PhD Courses in Denmark

Vergnaud’s Theory of Conceptual Fields (TCF): Understanding and utilizing TCF in a networking perspective

Graduate School, Arts at Aarhus University


Gerard Vergnaud is a student of Piaget, and Vergnaud’s TCF relies on Piaget’s notion of schemes. This PhD course will concern the use of TCF in relation to the doctoral students’ own topics in various areas of mathematics education. The course consists of three parts:

Part I is a general introduction to Vergnaud’s TCF. This part involves lectures and discussions aiming at increasing the students' understanding of the theory in general as well as its potential relevance for the students own projects. The course literature for this part of the course will mainly consist of Vergnaud’s own texts, but will also be contrasted by a few texts of others (e.g. Piaget and Vygotsky). 

Part II focuses on the use of Vergnaud's TCF in theoretical constructions within other areas of mathematics education. Notably, within the instrumental approach, which is a theoretical framework concerning the use of digital technologies, and which explicitly includes Vergnaud’s notion of scheme. In addition, the relationship between Vergnaud's conceptual fields and mathematical competencies will be discussed along with other potential areas, e.g. misconceptions and mathematical difficulties.

Part III is a written assignment (in the format of an 8-page conference paper following the CERME template) about the interplay between Vergnaud's TCF and other theoretical constructs related to the doctoral students’ own projects, e.g. from a networking of theories perspective. This part also includes peer reviewing and providing feedback on other doctoral students’ assignments.


The doctoral course aims at:
- The students understanding Vergnaud's theory of conceptual fields, and that they can apply issues of mathematics education using the terminology from TCF.

- The students being able to use TCF as a new perspective on their own doctoral projects. 

- The students being able to compare, contrast and combine TCF and related perspectives with other theories of mathematics education, e.g. in a networking of theories perspective.  

Regarding the literature, the participants receive a list of references before the course begins.

Regarding the literature, the participants receive a list of references before the course begins. We prefer literature written by Gérard Vergnaud himself, but texts of other authors will also occur.


Part I:

Ahl, L., & Helenius, O. (2018). The role of language representation for triggering students’ schemes. In: J. Häggström, Y. Liljekvist, J. B. Ärlebäck, M. Fahlgren, O. Olande, Perspectives on deleveopment of mathematics teachers. Proceedings of MADIF11: the eleventh research seminar of the Swedish Society for Research in Mathematics Education, Karlstad, January 23–24, 2018.

Radford, L. (2005). The Semiotics of the Schema. In: M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and Sign – Grounding Mathematics Education. Festschrift for Michael Otte (pp. 137-152). Netherlands: Kluwer Academic Publishers.

Vergnaud, G. (1982). Cognitive and Developmental Psychology and Research in Mathematics Education: Some Theoretical and Methodological Issues. For the Learning of Mathematics, 3(2), 31-41

Vergnaud, G. (1988). Multiplicative Structures. In J. Hiebert & M. Behr (eds.), Number Concepts and Operations in the Middle Grades (pp. 141-161). Hillsdale, NJ: Lawrence Erlbaum

Vergnaud, G. (1989). La formation des concepts scientifiques. Relire Vygotski et débattre avec lui aujourd'hui. Enfance, 42(1-2), 111-118. doi : 10.3406/enfan.1989.1885

Vergnaud, G. (1996). Some of Piaget's fundamental ideas concerning didactics. Prospects, 26(1), 183-194. doi:10.1007/BF02195617

Vergnaud, G. (1997). The Nature of Mathematical Concepts. In T. Nunes & P. Bryant (Eds.), Learning and Teaching Mathematics - An International Perspective (pp. 5-28). Hove, UK: Psychology Press.

Vergnaud, G. (1998). A comprehensive theory of representation for mathematics education. Journal of Mathematical Behavior, 17(2), 167-181. doi:10.1016/S0364-0213(99)80057-3

Vergnaud, G. (1998). Towards a Cognitive Theory of Practice In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics Education as a Research Domain: A Search for Identity (pp. 227-240). Great Britain: Kluwer Academic Publishers.

Vergnaud, G. (2009). The Theory of Conceptual Fields. Human Development, 52(2), 83-94. doi:10.1159/000202727

Supplementary literature for Part I

Schliemann, A. D., & Nunes, T. (1990). A situated schema of proportionality. British Journal of Developmental Psychology, 8(3), 259-268. doi:10.1111/j.2044-835X.1990.tb00841

Part II

Artigue, M. (2002). Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work. International Journal of Computer for Mathematical Learning, 7(3), 245-274.

Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses. A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives, Educational Studies in Mathematics, 82, 23-49

Gomes, A. S., & Vergnaud, G. (2004). On the Learning of geometric concepts using Dynamic Geometry Software. Revista Novas Tecnologias na Educação, 2(1)

Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3), 199-218

Niss, M., &, Højgaard, T. (2019). Mathematical competencies revisited, Educational Studies in Mathematics, 102, 9-28.

Trouche, L. (2003). From Artifact to Instrument: Mathematics Teaching Mediated by Symbolic Calculators, in P. Rabardel, & Y. Waern (Eds), special issue of Interacting with Computers, 15(6), 783-800

Rabardel, P., & Samurçay, R. (2001). From Artifact to Instrument-Mediated Learning, Invited Conference, New challenges to research on learning,

International symposium organized by the Center for Activity Theory and Developmental Work Research, University of Helsinki, March 21-23

Trouche, L., Gueudet, G., & Pepin, B. (2018). Documentational approach to didactics. In S. Lerman (Ed.), Encyclopedia of Mathematics Education. N.Y.: Springer. doi:10.1007/978-3-319-77487-9_100011-1

Vergnaud, G. (2013). Conceptual Development and Learning, Revista Qurriculum, 26, 39-59.

Vérillon, P., & Rabardel, P. (1995). Cognition and Artefact: a contribution to the study of thought in relation to instrumented activity, European Journal of Psychology in Education, 9(3), 77-101

Supplementary literature for Part II

Trouche, L., Drijvers, P., Gueudet, G., & Sacristán, A. (2013). Technology-Driven Developments and Policy Implications for Mathematics Education. In M. A. K. Clements, A. Bishop, C. Keitel-Kreidt, J. Kilpatrick, F.K.S. Leung (Eds.), Third International Handbook of Mathematics Education (pp. 753-789). New York, NY: Springer

Guin, D., & Trouche, L. (2002). Mastering by the teacher of the instrumental genesis in CAS environments: necessity of intrumental orchestrations. ZDM: The International Journal on Mathematics Education, 34(5), 204-211 DOI: 10.1007/BF02655823

Geraniou, E. & Jankvist, U.T. (2019) Towards a definition of “mathematical digital competency”. Educ Stud Math, 102 (1), 29-45.

Target group:

The PhD course is relevant for all doctoral students within the field of mathematics education




We divide the course into three parts. The main lectures in all parts of the course are Uffe Thomas Jankvist, AU, and Morten Misfeldt, KU.

In Part I of the course, guest lectures will be: 
- Researcher, Ola Helenius, Nationellt Centrum för Matematikutbildning, Göteborgs Universitet, who is the Nordic person who probably knows most about Vergnaud’s work.
- Licentiate Linda Ahl, Stockholm University, Sweden, who has studied Vergnaud extensively and also recently conducted an interview with him about his work.

In Part II of the course, guest lectures will be:
- Prof. Luc Trouche, École Normale Supérieure de Lyon, Frankrig, who is the originator of the instrumental approach, and who has worked with Vergnaud’s theory of conceptual fields for decades.

Both part I and part II include lectures from the guest lecturers, plenum discussions, group work and presentations by students.

Part III consists of students' autonomous work and peer review (the students give and receive feedback). This takes place in an online environment, e.g. PeerGrade ( 


Participation in part I and II gives 2 ECTS credits for each part, and thereby, these parts require 100 hours of work in total.
The third part involves a written assignment, continuous feedback, and peer reviewing. Participation in part III gives 1 ECTS credits and thus requires 25 hours of work.


The main lecturers: Uffe Thomas Jankvist, course organiser, ( and Morten Misfeldt (
In Part I of the course, guest lecturers will be: 
- Researcher, Ola Helenius, Nationellt Centrum för Matematikutbildning, Göteborgs Universitet, Sweden
In Part II of the course, the guest lecturer will be:
- Prof. Luc Trouche, École Normale Supérieure de Lyon, Frankrig

Dates and time:

Part I: 4-5 November 2020 
Part II: 2-3 December 2020 
Part III: The written assignment has to be handed by 15 January, 2021

Part I and part II are all-day course days from 09:00 - 16:00 -  Part III consists of handing in a written assignment


Campus Emdrup/Copenhagen, Tuborgvej 164, 2400 Copenhagen NV

Wednedsday 4 November: room D166 from 9-16

Thursday 5 November: room A214 from 9-16

Wednedsday 2 December: room A214 from 9-16

Thursday 3 December: room A214 from 9-16

Application deadline:

The application deadline for participation in only part I is no later than 2 October 2020.

The application deadline for participation in only part II is no later than 1 November 2020.

Please apply for a spot via