Studentsâ€™ Ability and Achievement in Recognizing Multiple Representations in Algebra
Keywords:ability, recognize, multiple-representation, achievement, algebra
Secondary school students often demonstrate a degree of proficiency manipulating algebraic symbols, when learning linear relationships with one unknown. They, in some cases and when encouraged verbalize and explain the steps taken, thereby demonstrating awareness of the procedures with symbols according to fixed rules. It is well known that correct procedural skills are not always supported by conceptual understanding. Previous research suggests that one of the indicators of conceptual understanding is the ability to structurally recognize the same relationship posed through multiple representations. The purpose of this study is to examine the relationship between secondary school studentsâ€™ achievement on standardized test and their ability to recognize structurally the same relationship presented in different forms and their ability to solve problems involving linear relationships with one unknown presented in different ways. The study was conducted with a large sample size (N=300), of senior secondary school class two(SS2) students from Bwari Area Council of Federal Capital Territory, Abuja, Nigeria, using questions drawn from past Educational Resource Centerâ€™s (ERC) past promotion examination questions. It was observed that there were positive but weak correlations (Ï=0.114) between, the studentsâ€™ examination scores and their ability to identify the same relationship posed in different modalities, the studentsâ€™ examination scores and their ability to solving problems in all problem sets, the studentsâ€™ solved problems posed in different modalities and their ability to identify the same relationship posed in different modalities, using both Pearson and Spearmanâ€™s correlations. It is recommended that teachers should emphasize multiple-representation in algebra in the classes they teach.
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, 1â€“36.
Dubinsky, E. and M. McDonald. (1991). APOS: A Constructivist Theory of Learning in Undergraduate Mathematics
Education Research. New ICMI Study Series, Kluwer Academic Press, (pp. 275-282).
Herscovics, N. (1996). The construction of conceptual schemes in mathematics. In L. Steffe (Ed.), Theories of
mathematical learning (pp. 351-380). Mahwah, NJ: Erlbaum.
Langrall, C. W. and Swafford, J. O. (1997). Grade six studentsâ€Ÿ use of equations to describe and represent problem
situation. Paper presented at the American Educational Research Association, Chicago, IL.
Panasuk, R. (2010). Three-phase ranking framework for assessing conceptual understanding in algebra using multiple
representations, EDUCATION, 131 (4),
Niemi, D. (1996). Assessing Conceptual Understanding in Mathematics: Representations, Problem Solutions,
Justifications, and Explanations. The Journal of Educational Research, 89(6), 351-363.
Herbert, K. and R. Brown. (1997). Patterns as tools for algebraic reasoning.Teaching Children Mathematics 3 (February),
Driscoll, M. (1999). Fostering algebraic thinking. A guide for teachers grade 6- 10. Portsmouth, NH, Heinemann.
Swafford, J. O. and Langrall. C. W. (2000). Grade 6 students' pre- instructional use of equations to describe and represent
problem situations. Journal for Research in Mathematics Education, 31(1), 89-112.
Vance, J. (1998). Number operations from an algebraic perspective. Teaching Children Mathematics 4 (January), 282-
Kaput, J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.), Research
issues in the teaching and learning of algebra (pp. 167-194).Reston, VA: NCTM.
Seeger, F. (1998). Discourse and beyond: on the ethnography of classroom discourse. In A. Sierpinska (Ed.), Language
and communication in the mathematics Classroom. Reston,VA: NCTM.
Vergnaud, G. (1997). The nature of mathematical concepts. In P. Bryant (Ed.), Learning and Teaching Mathematics.
East Sussex: Psychology Press. Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Belknap Press. Pape, S. J. and Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical
understanding. Theory into Practice, 40(2), 118- 125.
Hiebert, J. and T. Carpenter. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of
research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
Goldin, G. and Shteingold, N. (2001). System of mathematical representations and development of mathematical
concepts. In F. R. Curcio (Ed.), The roles of representation in school mathematics:2001 yearbook. Reston, VA: NCTM. Pirie, S. E. B. (1998). Crossing the gulf between thought and symbol: Language as steppingstones. In H.
Steinbring, M., G. B. Bussi and A.
Ainsworth, S., Bibby, P., and Wood, D. (2002). Examining the Effects of Different Multiple Representational Systems in
Learning Primary Mathematics.The Journal of the Learning Sciences, 11, 25-61.
Diezmann, C. M. (1999). Assessing diagram quality: Making a difference to representation. In J.M. Truran & K. M.
Truran (Eds.), Proceedings of the 22nd Annual Conference of Mathematics Education Research Group of
Australasia (pp. 185-191), Adelaide: Mathematics Education Research Group of Australasia.
Diezmann, C. M. and English, L. D. (2001). Promoting the use of diagrams as tools for thinking. In A. A. Cuoco (Ed.),
National Council of Teachers of Mathematics Yearbook: The role of representation in school mathematics
(pp.77-89). Reston, VA: NCTM.
Lowrie, T. (2001). The influence of visual representations on mathematical problem solving and numeracy performance.
In B. Perry (Ed.), Numeracy and Beyond (Vol. 2). Sydney: MERGA.
Piaget, J. (1970). Genetic epistemology. New York: Columbia University Press.
Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In
E. Dubinsky and G. Harel (Eds.), The Concept of Functionâ€”Aspects of Epistemology and Pedagogy, MAA Notes.
Panasuk, R. (2006). Multiple representations in algebra and reducing level of
abstraction.Unpublished instrument. University of Massachusetts Lowell, MA
Beyranevand, M. (2010). Investigating mathematics studentsâ€Ÿ use of multiple representations when solving linear
equations with one unknown. Unpublished doctoral dissertation, University of Massachusetts, Lowell.
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on
mathematics teaching and learning (pp. 390- 419). New York: Mac-Millan Publishing Company.
Mosley, B. (2005). Studentsâ€Ÿ early mathematical representation knowledge: The effects of emphasizing single or
multiple perspectives of the rational number domain in problem solving. Educational Studies in Mathematics, 60,
Van Essen, G., and Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word problems. Journal of
Educational Research, 83(6), 301-312.
Larkin, J. H., and Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11,
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