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I read Innumeracy by John Allen Paulos and would like to share more up-to-date and relevant examples of innumeracy to motivate my grade 8, 9 & 10 students. Are there any websites, blogs, books, etc. with lots of examples of innumeracy in the form of pictures, reporting, news articles, etc.?

Here are just 2 examples of what I'm thinking about, I just want to find LOTS more:

UPDATE: Some MESE members recently started a new site devotedd to innumeracy:

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I feel pretty bad about pasting this but – Steven Gubkin Apr 12 '14 at 19:26
The "infographic" on the cover of USA Today every day is frequently nonsensical or misleading. – Eric Lippert Apr 12 '14 at 19:57
Let your students read a random newspaper, and check whether the percentages, fractions and the like are right. It's an interesting exercise, and very rewarding towards their self-esteem, since they will find mistakes for sure. – Quora Feans Apr 13 '14 at 8:11
Reactions to Marilyn Vos Savant's presentation of The Monty Hall Problem: – David Ebert Apr 13 '14 at 15:13
The problem with the Monty Hall problem is that tiny differences in the way that the problem is posed make significant differences to what the correct answer is, and most people are not quite clever enough to handle that. They read the problem and the answer, then try to test someone who is supposedly good at maths, but pose a slightly different problem and feel good about themselves when they get a different answer than what they read as the correct answer. – gnasher729 Apr 13 '14 at 17:01

17 Answers 17

enter image description here

A recent Times article titled Americans Are Bad at Math, but It’s Not Too Late to Fix offered an example -

A&W's "Third Pounder hamburger failed to catch on because

During focus groups, the company discovered that customers believed they were getting less meat. Because the “3” in ⅓ was smaller than “4” in ¼, “customers believed they were being overcharged.”

If this is not classic innumeracy, I don't know what is.

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Thanks; I've moved it. – Andreas Blass Jan 17 at 15:46

There's the Verizon "0.002 cents versus 0.002 dollars" mishap, wherein an unhappy customer calls to complain that he was billed 0.002 $/kB after being told the rate is 0.002 cents/kB. The confusion is perhaps deeper than expected.

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...So this is pretty funny, but I'm not sure whether this is "innumeracy" in action or just a misunderstanding of terminology. It reminds me of confusion around 0.05% being interpreted as 5% (since 0.05 = 5%). But perhaps all of these are examples of what is considered innumeracy. – Benjamin Dickman Apr 14 '14 at 11:56
@BenjaminDickman Is this any better:… – David Steinberg Apr 14 '14 at 15:41
Was Verizon found at fault or the guy? – Jeff-Inventor ChromeOS Aug 11 '14 at 3:24
@Jeff-InventorChromeOS: Verizon admitted the error, and credited back the fees in question:… – Daniel R. Collins Jan 16 at 1:36

Re-Edit (Jan. 2016): Perhaps this does not quite qualify, but I was rather surprised to spot the following question (#6 in the image below) in a recent airplane Mensa quiz:

enter image description here

(Side-note: #2 part 1 has two answers, as does part 4.)

Take a moment to solve #6 before reading on.

Here are the "answers" from that same page (American Way, Jan 2016):

enter image description here

The description is not wrong, but it is a bit different from the phrasing I had in mind (adding 18).

Edit (Jan. 2015): After seeing MESE 7200, a quick google yielded the following from a blog called 360:

enter image description here

Looking at your examples, I recall a story that had been trending online for quite some time: It alleged that Samsung paid Apple about \$1 billion (USD) in nickels (\$0.05 coins) as carried by "30 trucks."

After seeing this shared on facebook far too many times, I posted the following:

enter image description here

It seems the story goes back at least to August of 2012, and was also debunked by Snopes.

(The Snopes estimate is 2,755 eighteen wheelers, but is based off of the judgement being \$1.05 billion, as opposed to the \$1 billion claim in the image above. Scaling my estimation up by 5% gives 2,625 eighteen wheelers. So, the numbers are pretty close. In any event: There is some "innumeracy" here.)

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Yes! More of this! – David Ebert Apr 12 '14 at 17:49
Fact is, the lawsuit in question is now at the next higher court and no money has exchanged hands at all at this point in time. – gnasher729 Apr 13 '14 at 16:58
@gnasher729 Mathematically, though, the salient point is that 30 trucks (even huge ones) would not be able to deliver that many nickels without numerous (around 90...) trips back and forth. – Benjamin Dickman Apr 13 '14 at 19:26
There's also the material limitation of needing the entire production of nickel metal in Madagaskar in 2011 and the entire production of copper in Portugal from 2006 to manufacture this. Also, the treasury only minted about 500,000,000 coins in 2010 (and less than 1/5th of that in 2009), so you'd easily need over 40 years of nickel production to just make that many coins. – Nzall Apr 14 '14 at 11:05
About the deliciousness example: Maybe the original (and final) deliciousness was $0$. Or maybe each pack used to contain only $2/3$ of a bar. – Andreas Blass Jan 17 at 15:45

Let me offer a different type of response, a student's answer to a problem.

The question offered the height of a building, the equation for distance of a falling object, and asked to calculate the time till a rock dropped off the building would fall to the ground. The student used his calculator and the answer was 900 seconds. I asked if that was right, and tried to get him to apply common sense. 900 sec = 15 minutes. Do you think you can see your friend drop the rock, go to Starbuck's, get a coffee, and step back out before it hits the ground? Of course not. His answer was off by a factor of 100.

I'd read Innumeracy a long time ago, but recall that this was one of the author's lessons, the ability to estimate orders of magnitude as being correct or way off. Part of my goal is to ask students if the answer makes sense, in cases where it's not just numbers but real life situations.

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True story: I ordered new carpet flooring for a room in my house. The length of the room was 13 foot 11 inches. The employee took his calculator and typed "13.11 x 30.48 =" to convert into centimetres. I didn't actually manage to convince him of his mistake, but had to ask for a more experienced colleague. Would have been a nasty surprise if I hadn't noticed and they had delivered a piece of carpet 10 inch too short.

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"How would you type in 13 feet and 12 inches?" – Chris Cunningham Apr 13 '14 at 17:46
13.12 :-( Seriously, I told him that 13 foot 11 inch is almost 14 foot which is more than 4.20 metre, not under four metre. Didn't get more than a blank stare. – gnasher729 Apr 13 '14 at 22:47
Well, 13.11*2 = 26.22, But isn't 'just under 14' * 2 'just under 28' and not 'just over 26'? Meters? Never heard of them. – JoeTaxpayer Apr 14 '14 at 1:01

Common accounts in popular press and TV and on-line about "the rate of increase of X is slowing", with varying interpretations, all too often mistaking this for X itself decreasing, etc.

As in "unemployment" or "inflation" or "debt" or ...

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Recent meme that spending 360 million dollars to give 317 million people health care means that you're spending over 1 million dollars person:

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@David Steinberg: Good lord! – Daniel R. Collins Oct 7 '15 at 3:05

A lot of the xkcd "what if" posts, for example this one about hitting golf balls off a spaceship in order to reach escape velocity, seem surprising to me in part, I suspect, because of my own innumeracy. (It turns out, in this case, you might well need a bag of golf balls about 100 billion miles in diameter...)

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A YouTube video of an Illustrious Senator Talking about the cost of health care, 500 trillion dollars. This is more than all the world's wealth, and nearly 8 times all the wealth in the US. I guess he meant Billion, but in Washington, no one is listening anyway.

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enter image description here

This was making the rounds on Twitter this week. There's so much wrong with this, I don't know where to start. It certainly goes back to the manipulation of large numbers, which people in general tend to struggle with.

(Note - in the US, our lottery hit a record $1.5 billion dollar prize. Our billion has 9 zeros.)

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There was this story from 2009, where a town mistakenly believed it had passed a measure which required support from two-thirds of the voters, when it only had support from 66%. (The town accountant (!) calculated two-thirds by multiplying by .66.)

This seems to be a common problem. One of my colleagues was consulted by a representative of the Pennsylvania judicial system to sort out a standing dispute about how to calculate fees which were capped by law at one-third: some judges multiplied by .33, others by .333, and they wanted to know what was right. (They were allegedly impressed by the suggestion that they could simply divide by three.)

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One example that annoys me is when science stories in newspapers (especially stories about high energy physics or astronomy) insist on writing out large numbers, such writing $1,000,000,000,000,000,000,000,000,$ or writing things like trillion trillion, when neither is very useful to a reader with a high school education and neither would make any sense to anyone else. Why not write $10^{24}?$ Scientific notation is taught (in the U.S.) to students who have not yet begun the study of school algebra, and it is used in high school science classes.

Below are two other examples that I've previously posted about in the past.

Example 1: Atlanta mayor: In resettling evacuees, FEMA no help, CNN news article, 14 October 2005. [I previously posted a different version in this 15 October 2005 sci.math post.]

CNN anchor Miles O'Brien on Friday spoke about the challenges facing one city with Atlanta Mayor Shirley Franklin. Atlanta took in 42,000 families fleeing the disaster.

O'BRIEN: All right, let's talk about this, 42,000 families. You're a big city. It's a prosperous city, but that still puts a burden on the city, doesn't it?

FRANKLIN: Well, it certainly does, but I don't think it's a burden that FEMA [Federal Emergency Management Agency] can't help us to address.

The Congress and the president have allocated 62 billion [dollars]. Our estimates are that a family needs assistance for about six months in order to stabilize themselves and that would cost about 11,000 [dollars] per family. The city of Atlanta can't absorb that cost, but we can certainly work with FEMA, if they were willing, to help families get resettled in the city and the metropolitan area.

O'BRIEN: So 11,000 times 42,000. I can't do that kind of math on the fly here. But how much of that money have you seen?

Note: This is trivially estimated via $1$ times $4$ followed by $4+4=8$ zeros, or $400$ million. By missing this, and hence also the fact that $60$ billion divided by $400$ million is $(1.5)(100)=150,$ Miles O'Brien (a well known broadcast news journalist who, incidentally, specializes in science, technology, and aerospace reporting) missed a good opportunity to make a point I suspect he would have liked to make.

Example 2: Formerly posted in this 8 February 2009 math-teach post at Math Forum:

I wonder if the author of the article below has any awareness of just how mathematically illiterate one of the comments below makes professional news writers sound. The author writes "CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos" for something that any college-bound middle school student should be able to do, even without a calculator. Although some of Paulos' comments are nice, especially his speculation "People tend to lump [million, billion, trillion] together, perhaps because they rhyme", going to him in order to check McConnell's numbers is like asking a university linguist for the correct spelling of the word "especially".

I mention this because I've seen many examples of this over the years, especially newspaper writers consulting mathematicians for something that is nothing more than an easy high school level probability or combinatorics problem (easier than many of the problems in standard precalculus and college algebra texts). I don't know whether the reporters really don't know how to work these problems (like checking McConnell's numbers below) or whether they are just using the occasion to get some possibly interesting remarks from someone well known and don't realize how stupid their rationale sounds to a large percentage of their readership.

"Numb and number: Is trillion the new billion?" by Christine Romans CNN's American Morning

"To put a trillion dollars in context, if you spend a million dollars every day since Jesus was born, you still wouldn't have spent a trillion," McConnell said.

CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos.

"A million dollars a day for 2,000 years is only three-quarters of a trillion dollars. It's a big number no matter how you slice it," Paulos said. Here's another way to look at it.

"A million seconds is about 11.5 days. A billion seconds is about 32 years, and a trillion seconds is 32,000 years," Paulos said. "People tend to lump them together, perhaps because they rhyme, but if you think of it in terms of a jail sentence, do you want to go to jail for 11.5 days or 32 years or maybe 32,000 years? So, they're vastly different, and people generally don't really have a real visceral grasp of the differences among them."

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A police (TV) show recently talked about suicide. The policeman said in the US there was a suicide every 40 seconds. Knowing there are 525,600 minutes (from a song in the Broadway musical Rent)in a year, I paused the TiVo and told my wife the statistic was wrong, there aren't 750,000 suicides a year. In fact, the number is closer to 40,000. The show's writers were off by nearly a factor of 20. – JoeTaxpayer Dec 14 '14 at 1:12
@JoeTaxpayer: Here's something I came across this last Friday (today is Monday). (For future readers, see here also.) Now I can somewhat sympathize with the cube root of 125 question, since if you haven't heard or dealt with cube roots in 30 or more years you'll likely not know, especially when put on the spot like this, but look further down for where England's Chancellor George Osborne did the same thing with a child asking him what $7 \times 8$ is. – Dave L Renfro Dec 15 '14 at 14:34
The author of the New Yorker's article earlier this year on the work of Yitang Zang starts like this: "I don’t see what difference it can make now to reveal that I passed high-school math only because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations and x’s and y’s. On test days, I sat next to... smart boys whose handwriting I could read—and divided my attention between his desk and the teacher’s eyes." I rant more about this here: – Daniel R. Collins Oct 7 '15 at 3:03
@Daniel R. Collins: After reading your blog post, I thought you might enjoy my 2010 pi day rant‌​. – Dave L Renfro Oct 7 '15 at 15:02
@Dave L Renfro: That's great! If you don't mind, I'm going to link to that from my blog when I get a chance. – Daniel R. Collins Oct 7 '15 at 22:22

Let me mention a bit of (somewhat) good news in this area. Some years ago, Wal-Mart sold cardboard shipping boxes, labeled as 14" by 14" by 14". They were also labeled in metric units; since an inch is 2.54 centimeters, the dimensions were printed as 35.56 cm. I used to make fun of that --- dimensions of a shipping box accurate to a tenth of a millimeter. The good news is that now those boxes, still 14" on a side, are labeled 35.5 cm. It still seems to claim an unwarranted level of precision, but not as absurdly unwarranted as before.

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A general answer of "every time the unit pricing makes no sense." The specific example:

enter image description here

The product is identical. Presale, 12 for 2.29, 24 for 3.99. Fair enough, a bit of a discount to buy twice the number of pencils. Now, the unit price for the 24 pack is 50% more than the unit price of the 12 pack. I wonder how many of the large pack they'll sell this summer.

All too often I've seen a 1lb package of a grocery item sell for 1.99, and the 2lb package of the identical item, 3.99. But this is the first I've seen the side by side unit price so off.

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Note that technically, the store is not selling individual pencils, it is selling boxes of pencils. If there is an over-supply of boxes of 12 and an under-supply of boxes of 24, I think it makes good business sense to sell the boxes of 12 at a lower "unit price" than the boxes of 24. – Joel Reyes Noche Aug 11 '14 at 1:15
Also, it seems you are assuming that one box of 24 contains exactly the same type of pencils as two boxes of 12. This may not be the case. For example, the box of 12 could have pencils of 12 different colors and the box of 24 could have pencils of 24 different colors. That is, it is possible that the box of 24 has a, say, gold colored pencil and the box of 12 does not. (It may not be so in this particular case, but it may be so in other cases of differing unit prices.) – Joel Reyes Noche Aug 11 '14 at 1:19
Joel - great points. In this case, I happen to have both a 12 and 24 box, bought, presale, and grabbed the last boxes. The first 12 are identical, but the back row in the 24 box does have different colors. There's no added value to me, as I offer these to my students who are trying to produce 3 or 4 color graphs, but I can see other uses where the extra colors are needed. Good call. – JoeTaxpayer Aug 11 '14 at 2:36
Thanks. Also, +1 because I agree that in other cases the objects being sold will all be similar and so there will be cases where very different unit prices won't make sense. – Joel Reyes Noche Aug 11 '14 at 3:06
With an item of food I used to buy, the larger packet was significantly worse value than the smaller one. So much so, that even when the bigger bag was discounted and the smaller one wasn't, it was still better value to buy the smaller one. And here smaller was better, as an open packet would go off, where an unopened one wouldn't. – Jessica B May 30 '15 at 12:01

I would check out some of the work of Edward Tufte ( His book The Visual Display of Quantitative Information is replete with examples of deliberate and accidental mis-use of graphs which mislead readers. I know this is going to sound like a commercial. I have no connection to the author in any way. Unfortunately, I'm only aware of his hardbound books. It's not easy to find online examples for you to look through. You may be able to find copies at a local library. In the book mentioned above, chapter 2 "Graphical Integrity," shows many practical examples of misleading graphs taken from pictures, newspapers, articles, etc.

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Paulos defines innumeracy as "an inability to deal comfortably with the fundamental notions of number and chance" (Paulos, 1988, p. 3). I read (and enjoyed) his book back when I was primarily a software developer and didn't know enough to ask why he wrote the definition that way.

The first example he gives of innumeracy is a newscaster concluding that the weekend had a 100% chance of rain if both Saturday and Sunday had a 50% chance. But he also found that a fellow viewer of the weather report found no problem with the conclusion. Today I would ask Paulos: "What evidence do you have of the level of comfort felt by the participants in your example?" And if he told me "That's not what I meant" I would ask why he wrote it that way if he didn't mean it.

Beside the point? It is not. Definitions are important in our understanding of mathematics as well as in our understanding people (and education).

The book is really mostly about the types of errors similar to this weathercaster's. In this case, conclusions that don't follow mathematically from the available data. Some people collect examples of this sort of thing. I credit user Michael-E2 for bringing this collection to my attention: Collected Forsooths of the newsletter of Royal Statistics Society.

But there is a problem here. So little context is given with these examples that it is impossible to determine whether the errors are truly the result of mathematical errors, some different sort of error, or no error at all.

How bad a problem is this? For education purposes, I say it is no problem at all, depending on the lesson and the point you are trying to make. In my view, mathematics in the world is a sense-making endeavor. An education gives your students a greater capability to make sense of more of the world, allowing the world to actually make more sense to them. And that includes the things people say -- also including the occasional nonsensical-sounding mathematical claim.

How useful is it to be given a list of known mathematical errors? It is certainly of some use. But is not it also of good use to be given a possible error and to have to argue for what it could mean? In some cases, these errors may not reflect realty, but they could have some meaning. How do we use our mathematical understanding to make sense of the situation?

At some point, someone said something; at another point, someone used mathematical knowledge to determine "that was a problematic statement." How do we choose sides? Can we narrow the choices of what the statement could mean? What do we rely on? What arguments can we make?

I think this would be a valuable educational activity: give students statements and have them argue about the mathematical reasoning used. And what they can make sense of. And how they support their own view of the situation. These are types of mathematical reasoning that appear in standards documents, but also are what they will have to rely on if they use math to make sense of their world. And of course, what they will do to convince themselves and others of what they know.

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@JoelReyesNoche - Thanks. I must have gotten interrupted. – JPBurke Jan 28 at 23:15
JP - on re-reading your answer, I'd make two points. It was Douglas Hofstadter that coined the term, and I think that Paulos first book on the topic was written in a way that was a bit scattered. Random examples, some of which that were meta-equivalent to errant apostrophes and commas, which are not quite symptoms of illiteracy. – JoeTaxpayer Jan 30 at 19:15
I addressed Paulos' definition because the original question was framed in terms of Paulos' book, so his definition seems at least somewhat relevant for talking about the types of examples that David was looking for. But thanks for the additional detail! Are you aware of where Hofstadter coined the term? I'm only familiar with one of his works. – JPBurke Feb 1 at 6:29
Got it. He had a series of articles in Scientific American titled Mathematical Games. A May 1982 article "On Number Numbness" was included in a 1985 compilation (Metamagical Themas). "It is fashionable for people to decry the appalling illiteracy of this generation, particularly its supposed inability to write grammatical English. But, what of the appalling innumeracy of most people, old and young, when it comes to making sense of the numbers that, in point of fact, and whether they like it or not, run their lives?" I have the book on my shelf. Innumeracy was just a small bit of the book. – JoeTaxpayer Feb 2 at 12:53
@JoeTaxpayer Excellent, thank you. I am familiar with that quotation, but I didn't know the origin. I appreciate the reference! – JPBurke Feb 3 at 15:46

This was told in an interview I read with a successful businessman (a dane). He told that he learned business from his grandfather. They sold roses on the street, his grandfather told him to offer "a rose for 75 øre (that is, cent of a krone), three for only 2.50 kroner".

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You offer a price break to poor people who can only afford to buy one. – Gerald Edgar Jan 26 at 13:22
This reminds me of a story - a woman sees a beautiful chair in a woodworkers shop and asks how much a beautiful chair costs. \$500. How much for four? $2500 the owner says. She's a bit incredulous, until he explains - it was fun to build that first one, another 3 of that design is just work. – JoeTaxpayer Jan 27 at 1:01

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