Written by Len Asimow, head of the Department of Mathematics at Robert Morris University and founding Director of the Actuarial Science Program, currently designated a Center of Actuarial Excellence by the SOA:
Jared can complete a standard worksheet of old actuarial exam problems in 3 hours. Jamie takes 6 hours do a worksheet and a half. Working together, how long would it take for Jared and Jamie to complete 7 standard worksheets.
For 99% of humanity the correct answer is, “Who cares?” Nearly everyone recognizes that the infamous “work problem” of high-school algebra classes has no practical application in real life. Seemingly, its only purpose is to confound and discourage students, most of whom correctly conclude that they have much better things to fritter away their youth contemplating.
G.V. Ramanathan is a professor of mathematics and statistics, and also the author of a popular study guide for Actuarial Exam P/1. A few years ago he wrote a widely cited and highly controversial op-ed for the Washington Post in which he argues that the importance of math is wildly overblown. He writes, “How much math do you really need in everyday life? Ask yourself that – and also the next 10 people you meet, say your plumber, your lawyer, your grocer, your mechanic, your physician or even a math teacher. … Most adults have no contact with math at work, nor do they curl up with an algebra book for relaxation.”
From the Soviet launch of Sputnik in 1957 to the present day there has been an atmosphere of national crisis concerning the critical need to improve math education and ramp up the output of scientists and engineers. We’ve seen the New Math campaign, the famous Nation at Risk report, the New New Math, No Child Left Behind, Common Core, etc., etc. Granted, the near universal insistence that math is important and that the more the better, has been very good for university math faculties. As long as math is considered an indispensable tool for daily life math departments will be largely protected from shrinkage or, gasp!, elimination. But over all the years and with all the money spent there has been, if anything, an increase in general innumeracy.
The fast-food chain A&W once came out with a new burger to challenge the McDonald’s Quarter Pounder. The new burger had a third of a pound of beef, and was preferred in blind taste tests. However the roll-out failed badly. Turns out people couldn’t understand why they should pay the same for a 1/3 of a pound as they were paying at McDonald’s for a 1/4 of a pound. Go figure. Did A&W mobilize the education industry to improve the teaching of fractions and their relative sizes? Of course not. They simply accepted the inability of the general public to grasp a simple math concept and changed the marketing plan accordingly. Maybe it’s time for the educational establishment to do the same.
But then there is the relatively small subset of people who actually enjoy working on challenging math problems – no matter how impractical – and devising clever solutions. For those with the aptitude, background and fortitude to do math there are certainly great opportunities in the STEM fields. Among this group you are likely to find present and future actuaries, like yourselves. Did you benefit from working through artificial and unrealistic word problems in algebra class? Probably more than you realize, as we shall soon see.
Obviously, math skills are important in actuarial science. Passing exams requires a mastery of certain standard techniques learned in calculus and statistics classes. You must also be adept in translating prose (as in story problems) into symbols and equations so that those techniques can be brought to bear in finding solutions.
But acquiring math skills is different from learning mathematics. I’m speaking here of pure mathematics, that rarified world of abstract theorizing, precise definitions and theorems with rigorous proofs. The first exposure to this type of rigor is usually in high-school with Euclidean geometry. Unfortunately, that may also be the last exposure, even for actuaries.
Student actuaries can and do succeed without ever taking pure math courses such as real analysis and modern algebra. Still, someone on the level of an actuary should at least have an appreciation of the role that rigor and precision play in the tools that they use. Learning these things helps students to understand the theories and to see the big picture; it facilitates seeing patterns and making connections between seemingly disparate ideas. But, most students seem to confuse rigor with rigor mortis, finding it stultifying and deadly. That seems a shame since there is a great deal of beauty in the world of mathematics, but you must dig deep and ponder hard to appreciate it. It’s not like music or art where the beauty is readily apparent to the ear or eye.
Speaking of patterns and connections, let’s return to Jared and Jamie and see how they are getting along with their worksheets. Perhaps you recall that the trick to such problems is to convert the completion times into completion rates (in this case worksheet completion rates per hour) by taking reciprocals. It makes no sense to add the times, but rates are additive, at least under the assumption that Jared and Jamie work independently, and so forth. So now, let’s rephrase the question. Suppose that the completion of a worksheet is the defining event in a Poisson process. Calculate the expected time to complete 7 worksheets if Jared and Jamie work together.
This could easily be a question on the Actuarial Exam P/1, and thus by definition it is practical, at least for actuaries. Jared and Jamie working together constitutes a Poisson process with a worksheet completion rate of:
So you see, “work problems” have their uses after all.