John Allen Paulos's Innumeracy is one of those classics of the field that I've never gotten around to reading. I've been thinking more about these sorts of issues recently, though, so when the copy I bought a few years ago turned up in our recent book-shuffling, I decided to give it a read.

Unfortunately, I probably would've been a lot more impressed had I read it when it first came out in 1988. Most of the examples used to illustrate his point that people are generally very bad with numbers are exceedingly familiar. They appear in How to Lie With Statistics, and the recent The Drunkard's Walk by Leonard Mlodinow, and a bunch of other books and articles.

It's hard to beat Paulos's description of the core problem, though:

Innumeracy, an inability to deal comfortably with the fundamental notions of number and chance, plagues far too many otherwise knowledgeable citizens. The same people who cringe when words such as "imply" and "infer" are confused react without a trace of embarrassment to even the most egregious of numerical solecisms. I remember once listening to someone at a party drone on about the difference between "continually" and "continuously." Later that evening we were watching the news, and the TV weathercaster announced that there was a 50 percent chance of rain for Saturday and a 50 percent chance for Sunday, and concluded that there was therefore a 100 percent chance of rain that weekend. The remark went right by the self-styled grammarian, and even after I explained the mistake to him, he wasn't nearly as indignant as he would have been had the weathercaster left a dangling participle. In fact, unlike other failing which are hidden, mathematical illiteracy is often flaunted: "I can't even balance my checkbook." "I'm a people person, not a numbers person." Or "I always hated math."

Paulos clearly and concisely identifies all the major sources of innumeracy in dealing with probability and statistics, excessive personalization and a kind of misplaced romanticism chief among them. He doesn't go into as much detail as some other treatments of the subject, opting for a more typically terse mathematician's approach, but there's a sort of spare elegance to his presentation.

I would've liked to see more documentation of the problems of innumeracy-- how many people have a functional grasp of numbers, what are the policy consequences, what are the solutions that might be attempted-- but that's sort of ahistorical, based on reading other treatments of the same mathematical issues recently. I'd still like some good numbers on the subject, if anyone knows a source.

The other striking thing about this book is how little has changed. This was written when Reagan was President, and yet the concrete examples he gives still apply perfectly well. Despite twenty-odd years of people pointing to the problem, nothing has gotten any better. Of course, it's not clear that things have gotten any **worse**, so there's that to cling to at least...

If you've read other books on the subject recently, you've probably already seen all the examples he uses covered in greater detail. If you haven't read about the problems of mathematical illiteracy before, though, you won't find a more concise and readable outline of the basic problems.

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Of course, it's not clear that things have gotten any worse, so there's that to cling to at least...I'd beg to differ here. Two words: negative amortization.

You and I are numerate, so we understand that taking out a loan where the payment does not cover the interest (let alone the principal) is likely to be a bad idea. A major factor in the real estate bubble was the fact that banks and mortgage brokers started issuing such loans. They're usually called Option ARMs because they offer several payment level options (among them interest only, 30-year amortizing, and 15-year amortizing), but by far the most commonly chosen option (and all too often the option used to determine that the borrower was "qualified" for the loan) was a negatively amortizing payment. I've owned my present house for almost ten years (with a fixed rate mortgage, thank you), but had I been buying in the last few years, the only reason I would not have walked away from such a loan would be that I would have been running. It's because of these idiotic loans that the credit crisis is likely to continue for at least another couple of years or so.

I have read it back in 1992 or so and like it very much at the time. I wonder if I'd still like it as much today. I often recommend it to people, though.

An amazing book by an amazing man. You might now try his other books. He is SO very right. Of course we like books by Leonard Mlodinow -- how to top doing Physics with Gell-Mann and Feynman, then dropping out to write for Star Trek TNG?

Math below Calculus is a good thing for a teacher to experience. I taught Algebra 1, Algebra 2, and Geometry for 5 semesters at Woodbury University, and a year of those (plus pre-Algebra and pre-Calculus and other courses) in middle schools and high schools in Pasadena. I am better for the experience. My wife has taught Algebra and Trigonometry to students there, to prepare them for the Physics courses that she teaches. Math for Physics. And, by the way, Physics for Architects.

Your primary role is to balance instruction (which is usually all that the naive think that teachers do), assessment, and management (lesson planning and classroom management). The assessment is not just about grading homework and exams; it is to determine the learning style of each unique student. What matters is NOT what you know, but what you can find out about what is going on in the head of the student. Give at least half credit on homework and exams under my mantra: if you can't write the equation, draw me a picture, or write me an English paragraph, but show my what you think. What the student knows that is right, build on, using their learning style.

What the student knows that is wrong, usually the result of a bad teacher in the past, you solve by regressing them to just before that mistake, and then rolling forwards on the right path. The student usually knows where and when they went off-track, and still resents the teacher who did that.

Read papers on Dyscalculia -- Mathematics Disorder. It is real, it is insidious, and in only 1/3 of the clinical cases is the cause neurological; the rest can be cured by good teaching. This is your chance to save lives!

Innumeracy is the tip of the iceberg. What kills is the resultant anti-Science worldview, and the muddled thinking plus plunging self-esteem of Dyscalculia. The minor flaws acknowledged, John Allen Paulos's Innumeracy should be required reading in every teacher's college in the USA. The rest of the developed world already knows.

Innumeracy is analogous to illiteracy, but what do we have for ignorance of the world? I once watched a newsreader reading the teleprompter, informing me about the planet of Jupiter called 'ten'.

The idiot who wrote the teleprompter script had read in "Io" and wrote out "IO". It seems that all teleprompters use all-caps, which is designed-in stupidity.

I am barely numerate myself, and owe what little familiarity with math I have to an astounding woman who taught Grade ten algebra to a class of students in a rural highschool in which the school had *forgotten* to include a grade nine algebra course for my year class - about 150 students.

She was the very stereotype of a middle aged math whiz, unable to keep her clothing straight, her hair under control (it was always wildly undone from its bun by the end of class), or chalk smears from her face. Her enthusiasm about math and the wonderful things that could be learned through understanding and using it was boundless and infectious. She crammed two years worth of algebra into one, and almost all of us did well in the finals.

Nevertheless, aware that the available teachers for the next two years were known to be unable to understand the subject they were supposed to be teaching, I opted to take another two years of Latin and Biology.

Innumeracy starts with ill-taught teachers.

The reason people are innumerate and bad at statistics is not because of a flaw in Platonic logic off in the distance on some mythical hill. The reason is that, despite all the groovy trigonometry a human uses to wave a multijointed limb around, there's no need for consciousness of numbers in evolution. The reason for innumeracy is biology. It may feel good to whinge away at Philistines, but the thing is is that humans use lossy computation and don't have calculators in the cranium. Language good, engineering who needs it, math who needs it. Now there is a RS for numeracy, but only recently.

Moral: There are 3 kinds of people in the world: Those who are good at math and those who aren't.

You should read today's Baldo comic strip. It's right on point. Here's a link, but it's only good for today (7/8/08): http://www.gocomics.com/baldo/

Despite twenty-odd years of people pointing to the problem, nothing has gotten any better. Of course, it's not clear that things have gotten any worse, so there's that to cling to at least...To be slightly puckish... do you have any numbers to back up either statement? That statement's a little odd, coming as it does right after your stated desire to have some more documentation of the problem...

Who oh who gets to decide what the set of knowledge everyone needs to know is? Sure having numeracy skills and literacy skills are important, but so is customer service and knowing what blood sugar means and what cholesterol can do to you. What about reading a contract and knowing how to unclog a drain? Where do you draw the line? What is sufficient numeracy? Whatever it is, scientists will say it is not enough because they work with it. But most people aren't going to be scientists. The great huge majority aren't going to be. I'd hazard a guess that 90% of college students won't ever do anything remotely related to science in the high math / high theory meaning of the term. R.N.s etc. need to compute. It's true. But I'd rather they use a calculator if in doubt. How much real numeracy ought real people do in a world full of computers?

On the other hand, on3 of the fastest growing professions on the planet is the field of accountancy. They have no great trouble finding candidates to take the degree programs which are growing rapidly. Accountants get paid. I see no one on the street corner selling "numeracy skills." If it is so important people can figure out how to reduce the coins in their pocket, why isn't anyone seeking it out? Ever notice that drugs get dealt in areas where the math scores are horrible? I wonder why.

@ #10: The Onion would disagree with you on that last bit.

As for your question about numeracy in a world of computers, I don't think computers make all that much difference. A calculator can make computing a tip trivial, and allows people with essentially zero math skill handle retail transactions; but it won't give somebody with no clue about statistics even a basic resistance to common forms of sloppy and/or evasive presentation. Computers can do math for you; but they can't understand math for you.