. Advanced Science Letters, 22(12), 4205–4208.

    Article  Google Scholar 

  • OECD. (2016). PISA 2015 assessment and analytical framework: Science, reading, mathematic and financial literacy, PISA. Paris: OECD Publishing. https://doi.org/10.1787/9789264255425-en

    Book  Google Scholar 

  • Orrantia, J., González, L. B., & Vicente, S. (2005). Un análisis de los problemas aritméticos en los libros de texto de Educación Primaria. Infancia y Aprendizaje, 28, 420–451.

    Article  Google Scholar 

  • Ponte, J. P., & Henriques, A. (2013). Problem posing based on investigation activities by university students. Educational Studies in Mathematics, 83(1), 145–156.

    Article  Google Scholar 

  • Puig, L. (1996). Elementos de resolución de problemas. Granada: Comares.

    Google Scholar 

  • Resnick, L. B., Cauzinille-Marmèche, E., & Mathieu, J. (1987). Understanding algebra. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics (pp. 169–203). Oxford: Clarendon Press.

    Google Scholar 

  • Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102–1118). Oxford: Oxford University Press.

    Google Scholar 

  • Rodríguez-Domingo, S., & Molina, M. (2013). De lo verbal a lo simbólico: Un paso clave en el uso del álgebra como herramienta para la resolución de problemas y la modelización matemática. In L. Rico, M. C. Cañadas, J. Gutiérrez, M. Molina, & I. Segovia (Eds.), Investigación en Didáctica de la Matemática. Homenaje a Encarnación Castro (pp. 111–118). Granada: Comares.

    Google Scholar 

  • Rodríguez-Domingo, S., Molina, M., Cañadas, M. C., & Castro, E. (2015). Errores en la traducción de enunciados algebraicos entre los sistemas de representación simbólico y verbal. PNA, 9(4), 273–293.

    Google Scholar 

  • Ruano, R. M., Socas, M., & 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.

    Google Scholar 

  • Schmidt, S., & Weiser, W. (1995). Semantic structures of one-step word problems involving multiplication or division. Educational Studies in Mathematics, 28(1), 55–57.

    Article  Google Scholar 

  • Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14, 19–28.

    Google Scholar 

  • Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27, 521–539.

    Article  Google Scholar 

  • Silver, E. A., Mamona-Downs, J., Leung, S., & Kenney, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.

    Article  Google Scholar 

  • Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7.

    Article  Google Scholar 

  • Star, J. (2005). Re-“conceptualizing” procedural knowledge in mathematics. Journal for Research in Mathematics Education, 36(5), 404–411.

    Google Scholar 

  • Stephens, A. (2003). Another look at words problems. The Mathematics Teacher, 96(1), 63–66.

    Google Scholar 

  • Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In P. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia.

    Google Scholar 

  • Van Harpen, X. Y., & Presmeg, N. C. (2013). An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge. Educational Studies in Mathematics, 83, 117–132.

    Article  Google Scholar 

  • Wheeler, D. (1989). Contexts for research on the teaching and learning of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 278–287). Reston: Lawrence Erlbaum Associates.

    Google Scholar 

Download references

Acknowledgements

This study was developed within the Spanish projects of Research and Development with reference codes EDU2013-41632-P and EDU2016-75771-P, financed by the Ministerio Español de Economía y Competitividad and FEDER funds.

Author information

Authors and Affiliations

  1. Departamento de Didáctica de la Matemática, Facultad de Ciencias de la Educación, Universidad de Granada, Campus de Cartuja s/n, 18071, Granada, Spain

    María C. Cañadas, Marta Molina & Aurora del Río

Authors
  1. María C. Cañadas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marta Molina
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aurora del Río
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to María C. Cañadas.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cañadas, M.C., Molina, M. & del Río, A. Meanings given to algebraic symbolism in problem-posing. Educ Stud Math 98, 19–37 (2018). https://doi.org/10.1007/s10649-017-9797-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-017-9797-9

Keywords

Search

Navigation