Skip to main content
SpringerLink
Log in

Students’ interpretations of mathematical statements involving quantification

  • Articles
  • Published:
Mathematics Education Research Journal Aims and scope Submit manuscript

Abstract

Mathematical statements involving both universal and existential quantifiers occur frequently in advanced mathematics. Despite their prevalence, mathematics students often have difficulties interpreting and proving quantified statements. Through task-based interviews, this study took a qualitative look at undergraduate mathematics students’ interpretations and proof-attempts for mathematical statements involving multiple quantifiers. The findings of this study suggest that statements of the form “There exists … for all …” (which can be referred to as EA statements) evoked a larger variety of interpretations than statements of the form “For all … there exists …” (AE statements). Furthermore, students’ proof techniques for such statements, at times, unintentionally altered the students’ interpretations of these statements. The results of this study suggest that being confronted with both the EA and AE versions of a statement may help some students determine the correct mathematical meanings of such statements. Moreover, knowledge of the structure of the mathematical language and the use of formal logic may be useful tools for students in proving such mathematical statements.

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

Similar content being viewed by others

References

  • Bagchi, A., & Wells, C. (1998a). On the communication of mathematical reasoning.Primus, 8(1), 15–27.

    Article  Google Scholar 

  • Bagchi, A., & Wells, C. (1998b). Varieties of mathematical prose.Primus, 8(2), 116–136.

    Article  Google Scholar 

  • Bullock, J. (1994). Literacy in the language of mathematics.The American Mathematical Monthly, 101(8), 735–743.

    Article  Google Scholar 

  • Burton, M. (1988). A linguistic basis for student difficulties with algebra.For the Learning of Mathematics, 8(1), 2–7.

    Google Scholar 

  • Cheng, P., Holyoak, K., Nisbett, R., & Oliver, L. (1986). Pragmatic versus syntactic approaches to training deductive reasoning.Cognitive Psychology, 18, 293–328.

    Article  Google Scholar 

  • Dubinsky, E. (1997). On learning quantification.Journal of Computers in Mathematics and Science Teaching, 16(2/3), 335–362.

    Google Scholar 

  • Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification.For the Learning of Mathematics, 8(2), 44–51.

    Google Scholar 

  • Dubinsky, E. & Yiparaki, O. (2000). On student understanding of AE and EA quantification. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.),CMBS issues in mathematics education (pp. 239–289). Providence, RI: American Mathematical Society.

    Google Scholar 

  • Epp, S. (1999). The language of quantification in mathematics instruction. In L. V. Stiff & F. R. Curcio (Eds.),Developing mathematical reasoning in grades K-12 (1999 Yearbook) (pp. 188–197). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Epp, S. (2003). The role of logic in teaching proof.The American Mathematical Monthly, 110, 886–899.

    Article  Google Scholar 

  • Ferrari, P. (2002). Understanding elementary number theory at the undergraduate level: A semiotic approach. In S. R. Campbell & R. Zazkis (Eds.),Learning and teaching number theory: Research in cognition and instruction (pp. 97–115). Westport, CT: Greenwood Publishing Group, Inc.

    Google Scholar 

  • Hersh, R. (1997). Math lingo vs. plain English: Double entendre.The American Mathematical Monthly, 104(1), 48–51.

    Article  Google Scholar 

  • Lay, S. (2001).Analysis with an introduction to proof. Upper Saddle River, NJ: Prentice Hall.

    Google Scholar 

  • Piatek-Jimenez, K. (2004). Undergraduate mathematics students’ understanding of mathematical statements and proofs. Doctoral dissertation, University of Arizona, Tucson.

    Google Scholar 

  • Pimm, D. (1988). Mathematical metaphor.For the Learning of Mathematics, 8(1), 30–34.

    Google Scholar 

  • Selden, J. & Selden, A. (1995). Unpacking the logic of mathematical statements.Educational Studies in Mathematics, 29, 123–151.

    Article  Google Scholar 

  • Tall, D. and Chin, E.T. (2002). University students’ embodiment of quantifier. In A. D. Cockburn & E. Nardi (Eds.),Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (pp. 273–280). Norwich, UK.

  • Walk, S. M. (2004). Mind your ∀’s and ∃’s.The College Mathematics Journal, 35(5), 362–369.

    Article  Google Scholar 

  • Wells, C. (2003).A handbook of mathematical discourse. Haverford, PA: Infinity Publishing.com.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, 214 Pearce Hall, 48859, Mount Pleasant, MI, USA

    Katrina Piatek-Jimenez

Authors
  1. Katrina Piatek-Jimenez
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Piatek-Jimenez, K. Students’ interpretations of mathematical statements involving quantification. Math Ed Res J 22, 41–56 (2010). https://doi.org/10.1007/BF03219777

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03219777

Keywords

Search

Navigation