
Re: What difficulties encountered by beginners in abstract algebra?
Posted:
Jan 20, 2013 2:39 AM


In article <slrnkf0p8c.ovq.hrubin@skew.stat.purdue.edu>, Herman Rubin <hrubin@skew.stat.purdue.edu> wrote:
> On 20130110, Michael Press <rubrum@pacbell.net> wrote: > > In article <c546d56db2a1417dbaa67e1e2721244e@googlegroups.com>, > > Dan Christensen <Dan_Christensen@sympatico.ca> wrote: > > >> On Monday, January 7, 2013 4:48:55 AM UTC5, William Elliot wrote: > >> > On Sun, 6 Jan 2013, porky_pig_jr@mydeja.com wrote: > > >> > > On Sunday, January 6, 2013 11:46:37 PM UTC5, Dan Christensen wrote: > > > > >> > > > I am working on some introductory notes for group theory. What > > >> > > > difficulties are typically encountered by math or science undergrads > > >> > > > in an introductory course on abstract algebra? > > > > >> > > The same kind of difficulties as moving from Calculus to > Analysis. Need > > >> > > set theory, need some intro to logic and proofs. > > > > >> > What do you mean need an introduction to logic and proofs? > > >> > I learned logic and proofs during my high school sophomore year > > >> > in the Euclidean geometry class. Where are they these days? > > > >> Studies have shown that proofwriting skills learned in one branch > of mathematics such as geometry may not be easily transferred to other > branches such as abstract algebra and analysis. > > >> 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. > > >> Ersoz also suggests that introductory geometry courses seldom present > many of the methods of proof used in more abstract courses Ñü methods > like proofs by induction, contrapositive or contradiction (p. 164). > http://140.122.140.1/~icmi19/files/Volume_1.pdf > > I agree that proofs by induction are not present in the > classical Euclidean geometry course. However, contrapossitive > and contradiction are present, although not to a large extent. > > The real problem is that many students do not even have an > opportunity to take a prooftype geometry course, and I have > been told by many colleagues that even the "college algebra" > taught in universities may well not cover induction. Nobody > who does not understand induction understands the integers; > it belongs in first grade, and to be used afterward. > > Here is how algebra can and should be taught early, and then > USED: > > A variable is a temporary name for something. > The same operation on equal erntities gives equal results. > > The rest is application of this. > > > This fails to mention that the proof writing skills > > learned in plane geometry are 100% transferrable to > > more advanced subjects. _Of course_ plane geometry > > takes some liberties, but they are _warranted_ > > liberties just as in all teaching a little bit of > > liberty with the way the subject actually works is > > warranted by the student getting a good start on it. > > Plane geometry texts have been written with these liberties > removed. But it makes little difference; learning how to > compute answers is of little value in understanding mathematics, > or just about anything else. Machines can do that well.
We do not seem to be talking about the same thing. In my second to last year of secondary school I took a plane geometry course in which we wrote proofs from the very start. This does not seem to be what you are talking about. What I learned in that class writing proofs all translated to proofs in further studies.
 Michael Press

