A recent column in the New York Times started with a math word problem, which I will oversimplify slightly here:
Sarah takes six hours to paint a fence, and John takes twelve hours to paint the same fence. How long will it take them to paint the same fence if they work together?
One thing that is really fun about this problem, I think, is that it turns out this is exactly like asking “what resistance do you get if you put a six-ohm resistor and a twelve-ohm resistor in parallel?”
The answer turns out to be your choice of four hours (if you talk about fence painting) or four ohms (if you talk about resistors in parallel). The formula is:
You can model Sarah and John as impediments that stand between you and your goal of a painted fence. John is a worse or more formidable impediment, a greater restriction on a flow rate toward the goal. (John carries out the task of fence-painting more slowly than Sarah does. We can think of him as a resistor of higher value.) If you make both of them available for the painting project, then there are two flow paths toward the goal. It is like two resistors in parallel offering two electrical current flow paths, together allowing more current to flow.
What’s slightly annoying is that the people who design these problems (in this case the author of the homework problem that troubled the writer of this column) never quite come out and state their many underlying assumptions. For example the people who design these problems are assuming that everything about the problem is linear and and they are assuming that everything about the problem is continuous. If you had two Sarahs working, they assume, the fence would get painted twice as fast (linearity). If you had one-third of a Sarah, they assume, it would take three times as long for the fence to get painted (continuity). In real life one has no choice but to consider that if, for example, you have two hundred people working on painting of the fence, it is simply not going to get painted two hundred times faster – in real life, the workers bump into each other and get in each others’ way, and so linearity is lost. In real life, some of the inputs are discontinuous (for example people are quantized), for example there is not really any such thing as “one-third of a Sarah”, at least not in a way that meaningfully contributes to a fence getting painted at all.
This particular column in the New York Times bears a headline How to Solve Any Problem Using Just Common Sense and spends most of its paragraphs talking about “Fermi problems”. A Fermi problem (Wikipedia article) is a problem where the goal (often more a teaching goal or brain-stretching goal than anything else) is to arrive at an estimate of an answer. How many piano tuners do you think there might be in Chicago, famous physicist Enrico Fermi might have asked? What Fermi had in mind is that you start with a guess about the population of the city, then make a guess how many households that might constitute, then a guess what fraction of homes are likely to have pianos in them. Then maybe a guess as to what fraction of the piano owners care enough about their pianos to try to keep them in tune. Then how often per year the tuning would likely need to take place. Then for a particular piano tuner to make a living at it, how many tunings the tuner would need to do per year. Voila! Maybe there are 170 or so piano tuners in Chicago, Enrico Fermi might have said with a smile.
The point here is not that we are trying to figure out exactly how many piano tuners there are. We are trying to develop a category of mental skills that might permit us to at least opine that if somebody were to say that there are only five piano tuners in Chicago, we can guess that they got it wrong in the direction of the number being too small. We are trying to develop a category of mental skills that might permit us to at least opine that if somebody were to say that there are fifty thousand piano tuners in Chicago, we can guess that they got it wrong in the direction of the number being way too large (unless on that day there happens to be a professional meeting of piano tuners taking place in Chicago).
Which brings me to a second thing that disappointed me about this column in the New York Times, namely that the writer seemed to think that this fence-painting math problem was an example of a Fermi problem. It simply is not. The fence-painting math problem is the kind of problem where you plug the number 8 and the number 12 into a calculator and you do some calculating, and eventually what you end up with is the number 4. There is no guessing involved, no estimating. There is an exact right answer and there are exact steps to get to the exact right answer (according to the author of the homework problem that vexed the writer of this column).
Which brings me to a third thing that disappointed me about this column in the New York Times. Yes, I was disappointed that it failed to recognize the unstated assumptions of linearity and continuity in the fence-painting problem that was meant to be the hook that would get the reader to read the remaining three-quarters of the column. Yes, I was disappointed that the writer was apparently holding out the fence-painting problem as supposedly an example of a Fermi problem when it was not. But the biggest disappointment for me came from the headline of the column. It is simply false to say that you can “solve any problem using just common sense”. Part of being wise is recognizing which categories of problems are ones for which a mere estimate is “good enough” and thus an application of “just common sense” is appropriate. But part of being wise is also recognizing which categories of problems are ones for which an exact answer is called for and nothing else will do. If we are disarming a bomb and the question is whether to cut the red wire or the blue wire, what we want is the exact correct answer, not an estimate of the correct answer. The fence-painting problem in the writer’s math homework is one for which the real situation is that a few clicks on a calculator permit arriving at the exact answer that the person who designed the question had in mind as the correct answer. The writer of the column never quite came out and said why the question was being posed, but in almost all textbooks the reason why such a problem would be posed is to see whether the student has the discipline to work out the math calculations that eventually arrive at the exact number 4. Not some estimate that is about four, but a calculation in a methodical way that arrives at the exact number 4.
Setting aside my disappointment with the column in the New York Times, let’s talk about the skills that we hope might be nurtured in today’s schoolchildren. Yes of course we hope they will learn the three R’s — reading, writing, and arithmetic. But we also hope that they will be encouraged to develop critical thinking skills, which includes estimating skills that permit arriving at guesses as to whether something that someone else has said is plausible or not. Such skills permit a person to pick and choose which statements deserve closer scrutiny, for example, in a world where there are not enough hours in a day to scrutinize everything that everybody says.
In the old days, people used slide rules. Part of using a slide rule was having to devote a corner of one’s brain to keeping track of the likely order of magnitude of “the answer”. What the slide rule provided was merely the significant digits of “the answer” and it was up to the user to work out “where the decimal point belongs” in the string of digits provided by the slide rule. The corner of the brain that a slide rule user employed to keep track of orders of magnitude for use of the slide rule was also helpful in catching order-of-magnitude mistakes made by others. Somebody might say “I think it will require 40 or so gallons of gas to drive to that destination” and if the speaker had gotten an order of magnitude wrong by putting a decimal point in the wrong place, a person who used slide rules frequently would almost instantly catch the mistake. If the true answer was much more likely to be 4 or 400, the slide rule user would be likely to catch it.
Nowadays nobody uses slide rules. Well, almost nobody except some oldsters who get nostalgic about it now and again.
But the estimation skills, the Fermi-problem-solving skills, they are just as important in today’s world as they were a generation ago or fifty years ago when Fermi posed his first Fermi problem. Whenever we are around our nieces and nephews, we owe it to ourselves to pose lots of Fermi problems to them, and spend time with them to talk about ways of arriving at answers. I take this to be the main thing the writer of the column hoped to communicate, and with this, I agree.