Joe Niederberger posted Dec 30, 2013 12:31 AM (http://mathforum.org/kb/message.jspa?messageID=9352905) - GSC's remarks follow: > > >A transitional course between school math and math > major math shouldn?t be about 10th grade logic. It > should be about analysis. > > Well, where I studied they had a 200 or 400 level > course called "advanced calculus" where you learn > calculus "all over again" (repeating your phrase!) > but this time in that mathematical "axiom, > definition, theorem, proof" style. If you hadn't been > exposed to that style before, you might be > overwhelmed. > > If Devlin's book could help "remediate" a lack of > having had a prior introduction to logic, (say in > 10th grade geometry, which most don't get these days) > then I'm all for it. I just find his avoidance of the > word "logic" interesting. I really do believe it has > something to do with his sense of marketing tactics. > > >After this course, if you have a genuine interest, > you would have to take a genuine introduction to > logic course, all over again, with all the necessary > development that this one lacks. > > Yes, such as that "advanced calculus" course. > > Here's a link (not to the text I used so long ago, > but jut to give n example): > http://www.math.harvard.edu/~shlomo/docs/Advanced_Calc > ulus.pdf > > Note how they start right out with a review of > mathematical logic. Its a pretty brisk intro though. > > Cheers, > Joe N > I've quickly glanced through a few pages of an e-copy of the "Advanced Calculus" book by Lynn H. Loomis and Shlomo Sternberg (downloaded from http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf). It has been a very frustrating exercise indeed! (Because of my slow Internet connection, I have to wait practically till the cows come home to get pages downloaded).
However: the book appears to be entirely sound, covers much the same ground as the book(s) we used for our calculus courses when I did engineering (years 2 and 3, if I recall in the 4-year engineering course. It was, in fact, I think my Year 3 Calculus course, taught by a difficult and VERY demanding - but truly brilliant - teacher (an excellent mathematician), Professor K. Venkatachalaiah Iyengar, that truly 'turned me onto' math).
I observe that I was already pretty good at math, and I found Venkatachalaiah Iyengar's teaching to be most exciting. A great many students found him way too difficult, and he'd make few or no concessions for them. Though I then found KVI's course to be excellent - just what I was looking for - I now believe (after having got myself some understanding of 'systems') that it was NOT the right way of teaching those engineering students. Robert Hansen would doubtless disagree, but I'm pretty certain that such courses have first of all to be aimed at the average student, one who may find many difficulties in math. The students with sound skills in math may be given 'special courses', such as those given by KVI.
I believe considerable further development is required of all such texts (and all such teaching), to enable (and ensure that) students are more *effectively* guided through such learning. All such books are, I claim, only at a very early stage of development as 'tools to aid learning' (of any discipline). There is a revolution yet to come in the way we transmit knowledge, enable understanding.
I observe that Professor Richard Hake has just asked the question "Can the Cognitive Impact of Calculus Courses Be Enhanced?" (and he has provided links to an interesting (and useful) 80-page paper with just that title. Without having studied via any detailed 'modeling' of the arguments in that paper, I believe he has NOT yet shown the way by which "the cognitive impact of calculus courses" may be enhanced - but I claim the answer is very definitely yes. For that matter, I don't believe Devlin has the needed answers either.
Note to Professor Bishop, in case he's reading this: Yes, the OPMS can help teachers and writers discover just how to "enhance the cognitive Impact of Calculus, or any other discipline for that matter.
I claim it would not be inordinately difficult to take up just such an aim I.e., actually "To enhance the cognitive impact of calculus [/or algebra; or geometry] courses".
What is required (and what is in general lacking today) is to handle the issue by exploring the 'learning + teaching dyad' in the specific respect of the discipline in question. Currently, most of these ventures are - despite all good intentions - only looking at it as 'teaching-in-itself'.
The 'good intentions' are amply demonstrated in the examples provided and the exercises/problems set (many of them excellent) - but no such course and no such book that I've seen has yet handled any such venture as a truly interactive collaboration between student(s)/ reader(s) and teacher (/or writer of book): every such exercise should be, in fact, a 'learning+teaching venture'.
The modeling tools that I discuss in the 'One Page Management System' (OPMS) can help develop the interactive nature of the 'learning+teaching venture'. Of course, it will be necessary to go somewhat beyond the 'model-meaning' of the transitive relationship "PRECEDES" in order to accomplish the above.
[If I know him right, Robert Hansen (RH) will now have something to say about OPMS being an "empty list-creating product", or words to that effect: this is a falsehood. Those who wish to examine the reality/ investigate the truth may check out the attachments to my post heading the thread "Democracy: how to achieve it?" - see http://mathforum.org/kb/thread.jspa?threadID=2419536. The prototype OPMS software is FREELY available on demand - contrary to the falsehood propagated by Robert Hansen that I have "solicited funds" for this software from people who have wanted to use it].