The title of this post is part of a quote from Mark Saul, current editor (June 2018) of the American Mathematical Society blog on mathematics education:
“… it is important for us to understand that the language of mathematics is a language of thought. And that thought is synonymous with intuitive thought. “
Mark’s post fits well with our focus on understanding actuarial formulas so as to set up patterns of thought. And those patterns of thought are based on, and help refine, our intuitive thought.
Let’s think about one of the simplest formulas in the time value of money: the compound interest formula.
The basic, and intuitive, idea of compound interest is that money is initially deposited in an interest-bearing account, and at certain fixed time intervals – for example, every month – the amount in the account is updated by adding interest earned, at a specified rate, over the previous time interval.
Usually for convenience these time intervals are described in terms of a year: annually, (every 12 months), semi-annually (every 6 months ), quarterly (every 3 months), monthly (every month), or in some similar way (weekly, or daily, for example).
This is a simple pattern of thought that forms the basis of compound interest.
Stating this simple pattern of thought in mathematical language is just a way of allowing us, or a calculator or computer, to do arithmetic on the names of things in these thought patterns.
For example, suppose we earn interest at a nominal annual rate of 8 %, compounded semi-annually.
What this means is that at the end each 6-month period our account is credited with interest at the rate of 8/2 % = 4%.
Let’s deposit $200 in our account and try to calculate how much we have in our account at the end of each year.
We can turn this language of thought patterns into mathematical language by, for example, denoting the amount in our account t years after our initial deposit of $200 by A(t).
We know that A(0) = 200 because we initially deposited $200 – that’s just the mathematical language way of expressing our intuitive understanding of the situation.
We also know that after 6 months we will earn interest on our initial $200 at the rate of 4%.
So after 6 months we will have our initial $200 plus another $200 × 4/100 = $8 giving us a total of $200 + 200 × 4/100 = $208.
At the end of the second 6 month period we will have earned interest not only on our original $200 deposit but also on the interest earned on that in the first 6 months: in other words, we will earn the following amount of interest at the end of 12 months:
giving us a total of
We’ve written this out in full, without initially simplifying by approximating as we go, because we want the mathematical language to fully reflect our thought patterns.
Using our name A(t) for the amount in our account after t years, we can follow that thought pattern above to see that
Surprisingly at first glance, it is actually easier to see how the mathematical language reflects our patterns of thought if we become more general and less concrete:
Suppose instead of $200 to open an account we just have an amount P (for “principal”), and suppose our nominal annual rate of interest is r (for instance with an nominal annual interest rate of 8% , r= 8/100= 0.08).
After 6 months our account has grown to:
This now becomes our account balance at the beginning of the next 6 months, so at the end of the second 6 month period our account has grown to:
If we use the same name A(t) to denote our account balance at the end of t years then A(0)=P and
The same reasoning – the same pattern of thought – tells us that
and – from this line of reasoning repeated – we can see that
We leave it to you to figure out how to alter this formula – this mathematical language way of expressing our intuitive and refined patterns of thought – when interest is compounded more than twice a year.
– The language of mathematics is a language of thought –
Gary Davis is a mathematics educator and mathematician with a Ph.D. in Mathematics from Monash University, Australia. He was instrumental in establishing the Data Science program at the University of Massachusetts Dartmouth and has periodically run a course in Financial Mathematics He has been the Boeing Distinguished Professor of Mathematics Education at Washington State University, Visiting Professor of Mathematics Education at Rutgers University in New Jersey, Visiting Professor in the Mathematics Education Research Centre, University of Warwick, UK, Professor of Education at the University of Southampton, UK, and Director of the Institute for Mathematics Education, LaTrobe University, Melbourne, Australia.
Gary lives in Massachusetts, USA, with wife Linda, dog Daisy and cat Bella.