## Abstract

This paper originates from our doctorate research project, in which we investigated graduate and undergraduate students’ errors and difficulties in vector space theory (VST). After a brief account of the whole study, we will focus on the analyses carried out through the lenses of two different theoretical frameworks: Fischbein’s Theory of Tacit Models and Sfard’s process/object duality theory. We also propose a retrospective reflection concerning the reasons why we chose to carry out our analysis according to more than one theoretical framework and why we chose those specific theoretical frameworks as well. Hence this contribution has a double focus: on the one hand it reports on the results drawn from our study on students’ difficulties in VST; on the other hand it presents a personal retrospective reflection on the use of theoretical frames within that study.

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## Notes

The Authors explicitly state that their survey “does not claim the title of an exhaustive review of research on the teaching and learning of linear algebra at the undergraduate level” (Dorier and Sierpinska, 2001, p. 255).

In this respect we acknowledge that selecting the excerpts, deciding whether to show them entirely or partially, deciding how to summarize them,… is already part of the data interpreting process.

Coordinate subspaces of \( {\mathbb{R}}^{n} \) are those spaces spanned by a subset of \( {\mathbb{R}}^{n} \) canonical basis.

Here and in the following, we use ‘natural’ after the mathematical common use of the term. We do not mean to refer to something ‘natural’ from a cognitive point of view.

More precisely: do there exist vectors

**v**_{ 1 },**v**_{ 2 },**v**_{ 3 },**v**_{ 4 }and scalars*α*_{1},*α*_{2},*α*_{3},*α*_{4}such that…? Of course the task might be rephrased in many other different ways.

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## Acknowledgments

This paper originates from my doctorate reserach project, and I would like to seize this occasion to thank my teacher and mentor Prof. M. A. Mariotti for her constant and inestimable help, guidance and encouragement. I also wish to thank the reviewers of this paper for their helpful comments.

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Research funded by MIUR (PRIN 2005019721) and the University of Siena: Meanings, conjectures, proofs: from basic research in mathematics education to curriculum (national coordinator: M. G. Bartolini Bussi).

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Maracci, M. Combining different theoretical perspectives for analyzing students’ difficulties in vector spaces theory.
*ZDM Mathematics Education* **40**, 265–276 (2008). https://doi.org/10.1007/s11858-008-0078-z

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DOI: https://doi.org/10.1007/s11858-008-0078-z