The Nature of Teaching…The Teaching of Nature

Drawn from a review in Nation, 10/31/11

At the end of World War II there were hundreds of thousands of unaccompanied children, described by one aid worker as “tired, wan, broken little old men and women,” who had forgotten or never knew how to play.

Sometimes these children were brought together in large camps, but most often they were separated into enclaves by nationality or ethnicity.  There was an issue of “indoctrination.”  Who has the right to teach the “right” cultural setting for such children?  Such concerns even led some to distrust that families would teach these children properly and advocated instead for communal arrangements, where children would live in an extended family of a children’s facility or kibbutz.

One aid worker with the UN testified to the hope and long hours spent “on individual children…endeavoring to revive their love for their family and country.”  Often, postwar nationalists, psychologists and workers for international charity organizations were given to bouts of frustration and despair, fearing that these children were neither malleable nor innocent.  From a different perspective, Anna Freud wrote:  “It is a common misunderstanding of the child’s nature which leads people to suppose that children will be saddened by the sight of destruction and aggression.” Far from being traumatized by such experiences, children were as likely to be scintillated by them.  “If we observe young children at play, we notice that they will destroy their toys, pull off the arms and legs of their dolls and or soldiers, puncture their balls, smash whatever is breakable, and will mind the result only because complete destruction of the toy blocks further play.”

Europe’s lost children were not only cherished as the hope of a new and brighter future, but also feared as the totalitarian henchmen of tomorrow.  One official with the largest Jewish child welfare organization ran a home for ‘Buchenwald boys’…boys who had survived the camp, one of whom was Elie Wiesel.  The official characterized them as “true psychopaths, cold and indifferent by nature, and that this was the reason they were able to survive.”  Others used such sentiments to argue against allowing these children to immigrate…the refugees had experienced trauma that was irreversible and would undermine the well-being of society. …

My imagination can give place to such fears; just as I can picture these children as needing nothing so much as love and the chance to be children.  There is a contradiction in policy, perhaps, but the human condition is full of such contradictions.  What would I have done?  Can I fear and offer love at the same time?  Is that what is called for?  What about respect and the invitation to join a common cause?  “There is a place for you,” I might say. “What do you think?”

(This post is drawn from an article in The Independent School)

Not long ago I sat with a bunch of students from different parts of the country, urban and rural, white, black and Hispanic.  They came from troubled communities and troubled schools.  We started with several activities to break the proverbial ice, including one where students paired off, interviewed one another and then with the whole group back together, they introduced their partner.  I was struck by the number of times they got it wrong.  They had the wrong name for their partner or her school, or maybe they got the name of a favorite movie wrong.

Why?

I’ve been a high school science teacher for over 25 years.  I had long been prepared to expect that I might not be understood; so when I started to work with inner city kids, I worked hard to engage them in conversations so that we could all be on the same page.  When a student offered a comment or question that I did not understand, I would apologize and ask him or her to please say it again.  They were patient and put up with this old guy.

I assumed that any trouble I had understanding them came from the linguistic/cultural distance between us.  But when so many kids were getting it wrong that afternoon, I started to see something else.  There were few linguistic/cultural divides amongst these kids.  They were the same age, immersed in the same pop culture and the same poverty.  Why did they not get it right?  Perhaps, these kids were not speaking to one another with the confident expectation that they would be understood…

Certainly, if we learn to look one another in the eye; if we learn to weigh our words and take their measure from the reaction we get; if we are empowered by that reaction, then when that doesn’t happen, perhaps we learn a different lesson…perhaps we learn to get by, to cope with less than clarity.

Another Anecdote:

Hans Furth, the author of Thinking Without Language (1966), a book about the deaf, was fond of making this observation:  that of all the natural groups within American society, groups defined by profession, by race, religion, etc., the healthiest mentally were the deaf.  Why would this be so?  Why would being deaf mean you were less likely to suffer from schizophrenia or to be institutionalized?  The answer he favored had to do with the sign language.  The argument goes like this…

The gestures of the sign language are everywhere complemented by facial expression.  This is because the signs themselves are open to interpretation that is completed by cues for intensity and the like which come across via the eyes, a nod of the head, etc.  For example, there are several words in English which indicate degree of obligation…ought, should, must.  But in the sign language, there is just one sign…you crook your index finger, somewhat like the way you might hold a pencil.  You indicate the degree of obligation by the intensity of the gesture, ranging from a slight pecking motion to an emphatic sweeping motion. As a result, the deaf look at one another squarely as they talk.  At the same time, as a speaker, you are more energetically committed to what you are saying and so it is difficult to deceive.  That is, the deaf are more straightforward with one another, which contributes to good mental health.

Talk is healthy; it’s important, and for too many it is not happening.  We know, for example, from considerable research that low-income children start to lag behind in vocabulary growth compared to middle class children by the age of three.  It is not a matter of exposure to the spoken word.  Having a television on in the background doesn’t do the trick.  There’s got to be a real person in the room.  But even here there is an important caveat.  It is not the amount of talk children are exposed to.  That is, children of low income parents who were very talkative still lag behind their middle class peers.  In their study, Pan et al (2005) point to a couple of considerations, most especially the degree of negative talk.  Low income parents used more prohibitions, discouragements, and directives than middle or upper middle class parents.  Such considerations heighten the importance of talk for school aged children, but we are sorely disappointed when we visit school classrooms where all too often we find a steady pulse of the wrong talk…talk with just this negative quality; talk that is about doing what you are told.

Anecdote number three:

            Back when I was in graduate school I found myself teaching at the University of Leeds for a year.  I had a lovely time.  Here are two of the more important stories I brought back to the States with me.  One came from the grading form used by the university, where students were listed in two columns as either regular or mature.  Intrigued, I inquired as to what constituted maturity and learned that it referred to those who has interrupted their trajectories as students and had gone off to work or perhaps to travel.  Presumably the university felt that such students returned to their schooling with a greater appreciation for their studies.  A perceptive judgement.

The other story is also about schools.  A senior professor at Leeds took me under his wing.  He was a lovely guy and I can still recall many of our conversations.  One morning as we walked to school he told me about a television program the evening before, where reporters interviewed students from the Westminster School.  Westminster is what we would call a private or independent school and can trace its history all the way back to the late 12^th century.  What Jerry Ravetz had found so striking was not that these young men and women were so bright, but how they took it as perfectly natural that adults would be interested in what they had to say.  This is the complement to what I had seen with students from troubled schools.  I have no doubt that we come hard-wired for language, á la Chomsky’s deep structures, but evidently our capacity to connect to people and to speak directly is learned.  So much of what we become is formed by the interest, or neglect, that others exercise.

Anecdote four:

Let’s look at the standard approach to classroom management.   Class time is choreographed.  Students come in and a warm-up task or drill is on the board, along with a timer that is projected.  Students have five minutes, make that four minutes and fifty-five seconds, to finish this assignment.  Meanwhile the teacher works the room, checking off those who have their homework.  When the five minutes has elapsed, these papers are passed forward and students retrieve books from the back of the room.  They are to read the first four pages of chapter 6.  The timer shows ten minutes.  Then they collect into small groups and work on certain problems.  Time again directs the dance.  An entire hour and a half is consumed this way, with nary a moment for a group discussion.  There are worksheets, exercises like coloring the nucleotide sequences of a stretch of DNA, or labeling items on a map of Europe in the 15^th century.  But there is no talk, no utterances that are whole thoughts or sentences strung together to explain a point of view or to frame a question…save the occasional request for permission to sharpen a pencil or go to the bathroom.  Students offer answers to questions posed, but they are the spoken equivalent of fill-in-the-blank questions and a word or phrase is all that is called for in reply.

Anecdote 4.1:

Such answers, are not really talk.  You can tell because no one is listening.  I am sitting in a physics classroom.  Students have been given a worksheet and the same sheet is projected onto the white board at the head of the class.  Diagram one shows a weight on a table attached to a spring.  The spring is compressed.  What do we call this the sheet and the teacher ask?  One student suggests it is momentum; another that it is newtons; yet another that it is kilograms.  No one is trying to figure out why the preceding answers were wrong.  They are calling out technical terms until the right one is spotted by the teacher.  “Potential energy,” says a student, and they move onto the second diagram.

An Anecdote about Ivanhoe:

            My uncle Marty was a great story teller.  One night he told me the story of Ivanhoe and I loved it so that I went to the library to read it for myself.  It was my first book from the adult section of the library and I was a bit trepidatious, not quite sure it would be OK.  I found it and checked it out, no problem.  When I started to read it, however, I was confused.  There was this swine herd, Garth, and he was walking across the English countryside.  No knights in shining armor, no Ivanhoe.  I took the book back, looking for the real Ivanhoe but all I could find was the one by Sir Walter Scott.  So I tried it again.  This time I must have read fifty or sixty pages.  Still all you got was Garth.  I couldn’t figure it out. Then I cam at it a third time, and somewhere around page 75 it all started to happen.  Garth, by the way, becomes Ivanhoe’s squire.  I loved that book, and I always remembered how you had to give an author a good 75 pages to get the story going.  If you didn’t, you might miss the best story ever.

I taught most of my career at the Park School and the students there were comfortable giving me my 75 pages.  I’d start a story and they figured I would bring it around to the topic at hand in time; though it might not be clear just how I was going to do that.  They were comfortable with the idea of trying to figure out where I might be going.  When I started teaching at the Baltimore Freedom Academy, it was clear I was not getting that lengthy grace period.  I only had a couple of pages, as it were.  We talked about it.  I told them the Ivanhoe story, and I talked about the value of listening and playing with the possibilities, and we talked about the difference between teaching where you tell students what they are supposed to know and teaching where you invite them to think about something that is puzzling.  Garth and I did a lot of walking.

Teaching is not just what we have to say.  It’s not about being thoughtful and clear.  If that were so, then textbooks would be far more effective.  The learning we seek thrives on the connection between the material and the open mind of the student.  That connection requires that we listen better…whether we are talking to a partner at a summer gathering or to a group of students.  If we would do that, who knows how much growth would follow.

References:

Furth, Hans. 1966.  Thinking Without Language: Psychological Implications of deafness. New York: Free Press.

Pan, Barbara, Meredith Rowe, Judith Singer, & Catherine Snow, 2005, “Maternal Correlates of Growth in Toddler Vocabulary Production in Low-income Families.” Child Development, 76(4), 763-782.

Last week we sketched some data on AP testing and closed by asking: how does it all add up?  This turns out to be difficult because a key number is missing: the percentage of students who actually took exams.  That is important for me, because I am centrally concerned with whether AP courses are being offered to meet the needs of students or of school systems.  It is possible that only 3.5 % of the students took an AP course and that each of them passed the exam.  But suppose 25 % of the city’s students are taking AP courses.  Since 3.5 % passed at least one exam; that means roughly 1 out of every 8 in those classes got a 1 or a 2 on the exam.  In other words, 7/8ths of the students in  AP courses would have failed the exam.  Why would we do that?  Isn’t that just hitting these students over the head?  If they are strong enough to be seen as potential for an AP course, give them a course that meets their needs…one that will strengthen their analytical skills and expand their horizons.

Perhaps we are convinced that that AP stands for excellence and that it is important to make excellence available.  Fair enough.  If our students are failing at this opportunity, then we need to re-examine what excellence means in our communities and we especially need to re-examine why we think AP stands for excellence.

The National Research Council (NRC) is not convinced AP is the right way to go.  The NRC is an arm of the National Academy of Sciences designed to help the Academy meets its formal obligation to advise the government on relevant issues.  The NRC has produced research-based documents on issues ranging from nuclear arms to environmental problems, from the state of the national electrical grid to standards in science education.  It also produced a study of advanced study programs, called Learning and Understanding.  This study was sharply critical of both AP and the International Baccalaureate program, chiefly because they tended to take students across far too much material in a superficial manner.  It recommends, instead, that courses use inquiry opportunities to enable students to come to a deeper and fuller appreciation of material.

So here is the central irony.  Our schools are pushing AP study when the research suggests it is misleading our students, failing to give them a solid appreciation of the material.  On top of that it would appear that far too many of the students taking these courses are failing to master the masses of material they involve.  It strikes me as much like fast food or processed food…the bulk is there, but not the quality.  So, is it progress if our students do well?  No. But that they are not doing well is not a good sign either.  We have to change the diet.

I live in an East Coast city which is struggling with a host of problems…an all too familiar litany of deeply disturbing consequences of long-term poverty: unemployment, drugs, crime and schools where too many are doing far too poorly.  As a high school teacher I would often talk to students about college and what a good idea it was:  lots of kids, a small group of earnest adults, and good libraries.  It’s the way I feel about schooling; the way schooling was for me from my first experiences in Chicago at Lawson Elementary (?) through to graduate school…and on through my years as a teacher.  But how very different things must be if so many of our children are choosing to leave school first chance they get.  Just look at the basics.  On the one hand you could be inside where it is warm and dry when it is cold and wet outside, surrounded by kids your age, and the task before you is to come to a fuller appreciation of our culture and its many roots reaching back to classical antiquities in Greece, the Middle East, India, China and everywhere…and indeed reaching all the way back to ice ages, migrations out of East Africa, to the flourishing of mammals some 70 million years ago, even to the first cellular organisms in primordial seas.  Or, you could be standing out on a corner.  What are we doing that so many choose the corner?

Now we come to the recent news story…a story about AP testing.

First let’s try to make sense of the data.  This can be very tricky.  I remember being very confused by data on student drop out rates.  Then I realized that instead of talking about the percentages of students who had started out as freshmen but had not graduated, they meant the number in the senior class who had failed to graduate…a much smaller figure.

So back to AP tests.  The article is reporting on the percentages of students who graduated in 2010 and had passed at least one AP exam with a 3, 4, or 5.  This is not the percentage of students who took the test and passed, but the percentage out of the graduating class who had passed at least one.  Unless everyone had taken an AP exam, this number is inherently smaller than the percentage that actually took an exam and passed.  I assume the value of this number is for comparison purposes.  It is getting at the over-all “strength” of the graduating student body.  I put strength in quotation marks because I question the value of AP programs, but we’ll get to that later.

Out of the more than 30 high schools in the city, most had 0 % passing rates for the graduating classes.  One school had a passing rate of 0.7 % which corresponds to one student out of every 142 in the graduating class passing at least one exam.  For the city as a whole 3.5 % passed at least one.

One other number strikes me as significant: the number of students taking AP exams has doubled over the last decade.

How does this all add up?

I live in an East Coast city which is struggling with a host of problems…an all too familiar litany of deeply disturbing consequences of long-term poverty: unemployment, drugs, crime and schools where too many are doing far too poorly.  As a high school teacher I would often talk to students about college and what a good idea it was:  lots of kids, a small group of earnest adults, and good libraries.  It’s the way I feel about schooling; the way schooling was for me from my first experiences in Chicago at Lawson Elementary (?) through to graduate school…and on through my years as a teacher.  But how very different things must be if so many of our children are choosing to leave school first chance they get.  Just look at the basics.  On the one hand you could be inside where it is warm and dry when it is cold and wet outside, surrounded by kids your age, and the task before you is to come to a fuller appreciation of our culture and its many roots reaching back to classical antiquities in Greece, the Middle East, India, China and everywhere…and indeed reaching all the way back to ice ages, migrations out of East Africa, to the flourishing of mammals some 70 million years ago, even to the first cellular organisms in primordial seas.  Or, you could be standing out on a corner.  What are we doing that so many choose the corner?

Now we come to the recent news story…a story about AP testing.

First let’s try to make sense of the data.  This can be very tricky.  I remember being very confused by data on student drop out rates.  Then I realized that instead of talking about the percentages of students who had started out as freshmen but had not graduated, they meant the number in the senior class who had failed to graduate…a much smaller figure.

So back to AP tests.  The article is reporting on the percentages of students who graduated in 2010 and had passed at least one AP exam with a 3, 4, or 5.  This is not the percentage of students who took the test and passed, but the percentage out of the graduating class who had passed at least one.  Unless everyone had taken an AP exam, this number is inherently smaller than the percentage that actually took an exam and passed.  I assume the value of this number is for comparison purposes.  It is getting at the over-all “strength” of the graduating student body.  I put strength in quotation marks because I question the value of AP programs, but we’ll get to that later.

Out of the more than 30 high schools in the city, most had 0 % passing rates for the graduating classes.  One school had a passing rate of 0.7 % which corresponds to one student out of every 142 in the graduating class passing at least one exam.  For the city as a whole 3.5 % passed at least one.

One other number strikes me as significant: the number of students taking AP exams has doubled over the last decade.

How does this all add up?

We have considered several examples of naïve questioning:

• Where does the sun go at night? And how can the moon reflect the sun’s light when it’s not even there at night?
• Are rocks alive and do plants move?
• How long before my keys land in the clouds?
• How can an equation be true, if it’s never actually been the case?
• How could the atom, which was invented (because it could not be discovered), have become an object?

By asking these questions we move to the heart of what the material is really saying about the world we live in.  It pushes students to make sense of things, allowing them to exercise their critical judgement, just as when Calvin said to his father ‘Yes, but if the sun spends the night in Arizona, how come it rises in the East?’

It is possible to see the role of education as providing basic skills and basic facts; but that is not what we would suggest.  Instead, we should jump right in.  In math and the sciences we should expose children to the beauties of nature, its wonders, and most especially its puzzles.  We need to listen to their questions and their musings, tease out their notions, and explore and test them.  From butterflies to dinosaurs to building with blocks, students should explore and examine, conjecture, compare, take measure, draw and draw conclusions. That is very different from the passive witnessing of material that too often marks our work.

Let me close by going back to ancient Greece for a moment.  The Greek word “physis” is the root of such words as physics and physicians.  “Physis,” it turns out, was an agricultural term referring to the internal push of a plant as it breaks through the soil.  There was a corresponding word in Latin, “natura,” again referring to the push of a seedling breaking through the soil and asserting its place in the scheme of things.  To seek the nature of things was thus to seek their internal push, what made them what they are.  We preserve this vital sense when we speak of someone’s nature in explaining their behavior.  It is, I believe, just this push we cultivate when we teach our children the right way:  a growing sense for their own voice, for the story they can tell of how the world came to be the way it is and how it works.

So the basic idea with the naïve perspective when you are teaching intro physics, high school or college, is that you draw students attention to the authority behind an expression:  why would we think that this is the case?  And it turns out that many times the foundation is idealized…drawn from first principles rather than a data set.  I think of this as “naïve” because it steps back a bit and allows everyone to take stock of just what they are doing…just what is going on.

Atoms:

Perhaps we may look at one more bit of a naïve perspective on the sciences.  Again, the point here is to frame your exploration of these concepts by engaging the root understandings your students bring to the classroom.  Take the atom.  The atom is not large enough to trip over.  It cannot be picked up and looked at.  It can’t be placed on a scale and weighed.  It can’t, in short, be encountered.  How then was it discovered?  It wasn’t.  It was invented.  A product of our imagination, it first appeared on the horizon of intellectual discourse over  2,000 years ago — 2,000 years, that is, before it hovered over the horizon at Hiroshima.

There’s a host of concepts invented by the human imagination, notions of truth, beauty, justice, god, and the like.  But these concepts are only indirectly agents.  We may take to the streets or the battlefield in the name of our god or a notion of what is right, but it is the sword in our hands that does the slaying.  But when the bomb bay doors of the Enola Gay opened what tumbled out was not a set of books and theories, but a bomb, a bomb whose energies came from within the atom. We might think of this as a ‘Papa Smurf’ moment, a moment of real magic, where words and notions, mere incantations brought something into being.

That 2,000 years of argument and analysis which led from the critical thought of ancient Greece to the extra-ordinary destructive might of nuclear weapons is a complex tale, one which reaches out well beyond the confines of technical science.  At its center is that extra-ordinary moment when the atom ceased to be simply a notion invented and became, instead, an object –a thing discovered.  (Note: this is precisely the story I have been exploring in the Tuesday posts on the atom.)

It is unclear to me that this “magic” has occurred at any other time.  How remarkable, what a statement it makes about being human, that it would lead to a whole new category of arms…weapons of mass destruction.  For at this moment there appeared not only a new object, but also a new potential, a new promise in the structure of things.  This potential itself became an important phenomenon as a promise both for the liberation of humankind and for its destruction.  It is a moment our world has yet to escape, and one we must work to more fully understand.

Such work is not idle.

The topic on the table is geometric as opposed to algebraic derivations.  Geometry begins with a small number of ideas that just seem true…axioms and postulates.  We then set about seeing what other truths follow from these initial assumptions.  What I think of as the geometric character of physics is just this: you take certain notions as the case, and you see what follows.

There’s a really good example of this from kinematics, the study of motion.  The idea that you can get distance travelled by multiplying speed times time was worked out a long, long time ago.  But people understood that you don’t always go at a steady rate, so they wanted to know how far you travelled if your speed varied…if you accelerated.  But they couldn’t solve this problem for centuries.  The guy who solved this problem back in the 14^th century (?), Nicholas Oresme, did not solve it by collecting data and working it out from there.  Instead he drew a rectangle.  The area of a rectangle is the product of the sides, say a x b, where a is the height and b is the base.  If we let a represent speed and b represent time, then the area of the rectangle corresponds to the distance.  Now that all by itself is neat because here you have a patch, the area of a rectangle and it corresponds to a line, the distance an object traveled.

Now the kicker.  Imagine a trapezoid.  The base is still b, but the two heights are different, a₁ is shorter, say, than a₂.  If we let a₁ represent the initial speed and a₂ the final speed, then, Oresme argued, the area of the trapezoid would represent the distance travelled while the object accelerated.  The expression for the area of a trapezoid was well-known and so you get the kinematics equation:  x = ½(v₁ + v₂)t.  This equation was true because it made sense, not because of any data.  In fact, if you did an experiment, you judged how good your apparatus was by how close you came to Oresme’s equation.

It turns out that a lot of physics expressions are derived in this way.  We start with some ‘basic’ notions and see where they take us.  Then we go to the lab, try them out and see how close things are.  Then, assuming it is pretty close, we consider the factors that might account for the differences between what we know is the right answer and the way things actually happened.  If the discrepancy between theory and data is too great, people get uncomfortable and now you have an interesting area for further study, tuning up relevant ‘basic’ notions, introducing heretofore unappreciated factors, and maybe devising some new experiment that would be a better window into the way things work.

More naïve physics next week.

Last week we found a standard scientific expression, something we would call the right answer, and yet it has never actually been the case. This opens the door to an interesting set of questions. The most pressing of which are: if it didn’t come from the data where did it come from and what is its authority?
Let’s look at where it came from because it takes us to the heart of physics. The many divisions of the sciences, geology, chemistry, biology, etc. are delineated chiefly by their domain…the earth, the properties of matter, the nature of living things. But physics is different. There is no particular patch of nature that is the domain of physics. What distinguishes physics is its approach to things, and the expression for the period of a pendulum is a lovely example.
Typically, we think of science as collecting data and trying to come up with an effective way to represent it. So we might have students swing pendula of different lengths and try to derive an expression. This is what we may call an algebraic approach: what function best satisfies the data. But what best characterizes physics is a geometric approach.
Whenever I think of geometry I am reminded of a delightful episode from the time I taught high school geometry. We had come to a theorem about isosceles triangles: that the angle bisector of the vertex angle of an isosceles triangle is also a median. A student volunteered to go to the board and prove the proposition. As he walked to the front of the class, he picked up a meter stick. He then proceeded to measure the two segments of the base that had been divided by the extended angle bisector. One was larger than the other. Therefore the theorem was false. The angle bisector was not a median.
It was lovely. And we then went on to explore how there were two triangles –the one on the board and the one in your head. Geometry did not examine the triangles we could draw, only the ones in our heads. None of the theorems actually worked. They were only true in an ideal sort of way. You couldn’t even draw a line; since a line has only one dimension, its length, and so would have no thickness.
This was a great discussion and we came back to it many times. And it fits physics exactly. Just like with pendula, these are truths that never actually happen. The expression in physics comes from a geometric derivation, a notion we can consider next week.

The first two examples of naïve perspective have been directed at younger students.  Here’s a couple for older students.

Pendulums: The heart of the naïve perspective is to re-discover the magic in the science we teach.  And it’s not only basic matters in elementary school.  Take pendulums.  I often played with pendulums across my schooling, learning such key bits as the period of a pendulum does not depend on the weight of the pendulum bob, and more intriguingly that it also doesn’t depend on the vigor of its swing.  Let it sweep out a long arc and it takes no more or less time than just a little swing back and forth.  That’s why you had pendulum clocks; they didn’t slow down as they wound down.

The whole thing gets delightful when we look at the expression for the period of a pendulum:  T = 2π√l/g .  All intro physics book use this expression.  It’s a standard experiment for students.  You have them time the swing for different lengths and plug the results into the expression, and if you did it right your numbers work.

But it is a most curious expression and it is a lot more valuable to take a slightly different tack.  Don’t give the students the right answer first.  Have them measure the times for different lengths of string.  Then ask them to derive an expression for the relationship.  This a great exercise for calculators these days, because if you stick too close to the data you get very arcane cubic expressions.  This, in itself, is a good discussion, but it gets better.

Whatever expression they get, it will lack the trappings of the standard expression.  They are playing with the relationship between l and T, and there is no reason to invoke or introduce the acceleration due to gravity, “g”.  How did it get in there?  Clearly gravity has a lot to do with pendulums going back and forth, but still…where did “g” come from –all you have are numbers on “l” and “T”?  But if you look carefully there’s another more pressing issue; there’s an irrational number right in the middle there – π.  It is not unreasonable for there to be constants in such cases, but how could the constant be an irrational number? That means no one has ever measured the period of a pendulum to be what we say the period of a pendulum is.  You can’t.  Nor can you get it out of some average of results.  The simple fact is: what we know to be the right answer never actually happens.

This is a really critical matter to talk about with students who are serious about what they are learning in school.  Oftentimes, students take the simple bromides teachers and textbooks offer them as the truth with a capital “T”.  Science is about careful, objective observation and just a matter of facts.  But the sciences are much richer and more complicated than that.  Pascal said that the heart has reasons that reason knows not… which is lovely.  We might alter that a bit and suggest that science has reasons that facts know not.  In this case, something is understood as true, even though it never happens.