Innumeracy: Mathematical Illiteracy and Its Consequences

The problem that resulted in this book is far-reaching: the public simply doesn't understand mathematics. Statistics, ranging from a 10%-off sale to the sort found in opinion polls, are unfathomable to the general populace. Probability, especially in the context of gambling, is understood by only a scant handful of people. The list of misunderstood mathematics is nearly endless.
In the first few chapters of the book, Paulos describes various issues that the innumerate (that is, those who don't understand numbers and math) often have issues understanding. He describes the issue to a reasonable level of detail, then derives answers for them. Don't let the use of the word 'derive' scare you off: the answers are readable and readily understandable to a general audience. In some cases, if you're really rusty, you might need to read them a second time to grasp the solution.
Later chapters, however, are not written for the innumerate. They are attempts to convince the reader that mathematical education needs to be improved. I think that everyone agrees that education should be improved, but he offers suggestions that are impractical or nonsensical.
Ultimately, the problem of this book is a lack of focus. Paulos could have written either a book that tackles basic mathematical issues that the general public doesn't understand, or he could have written a book that describes the consequences of innumeracy. He tried to do both, and stuffed both topics into a single slim volume. In doing so, he shortchanges both audiences. The result is a book that is good, but does not fully address the needs of anyone.
If you find yourself uncomfortable with mathematics, pick up a copy of this book and read up to chapter five. If you are comfortable with mathematics and are looking for fodder to prove the point that improving mathematical knowledge at any level is productive, this book will not serve your purpose.

In this short book, Paulos does an outstanding job of pointing out what lack of number intimacy can do to a person. The anecdotes are outstanding, especially the ones on large numbers and on probability. For example, he shows how one is fooled by probability: If we have 23 people in a room, what is the probability that two of them have the same birthday? 50%!! Very conterintuitive.
The author also tries to understand why it is almost considered acceptable for a person to admit that one is "bad with numbers", while it not being ok to be "bad with words". The realm of psychology is not his forte, but the ideas he points to are interesting.
Overall, this is an easy to read book, much easier even to one literate with numbers. I was done with it in 3 hours, and was left wanting more, so much so that I am now buying some more of his works. If they are half as good as Innumeracy, then they will be good enough.

I bought this book after it was referenced in the very well-written The Panic Virus (Seth Mnookin), which I'd highly recommend. I was curious about a professional's take on math skills in the general population. There's plenty of good insight and information, but it's like the author couldn't make up his mind if he wanted to rant or educate. He covers a number of numerical fallacies and adds explanations, but often it feels like the explanation would come up short to anyone who really didn't understand the particular issue being covered. I'm quite good with math myself so it was easy for me to follow everything, but parts of the book came off more as a "we smart people get why that's so dumb". I think the book could be improved by either tightening it up and focusing on a more advanced reader group or by really fleshing out the explanations to try and really help readers who are not good at math and actually suffer from the innumeracy/ies he describes. He could probably have done both even, if he set up chapters with a high-level concept, issue and effects section, then an educational/explanation section that the advanced reader could easily skip over.

Lots of good, but could really use a revision or two.

I found this book to be about as deep as morning mist. The authors points, supposedly illustrating "innumeracy" more illustrative of the authors lack of depth. Most children I know realize that the length of a calendar day does not change with the seasons. Yet the author chooses to spend far too many words agonizing about how "people" - the innumerati - believe that the day is shorter in winter because of the poetic use "the days grow shorter in winter". Clearly the days that grow shorter are the daylight hours. Most native speakers of english know this even when they use they use the phrase the author finds so objectionable. This use of day can also be found in the expression "the day" and is alluded to in "the night". In fact the author's pedantic rant is exactly as offensive to clear thinking as the astronomers objection to Pink Floyd's Dark Side of the Moon. Again, clearly the dark side is the metaphorical dark side, the one we do not see. In ignoring this in both cases, the argument sets up a straw man to knock down. Only a very shallow mind finds solace knocking down straw men.

In reading this book I got the sense that the author wanted to show us just how smart he is. At times I felt that the text overflowed with intellectual elitism ad nauseum, and while it contained lots of examples, it offered very little in the way of (numerical) enlightenment.

Two notes of caution are in order when reading this book. First, Innumeracy is perhaps the first of several books on the subject of mathematics and numerical befuddlement among the general population written by Mr. Paulos, and he tends to recycle much of his material from book to book. So, if you have read this little book, you have pretty much read most of his other books. Second, most of the examples in Innumeracy do not qualify as truly original content, and some readers may have run into most of them in other math texts in high school and college.

While it is true that numbers and statistics require interpretation, the author should also bear in mind that both have a context, and when viewed outside of their proper context, any number of questionable and downright false interpretations can result. It is really no difficult matter to compute, manipulate or convert numbers, but gaining insight from these treatments is the real challenge. This requires critical thinking

That said, in the first chapter, the author plays a bit with measurements, and shows that we can for example calculate the length of our arms in miles if it suits us. However, I must ask if it is appropriate to calculate my arm length in miles when inches work just fine. Just because we can do something does not necessarily imply that we should do it, or that it is even appropriate, and I do not think that the author learned that particular lesson (which incidentally is a lesson many scientists never seem to learn).

The second chapter delves into chance, coincidence and probability in everyday life. He muses on how coincidence is often mistaken for fate (or worse, divine intervention), and how we try to derive profound meaning from events due to pure chance. The chapter is a fair, witty, but somewhat dry take on a typically mundane topic.

The third chapter on pseudoscience is probably the best part of the book. Here, he skewers all manner of foolishness, from astrology and numerology to quack cures and fringe science with good humor. Moreover, his peculiar mathematically oriented humor would find its most appropriate home here.

By the fourth chapter, the real problems of the text begin to show themselves in earnest. First, Mr. Paulos doesn't bother to give references for certain `facts'. Second, he tends to make claims and then offer no support for them. Third, and this is probably the most damaging of all, he references the work of Tversky and Kahneman (the founders of Prospect Theory and later Nobelists for their contribution to economic science), but does not seem to be aware of the fact that most of their studies suffered from a lack of what statisticians used to call external validity. That is to say, the conclusions made from the study, while being for the most part reliable and truthful for the participants (internal validity), are not necessarily true for the greater population at large which you may want to make conclusions about.

Although the fifth chapter, titled `Statistics, Trade-offs and Society' starts off with the right idea in mind, it finally falls flat when he reaches statistical significance and Type I and II errors. People who read about the latter for the first time may come away either confused, which is bad, or thinking that they know something when they really do not, which is even worse. As this chapter is most likely the most relevant to the vast majority of lay readers, given that it deals with both statistics in the news and everyday life and some of the mass public's favorite bogeymen- cancer, heart disease and what not, a lack of references for further enlightenment demonstrates yet another glaring weakness in this text.

Finally, in the last chapter, we are treated to a fairly decent serving of thoughtful and intelligent prose of such great clarity and candor that I for one was amazed that a nutty math professor could produce. How unfortunate it was that such fleeting moments of clarity and beauty were so infrequent in this text, and having read three other books by Paulos, I can say with force that he has failed to build upon such an outstanding snippet of quality writing.

In sum, you may be able to improve your SAT or GRE score with this book, but not much else.

Paulos(P)has written an interesting book.The current 2001 edition is a moderate improvement over the original 1988 edition.Paulos shows that Americans, as a whole, lack mathematical,statistical,and probabilistic expertise.Most of the examples are well chosen.Unfortunately,his discussions of a) the behavioral work of psychologists Tversky and Kahneman and b)the law of large numbers,the central limit theorem,and the normal probability distribution,are not satisfactory.Tversky and Kahneman problems generally attempt to trick the decision maker by incorporating what Tversky and Kahneman think is irrelevant information ,phrased in vague and ambiguous fashion,in addition to the information that allows a decision maker to compute marginal,conditional,and joint probabilities.Many experimental subjects,based on their own personal life experiences,view this additional "irrelevant" information as relevant.Paulos appears to be ignorant of the important work done in decision making under both risk and uncertainty(ambiguity)by Allais,Ellsberg,Keynes, and Mandelbrot.Consider the battlefield problem put forth as an example of decision making under ambiguity or uncertainty(Paulos,2001,pp.116-117;1988,pp.86-87)by Tversky and Kahneman.In such a situation ,it is not possible to specify the exact,precise probabilities and outcomes that Tversky and Kahneman(T-K) claim the scouts report back to the general.Many decision makers interpret these probabilities and outcomes as limiting results only.This creates problems because intervals are used implicitly by individual decision makers faced with this experimental problem,which is very similar to the rare Asian disease problem where the specification of exact probabilities also makes no sense at all.T-K are mixing up problems of Allais type risk(certainty-reflection problems)with problems of ambiguity and uncertainty,where the specification of exact,precise single number probabilities makes no sense.P's discussion of the law of large numbers and central limit theorem(pp.156-160) ,which is the foundation used to justify the assumption of the normality of the probability distribution,should have made it clear that researchers in social science,liberal arts,education,business,and economics should take the additional step and check their data using some type of goodness of fit test,such as the Chi Square test.The problem here is that much economic and social science phenomena is not continuous and independent of past occurrences over time.The data is not uniform,homogeneous,or stable due to the existence of what Joseph Schumpeter called "regular irregularities" over the long run generated by innovations in technology and financial instruments.Social science is not social physics.Benoit Mandelbrot has demonstrated that the price data generated over time in all financial markets is NOT normally distributed,due to short and long run dependence,discontinuity,jumps in the data,etc.It turns out that none of the founders of econometrics tested any of their time series data for normality.The normal distribution is sound for the data encountered in most of the life and physical sciences.This is not so for the social sciences or liberal arts fields.

Someone recommended this book to me and I can't remember who it was; it turn there are certain people I know to whom I would do the same, and others that I would warn off.
If you are earnestly interested in learning some practical math yet utterly uninitiated in numerical ways this may be the book for you. If, however, you are firmly stuck in your innumerate ways, I doubt that this book is compelling or shocking enough to convince you otherwise. If you are numerate, but curious about how the other half lives, you will need to manage bouts of boredom sitting in the choir while Paulos preaches. I mostly fall into the last catergory, yet I managed to find some revelations and some interesting bits here and there. Also, the author has a friendly, conversational style with a touch of irreverance -- I appreciate that.
Yet I nearly gave up on this book before I reached the halfway point, and I RARELY give up on books. What pulled me through is the author's excellent advice from the foreword: feel free to skip the bits that are too complicated for the novitiate or too obvious for the adept. A generous gift from the author -- take advantage of it and you will enjoy the book all the more. :)
This book focuses heavily on statistics although it does touch on a number of other flavors of math, including fractions and magnitudes. Still, the best concrete examples come from stats, yet I am sure that better books must exist for providing the "gee-whiz! I didn't realize what a boob I was for not realizing X, Y and Z about real life statistics" revelations that may shake the sluggish right brain of the innumerate. This book has the advantage of being thin, though, and it does fit nicely into one's pocket. ;) The book also comments on potential social factors that turn budding math whizzes into the innumerate masses -- I didn't expect it, and it is refreshing.

The author goes through a large number of real-world examples of applied mathematics, that I (a retired engineer) had thought were like breathing out, breathing in. When I shared a couple of them with my adolescent grand-children, I found some enthusiastic "aha!!" moments. Huh...

John Allen Paulos passionately explains in a personal essay the probabilistic nature of life. He believes that too often people are innumerate and have many misconceptions about math. He enumerates these fallacies in the preface (Paulos, xii,xiv):
1. Math is nothing more than computation;
2. Math is a completely hierarchical subject;
3. Storytelling is not effective as an educational tool in mathematics;
4. Mathematics is only for a few;
5. Math numbs or limits our freedom.
Since Paulos studied philosophy as an undergrad and mathematics in graduate school, he respectfully acknowledges the duality of the two disciplines. Throughout the book he makes comparisons such as "The romantics believe that a concern with numbers numbs us to the big questions of grandeur of sunsets and waterfalls" (Paulos, 179).
Paulos provides real world examples of innumeracy in subjects like stock scams, choice of spouse, newspaper psychics, diet, medical claims, lottery, astrology, drug testing, insurance and law. At times the reader must patiently muster through detailed proofs. But, don't worry as Paulos promises "there won't be a test" (Paulos, xiv). Be patient, just rest and relax while reading "Innumeracy" then will you thoroughly enjoy it. If you are among the innumerate then simply read through the math formulas without a concern for grasping the conceptual depth. John Allen Paulos is flexing his numerate muscles giving Innumeracy greater strength.
The author's aim is to make each of us less intimidated by the numbers. He begins by familiarizing the reader with the basis of counting: how many people are in the US; how many people in the US are under 18, how many die each year; how many cigarettes are smoked each year; and how many words are there in a good sized novel?
Many of us believe incorrectly in the magic of coincidences. However, there are no coincidences instead statistical probabilities. For example in a room of 23 random people - 2 or more will have the same birthday. Another example is the old adage "everything comes in threes" (Paulos, 69). Again this is just statistical probability since if you wait long enough things will come in sets of three. Another misconception is the "why me?" attitude toward misfortunes. Bad things happen periodically, the probability is that they're going to happen to somebody, why not you?
"The tendency to attribute meaning to phenomena governed only by chance is ubiquitous" (Paulos, 114). The fruitful example the author uses to illustrate this concept is regression to the mean. Regression to the mean is the tendency for an extreme value of random quantity followed by a value closer to the mean. In this case he provides the example of very intelligent people who may have intelligent offspring, yet generally will be more likely to be less intelligent than their parents.
John Allen Paulos has an affinity to number ID over name as a means of identification since no two people can have the same ID number. However, he is appalled by the use of an excessive number of digits. In being numerate one would know that an identification number with 9 digits will have 109 (one billion) possible different numbers.
Innumeracy covers relative risk, which is a significantly large issue for Public Health and the future of health care. This interesting passage in the book proposes that a logarithmic safety index be used. In this way we would be measuring the relative safety against the relative danger adding on the likelihood of the endeavor.
Innumeracy is not a romantic beach read. Its meager attempt to be sexy is the argument over the "bell-curve" where "the urge to average can be seductive" (Paulos, 169). The appeal of this book comes from the nature of the author's experience in having studied philosophy, literature and mathematics. Paulos is a believer in the human condition, specifically the tendency to want it all. Numeracy will help people understand that there are going to be trade-offs. A romantic at heart he adamantly opposes the view that mathematics is esoteric and those that study numbers are "coldly rational". The author's desire is to stimulate numeracy and enhance the appreciation of the probabilistic nature of life. John Allen Paulos is a best selling author, public speaker, columnist for [...] and Guardian and a professor of math at Temple. He offers his web address and welcomes feedback [...]

paulos is full of himself. after a couple of pages, i got tired of reading his jibberish which amounted to nothing more than him showing off his vocabulary. i found the book choppy and hard to follow. regret buying the book, at least it was paperback and cheap. stay away.... dont buy the book.