## 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, access via your institution.

## References

Bagchi, A., & Wells, C. (1998a). On the communication of mathematical reasoning.

*Primus, 8*(1), 15–27.Bagchi, A., & Wells, C. (1998b). Varieties of mathematical prose.

*Primus, 8*(2), 116–136.Bullock, J. (1994). Literacy in the language of mathematics.

*The American Mathematical Monthly, 101*(8), 735–743.Burton, M. (1988). A linguistic basis for student difficulties with algebra.

*For the Learning of Mathematics, 8*(1), 2–7.Cheng, P., Holyoak, K., Nisbett, R., & Oliver, L. (1986). Pragmatic versus syntactic approaches to training deductive reasoning.

*Cognitive Psychology, 18*, 293–328.Dubinsky, E. (1997). On learning quantification.

*Journal of Computers in Mathematics and Science Teaching, 16*(2/3), 335–362.Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification.

*For the Learning of Mathematics, 8*(2), 44–51.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.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.Epp, S. (2003). The role of logic in teaching proof.

*The American Mathematical Monthly, 110*, 886–899.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.Hersh, R. (1997). Math lingo vs. plain English: Double entendre.

*The American Mathematical Monthly, 104*(1), 48–51.Lay, S. (2001).

*Analysis with an introduction to proof*. Upper Saddle River, NJ: Prentice Hall.Piatek-Jimenez, K. (2004). Undergraduate mathematics students’ understanding of mathematical statements and proofs. Doctoral dissertation, University of Arizona, Tucson.

Pimm, D. (1988). Mathematical metaphor.

*For the Learning of Mathematics, 8*(1), 30–34.Selden, J. & Selden, A. (1995). Unpacking the logic of mathematical statements.

*Educational Studies in Mathematics, 29*, 123–151.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.Wells, C. (2003).

*A handbook of mathematical discourse*. Haverford, PA: Infinity Publishing.com.

## Author information

### Authors and Affiliations

## Rights 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

Issue Date:

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

### Keywords

- Mathematical Statement
- Multiple Quantifier
- Universal Quantifier
- Mathematical Language
- Existential Quantifier