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!]

Math, nature, nurture, and effort

The Atlantic ran a story last week by Miles Kimball and Noah Smith called "The Myth of 'I'm Bad at Math.'"  In it, we get the hopeful message that people who have claimed all along that they are "bad at math" may not be, that ability at mathematics comes from hard work, not genetics.

(Photograph courtesy of AdamK and the Wikimedia Commons)

They cite a number of sources (and their own experience with educating students) in supporting their assertion.  The most interesting evidence comes from a study at Columbia University by Lisa Blackwell, Kali Trzesniewski, and Carol Dweck, which showed that students who agreed with the statement "You can greatly change how intelligent you are" achieved higher grades than those who agreed with the statement "You have a certain amount of intelligence, and you can't really do much to change it."  Further, convincing students who agreed with the second statement that intelligence was actually under their control had the effect of raising their grades -- and their self-confidence.

On one level, this is hardly surprising.  No one seriously believes that intelligence, or even a more limited slice of it -- like mathematical ability -- is entirely inborn.  We all know examples of people who seem to have a great deal of talent but who are lazy and never develop it.  They cite the Japanese culture as one in which hard work is valued above innate talent, and imply that this is one of the reasons Japanese children score, on average, better than American children on math assessments.  Kimball and Smith state, in their closing paragraph,

Math education, we believe, is just the most glaring area of a slow and worrying shift. We see our country moving away from a culture of hard work toward a culture of belief in genetic determinism. In the debate between “nature vs. nurture,” a critical third element—personal perseverance and effort—seems to have been sidelined. We want to bring it back, and we think that math is the best place to start.

And while I agree with their general conclusion -- that everyone could probably do with putting out a great deal more effort -- I can't help but think that Kimball and Smith are overstating their case.

I have a 27-year-long baseline of watching students attempting to master technical concepts, and there is a difference in the native ability students bring to bear on the topics they are trying to learn.  I still remember one young lady, in one of my AP Biology classes years ago, who spent many frustrated hours attempting to master statistical genetics, and who failed fairly catastrophically.  Her habit of hard work, and an excellent ability with verbal information, led to success in most of the other areas we studied -- in which a capacity for remembering names of things, and the connections between them, matter more than a quantitative sense.  But in statistical genetics, where you have to be able to understand how numbers work on a very fundamental level, that combination of hard work and verbal ability didn't help.

I recall her saying to me one day, after an hour-long fruitless attempt to understand how the Bateson-Punnett method of mapping genes works, "I guess I just have a genetics-proof brain."

In no activity during the year in my introductory biology class do I notice this dichotomy between the math brains and the math-proof brains more than the one we did last week.  It's a common lab, and I bet many of you did it, when you were in high school.  Cubes of raw potato (or some other absorbent material) of different sizes are soaked in iodine solution (or some other dye), and after a given time, they're cut in half to see how far the dye has diffused into the cubes.  After a series of calculations, the far-reaching (and rather counter-intuitive) conclusion is arrived at -- that small cubes have a much larger ratio of surface area to volume than big ones do, and as a result, diffusion is way less efficient for big cubes.  This is one of the reasons that the cells of a whale, a human, and a mouse are all about the same size (really freakin' small) -- any larger, and transport would be hindered by their low surface-area-to-volume ratio.

The calculations aren't hard, but I see many kids losing the forest for the trees.  Quickly.  Which kids get lost seems to have little to do with effort level, and almost nothing to do with verbal ability.  I can typically divide the class into two sections -- the group that will get the concept quickly and easily (usually with a delighted, "Oh!  Wow!  That's cool!"), and the group that after slogging their way through the calculations, still don't see the point -- sometimes, not even after I explain it to them.  Which are in which group seems to have nothing to do with their grades on prior tasks -- or with the effort they exert.

It's ironic that nearly simultaneously with the article in The Atlantic, a paper was published in PNAS (The Proceedings of the National Academy of Sciences) by Ariel Starr, Melissa Libertus, and Elizabeth Brannon, of Duke University.  Entitled "Number Sense in Infancy Predicts Mathematical Ability in Childhood," the study by Starr et al. tells us something fascinating -- that a "preverbal number sense" in infants, who have never manipulated numbers before, predicts their score on standardized math assessments three years later.

Here's how Rachel Nuwer of Science Now describes the experiment:

The researchers showed the babies opposing images of two sets of dots that flashed before them on a screen. One side of the screen always contained 10 dots, which were arranged in various patterns. The other side alternated between 10 and 20 dots, also arranged in various patterns. The team tracked the infants’ gaze—a common method for judging infant cognition—to see which set of dots they preferred to watch. Babies prefer to look at new things to old things, so the pattern of dots that flashed between arrays of 10 and 20 should appear more interesting to infants because the dots were changing not just in position, but in number. Both screens changed dot position simultaneously, so in theory, the flashing pattern changes were equally distracting. If an infant indicated that she picked up on the difference in dot numbers by preferentially staring at the 10- and 20-dot side of the screen, the researchers concluded that her intuitive number sense was at work.

Three years later, the children who achieved the best scores on preschool math assessments were, to a great degree, the ones who had shown innate mathematical sense as infants.

Now, I don't want to imply that hard work isn't important; there's a lot to be gained by effort, and I suspect that even my long-ago student with the "genetics-proof brain" would have gotten it had she persisted.  But Kimball and Smith's assertion, that hard work can trump innate ability, may simply be factually incorrect.  The bottom line may be that perhaps everyone can learn differential calculus, but the hard-wiring of our brains is probably different enough that for some of us, the effort and time that would be required would probably represent the limit of an exponential function as t approaches infinity.