Series and Parallel Circuits — an unhelpful dichotomy?

Anakin Skywalker and Obi Wan Kenobi discuss the possible unhelpfulness of the concept of ‘series circuits’ and ‘parallel circuits

Are physics teachers following the Way of the Sith? Are we all crossing over to the Dark Side when we talk about ‘series circuits’ and ‘parallel circuits’?

I think that, without meaning to, we may be presenting students with what amounts to a false dichotomy: that all circuits are either series circuits or parallel circuits.

Venn diagram showing the false dichotomy view of series and parallel circuits

The actual situation is more like this:

A Venn diagram showing a more nuanced and realistic view of series and parallel circuits

The confusion may stem from our usage of the word ‘circuit’: are we referring holistically to the entire assemblage of components (highlighted in red) or the individual ‘complete circuits’ (highlighted in green and blue)?

Will the actual ‘circuit’ please stand up? The red circuit is a hybrid circuit, the green circuit is a series circuit, and the blue circuit shows a single resistor in series or parallel with cell (depending on how you look at it)

How to avoid the false dichotomy

I think we should always refer to components in series or components in parallel rather than ‘series circuits’ or ‘parallel circuits’.

Teaching components in parallel using the ‘all-in-a-row’ circuit convention

I’ve written before about what I think is the confusing ‘hidden rotation’ present in normal circuit diagrams. I find redrawing circuit diagrams using the ‘all-in-a-row’ convention useful for explaining circuit behaviour. For simplicity, we’ll assume that all the resistors in the diagrams that follow have a resistance of one ohm.

This can be shown using the Coulomb Train Model like this (coulombs pictured as moving clockwise):

The reason the voltmeter across the cell reads +1.5 V is that energy is being transferred from the chemical energy store of the cell *into* the coulombs. The reason the voltmeter reads -1.5 V across the resistor is that energy is being transferred *from* the coulombs and into the thermal energy store of the resistor.

The current passing through the resistor using I = V/R = 1.5 V / 1 = 1.5 amperes.

Now let’s apply this convention when two resistors are in parallel.

This can be represented using the Coulomb Train Model like this:

I think it’s far clearer that ammeter W is measuring the total current in the circuit while X and Y are measuring the ‘part-current’ passing through R1 and R2 using this convention. (Note: we are assuming that each resistor has a resistance of one ohm.)

Each resistor has a potential difference of -1.5 V because 1.5 J of energy is being shifted from each coulomb as they pass through each resistor.

Also, it is clearer that the cell’s chemical energy store is being drained more quickly when there are two resistors in parallel: two coulombs have to be filled with 1.5 J of energy for each one coulomb in the single resistor circuit.

Thinking about current, the total current in the circuit is 3.0 amperes; so the resistance R = V / I = 1.5 / 3.0 = 0.5 ohms. So two resistors in parallel have a smaller resistance than a single resistor — this is a result that is well worth emphasising for students as so many of them find this completely counterintuitive!

Teaching components in series using the all-in-a-row convention

This circuit can be represented using the Coulomb Train Model like this:

The pattern of potential difference can be explained by looking at the orange ‘energy levels’ carried by each coulomb.

A current of one amp is one coulomb passing per second, so we can see that an ammeter reading would have the same value wherever the ammeter is placed in the circuit.

But look closely at R1: it only has 0.75 V of potential difference across. From I = V/R = 0.75 / 1 = 0.75 amperes.

This means that the total resistance of the circuit from R = V/I is, of course, 2 ohms.


I regret to say that I have probably been teaching ‘series circuits’ and ‘parallel circuits’ on autopilot for much of my career; the same may even be true of some readers of this blog(!)

The Coulomb Train Model has been considered in depth in previous blogs, but I think it’s a good model to encourage students to use their physical intuition (aka ’embodied cognition’) to understand electric circuits.

Whether you agree with the suggested outlines above or not, I hope that it has given you some fruitful food for thought.

Circuit Diagrams: Lost in Rotation…?

Is there a better way of presenting circuit diagrams to our students that will aid their understanding of potential difference?

I think that, possibly, there may be.

(Note: circuit diagrams produced using the excellent — and free! — web editor at

Old ways are the best ways…? (Spoiler: not always)

This is a very typical, conventional way of showing a simple circuit.

A simple circuit as usually presented

Now let’s measure the potential difference across the cell…

Measuring the potential difference across the cell

…and across the resistor.

Measuring the potential difference across the resistor

Using a standard school laboratory digital voltmeter and assuming a cell of emf 1.5 V and negligible internal resistance we would get a value of +1.5 volts for both positions.

Let me demonstrate this using the excellent — and free! — pHET circuit simulation website.

Indeed, one might argue with some very sound justification that both measurements are actually of the same potential difference and that there is no real difference between what we chose to call ‘the potential difference across the cell’ and ‘the potential difference across the resistor’.

Try another way…

But let’s consider drawing the circuit a different (but operationally identical) way:

The same circuit drawn ‘all-in-a-row’

What would happen if we measured the potential difference across the cell and the resistor as before…

This time, we get a reading (same assumptions as before) of [positive] +1.5 volts of potential difference for the potential difference across the cell and [negative] -1.5 volts for the potential difference across the resistor.

This, at least to me, is a far more conceptually helpful result for student understanding. It implies that the charge carriers are gaining energy as they pass through the cell, but losing energy as they pass through the resistor.

Using the Coulomb Train Model of circuit behaviour, this could be shown like this:

+1.5 V of potential difference represented using the Coulomb Train Model
-1.5 V of potential difference represented using the Coulomb Train Model. (Note: for a single resistor circuit, the emerging coulomb would have zero energy.)

We can, of course, obtain a similar result for the conventional layout, but only at the cost of ‘crossing the leads’ — a sin as heinous as ‘crossing the beams’ for some students (assuming they have seen the original Ghostbusters movie).

Crossing the leads on a voltmeter

A Hidden Rotation?

The argument I am making is that the conventional way of drawing simple circuits involves an implicit and hidden rotation of 180 degrees in terms of which end of the resistor is at a more positive potential.

A hidden rotation…?

Of course, experienced physics learners and instructors take this ‘hidden rotation’ in their stride. It is an example of the ‘curse of knowledge’: because we feel that it is not confusing we fail to anticipate that novice learners could find it confusing. Wherever possible, we should seek to make whatever is implicit as explicit as we can.


A translation is, of course, a sliding transformation, rather than a circumrotation. Hence, I had to dispense with this post’s original title of ‘Circuit Diagrams: Lost in Translation’.

However, I do genuinely feel that some students understanding of circuits could be inadvertently ‘lost in rotation’ as argued above.

I hope my fellow physics teachers try introducing potential difference using the ‘all-in-row’ orientation shown.

The all-in=a-row orientation for circuit diagrams to help student understanding of potential difference

I would be fascinated to know if they feel its a helpful contribition to their teaching repetoire!

Misconceptions and p-prims at ResearchED 2021

Many thanks to all those who attended my talk on “Dealing with Misconceptions: the p-prim and refining raw intuitions approach” and for the stimulating discussions afterwards!

And especially huge thanks to Bill Wilkinson for his help in sorting out some tech issues!

The PowerPoint can be downloaded below.

You can watch a short summary of the talk here.

The references are below with links to freely available copies (where I’ve been able to find them).

I think Redish and Kuo (2015) is an excellent introduction to the Resources Framework.

DiSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49–70). Lawrence Erlbaum Associates, Inc.

DiSessa, A. A. (1993). Toward an epistemology of physics. Cognition and instruction10(2-3), 105-225.

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics68(S1), S52-S59.

Nelmes, A. (2004). Putting conceptions in their place: using analogy to cue and strengthen scientifically correct conceptions.

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

Booklet for teaching the Coulomb Train Model

At the bottom of the post are some links to a student booklet for teaching part of the electricity content for AQA GCSE Physics / AQA GCSE Combined Science using the Coulomb Train Model.

Extract from booklet

I have believed for a long time that the electricity content is often ‘under-explained’ at GCSE: in other words, not all of the content is explicitly taught. I have deliberately have gone to the opposite extreme here — indeed, some teachers may feel that I have ‘over-explained’ too much of the content. However, the booklets are editable so feel free to adapt!

I think the booklet is suitable for teacher-led instruction as well as independent study — I would love to hear how your students have responded to it.

The animations will be ‘live’ for the Google Docs and MS Word versions, but will be frozen for the PDF version. They can be cut and pasted into Powerpoint or other teaching packages (but please note that in some versions of PPT, the animations will appear frozen until you go into presenter mode).

Please feel free to download, use and adapt as you see fit. It is released under the terms of the Creative Commons Attribution License CC BY-SA 4.0 (details here), so please flag if you see versions being sold on TES or similar websites.

The remaining content for AQA electricity will be released (fingers crossed) over the next couple of months.

Feedback and comments (hopefully mainly positive) always welcome….

Why we wrote ‘Cracking Key Concepts in Secondary Science’

From the Introduction

“We strongly believe that the central part of any science lesson or learning sequence is a well-crafted and executed explanation.

“But we are also aware that many – if not most – teachers have had very little training in how to actually go about crafting or executing their explanations. As advocates of evidence-informed teaching, we hope to bring a new perspective and set of skills to your teaching and empower you to take your place in the classroom as the imparter of knowledge.

“We do, however, wish to put paid to the suspicion that we advocate science lessons to be all chalk and talk: we strongly urge that teachers should use targeted and interactive questioning, model answers, practical work, guided practice and supported individual student practice in tandem with ‘teacher talk’. There is a time when the teacher should be a ‘guide on the side’ but the main focus of this book is to enable you to shine when you are called to be a science ‘sage on the stage’.

[…] “For many years, it seems that teacher explanation has been taken for granted. In a nation-wide focus on pedagogy, activity, student-led learning and social constructivism, the role of the teacher in taking challenging material and explaining it has been de-emphasised, with discovery, enquiry, peer-to-peer tuition and ‘figuring things out for yourself’ becoming ascendant. Not only that, but a significant number of influential organisations and individuals championed the cause of ‘talk-less teaching’ where the teacher was relegated to a near-voiceless ‘guide on the side’, sometimes enforced by observers with a stopwatch and an inflexible ‘teacher talk’ time limit.

“We earnestly hope that such egregious excesses are now a thing of the past; but we must admit that all too often, the mistakes engendered by well-meaning edu-initiatives live on, while whatever good they achieved lies composting with the CPD packs from ancient training days. Even if they are a thing of the past, there has been a collective deskilling when it comes to the crafting of a science explanation – there is little institutional wisdom and few, if any, resources for teachers to use as a reference.”

And that is one reason why we wrote the book.

What follows is an example of how we discuss a teaching sequence in the book.

Viewing waves through the lens of concrete to abstract progression

Many students have a concrete idea of a wave as something ‘wavy’ i.e. something with crests and troughs. However, in a normal teaching sequence we often shift from a wave profile representation to a wavefront representation to a ray diagram representation with little or no explanation — is it any wonder that some students get confused?

I have found it useful to consider the sequence from wave profile to wavefront to ray as representations that move from the concrete and familiar representation of waves as something that looks ‘wavy’ (wave profile) to something that looks less wavy (wavefront) to something more abstract that doesn’t look at all ‘wavy’ (ray diagram) as summarised in the table below.

Each row of the table shows the same situation represented by different conventions and it is important that students recognise this. You can quiz students to check they understand this idea. For example:

  • Top row: which part of the wave do the straight lines in the middle picture represent? (The crests of the waves.)
  • Top row: why are the rays in the last picture parallel? (To show that the waves are not spreading out.)
  • Middle row: compare the viewpoints in the first and middle picture. (The first is ‘from the side’, the middle is ‘from above, looking down.’)
  • Middle row: why are the rays in the last picture not parallel? (Because the waves are spreading out in a circular pattern.)

Once students are familiar with this shift in perspective, we can use to explain more complex phenomena such as refraction.

For example, we begin with the wave profile representation (most concrete and familiar to most students) and highlight the salient features.

Next, we move on to the same situation represented as wavefronts (more abstract).

Finally, we move on to the most abstract ray diagram representation.

‘Cracking Key Concepts in Secondary Science’ is available in multiple formats from Amazon and Sage Publishing. You can also order the paperback and hardback versions direct from your local bookshop 🙂

We hope you enjoy the book and find it useful.

STOP PRESS! 25% discount!

This is only available if you order directly from SAGE Publishing before 31/12/2021 and some terms and conditions apply (see SAGE website).

  1. Go to
  2. Search for ‘Cracking Key Concepts’
  3. Enter the discount code ‘UK21AUTHOR’ at the checkout.
  4. Wait for your copy to be delivered post-haste by Royal Mail.
  5. Enjoy!

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.

The Coulomb Train Revisited (Part 2)

In this post, we will look at understanding potential difference (or voltage) using the Coulomb Train Model.

This is part 2 of a continuing series. You can read part 1 here.

The Coulomb Train Model (CTM) is a helpful model for both explaining and predicting the behaviour of real electric circuits which I think is suitable for use with KS3 and KS4 students (that’s 11-16 year olds for non-UK educators).

To summarise what has been discussed so far:

Modelling potential difference using the CTM

Potential difference is the ‘push’ needed to make electric charge move around a closed circuit. On the CTM, we can represent the ‘push’ as a gain in the energy of the coulomb. (This is consistent with the actual definition of the volt V = E/Q, where one volt is a change in energy of one joule per coulomb.)

How can we observe this gain in energy? Simple, we use a voltmeter.

Kudos to for the lovely circuit diagrams

On the CTM, this would look like this:

What the voltmeter does is compare the energy contained by two coulombs: one at A and the other at B. The coulombs at B, having passed through the 1.5 V cell, each have 1.5 joules of energy more than than the coulombs at A. This means that the voltmeter in this position reads 1.5 volts. We would say that the potential difference across the cell is 1.5 V. (Try and avoid talking about the potential difference ‘through’ or ‘of’ any part of the circuit.)

More potential difference measurements using the CTM

Let’s move the voltmeter to a different position.

On the CTM, this would look like this:

Let’s make the very reasonable assumption that the connecting wires have zero resistance. This would mean that the coulombs at C have 1.5 joules of energy and that the coulombs at D have 1.5 joules of energy. They have not lost any energy since they have not passed through any part of the circuit that actually has a resistance. The voltmeter would therefore read 0 volts since it cannot detect any energy difference.

Now let’s move the voltmeter one last time.

On the CTM, this would look like this:

Notice that the coulombs at F have 1.5 fewer joules than the coulombs at E. The coulombs transfer 1.5 joules of energy to the bulb because the bulb has a resistance.

Any part of the circuit that has non-zero resistance will ‘rob’ coulombs of their energy. On this very simple model, we assume that only the bulb has a resistance and so only the bulb will ‘push back’ against the movement of the coulombs and cost them energy.

Also on this simple model, the potential difference across the bulb is identical to the potential difference across the cell — but this is not always the case. For example, if the wires had a small but non-negligible resistance and if the cell had an internal resistance, but these would only come into play at A-level.

The bulb is shown as ‘flashing’ on the CTM to provide a visual cue to help students mentally model the transfer of energy from the coulombs to the bulb. In reality, instead of just one coulomb transferring a largish ‘chunk’ of energy, there would be approximately 1.25 billion billion electrons continuously transferring a tiny fraction of this energy over the course of one second (assuming a d.c. current of 0.2 amps) so we wouldn’t see the bulb ‘flash’ in reality.

How do the coulombs ‘know’ how much energy to drop off?

This section is probably more of interest to specialist physics teachers, but all are welcome.

One frequent criticism of donation models like the CTM is how do the coulombs ‘know’ to drop off all their energy at the bulb?

The response to this, of course, is that they don’t. This criticism is an artefact of an (arguably) over-simplified model whereby we assume that only the bulb has resistance. The energy carried by the coulombs according to this model could be shown as a sketch graph, and let’s be honest it does look a little dodgy…

But, more accurately, of course, the energy loss is a process rather than an event. And the connecting wires actually have a small resistance. This leads to this graph:

Realistically speaking, the coulombs don’t lose all their energy passing through the bulb: they merely lose most of their energy here due to the process of passing through a high resistance part of the circuit.

In part 3 of this series, we’ll look at how resistance can be modelled using the CTM.

You can read part 3 here.

Put Not Your Trust In Pyramids

Put not your trust in princes.

Psalm 146, KJV

Triangles and pyramids do to teachers what catnip does to cats.

Translation from Lolcat: ‘More triangles, please!’

Put just about any idea in the form of a three-sided polygon and watch teachers adopt it en masse as an article of faith. And, boy, have we as a profession unquestionably and uncritically adopted some stinkers.

What follows is a countdown, from least-worst to worst (in my estimation), of what I would collectively call . . . [PLAYS SINISTER ORGAN NOTES, ACTIVATES VOICE-ECHO] . . . PERFIDIOUS PYRAMIDS!

Number 4: Maslow’s Hierarchy of Needs

Why bring this one up? Firstly, Maslow never put his hierarchy in the form of a pyramid. This implies that all of a student’s ‘Deficiency Needs’ must be met before the ‘Being (growth) Needs’ can be addressed; Maslow was more nuanced in his original writings.

The analogy of psychological needs to vitamins was drawn by Maslow. Like vitamins, each of the needs is individually required, just as having much of one vitamin does not negate the need for other vitamins. All needs should independently contribute to subjective well being.

Tay and Diener 2011

Secondly, the methodology by which Maslow arrived at the characteristics of a ‘self-actualized’ person was by looking at the writings and biographies of a number of people (including Albert Einstein and Mother Theresa) whom he considered to be ‘self-actualized’: this is a qualitative and subjective approach that would seem highly open to personal bias and hard to characterise as ‘scientific’ (McLeod 2018).

Number 3: Bloom’s Taxonomy

As with Maslow, there is little to argue with the intent behind Bloom’s Taxonomy, which was an attempt to classify educational objectives without — repeat, without — arranging them into a formal hierarchy.

Bloom’s Taxonomy is often missapplied in education because the ‘higher’ levels are deemed more desirable than the ‘lower’ levels.

As Sugrue (2002) notes:

It was developed before we understood the cognitive processes involved in learning and performance. The categories or ‘levels’ of Bloom’s taxonomy … are not supported by any research on learning.

And, sadly,

the popular misinterpretation of the taxonomy has led to a multi-generational loss of learning opportunities. It is a triumph of philosophy over science, of populism over rigour, of politics over responsibility.

James Murphy, The False Dichotomy

Number 2: Formula Triangles

The main issue with formula triangles is that they are a replacement for algebra rather than a system or scaffold for supporting students in learning how to manipulate equations.

Koenig (2015) makes some trenchant criticisms of forumula triangles, as does Southall (2016) who argues that they are a form of ‘procedural’ teaching rather than the demonstrably more effective ‘conceptual’ teaching. Conceptual teaching encourages students to understand why a particular technique is used rather than applying it as a ‘magic’ formula. Borij, Radmehr and Font (2019) also have an interesting and nuanced discussion on these types of teaching (in the context of learning calculus).

Workable alternatives to formula triangles are the FIFA and EVERY systems.

But the winner for the educationally worst pyramid or triangle is . . . [DRUM ROLL]

NUMBER 1: The Learning Pyramid

To put it bluntly, there is no research to support the percentage retention rates claimed on any version of this pyramid.

Modern versions of the learning pyramid seem to be based on Edgar Dale’s ‘Cone of Experience’ first published in 1946 in his influential book Audio-Visual Teaching Techniques.

Dale’s Cone of Experience as presented in the 1954 edition (with ‘Television’ added from previous versions). From Lalley and Miller 2007

Dale’s main argument was to encourage

the use of audio-visual materials in teaching – materials that do not depend primarily upon reading to convey their meaning. It is based upon the principle that all teaching can be greatly improved by the use of such materials because they can help make the learning experience memorable…this central idea has, of course, certain limits. We do not mean that sensory materials must be introduced into every teaching situation. Nor do we suggest that teachers scrap all procedures that do not involve a variety of audio-visual methods

Dale 1954 quoted by Lalley and Miller 2007

The peculiarly neat percentage increments in retention rates on the learning pyramid are first found in Treichler (1967). As Letrud and Hernes (2018) note:

Treichler asserted that these numbers came from studies, but he did not say where they could be found. […] A set of learning modalities similar to those distributed by Treichler were at some point fused with a misreading of Edgar Dale’s Cone of experience as a hierarchy of learning modalities, and these early categories were supplemented and partly replaced with categories of presentation modalities like “audiovisual”, “demonstrations”, and “discussion groups”.

The final word is perhaps best left to Lalley and Miller:

The research reviewed here demonstrates that use of each of the methods identified by the pyramid resulted in retention, with none being consistently superior to the others and all being effective in certain contexts. A paramount concern, given conventional wisdom and the research cited, is the effectiveness and importance of reading and direct instruction, which in many ways are undermined by their positions on the pyramid. Reading is not only an effective teaching/learning method, it is also the main foundation for becoming a “life-long learner”


Borji, V., Radmehr, F., & Font, V. (2019). The impact of procedural and conceptual teaching on students’ mathematical performance over time. International Journal of Mathematical Education in Science and Technology, 1-23.

Dale, E. (1954). Audio-visual methods in teaching (2 ed.). New York: The Dryden Press.

Koenig, J. (2015). Why Are Formula Triangles Bad? Education In Chemistry, Royal Society of Chemistry.

Lalley, J., & Miller, R. (2007). The learning pyramid: Does it point teachers in the right direction. Education128(1), 16.

Letrud, K., & Hernes, S. (2018). Excavating the origins of the learning pyramid myths. Cogent Education5(1), 1518638.

McLeod, S. (2018). Maslow’s hierarchy of needs. Simply psychology1, 1-8.

Southall, E. (2016). The formula triangle and other problems with procedural teaching in mathematics. School Science Review97(360), 49-53.

Sugrue, B. Problems with Bloom’s Taxonomy. Presented at the International Society for Performance Improvement Conference 2002

Tay, L., & Diener, E. (2011). Needs and subjective well-being around the world. Journal of personality and social psychology101(2), 354.

Treichler, D. G. (1967). Are you missing the boat in training aids? Film and Audio-Visual Communication, 1(1), 14–16, 28–30,48.

Cornell versus Ebbinghaus

Most of us are only too familiar with the mordant truth of Shakespeare’s observation that “Old men forget, yet all shall be forgot”. In fact, things are generally even worse than the Bard suggests: everyone forgets, all the time.

In time, all shall indeed be forgot.

This was established experimentally by Hermann Ebbinghaus in 1880. The graph below shows Ebbinghaus’ original results with some more recent replications (from Murre and Dros 2015).

Diagram from

However, there is a workaround or “hack” that allows us to beat the Ebbinghaus curve of forgetfulness.

The Power of Review

Diagram from Chun and Heo 2018. (Top annotations with coloured circles added to original.)

If the content is reviewed at regular intervals, not only do we remember more but the review process also slows down the rate at which knowledge decays.

Cornell notes as a structure for regular review

‘Cornell notes’ is a two column note-taking system developed by Cornell University Professor of Education Walter Pauk (1974). (See also this link.)

I developed its use in Physics classes with a mind to defeating the Ebbinghaus forgetting curve using this template (click on the link to download a blank printable pdf version).

Example of Cornell style notes on the photoelectric effect

Step 1 Students write notes

In the lesson, students complete the sections highlighted in red but they should leave the other sections blank. This can be a bit of struggle with some students, but is actually a vital part of the process.

Then the students wait 24 hours.

The first couple of times you try this with a class, it might be worth insisting that all students hand in their incomplete Cornell notes at this point just to make sure they follow the process correctly. As students learn to appreciate the effectiveness of the process, you can trust them to follow it without taking control of their work (hopefully!)

Step 2 Students complete the Questions / Key Words section

After a pause of 24 hours, students then complete the section highlighted in green. Of course, they have to thoroughly review and think hard about the material in the notes section to do this, and in Daniel Willingham’s resonant phrase: “Memory is the residue of thought.”

Then, wait a further 48 hours. (Again, the first couple of times you do this with a class, you may want to take in the incomplete Cornell notes to make sure the process is followed correctly: many students seem to find it impossible to “let it be”!)

Step 3 Students complete the summary section

48 hours after completing the Questions / Key Words section, students complete the Summary section.

Students often find writing the Summary the hardest part of the process and usually need the most support with this section. The limited space forces concision and an intense focus on the most important concepts — which, of course, is no bad thing in itself!

As an addition to step 3 and following Cho (2011), writing a Reflection on the back of the Cornell notes sheet can be useful to encourage retention. The Reflection is intended to elicit or memorialise an emotional reaction to the content. The context of this could be “Big Picture”, professional, historical or personal.

Students are encouraged to select one context and write something that has emotional resonance for them. Examples relevant to the photoelectric effect (see above) might be:

  • “Big picture”: The photoelectric effect is the basis of all light detection technology. Without the science of the photoelectric effect, the fibre optic data networks on which our interconnected society depends would be not only impossible but unthinkable.
  • Professional: As an electronic engineer, I would use the photoelectric effect to design super-sensitive electronic cameras that can be used with large aperture telescopes to build up — photon by photon — images of galaxies that are so distant that their light left them four and a half billion years before the Sun formed.
  • Historical: Einstein’s 1905 paper on the photoelectric effect was one of the trio of papers published in his “Annus Miriablis” (“Miracle Year”). In the other two he outlined the theory of Special Relativity and used Brownian motion to prove the existence of atoms. Historians of science say that any one of the three would have been enough to secure his reputation as one of the most important physicists of the 20th Century!
  • Personal: I thought this was one of the most mathematically challenging topics that we have covered so far in Physics. I am really pleased that I can successfully handle the algebra but also have a good understanding of the physical meaning of all the terms.

Step 4 Independent Review

This can be as simple as covering the red section 1 with a piece of paper and using the Questions and Key Words section as a cue to recall the hidden content.


This was run as a pilot project in Y12 with A-level Physics students. In Y13, they were taught by different teachers who did not use the adapted system. About one quarter of the students who had been taught the process were still using it for Y13 revision and were enthusiastic about how much they felt it boosted their recall of content and understanding.

Some research (e.g. Ahmad 2019) suggests learning gains for students who use the traditional (non-adapted) Cornell notes system. Interestingly, Jacobs (2008) suggests a large improvement in “higher level question” scores for Cornell notes students (again, not the adapted Cornell notes version outlined above).


Ahmad, S. Z. (2019). Impact of Cornell Notes vs. REAP on EFL Secondary School Students’ Critical Reading Skills. International Education Studies12(10), 60-74.

Cho, J. (2011). Improving science learning through using interactive science notebook (ISN). In P. Gouzouasis (Ed.), Pedagogy in a new tonality (pp. 149-166). Rotterdam, the Netherlands: Sense Publishers.

Chun, B. A., & Heo, H. J. (2018). The effect of flipped learning on academic performance as an innovative method for overcoming Ebbinghaus’ forgetting curve. In Proceedings of the 6th International Conference on Information and Education Technology (pp. 56-60).

Jacobs, K. (2008). A comparison of two note taking methods in a secondary English classroom. Proceedings of the 4th Annual GRASP Symposium, Wichita State University, 2008 (pp. 119-120).

Murre, J. M., & Dros, J. (2015). Replication and analysis of Ebbinghaus’ forgetting curve. PloS one10(7), e0120644.

Pauk, W. (1974). How to study in college. Boston: Houghton Mifflin.