CAS Assisted Proofs in Upper Secondary School Mathematics Textbooks

Authors

https://doi.org/10.17583/redimat.2019.3315

Downloads

Abstract

This article addresses the didactical effects of CAS assisted proofs in Danish upper secondary mathematics textbooks as a result of the 2005 reform that introduced CAS as a part of the upper secondary level curriculum (and examinations). Based on a reading of 33 upper secondary school mathematics textbooks, 38 instances of CAS assisted proofs are identified in ten different textbooks. The CAS based proofs in these textbooks are of three types: complete outsourcing of the proof to CAS; partial outsourcing of the proof to CAS; and additional verification of the proof’ correctness by CAS. Analyses of examples of each of these types are provided. The analyses draw on theoretical constructs related to both proofs and proving (e.g. proof schemes) and to use of digital technologies in mathematics education (lever potential, blackboxing, instrumental genesis). In particular, the analyses make use of a distinction between epistemic, pragmatic and justificational mediations. Results suggest both potential problems with using CAS as an integrated part of deductive mathematical proofs in textbooks, since it appears to promote undesired proof schemes with the students, and difficulties with understanding these problems using the constructs of epistemic and pragmatic mediations that are often adopted in the literature regarding CAS use in mathematics teaching and learning.

Downloads

Download data is not yet available.

References

Appel, K. & Haken, W. (1977). Solution of the four color map problem. Scientific American, 237(4), 108-121.

Google Scholar Crossref

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 Computers for Mathematical Learning, 7(3), 245–274.

Google Scholar Crossref

Artigue, M. (2010). The future of teaching and learning mathematics with digital technologies. In: C. Hoyles and J.-B. Lagrange (Eds.), Mathematics Education and Technology-Rethinking the Terrain (pp. 463–475). New York, USA: Springer.

Google Scholar Crossref

Author 2 & Author 1 (in press).

Google Scholar Crossref

Borwein, J. (2005). The experimental mathematician: the pleasure of discovery and the role of the proof. International Journal of Computers for Mathematical Learning, 10, 75-108.

Google Scholar Crossref

Buchberger, B. (2002). Computer algebra: the end of mathematics?. ACM SIGSAM Bulletin, 36(1), 3-9.

Google Scholar Crossref

Carstensen, J., Frandsen, J. & Studsgaard, J. (2007). Mat A3, stx. Aarhus: Systime.

Google Scholar Crossref

Carstensen, J., Frandsen, J. & Studsgaard, J. (2009). Mat C til B, stx. Aarhus: Systime.

Google Scholar Crossref

Clausen, F., Schomacker, G. & Tolnø, J. (2007). Gyldendals Gymnasiematematik. Grundbog A. København: Gyldendalske Boghandel, Nordisk Forlag A/S.

Google Scholar Crossref

Dana-Picard, T. (2005). Some reflections on CAS assisted proofs of theorems. International Journal for Technology in Mathematics Education, 12(4), 165-171.

Google Scholar Crossref

De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.

Google Scholar Crossref

Dreyfus, T. (1994). The role of cognitive tools in mathematics education. In: R. Biehler, R. W. Scholz, R., Strässer and B. Winkelmann (Eds.), Didactics of Mathematics as a Scientific Discipline (pp. 201–211). Dordrecht, The Netherlands: Kluwer.

Google Scholar Crossref

Dreyfus, T. (1999). Why Johnny can’t prove. Educational Studies in Mathematics, 38(1-3), 85–109.

Google Scholar Crossref

Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In: P. Boero (Ed.) Theorems in school: From history, epistemology and cognition to classroom practice (pp. 137-161). Rotterdam: Sense Publishers.

Google Scholar Crossref

Education Committee of the EMS (2011). Do theorems admit exceptions? Solid findings in mathematics education on empirical proof schemes. Newsletter of the European Mathematical Society, 82, 50-53.

Google Scholar Crossref

Flynn, P. & MacCrae, B. (2001). Issues in assessing the impact of CAS on mathematics examinations. Proceedings of 24th Conference of the Mathematics Education Research Group of Australasia (pp. 222-230). Retrieved from:

Google Scholar Crossref

https://www.merga.net.au/documents/RR_Flynn&McCrae.pdf

Google Scholar Crossref

Hanna, G. (1989). Proofs that prove and proofs that explain. In: G. Vergnaud, J. Rogalski and M. Artigue (Eds.), Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 45-51). Paris: CNRS.

Google Scholar Crossref

Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.

Google Scholar Crossref

Harel, G. & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In: F. K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 805-842). Charlotte, NC: Information Age Publishing.

Google Scholar Crossref

Hoyles, C. (2014). Solid findings in mathematics education: the influence of the use of digital technology on the teaching and learning of mathematics in schools. Newsletter of the European Mathematical Society, 91, 49-51.

Google Scholar Crossref

Hoyles, C., & Lagrange, J. B. (2010). Mathematics education and technology: Rethinking the terrain. Berlin: Springer.

Google Scholar Crossref

Jankvist, U. T. & Misfeldt, M. (2015). CAS-induced difficulties in learning mathematics? For the Learning of Mathematics, 35(1), 15-20.

Google Scholar Crossref

Jankvist, U. T., Misfeldt, M. & Marcussen, A. (2016). The didactical contract surrounding CAS when changing teachers in the classroom. REDIMAT Journal of Research in Mathematics Education, 5(3), 263-286.

Google Scholar Crossref

Johansen, M. W. & Misfeldt, M. (2016). Computers as a Source of A Posteriori Knowledge in Mathematics. International Studies in the Philosophy of Science, 30(2), 111-127.

Google Scholar Crossref

Laborde, C. & Strässer, R. (2010). Place and use of new technology in the teaching of mathematics: ICMI activities in the past 25 years. ZDM - International Journal on Mathematics Education, 42(1), 121–133.

Google Scholar Crossref

Lagrange, J. (2005). Using symbolic calculators to study mathematics: the case of tasks and techniques. In: D. Guin, K. Ruthven and L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 113–135). New York, NY: Springer.

Google Scholar Crossref

Lavicza, Z. (2010). Integrating technology into mathematics teaching at the university level. ZDM - International Journal on Mathematics Education, 42(1),105–119.

Google Scholar Crossref

MacAogáin, E. (2000). Assessment in the CAS age: An Irish perspective. Paper presented at the 6th ACDCA Summer Academy, 2000.

Google Scholar Crossref

McEvoy, M. (2008). The epistemological status of computer-assisted proofs; Philosophia Mathematica, 16(3), 374–387.

Google Scholar Crossref

Misfeldt, M. & Zacho, L. (2016). Supporting primary-level mathematics teachers’ collaboration in designing and using technology-based scenarios. Journal of Mathematics Teacher Education, 19(2-3), 227-241.

Google Scholar Crossref

Nabb, K. A. (2010) CAS as a restructuring tool in mathematics education. In: P. Bogacki (Ed.) Electronic Proceedings of the 22nd International Conference on Technology in Collegiate Mathematics (pp. 247-259). Retrieved from: http://archives.math.utk.edu/ICTCM/v22.html

Google Scholar Crossref

Niss, M. (1999). Aspects of the nature and state of research in mathematics education. Educational Studies in Mathematics, 40(1), 1-24.

Google Scholar Crossref

Ottesen, S. T. (2009). Relating university mathematics teaching practices and students’ solution processes. PhD Thesis. Tekster fra IMFUFA, no. 463. Roskilde: Roskilde University. Retrieved from: http://milne.ruc.dk/imfufatekster/pdf/463.pdf

Google Scholar Crossref

Rabardel, P. & Bourmaud, G. (2003). From computer to instrument system: a developmental perspective. Interacting with Computers, 15(5), 665–691.

Google Scholar Crossref

Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135-151.

Google Scholar Crossref

Trouche, L. (2005). Instrumental genesis, individual and social aspects. In: D. Guin, K. Ruthven and L. Trouche (Eds.), The Didactical Challenge of Symbolic Calculators: Turning a Computational Device into a Mathematical Instrument (pp. 197–230). New York, NY: Springer.

Google Scholar Crossref

Tymoczko, T. (1979). Four-color problem and its philosophical significance. Journal of Philosophy, Inc., 76(2), 57–83.

Google Scholar Crossref

UVM (2013). Bekendtgørelse om uddannelsen til studentereksamen. Matematik A, stx, bilag 35. København: Undervisningsministeriet (UVM).

Google Scholar Crossref

https://www.retsinformation.dk/Forms/R0710.aspx?id=152507#Bil35

Google Scholar Crossref

Verillon, P. & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77-101.

Google Scholar Crossref

Vinner, S. (2007). Mathematics education: procedures, rituals and man’s search for meaning. Journal of Mathematical Behavior, 26, 1–10.

Google Scholar Crossref

Winsløw, C. (2003). Semiotic and discursive variables in cas-based didactical engineering. Educational Studies in Mathematics, 52(3), 271-188.

Google Scholar Crossref

Downloads

Published

2019-10-24

Almetric

Dimensions

Issue

Section

Articles