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.
Highlights
This article addresses the didactical effects of Computer Algebra System (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
In the third example CAS does not enter into the actual proof, but only when this is complete as a means for “checking” the correctness of the already proven mathematical result
Regarding the cases of very classical mathematics, as we have seen in the CAS assisted proofs from upper secondary school textbooks, it might not be a productive uncertainty to foster in the mind of the students
Summary
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). A s part of a reform of the Danish upper secondary school in 2005, CAS entered in at all levels of the upper secondary mathematics program It entered into the final written national examinations, into the classroom teaching and into the textbooks. An instrument for underpinning conceptual understanding, the actual implementation of CAS into the mathematics program was pretty much left up to the schools, the teachers, and not least the textbook authors. This left the textbook authors of more than one textbook system to invent the notion of “CAS proofs”1 - or “CAS assisted proofs” as termed by Dana-Picard (2005). Neither seems to be the case in the example above, where no means for reasoning and REDIMAT 8(3)
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