Mathematics Revisited, This Time Using Fermi’s Problem
September 12, 2021 at 6:42 p.m.
By Max [email protected]
As someone who is semi- innumerate (lacking many math skills), I remain enamored with mathematics of all descriptions including statistics and calculus. The latter was defined by humorist Dave Barry as “the branch of mathematics that is so scary it causes everybody to stop studying mathematics.”
My library at home is filled with books of all descriptions about the history of mathematics, the birth of numbers, the joy of numbers, statistics, biostatistics, the science of measurement, calculus, precalculus and numerous others. I also have DVDs from Great Courses on the history of mathematics.
Despite this vast collection, I remain confounded by the subject but more appreciative of it. I view it as a lingering challenge despite my advanced age and I encourage all people young and old to become more familiar with the vast number of benefits derived from being more numerate.
A good way to start is to read "Innumeracy," a book written by John Allen Paulos. His wonderful book clarifies just how much innumeracy pervades all of our lives. Parents should also encourage their children to consider a career in mathematics, whether it be teaching, science, engineering, statistics, quality control, construction work or annuities. There will always be opportunities for work in any of these fields.
I thought about math because of a train derailment in Washington state where there were a number of fatalities. When it occurred I wondered what calculations had been used to establish speed limits for trains negotiating curves. It was not difficult of find.
Not surprisingly, a study at the University of Florida considered maximum velocity, railway track embankment, track curvature, centrifugal forces, friction and other factors to establish derailment speed. All of the various factors were determined using mathematical formulas and proved once again the ancient wisdom that “Mathematics is the way to understand the universe.”
Since the days of Galileo and Newton, math has nurtured science. Unfortunately, the United States is far behind other nations in mathematics and math teachers are in short supply. Studies have shown that American students score significantly lower than students worldwide in math achievement, ranking 25th among 34 countries.
One reason may be that many of us agree with Dave Barry. In an earlier article I included a few examples where simple math can actually be surprisingly useful. They may even provide the incentive to learn more about the essentials and is available in one of the chapters in my book “Science Snippets.”
One excellent exercise for the family to tackle and enjoy is what is known as Fermi’s problem. It is a riddle based on estimation and named after the physicist Enrico Fermi (1901-1954). (He was known for his ability to make good approximate calculations with little or no actual data.)
One classic example, and one often used, is to guess how many piano tuners there are in Chicago. Without looking and calling your cell phone for an answer you could start by guessing the population of that city. Using 2.5 million to start, and assuming the average household has four people, we would have 625,000 households. Predict that one out five has a piano; that gets us to about 125,000 pianos. Let’s say that all get tuned once a year. Now the question is how many pianos a tune can service annually. You can guess that one can tune three pianos per day. Multiply by five times a week, for 50 weeks a year, and that comes out to be about 750 pianos per tuner per year. Divide the number of pianos (125,000) by 750, and you get roughly 170 tuners in Chicago. The goal isn’t knowing the exact number but rather being able to estimate the right order of magnitude using nothing but common sense.
According to a recent article in the New York Times magazine written by Caroline Chen, contemplating Fermi problems keeps one curious about the world and how things relate to one another. Examples of similar problems can be found on the internet.
Final Thoughts
One article I read suggests starting the first week of the eighth grade with Fermi problems. This could set the tone for a positive academic environment and community of learners, which foster both mathematical and affective processes. The Fermi problems and social skills introduced that week combine to generate the kind of involvement and thinking processes that are at the root of quantitative literacy. However, solving Fermi problems needn’t be limited to schools, they can teach all of us how to be more creative and to work as a team.
Max Sherman is a medical writer and pharmacist retired from the medical device industry. His new book “Science Snippets” is available from Amazon and other book sellers. It contains a number of previously published columns. He can be reached by email at [email protected].
As someone who is semi- innumerate (lacking many math skills), I remain enamored with mathematics of all descriptions including statistics and calculus. The latter was defined by humorist Dave Barry as “the branch of mathematics that is so scary it causes everybody to stop studying mathematics.”
My library at home is filled with books of all descriptions about the history of mathematics, the birth of numbers, the joy of numbers, statistics, biostatistics, the science of measurement, calculus, precalculus and numerous others. I also have DVDs from Great Courses on the history of mathematics.
Despite this vast collection, I remain confounded by the subject but more appreciative of it. I view it as a lingering challenge despite my advanced age and I encourage all people young and old to become more familiar with the vast number of benefits derived from being more numerate.
A good way to start is to read "Innumeracy," a book written by John Allen Paulos. His wonderful book clarifies just how much innumeracy pervades all of our lives. Parents should also encourage their children to consider a career in mathematics, whether it be teaching, science, engineering, statistics, quality control, construction work or annuities. There will always be opportunities for work in any of these fields.
I thought about math because of a train derailment in Washington state where there were a number of fatalities. When it occurred I wondered what calculations had been used to establish speed limits for trains negotiating curves. It was not difficult of find.
Not surprisingly, a study at the University of Florida considered maximum velocity, railway track embankment, track curvature, centrifugal forces, friction and other factors to establish derailment speed. All of the various factors were determined using mathematical formulas and proved once again the ancient wisdom that “Mathematics is the way to understand the universe.”
Since the days of Galileo and Newton, math has nurtured science. Unfortunately, the United States is far behind other nations in mathematics and math teachers are in short supply. Studies have shown that American students score significantly lower than students worldwide in math achievement, ranking 25th among 34 countries.
One reason may be that many of us agree with Dave Barry. In an earlier article I included a few examples where simple math can actually be surprisingly useful. They may even provide the incentive to learn more about the essentials and is available in one of the chapters in my book “Science Snippets.”
One excellent exercise for the family to tackle and enjoy is what is known as Fermi’s problem. It is a riddle based on estimation and named after the physicist Enrico Fermi (1901-1954). (He was known for his ability to make good approximate calculations with little or no actual data.)
One classic example, and one often used, is to guess how many piano tuners there are in Chicago. Without looking and calling your cell phone for an answer you could start by guessing the population of that city. Using 2.5 million to start, and assuming the average household has four people, we would have 625,000 households. Predict that one out five has a piano; that gets us to about 125,000 pianos. Let’s say that all get tuned once a year. Now the question is how many pianos a tune can service annually. You can guess that one can tune three pianos per day. Multiply by five times a week, for 50 weeks a year, and that comes out to be about 750 pianos per tuner per year. Divide the number of pianos (125,000) by 750, and you get roughly 170 tuners in Chicago. The goal isn’t knowing the exact number but rather being able to estimate the right order of magnitude using nothing but common sense.
According to a recent article in the New York Times magazine written by Caroline Chen, contemplating Fermi problems keeps one curious about the world and how things relate to one another. Examples of similar problems can be found on the internet.
Final Thoughts
One article I read suggests starting the first week of the eighth grade with Fermi problems. This could set the tone for a positive academic environment and community of learners, which foster both mathematical and affective processes. The Fermi problems and social skills introduced that week combine to generate the kind of involvement and thinking processes that are at the root of quantitative literacy. However, solving Fermi problems needn’t be limited to schools, they can teach all of us how to be more creative and to work as a team.
Max Sherman is a medical writer and pharmacist retired from the medical device industry. His new book “Science Snippets” is available from Amazon and other book sellers. It contains a number of previously published columns. He can be reached by email at [email protected].
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