# Middle School Students’ Conceptual Understanding of Equations: Evidence from Writing Story Problems

## https://doi.org/10.4471/ijep.2014.13

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## Abstract

This study investigated middle school students’ conceptual understanding of algebraic equations. 257 sixth- and seventh-grade students solved algebraic equations and generated story problems to correspond with given equations. Aspects of the equations’ structures, including number of operations and position of the unknown, influenced students’ performance on both tasks. On the story-writing task, students’ performance on two-operator equations was poorer than would be expected on the basis of their performance on one-operator equations. Students made a wide variety of errors on the story-writing task, including (1) generating story contexts that reflect operations different from the operations in the given equations, (2) failing to provide a story context for some element of the given equations, (3) failing to include mathematical content from the given equations in their stories, and (4) including mathematical content in their stories that was not present in the given equations. The nature of students’ story-writing errors suggests two main gaps in students’ conceptual understanding. First, students lacked a robust understanding of the connection between the operation of multiplication and its symbolic representation. Second, students demonstrated difficulty combining multiple mathematical operations into coherent stories. The findings highlight the importance of fostering connections between symbols and their referents.

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## References

Baroody, A., & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition & Instruction, 1, 321-339.

Google Scholar CrossrefBarr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory & Language, 68, 255–278.

Google Scholar CrossrefBates, Maechler, Bolker, & Walker, 2014 Bates, D., Maechler, M., Bolker, B., & Walker, S. (2014). lme4: Linear mixed-effects models using Eigen and S4 (Version 1.0-6) [R package] Available from http://CRAN.Rproject.org/package=lme4

Google Scholar CrossrefCarpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Google Scholar CrossrefClement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16-30.

Google Scholar CrossrefCrooks, N. M. & Alibali, M. W. (in press). Defining and measuring conceptual knowledge of mathematics. Developmental Review.

Google Scholar CrossrefDixon, J. A., Deets, J. K., & Bangert, A. (2001). The representations of the arithmetic operations include functional relationships. Memory & Cognition, 29, 462-477.

Google Scholar CrossrefDixon, J. A., & Moore, C. F. (1996). The developmental role of intuitive principles in choosing mathematical strategies. Developmental Psychology, 32, 241-253.

Google Scholar CrossrefHeffernan, N., & Koedinger, K. R. (1997). The composition effect in symbolizing: The role of symbol production versus text comprehension. In M. G. Shafto & P. Langley (Eds.), Proceedings of the Nineteenth Meeting of the Cognitive Science Society. Mahwah, NJ: Lawrence Erlbaum Associates.

Google Scholar CrossrefHeffernan, N., & Koedinger, K. R. (1998). A developmental model for algebra symbolization: The results of a difficulty factors assessment. In M. A. Gernsbacher & S. J. Derry (Eds.), Proceedings of the Twentieth Annual Conference of the Cognitive Science Society (pp. 484-489). Mahwah, NJ: Lawrence Erlbaum Associates.

Google Scholar CrossrefHerscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59-78.

Google Scholar CrossrefHiebert, J., & Wearne, D. (1996). Instruction, understanding and skill in multidigit addition and subtraction. Cognition & Instruction, 14, 251-283.

Google Scholar CrossrefKaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K–12 curriculum. In National Council of Teachers of Mathematics, Mathematical Sciences Education Board, & National Research Council (Eds.), The nature and role of algebra in the K–14 curriculum: Proceedings of a National Symposium (pp. 25–26). Washington, DC: National Academies Press.

Google Scholar CrossrefKaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. A. Romberg (Eds.), Mathematical classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Lawrence Erlbaum Associates.

Google Scholar CrossrefKaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: Lawrence Erlbaum Associates.

Google Scholar CrossrefKaput, J. J., Blanton, M. L., & Moreno, L. (2008). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 19-55). New York: Lawrence Erlbaum Associates.

Google Scholar CrossrefKenney, P. A., & Silver, E. A. (1997). Results from the sixth mathematics assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.

Google Scholar CrossrefKilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Google Scholar CrossrefKnuth, E. J., Stephens, A. C., McNeil, N. M. & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37, 297-312.

Google Scholar CrossrefKoedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving. Cognitive Science, 32, 366-397.

Google Scholar CrossrefKoedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences. 13(2), 129-164.

Google Scholar CrossrefKoehler, J. L. (2004). Learning to think relationally: Thinking relationally to learn. Unpublished dissertation, University of Wisconsin-Madison, Madison, WI.

Google Scholar CrossrefLappan, G., Fey, J. T., Fitzgerald, W. M. Friel, S. N., & Phillips, E. D. (1998). Connected Mathematics. Dale Seymour Publications.

Google Scholar CrossrefMacGregor, M., & Stacey, K. (1993). Cognitive models underlying students' formulation of simple linear equations. Journal for Research in Mathematics Education, 24, 217-232.

Google Scholar CrossrefMcNeil, N. M., Weinberg, A. Stephens, A. C., Hattikudur, S., Asquith, P., Knuth, E. J., & Alibali, M. W. (2010). A is for Apple: Mnemonic symbols hinder students’ interpretations of algebraic expressions. Journal of Educational Psychology, 102(3), 625-634.

Google Scholar CrossrefNational Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.

Google Scholar CrossrefRAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: A strategic research and development program in mathematics education. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.

Google Scholar CrossrefRittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175-189.

Google Scholar CrossrefRittle-Johnson, B. & Schneider, M. (2014). Developing conceptual and procedural knowledge of mathematics. To appear in R. Cohen Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition. Oxford University Press.

Google Scholar CrossrefRittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362.

Google Scholar CrossrefRussell, S. J., Schifter, D., & Bastable, V. (2011). Connecting arithmetic to algebra: Strategies for building algebraic thinking in the elementary grades. Portsmouth, NJ: Heinemann.

Google Scholar CrossrefSchifter, D. (1999). Reasoning about operations: Early algebraic thinking in grades K-6. In L. V. Stiff (Ed.), Developing mathematical reasoning in grades K-12 (pp. 62-81). Reston, VA: National Council of Teachers of Mathematics.

Google Scholar CrossrefSidney, P. G. & Alibali, M. W. (2013). Making connections in math: Activating a prior knowledge analogue matters for learning. Journal of Cognition and Development, in press.

Google Scholar CrossrefSowder, L. (1988). Children's solutions of story problems. Journal of Mathematical Behavior, 7, 227-238.

Google Scholar CrossrefStephens, A. C. (2003). Another look at word problems. Mathematics Teacher, 96, 63-66.

Google Scholar CrossrefSwafford, J., & Langrall, C. (2000). Grade 6 students' preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31, 89-112.

Google Scholar CrossrefFigure Captions

Google Scholar CrossrefFigure 1. Percent of participants who succeeded on the equation-solving task for each operation or operation combination and each position of the unknown.

Google Scholar CrossrefFigure 2. Percent of participants who succeeded on the on the story-writing task for each operation or operation combination and each position of the unknown.

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*International Journal of Educational Psychology*,

*3*(3), 235–264. https://doi.org/10.4471/ijep.2014.13

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