College Math Teaching

August 21, 2015

Just shoot me. Now. (personal…and my screw up…)

The upcoming semester line up:

1. My original schedule called for science/engineering calculus II, “business calculus” I, and numerical analysis. No, my specialty is pure math (topology) but I got roped into teaching this course a few years ago, and because no one complained…guess who is stuck with it now? 🙂 (and yes, as an undergraduate, and once as a part time graduate student, I made C’s in this class)

2. But a part time faculty who taught our actuarial mathematics classes got called away so..I said “if you can’t find anyone else…” and so I got stuck with that class (in lieu of the “business calculus” class)

3. I get an e-mail about the class; I read the Fall 2014 syllabus and so prepare based on that (since it is Fall 2015)..but the topics for that class go “spring, fall, off, fall, spring” instead of “fall/spring”…so I had prepared FOR THE WRONG CLASS and ordered THE WRONG BOOK.

I caught that a week prior to classes starting..hence frantic e-mail to the department chair and secretary….and I’ll have to really work some this weekend.

Fortunately much of the stuff in this topic (“life contingencies”) is like reliability engineering and I’ve had that class. Things like the “survival function” and “bathtub curve” are familiar to me. The mathematics won’t be hard; I’ll have to focus my self study on definitions and notation.

Still, this is more interesting than I’d hoped that it would be.

The positive: I’ll have learned some new mathematics (I always learn something new every time I teach numerical analysis) and new applications of mathematics (the life contingencies …and I’ve already learned a little bit of interest theory by preparing for the class that I thought that I was teaching..)

The university
But our university (6000 students, 5000 undergraduates) is suffering from an enrollment slump for the second year in a row. We are at about 85-88 percent of what would be a “healthy” enrollment. The place is in turmoil; we lost our athletic director, provost (left) and president, the flagship basketball team has hit rock bottom and things are in disarray.

So we have a second “small” class but this time: enrollments are UP in our remedial sections. UP. And many who couldn’t place into our regular calculus sequence have been admitted…by …engineering. Seriously. They are hurting that badly, and when they hurt, WE hurt.

I’ll be shielded from much of that in the classroom because of the classes I am teaching BUT with these changes come “changes in major”; we are going to try to make our major easier to navigate by trying to maximize flexibility by making our “required courses” less prerequisite dependent. It will water down the major somewhat but hopefully make it more likely that we keep a major.

Oh well…this is what I get for taking my Ph. D. in pure mathematics instead of applied. 🙂

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August 1, 2015

Interest Theory: discounting

Filed under: applied mathematics, elementary mathematics — Tags: , — collegemathteaching @ 10:29 pm

Some time ago, I served in the U. S. Navy. The world “Navy” was said to be an acronym for Never Again Volunteer Yourself. But I forgot that and volunteered to teach a class on Mathematical interest theory. That means, of course, I have to learn some of this, and so I am going over a classic text and doing the homework.

The math itself is pretty simple, but some of the concepts seem strange to me at this time. So, I’ll be using this as “self study” prior to the start of the semester, and perhaps I’ll put more notes up as I go along.

By the way, if you are interested in the notes for my undergraduate topology class, you can find them here.

Discounting: concepts, etc. (from this text) (Kellison)

Initial concept:

Suppose you borrow 100 dollars for one year at 8 percent interest. So at time 0 you have 100 dollars and at time 1, you pay back 100 + (100)(.08) = 108.
Now let’s do something similar via “discounting”. The contract is for 100 dollars and the rate is an 8 percent discount. The bank takes their 8 percent AT THE START and you end up with 92 dollars at time zero and pay back 100 at time 1.

So the difference is: in interest, the interest is paid upon pay back, and so the amount function is: $A(t) = (1+it)A(0)$. In the discount situation we have $A(1)(1-d(1)) = A(0)$ where $d$ is the discount rate. So the amount function is $A(t) = \frac{A(0)}{1-dt}$ where $t \in [0, \frac{1}{d})$

If we used compound interest, we’d have $A(t) = (1+i)^tA(0)$ and in compound discount we’d have $A(t) = \frac{A(0)}{(1-d)^t}$

This leads to some interesting concepts.

First of all, there is the “equivalence concept”. Think about the above example: if getting 92 dollars now lead to 100 dollars after one period, what interest rate would that be? Of course it would be $\frac{8}{92} = .087$. So what we’d have is this: $i = \frac{d}{1-d}$ or $d = \frac{i}{1+i}$.

Effective rates: this is only of interest in the “simple interest” or “simple discount” situation.

Let’s start with simple interest. The amount function is of the form $A(t) = (1 +it)A(0)$. The idea is that if you invest, say, 100 dollars earning, say, 5 percent simple interest (NO compounding), then in one year you get 5 dollars of interest, 2 years, 10 dollars of interest, 3 years 15 dollars of interest, etc. You can see the problem here; say at the end of year one your account was worth 105 dollars and at the end of year 2, it was worth 110 dollars. So, in effect, your 105 dollars earned 5 dollars interest in the second year. Effectively, you earned a lower rate in year 2. It got worse in year 3 (110 earned only 5 dollars).

So the EFFECTIVE INTEREST in period $n$ is $\frac{A(n) - A(n-1)}{A(n-1)} = \frac{1 + ni)-(1+(n-1)i)}{1+(n-1)i}=\frac{i}{1+(n-1)i}$ which you can see goes to zero as $n$ goes to infinity.

Effective discount works in a similar manner, though we divide by the amount at the end of the period, rather than the beginning of it:

$\frac{A(n)-A(n-1)}{A(n)} = \frac{\frac{1}{1-nd} - \frac{1}{1-(n-1)d}}{\frac{1}{1-nd}} = \frac{d}{1-(n-1)d}$