Skip to main content
SpringerLink
Log in

College developmental math students’ knowledge of the equal sign

Educational Studies in Mathematics Aims and scope Submit manuscript

Cite this article

Abstract

Decades of research have documented young students’ misinterpretations of the equal sign and the impediments these present for children’s mathematical development. Much less is known about individual differences in adults’ knowledge of the equal sign. We assessed 182 college students from developmental math courses and present analyses from a subset of items on a previously validated assessment of mathematical equality. College students exhibited variability in their interpretations of the equal sign, and these individual differences were related both to their solutions for open equations with variables and to their interpretations of an algebraic expression. Students who provided a relational interpretation of the equal sign were most successful on target algebra problems and students who only provided an operational interpretation of the equal sign were least successful. These results show that misinterpretations of the equal sign can persist well beyond elementary school and can provide a critical indicator of broader mathematical reasoning among college students.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  • ACT. (2018). The Condition of College and Career Readiness: National 2018. ACT. Retrieved from https://www.act.org/content/dam/act/unsecured/documents/cccr2018/National-CCCR-2018.pdf.

  • Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9, 221–247. https://doi.org/10.1080/10986060701360902

    Article  Google Scholar 

  • Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equal sign. Mathematics Teaching, 92, 13–15.

    Google Scholar 

  • Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Lsler, I., & Kim, J. S. (2015). The development of children’s early algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46, 39–87. https://doi.org/10.5951/jresematheduc.46.1.0039

    Article  Google Scholar 

  • Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G. (2015). A specific misconception of the equal sign acts as a barrier to children’s learning of early algebra. Learning and Individual Differences, 38, 61–67. https://doi.org/10.1016/j.lindif.2015.01.001

    Article  Google Scholar 

  • Capraro, R. M., Capraro, M. M., Yetkiner, Z. E., Özel, S., Kim, H. G., & Küçük, A. R. (2010). An international comparison of grade 6 students’ understanding of the equal sign. Psychological Reports, 106, 49–53.

    Article  Google Scholar 

  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Charles, R. I. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. National Council for Supervisors of Mathematics: Journal of Mathematics Education Leadership, 8, 9–24.

    Google Scholar 

  • Chesney, D. L., McNeil, N. M., Brockmole, J. R., & Kelley, K. (2013). An eye for relations: Eye-tracking indicates long-term negative effects of operational thinking on understanding of math equivalence. Memory and Cognition, 41, 1079–1095. https://doi.org/10.3758/s13421-013-0315-8

    Article  Google Scholar 

  • Chimoni, M., Pitta-Pantazi, D., & Christou, C. (2018). Examining early algebraic thinking: Insights from empirical data. Educational Studies in Mathematics, 98, 57–76.

    Article  Google Scholar 

  • Collis, K. F. (1974). Cognitive development and mathematics learning. Paper presented at Psychology of Mathematics Education Workshop, Center for Science Education, Chelsea College, London.

  • Diamond, A., & Kirkham, N. (2005). Not quite as grown up as we like to think: Parallels between cognition in childhood and adulthood. Psychological Science, 16, 291–297. https://doi.org/10.1111/j.0956-7976.2005.01530.x

    Article  Google Scholar 

  • Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232–236.

    Google Scholar 

  • Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9, 19–26.

    Google Scholar 

  • Freiman, V., & Lee, L. (2004). Tracking primary students’ understanding of the equality sign. In M. Hoines & A. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education 2:415–422). Bergen, Norway: PME.

  • Fyfe, E. R., Matthews, P. G., Amsel, E., McEldoon, K. L., & McNeil, N. M. (2018). Assessing formal knowledge of math equivalence among algebra and pre-algebra students. Journal of Educational Psychology, 110, 87–101. https://doi.org/10.1037/edu0000208

    Article  Google Scholar 

  • Fyfe, E. R., & Rittle-Johnson, B. (2017). Mathematics practice without feedback: A desirable difficulty in a classroom setting. Instructional Science, 45, 177–194. https://doi.org/10.1007/s11251-016-9401-1

    Article  Google Scholar 

  • Henrich, J., Heine, S. J., & Norenzayan, A. (2010). Most people are not WEIRD. Nature, 446, 29. https://doi.org/10.1038/466029a

    Article  Google Scholar 

  • Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59–78.

    Article  Google Scholar 

  • Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38, 258–288.

    Google Scholar 

  • Jones, I., Inglis, M., Gilmore, C., & Dowens, M. (2012). Substitution and sameness: Two components of a relational conception of the equals sign. Journal of Experimental Child Psychology, 113, 166–176. https://doi.org/10.1016/j.jecp.2012.05.003

    Article  Google Scholar 

  • Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326. https://doi.org/10.1007/BF00311062

    Article  Google Scholar 

  • 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 

  • Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7, 23–26 http://www.jstor.org/stable/30213397

    Google Scholar 

  • 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, 195–217. https://doi.org/10.1080/07370000801980845

    Article  Google Scholar 

  • Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39–65.

    Article  Google Scholar 

  • MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation. Educational Studies in Mathematics, 33, 1–19. https://doi.org/10.1023/A:1002970913563

    Article  Google Scholar 

  • Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43, 316–350. https://doi.org/10.5951/jresematheduc.43.3.0316

    Article  Google Scholar 

  • McNeil, N. M., & Alibali, M. W. (2005a). Knowledge change as a function of mathematics experience: All contexts are not created equal. Journal of Cognition and Development, 6, 285–306. https://doi.org/10.1207/s15327647/jcd0602_6

    Article  Google Scholar 

  • McNeil, N. M., & Alibali, M. W. (2005b). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 1–17. https://doi.org/10.1111/j.1467-8624.2005.00884.x

    Article  Google Scholar 

  • McNeil, N. M., Hornburg, C. B., Devlin, B. L., Carrazza, C., & McKeever, M. O. (2017). Consequences of individual differences in children’s formal understanding of mathematical equivalence. Child Development. https://doi.org/10.1111/cdev.12948

    Article  Google Scholar 

  • McNeil, N. M., Rittle-Johnson, B., Hattikudur, S., & Petersen, L. A. (2010). Continuity in representation between children and adults: Arithmetic knowledge hinders undergraduates’ algebraic problem solving. Journal of Cognition and Development, 11, 437–457. https://doi.org/10.1080/15248372.2010.516421

    Article  Google Scholar 

  • McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., & Alibali, M. W. (2010). A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102, 625–634. https://doi.org/10.1037/a0019105

    Article  Google Scholar 

  • National Center for Education Statistics. (2015). The nation’s report card, National Center for education statistics, Institute of Education Sciences, U. Washington, DC: S. Department of Education. Retrieved from https://www.nationsreportcard.gov

  • Provasnik, S., Malley, L., Stephens, M., Landeros, K., Perkins, R., & Tang, J. H. (2016). Highlights from TIMSS and TIMSS advanced 2015. U.S. Department of Education. Retrieved from: https://nces.ed.gov/pubs2017/2017002.pdf

  • Renwick, E. (1932). Children’s misconceptions concerning the symbols for mathematical equality. British Journal of Educational Psychology, 2, 173–183. https://doi.org/10.1111/j.204408279.1932.tb02743.x

    Article  Google Scholar 

  • Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. (2011). Assessing knowledge of mathematical equivalence: A construct modeling approach. Journal of Educational Psychology, 103, 85–104. https://doi.org/10.1037/a0021334

    Article  Google Scholar 

  • Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191–228. https://doi.org/10.1007/978-94-017-2057-1_4

    Article  Google Scholar 

  • Simsek, E., Xenidou-Dervou, I., Karadeniz, I., & Jones, I. (2019). The conception of substitution of the equals sign plays a unique role in students’ algebra performance. Journal of Numerical Cognition, 5, 24–37.

    Article  Google Scholar 

  • Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teacher, 77, 20–26.

    Google Scholar 

  • Steinberg, R. M., Sleeman, D. H., & Ktorza, D. (1990). Algebra students’ knowledge of equivalence of equations. Journal for Research in Mathematics Education, 22, 112–121.

    Article  Google Scholar 

Download references

Funding

Parts of this research were conducted while Fyfe was financially supported by Institute of Education Sciences, U. S. Department of Education, training grant R305B130007 as part of the Wisconsin Center for Education Research Postdoctoral Training Program.

Author information

Authors and Affiliations

  1. Department of Psychological and Brain Sciences, Indiana University, 1101 E. 10th Street, Bloomington, IN, 47405, USA

    Emily R. Fyfe

  2. Department of Educational Psychology, University of Wisconsin-Madison, Madison, WI, USA

    Percival G. Matthews

  3. Department of Psychology, Weber State University, Ogden, UT, USA

    Eric Amsel

Authors
  1. Emily R. Fyfe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Percival G. Matthews
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric Amsel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emily R. Fyfe.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fyfe, E.R., Matthews, P.G. & Amsel, E. College developmental math students’ knowledge of the equal sign. Educ Stud Math 104, 65–85 (2020). https://doi.org/10.1007/s10649-020-09947-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-020-09947-2

Keywords

search

Navigation