Thermal Energy and Internal Energy

The AQA GCSE Science specification calls for students to understand and apply the concepts of not only thermal energy stores but also internal energy. What follows is my understanding of the distinction between the two, which I hope will be of use to all science teachers.

My own understanding of this topic has undergone some changes thanks to some fascinating (and ongoing) discussions via EduTwitter.

What I suggest is that we look at the phenomena in question through two lenses:

  • a macroscopic lens, where we focus on things we can sense and measure directly in the laboratory
  • a microscopic lens, where we focus on using the particle model to explain phase changes such as melting and freezing.

Thermal Energy Through the Macroscopic Lens

Screenshot 2019-04-14 at 14.29.39.pngThe enojis for thermal energy stores (as suggested by the Institute of Physics) look like this (Note: ‘enoji’ = ‘energy’ + ’emoji’; and that the IoP do not use the term):Screenshot 2019-04-14 at 14.22.10.png

In many ways, they are an excellent representation. Firstly, energy is represented as a “quasi-material entity” in the form of an orange liquid which can be shifted between stores, so the enoji on the left could represent an aluminium block before it is heated, and the one on the right after it is heated. Secondly, it also attempts to make clear that the so-called forms of energy are labels added for human convenience and that energy is the same basic “stuff” whether it is in the thermal energy store or the kinetic energy store. Thirdly, it makes the link between kinetic theory and thermal energy stores explicit: the particles in a hot object are moving faster than the particles in the colder object.

However, I think the third point is not necessarily an advantage as I believe it will muddy the conceptual waters when it comes to talking about internal energy later on.

If I was a graphic designer working for the IoP these are the enojis I would present:Screenshot 2019-04-14 at 15.10.50.png

In other words, a change in the thermal energy store is always associated with a temperature change. To increase the temperature of an object, we need to shift energy into the thermal energy store. To cool an object, energy needs to be shifted out of the thermal energy store.

This has the advantage of focusing on the directly observable macroscopic properties of the system and is, I think, broadly in line with the approach suggested by the AQA specification.Screenshot 2019-04-14 at 15.32.13.png

Internal Energy Through the Microscopic Lens

Screenshot 2019-04-14 at 15.27.25.png

Internal energy is the “hidden” energy of an object.

The “visible” energies associated with an object would include its kinetic energy store if it is moving, and its gravitational potential energy store if it is lifted above ground level. But there is also a deeper, macroscopically-invisible store of energy associated with the particles of which the object is composed.

To understand internal energy, we have to look through our microscopic lens.

The Oxford Dictionary of Physics (2015) defines internal energy as:

The total of the kinetic energies of the atoms and molecules of which a system consists and the potential energies associated with their mutual interactions. It does not include the kinetic and potential energies of the system as a whole nor their nuclear energies or other intra-atomic energies.

In other words, we can equate the internal energy to the sum of the kinetic energy of each individual particle added to the sum of the potential energy due to the forces between each particle. In the simple model below, the intermolecular forces between each particle are modelled as springs, so the potential energy can be thought as stretching and squashing the “springs”. (Note: try not to talk about “bonds” in this context as it annoys the hell out of chemists, some of whom have been known to kick like a mule when provoked!)

Screenshot 2019-04-14 at 16.02.55.png

We can never measure or calculate the value of the absolute internal energy of a system in a particular state since energy will be shifting from kinetic energy stores to potential energy stores and vice versa moment-by-moment. What is a useful and significant quantity is the change in the internal energy, particularly when we are considering phase changes such as solid to liquid and so on.

This means that internal energy is not synonymous with thermal energy; rather, the thermal energy of a system can be taken as being a part (but not the whole) of the internal energy of the system.

As Rod Nave (2000) points out in his excellent web resource Hyperphysics, what we think of as the thermal energy store of a system (i.e. the sum of the translational kinetic energies of small point-like particles), is often an extremely small part of the total internal energy of the system.Screenshot 2019-04-14 at 16.31.55.png

AQA: Oops-a-doodle!

My excellent Edu-tweeting colleague @PhysicsUK has pointed out that there is indeed a discrepancy between the equations presented by AQA in their specification and on the student equation sheet.

If a change in thermal energy is always associated with a change in temperature (macroscopic lens) then we should not use the term to describe the energy change associated with a change of state when there is no temperature change (microscopic lens).

@PhysicsUK reports that AQA have ‘fessed up to the mistake and intend to correct it in the near future. Sooner would be better than later, please, AQA!

Screenshot 2019-04-14 at 16.44.38.png

References

Nave, R. (2000). HyperPhysics. Georgia State University, Department of Physics and Astronomy.

 

 

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It is F=ma, you know

“It is cheese. (Caerphilly.)”

Main research finding of the first manned Welsh mission to the Moon, as reported by Max Boyce c. 1974.

A Brief History of F=ma

When writing A Brief History Of Time, it is said that a literary agent warned Stephen Hawking that each mathematical equation that he included in the final draft would halve its eventual sales.

And one of the complex equations that Hawking wished to include? A summary of Newton’s 2nd Law: F=ma.

In other words, the force F acting on an object is equal to the mass m of the object multiplied by the acceleration experienced by the object.

(In the end, Hawking opted to include only Einstein’s E=mc2 in what turned out to be the ultra bestselling A Brief History of Time.)


F=ma is Newton’ s 2nd Law: NOT!!!

@SciByDegrees wrote an interesting post arguing that the old Physics teacher’s standby of summarising Newton’s Second Law of Motion (N2) as F=ma is wrong.

The gist of his argument (and it’s a hard argument to counter) is that F=d(mv)/dt is a far better expression of the law than the F=ma version because it covers a wider range of circumstances.

This states N2 in terms of momentum, where momentum is the product of mass m multiplied by the velocity v. More exactly, it says that force acting on an object is equal to the object’s rate of change of momentum: or, if you prefer, the change in momentum divided by the time taken for the change is equal to the force.

This is the version of N2 stated in most dictionaries of Physics. For example, the Oxford Dictionary of Physics (2015) p. 383.

I know, because on reading @SciByDegrees’ post I immediately looked up N2 in the Dictionary with the express intention of countering the argument. Imagine my consternation and horror when I found that I was wrong. (Actually, not that much consternation and horror: I am fairly inured to being wrong as it happens fairly often…)

The argument suggests that, just like V=IR is not a statement of Ohm’s Law unless R has a fixed value (like a fixed length of wire at a constant temperature), F=ma is not a sufficient statement of N2 unless the mass m is constant.

For example, if we consider a rocket capable of producing a steady 1000 N of thrust; at t=0 its mass is (say) 10 kg so its acceleration is 100 m/s2. However at t=1 s its mass has decreased by 1 kg so the acceleration is now 111 m/s2 even though the thrust is still 1000 N so obviously F is not proportional to m so F does not equal ma in this situation.


Feynman as the new Aristotle

Richard Feynman (1965) wrote along similar lines in his justly famous Lectures on Physics:

Thus at the beginning we take several things for granted. First, that the mass of an object is constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their masses add. These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless. For example, suppose the mass varied inversely as the velocity; then the momentum would never change in any circumstance, so the law means nothing unless you know how the mass changes with velocity. At first we say, it does not change.

However, I think Feynman is considerably oversimplifying what Newton said here. Dare one suppose that Feynman, who had an enviable natural facility for talking intelligently and arrestingly about nearly any subject under the Sun, had perhaps skimped a little on his background reading?

Incidentally, does anyone think that physicists (especially physics educators — myself included) are beginning to treat Feynman as the medieval scholastics are reputed to have treated Aristotle? That is to say, he is regarded as the final word on everything; or, at least, everything physics-related in the case of Feynman.


What Would Newton Do?

George Smith (2008) points out that:

The modern F=ma form of Newton’s second law nowhere occurs in any edition of the Principia [ . . . ] Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces.

This, I think, supports my contention that F=ma is as good a modern reformulation of Newton’s 2nd Law as any other.

If we go back to the rocket example, the instantaneous acceleration at t=0 and t=1 s can be calculated using F=ma (provided we take account of the change in m, of course). In effect, we are considering the change in motion due an instantaneous impulse here.

Please note that I would cheerfully concede that F=d(mv)/dt would yield a better and more productive analysis of rocket motion if we are considering the continuous action of the force over time rather than at isolated instants.

The analogy with V=IR is useful here. V is always equal to I times R but V is only directly proportional to I over a continuous range of values of I for a limited set of conductors we call Ohmic conductors whose resistance R is fixed over a range of physical conditions. Likewise, F is always equal to m times a but F is only directly proportional to a for a continuous range of values of a when we are considering a system whose mass is fixed.

As V=IR is neutral with respect to whether R is fixed is not, I believe that F=ma is neutral with respect to whether m is fixed or not.


Will the Real Second Law Please Stand Up?

What is Newton’s Second Law? Is it a definition of force? Is it a definition of mass? Or is it an empirical proposition linking force, mass and acceleration?

Brian Ellis (1965) argues that it partakes of all three:

Consider how Newton’s second law is actually used. In some fields it is unquestionably true that Newton’s second law is used to define a scale of force. How else, for example, can we measure interplanetary gravitational forces? But it is also unquestionably true that Newton’s second law is sometimes used to define a scale of mass. Consider, for example, the use of mass spectrography. And in yet other fields, where force, mass and acceleration are all easily and independently measurable, Newton’s second law of motion functions as an empirical correlation between these three quantities. Consider, for example, the application of Newton’s second law in ballistics and rocketry [ . . .] To suppose that Newton’s second law of motion, or any law for that matter, must have a unique role that we can describe generally and call the logical status is an unfounded and unjustifiable supposition.

In some senses, I suppose we might like those unfortunate nations in Gulliver’s Travels who fought a long and bitter war over the question of whether one should eat a boiled egg from the pointed or rounded end:

During the course of these troubles, the emperors of Blefusca … accusing us of making a schism in religion, by offending against a fundamental doctrine of our great prophet Lustrog, in the fifty-fourth chapter of the Blundecral (which is their Alcoran). This, however, is thought to be a mere strain upon the text; for the words are these: ‘that all true believers break their eggs at the convenient end.’ And which is the convenient end, seems, in my humble opinion to be left to every man’s conscience.


REFERENCES

Ellis, Brian. “The origin and nature of Newton’s laws of motion.” Beyond the edge of certainty (1965): 29-68.

Feynman, R. P., Leighton, R. B., & Sands, M. (1965). The Feynman Lectures on Physics; vol. 1 (Accessed from http://www.feynmanlectures.caltech.edu/I_09.html on 11/4/19)

Smith, George, “Newton’s Philosophiae Naturalis Principia Mathematica“, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2008/entries/newton-principia/&gt;.ewto

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Ut tensio sic vis

And if two men ride of a horse, one must ride behind.

Shakespeare, Much Ado About Nothing

Sir Isaac Newton stands in popular estimation as the foremost intellect of his age; or perhaps, of any age. If a person is never truly dead while their name is spoken, then Sir Isaac stands with us still: partially overshadowed by Einstein at the dawn of the twentieth century, maybe, but never totally eclipsed.

But in the roiling intellectual cauldron of the Age of Enlightenment, even such a venerable polymath as Newton had some serious competition. As Newton himself modestly observed in a letter to a contemporary in 1676: “If I have seen a little further it is by standing on the shoulders of Giants.”

Except that one interpretation has it that the letter was not intended to be modest, but was rather a combative dig at the man to whom it was addressed: Robert Hooke, a man of but “middling” stature and, as a result of a childhood illness, also a hunchback. Not one of the “Giants” with broad philosophic shoulders to whom Newton felt indebted to, then.

Robert Hooke, as painted by Rita Greer in 2007. The painting is based on contemporary descriptions of Robert Hooke. No undisputed contemporary paintings or likenesses of Hooke have survived, possibly because of malicious intent on the part of Newton.

In popular estimation, therefore, it would appear that Hooke is fated always to sit behind Newton. At GCSE and A-level, students learn of Newton’s Laws of Motion, the eponymous unit of force, and his Law of Universal Gravitation.

And what do they learn of Hooke? They learn of his work on springs. They learn of “Hooke’s Law”: that is, the force exerted by a spring is directly proportional to its extension.

Ut tensio, sic vis.

[As extension, so is the force.]

— Robert Hooke, Lectures de Potentia Restituvia [1678]

Newton has all the laws of motion on Earth and in Heaven in the palm of his hand, and Hooke has springs. Perhaps, then, Hooke deserves to be forever second on the horse of eternal fame?

But look closer. To what objects or classes of object can we apply Hooke’s Law? The answer is: all of them.

Hooke’s Law applies to everything solid: muscle, bone, sinew, concrete, wood, ice, crystal and stone. Stretch them or squash them, and they will deform in exact proportion to the size of the force applied to them.

That is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward.

The major point being that Hooke’s Law is as universal as gravity: it is baked into the very fabric of the universe: it is a direct consequence of the interactions between atoms.

Graph of interatomic force against distance between two atoms. Hooke’s Law applies in the red circle.

Now before I wax too lyrical, it must be pointed out that Hooke’s Law is a first-order linear approximation: it fails when the deforming force increases beyond a certain limit, and that limit is unique to each material. But within the limits of its domain indicated by the red circle above, it reigns supreme.

How do you calculate how much a steel beam will bow when a kitten walks across it? Hooke’s Law. How could we model the stresses on the bones of a galloping dinosaur? Hooke’s Law. How can we calculate how much Mount Everest bends when it is buffeted by wind? Hooke’s Law.

Time to re-evaluate the seating order on Shakespeare’s horse, mayhap?

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Why does kinetic energy = 1/2mv^2?

Why does kinetic energy Ek=½mv2?

Students and non-specialist teachers alike wonder: whence the half?

This post is intended to be a diagrammatic answer to this question using a Singapore Bar Model approach: so pedants, please avert your eyes.

I am indebted to Ben Rogers’ recent excellent post on showing momentum using the Bar Model approach for starting me thinking along these lines.

Part the First: How to get the *wrong* answer

Imagine pushing an object with a mass m with a constant force F so that it accelerates with a constant acceleration a so that covers a distance s in a time t. The object was initially at rest and ends up moving at velocity v.Screenshot 2019-03-09 at 14.24.59.png

(On the diagram, I’ve used the SUVAT dual coding conventions that I suggested in a previous post.)

So let’s consider the work done on the object by the force:

Step 1:    work done = force x distance moved in the direction of the force

Step 2:    W= F x s

But remember s = v x t so:

Step 3:    W= F x vt

And also remember that F = m x a so:

Step 4:    W= ma x vt

Also remember that a = change in velocity / time, so a = (v – 0) / t = v / t.

Step 5:        Wd = (v / t) x vt

The ts cancel so:

Step 6:    W= mv2

Since this is the work done on the object by the force, it is equal to the energy transferred to the kinetic energy store of the object. In other words, it is the energy the object has gained because it is moving — its kinetic energy, no less: E= mv2.

On a Singapore Bar Model diagram this can be represented as follows:

Screenshot 2019-03-09 at 15.14.17

The kinetic energy is represented by the volume of the bar.

But wait: Ek=mv2!?!?

That’s just wrong: where did the half go?

Houston, we have a problem.

Part the Second: how to get the *right* answer

The problem lies with Step 3 above. We wrongly assumed that the object has a constant velocity over the whole of the distance s.

Screenshot 2019-03-09 at 17.35.43.pngIt doesn’t because it is accelerating: it starts off moving slowly and ends up moving at the maximum, final velocity v when it has travelled the total distance s.

So Step 3 should read:

But remember that s = (average velocity) x t.

Because the object is accelerating at a constant rate, the average velocity is (v + u) / 2 and since u = 0 then average velocity is v / 2.

Step 3:    Wd= F x (v / 2) t

And also remember that F = m x a so:

Step 4:    Wd= ma x (v / 2) t

Also remember that a = change in velocity / time, so a = (v – 0) / t = v / t.

Step 5:        Wd = (v / t) x (v / 2) t

The ts cancel so:

Step 6:    Wd= ½mv2

Based on this, of course, E= ½mv2
(Phew! Houston, we no longer have a problem.)

Screenshot 2019-03-09 at 17.58.45.png

Using the Bar Model representation, the volume of the bar which is above the blue plane represents the kinetic energy of an object of mass m moving at a velocity v.

The reason it is half the volume of the bar and not the full volume (as in the incorrect Part the First analysis) is because we are considering the work done by a constant force accelerating an object which is initially at rest; the velocity of the object increases gradually from zero as the force acts upon it. It therefore takes a longer time to cover the distance s than if it was moving at a constant velocity v from the very beginning.

So there we have it, E= ½mvby a rather circuitous method.

But why go “all around the houses” in this manner? For exactly the same reason as we might choose to go by the path less travelled on some of our other journeys: quite simply, we might find that we enjoy the view.

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Dual-coding SUVAT Problems

The theory of dual coding holds that the formation of mental images, in tandem with verbal processing, is often very helpful for learners. In other words, if we support verbal reasoning with visual representations, then better learning happens.

Many years ago, I was taught the dual coding technique outlined below to help with SUVAT problems. Of course, it wasn’t referred to as “dual coding” back then, but dual coding it most definitely is.

I found it a very useful technique at the time and I still find it useful to this day. And what is more, it is in my opinion a pedagogically powerful procedure. I genuinely believe that this technique helps students understand the complexities and nuances of SUVAT because it brings many things which are usually implicit out into the open and makes them explicit.

SUVAT: “Made darker by definition”?

BOSWELL. ‘He says plain things in a formal and abstract way, to be sure: but his method is good: for to have clear notions upon any subject, we must have recourse to analytick arrangement.’

JOHNSON. ‘Sir, it is what every body does, whether they will or no. But sometimes things may be made darker by definition. I see a cow, I define her, Animal quadrupes ruminans cornutum. But a goat ruminates, and a cow may have no horns. Cow is plainer.

— Boswell’s Life of Johnson (1791)

As I see it, the enduring difficulty with SUVAT problems is that such things can indeed be made darker by definition. Students are usually more than willing to accept the formal definitions of s, u, v, a and t and can apply them to straightforward and predictable problems. However, the robotic death-by-algorithm approach fails all too frequently when faced with even minor variations on a theme.

Worse still, students often treat acceleration, displacement and velocity as nearly-synonymous interchangeable quantities: they are all lumped together in that naive “intuitive physics” category called MOVEMENT.

The approach that follows attempts to make students plainly see differences between the SUVAT quantities and, hopefully, as make them as plain as a cow (to borrow Dr Johnson’s colourful phrasing).

Visual Symbols for the Dual-coding of SUVAT problems

Screenshot 2018-12-25 at 12.02.38.png

1.1 Analysing a simple SUVAT problem using dual coding

Problem: a motorcycle accelerates from rest at 0.8 m/s2 for a time of 6.0 seconds. Calculate (a) the distance travelled; and (b) the final velocity.

Screenshot 2018-12-25 at 12.09.42.png

Please note:

  1. We are using the AQA-friendly convention of substituting values before rearrangement. (Some AQA mark schemes award a mark for the correct substitution of values into an expression; however, the mark will not be awarded if the expression is incorrectly rearranged. Weaker students are strongly encouraged to substitute before rearrangement, and this is what I model.)
  2. A later time is indicated by the movement of the hands on the clock.

So far, so blindingly obvious, some might say.

But I hope the following examples will indicate the versatility of the approach.

1.2a Analysing a more complex SUVAT problem using dual coding (Up is positive convention)

Problem: A coin is dropped from rest takes 0.84 s to fall a distance of 3.5 m so that it strikes the water at the bottom of a well. With what speed must it be thrown vertically so that it takes exactly 1.5 s to hit the surface of the water?

Screenshot 2018-12-25 at 14.33.25.png

Another advantage of this method is that it makes assigning positive and negative directions to the SUVAT vectors easy as it becomes a matter of simply comparing the directions of each vector quantity (that is to say, s, u, v and a) with the arbitrarily selected positive direction arrow when we substitute values into the expression.

But what would happen if we’d selected a different positive direction arrow?

1.2b Analysing a more complex SUVAT problem using dual coding (Down is positive convention)

Problem: A well is 3.5 m deep so that a coin dropped from rest takes 0.84 s to strike the surface of the water. With what speed must it be thrown so that it takes exactly 1.5 s to hit the surface of the water?

Screenshot 2018-12-25 at 14.43.42.png

The answer is, of course, numerically equal to the previous answer. However, following the arbitrarily selected down is positive convention, we have a negative answer.

1.3 Analysing a projectile problem using dual coding

Let’s look at this typical problem from AQA.

Screenshot 2018-12-25 at 14.50.12.png

We could annotate the diagram like this:

Screenshot 2019-01-03 at 18.30.09.png

Guiding our students through the calculation:

Screenshot 2019-01-03 at 18.34.19.png

Just Show ‘Em!

Some trad-inclined teachers have embraced the motto: Just tell ’em!

It’s a good motto, to which dual coding can add the welcome corollary: Just show ’em!

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The famous phrase is, of course, from physicist Eugene Wigner (1960: 2):

My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.

Further exploration of the above problem using dual coding can, I believe, give A-level students a glimpse of the truth of Wigner’s phrase.

This Is The Root You’re Looking For

In the calculation above, we found that when s = -1.8 m, v could have a value of plus or minus 6.90 m/s. Since we were interested in the velocity of the kite boarder at the end of the journey, we concluded that it was the negative root that was significant for our purposes.

But does the positive root have any physical significance? Why yes, it does. It indicates the other possible value of v when s = -1.8 m.

The displacement was -1.8 m at only one point on the real journey. However, if the kite boarder had started their projectile motion from the level of the water surface instead of from the top of the ramp, their vertical velocity at this point would have been +6.9 m/s.

Screenshot 2019-01-04 at 14.24.14.png

The fact that the kite boarder did not start their journey from this point is immaterial. Applying the mathematics not only tells us about their actual journey, but all other possible journeys that are consistent with the stated parameters and the subset of the laws of physics that we are considering in this problem — and that, to me, borders enough on the mysterious to bring home Wigner’s point.

And finally…

Screenshot 2019-01-04 at 15.11.47.png

This information allows us to annotate our final diagram as below (bearing in mind, of course, that the real journey of the kite boarder started from the top of the ramp and not from the water’s surface as shown).

Screenshot 2019-01-04 at 15.14.20.png

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Wigner 1960: 9

Reference

Wigner, E. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics; Vol. 13, No. 1.

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My Best Blogs of 2018

I am very pleased to say that I got my best annual viewing figure ever: just over 9000 views in total beating a previous best of 7000.

Small beer for some in the edu-blogosphere perhaps, but I am well chuffed.

And my most popular blogs were (in reverse order):

  • 5) IoP Energy for Busy Teachers. This is yet another of my periodic tilts at the Institute of Physics’ revised schema for teaching energy, including some attempted humour.
  • 4) The FBI and Gang Sign for Physicists. I am am at a loss to explain why this random stream-of-edu-conciousness post from 2016 seems to get a slow but steady stream of readers, mainly from the U.S.A. for some reason
  • 3) Teaching Electric Circuits? Climb On Board The Coulomb Train! This, I have to admit, is one of my personal favourites. Although a persuasive case can be made for the rope model (I’m looking at you, @PhysicsUK and this), the CTM (Coulomb Train Model) is still the best IMHO. For example, which one would be the best when we’re considering RMS values, huh?
  • 2) Two posts on applying the Singapore Bar Model to GCSE Science Topics and ditching those horrible, horrible formula triangles. The first was on Magnification and the second on Kinetic Energy.
  • 1) And my all-time most-viewed blog post is … MARKOPALYPSE NOW!!! (And thanks to Adam Boxer’s A Chemical Orthodoxy for a link that generated many of the views.)

Bubbling under, we have my contribution to the #CurriculumInScience symposium, Using P-Prims For Fun And Profit. I’m hoping this will get a few more views in the New Year.

And on that note: thanks for reading this far and Happy New Year everyone.

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Using P-prims For Fun And Profit

This is my contribution to the #CurriculuminScience symposium. You can read the first contribution from Ruth Walker here. The next contribution from Jasper Green can be found here.

“She said she was going to join a church as soon as she decided which one was right. She never did decide. She did develop a terrific hankering for a crucifix, though. And she bought one from a Santa Fe gift shop during a trip the little family made out West during the Great Depression. Like so many Americans, she was trying to construct a life that made sense from things she found in gift shops.

— Kurt Vonnegut, Slaughterhouse-Five [emphasis added]

Introduction

It was never supposed to be like this, of course. Many of the great thinkers of the past conceived of the human mind as a vast pyramid: either an inverted pyramid resting on an apex consisting of a single, unfalsifiable thought such as “I think therefore I am” as Rationalists such as Descartes posited; or, alternatively, as a pyramid resting on a base of simple sense-impressions as Empiricists such as Locke suggested.

pyramid

The examples of entities that exist are taken from Douglas Adams’ The Hitchhiker’s Guide To The Galaxy, where the sentient computer Deep Thought started with “I think therefore I am” and deduced the existence of rice pudding and income tax before anyone managed to switch it off.

The truths emerging from modern cognitive science indicate that things are a good deal more complicated and messier than either the Rationalists or Empiricists supposed.

In fact, all of us are closer to Mrs Pilgrim in Vonnegut’s Slaughterhouse-Five than we would generally like to admit: the uncomfortable truth is that we are all closer to opportunistic concept-grubbing, “gift shop”-magpies than the systematic pyramid-masons of either Rationalist or Empiricist thought. Each and every one of us is, to a greater or lesser degree, “trying to construct a life that makes sense” from random things that we find lying around in real or metaphorical gift shops.

Perhaps (of all people!) Dashiell Hammett put it best:

“Nobody thinks clearly, no matter what they pretend. Thinking’s a dizzy business, a matter of catching as many of those foggy glimpses as you can and fitting them together the best you can. That’s why people hang on so tight to their beliefs and opinions; because, compared to the haphazard way in which they arrived at, even the goofiest opinion seems wonderfully clear, sane, and self-evident. And if you let it get away from you, then you’ve got to dive back into that foggy muddle to wangle yourself out another to take its place.”
— Dashiell Hammett, The Dain Curse

Defeat From The Jaws of Victory: “Here’s to you, Mrs Pilgrim.”

Andrea diSessa (1996) recounts a series of interviews with “J”, a freshman undergraduate student of Physics at university. During one interview, J was asked to explain the physics of throwing a ball up into the air. She recounted a near-perfect, professional physicist-level analysis of the phenomenon, noting (correctly) that after the ball leaves the hand the “only force acting on it is gravity”. However, when diSessa asked the seemingly innocuous question about what happens at the peak of the toss:

Rather than produce a straightforward answer, J proceeded to reformulate her description of the toss. The reformulation is not instantaneous . . . Strikingly she winds up with an “impetus theory” account of the toss. “Your hand imparts a force that at first overcomes gravity, but gradually dies away. At the peak, there a balance of forces, which is broken as the internal force fades further and gravity takes over.”

In other words, even a student of Physics, educated to a much higher level of domain-specific knowledge than the typical layperson, can be persuaded to retreat back into the ”foggy muddle” with surprising ease. In other words, even the very best of us can snatch defeat from the jaws of victory all too easily.

diSessa (1988) explains this and similar models as part of the KIP model (Knowledge in Pieces). For example:

intuitive physics is a fragmented collection of ideas, loosely connected and reinforcing, having none of the commitment or systematicity that one attributes to theories.

The basic “atom” or building block of this empirical model is the p-prim or phenomenological primitive.

P-prims are elements of intuitive knowledge that constitute people’s “sense of mechanism”, their sense of which happenings are obvious, which are plausible, which are implausible, and how one can explain or refute real or imagined possibilities. [diSessa 2018: 69]

P-prims are abstractions of familiar events that come to serve as explanations as they are applied to a wider range of contexts. The p indicates that they originate from the phenomenologically-rich and lived experience of human beings; the prim indicates that they are primitive in the sense that they sufficient explanations of phenomena. Once a p-prim is invoked, usually no further explanation is required or possible: “That’s just the way it is.” Examples of p-prims suggested by diSessa [1996: 716] are:

  • The “Ohm’s Law” p-prim: the idea that an outcome increases as a “force” increases, but decreases as the “resistance” increases.
  • The “Balance and Equilibrium” p-prim: systems which are “in balance” will be stable; systems which are “out of balance” will naturally and spontaneously return to equilibrium.
  • The “Blocking and Guiding” p-prim: solid and stable objects can stop objects moving without applying a force; tubes and railway tracks can also “guide” moving objects without applying any force.
  • The “Dying Away” p-prim: lack of motion or activity is the natural state of inanimate objects; if disturbed, they will naturally return to this state as the perturbation “dies away’

P-prims are subconceptual: they comprise a fluid and changeable layer below concepts and beliefs. Humans may have hundreds if not thousands of p-prims. There is no strict hierarchy: we may shift from one p-prim to another with simply a shift of attention. Where multiple p-prims conflict, one facet of the situation may cue the application of a particular p-prim rather than another. [see diSessa 1996: 715]

The Wrath of Kuhn: “So You Say You Want a Revolution?”

In his hugely-influential The Structure of Scientific Revolutions (1970), Thomas Kuhn suggested that scientific progress had two distinct phases:

  • Normal Science, where essentially scientists engaged in puzzle-solving activity but where the guiding paradigm or disciplinary matrix of the science is more or less accepted without question. An example might be pre-Copernican astronomy where astronomers made observations and predictions without questioning the geocentric model of the Solar System;
  • Revolutionary Science, where scientists realise their previously-successful paradigm is no longer able to adequately explain observed phenomena. An example might be the rejection of the Newtonian paradigm and the acceptance of Einsteinian relativistic physics in the early 1900s.

Scientific progress was thus viewed as a gestalt switch between two incommensurable systems of knowledge. One either sees a “Newtonian”-duck, or a “Relativistic”-rabbit. One cannot see both simultaneously.

gestalt.png

One can either see a duck or a rabbit: but not both at the same time.

Kuhn’s work was immensely influential (perhaps overly influential) in a number of spheres; in the context of education, the heady seductiveness of Kuhn’s approach directly influenced what diSessa [2014: 5] dubs the “misconceptions movement”.

Broadly speaking, proponents thought that students had deeply entrenched but false beliefs. The solution seemed obvious: these false beliefs were barriers to learning that had to be rooted out and overcome (c.f. the Ohm’s Law p-prim above!) . Students had to be persuaded to ditch their false beliefs and accept the correct ones.

But what was the nature of these false beliefs? diSessa [2014:7] argues that some like Carey (1985) drew explicit parallels with Kuhn’s work, arguing that children undergo a paradigm-shift at about 10-years-old when they recognise that inanimate objects do not have intentions and begin to think of “alive” as describing a set of mechanistic processes. Others (argues diSessa) like McCloskey (1983) supposed that students begin school physics with a well-formed, coherent and articulate theory (with parallels to early medieval scientists such as Buridan and Galileo) that directly competes with and interferes with their acceptance of Newtonian physics.

However, all of these approaches can be categorised as being part of the “Misconceptions Movement”.

Yin vs. Yang: Positive and Negative Influences of the Misconceptions Movement

A positive influence of misconceptions studies was bringing the importance of educational research into practical instructional circles. Teachers saw vivid examples of students responding to apparently simple conceptual questions in incorrect ways. Poor performance in response to basic questions, often years into instruction, could not be dismissed.

[diSessa 2014: 6]

Another hugely positive influence of Misconceptions research was that it showed that students were not “blank slates” and that prior knowledge had a strong influence on future learning.

However, according to diSessa the misconceptions movement also had some pernicious negative influences:

  • It emphasised the negative contributions of prior knowledge: it almost exclusively characterised prior knowledge as either false or unhelpful which led to “conflict” models of instruction. Ironically, the explicit detailing of “wrong” ideas in order to “overcome” them led to them being strengthened for some students.
  • How learning was possible was not a matter that was often discussed in detail. The depth, coherence or strength of particular misconceptions was not always assessed: were they simply isolated beliefs or coherent theories of a similar nature to those held by working scientists? As a result, practical guidance on how to teach particular concepts was not always forthcoming.

Tourist: “Is This Way To Amarillo?” Local: “Well, I wouldn’t start from here if I were you.”

As a working Physics teacher, one of the most useful teaching tools that I’ve begun using as a result of becoming aware of diSessa’s work, is that of a bridging analogy. This approach was outlined by Hammer 2000: S54-55. For example, how can we successfully introduce the idea of a normal reaction force, say in the context of a book resting on the surface of a table?

Students often invoke the “blocking” p-prim in this context. The table passively “blocks” the action of gravity — and that’s all there is to it.

However, a bridging analogy can be used here. Show an object resting on (and compressing) a spring; identify the forces acting on the object. Because the spring is an “active” component in this situation, students can accept that pushing down on it produces an upward “reaction force”. One can then extend this to (say) a student sitting on a plank (which “bows” slightly with their weight) and then apply it to more stable structure such as a table which exhibits no visible “bowing”.

I have found such approaches to be the most productive: in other words, we aim to work around the p-prim rather than attacking the p-prim head on, and along the way we try to get our students to activate more helpful p-prims that have more direct applicability to the context.

As teachers, we only very rarely have the luxury of choosing our students’ starting points. There is no “Well, if you want to get where you’re going, I wouldn’t start from here if I were you.”

We are teachers. Whatever the situation, we start from where our students start. Ladies and gentlemen, we start from here.

References

Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press/Bradford Books

diSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49-70). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.

diSessa, A. A. (1996). What do” just plain folk” know about physics. The handbook of education and human development: New models of learning, teaching, and schooling, 709-730. [Accessed from http://www.staff.science.uu.nl/~savel101/fi-msecint/literature/disessa1996.pdf on 22/10/18]

DiSessa, A. A. (2014). A history of conceptual change research: Threads and fault lines. [Accessed from https://escholarship.org/uc/item/1271w50q on 22/10/18]

diSessa, A. A. (2018). A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning. In Invited Lectures from the 13th International Congress on Mathematical Education (pp. 65-84). Springer International Publishing. [Accessed from https://link.springer.com/chapter/10.1007/978-3-319-72170-5_5 on 22/10/18]

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics, 68(S1), S52-S59 [Accessed from http://oap.nmsu.edu/JiTT_NMSU_workshop/pdfs/StudentResourcesHammer.pdf on 22/10/18]

McCloskey, M. (1983). Naive theories of motion. In D. Gentner and A. Stevens (Eds.) Mental Models (pp. 299-323). Hillsdale, NJ: Lawrence Erlbaum Associates.

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