Nucleus, Image and Composition of Linear Transformations: a graphical approach in computational enviroment


  • Monica Karrer Centro Universitário da FEI
  • Renata Caroline Mota Santos Centro Universitário da FEI



Álgebra Linear; Transformações Lineares; GeoGebra; Representações Semióticas; Design Experiment.


The purpose of this article is to present a teaching experiment on the content of linear transformations in the plane and the results of its application in a pair of students of Mechanical Engineering course. Difficulties of the students in Linear Algebra discipline concerning its formal aspect are pointed by Aydin, Hannah, et al. and Isik, furthermore, new pedagogical practices which considers this problematic are required. In this context, elaborating and applying a teaching approach to geometrical linear transformations is aimed at bringing innovative learning and teaching perspectives. This experiment essentially regards the concepts of composition, nucleus and image and also integrates the GeoGebra software to explore the relations among representations of different registers such as graphic, algebraic, figural and of natural language. The study is grounded on the registers of semiotic representations provided by Duval's theory and on a technology according to the views of Drijvers, Baki, Hoyles and Noss and Borba and Penteado, in order to allow different developments from those reached by paper and pencil environment. For the experiment preparation and conduction, it is used some aspects of the Design Experiment by Cobb et al., a specific method to build an innovative approach to the mathematics education. This research has a qualitative character and the results revealed that providing a favorable environment to the students to coordinate representations of different registers, there was progress in understanding the concepts related to linear transformations, mainly, nucleus and image. The autonomy in building concepts is privileged by the GeoGebra software utilization, regarding its dynamic character, the students could test their conjectures in its environment.


AYDIN, S. The factors effecting teaching Linear Algebra. In: World Conference on Educational Sciences. 2009. In: WORLD CONFERENCE ON EDUCATIONAL SCIENCES, 2009, Nicosia, North Cyprus. Procedia - Social and Behavioral Siences. Nicosia, North Cyprus: ELSEVIER LTDA, 2009, p. 1549 – 1553.

BAKI, A. Integration of Technology into Mathematics Teaching: past, presente and future. In: SUNG JE CHO (ed.). Selected Regular Lectures from the 12th International Congress on Mathematical Education. Switzerland: Springer International Publishing, 2015. p. 17-26.

BARDIN, L. Análise de conteúdo. Lisboa: Edições 70, 1977

BORBA, M.; PENTEADO, M.G. Informática e Educação Matemática. São Paulo: Autêntica, 2010.

COBB, P.; CONFREY, J.; DISESSA, A.; LEHRER, R.; SCHAUBLE, L. Design experiments in education research. Educational Researcher, v.32, n.1, p. 9-13, 2003.

DUVAL, R. Sémiosis et pensée humaine. Berna: Peter Lang, 1995.

DUVAL, R. Basic Issues for Research in Mathematics Education. In: CONFERENCE OF THE INTERNATIONAL GROUP FOR THE PSYCHOLOGY OF MATHEMATICS EDUCATION, 24, 2000, Hiroshima. Proceedings of the 24th PME. Hiroshima: Department of Mathematics Education Hiroshima University, 2000. p. 55-69.

DUVAL, R. A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, Springer, v. 61, p. 103-131, 2006.

DUVAL, R. Ver e ensinar a matemática de outra forma: entrar no modo matemático de pensar os registros de representações semióticas. São Paulo: PROEM, v. 1, 2011.

DRIJVERS, P. Digital Technology in Mathematics Education: why it works (or doesn’t). In: SUNG JE CHO (ed.). Selected Regular Lectures from the 12th International Congress on Mathematical Education. Switzerland: Springer International Publishing, 2015. p. 135-151.

HANNAH, J.; STEWART, S. THOMAS, M. Teaching Linear Algebra: one lecturer’s engagement with students. Mathematics: traditions and new practices. Austrália: AAMT & MERGA, 2011.

ISIK, A. et al. Linear Algebra from students’ perspectives. Middle eastern & African Journal of Educational Research, 2014. p. 29 – 40.

KARRER, M. Articulação entre Álgebra Linear e Geometria: um estudo sobre as transformações lineares na perspectiva dos registros de representação semiótica. São Paulo, 2006. Tese (Doutorado em Educação Matemática) – Programa de Estudos Pós-Graduados em Educação Matemática, Pontifícia Universidade Católica de São Paulo. (372p.)

KELLY, A.; LESH, R.A.; BAEK, J.Y. Handbook of Design Research Methods in Education. Innovations in Science, Technology, Engineering and Mathematics Learning and Teaching, 2010. Disponível em /dp/0805860592. Acesso em: 04 jul. 2017.

NOSS, R.; HOYLES, C. The technological mediation of Mathematics and its learning. Human Development: giving meaning to Mathematical signs: Psychological, Pedagogical and Cultural Processes, Basel, v. 52, n. 2, p. 129-147, 2009.

PONTE, J. P. Estudos de caso em Educação Matemática. Bolema, n. 25, p. 105-132, 2006



How to Cite

KARRER, M.; SANTOS, R. C. M. Nucleus, Image and Composition of Linear Transformations: a graphical approach in computational enviroment. Research, Society and Development, [S. l.], v. 7, n. 10, p. e9710405, 2018. DOI: 10.17648/rsd-v7i10.405. Disponível em: Acesso em: 27 sep. 2021.