Tag Archives: innumeracy

Innumeracy and Sharks

Here’s a brief snippet from a conversation about the recent spate of shark attacks in North Carolina as I heard yesterday morning (approx 6AM, 7/4/15) on CNN.

George Burgess (Director, Florida Program for Shark Research):  “One thing is going to happen and that is there are going to be more [shark] attacks year in and year out simply because the human population continues to rise and with it a concurrent interest in aquatic recreation.  So one of the few things I, as a scientist, can predict with some certainty is more attacks in the future because there’s more people.”

Alison Kosik (CNN anchor):  “That is scary and I just started surfing so I may dial that back a bit.”

This marks another great teaching moment spinning out of innumeracy in the media.  I plan to drop just those two paragraphs on my classes when school restarts this fall and open the discussion.  I wonder how many will question the implied, but irrational probability in Kosik’s reply.


Burgess argued elsewhere that

Increased documentation of the incidents may also make people believe attacks are more prevalent.  (Source here.)

It’s certainly plausible that some people think shark attacks are more common than they really are.  But that begs the question of just how nervous a swimmer should be.


CNN–like almost all mass media, but not nearly as bad as some–shamelessly hyper-focuses on catchy news banners, and what could be catchier than something like ‘Shark attacks spike just as tourists crowd beaches on busy July 4th weekend”?  Was Kosik reading a prepared script that distorts the underlying probability, or was she showing signs of innumeracy? I hope it’s not both, but neither is good.


So just how uncommon is a shark attack?  In a few minutes of Web research, I found that there were 25 shark attacks in North Carolina from 2005-2014.  There was at least one every year with a maximum of 5 attacks in 2010 (source).  So this year’s 7 attacks is certainly unusually high from the recent annual average of 2.5, but John Allen Paulos reminded us in Innumeracy that [in this case about 3 times] a very small probability, is still a very small probability.

In another place, Burgess noted

“It’s amazing, given the billions of hours humans spend in the water, how uncommon attacks are,” Burgess said, “but that doesn’t make you feel better if you’re one of them.”  (Source here.)

18.9% of NC visitors went to the beach (source) .  In 2012, there were approximately 45.4 million visitors to NC (source).  To overestimate the number of beachgoers, Let’s say 19% of 46 million visitors, or 8.7 million people, went to NC beaches.  Seriously underestimating the number of beachgoers who enter the ocean, assume only 1 in 8 beachgoers entered the ocean.  That’s still a very small 7 attacks out of 1 million people in the ocean.  Because beachgoers almost always enter the ocean at some point (in my experiences), the average likely is much closer to 2 or fewer attacks per million.

To put that in perspective, 110,406 people were injured in car accidents in 2012 in NC (source).  The probability of getting injured driving to the beach is many orders of magnitude larger than the likelihood of ever being attacked by a shark.


Alison Kosik should keep up her surfing.

If you made it to a NC beach safely, enjoy the swim.  It’s safer than your trip there was or your trip home is going to be.  But even those trips are reasonably safe.

I certainly am not diminishing the anguish of accident victims (shark, auto, or otherwise), but accidents happen.  But don’t make too much of one either.  Be intelligent, be reasonable, and enjoy life.

In the end, I hope my students learn to question facts and probabilities.  I hope they always question “How reasonable is what I’m being told?”

Here’s a much more balanced article on shark attacks from NPR:
Don’t Blame the Sharks For ‘Perfect Storm’ of Attacks In North Carolina.

Book suggestions:
1)  Innumeracy, John Allen Paulos
2) Predictably Irrational, Dan Ariely

Next Steps from a Triangle

Watching the news a couple mornings ago, an impossible triangle appeared on the screen.  Hopefully some readers might be able to turn some first ideas a colleague and I had into a great applied geometry lesson.  What follows are some teacher thoughts.  My colleagues and I hope to cultivate classes where students become curious enough to raise some of these questions themselves.



At first glance, the labeling seems off.  In Euclidean geometry, the Triangle Inequality says the sum of the lengths of any two sides of a triangle must exceed the length of the third side.  Unfortunately, the shorter two sides sum to 34 miles, so the longest side of 40 miles seems physically impossible.  Someone must have made a typo.  Right?

But to dismiss this as a simple typo would be to miss out on some spectacular mathematical conversations.  I’m also a big fan of taking problems or situations with prima facie flaws and trying to recover either the problem or some aspects of it (see two of previous posts here and here).


Without confirming any actual map distances, I first was drawn to the vagueness of the approximated side lengths.  Was it possible that this triangle was actually correct under some level of round-off adjustment?  Hopefully, students would try to determine the degree of rounding the graphic creator used.  Two sides are rounded to a multiple of 10, but the left side appears rounded to a nearest integer with two significant digits.  Assuming the image creator was consistent (is that reasonable?), that last side suggests the sides were rounded to the nearest integer.  That means the largest the left side could be would be 14.5 miles and the bottom side 20.5 miles.  Unfortunately, that means the third side can be no longer than 14.5+20.5=35 miles.  Still not enough to justify the 40 miles, but this does open one possible save.

But what if all three sides were measured to the nearest 10 instead of my assumed ones place?  In this case the sides would be approximately 10, 20, and 40.  Again, this looks bad at first, but a 10 could have been rounded from a 14.9, a 20 from a 24.9, making the third side a possible 14.9+24.9=39.8, completely justifying a third side of 40.    This wasn’t the given labeling, but it would have potentially saved the graphic’s legitimacy.


Is there another way the triangle might be correct?  Rarely do pre-collegiate geometry classes explore anything beyond Euclidean geometry.  One of my colleagues, Steve, proposed spherical geometry:

Does the fact that the earth is round play a part in these seemingly wrong values (it turns out “not really”… Although it’s not immediately clear, the only way to violate the triangle inequality in spherical geometry is to connect point the long way around the earth. And based on my admittedly poor geographical knowledge, I’m pretty sure that’s not the case here!)


Perhaps students eventually realize that the distances involved are especially small relative to the Earth’s surface, so they might conclude that the Euclidean geometry approximation in the graphic is likely fine.

Then again, why is the image drawn “as the crow flies”?  The difficult mountainous terrain in upstate New York make surface distances much longer than air distances between the same points.  Steve asked,

in the context of this problem (known location of escaped prisoners), why is the shortest distance between these points being shown? Wouldn’t the walking/driving distance by paths be more relevant?  (Unless the prisoners had access to a gyrocopter…)

The value of a Euclidean triangle drawn over mountainous terrain has become questionable, at best.


I suspect the triangle awkwardly tried to show the distances the escapees might have traveled.  Potentially interesting, but when searching for a missing person in the mountains–the police and news focus at the time of the graphic–you don’t walk the perimeter of the suspected zone, you have to explore the area inside.

A day later, I saw the search area around Malone, NY shown as a perfect circle.  (I wish I had grabbed that image, too.).  Around the same time, the news reported that the search area was 22 square miles.

  • Was the authorities’ 22 measure an approximation of a circle’s area, a polygon based on surface roads, or some other shape?
  • Going back to the idea of a spherical triangle, Steve hoped students would ask if they could “compute that from just knowing the side lengths? Is there a spherical Herons Formula?”
  • If the search area was a more complicated shape, could you determine its area through some sort of decomposition into simpler shapes?  Would spherical geometry change how you approach that question?  Steve wondered if any students would ask, “Could we compute that from just knowing the side lengths? Is there a spherical Herons Formula?
  • At one point near the end of the search, I hear there were about 1400 police officers in the immediate vicinity searching for the escapee.  If you were directing the search for a prison escapee or a lost hiker, how would you deploy those officers?  How long would it take them to explore the entire search zone?  How would the shape of the potential search zone affect your deployment plan?
  • If you spread out the searchers in an area, what is the probability that an escapee or missing person could avoid detection?  How would you compute such a probability?
  • Ultimately, I propose that Euclidean or spherical approximations seriously underestimated the actual surface area?  The dense mountainous terrain significantly complicated this search.  Could students extrapolate a given search area shape to different terrains?  How would the number of necessary searchers change with different terrains?
  • I think there are some lovely openings to fractal measures of surface roughness in the questions in the last bullet point.


Ultimately, we hope students would ask

  • What caused the graphic’s errors?  Based on analyses above and some Google mapping, we think “a liberal interpretation of the “approximately” label on each leg might actually be the culprit.”  What do the triangle inequality violations suggest about round-off errors or the use of significant digits?
  • The map appeared to be another iteration of a map used a few days earlier.  Is it possible that compounded rounding errors were partially to blame?
  • Surely the image’s designer new the triangle was an oversimplification of the reality.  Assuming so, why was this graphic used anyway?  Does it have any news value?  Could you design a more meaningful infographic?


Many thanks to Steve Earth for his multiple comments and thoughts that helped fill out this post.

Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children.


I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.