I recently took a 60 minute, multiple-choice problem-solving test, in which, realistically, once you have completed all the background reading and scanned over the provided tables and charts, you have about 1.5 minutes per question. It appears there is an assumption that problem-solving is ingrained in the very DNA of scientists and engineers. Perhaps mine is faulty, or missing, for, when confronted with the problem of finding “the difference, in percentage points, between the December profit margin and the average profit margin for the other months”, I didn’t solve it. The direct approach would have been to take the difference between the two numbers, and divide the answer by the average over 12 months to find the percent difference. Instead, I opted for the alternative approach, which included hyperventilating, and pleading repetition of the key words (“Difference? Percentage points? Profit margin? Average?) in the hope that the mythical math gods might respond in ethereal, echoey tones with something like “The answer, my child, is B, 6.4%”. After the test –which went woefully –I went into the office and casually solved a bunch of problems that would generally be considered to be at least a few orders of magnitude more difficult than finding the difference, in percentage points, between the December and Jan-Nov average profit margins.
How? I wondered. Is it because “everyday” maths, like that in the problem-solving test, is no longer something we care to do by hand, and thereby, in the modern age of technology, takes much longer than 1.5 minutes? The amusing anecdotes of other PhD students who took the test (“Yeah, the other day, I wanted to find out what 1000 divided by 200 was, and I brought the calculator out”, or, “I do my fractions on Google”) seemed to confirm this. But, having said that, I also realised that the bunch of complex problems I solved in the office were almost entirely solved by sophisticated software –I just (to oversimplify slightly) “pressed the button”. It did make me wonder if, through the use of technology we are getting dumber, or if instead, by employing technology to do the more mundane stuff, we are cleverly freeing up more processing power to think about the bigger picture? Am I just making up excuses for what is an inexcusable lack of skills that are fundamental to my chosen field?
My answer to this has two parts. Firstly, computers are good at computing. They’re much better than brains. I could give a computer 10000 sums and it would return the answer to all 10000 of them in a second. My brain, on the other hand, would do the sums one by one, probably get bored, and then instruct me to go over to the fridge, eat something, read the news, watch an episode of Family Guy, and maybe only then, if at all, would it return to the sums. Brains are much better at deductive reasoning. Once all the boring computing is done, the brain can deduce meaning, and the next course of action. A computer will tell me that if I have 2 apples and Jon gives me another 3 apples, I have 5 apples. It can even tell me that, as I have 5 apples, I have enough apples for an apple pie (though only if I tell it, in computer-speak, that if x >= 4 apples, an apple pie can be made). But even if the computer does alert me to the event that an apple pie can be made, it won’t tell me the myriad alternative options, such as sharing my apples with four of my friends, or greedily keeping my apples so that I could have one per day for 5 days. Or that I could use the apples as window-smashing projectiles, which would allow opportunistic plundering during the London riots. So, to summarise, computing is best left to computers, while deductive reasoning is better done by brains.
On the other hand –and this is the second part of the two-part answer –I am compelled to weigh up the advantage of having a computer handy for all computing needs. Would having a calculator really have saved me in the test? Or would I have been far better off having remembered some elementary (but clever) tricks with numbers, which make calculating by hand faster than whipping out the Casio? Let us take the “difference, in percentage points” example I mentioned at the beginning. Say the profit margin for December is $4.88m, while the average for all other months is $3.37m, and we want to know the percent difference between them.
The options are,
A) 20%
B) 30%
C) 40%
D) 50%
E) None of the above
This may be laughable to others, but the sheer presence of the decimal point is enough to make some of us sweat. In fact, before you read through the detailed break-down of the problem below, try and do it as fast as you can, and time yourself (be honest!).
So, we need to take the difference between the two, then divide the answer by the average to find the percent difference. When you write it out on paper it looks like
4.88-3.37=1.51, which we will call “A”. Easy enough.
Now, the average of the 2 numbers is NOT (4.88+3.37)/2=8.25/2=4.125, although, I wouldn’t blame you if you thought it is. We have to remember that $3.37m is itself an average covering 11 months of the year, so to find the average from Jan-Dec, we have to multiply $3.37 by 11, add $4.88, then divide by 12 to get the average. So, that’s
3.37×11=37.17. Now divide this by 12.
Shit! You have to divide an ugly-looking decimal by a number that doesn’t neatly divide into it! At this stage, as the answers are multiple-choice, it pays to do a bit of rounding-off, because your answer is likely to be close to only one of the 4 options. This is a crucial time-saving step!
37.17 is effectively 37.
12 goes into 37 3 times, remainder 1.
We’ll forget the remainder and stick with an average of $3m across 12 months, and we’ll call this “B”.
Now, we need to divide A by B to find the percentage difference. So, on paper, this looks like
1.51/3.
At this point, you might have a mini freak-out when working under pressure, and yearn for the calculator/mobile phone/Google. How do I divide 1.51, a little decimal number, by 3, a big number? By moving the decimals to suit us, of course! To begin with, 1.51 is effectively 1.5, and we will turn 1.5 into 15, while remembering that we have just moved the decimal one spot to the right and have to return it to its place at the end of the calculations. So, 15/3 = 5, and with the decimal placed back, it turns into 0.5, which gives us 50%, allowing us to confidently choose (D).
If we had done the computation meticulously without any rounding, we would have come to an exact answer of 0.487489911, which would have taken much longer, and we still would have picked (D).
That brings us to the end of the problem. While the explanation above appears long, with all the right number tricks, the answer comes out quickly – certainly in less than 1.5 minutes!
The verdict? Yes, computers are still better at computing than brains, and I can imagine some people proclaiming, with a snort and a smirk “Ha! It was so much faster on the calculator, and I even factored in the time it took to get it from the other room!” While this may be true, the reality is that there are situations where whipping out a calculator is socially unacceptable, or others where one is simply not available –in such situations, our brain is well-equipped for the task, precisely because it is so good at reasoning. So good, that it will round off numbers and shift decimals – anything to make the computation quicker and easier, whilst knowing that the answer will be reasonably close to only one of the provided options.
On a final note, I have learned through writing this post that “mundane computing”, as you might call it, is challenging enough to warrant a plethora of online resources. If you’re ever in need, this one and its relatives are some of my favourites (not least because of the background music). Happy problem-solving!
by alisa
Bang on.