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Random Musings

On this page you will find random musings on various aspects of mathematics. There is not set schedule for these thoughts. Keep coming back here often. I hope that you find it interesting.
Wojciech Komornicki
Chair, Department of Mathematics

Vol 7: Getting Ready for a new semester

It is Sunday, the first of September 2013. Tomorrow is Labor Day and classes at Hamline start in 3 days. The summer has been busy and rewarding. Four students

• Yutong Bao
• Josiah Biernat
• Yu Chen
• David Schmitz
worked for ten weeks investigating the geometry of two-dimensional surfaces and homological algebra. Some of them had never heard of these concepts but towards the end of the summer they could tell you with confidence about exact sequences and derived modules and Tor and Hom and all sorts of other things. Some of these students are now starting only their second year of college education. Most are planning to continue their investigations since they realize that they have only scratched the surface (no punn intended.).

I should add that the work of these students for this summer is not yet finnished since they are now busy writing up what they had done and what they have learned. These reports will be posted on thse pages as soon as they are available.

But that is the past. Every semester as I get ready for classes I forget all the frustrations of the past year, all the blunders I made, all the self-recrimination that something in a class could have been done better and look forward with enthusiasm for the start of classes. I will be teaching two sections of first semester Calculus and a section of Mathematical Analysis. (My favorite definition of Mathematical Analysis is that it is Calculus done from a mathematical point of view.) In the mathematical community -- this means faculty at various institutions -- there has been a lot of discussion of what we should teach in a first Calculus course and of course how we should teach it. Is Calculus just about computing derivatives and integrals. To put this in another context, one could design a course in which students learn to be perfect spellers. Graduates of this course would outshine all the winners of the National Spelling Bee. What if these perfect spellers could not define most of the words that they could spell? To come back to mathematics, should a student, after a first Calculus course, know that \[ \mathbf{e}^{\mathbf{i}\pi} = -1 \] Before we look at the question, let's look at this formula. In one simple formula we incorporate four of the most important numbers (whatever that means) in mathematics: \( \mathbf{e}, \mathbf{i}, \pi \text{ and } -1 \). This is beautiful -- and yes it has something to to with calculus. And so I say yes, someone who has a Calculus course under their belt should understand this formula. It has much more to do with understanding the concepts of calculus rather then just doing the calculations.

So the question that I have for myself, and for my fellow mathematicians, what do we have to do in our courses to convey understanding and not just calculation.


Past Musings:
18 May 2013 | 15 May 2012 | 15 Apr 2012
21 Mar 2012 | 27 Jan 2012 | 16 Jan 2012