, Volume 51, Issue 7, pp 1083–1095 | Cite as

Mathematics students talking past each other: emergence of ambiguities in linear algebra proof constructions involving the uniqueness quantification

  • Ann Sophie StuhlmannEmail author
Original Article


This paper examines proof constructions in group work in the field of linear algebra teaching at the university level. Studies have shown that students at tertiary level have difficulties in understanding different kinds of quantifiers, which are fundamental in linear algebra proof constructions. This study investigates how two student groups, with a tutor involved in one of the groups, construed meaning in the context of proving unique existence of the adjoint endomorphism. The students and the tutor used certain words and phrases in the context of mathematical uniqueness differently. The study analyses from an interactionist standpoint how these ambiguities emerged. The results indicate that due to different background understandings of mathematical uniqueness students attributed different meanings to certain words and expressions, which prevented the students from negotiating a consensus during the proving process.


Collaborative proof construction Linear algebra teaching Unique existential quantification Interactionist perspective Ambiguity 



  1. Barwell, R. (2005). Ambiguity in the mathematics classroom. Language and Education,19(2), 117–125.CrossRefGoogle Scholar
  2. Bauersfeld, H., Krummheuer, G., & Voigt, J. (1988). Interactional theory of learning and teaching mathematics and related microethnographical studies. In H.-G. Steiner & A. Vermandel (Eds.), Foundations and methodology of the discipline mathematics education (pp. 174–188). Antwerp: University of Antwerp.Google Scholar
  3. Blanton, M., & Stylianou, D. (2014). Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs. The Journal of Mathematical Behavior,34, 76–98.CrossRefGoogle Scholar
  4. Bosch, S. (2014). Lineare algebra. Berlin: Springer.Google Scholar
  5. Britton, S., & Henderson, J. (2009). Linear algebra revisited: An attempt to understand students’ conceptual difficulties. International Journal of Mathematical Education in Science and Technology,40(7), 963–974.CrossRefGoogle Scholar
  6. Cobb, P., & Bauersfeld, H. (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  7. Dorier, J.-L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics,29(2), 175–197.CrossRefGoogle Scholar
  8. Dorier, J.-L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton (Ed.), The teaching and learning of mathematics of university level: An ICMI study (pp. 255–273). Dordrecht: Kluwer Academic Publishers.Google Scholar
  9. Dubinsky, E. (1997). On learning quantification. Journal of Computers in Mathematics and Science Teaching,16(2/3), 335–362.Google Scholar
  10. Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification. For the Learning of Mathematic,8(2), 44–51.Google Scholar
  11. 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: American Mathematical Society.Google Scholar
  12. Forman, E., & Cazden, C. (1985). Exploring Vygotskian perspectives in education: The cognitive value of peer construction. In J. V. Wertsch (Ed.), Culture, communication, and cognition (pp. 323–346). Cambridge: Cambridge University Press.Google Scholar
  13. Fukawa-Connelly, T. (2012). Classroom sociomathematical norm for proof presentation in undergraduate in abstract algebra. Journal of Mathematical Behavior,31(3), 401–416.CrossRefGoogle Scholar
  14. Goffman, E. (1974). Frame analysis. An essay on the organization of experience. Cambridge: Harvard University Press.Google Scholar
  15. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–270). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  16. Krummheuer, G., & Naujok, N. (1999). Grundlagen und Beispiele interpretativer Unterrichtsforschung. Opladen: Leske + Budrich.CrossRefGoogle Scholar
  17. Lang, S. (1987). Linear algebra. New York: Springer.CrossRefGoogle Scholar
  18. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics,27, 249–266.CrossRefGoogle Scholar
  19. Mueller, M., Maher, C., & Yankelewitz, D. (2009). A framework for analyzing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics,80(3), 1–19.Google Scholar
  20. Piatek-Jimenez, K. (2010). Students’ interpretations of mathematical statements involving quantification. Mathematics Educational Research Journal,22(3), 41–56.CrossRefGoogle Scholar
  21. Remillard, K. (2009). The mathematical discourse of undergraduate mathematics majors: The relation to learning proof and establishing a learning community. Unpublished doctoral dissertation. Indiana: Indiana University of Pennsylvania.Google Scholar
  22. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press.Google Scholar
  23. Schütte, M. (2014). Language-related learning of mathematics: A comparison of kindergarten and primary school as places of learning. ZDM Mathematics Education,46(6), 923–938.CrossRefGoogle Scholar
  24. Schütte, M., Friesen, R.-A., & Jung, J. (2019). Interactional analysis: A method for analysing mathematical learning processes in interactions. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education. ICME-13 monographs (pp. 101–129). Cham: Springer.CrossRefGoogle Scholar
  25. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education,34, 4–36.CrossRefGoogle Scholar
  26. Shipman, B. (2013). On the meaning of uniqueness. Problems, Resources, and Issues in Mathematics Undergraduate Studies,23(3), 224–233.Google Scholar
  27. Stewart, S., & Thomas, O. J. M. (2019). Student perspectives on proof in linear algebra. ZDM Mathematics Education. Scholar
  28. Stylianides, G., Stylianides, A., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). Reston: National Council of Teachers of Mathematics.Google Scholar
  29. Thompson, P. W. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 57–93). New York: Springer.CrossRefGoogle Scholar
  30. Uhlig, F. (2002). The role of proof in comprehending and teaching elementary linear algebra. Educational Studies in Mathematics,50, 335–346.CrossRefGoogle Scholar
  31. Vinner, S. (1997). Scenes from linear algebra classes. In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.), Resources for teaching linear algebra, MAA Notes (Vol. 42, pp. 155–171). Washington: Mathematical Association of America.Google Scholar
  32. Voigt, J. (1994). Negotiation of mathematical meaning and learning mathematics. Educational Studies in Mathematics,26, 275–298.CrossRefGoogle Scholar
  33. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics,48, 101–119.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.Universität HamburgHamburgGermany

Personalised recommendations