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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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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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Teaching Electric Circuits? Climb On Board The Coulomb Train!

I’ve said it before and I’ll say it again: teaching electric circuits is hard.

Providing your students with a conceptual model can, in my opinion, be immensely helpful. In recent years, I have used what I call the Coulomb Train Model (CTM). It is essentially a variation on the “stiff chain” analogies that some researchers have argued as being particularly useful in developing students’ understanding.

One reason why I like the CTM is that it provides a physical picture to aid students’ comprehension of many of the electrical equations needed at GCSE.

Of course, any analogy or model will have its flaws, but on the whole I think the CTM has fewer than many of its rivals!

Essentially, the CTM pictures an electric circuit as a continuously moving chain of postively-charged “trucks” called coulombs that carry energy from the cell to (say) the bulb. In the diagram below, they should be pictured as moving clockwise.

The coulomb is, of course, the S.I. unit of electric charge, so rest assured that there is method in the apparent madness of naming our “trucks” with a word that would be unfamiliar to most of our students.

Charge flow = current x time

Charge flow = number of coulombs that pass a given point in time.

Current = number of coulombs that pass by in one second (i.e. current = charge flow / time).

In other words, an ammeter counts the coulombs passing by in one second. The ammeter only “sees” the coulombs and does not register how much (or how little) energy each one contains. Therefore current I1 and current I2 are equal.

The ammeters are shown as being semi-transparent in order to provide a visual cue that they are low resistance devices.

Energy transferred = charge flow x potential difference

On the CTM, potential difference can be pictured as energy being added to, or removed from, each coulomb.

For example, if one joule is removed from each coulomb as they pass through the bulb, the potential difference across the bulb is one volt. If one joule is added to each coulomb as they pass through the cell, then the potential difference (or e.m.f.) across the cell is one volt.

The total energy transferred from (say) ten coulombs passing through the bulb would be charge flow (10 coulombs) x potential difference (1 volt) = 10 joules.

The white gloves on the voltmeter are intended to be reminiscent of the white gloves of a snooker referee.

The intention is to disrupt the flow of the coulombs as little as possible and so this is a visual reminder that a voltmeter is a high resistance instrument.

To emphasise the fact that potential difference is an “energy difference”, challenge students to predict the reading on this voltmeter.

The potential difference V3 is, of course, zero since there is no transfer of energy to or from the coulombs.

Current in Series and Parallel Circuits

I think the CTM can be really effective in allowing students to a see a comprehensible physical analogue of the circuits.

For example, I3 = I4 = I5 = 0.5 amps; I6 = I11 = 2 amps; and I7 = I8 = I9 = I10 = 1 amp.

Potential difference in series and parallel circuit

Equally, I think the CTM can give a comprehensible physical picture of the situation.

In this case (assuming the the cell has a p.d. of 1 V and the bulbs are identical), V4 = V5 = 0.5 V.

In the parallel circuit, each bulb tranfers one joule of energy from each bulb, and so the potential difference across each bulb is one volt.

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Ohm From Ohm

Amongst the myriad inconveniences and troubles of a Physics teacher’s life, the choice of the symbols commonly used to represent voltage, current and resistance, must surely rank in the top ten.

V is for voltage in volts, V

Well, OK, that’s sensible enough. On a good day, I may even remember to call it “potential difference”. The sage advice of Never use two words when one will do is commonly accepted by everyone; however, Physics teachers have, as a profession, decided to go it alone and completely ignore this tired old saw. Thus, voltage is become potential difference because of — erm, reasons (?)

One can only hope that everyone got the memo . . .

R is for resistance in ohms, Ω

R for resistance? That’s fairly sensible too.

“But what’s that weird squiggly thing, Sir?”

“Ah, you mean the Greek letter omega? Because Physics is soooo enormous that the measly 26 letters of the Latin alphabet ain’t big enough for it…”

I is for current in amps, A

“WTφ? Are you taking the πΣΣ, Sir?”

“I know, I know! Look, if it helps, think of it as short for intensité du courant . . . Wait, don’t leave! Stop, I have many more fun Physics facts to teach you! Look, here’s a picture of Richard Feynman playing his bongo drums — nooooooooo!”

Ohm’s Law: or is it more a sort of guideline?

Let’s start with a brief statement of Ohm’s Law:

V = I R

Except, that’s not Ohm’s Law; it’s actually the definition of resistance:

R = V / I

There is not a single instance where it is not true by definition. The value of resistance will always be equal to the ratio of the potential difference and the current.

Think of it like this. At room temperature, 1 V of potential difference can push (say) 0.5 A of current through the wire in a filament bulb. (I just love that retro 1890s tech, don’t you?)

This means it has a resistance of 1/0.5 = 2 ohms. However, bump up the potential difference to 6 V and the current is (say) 0.75 A. This means that is has a resistance of 6/0.75 = 8 ohms. Its resistance has changed because it has become hotter. In other words, its resistance is not constant.

Ohm’s Law is perhaps most simply stated as:

The potential difference is directly proportional to the current over a range of physical conditions (including temperature).

Using standard symbols:

V α I

or, taking R’ as a constant of proportionality:

V = I R’

You do see the difference, don’t you?

In the first example, R is not a constant value for a given range of physical conditions: for example it can get higher as the temperature increases.

In the second, R’ is constant over a range of temperatures and other physical conditions.

And so there we have it: V=IR can be a perfectly valid statement of Ohm’s Law, provided it is specified that R is constant. If one does not do that, then all bets are off…

In the meantime, here’s another picture of Richard Feynman playing the bongo drums. Enjoy!

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Lottie and Lorentzian Length Contraction

@_youhadonejob tweeted this textbook picture with the amusing and sardonic comment “Little girl in this textbook is 5 m tall”.

I liked @jim_henderson60’s take on this when he tweeted: “You see. Physics helps us all grow tall.”

But then I started thinking, what if the 5 m measuring stick was in an inertial frame moving past Lottie’s inertial frame at a substantial fraction of light speed? (In my head, I named the girl “Lottie”, although “Alice” would be more in the more usual tradition of SR* pedagogy, I guess.)

The illustration could represent that single instant at which both ends of the 5 m ruler were precisely opposite Lottie’s head and feet as its inertial frame whizzed by hers…

A quick calculation indicated that Lorentz length contraction could indeed account for the relative measurements on the illustration if v = 0.97c

Of course, Lorentz length contraction is a two way street. From the 5 m ruler’s inertial frame, length contraction would make Lottie appear even shorter than her compact 1.2 m. Given that v = 0.97c, I calculate that she would appear only 0.29 m tall.

Correction: not appear. She would genuinely be only 0.29 m tall when viewed from that inertial frame, just as the 5 m rule would genuinely be only 1.2 m long when viewed from Lottie’s inertial frame.

We live in an universe where everything is indeed relative. However, for most of us that takes a fair amount of getting used to…

*SR = special relativity. My brain is currently too small to handle GR (general relativity).

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