Know Thyself

Hypothetical:  A child in your class says, "I hate math! I'm not good at math, and I never will be!" What experiences do you think this child has had in his/her life that contributes to this child's feelings about his or herself, math, and his or her relationship to math?

I think some students who say things like this have been made to feel like “I can’t” is a good excuse.  They’ve heard and used it so many times (and found it to be an acceptable excuse), that it is just easier to say “I can’t” than to just “do.”  Instead of being awed by learning a new subject or topic, the child has it “tuned out” before even trying his/her hand at it.  This is not always the case.  Some children who grow up in environments where an education is not a premium, may be told that they aren’t “smart enough” or are made to feel like being interested in learning is not high on a list of priorities . . . so why try in the first place?  I also look at it from my point of view as a student.  When I was younger, I used to think that I could not do math . . . “I just wasn’t a math person.”  I liked to read and write, and that was it.  In all honesty, I was lazy and didn’t want to take the time to do my work.  Since I barely did my homework and daydreamed through class, I did horribly on tests.  It was like a self-fulfilling prophesy.  I said I can’t, and when I didn’t . . . it was reinforced.  I did just enough to get by all through high school.  Even my first few semesters of college . . . I was more interested in having a good time than doing my work.  It wasn’t until meeting my first wife, that I took education seriously.  She was a biomedical engineering major, and she was brilliant.  I remember telling her how poorly I had performed in school up to that point and being so ashamed.  When we started dating, I made it my goal to prove to her (and myself) that I was not a bad student.  I simply applied myself, and the light came on.  I stopped thinking of math in terms of “problems” to be solved . . . I started thinking of it as a puzzle to be solved.  I know it’s just semantics, but it worked for me.  A problem sounds like something I want nothing to do with, but a puzzle . . . I can wrap my brain around that.  I’ve never looked back.  When she passed away, I left school, but I never stopped trying to learn new things.  I wasn’t learning in an academic setting, but I found independent study and research to be just as rewarding.  When my daughter was born, I was bound and determined to help her realize the joys of learning.  It is my hope that my new-found (relatively in the grand scheme of things) love of education and edification will somehow spark that love in her.        

(Here's a short song that this post made me think about . . . enjoy.)

Idealization in the Face of Aristotelianism

I approached this article, Pendulum Motion: The Value of Idealization in Science, with trepidation.  I honestly thought that 5 pages about the pendulum were probably about 4 pages too long.  How wrong I was.  The opening paragraph made me aware of my ignorance . . . “despite its modest appearances, the pendulum has played a significant role in the development of Western science, culture, and society.”  I expected some scientific applications to be discussed, but having an effect on culture and society?  I was quickly realizing just how wrong my original assumptions were.   Beyond the advantages that science (and the world consequently) gained from Galileo’s study of the pendulum, this piece was aimed at a bigger picture.  The pendulum was key to the ideas being introduced here, but it was really just a vehicle for the true point of the article . . . the importance of idealization in the face of empirical evidence.

My favorite line from the article is the following from Nancy Cartwright, “. . . if the laws of physics are interpreted as empirical or phenomenal, generalizations, then the laws lie.”  Let me begin by stating the meaning of empirical.  Empirical is defined by Merriam-Webster as, “Relying on experience or observation alone often without due regard for system and theory.”   If Galileo had not stepped “outside the box” and questioned the widely held beliefs of the Aristotelian philosophy of observation, science and the world might look quite different than it does today.  The article points this out quite plainly, “. . . the seventeenth century’s analysis of pendulum motion illustrates a different way of thinking that is he methodological heart of the Scientific Revolution . . . the larger methodological struggle between Aristotelianism and the new science.  This struggle is in large part about the legitimacy idealization in science, and the utilization of mathematics in the construction and interpretation of experiments.”  Galileo was making waves that would be felt for centuries to come.  An article found in the Journal of Science Education and Technology, Vol. 8, No. 2, 1999 titled The Role of Idealization in Science and Its Implications for Science Education by Mansoor Niaz stated the following, “As compared to Aristotle and the Middle Ages, when ideal situations were thought to be incommensurable with reality, Galileo makes a sharp break with this tradition by pointing out that, ‘. . . the limiting case, even where it did not describe reality, was the constitutive element in its explanation’” 

How wrong I was indeed.  If not for Galileo’s study of the pendulum, the way we tell time would be vastly different (and apparently off by as much as “plus or minus half-an-hour per day,” which is no trivial amount).  Without these advances in telling time, one could surmise that the Scientific Revolution may have come much later . . .  if it came at all.  This article does an expert job at pointing out the importance of idealization in the face of the unknown.  It is thanks to idealization, more so than the pendulum, that science has advanced as far as it has today . . . and the reason why it will continue to advance.  Thanks to Galileo’s work, Christiaan Huygens was able to continue research into accurate time measurement and create the first pendulum clock, which was accurate to plus or minus one minute per day and eventually plus or minus one second per day.  The world of science opened up in the presence of accurate time measurement.  The article finishes by giving us a glimpse at what accurate timing gave the world of science, culture, and society, “This then launched the era of precision timekeeping that enabled western science to rapidly progress.  It also enabled the longitude problem to be solved, facilitating global exploration, trading and conquest by the European maritime powers.”   

A Beam of Light

My wife and I went to visit a church with friends today.  It was one of those new, "hip" churches where the auditorium looked like a miniature concert arena complete with dramatic lighting and stadium-style seating (they did indeed kick out the Jesus-jams, so to speak) , and the minister wore blue jeans and a goatee.  As he spoke, I began to notice the spotlights on the ceiling pointing towards the stage.  The ceiling and tops of the walls were jet black and completely unlit (save for the spotlights shining on the stage), but  the walls became lighter in color as they approached the floor.  It made me think of Idea #2 from the class discussion  . . . "The hallway (or box) wasn’t dark enough to allow us to see the beam. Some said that a darker room would allow us to see the path of light, because there would be more contrast."  Coming out of the spotlight on the ceiling, one could clearly make out the beam of light coming from it.  However, as the walls became lighter, the beam of light seemed to disappear, and one could clearly see the area of the stage where the light came to a stop.  So I decided that after the service I would walk around and try to view the light from different vantage points.  As my perspective changed, so too did the length of the beam of light that was observable (which harkens back to Idea #4 which said, "Some said that whether you see the path of light, the source of light, or just lit areas depends on where you are standing and looking.").  How much of the beam was visible seemed to depend on how dark, or light, the area surrounding the light was.  In my mind, this rules out Idea #5 which states, "The path of light is only visible when it hits something in the air (like moisture, smoke, or dust).  The auditorium was not smoky or dusty (or moist for that matter).  It was simply darker towards the ceiling.  I could see how this could also prove Idea #1, because the light was much brighter than your average, every day flashlight.  It also lends credence to Idea #3 which states, "A more open space would allow us to see the path of light, because there wouldn’t be any reflections to interfere with the beam."  So the only clear answer I was able to surmise is that Idea #5 is the only idea that does not fit.