The Nature of Teaching…The Teaching of Nature

Back around the turn from the 17^th into the 18^th century there would have been no real science supply stores.  Most equipment would have been made by local artisans, glass blowers, blacksmiths and the like.  The problem facing researchers who wanted to explore patterns of changes in temperature and their effects was to set up a scale that everyone had access to.  They had to somehow normalize the data…find a base that everyone could set for themselves and then find another readily reproducible point.  Clearly this had not been 32 and 212; such numbers are too arbitrary.  What then?

The first thing to recognize is that 0° Fahrenheit is pretty cold, well below freezing.  Is there anything that happens on a pretty regular basis that is this cold?  There is a clue in the simple act of spreading salt to melt icy patches on your steps or sidewalk.  I’ve always been puzzled by this.  Spreading sand made sense.  It simply provided a grip for your step.  But salt isn’t going to lower the temperature.  How could it melt the ice?  It turns out, and you can try this at home, that salty water needs a colder temperature to freeze than clean water.  When we throw salt onto ice, we’re counting on the smallest crystals of salt mixing with odd bits of liquid water and so the surrounding ice finds itself not cold enough to stay frozen.

So, we may ask, at what temperature does salt water freeze?  It turns out that it depends on how salty the water is.  But if we use a salinity like that of ocean water, a sort of natural saltiness, then it freezes at 0°.  Pretty sweet.

With this as a baseline, Fahrenheit then looked around for a second ‘natural’ temperature.  What temperature would everyone have ready access to?  A clue is a temperature pretty close to 100°, the 98.6° of body temperature.  Fahrenheit had actually called this 100 and not 98.6.  His temperature scale had originally been a centigrade scale!  When scientists decided that freezing clean water was more reliable than the vagaries of ocean salinity and that the boiling point of clean water would be another equally secure and reproducible temperature…which happened to fall at 212°…that shifted things a bit and the ‘normal’ temperature for most of us most of the time became 98.6.

Reason restored.

Put an ice cube in the beaker at the bottom.  Does the level of the water in the tube change?  Not clear, but looks like nothing happened.  Now put the ice cube on the glass container at the top.  You will get a decided change right away.  The water will rise.  Try heating the water at the bottom…nothing.  Now just put your hand around the flask at the top…the water goes down.

What’s going on here?  Why would the water go down, when you add heat?  It takes a moment, but then kids realize the thermometer is upside down.  That it is really “rising” when the water goes down.  Now another problem arises…what is the thermometric fluid? That is, what is the substance that is expanding and contracting with changes in temperature?  It’s the air above the water.  The water is just there so that we can keep track of the changes the air is going through.  This is a neat realization, and students suddenly appreciate the mechanics of thermometers in ways they hadn’t before.

Next, we take the temperature of different items.  We first mark where the water level is.  Then we take the temperature of our hands.  How far down did the water get pushed?  Curiously, it won’t be the same for everyone.  Most of the thermometers in the classroom are quite different from one another and even though there is no reason to think the air trapped in the bulbs is any different, how much they rise and fall is largely a function of the proportions of the glassware.  The one thing they have in common is the starting point: for each thermometer, that is room temperature.  But that means the next day, the base line may well have moved.  The room may be warmer and so the water would drop less with your hands.

What we need is a temperature scale that could standardize measurements.  That’s the problem Fahrenheit solved.

We all know that the Fahrenheit scale is marked by those curious numbers, 32° for the temperature of freezing water and 212° for the boiling point.  But who would make a scale that runs from 32 to 212?  No one.

Years ago I was in a small town in England on the North Sea, Aldeborough.  It was a typical Spring day, cold and damp.  After walking a while along the pebbly beach, Roy suggested we head into town for a cup of tea.  That sounded like a good idea, so we turned our backs on the water and walked up to the road that ran parallel to the shore.  As we crossed, I noticed it was 7^th street.  I mention this because as we walked up the High Street the nest street was not 6^th but 8^th.  Who would lay out a town starting with the number 7?  No one.

I turned to Roy, our English friend and asked him what was up.  He smiled and asked me to look out into the inlet.  I was stunned to discover the tops of a couple of buildings quite a ways out into the water.  Aldeborough, an old town, was giving way to an encroaching sea.  First street was out there.

So, borrowing the morale of this tale, can we re-construct the rationale of the Fahrenheit scale?   It, too, was a centigrade scale.  Can you figure out what 0° Fahrenheit would have been?  Or 100°?

To go back to the three bowls of water, what we are looking for is some operational definition of the ‘hotness’ of the water that doesn’t depend on how it ‘feels’.  That is, you could put a rock in the water and then see how hot the rock becomes, but that would still be a matter of how it felt.  What you want is something that changes with things that are hotter.  The oldest device in this vein that I know of was developed by Galileo, and it turns out to be really easy to reproduce.  All you need is a simple set of glassware and some water.

A typical set-up uses a flask at the top with a one-hole stopper and a length of glass tubing running down to a beaker of water.  I would usually put out a range of options: Erlenmeyer and Florence flasks of different sizes, different lengths and diameters of tubing, and different sizes for the beaker at the base.  A former student who did this lab with his middle-school classes devised a delightful inexpensive and low-tech variation that is just fine.  It uses a small glass bottle, like a ‘Veryfine’ juice bottle.  Drill a hole through the plastic cap large enough for a transparent straw, and then use glue to seal the fit of the straw.  A plastic cup works just fine as a beaker.

The next step transforms the glassware into a thermometer.  First put water into the beaker so that the tubing (or straw) is standing in water, though not touching the bottom of the container.  By the way, you should put some food coloring into the water to make it easier to see.  We want the water to stand about midway up the glass tubing or straw.  Ask students how they might get this to happen.  They might come up with a good idea or two.  If not, ask what would happen if you heated the air in the flask or bottle at the top.  You might try it with a set-up of your own.  The air expands as it’s heated and pushes its way out the only opening…the bottom of the glass tube or straw.  If we used a candle to heat the flask and then pull it away, the air can now cool.  What happens now?  The air will contract and the water will rise up the tube.

This is a great time to talk about vacuums, and the difference between seeing a vacuum as sucking things up and surrounding air pressure as pushing things in.  I would usually offer the plausible maxim that “nothing is not an agent”.  If there is nothing to a vacuum, how could the nothingness actually pull anything?  This conversation would itself lead to all sorts of notions and side explorations with a vacuum pump.

But, let’s get back to the thermometer.  If you have guessed how much air escaped by counting bubbles, for example, and it was a good guess, the water will stop near the middle of the tube, and now you have a thermometer!  That is, you have a device that is sensitive to changes in how hot something is, but at which end?

But let’s leave this for next week…

Let’s pause to catch our breath…what have we seen?  The new electrical atom, an atom shrouded by an electrical force, could be both perfectly hard and capable of bouncing off of other atoms with perfectly elastic collisions.  The kinetic theory has earned its metaphysical foundations and so would become the platform for the development of the laws of thermodynamics.  But before we go on to sketch these events and arguments, it makes sense for us to go back to the classroom.  A core aspect of this study is its labs and the delightful way we can tap into common phenomena.   Let’s look at these for a bit.

The first thing we do is try to make a fire.  It turns out to be really hard. Students try rubbing sticks to together and it never works.  Even when they push things by making fire bows to increase the speed of the ‘drill’, it fails.  They also try striking flint to get sparks that would catch tinder and cause it to light.  These, too, always fail, though you can get some things to char a bit; unless you are very careful about your materials.

Magnifying lenses are much more successful, but glass-making requires pretty sophisticated fires; so clearly, this was not how fires were made early on.  The moral of these efforts is that fires were so difficult to produce and yet so valuable, that ancient peoples would have worked very hard to make sure they did not go out.  To keep the flame going would have been of paramount importance.  As we discussed earlier in this essay, so great was its import that civil society itself may well have been the consequence, not to mention the central role of fires on the altar and the emergence of the priest as keeper of the flame.

Once we have done this, the next question we explore is the difference between heat and hotness.  That there is an issue here is made delightfully clear with a simple experience.  Get three bowls of water.  To the left is a bowl of cold water, on the right –hot water.  In the middle put a mix of hot and cold, lukewarm water.  Now place your left hand in the cold water and your right hand in the hot water.  Let them acclimate to the feel of the two bowls and then put both hands simultaneously in the middle bowl.  Your hands will give you conflicting signals…your left hand will say the water is hot, your right hand will say it’s cold.  Since we have every reason to believe that the water doesn’t care how hot or cold your hands had been before and that it is what it is, we conclude that hands are not a good measure of what’s going on with the water.  We also begin to see that there is some quality to the water that we are seeking that has to do with degrees of hotness.  Hence the need for a thermometer…our next lab.

Yet, even as Herapath’s work was being rejected developments were afoot that would change the picture of the atom and solve the problem of collisions without invoking caloric.  In 1802 Dalton had imagined atmospheres of caloric surrounding each particle.  With the development of the electro-chemical theory of matter in the 1820’s and ‘30’s, the atmosphere of caloric was replaced by an atmosphere of electricity.  The fundamental particles of matter were still perfectly hard, but they were protected from actual contact in a collision by their electrical quality.

Herapath’s hypothesis would be revived in the 1840’s sans problem of hardness.  Here for example is Joule writing in 1848:

…since the hypothesis of Herapath, in which it is assumed that the particles of a gas are constantly flying about in every direction with great velocity, the pressure of the gas owing to the impact of the particles against any surface presented to them, is somewhat simple, I shall employ it…

What had only 30 years before been stymied by an impassable metaphysical barrier was now a simple hypothesis.  In this fashion, we find Herapath’s work built upon by such master physicists as Joule, Maxwell, Kelvin, and Clausius.  The kinetic theory had been rescued by the new electro-chemical atom…a complex, whirling, and clanking beast most fit for the new industrial age.

In a text on the kinetic theory, written in 1872, Maxwell pictures atoms colliding.  He begins by observing that the molecules of a gas are not acted upon by any sensible force for the greater part of the time, and so move in straight lines with uniform velocities.  When they move close enough to one another, the molecules act on one another with a force that grows as they approach and then weakens as they move apart.  He then adds: “the free motion of a molecule takes up much more time than that occupied by an encounter.”

And so, Maxwell sets aside the conflict between Herapath and the reasoning of Newton et al.  The bulk of the time a particle is effectively free, and not in collision.  Collisions are mediated by some force, unspecified and unproblematic.  A collision, or an encounter, was simply strong repulsive forces acting as particles approached one another and receded…at least that is all a collision had become.  It had been something very different when the young Herapath had laid out his argument.

Herapath first submitted his paper to the Royal Society in 1816.  They rejected his paper.  When, in 1821, his paper did appear in The Annals of Philosophy, there is little evidence of support or acceptance.  His work lay abandoned on the margins of physical science until the 1840’s.  Why did this happen when we know that so much of his work was spot on?

Some have looked at this episode as an instance where the customary open-minded objectivity of science had been distorted.  Shadows of elitism or intellectual cowardice, they surmise, had cast a dark shroud over Herapath’s work and prevented it from being evaluated properly.  That is, Herapapth’s rejection was an instance of scientists acting under petty influences and not as good scientists.

But maybe that was not the case.  Maybe we need to open our sense for what was at stake in Herapath’s work.  Let’s look at the actual criticism contemporaries offered and try to see what made his seemingly well-formed work so objectionable.

There were four published replies to Herapath’s work, expressing concern about various aspects…he was too speculative, relied too much on mathematical deductions…but above these objections stood their concern with his portrayal of the molecular world.  He envisioned perfectly hard particles moving freely, slamming into one another and bouncing elastically…without losing momentum or energy.  This was the rub.  Herapath, it turns out, had run smack into a long standing metaphysical controversy.  Perfect hardness logically barred elasticity.

Here’s how Newton had put it in his Principia near the end of the 17^th century:

For bodies which are either absolutely hard or so soft as to be void of elasticity, will not rebound from one another. Impenetrability makes them only stop.  If two equal bodies meet directly in vacuo, they will by the Laws of Motion stop where they meet, and lose all their Motion, and remain in rest, unless they be elastic, and receive new motion from their Spring.

Since it was assumed that atoms as the fundamental particles of matter were perfectly hard…how else could they maintain themselves…they could not bounce off of one another, but would simply ‘splat’ like a snowball against a wall.  Without a source of Spring, without a repulsive force like that of Caloric, Herapath’s vision of the hidden world of atoms and molecules was rooted in contradiction.

Herapath replied to his critics, urging that such metaphysical arguments be set aside.  Grant elastic collisions, grant that somehow momentum is exchanged and no motion is lost.  The proof, he urged, should lie in the proverbial pudding –in the fruitfulness of his analysis.  His strategy works.  The result is elegant mathematically and satisfies the data.  Surely that is enough.

But the critics did not agree.

We do not need to charge Herapath’s colleagues with intellectual cowardice, elitism, or the like.  We need simply to recognize that deep metaphysical notions about the way things must be had led reasonable scientists to deny the premises of Herapath’s work.  Though the molecular world is un-observable, it is clear that Herapath’s views of molecular collisions were un-acceptable and reasonably so.  It was not a case of ‘just the facts’.

Across the countryside of England there lies an ancient road, the Ridgeway, carved by the travels of the earliest Britons.  Connecting such sacred sites as Stonehenge and Silbury Mound, the Ridgeway today is hardly discernible in parts –a mere path across the fields –while elsewhere it underlies the concrete and asphalt of the motorway from London to Bristol.  Why is it that the Ridgeway, or Herapath as it is called in the West Country, has been treated so differently by history?  Why should some parts lay obscure, abandoned to the margins, whilst others have been built upon again and again, remaining vital –though of course transformed?

As with one Herapath, so with another.  John Herapath, also of the West Country, has left a similarly mixed legacy.  Some of his work was built upon, some not; and little of it was received with enthusiasm by his contemporaries.  What was at work here?  What were the laws of commerce and traffic that determined what would be built upon and what not?

Born near the turn into the 19^th century, Herapath participated in the scientific world of Dalton, Davy, Laplace, and Joule.  He also participated in the booming world of the railroad.  The railway journal he edited was successful; in time it would be named Herapath’s Journal.  As a young man Herapath threw himself into the debate over the nature of heat, and his entry effectively marked a new plateau for the notion that heat was not an ethereal substance, a material stuff, but a certain busy-ness within matter, a mode of motion.  This concept, the kinetic theory of heat, would become one of the core concepts of modern science, one that is still vital, though of course transformed.

Notable scientists before Herapath’s day had offered a notion of heat as a quality of motion, scientists such as Francis Bacon, Robert Boyle, and Daniel Bernoulli.  And in his own day there were those who leaned this way:  Humphrey Davy, president of the Royal Society, Count Rumford, the founder of the Royal Institution, Thomas Young one of the first to propose a wave theory of light, each rejected the caloric theory and sought an alternative in some quality of internal agitation of the particles of matter.

But Herapath went further than they had.  He proposed that the molecular world was a chaos, with particles of a gas, for example, moving freely, in straight lines, colliding with one another.  These collisions, he went on, should be taken to be perfectly elastic, like billiard balls in an ideal world, and he drew upon the mathematical expressions that had been worked out for such collisions.  He then proposed that the temperature of a body reflects the average value of a particular kinematic quality, momentum.  The modern theory links temperature to kinetic energy, not momentum, and there is a difference here, but Herapath was able to derive the general gas laws in a manner closely paralleling modern analyses.  With this as the core of his theory, he went on to consider a host of heat phenomena: specific heats, latent heats, heats of chemical reaction, etc.  Notably, his explanations of diffusion and of adiabatic phenomena, which followed naturally from his principles, addressed major issues then confronting the caloric theory.  His insights and the over-all form of his analysis lay at the foundations of the modern understanding.

This was a most impressive grasp of the truth with a capital “T”.  So much so that we are left with a most curious problem:  How is it that so much right-thinking, so much of what we know to be good science, the facts and perspectives we accept today, how could they have been found false?  For that is what happened.  The scientific community rejected his views.

After the nature of science and content, we come to the third leg of our platform, pedagogy.  I understand science to be essentially about making sense of things, and so that is my aim as I teach: that students would understand the material, be able to give an account of concepts and theories and the things they explain.  I understand that a crucial resource in coming to understand scientific concepts and theories is the history of science, which offers us insights into what it really means to buy into these concepts and theories.  But we still have to teach it.  How will we do so?

The leading feature of the little sketch we traced was the puzzle:  is heat a dancer or a dance?  There is a tendency to tell students what they need to know, instead of asking them how they might make sense of things.  In an intriguing book, A is for Ox Barry Sanders talks about the importance of talk: “Young children need to feel lost, confused, and bewildered enough to concoct their own stories in order to climb out of tight situations.  They need to string together narrative threads from here and there to reach meaning in their lives.”

As with the development of children, so too with the development of students.  Telling students what they need to know strikes me as too passive.  Students do not uncover.  That is the work of the text and the teacher.  There is no driving perplexity and so no push to string together a narrative that explains things.

By framing our study with the option, dance or dancer, we move away from telling students the answer.  In effect, the alternative to teaching students the answer is to teach them two answers and to invite them to sort out their strengths and weaknesses, the issues that critically distinguish them.

This respect for the role of problems sets up another clear aspect of our sketch.  The lab investigations were framed by problems rather than protocols.  Instead of carefully laying out what students were to do, I worked to carefully lay out an issue and invited them to figure out how they might resolve it.  I had a colleague once who would regularly stay late working on labs.  I’d stop by to see what he was up to and each time it was clear that he was the one who was really doing the lab.  All his students would do is turn on some apparatus and collect the numbers off the dials.  The effort to make nature answer a question we pose; that is a neat trick.  To go back to dancers and dances, I would ask how they could show whether there had been a change in weight when you added heat? They designed the approach.  And when they systematically would find that the water lost weight, they designed some follow up investigation that would try to control for evaporation.  It was a student, by the way, who suggested comparing equal volumes of hot and cold water.

We can see the value of wrong answers in our discussion with students about dancers and dances and how we might understand the nature of fire and of heat.  The goal is that they come to appreciate the modern understanding, the kinetic Theory, where heat is a dance. But we get there via a century long debate between those who favored the Caloric theory, a materialist theory which envisioned heat as a dancer, and those who saw heat as a dance.

We might think of this as a variation on a 19^th century biological notion: ontogeny recapitulates phylogeny…that the life of an individual entity re-traces its developmental history.  Our version is something like this: effective understanding re-traces the developmental history of the concept.  That is, the best way to get students to appreciate a given concept is to take them through key issues in the development of that concept.

The Caloric theory was the prevailing understanding of the nature of heat across the hundred years from the middle of the 18^th to the middle of the 19^th century.  A key consequence of the Caloric Theory was the idea that heat would be conserved.  The success of the conservation of heat principle affirmed the Caloric Theory, and Joseph Black used it to frame the curious notion that not all heat was hot.  There was a latent caloric, as well as a sensible caloric.  Latent caloric was the heat materials absorbed as they melted or boiled; what we now call heats of fusion and evaporation.  A gram of ice at 0° Centigrade, for example, absorbs 80 calories of heat as it melts to become a gram of liquid water still at 0°.

There is an intriguing illustration of how the Caloric theory explained rather puzzling heat phenomena that comes from a Cambridge physicist, J.B. Emmett in 1820.  In the old days, before matches, blacksmiths could either keep their fires smoldering which consumed lots of fuel, or they had to come up with a good way to start a fire.  They did…an iron nail!  If you strike a nail hard enough, it will glow red hot and can be used to ignite flammable material like straw.  Now here is the neat part.  After the nail is red hot, you need to put it back into the fire for a while.  If you just let the nail cool, you won’t be able to pound the nail up to red hot again, no matter how hard you hit it!  This was clearly problematic for those who saw heat as a dance.  Why wouldn’t the nail become red hot again? For the Caloric theory, this was more straightforward.  Pounding a nail drives out its caloric, and the surface becomes red hot.  Putting it into the flame allows it to replenish its inner store, ready for the next occasion.  Just cooling it means that there is not enough caloric inside to pound out next time to the point where it could become red hot.  That is, the heat is not generated by the pounding, but is released by the pounding from an inner store.  Heat is a material substance.

Isn’t the human imagination wonderful?

Linking the wonders of science to the human imagination changes the way we should think about engaging our students…less a matter of the proportions of nature than the outrageous proportions of how we understand these things.  Furthermore, if I am aiming for students to make sense of things, I have found that I cannot live in a house made of right answers.  This may seem a bit bizarre, but perhaps I can explain myself.

There is a tendency to think that as we teach science at each different grade level we are laying a solid foundation and then adding to it carefully, brick by brick.  But I prefer a different construction metaphor, one borrowed from the tale of the three little pigs.  When they are young the idea is to help children build a house reflecting how they understand the world.  We don’t need to clutter it with atoms or chemical formulas, with curious notions like the earth is a planet and the sun is standing still even as we watch it move across the daytime sky.  This will be a house of straw.  It will be tested and like Jesse they will see that their house is wanting, that it was not strong enough.  So we help them build another house.  This one of sticks.  It will be stronger, but still it will be wanting and so they will go on re-building until they make that house of bricks, the house that is still standing so far.  But we should not forget that the modern house of science is a most bizarre house, built not of atoms but of quarks and other curious entities guided by quantum mechanics and relativity and perhaps held together by strings and who knows what else.  It is a house that is a lifetime in coming, a lifetime of building and re-building.  Given the proportions of this house, we realize that we haven’t been teaching the right answers in the first place!  We’ve been letting a curious attachment to a certain manner of wrong answer clutter things.

Stop and think about other areas of school study, like learning how to read.  We may have in mind that our high school graduates will read Shakespeare or Faulkner, but we do not teach children the vocabulary they will need then.  We give them good stories, age-appropriate stories.  We may start with Ezra Jack Keats and move on to Beverly Cleary and after Cleary, there’s Walter Dean Meyers, or perhaps Rudyard Kipling, I loved Kipling.  The point is we nurture their growth as readers.  Let’s do the same in the sciences.