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.
#### Vol 7: Getting Ready for a new semester

**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.).

Wojciech Komornicki

Chair, Department of Mathematics

Chair, Department of Mathematics

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 • Josiah Biernat

• Yu Chen

• David Schmitz

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.