Problem-Solving Under Pressure

Near the end of a long morning building a small table, I encountered the following simple geometry problem:  I needed to cut four small rectangles from a square of self-adhesive rubber to serve as the feet of the table’s legs.  So I cut the square into four equal strips, lopped off the end of eachsquares 1and had my feet.four feet

All well and good, but I missed the superior solution that any decent problem solver should have seen immediately:

better solution

This solution would have left me with one long rectangular remainder, as opposed to four small square remainders.

After working on the table for a while, I was mentally and physically drained, and I think this affected my ability to see the better solution.  I guess it makes sense that being tired [and frustrated!] would negatively impact one’s ability to solve problems.

It’s interesting to think about how our physical, mental, and emotional states can affect our problem-solving abilities.  And I think this suggests that problem-solving stamina is something we might want to work on.

By patrick honner, ago

How Do You Study Extinction? Commit Ecocide

E.O. WilsonI watched “Lord of the Ants” on PBS the other night, a documentary about biologist E.O. Wilson.  Wilson possesses the characteristics of the great natural scientist:  a never-ending fascination with the world, the persistence to keep asking questions and to keep looking for answers, and the discipline to focus on and master a specific domain.  Wilson’s impact has been both deep and broad, and he’s even been at the center of a scientific-political-cultural controversy–another benchmark of greatness.

“Lord of the Ants” tells the story of his scientific life–past, present, and future–and it is viewable here.  In Wilson’s story, a couple of cool math-y things caught my attention.

Wilson and Daniel Simberloff, a mathematician-turned-biologist, were interested in studying how ecosystems re-populate after extinction, so they fumigated a small island in the Florida Keys and watched what happened.

In particular, they wanted to know how re-population depends on the area of the region, and its distance from the “mainland”.   Furthermore, they wanted to see if the same number of different species would return, if the same, or different, species would return, and if the relative populations of the various species would return to pre-extermination levels.

Later, Wilson goes on to describe an “Iron Law of Ecology”, namely that a 10-fold increase in habitat doubles the number of species that can be supported there.  This quantitative analysis is obviously very useful for naturalists arguing in favor of preserving more and more natural habitat.

By patrick honner, ago

Who Tests the Testers?

bell curveIt’s tricky business, curving state exams.

An audit by Harvard researchers compared student results on NY State exams (Regents, et al) with corresponding national exams, and it seems that much of the “progress” made by NY students over the past few years was probably illusory.

There are several telling statistics in the report, but none clearer than this:  in 2007, the minimum score on the NY state math exam corresponded to the 36th percentile nationwide.  In 2009, the minimum score on the NY state math exam corresponded to the 19th percentile nationwide.  This effectively defined proficiency as “do better than 19 percent of students across the country”.

In theory, curves for tests can drop if exams get harder, but no one with any knowledge of NY State math exams would make that argument.  Indeed, these exams have been getting easier and easier to pass.  For example, to pass the Integrated Algebra Regents Exam in 2009, a student only needed 30 raw points out of 88.  A passing score of 34% seems pretty low to begin with, but keep in mind that a student guessing randomly on the multiple choice questions alone should get about 1/4 of the questions right, which amounts to 15 points.  Halfway to proficiency.

By patrick honner, ago

Encrypting Secret Messages in Pictures

This is a very cool (and somewhat technical) description of a simple way to use Mathematica to hide a secret message in an image file.

http://blog.wolfram.com/2010/07/08/doing-spy-stuff-with-mathematica/

The basic idea is that you erase the last digit (in binary) of each pixel’s “color channel”, and then use that spot to store part of the secret message.  Given an image, you can then recover the secret message by looking at that last digit in each channel.

This process does change the image somewhat, but not in a way that the normal human eye (with normal viewing equipment) would ever notice.  In the post, the author subtracts the adjusted image from the original:

Chicken Difference

To most viewers, the difference looks like this:

Chicken Diff 1 final

Only under extreme contrast can you actually see the real difference in the two.  Here, we see the invisible ink reappear!

Chicken Diff 2 final

All that’s left is to decode the colors and read your message.  But be careful–sending and receiving secret messages might get you some unwanted attention.

By patrick honner, ago

Maybe Monkeys Can’t Do Algebra

giuseppe monkeyThere is a controversy brewing around Harvard scientist Marc Hauser, his research students, and his published results.  Some of Hauser’s work–which apparently focuses on the cognitive ability of non-human primates–has come under scrutiny, and one of his articles has been retracted.

According to the abstract, the retracted article (“Rule-Learning By Cotton Top Tamarins“) discussed how tests usually performed on human infants were used to “assess whether cotton-top tamarin monkeys can extract abstract algebraic rules”.  Obviously the results of the study are now in question, but it’s a fascinating idea nonetheless.

By patrick honner, ago
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