Mathematical stumbles

In the first part of my teaching career, I taught mainly physics and math, before switching to biology (which I then taught for the rest of my 32 years).  During my time as a physics and math teacher, I was fascinated by the number of students who didn't seem to be able to think numerically.  Some of them were quite skilled at equation manipulation, and so got good grades on quizzes.  The trouble started when they punched something into their calculator wrong, and got an answer that was wildly off -- and then didn't recognize that anything was amiss.

Probably the most extreme example of this was a girl in my physics class.  While we were studying electrostatics, there was a problem set up that was intended to lead you in the end to a value for the mass of an electron.  Well, she entered the numbers wrong, or divided when she was supposed to multiply, or some other simplistic careless error -- and got an answer of 86 kilograms.

She called me over, because when she checked her answer against the accepted value, it wasn't the same.  (Really not the same.  The mass of an electron is about 9 x 10^-31 kilograms -- a decimal point, followed by thirty zeroes, ending with a nine.)

"I must have done something wrong," she said.

I laughed and said, "Yeah, that's kind of heavy for an electron."

She gave me a baffled look and said, "It is?"

I thought she was kidding, but it became obvious quickly that she wasn't.  She knew 86 kilograms wasn't the number in the reference tables, but she honestly had no idea how far off she was.

"86 kilograms is almost two hundred pounds," I said.

She went, "Oh."

I saw this kind of thing over and over, and the problem became worse when you threw scientific notation into the mix, which I suspect was part of the problem with my student.  It was all too common for students to believe that whatever came out of the calculator must be right -- many of them seemed to have no ability to give an order-of-magnitude check of their answers to see if they even made sense given the parameters of the problem they were trying to solve.

[Image is in the Public Domain]

It's easy for those of us who are mathematically adept to be feeling a little smug right now.  But what is interesting is that if you change the context of the question, all of us start having similar troubles -- even expert mathematicians.

A group of psychologists at the Université de Genève set up two different sorts of (very simple) math problems, one of which requires you to think in sets, the other in linear axes.  Here's an example of each:

• Set thinking:  Jim has fourteen pieces of fruit in his shopping basket, a combination of apples and pears.  John has two fewer pears than Jim, but the same number of apples.  How many total pieces of fruit does John have?
• Axes thinking:  When Jane stands on a tall ladder, she can reach a spot fourteen feet high on the side of a house.  Jane is the same height as her twin sister Jill.  If Jill stood on the same ladder, but on a step two feet lower down, how high could she reach?

Both of these problems have the same parameters.  There are pieces of information missing (in the first, the number each of apples and pears Jim has; in the second, Jane's height and the height of the step she's standing on).  In each case, though, the missing information is unnecessary for solving the problem, and in each the solution method is (the same) simple subtraction -- 14 - 2 = 12.

What is extraordinary is that when asked to solve the problems, with an option to answer "no solution because there is insufficient information," people solved the axes problems correctly 82% of the time, and the sets problems only 47% of the time!

Even more surprising were the results when the same problems were given to expert mathematicians.  They got 95% of the axis problems correct -- but only 76% of the sets problems!

I found these results astonishing -- almost a quarter of the mathematicians thought that the information in the "apples and pears" problem above, and others like it, was insufficient to answer the question.

"We see that the way a mathematical problem is formulated has a real impact on performance, including that of experts, and it follows that we can't reason in a totally abstract manner," said Emmanuel Sander, one of the researchers in the study.

"One out of four times, the experts thought there was no solution to the problem even though it was of primary school level," said Hippolyte Gros, another of the authors of the paper, which appeared in the journal Psychonomic Bulletin and Review.  "And we even showed that the participants who found the solution to the set problems were still influenced by their set-based outlook, because they were slower to solve these problems than the axis problems...  We have to detach ourselves from our non-mathematical intuition by working with students in non-intuitive contexts."

What this shows is that the inability to think numerically -- what researchers term innumeracy -- isn't as simple as just a stumbling block in quantitative understanding.  Presumably expert mathematicians aren't innumerate (one would hope not, anyway), but there's still something going awry with their cognitive processing in the realm of sets that does not cause problems with their thinking about linear axes.  So it's not a mental math issue -- the mental math necessary for both problems is identical -- it's that somehow, the brain doesn't categorize the two different contexts as having an underlying similarity.

Which I find fascinating.  I'd love to have the same experiment run while the participants are hooked to an fMRI machine, and see if the regions of the brain activated in sets problems are different from the parts in axes problems.  I'd bet cold hard cash they are.

However, it still probably wouldn't answer what was amiss with the student who had the 86 kilogram electron.

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This week's Skeptophilia book of the week is both intriguing and sobering: Eric Cline's 1177 B.C.: The Year Civilization Collapsed.

The year in the title is the peak of a period of instability and warfare that effectively ended the Bronze Age.  In the end, eight of the major civilizations that had pretty much run Eastern Europe, North Africa, and the Middle East -- the Canaanites, Cypriots, Assyrians, Egyptians, Babylonians, Minoans, Myceneans, and Hittites -- all collapsed more or less simultaneously.

Cline attributes this to a perfect storm of bad conditions, including famine, drought, plague, conflict within the ruling clans and between nations and their neighbors, and a determination by the people in charge to keep doing things the way they'd always done them despite the changing circumstances.  The result: a period of chaos and strife that destroyed all eight civilizations.  The survivors, in the decades following, rebuilt new nation-states from the ruins of the previous ones, but the old order was gone forever.

It's impossible not to compare the events Cline describes with what is going on in the modern world -- making me think more than once while reading this book that it was half history, half cautionary tale.  There is no reason to believe that sort of collapse couldn't happen again.

After all, the ruling class of all eight ancient civilizations also thought they were invulnerable.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]

We've got your number

As a science teacher, one of the things I find fascinating and perplexing is the phenomenon of innumeracy.

An innumerate person is someone who doesn't understand numbers.  We're not talking about simple ignorance of algebra, here; we're talking about someone who has no fundamental comprehension of quantity.

[image courtesy of the Wikimedia Commons]

As an example, take a student of mine who took physics with me, perhaps 25 years ago.  We were studying electrical force, and there was a problem set up that allowed you, with a few given parameters, to calculate the mass of an electron.  So after working for a time, this kid raised her hand, and asked, "Is this the right answer?"

She'd gotten the answer "36 kilograms."

Now, I'll point out from the get-go that she'd made a simple computational error -- divided when she should have multiplied.  What struck me is that she had no idea her answer was wrong.  When I said, "Doesn't your answer seem a little large, for an electron?" she replied, "Is it?  It's what my calculator said."

What's curious about innumerate people is that they're frequently quite good at rote cookbook math -- they can follow lists of directions like champs.  But they have no real sense of numbers, so they have no way to tell if they've gotten the wrong answer.

What's also interesting is that there are people who are pretty competent with small numbers, but lose it entirely with large numbers.  An exercise I used to do with my physics students to help correct this -- which, allow me to say up front, wasn't particularly successful -- was to have them do order-of-magnitude estimation problems.  Within an order of magnitude, how many ping-pong balls would it take to fill the classroom?  How many 1'x1' floor tiles are in the entire school?  How many telephone books in a stack would it take to reach from the Earth to the Moon?  And so on.  Once again, these kids could do the problems, once you'd established a protocol for how to solve them; but I don't think they really got any better at understanding magnitudes through doing it than they had to start with.

Now, scientists at Imperial College in London have gained an insight into why this big-versus-small number comprehension issue might exist; they have found that big and small numbers are processed in different parts of the brain.

The study, led by Qadeer Arshad of the Department of Medicine, said that the idea for the study came from studying victims of strokes whose damage interfered with very specific abilities apropos of number processing. "Following early insights from stroke patients we wanted to find out exactly how the brain processes numbers," Arshad said.  "In our new study, in which we used healthy volunteers, we found the left side processes large numbers, and the right processes small numbers.  So for instance if you were looking at a clock, the numbers one to six would be processed on the right side of the brain, and six to twelve would be processed on the left."

The team then used a procedure to activate one side of the brain more than the other, and asked the volunteers to do various estimation tasks.  Interestingly, people had a systematic tendency to err in a opposite directions depending on which side of the brain was stimulated.  "When we activated the right side of the brain, the volunteers were saying smaller numbers," Arshad explained.  "For instance, if we asked the middle point of 50-100, they were saying 65 instead of 75.  But when we activated the left side of the brain, the volunteers were saying numbers above 75."

Apparently, the context of the task was also critical.  "If someone was looking at a range of 50-100 then the number 80 will probably be processed on the left side of the brain," Arshad explained.   "However, if they are looking at a range of 50-300, then 80 will now be small number, and processed on the right."

Which at least gives a preliminary explanation of why there are students who do just fine with manipulating small numbers, but fall apart completely when dealing with large ones.  I deal with kids for whom 10,000 years ago, 1,000,000 years ago, and 1,000,000,000 years ago all sound about the same -- "big" -- making it difficult to give them any real sense of the time scale of evolutionary biology.

Anyhow, I think the study by Arshad et al. is fascinating, and gives us a further window into understanding how our brains work.  Which is all to the good.  Although it still doesn't quite answer how someone could think that a 36 kilogram electron sounds reasonable.

Bad math, overweight electrons, and zombie cats

Over the 26 years that I've been a public school teacher, I've noticed an interesting (and worrisome) phenomenon: some people seem to have no understanding of numbers.

I'm not saying they're bad in math classes.  In my experience, it's not the same thing at all.  These people often can learn to manipulate mathematical expressions by memorizing the rules, similar to the way you might memorize a noun declension pattern if you were learning Latin.  And they can usually use calculators quite well.  But once the calculator spits out a number, they have no idea if it's right, or even sensible.

I saw this more often in my first five years of teaching, when I taught physics in addition to my current subject, biology.  Physics requires a great deal of sometimes abstruse mathematics, and also dealing with quantitative concepts for which most people don't have a good numerical referent (e.g. charge, torque, frequency).  But you would think things like mass and velocity would be easier, right?  I distinctly recall assigning my students a problem in electromagnetic forces, the ultimate aim of which was to have them calculate the mass of an electron.  One young lady came out with 36 kilograms, and called me over to "check to see if it was right."

I looked at her in some disbelief.  I saw what she'd done, and it was a simple goof, the kind any of us could have done; she'd pressed the "divide" key instead of the "multiply" key on her calculator for one of the terms.  What amazed me was that she didn't look at the answer and immediately recognize that it was wrong.  I said to her, "Doesn't that seem a little... massive, for an electron?"

She shrugged and said, "Is it?"

I said, "36 kilograms is about 80 pounds or so."

She said, "Oh."

I hasten to add that this was an intelligent, articulate young lady, who actually was trying to understand.  She just seemed unable to look at a number, and recognize if it was in the ballpark based on an understanding of what numbers mean.  There was some quantitative common sense that she apparently lacked.

I've seen many other example of this sort of innumeracy.  The young man in introductory biology class who measured the diameter of an amoeba under a microscope, and came up with "114.7 meters."  A very earnest physics student who calculated the speed of revolution of the Moon around the Earth as 9.6 x 10^7 meters/sec -- just shy of the speed of light.  (And it gives me some cause for concern that this last-mentioned young man was bent on becoming an architect.)

I bring this up because having a sense of quantity is absolutely critical to everyday life.  Even if the majority of us don't need to worry about such things as the mass of an electron, having a general concept of what numbers mean -- not just how to manipulate them -- is pretty important.  But because a lot of people lack this skill, they become much more easily persuadable by bad thinking.  If you throw a statistic, graph, or data set at an innumerate person, they often will accept it without question -- and, worse, they won't recognize it if you're lying to them.

I ran into a particularly egregious example of this in this week's edition of our village's newspaper, The Trumansburg Free Press.  The Trumansburg Free Press costs 75 cents, but I guess the name The Trumansburg Three-Quarters-Of-A-Buck Press does sound a little clunky.  Be that as it may, this week there was an article that caught my eye, because it was about a subject that is near and dear to my heart; the problem of cats, both pet and feral, killing native songbirds.

The author of the article, Glynis Hart, went out of her way to be even-handed and "journalistic," presenting arguments from scientists at the Cornell Laboratory of Ornithology (who, understandably, would like to see pet cats never let outside, and feral cats eliminated as a noxious exotic predator) set against those of groups like Alley Cat Allies, who are supportive of outdoor cats in all forms.  Now, myself, I tend to agree with the former -- and I'm not only a birder but a cat owner.  I just see no earthly reason why cats, which are, after all, an introduced species, should have to go outside.  The toll on native bird species is undeniable -- even if we can quibble about exactly how many birds are killed yearly, and unnecessarily, no one denies that the number is large.

So, anyway, I was reading along, and then I got to the following line:  "Alarmists like to cite cat's [sic] amazing fertility to tell you that one pair of cats can produce another 400,000 in two years."

I had to pick the newspaper up off the floor after reading that.  Now, to her credit, Hart goes on to tell us that this statistic "isn't true," but what she doesn't give you is (1) any idea of how not true this is, and (2) the correct sense, that even the most virulently anti-cat, pro-bird person in the world would never claim such an obviously idiotic statistic.

So, anyhow, I decided to find out how fast cats would have to reproduce in order for two cats to generate a total of 400,000 in two years.

After a bit of number crunching and use of logarithms and scientific calculators, I came up with an answer.  If you assume that every cat mating produces a litter, and each litter is made up of five kittens all of whom survive to adulthood and reproduce themselves, it would take a little less than twelve generations to generate 400,000 cats from one pair.  Cram that into two years, and you're talking about one generation every two months.

Yup.  This would require kittens to go from birth to sexual maturity in a little over eight weeks, which makes me wonder if Ms. Hart has ever actually seen an eight-week-old kitten.  But the problem runs deeper than this, because we're not starting from just two cats in the world.  There are millions of cats already, all apparently reproducing at a rate we more commonly associate with fruit flies.  So if this was true, we would be hip deep in felines, and LOLCats would not be a laughing matter.  It would be viewed more like people now view zombie movies.

"Bolt the front door, and lock up the cabinet with the canned salmon, Edith!  I just looked down Maple Street, and the cats is coming!!!"

As with my long-ago physics student whose electrons needed to join Weight Watchers, I'm sure Ms. Hart's rapidly reproducing kitties was just due to a simple mistake -- a miscalculation, miscopied number, or the like.  What bothers me is not that she made the error, something any of us can do, but that no one -- not her, not an editor or a proofreader -- caught it, recognized that it was impossible.

And lest you think that such errors have no long-lasting consequences, allow me to point out that the whole idiotic "chemtrails conspiracy" was launched when a reporter at KSLA News (Shreveport, Louisiana) reported that dew collected in bowls after an unusually persistent jet contrail had 6.8 parts per million concentrations of barium -- well into the dangerous zone -- when the actual amount was 68 parts per billion.  Even though they retracted the claim, and publicly stated that the correct figure was a hundred times smaller than their original number, the first value given still is quoted as the real one in conspiracy theory websites.  KSLA was "pressured to change the value" by Evil Government Agents.  And that little numerical slip-up continues to haunt us, lo unto this very day.  [Source]

So, anyway.  I'm not sure what, if anything, we can do about this one.  As I mentioned earlier, the solution isn't in taking more math classes, because taking math -- or even physics -- is no guarantee against trusting whatever comes out of your calculator, whether or not it makes sense or is even in the realm of possibility.  But it does generate one cautionary note: be careful when you're reading anything that uses statistics.  Don't assume that the numbers are telling you what they seem to be -- or that the numbers are even right.  As usual, the watchword around here is to keep your brains engaged.  At least until the Zombie Cats arrive and want to eat them.  After that, you're on your own.