The Coulomb Train Model Revisited (Part 4)

In this post, we will look at parallel circuits.

The Coulomb Train Model (CTM) is a helpful model for both explaining and predicting the behaviour of real electric circuits which I think is useful for KS3 and KS4 students.

Without further ado, here is a a summary.

This is part 4 of a continuing series. (Click to read Part 1, Part 2 or Part 3.)

The ‘Parallel First’ Heresy

I advocate teaching parallel circuits before teaching series circuits. This, I must confess, sometimes makes me feel like Captain Rum from Blackadder Two:

The main reason for this is that parallel circuits are conceptually easier to analyse than series circuits because you can do so using a relatively naive notion of ‘flow’ and gives students an opportunity to explore and apply the recently-introduced concept of ‘flow of charge’ in a straightforward context.

Redish and Kuo (2015: 584) argue that ‘flow’ is an example of embodied cognition in the sense that its meaning is grounded in physical experience:

The thesis of embodied cognition states that ultimately our conceptual system grounded in our interaction with the physical world: How we construe even highly abstract meaning is constrained by and is often derived from our very concrete experiences in the physical world.

Redish and Kuo (2015: 569)

As an aside, I would mention that Redish and Kuo (2015) is an enduringly fascinating paper with a wealth of insights for any teacher of physics and I would strongly recommend that everyone reads it (see link in the Reference section).

Let’s Go Parallel First — but not yet

Let’s start with a very simple circuit.

This is not a parallel circuit (yet) because switch S is open. Resistors R1 and R2 are identical.

This can be represented on the coulomb train model like this:

Five coulombs pass through the ammeter in 20 seconds so the current I = Q/t = 5/20 = 0.25 amperes.

Let’s assume we have a 1.5 V cell so 1.5 joules of energy are added to each coulomb as they pass through the cell. Let’s also assume that we have negligible resistance in the cell and the connecting wires so 1.5 joules of energy will be removed from each coulomb as they pass through the resistor. The voltmeter as shown will read 1.5 volts.

The resistance of the resistor R1 is R=V/I = 1.5/0.25 = 6.0 ohms.

Let’s Go Parallel First — for real this time.

Now let’s close switch S.

This is example of changing an example by continuous conversion which removes the need for multiple ammeters in the circuit. The changed circuit can be represented on the CTM as shown

Now, ten coulombs pass through the ammeter in twenty seconds so I = Q/t = 10/20 = 0.5 amperes (double the reading in the first circuit shown).

Questioning may be useful at this point to reinforce the ‘flow’ paradigm that we hope students will be using:

  • What will be the reading if the ammeter moved to a similar position on the other side? (0.5 amps since current is not ‘used up’.)
  • What would be the reading if the ammeter was placed just before resistor R1? (0.25 amps since only half the current goes through R1.)

To calculate the total resistance of the whole circuit we use R = V/I = 1.5/0.5 = 3.0 ohms– which is half of the value of the circuit with just R1. Adding resistors in parallel has the surprising result of reducing the total resistance of the circuit.

This is a concrete example which helps students understand the concept of resistance as a property which reduces current: the current is larger when a second resistor is added so the total resistance must be smaller. Students often struggle with the idea of inverse relationships (i.e. as x increases y decreases and vice versa) so this is a point well worth emphasising.

Potential Difference and Parallel Circuits (1)

Let’s expand on the primitive ‘flow’ model we have been using until now and adapt the circuit a little bit.

This can be represented on the CTM like this:

Each coulomb passing through R2 loses 1.5 joules of energy so the voltmeter would read 1.5 volts.

One other point worth making is that the resistance of R2 (and R1) individually is still R = V/I = 1.5/0.25 = 6.0 ohms: it is only the combined effect of R1 and R2 together in parallel that reduces the total resistance of the circuit.

Potential Difference and Parallel Circuits (2)

Let’s have one last look at a different aspect of this circuit.

This can be represented on the CTM like this:

Each coulomb passing through the cell from X to Y gains 1.5 joules of energy, so the voltmeter would read 1.5 volts.

However, since we have twice the number of coulombs passing through the cell as when switch S is open, then the cell has to load twice as many coulombs with 1.5 joules in the same time.

This means that, although the potential difference is still 1.5 volts, the cell is working twice as hard.

The result of this is that the cell’s chemical energy store will be depleted more quickly when switch S is closed: parallel circuits will make cells go ‘flat’ in a much shorter time compared with a similar series circuit.

Bulbs in parallel may shine brighter (at least in terms of total brightness rather than individual brightness) but they won’t burn for as long.

To some ways of thinking, a parallel circuit with two bulbs is very much like burning a candle at both ends…

More fun and high jinks with coulomb train model in the next instalment when we will look at series circuits.

You can read part 5 here.


Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemologyScience & Education24(5), 561-590.

The Coulomb Train Model Revisited (Part 3)

In this post, we will look at explaining electrical resistance using the Coulomb Train Model.

This is part 3 of a continuing series (click to read part 1 and part 2).

The Coulomb Train Model (CTM) is a helpful model for both explaining and predicting the behaviour of real electric circuits which I think is useful for KS3 and KS4 students.

Without further ado, here is a a summary.

A summary of the Coulomb Train Model

Representing Resistance on the CTM

To measure resistance, we would set up this circuit.

We can represent this same circuit on the CTM as follows:

If we count how many ‘coulombs’ (grey trucks) pass one point in a certain time then on this animation we get 5 coulombs in 20 seconds.
This is equivalent to a current of
5 coulombs / 20 seconds = 0.2 coulombs per second = 0.2 amperes.

This way of thinking about current is consistent with the formula charge flow = current x time or Q=It which can be rearranged to give I=Q/t.

We have used identical labels on the circuit diagram and the CTM animation to encourage students to view them as different representations of a real situation. The ammeter at X would read 0.2 amps. We could place the ammeter at any other point in the circuit and still get a reading of 0.2 amps since ammeters only ‘count coulombs per second’ and don’t make any measurement of energy (represented by the orange substance in the trucks).

However, the voltmeter does make a measurement of energy: it compares the energy difference between a single coulomb at Y and a single coulomb at Z. If (say) 1.5 joules of energy is transferred from each coulomb as it passes through the bulb from Y to Z then the voltmeter will read a potential difference (or ‘voltage’ if you prefer) of 1.5 volts.

This way of thinking about potential difference is consistent with the formula energy transferred = charge flow x potential difference or E=QV which we can rearrange to give V=E/Q.

So as you can see, one volt is really equivalent to an energy change of one joule for every coulomb (!)

We can calculate the resistance of the bulb by using R=V/I so R = 1.5/0.2 = 7.5 ohms.

Resistance is not futile . . .

Students sometimes have difficulty accepting the idea of a ‘resistor’: ‘Why would anyone in their right minds deliberately design something that reduces the flow of electric current?’ It’s important to explain that it is vital to be able to control the flow of electric current and that one of the most common electronic components in your phone or games console is — the humble resistor.

One of many resistors on a circuit board. The colour codes tell us the value of each resistor.

Teachers often default to explaining electric circuits using bulbs as the active component. There is a lot to recommend this practice, not least the fact that changes in the circuit instantaneously affect the brightness of the bulb. However, it vital (especially at GCSE) to allow students to learn about circuits featuring resistors and other components rather than just the pedagogically overused (imho) filament lamp.

Calculating the resistance of a resistor

Consider this circuit where we have a resistor R1.

This can be represented as a coulomb train model like this:

The resistor does not glow with visible light as the bulb does, but it would glow pretty brightly if viewed through an infra red camera since the energy carried by the coulombs is transferred to the thermal energy store of the resistor. The only way we can observe this energy shift without such a special camera is to use a voltmeter.

Let’s begin by analysing this circuit qualitatively.

  • The coulombs are moving faster in this circuit than the previous circuit. This means that the current is larger. (Remember: current is coulombs per second.)
  • Because the current is larger, R1 must have a smaller resistance than the bulb. (Remember: resistance is a quantity that reduces the current.)
  • The energy transferred to each coulomb is the same in each example so the potential difference of the cell is the same in both circuits. (Of course, V can be altered by adding a second cell or turning up the setting on a power supply, but in many circuits V is, loosely speaking, a ‘fixed’ or ‘quasi-constant’ value.)
  • Because the ‘push’ or potential difference is the same size but the resistance of R1 is smaller, then the same cell is able to push a larger current around the circuit.

Now let’s analyse the circuit quantitatively.

  • 5 coulombs pass a single point in 13 seconds so the current is 5/13 = 0.38 coulombs per second = 0.4 amperes. (Double the current in the bulb circuit.)
  • The resistance can be calculated using R=V/I = 1.5/0.4 = 3.75 ohms. (Half the resistance of the bulb.)
  • Each coulomb is being loaded with 1.5 J of energy as it passes through the cell. Since this is happening twice as often in the resistor circuit as the bulb circuit, the cell will ‘go flat’ or ’empty its chemical energy store’ in half the time of the bulb cell.

So there we have it: more fun and high jinks with the CTM.

I hope that I have persuaded a few more teachers that the CTM is useful for getting students to think productively and, more importantly, quantitatively using correct scientific terminology about electric circuits.

In the next installment, we will look at series and parallel circuits.

Teaching Newton’s Third Law

Newton’s First and Second Laws of Motion are universal: they tell us how any set of forces will affect any object.

The “Push-Me-Pull-You” made famous by Dr Doolittle

If the forces are ‘balanced’ (dread word! — saying ‘total force is zero’ is better, I think) then the object will not accelerate: that is the essence of the First Law. If the sum of the forces is anything other than zero, then the object will accelerate; and what is more, it will accelerate at a rate that is directly proportional to the total force and inversely proportional to the mass of the object; and let’s not forget that it will also accelerate in the direction in which the total force acts. Acceleration is, after all, a vector quantity.

So far, so good. But what about the Third Law? It goes without saying, I hope, that Newton’s Third Law is also universal, but it tells us something different from the first two.

The first two tell us how forces affect objects; the third tells us how objects affect objects: in other words, how objects interact with each other.

The word ‘interact’ can be defined as ‘to act in such a way so as to affect each other’; in other words, how an action produces a reaction. However, the word ‘reaction’ has some unhelpful baggage. For example, you tap my knee (lightly!) with a hammer and my leg jerks. This is a reaction in the biological sense but not in the Newtonian sense; this type of reaction (although involuntary) requires the involvement of an active nervous system and an active muscle system. Because of this, there is a short but unavoidable time delay between the stimulus and the response.

The same is not true of a Newton Third Law reaction: the action and reaction happen simultaneously with zero time delay. The reaction is also entirely passive as the force is generated by the mere fact of the interaction and requires no active ‘participation’ from the ‘acted upon’ object.

I try to avoid the words ‘action’ and ‘reaction’ in statements of Newton’s Third Law for this reason.

If body A exerts a force on body B, then body B exerts an equal and opposite force on body A.

The best version of Newton’s Third Law (imho)

In our universe, body B simply cannot help but affect body A when body A acts on it. Newton’s Third Law is the first step towards understanding that of necessity we exist in an interconnected universe.

Getting the Third Law wrong…

Let’s consider a stationary teapot. (Why not?)

This is NOT a good illustation of Newton’s Third Law…

We can reject this as an appropriate example of Newton’s Third Law for two reasons:

  • Reason 1: Force X and Force Y are acting on a single object. Newton’s Third Law is about the forces produced by an interaction between objects and so cannot be illustrated by a single object.
  • Reason 2: Force X and Force Y are ‘equal’ only in the parochial and limited sense of being merely ‘equal in magnitude’ (8.2 N). They are very different types of force: X is an action-at-distance gravitational force and Y is an electromagnetic contact force. (‘Electromagnetic’ because contact forces are produced by electrons in atoms repelling the electrons in other atoms.) The word ‘equal’ in Newton’s Third Law does some seriously heavy lifting…

Getting the Third Law right…

The Third Law deals with the forces produced by interactions and so cannot be shown using a single diagram. Free body diagrams are the answer here (as they are in a vast range of mechanics problems).

This is a good illustration of Newton’s Third Law

The Earth (body A) pulls the teapot (body B) downwards with the force X so the teapot (body B) pulls the Earth (body A) upwards with the equal but opposite force W. They are both gravitational forces and so are both colour-coded black on the diagram because they are a ‘Newton 3 pair’.

It is worth noting that, applying Newton’s Second Law (F=ma), the downward 8.2 N would produce an acceleration of 9.8 metres per second per second on the teapot if it was allowed to fall. However, the upward 8.2 N would produce an acceleration of only 0.0000000000000000000000014 metres per second per second on the rather more massive planet Earth. Remember that the acceleration produced by the resultant force is inversely proportional to the mass of the object being accelerated.

Similarly, the Earth’s surface pushes upward on the teapot with the force Y and the teapot pushes downward on the Earth’s surface with the force Z. These two forces form a Newton 3 pair and so are colour-coded red on the diagram.

We can summarise this in the form of a table:

Testing understanding

One the best exam questions to test students’ understanding of Newton’s Third Law (at least in my opinion) can be found here. It is a really clever question from the legacy Edexcel specificiation which changed the way I thought about Newton’s Third Law because I was suddenly struck by the thought that the only force that we, as humans, have direct control over is force D on the diagram below. Yes, if D increases then B increases in tandem, but without the weighty presence of the Earth we wouldn’t be able to leap upwards…

Licensed to Teach?

I miss the GTC. Actually, no, scrub that. What I really miss are the reports from the GTC hearings that used to appear in the TES (you know, when it still looked like a newspaper rather than an in-flight magazine for a budget airline).

What I found fascinating was the jaw-dropping chutzpah of some of the cases. Not only could I not conceive of doing some of the stuff reported myself, but I honestly thought that not a single colleague that I had worked with over the years could come anywhere close either. Quite frankly, much of it seemed too bizarre to be true, and yet there it was, reported in sober black and white.

[The man] who recommended a curacy as the best means of clearing up Trinitarian difficulties, that “[holy] orders” are a sort of spiritual backboard, which, by dint of obliging a man to look as if he were strait, end by making him so.
George Eliot, Carlyle’s Life of Stirling.

And now along comes Tristram Hunt with yet another cunning plan to put a spiritual backbone (or a “spiritual backboard”) into the profession with a system of licenses.

In principle, I have no objection to this. There are some individuals who should not be allowed to remain in the profession. If I put my mind to it, I could probably name one or two that I have worked with over the last twenty or so years who (in my opinion) should have been chucked out. But I would have to think about it.

So, in my personal experience at least, this is not a major problem. To be blunt, the sad truth is that natural wastage from the tough environment of modern teaching will take care of most of the wasters and no-hopers. The ones who stay — we few, we happy few! — generally really want to stay.

Let me hasten to add that not that everyone who leaves is a waster or no-hoper — some leave through a lack of support or the insanely inappropriate priorities of their school or line manager.

The devil will be, as always, in the detail. I think that what I dread is a bizarre set of professional expectations drafted by someone who thinks that, by dint of obliging a teacher to look as if he or she were strait, that it will end by making them so.

For example, one of the most excellent and engaging teachers that I know is also one of the scruffiest. It would be a pity if an ill-considered set of criteria forced eccentric individuals  such as him out of the profession because they didn’t always do up their top button (are you listening, Sir Michael?).

We can but hope, because (going on past experience, at least) the profession will only have a very limited say in drafting the licensing criteria.

“Hear me. I am your new president. From this day on, the official language of San Marcos will be Swedish. Furthermore, all citizens will be required to change their underwear every half hour. Underwear will be worn on the outside so we can check.”

— The President’s victory speech , from Bananas (dir. Woody Allen 1971)

I have this idea about licenses for teachers and moving from a system of letter grades to number grades…

The Woman Who Is Kicking the Hornets’ Nest

So, I’m reading  Seven Myths about Education.  Just like most of the rest of the teaching blogosphere, I suspect. And just like most of the rest of the teaching blogosphere, I have an opinion about it. Several, as a matter of fact. And since I am now about halfway through, I thought I’d share my thrupence’ worth.

To begin with, is Ms Christodoulou more like the boy who cried that the king had no clothes or the boy who cried wolf?

For my money, she is more the former than the latter. I think the estimable Ms Christodolou is calling time on some pretty dodgy ideas.

Some ideas are as ubiquitous and seemingly essential as air, but as Joseph Joubert correctly opined: “A thought is a thing as real as a cannonball”.  And in some circumstances, the wrong idea can be more dangerous than a large round metal ball travelling at close to the speed of sound.

Now teaching-wise, I have to confess that I have been around the block a few times. I am the definitive “old fart in the staffroom”. Like many old farts, I could bring myself to believe that oftentimes it is not what Ofsted actually said that was the main problem, but what all-too-many people thought that Ofsted said: some half-remembered, half-digested soundbite from some godforsaken half-decade-old CPD.

Christodoulou marshals some convincing evidence that often it is the actual demands of Ofsted that create the problem. It seems that Ofsted genuinely do not like didactic teaching, and we’re not just imagining it. Christodoulou presents some damning examples of the current vogue of trashing “teacher talk” from inspection reports. Whether Wilshaw will be able to rein in the “talk-less-teaching” rottweilers on his staff is open to debate. Large organisations can have a momentum as stubborn as supertanker and carry on going in the same direction for mile after mile, whatever the frantic signals from the wheelhouse say.

One of the passages that resonated most strongly with me was this:

For example, in a project that involved pupils writing any type of extended writing … I would provide them with a helpsheet summarising what they should put in each paragraph. […] Rather than breaking down the individual components required to write good reports and teaching those, I was asking students to write a report and then giving them a few cheats or hints about how to do it. It is rather like teaching pupils a few cheats or hints that would help them play a certain song on the piano, while neglecting to teach them the scales and musical notation.
— Location 1727, Kindle edition

Been there, done that, smugly uploaded the worksheet on to the TES Resources website…

She quotes psychologist Dan Willingham: “the most general and useful idea that cognitive psychology can offer teachers [is to] review each lesson plan in terms of what the student is likely to think about.”

Christodoulou argues that teaching (say) Romeo and Juliet by getting the students to make fingerpuppets of the main characters is counterproductive because the students spend more time thinking about making fingerpuppets rather than Romeo and Juliet. “That is not to say that … puppetmaking [is] unimportant. The problem is that this lesson . . . was supposed to be about Romeo and Juliet. If the aim of the lesson was … how to make a puppet, it would have been a good lesson. Not only do these types of lesson fail in their ultimate aims, but because they are so time-consuming, they also have a very significant opportunity cost.”

I agree with Christodoulou that direct instruction is often the most effective form of teaching. Now don’t get me wrong, I am not proposing that teachers spend the whole of the lesson talking at their charges. What I am saying is that students’ thinking should be channelled to engage as directly with the concepts being taught as possible. And at the heart of good teaching is clear, succinct, unhurried teacher talk.

The fingerpuppet stuff I have done, but only to pass observations. Sadly, honesty is not the best policy these days.

A while back, Arnold Schwarzenegger was The Terminator: robot on the inside, human on the outside.

Call me the The Didactor: steely-eyed, garrulous, “I’ve-got-a-banda-and-I’m-not-afraid-to-use-it” old-school (hah!) schoolteacher on the inside; cuddly, Ofsted-friendly, near-mute “lesson-facillitator” on the outside (readers of a certain generation are invited to think of a cross between Fingerbobs and Marcel Marceau).

Sigh, I wish. I got a 3 (“Requires improvement”) in my last lesson observation.

The secret of success is sincerity. Once you can fake that you’ve got it made.

— Jean Giraudoux

More sincere faking is required on my part, I feel.

Post the first…yay!

So, you’ve decided to join the blogger bandwagon? Why yes, I most decidedly have. The TEACHER blogger bandwagon, if you please. I have been inspired by a number of educational blogs that I really enjoy to have a go myself. After all, how hard can it be? (As the free school committee said to the education secretary.)

So, I have decided to put finger (singular, I am a lousy typist) to virtual keyboard and write my profound thoughts on erm…well, stuff, basically. And stick some Physicsy guff in here at some point. And some ill-informed comment, gossip, innuendo and vapid intellectual posturing to boot!

Attentive readers will note that my blog title is a Physics-themed homage to the immortal “1066 And All That” by Sellar and Yeatman. I hope to do for Physics teaching what they did for History teaching.

So there. Plus I will be rude about Michael Gove from time to time.

John Mortimer once wrote that he joined the swinging sixties “just as the tube doors were closing”. I hope that one day a fellow teacher blogger who is insanely jealous of my reader stats and influence will be as cutting about my entrance on to the blogging scene.