# Jerzy's Reviews > Innumeracy: Mathematical Illiteracy and Its Consequences

Innumeracy: Mathematical Illiteracy and Its Consequences

by John Allen Paulos

by John Allen Paulos

Most of the book is a collection of examples commonly seen in other pop math books: how a particular gambling game or con trick lets the house win most of the time; tricky things about Bayes' Theorem and Simpson's Paradox; how raising the price by 40% and then lowering the new price by 40% does not give you back the original price; the difference between statistical correlation and causation; etc.

I hoped the book would be an in-depth look at where innumeracy stems from and how to prevent it. There is a chapter about this, but it's not the meat of the book. He mentions standard things like poor math education, psychological blocks like "math anxiety", and popular misconceptions that math is just cold spiritless arithmetic. He does propose a few solutions here and there, like getting more non-mathematicians writing about math and highlighting the warmth and passion of the subject to get rid of negative stereotypes... but this is definitely not an overarching policy to improve the standing of math in this country like I'd been hoping.

But I do really like his idea of placing more emphasis on estimation in schools, and especially that people should build personal mental libraries of collections of things for every power of 10 up to at least a trillion. (In other words, you should be able to visualize how many is a thousand of something vs a million of something vs a trillion of something. For example, the stadium in our town seats 1,000 people; a wall nearby has 10,000 bricks; etc.) It would be handy for people to be able to judge for themselves whether or not a number cited in the newspaper is realistic.

Another cool idea is his (logarithmic) risk scale or safety scale. For example, if 1 out of every 5,300 Americans dies in a car crash each year, then driving a car has a low safety index of log(5300) = 3.7. If 1 out of 800 die due to smoking annually, then smoking has an even lower safety index of log(800) = 2.9. If only 1 in 5 million US kids is kidnapped each year, the safety index is a much higher 6.7, and so on. If newspapers and TV started to use this kind of scale, it would be an easier way for people to compare the relative risk of various activities.

I also liked his discussion of coincidences - for example, hearing in the morning that vivid details of your previous night's dream match what you hear on the news. Assuming that there's only a one-in-ten-thousand or one-in-a-million chance of this happening on a given night, over the course of a year in a big country like the USA you'd still get

The author goes on to bash more pseudoscience in detail; I agree with him but doubt that anybody who believes that stuff in the first place is going to be convinced otherwise by something as simple as facts and math. (Anyway, reasonable people often believe total crap too. It cracks me up that, at one point, phrenological exams were commonly a precondition of employment in big corporations!)

There's also an interesting comment about "winners" and "losers". A given coin toss has a 50% chance of landing on heads and 50% of tails, and in the long run if you toss a coin many many many times, the ratio (number of heads) / (total number of tosses) will approach 1/2. HOWEVER! That only applies to the ratio - the absolute difference between (number of heads) and (number of tails) is NOT guaranteed to approach zero. If an initial large absolute difference arises due to chance, it's not likely to go away. So if Harry is betting heads and Tom is betting tails, and after the first 100 tosses Harry just happens to be ahead 60 to 40, Harry is likely to stay ahead for a long time. The next 100 tosses are likely to split about 50-50, so he'd end up ahead 160-140, and so on; at 1000 tosses Harry's still most likely to be ahead 510-490. The ratio keeps getting closer to 1/2 (60/100 = .6, but 510/1000 = .51), but not the absolute difference. This doesn't mean that one side or the other is necessarily likely to get that far ahead - but if someone DOES, by pure chance, then they're likely to stay ahead. Perhaps in real life some people end up treated like "winners" or "losers" in general because they've ended up on the wrong side of the difference in wins; Harry here always seems to be ahead of Tom, even though Tom and Harry are each successful at only about half the things they attempt.

Another good section is about reward and punishment. Say that each of us tends to perform at some mean level on a particular task (for example, if I throw darts, assume I'll tend to hit near the bullseye 10 times out of 50). I may do particularly well or particularly poorly (40/50 or 0/50) in one session, but the next time I'm still most likely to be back around my mean score of 10/50. So if I do poorly today I'm likely to do better tomorrow; and if I do well today I'm likely to do worse tomorrow. This is called regression to the mean. Now, if we reward good performance and punish poor performance, and regression to the mean occurs, we are likely to assume that punishment causes improvement while praise causes a lapse - even if the punishment or reward had no effect on the next day's performance.

Finally, he also says mathematicians tend to have a particular sense of humor - they take things literally when they're not meant to be, or they take a premise to extremes with comical result. And indeed it makes sense that this kind of play is exactly what you do when solving math problems or coming up with proofs. See, Katie? My puns and bad jokes aren't pathological - I'm just studying!

I hoped the book would be an in-depth look at where innumeracy stems from and how to prevent it. There is a chapter about this, but it's not the meat of the book. He mentions standard things like poor math education, psychological blocks like "math anxiety", and popular misconceptions that math is just cold spiritless arithmetic. He does propose a few solutions here and there, like getting more non-mathematicians writing about math and highlighting the warmth and passion of the subject to get rid of negative stereotypes... but this is definitely not an overarching policy to improve the standing of math in this country like I'd been hoping.

But I do really like his idea of placing more emphasis on estimation in schools, and especially that people should build personal mental libraries of collections of things for every power of 10 up to at least a trillion. (In other words, you should be able to visualize how many is a thousand of something vs a million of something vs a trillion of something. For example, the stadium in our town seats 1,000 people; a wall nearby has 10,000 bricks; etc.) It would be handy for people to be able to judge for themselves whether or not a number cited in the newspaper is realistic.

Another cool idea is his (logarithmic) risk scale or safety scale. For example, if 1 out of every 5,300 Americans dies in a car crash each year, then driving a car has a low safety index of log(5300) = 3.7. If 1 out of 800 die due to smoking annually, then smoking has an even lower safety index of log(800) = 2.9. If only 1 in 5 million US kids is kidnapped each year, the safety index is a much higher 6.7, and so on. If newspapers and TV started to use this kind of scale, it would be an easier way for people to compare the relative risk of various activities.

I also liked his discussion of coincidences - for example, hearing in the morning that vivid details of your previous night's dream match what you hear on the news. Assuming that there's only a one-in-ten-thousand or one-in-a-million chance of this happening on a given night, over the course of a year in a big country like the USA you'd still get

**plenty**of people to whom this happens simply due to plain chance - not any sort of ESP or anything. So the fact that this has occasionally happened to you or someone you know should not be surprising in the least.The author goes on to bash more pseudoscience in detail; I agree with him but doubt that anybody who believes that stuff in the first place is going to be convinced otherwise by something as simple as facts and math. (Anyway, reasonable people often believe total crap too. It cracks me up that, at one point, phrenological exams were commonly a precondition of employment in big corporations!)

There's also an interesting comment about "winners" and "losers". A given coin toss has a 50% chance of landing on heads and 50% of tails, and in the long run if you toss a coin many many many times, the ratio (number of heads) / (total number of tosses) will approach 1/2. HOWEVER! That only applies to the ratio - the absolute difference between (number of heads) and (number of tails) is NOT guaranteed to approach zero. If an initial large absolute difference arises due to chance, it's not likely to go away. So if Harry is betting heads and Tom is betting tails, and after the first 100 tosses Harry just happens to be ahead 60 to 40, Harry is likely to stay ahead for a long time. The next 100 tosses are likely to split about 50-50, so he'd end up ahead 160-140, and so on; at 1000 tosses Harry's still most likely to be ahead 510-490. The ratio keeps getting closer to 1/2 (60/100 = .6, but 510/1000 = .51), but not the absolute difference. This doesn't mean that one side or the other is necessarily likely to get that far ahead - but if someone DOES, by pure chance, then they're likely to stay ahead. Perhaps in real life some people end up treated like "winners" or "losers" in general because they've ended up on the wrong side of the difference in wins; Harry here always seems to be ahead of Tom, even though Tom and Harry are each successful at only about half the things they attempt.

Another good section is about reward and punishment. Say that each of us tends to perform at some mean level on a particular task (for example, if I throw darts, assume I'll tend to hit near the bullseye 10 times out of 50). I may do particularly well or particularly poorly (40/50 or 0/50) in one session, but the next time I'm still most likely to be back around my mean score of 10/50. So if I do poorly today I'm likely to do better tomorrow; and if I do well today I'm likely to do worse tomorrow. This is called regression to the mean. Now, if we reward good performance and punish poor performance, and regression to the mean occurs, we are likely to assume that punishment causes improvement while praise causes a lapse - even if the punishment or reward had no effect on the next day's performance.

Finally, he also says mathematicians tend to have a particular sense of humor - they take things literally when they're not meant to be, or they take a premise to extremes with comical result. And indeed it makes sense that this kind of play is exactly what you do when solving math problems or coming up with proofs. See, Katie? My puns and bad jokes aren't pathological - I'm just studying!

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*Innumeracy*.