Mathematical proof - Almeida, 2000, 2003

[torquetum]

 Almeida, D. (2000). A survey of mathematics undergraduates’ interaction with proof: some implications for mathematics education. International Journal of Mathematics Education in Science and Technology, 31(6), 869-890.

 

The author ran a large (N = 473) quantitative study of students’ perceptions of proof and then a smaller qualitative sub-study of proof perceptions and practices. He lists several tensions: between novice and professional, between informality and formality, between state of becoming and state of being. “From an educational point of view the prototypical proof practices of beginning undergraduates, despite being imperfect, ought to be initially accommodated simply because the learner believes in their veracity—for the student it is a proof. A tension also arises in the process of convincing the student of the necessity of making the transition to formal proof” (p. 870). Hmm, reference to Ruthven and Coe (1994). I should dig out work by these two if I do a study of my own; it looks intriguing. This study has a nicely described methodology. The qualitative part included students judging the efficacy of proofs across a range of representations, such as visual or abstract. There were also interviews. There is lots of detail on the qualitative stuff, I won’t summarise it here. One point: he finds the 2nd year students have a deeper understanding of proof than the 1st years, which is an encouraging result. The qualitative results show that most students have poor proof perceptions and practices despite apparent agreement with the principles encountered in the questionnaire. The author speculates that at least part of the problem is the difference between how maths is taught and how original mathematics is constructed. The author closes with a suggestions related to development of concept images. The paper has a detailed Appendix which could be of use of one wanted to replicate the study.

 

 

Almeida, D. (2003). Engendering proof attitudes: can the genesis of mathematical knowledge teach us anything? International Journal of Mathematics Education in Science and Technology, 34(4), 479-488.

 

The author “reports on how proof attitudes could be inculcated in students by offering them a course design that is faithful, to some extent, to the historical genesis of modern mathematics.” (p. 479). As some of the papers already discussed, the author gives a brief rundown on the history of proof. “If the sequence for understanding mathematics [intuition, trial, error, speculation, conjecture, proof ] as espoused by MacLane also applies in mathematics learning as well as in research and if the theory of levels in proving is accepted then there are implications for proof in mathematics education” (p. 482). The author describes a study where the students manipulated descrined shapes in a computer package and had to come up with a conjecture based on their observations and then prove the conjecture. The results were interesting, with students coming up with conjectures and providing proofs across a range of formality. Interestingly, the students displayed positive proof perceptions and practices.

 

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.