Misconceptions and Learning Algebra

  • Julie L. BoothEmail author
  • Kelly M. McGinn
  • Christina Barbieri
  • Laura K. Young


Rather than exclusively focus on mastery of procedural skills, mathematics educators are encouraged to cultivate conceptual understanding in their classrooms. However, mathematics learners hold many faulty conceptual ideas—or misconceptions—at various points in the learning process. In the present chapter, we first describe the common misconceptions that students hold when learning algebra. We then explain why these misconceptions are problematic and detail a potential solution with the capability to help students build correct conceptual knowledge while they are learning new procedural skills. Finally, we discuss other potential implications from the existence of algebraic misconceptions which require further study. In general, preventing and remediating algebraic misconceptions may be necessary for increasing student success in algebra and, subsequently, more advanced mathematics classes.


Misconceptions Worked examples Learning from errors Conceptual knowledge Self-explanation 



Funding for the writing of this chapter was provided by the Institute of Education Sciences and U.S. Department of Education through Grant R305B150014 to Temple University, Grant R305B130012 to the University of Delaware, and Grant R305A100150 to the Strategic Education Research Partnership. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education.


  1. Adams, D. M., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., et al. (2014). Using erroneous examples to improve mathematics learning with a web-based tutoring system. Computers in Human Behavior, 36, 401–411.CrossRefGoogle Scholar
  2. Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, DC: US Department of Education.Google Scholar
  3. Alibali, M. W. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35(1), 127–145.CrossRefGoogle Scholar
  4. Alibali, M. W., Phillips, K. M. O., & Fischer, A. D. (2009). Learning new problem-solving strategies leads to changes in problem representation. Cognitive Development, 24, 89–101.CrossRefGoogle Scholar
  5. Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ under- standing of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning: An International Journal, 9(3), 249–272.CrossRefGoogle Scholar
  6. Baroudi, Z. (2006). Easing students’ transition to algebra. Australian Mathematics Teacher, 62(2), 28–33.Google Scholar
  7. Behr, M. J., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92, 13–15.Google Scholar
  8. Blanco, L. J., & Garrote, M. (2007). Difficulties in learning inequalities in students of the first year of pre-university education in Spain. Eurasia Journal of Mathematics, Science and Technology Education, 3, 221–229.Google Scholar
  9. Bodin, A., & Capponi, B. (1996). Junior secondary school practices. In A. J. Bishop et al. (Eds.), International handbook of mathematics education (pp. 565–614). London: Kluwer Academic.Google Scholar
  10. Booth, L. R. (1986). Difficulties in algebra. Australian Mathematics Teacher, 42(3), 2–4.Google Scholar
  11. Booth, J. L., Barbieri, C., Eyer, F., & Paré-Blagoev, J. (2014). Persistent and pernicious errors in algebraic problem-solving. The Journal of Problem Solving, 7, 10–23.CrossRefGoogle Scholar
  12. Booth, J. L., Cooper, L. A., Donovan, M. S., Huyghe, A., Koedinger, K. R., & Paré-Blagoev, E. J. (2015). Design-based research within the constraints of practice: AlgebraByExample. Journal of Education for Students Placed at Risk, 20(1–2), 79–100.CrossRefGoogle Scholar
  13. Booth, J. L., & Davenport, J. L. (2013). The role of problem representation and feature knowledge in algebraic equation solving. The Journal of Mathematical Behavior, 32(3), 415–423.CrossRefGoogle Scholar
  14. Booth, J. L., & Koedinger, K. R. (2008). Key misconceptions in algebraic problem solving. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 30th annual cognitive science society (pp. 571–576). Austin, TX: Cognitive Science Society.Google Scholar
  15. Booth, J. L., Koedinger, K. R., & Siegler, R. S. (2007). [Abstract]. The effect of prior conceptual knowledge on procedural performance and learning in algebra. In D. S. McNamara & J. G. Trafton (Eds.), Proceedings of the 29th annual cognitive science society. Austin, TX: Cognitive Science Society.Google Scholar
  16. Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.CrossRefGoogle Scholar
  17. Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A., Barbieri, C., Augustine, A. A., et al. (2015). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530–551.CrossRefGoogle Scholar
  18. Brown, G., & Quinn, R. J. (2006). Algebra students’ difficulty with fractions. Australian Mathematics Teacher, 62(4), 28–40.Google Scholar
  19. Cangelosi, R., Madrid, S., Cooper, S., Olson, J., & Hartter, B. (2013). The negative sign and exponential expressions: Unveiling students’ persistent errors and misconceptions. The Journal of Mathematical Behavior, 32(1), 69–82.CrossRefGoogle Scholar
  20. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.Google Scholar
  21. Catrambone, R. (1996). Generalizing solution procedures learned from examples. Journal of Experimental Psychology. Learning, Memory, and Cognition, 22(4), 1020–1031.CrossRefGoogle Scholar
  22. Catrambone, R. (1998). The subgoal learning model: Creating better examples so that students can solve novel problems. Journal of Experimental Psychology, 127(4), 355–376.CrossRefGoogle Scholar
  23. Chase, W. G., & Simon, H. A. (1973). Perception in chess. Cognitive Psychology, 4, 55–81.CrossRefGoogle Scholar
  24. Cheng-Yao, L., Yi-Yin, K., & Yu-Chun, K. (2014). Changes in pre-service teachers’ algebraic misconceptions by using computer-assisted instruction. International Journal for Technology in Mathematics Education, 21(3), 21–30.Google Scholar
  25. Chi, M. T. (2000). Self-explaining expository texts: The dual processes of generating inferences and repairing mental models. Advances in Instructional Psychology, 5, 161–238.Google Scholar
  26. Chi, M. T., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182.CrossRefGoogle Scholar
  27. Chi, M. T., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5(2), 121–152.CrossRefGoogle Scholar
  28. Clark, R. C., & Mayer, R. E. (2003). e-Learning and the science of instruction: Proven guidelines for consumers and designers of multimedia learning. San Francisco, CA: Jossey-Bass.Google Scholar
  29. Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), 16–30.CrossRefGoogle Scholar
  30. Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing. Focus on Learning Problems in Mathematics, 11, 77–87.Google Scholar
  31. Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly, 88, 286–290.CrossRefGoogle Scholar
  32. Clement, J., Narode, R., & Rosnick, P. (1981). Intuitive misconceptions in algebra as a source of math anxiety. Focus on Learning Problems in Mathematics, 3(4), 36–45.Google Scholar
  33. Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from
  34. Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79(4), 347–362.CrossRefGoogle Scholar
  35. Corder, S. P. (1982). Error analysis & interlanguage. Oxford: Oxford University Press.Google Scholar
  36. Crooks, N. M., & Alibali, M. W. (2013). Noticing relevant problem features: Activating prior knowledge affects problem solving by guiding encoding. Frontiers in Psychology, 4, 884. doi: 10.3389/fpsyg.2013.00884.CrossRefGoogle Scholar
  37. Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214.CrossRefGoogle Scholar
  38. Durkin, K., & Rittle-Johnson, B. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21–29.CrossRefGoogle Scholar
  39. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–236.Google Scholar
  40. Festinger, L. (1957). A theory of cognitive dissonance. Evanston, IL: Row Peterson.Google Scholar
  41. Gardella, F. J. (2009). Introducing difficult mathematics topics in the elementary classroom: A teacher’s guide to initial lessons. New York: Routledge, Taylor & Francis.Google Scholar
  42. Graesser, A. C. (2009). Inaugural editorial for journal of educational psychology. Journal of Educational Psychology, 101(2), 259–261.CrossRefGoogle Scholar
  43. Herscovics, N., & Kieran, C. (1980). Constructing meaning for the concept of equation. Mathematics Teacher, 73, 572–580.Google Scholar
  44. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., et al. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12–21.CrossRefGoogle Scholar
  45. Hilbert, T., Renkl, A., Schworm, S., Kessler, S., & Reiss, K. (2008). Learning to teach with worked-out examples: A computer-based learning environment for teachers. Journal of Computer Assisted Learning, 24(4), 316–332.CrossRefGoogle Scholar
  46. Kieran, C. (1979). Children’s operational thinking within the context of bracketing and the order of operations. In Paper presented at the third international conference for the psychology of mathematics education, Coventry, England.Google Scholar
  47. Kieran, C. (1980). Constructing meaning for non-trivial equations. In Paper presented at the annual meeting of the American educational research association, Boston, MA.Google Scholar
  48. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 318–326.CrossRefGoogle Scholar
  49. Kieran, C. (1985). The equation-solving errors of novice and intermediate algebra students. In Paper presented at the ninth international conference for the psychology of mathematics education, The Netherlands.Google Scholar
  50. Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lister (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Reston, VA: NCTM.Google Scholar
  51. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up. In Mathematics Learning Study Committee, Center for Education. Washington, DC: National Academy Press.Google Scholar
  52. Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13(9), 514–519.Google Scholar
  53. Knuth, 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
  54. Kuchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7, 23–26.Google Scholar
  55. Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children’s understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26(2), 195–217.CrossRefGoogle Scholar
  56. Liebenburg, R. (1997). The usefulness of an intensive diagnostic test. In P. Kelsall & M. de Villiers (Eds.), Proceedings of the third national congress of the association for mathematics education of South Africa (Vol. 2, pp. 72–79). Durban, South Africa: University of Natal.Google Scholar
  57. Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. The Journal of Mathematical Behavior, 14, 113–120.CrossRefGoogle Scholar
  58. Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in ‘filling’ the gap in children’s extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39, 131–147.CrossRefGoogle Scholar
  59. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  60. MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19.CrossRefGoogle Scholar
  61. Mazzocco, M. M. M., Murphy, M. M., Brown, E. C., Rinne, L., & Herold, C. H. (2013). Persistent consequences of atypical early number concepts. Frontiers in Psychology, 4, 1–9.CrossRefGoogle Scholar
  62. McNeil, N. M. (2008). Limitations to teaching children 2 + 2 = 4: Typical arithmetic problems can hinder learning of mathematical equivalence. Child Development, 79, 1524–1537.CrossRefGoogle Scholar
  63. McNeil, N. M. (2014). A change-resistance account of children’s difficulties understanding mathematical equivalence. Child Development Perspectives, 8(1), 42–47.CrossRefGoogle Scholar
  64. McNeil, N. M., & Alibali, M. W. (2004). You’ll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28(3), 451–466.Google Scholar
  65. McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883–899.CrossRefGoogle Scholar
  66. McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., et al. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24(3), 367–385.CrossRefGoogle Scholar
  67. McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., et al. (2010). A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102(3), 625–634.CrossRefGoogle Scholar
  68. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: US Department of Education.Google Scholar
  69. Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.CrossRefGoogle Scholar
  70. Ohlsson, S. (1996). Learning from performance errors. Psychological Review, 103(2), 241.CrossRefGoogle Scholar
  71. Pinchback, C. L. (1991). Types of errors exhibited in a remedial mathematics course. Focus on Learning Problems in Mathematics, 13(2), 53–62.Google Scholar
  72. Prather, R. W. (2012). Implicit learning of arithmetic regularities is facilitated by proximal contrast. PLoS One, 7(10), e48868. doi: 10.1371/journal.pone.0048868.CrossRefGoogle Scholar
  73. Pugalee, D. (2010). Extending students’ development of proportional reasoning. In Paper presented at the regional meeting of the national council of teachers of mathematics, New Orleans, LA.Google Scholar
  74. Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23(1), 90–108.CrossRefGoogle Scholar
  75. Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175–189.CrossRefGoogle Scholar
  76. Rittle-Johnson, B., & Siegler, R. S. (1998). The relationship between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). East Sussex, England: Psychology Press.Google Scholar
  77. Rittle-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.CrossRefGoogle Scholar
  78. 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(3), 561.CrossRefGoogle Scholar
  79. Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Mathematics Teacher, 74, 418–420.Google Scholar
  80. Rowntree, R. V. (2009). Students’ understandings and misconceptions of algebraic inequalities. School Science and Mathematics, 109(6), 311–312.CrossRefGoogle Scholar
  81. Roy, M., & Chi, M. T. H. (2005). The self-explanation effect in multimedia learning. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 271–286). New York: Cambridge University Press.CrossRefGoogle Scholar
  82. Schwonke, R., Renkl, A., Krieg, C., Wittwer, J., Aleven, V., & Salden, R. (2009). The worked-example effect: Not an artefact of lousy control conditions. Computers in Human Behavior, 25(2), 258–266.CrossRefGoogle Scholar
  83. Seng, L. K. (2010). An error analysis of Form 2 (Grade 7) students in simplifying algebraic expressions: A descriptive study. Electronic Journal of Research in Educational Psychology, 8(1), 139–162.Google Scholar
  84. Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, 481–520.CrossRefGoogle Scholar
  85. Siegler, R. S. (2002). Microgenetic studies of self-explanations. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 31–58). New York: Cambridge University.CrossRefGoogle Scholar
  86. Stacey, K., & MacGregor, M. (1997). Ideas about symbolism that students bring to algebra. Mathematics Teacher, 90(2), 110–113.Google Scholar
  87. Stagylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518.CrossRefGoogle Scholar
  88. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.Google Scholar
  89. Stephens, A. C. (2005). Developing students’ understandings of variable. Mathematics Teaching in the Middle School, 11(2), 96–100.Google Scholar
  90. Swan, M. (2000). Making sense of algebra. Mathematics Teaching, 171, 16–19.Google Scholar
  91. Sweller, J. (1999). Instructional design in technical areas. Camberwell, VIC, Australia: ACER Press.Google Scholar
  92. Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89.CrossRefGoogle Scholar
  93. Trafton, J. G., & Reiser, B. J. (1993). The contributions of studying examples and solving problems to skill acquisition. In M. Polson (Ed.), Proceedings of the 15th annual conference of the cognitive science society (pp. 1017–1022). Hillsdale, NJ: Erlbaum.Google Scholar
  94. Tsamir, P., & Bazzini, L. (2004). Consistencies and inconsistencies in students’ solution to algebraic ‘single-value’ inequalities. International Journal of Mathematical Education in Science and Technology, 55, 793–812.CrossRefGoogle Scholar
  95. Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Schulte (Eds.), The ideas of algebra, K-12 (pp. 8–19). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  96. Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on student performance on, understanding of, linear equations and linear inequalities. Mathematical Thinking and Learning, 8, 113–147.CrossRefGoogle Scholar
  97. Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181–209.CrossRefGoogle Scholar
  98. Vamvakoussim, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467.CrossRefGoogle Scholar
  99. Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). New York: Pearson Education.Google Scholar
  100. Van Dooren, W., Lehtinen, E., & Vershcaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1–4.CrossRefGoogle Scholar
  101. Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of pre-service teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351.CrossRefGoogle Scholar
  102. Vlassis, J. (2002). About the flexibility of the minus sign in solving equations. In A. Cockburn & E. Nardi (Eds.), Proceeding of the 26th conference for the international group of the psychology of mathematics education (pp. 321–328). Norwich, UK: University of East Anglia.Google Scholar
  103. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469–484.CrossRefGoogle Scholar
  104. Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24(4), 535–585.CrossRefGoogle Scholar
  105. Warren, E. (2006). Comparative mathematical language in the elementary school: A longitudinal study. Educational Studies in Mathematics, 62, 169–189.CrossRefGoogle Scholar
  106. Watson, J. (1990). Research for teaching. Learning and teaching algebra. Australian Mathematics Teacher, 46(3), 12–14.Google Scholar
  107. Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10–17.Google Scholar
  108. Xin, Y. P., Wiles, B., & Lin, Y. (2008). Teaching conceptual model-based word problem story grammar to enhance mathematics problem solving. The Journal of Special Education, 42(3), 163–178.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Julie L. Booth
    • 1
    Email author
  • Kelly M. McGinn
    • 2
  • Christina Barbieri
    • 2
  • Laura K. Young
    • 2
  1. 1.Psychological Studies in EducationTemple UniversityPhiladelphiaUSA
  2. 2.College of Education and Human Development, University of DelawareNewarkUSA

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