# An assessment of the sources of the reversal error through classic and new variables

## Abstract

We present two empirical studies with 241 and 211 pre-service teachers that evaluate the explanatory power of word order matching and static comparison as models for the reversal error. We used tasks consisting of generating an algebraic equation representing a comparison given in a verbal statement. We introduce the types of magnitude involved in the statement as variables of analysis, something that was not previously tackled in previous works. Our results show that there are no statistical differences in the production of reversal errors depending on the information included in the name used to designate the variable, and that there are statistical differences depending on the syntactic configuration as well as the type of magnitude involved in the statement. The interpretation of these results indicates that both word order matching and static comparison have some potential as explanatory models for the reversal error, and that neither one of them, alone, is enough to completely explain the phenomenon.

## Keywords

Word problem solving Algebra Reversal error Magnitude## References

- Brown, D. E., & Clement, J. (1989). Overcoming misconceptions via analogical reasoning: Abstract transfer versus explanatory model construction.
*Instructional Science, 18*, 237–261.CrossRefGoogle Scholar - Castro, E. (1995).
*Niveles de comprensión en problemas verbales de comparación multiplicativa [Levels of comprehension in multiplicative comparison word problems]*. Granada: Comares.Google Scholar - Clement, J. J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception.
*Journal for Research in Mathematics Education, 13*(1), 16–30.CrossRefGoogle Scholar - Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics.
*The American Mathematical Monthly, 88*(4), 286–290.CrossRefGoogle Scholar - Cohen, E., & Kanim, S. E. (2005). Factors influencing the algebra reversal error.
*American Journal of Physics, 73*(11), 1072–1078.CrossRefGoogle Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics, 61*, 103–131.CrossRefGoogle Scholar - Filloy, E., Rojano, T., & Solares, A. (2010). Problems dealing with unknown quantities and two different levels of representing unknowns.
*Journal for Research in Mathematics Education, 41*(1), 52–80.Google Scholar - Fisher, K. J., Borchert, K., & Bassok, M. (2011). Following the standard form: Effects of equation format on algebraic modeling.
*Memory & Cognition, 39*(3), 502–515.CrossRefGoogle Scholar - Fisher, K. M. (1988). The students-and-professors problem revisited.
*Journal for Research in Mathematics Education, 19*(3), 260–262.CrossRefGoogle Scholar - Gómez-Ferragud, C. B., Solaz-Portolés, J. J., & Sanjosé, V. (2013). Analogy construction and success in mathematics and science problem-solving: A study with secondary students.
*Revista de Psicodidáctica, 18*(1), 81–108.CrossRefGoogle Scholar - González-Calero, J. A., Arnau, D., & Laserna-Belenguer, B. (2015). Influence of additive and multiplicative structure and direction of comparison on the reversal error.
*Educational Studies in Mathematics, 89*, 133–147.CrossRefGoogle Scholar - Kaput, J. (1987). Towards a theory of symbol use in mathematics. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 159–195). Hillsdale, NJ: Erlbaum.Google Scholar - Kieran, C. (1981). Concepts associated with the equality symbol.
*Educational Studies in Mathematics, 12*, 317–326.CrossRefGoogle Scholar - Kintsch, W. (1998).
*Comprehension. A paradigm for cognition*. Cambridge: Cambridge University Press.Google Scholar - Landy, D., Brookes, D., & Smount, R. (2014). Abstract numeric relations and the visual structure of algebra.
*Journal of Experimental Psychology: Leaning, Memory, and Cognition, 40*(5), 1404–1418.Google Scholar - Lewis, A. B., & Mayer, R. E. (1987). Students’ miscomprehension of relational statements in arithmetic word problems.
*Journal of Educational Psychology, 79*(4), 363–371.CrossRefGoogle 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.CrossRefGoogle Scholar - Matz, M. (1982). Towards a process model for high school algebra errors. In D. Gleeman & J. S. Brown (Eds.),
*Intelligent tutoring systems*(pp. 25–50). New York: Academic Press.Google Scholar - Maxwell, S. E., & Delaney, H. D. (2004).
*Designing experiments and analyzing data*(2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Nesher, P., Hershkovitz, S., & Novotna, J. (2003). Situation model, text base and what else? Factors affecting problem solving.
*Educational Studies in Mathematics, 52*(2), 151–156.CrossRefGoogle Scholar - Presmeg, N. (2006). Semiotics and the “connections” standard: Significance of semiotics for teachers of mathematics.
*Educational Studies in Mathematics, 61*, 163–182.CrossRefGoogle Scholar - Rosnick, P. (1981). Some misconceptions concerning the concept of variable.
*Mathematics Teacher, 74*, 418–420.Google Scholar - Rosnick, P., & Clement, J. (1980). Learning without understanding: The effect of tutoring strategies on algebra misconceptions.
*Journal of Mathematical Behaviour, 3*(1), 3–27.Google Scholar - Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. J. Behr (Eds.),
*Number concepts and operations in the middle grades*(pp. 41–52). Hillsdale, NJ: Erlbaum.Google Scholar - Sung-Hee, K., Phang, D., An, T., Ji Soo, Y., Kenney, R., & Uhan, N. (2014). POETIC: Interactive solutions to alleviate the reversal error in student–professor type problems.
*International Journal of Human-Computer Studies, 72*, 12–22.CrossRefGoogle Scholar - Wilcox, R. R. (2005).
*Introduction to robust estimation and hypothesis testing*. New York: Elsevier Academic Press.Google Scholar - Wollman, W. (1983). Determining the sources of error in a translation from sentence to equation.
*Journal for Research in Mathematics Education, 14*, 169–181.CrossRefGoogle Scholar