Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing

Elena Fernández-Millán; Marta Molina

doi:10.29333/iejme/649

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Elena Fernández-Millán, Marta Molina

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Abstract

Through the task of problem posing, we inquire into students’ conceptual knowledge of algebraic symbolism developed in compulsory secondary education. We focus on identifying the characteristics of equations and systems of equations that made the problem posing task difficult for the students and analyzing the meanings that they gave to the operations contained in the expressions. To collect the data we used two questionnaires in which students were asked to pose problems that could be solved by using given equations or system of equations. In the second questionnaire a specific meaning for the unknowns in the given expression was suggested. The results complement those of a previous study. Students evidence a good conceptual knowledge of algebraic symbolism when meanings for the unknowns are suggested. Decimal numbers and an equation including parenthesis and multiplication of unknowns are the main elements that made some weaknesses in students’ knowledge to surface. The results are more promising. They suggest the potential for compulsory algebra instruction to develop students’ conceptual knowledge, although greater attention should be paid to the semantic aspects of algebra if students are to access such knowledge unaided.

Keywords

Algebraic symbolism
conceptual knowledge
equations
Problem posing
unknown

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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

INT ELECT J MATH ED, Volume 12, Issue 3, October 2017, 799-826

https://doi.org/10.29333/iejme/649

Publication date: 27 Nov 2017

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References

Álvarez, I. & Gómez-Chacón, I. M. (2015). Understanding the algebraic variable: Comparative Study of Mexican and Spanish students. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1507-1529
Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 1(3), 24-35.
Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the learning of mathematics, 25(2), 42-47.
Arnau, D. & Puig, L. (2013). Actuaciones de alumnos instruidos en la resolución algebraica de problemas en la hoja de cálculo y su relación con la competencia en el método cartesiano. Enseñanza de las Ciencias, 31(3), 49-66. DOI: https://doi.org/10.5565/rev/ec/v31n3.967
Bills, L. (2001). Shifts in the Meanings of Literal Symbols. In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the Twenty-fifth PME International Conference (pp. 2-161, 2-168). Utrecht, The Neatherlands: PME.
Booth, L.R. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-Nelson.
Capraro, M. & Joffrion, H. (2006). Algebraic Equations: Can Middle-School Students Meaningfully Translate from Words to Mathematical Symbols? Reading Psychology, 27(2-3), 147-164.
Carpenter, T. & Moser, J. (1982). The development of addition and subtraction problem solving skills. In T. Carpenter, J. Moser & T. Romberg (Eds.), Addition and subtraction: Developmental perspective (pp. 9-24). Hillsdale, N. Jersey: Lawrence Erlbaum Associates.
Castro, E. (2001). Multiplicación y división. In E. Castro (Ed.), Didáctica de la matemática en educación primaria. Madrid, España: Síntesis, pp. 203-230.
Castro, E. & Castro, E. (1997). Representaciones y modelización. In L. Rico (Ed.), La educación matemática en la enseñanza secundaria (pp. 95-124). Barcelona, España: Horsori.
Castro, A., Prat, M. & Gorgorió, N. (2016). Conocimiento conceptual y procedimental en matemáticas: su evolución tras décadas de investigación. Revista de Educación, 374, 43-68
Cedillo, T. E. (2001). Toward an algebra acquisition support system: A study based on using graphic calculators in the classroom. Mathematical Thinking and Learning, 3, 221–259.
Cerdán, F. (2010). Las igualdades incorrectas producidas en el proceso de traducción algebraico: un catálogo de errores. PNA, 4(3), 99-110.
Chalouh, L. & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In A.F. Coxford & A.P. Shulte (Eds.), The ideas of algebra K-12 (pp. 33-42). Reston, Virginia: NCTM.
Chappell, M. F. (2001). Creating connections: Promoting algebraic thinking with concrete models. Mathematics Teaching in the Middle School, 7, 20–25.
Crooks, N. & Alibali, M. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344-377
Fernández-Millán E. y 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. DOI: http://dx.doi.org/10.5565/rev/ensciencias.1455
Ferrucci, B. J., Kaur, B., Carter, J. A. & Yeap, B. (2008). Using a model approach to enhance algebraic thinking in the elementary school mathematics classroom. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 2008 Yearbook (pp. 195–209). Reston, VA: National Council of Teachers of Mathematics.
Filloy, E. & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the learning of Mathematics, 9(2), 19-25.
Filloy, E., Rojano T. & Puig, L. (2008). Educational algebra. A theoretical and empirical approach. New York, NY: Springer.
Fujii, T. & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 2008 Yearbook (pp. 127–140). Reston, VA: National Council of Teachers of Mathematics.
Furinghetti, F. & Paola, D. (1994). Parameters, unknowns and variables: a little difference? In J.P. da Ponte & J.F. Matos (Eds.), Proceedings of the XVIII International Conference for the Psychology of Mathematics (vol. 2, pp. 368-375). Lisboa, Portugal: Universidad de Lisboa.
Gómez, P. (2007). Desarrollo del conocimiento didáctico en un plan de formación inicial de profesores de matemáticas de secundaria. PhD Thesis. Granada: Universidad de Granada.
González-Calero, J. A., Arnau, D. & Puig, L. (2013). Dificultades en la construcción de nombres de cantidades durante la resolución algebraica de problemas verbales por estudiantes de primaria. In A. Berciano, G. Gutiérrez, A. Estepa, & N. Climent (Eds.), Investigación en Educación Matemática XVII (pp. 301-310). Bilbao, Spain: Sociedad Española de Investigación en Educación Matemática.
Herscovics, N. & Kieran, C. (1980). Constructing meaning for the concept of equation. Mathematics Teacher, 73(8), pp. 572-580.
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.
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 29th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 145–152). Melbourne, Australia: University of Melbourne.
Kaput, J. (1989). Linking representations in the symbolic systems of algebra. In S. Wagner & C. Kieran (Eds.), Research agenda for mathematics education: Research issues in the learning and teaching of algebra (pp. 167-194). Reston, VA: NCTM.
Küchemann, D.E. (1981). Algebra. In K.M. Hart, M.L. Brown, D.E. Kuchemann, D. Kerslake, G. Ruddock & M. McCartney (Eds.), Children’s understanding of mathematics: 11-16 (pp. 102-119). London, Reino Unido: John Murray,
Lin, P. J. (2004). Supporting teachers on designing problem-posing tasks as a tool of assessment to understand students' mathematical learning. In M. Hoines y A. Fuglestad (Eds.), Proceedings of the 28th annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 257-264). Bergen, Noruega: Bergen University College.
Mestre J. P. (2002). Probing adults' conceptual understanding and transfer of learning via problem posing. Journal of Applied Developmental Psychology, 23(1), 9-50.
Mitchell, J. M. (2001). Interactions between natural language and mathematical structures: the case of “wordwalking”. Mathematical Thinking and Learning, 3(1), 29-52.
Molina, M. (2014). Traducción del simbolismo algebraico al lenguaje verbal: indagando en la comprensión de estudiantes de diferentes niveles educativos. La Gaceta de la RSME, 17(3), 559-579.
Molina, M., Rodríguez-Domingo, S., Cañadas, M.C. & Castro, E. (2016). Secondary School Student’s Errors in the Translation of Algebraic Statements. International Journal of Science and Mathematics Education, 13(3), 1-20.
Ng, S. F. & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313.
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. http://dx.doi.org/10.1174/021037005774518929
Resnick, L.B., Cauzinille-Marmeche, E. & Mathieu, J. (1987). Understanding algebra. In J.A. Sloboda & D. Rogers (Eds.), Cognitive process in mathematics (pp. 169-203). Oxford, Reino Unido: Clarendon Press.
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. 1118-1134). Oxford, UK: Oxford University Press
Rittle-Johnson, B. & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561–574. doi: 10.1037/0022–0663.99.3.561.
Rodríguez-Domingo, S. (2015). Traducción entre los sistemas de representación simbólico y verbal: un estudio con alumnado que inicia su formación algebraica en secundaria. PhD thesis. Granada: Universidad de Granada.
Ross, A. & Willson, V. (2012). The Effects of Representations, Constructivist Approaches, and Engagement on Middle School Students’ Algebraic Procedure and Conceptual Understanding. School, Science and Mathematics, 112(2), 117-128.
Sheikhzade, M. (2008, July). Promoting skills of problem-posing and problem- solving in making a creative social studies classroom. Presented at 4th Global Conference, Oxford. Available at http://www.inter-disciplinary.net/ati/education/cp/ce4/Sheikhzade%20paper.pdf.
Stoyanova, E. & Ellerton, N. F. (1996). A framework for research into students' problem posing in school mathematics. In P. C. Clarkson (Ed.), Technology in mathematics education (Proceedings of the 19th annual conference of the Mathematics Education Research Group of Australasia) (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia.
Usiskin Z. (1988). Conceptions of School Algebra and Uses of Variables. In Coxford A.F. & Shulte A. P. (Eds.), The Ideas of Algebra K-12 (pp. 8-19). VA: National Council of Teachers of Mathematics.
Vega-Castro, D., Molina, M. & Castro, E. (2012). Sentido estructural de estudiantes de bachillerato en tareas de simplificación de fracciones algebraicas que involucran igualdades notables. Relime, 15(2), 233–258.
Wagner, S. (1981). An analytical framework for mathematical variables. In Equipe de Recherche Pedagogique (Eds.), Proceedings of the 5th International PME Conference (pp.165-170). Grenoble, France: PME.

How to cite this article

APA

Fernández-Millán, E., & Molina, M. (2017). Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing. International Electronic Journal of Mathematics Education, 12(3), 799-826. https://doi.org/10.29333/iejme/649

Vancouver

Fernández-Millán E, Molina M. Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing. INT ELECT J MATH ED. 2017;12(3):799-826. https://doi.org/10.29333/iejme/649

AMA

Fernández-Millán E, Molina M. Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing. INT ELECT J MATH ED. 2017;12(3), 799-826. https://doi.org/10.29333/iejme/649

Chicago

Fernández-Millán, Elena, and Marta Molina. "Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing". International Electronic Journal of Mathematics Education 2017 12 no. 3 (2017): 799-826. https://doi.org/10.29333/iejme/649

Harvard

Fernández-Millán, E., and Molina, M. (2017). Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing. International Electronic Journal of Mathematics Education, 12(3), pp. 799-826. https://doi.org/10.29333/iejme/649

MLA

Fernández-Millán, Elena et al. "Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing". International Electronic Journal of Mathematics Education, vol. 12, no. 3, 2017, pp. 799-826. https://doi.org/10.29333/iejme/649