calculator

The Calculator

When I teach a finance unit in an introductory math course, I have to introduce the idea of compounded interest and from there I talk about annuities and mortgages and so forth. Part of the purpose of the finance unit is to develop some financial literacy:

  • what goes into a home purchase?
  • what is an annuity?
  • what is a retirement plan?
  • what is a stock?
  • what is a bond?
  • how do you buy stocks and bonds?
  • what is an interest rate?
  • what happens when we don’t pay our credit card balance in full?
  • how much per year is a $20/hr job?
  • will a full-time job at $20/hr be enough to support the lifestyle that one wants?
  • what kind of lifestyle changes do we have to make if our salary cannot support the lifestyle that we want?
  • how do we know if the car loan the dealer is proposing is good?
  • what kind of an impact will an extra $\(N\) have on the repayment period of a mortgage?
  • what is PMI?
  • what is a down payment?
  • how do taxes work?
  • what is meant by “risk-to-reward”?

Some of the questions above are simple facts that we can look up for a cursory understanding of the topics. Once we’ve been able to agree on the vocabulary, we start discussing the concepts and principles underlying personal financial planning. Eventually, we move from general discussion to application.

Application!

One of the universal criticisms about what is lacking in math education! “Why do I gotta learn this, if I’m never gonna use it?” Well here it is. Personal finance. But of course, we have a problem with the marketing, semantics, and the classification of what is considered Mathematics. This isn’t math, this is finance. Pshaw. It’s math in a finance context. If we are applying mathematics, then more often than not there is an entire topic in a given field of study named around that application of mathematics. If we don’t want to accept that this type of application is mathematics, then we have to let mathematics stands on its own and the complaints about the lack of “real world” applications in math education have to stop.

Why am I droning on about context and math education with a title like “The Calculator”? It comes down to this: in the applied versions of mathematics, (numerical) calculation is almost an eventuality when applying theory (that theory can be computational theory, for example). However, the calculation itself is viewed as a simple mechanical task not worth expending copious hours doing by hand, especially today, given the ubiquity of computing tools. This mentality tends to trickle down to basic (applied) math courses like finance. From here, there is a lateral cry towards the non-applied but elementary math courses, giving us complaints from adults to the tune of, “Why do kids have to memorize the times tables? In my job, I’ve never had to mentally compute anything, I just put in my spreadsheet.” I’m willing to grant meaningful debate with respect to how kids are forced to memorize certain facts, but I don’t see any meaningful argument for why something as elementary as knowing at least the single digit times tables is an awful task for kids to have to do. Arguably, it is, at minimum, an eight by eight grid of “facts”.

I don’t hear many complaints about having to memorize the alphabet in the order in which it is taught. After all, when is the last time I’ve actually had to alphabetize anything by hand? I just use Python’s sort command, for example. And who uses a paper dictionary any more? Just type in “define: ” followed by your word into Google and you’ll be given a definition. Maybe I’d have to leaf through a book’s index if that book is actually on paper, but even then the list of words to leaf through isn’t large and they are already clumped together.

We can argue that the alphabet is “different” from numbers because we use letters more often than we use numbers. But I think this is actually a mistake in classification. The English alphabet, for example, should be ['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z']. If you’re curious, I just typed the following into Python:


>>> x = list("qwertyuiopasdfghjklzxcvbnm1234567890")
>>> x.sort()
>>> x
['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z']

(See! I didn’t have to do any alphabetizing. Python did it all for me!)

And much the same way that we have to learn phonics, so must we have to learn the times tables, addition. subtraction, and long division.

This brings me to the calculator and the general question about whether there is a necessity today for any hand calculation whatsoever. A basic scientific calculator can do all the laborious calculation for us and few “real world” calculations require an urgency in calculation for something as simple as \(6 \times 8\), so why bother learning those? Why not focus on “concepts”?

While these are meaningful questions, the conversation quickly becomes polarized with an “either this or that” view. I yield to the argument that students are not really taught the ideas and fundamental principles underlying the mechanics and some of the mathematical objects they are asked to work with. But this doesn’t mean that we abandon the mechanics wholesale and teach only “concepts”. The concepts, especially, in the elementary math coursework, are not so grandiose and complicated that they require months of explanation. The concepts are relatively simple. The hard part actually is getting used to the mechanics and learning to read mathematics. It’s a type of language acquisition. I discuss this more fully here.

Without further ado, let me show you what I see with college students — students who have gone through approximately ten years of math for the masses.

Basic Finance

After explaining the concepts of compounded interest, I ask students to start doing some calculation. The purpose of the computation is to apply the concepts and to give students a sense of number with respect to exponential growth. In a basic skills course, I don’t require that students memorize the formula or even be able to derive it, but rather understand how to use it. Forced memorization leads to math misery. And requiring students to derive the formula is, unfortunately, too tall an order for a low level, introductory course. Thus, students are given $$P(t) = P_{0}\Big(1 + \frac{r}{n}\Big)^{nt}$$ with \(P_{0}\) as the initial principal amount, \(r\) the interest rate, \(n\) the compounding frequency (eg, if monthly \(n = 12\)), and \(t\) time in years. This formula is about as basic as it can get if we want to meaningfully look at real world problems.

Now, a basic question:

If you have a $1000 credit card bill which you can’t pay, then if the annual interest rate on your credit card is 18% compounded monthly and assuming that the credit card doesn’t charge late or any other fees for non-payment, how much will you owe in one month? two months? one year? Also assume you make no other charges on the credit card.

For a question like this, it is unreasonable to require that students do hand calculations. (Though perhaps for the one month case, it can be done by hand.) Thus, bring out the calculator.

All that needs to be done is for students to correctly input \(P_{0} = 1000\), \(r = .18\), \(n = 12\), t = 1, 2, or 12 \(nt = 1, 2, \mbox{ or } 12\) into the formula for \(P(t)\). Thanks to a reader for pointing out the typo! (See comments)

But what happens?

  • Professor Shah, I got an error! — Yes, you have to press the multiplication button before the open parenthesis, the calculator won’t understand “1000(”
  • Dr. Shah, how do I do the numbers on the top? — Use the exponentiation functionality on your calculator. — Where is that?
  • Here’s my answer for one month: $1000.015 — Does one and a half penny of interest seem like a reasonable answer? — I guess so. That’s what I got when I plugged all the numbers in.
  • Here’s my answer for two months: $1000.00125 — So you owe less money after two months of not paying?
  • Here’s my answer for one year: 8.1735201026527e+23, but I got an “e”, is that an error? — Take a look at how you entered the numbers.

And the list goes on. I want to assign blame. Whose fault is this? These are college students! A different batch every semester! How is this possible? They don’t know basic arithmetic because the calculator exists, but then they go to use their calculator and get nonsense answers! Or worse yet, they have the calculator and have no idea how to use it!

So that I don’t give a completely skewed picture of my classroom, I do acknowledge that I have students who do know how to use the calculator and who know basic arithmetic. I rarely have the student who knows how to use a calculator but doesn’t know basic arithmetic. I also rarely see the student who knows basic arithmetic but is unable to use the calculator. In other words, my class is bimodal in this respect: if they can use a calculator, then they also know basic arithmetic or if they don’t know basic arithmetic, they don’t know how to use a calculator.

So, whose fault is this anyway? Their math teachers over the years? The student him/herself for never taking control of their education? The mass education system for letting students skate through with barely a basic understanding of what is arguably useful mathematics? I blame everyone, including the parents. And blaming everyone makes me unpopular. But who cares. This is craziness! After almost ten years of going through what is, in my opinion, a reasonably good math curriculum, we regularly, in non-negligible number, and across the country (world?) have students in college with less than sixth grade proficiency in basic mathematics?

It’s one thing to get a nonsense answer, but it’s another thing to believe it hook, line, and sinker. Yup, the calculator said so. There are conceptual mistakes that students make with respect to applying the formula, but those are to be expected the first few times around with an unfamiliar subject. But accepting that the interest on a $1000 loan with an annual interest rate of 18% compounded monthly is one and a half pennies after one month? That’s a lack of understanding at a conceptual and at a mechanical level. Can I borrow money from them?

It’s not clear to me if knowing how to do pencil-paper arithmetic is a prerequisite for using the calculator, but in my experience it sure as heck makes a huge difference in understanding the tool. This applies to me, too. Before I sit down to program anything quantitative, I often work out a few simple test cases on paper. I do this because it helps to get a feel for how the calculations ought to work.

I can argue that using a calculator is a type of programming. The student has to know what a correct sequence of inputs must be in order to get a result that is inline with the intended calculation. It helps immensely if the student has done hand calculations enough to know how to use an automated tool.

All this was from a basic question about compounded interest. When they are asked to figure out the initial principal balance on an investment growing at a fixed interest rate after a certain number of months, they have to “do Algebra” to obtain $$P_{0} = \frac{P(t)}{(1 + \frac{r}{n})^{nt}}$$

Ha! This one is hopeless given the numerous fractions involved.

It’s a funny type of arrogance that people hold with respect to what is “meaningful” mathematics. Arithmetic (and Algebra) is (are) foundational to more advanced mathematics. But somehow, our kids are too special to have to stoop to doing hand calculations. As if the calculation were a type of manual labor that were unworthy of their time.

At a fundamental level I have no qualms with calculators and their presence in a math classroom. But the calculator is a tool. It is not a replacement for knowledge and understanding. One of the benefits (side effects, if you want to call it that) of learning to do hand calculations and a little bit of mental arithmetic is that these things help to develop a basic sanity check when using the calculator. When \(1 + \frac{.18}{12}\) comes out as \(.0983333\ldots\) the student ought to know that something crazy just happened and not just say, “Welp, if that’s what the calculator says, that’s what the calculator says.”


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