In <Pine.NEB.email@example.com>, on 01/09/2013 at 07:21 PM, William Elliot <firstname.lastname@example.org> said:
>On Mon, 7 Jan 2013, Dan Christensen wrote:
>> Studies have shown that proof-writing skills learned in one >> branch of mathematics such as geometry may not be easily >> transferred to other branches such as abstract algebra and >> analysis.
I'm always suspicious of such paraphrases. I wonder exactly what it is that studies have shown. I suspect that they were not tracking students moving from a rigorous presentation of one branch of Mathematics to a rigorous presentation of a different branch.
>> F. A. Ersoz (2009) suggests that the many informal "axioms" of >> Euclidean geometry, as usually taught, are based largely on >> personal intuition and imagination (p. 163). While this may >> serve as a productive basis for some discussion, it can blur >> the boundary between the formal and informal, and lead to >> confusion as to what constitutes a legitimate proof in other >> domains (branches) of mathematics.
>Personal intuition isn't needed for algebra?
That's not how I read it. He seems to be saying that intuition without rigor leads to confusion, not that intuition plus rigor does.
>> Ersoz also suggests that introductory geometry courses seldom >> present many of the methods of proof used in more abstract >> courses öÇ methods like proofs by induction, contrapositive >> or contradiction (p. 164). >> http://220.127.116.11/~icmi19/files/Volume_1.pdf
>Yes, geometry is mostly constructive while algebra or caluulus is >an opportunity to extend logically skills learned in Eucidean >geometry.
Even The Elements used more tools than just construction, while Algebra definitely relies on construction in various proofs. As for Calculus, it's usually taught as an Engineering course, even though it is not labelled as such.
I'd love to see high schools using Hilbert's Grundlagen der Geometrie as the text for Geometry, and then studying how the graduates adapt to reasoning in Algebra.
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