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.


Reference

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 https://www.circuit-diagram.org/editor/ 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”

References

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 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0120644

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.

Conclusion

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).

References

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. https://doi.org/10.1007/978-94-6091-669-4_10

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.

Electric Motors Without The Left Hand Rule

There is little doubt that students find understanding how an electric motor works hard.

What follows is an approach that neatly sidesteps the need for applying Fleming’s Left Hand Rule (FLHR) by using the idea of the catapult field.

The catapult field is a neat bit of Physics pedagogy that appears to have fallen out of favour in recent years for some unknown reason. I hope to rehabilitate and publicise this valuable approach so that more teachers may try out this electromagnetic ‘road less travelled’.

Screenshot 2020-02-21 at 09.48.42

(Incidentally, if you are teaching FLHR, the mnemonic shown above is not the best way to remember it: try using this approach instead.)

The magnetic field produced by a long straight conductor

Moving electric charges produce magnetic fields. When a current flows through a conductor, it produces a magnetic field in the form of a series of cylinders centred on the wire. This is usually shown on a diagram like this:

Screenshot 2020-02-21 at 09.59.26.png

If we imagine looking down from a point directly above the centre of the conductor (as indicated by the disembodied eye), we would see a plan view like this:

Screenshot 2020-02-21 at 10.01.22.png

We are using the ‘dot and cross‘ convention (where an X represents an arrow heading away from us and a dot represents an arrow heading towards us) to easily render a 3D situation as a 2D diagram.

The direction of the magnetic field lines is found by using the right hand grip rule.

Screenshot 2020-02-21 at 10.38.32.png

The thumb is pointed in the direction of the current. The field lines ‘point’ in the same direction as the fingers on the right hand curl.

3D to 2D

Now let’s think about the interaction between the magnetic field of a current carrying conductor and the uniform magnetic field produced by a pair of magnets.

In the diagrams below, I have tried to make the transition between a 3D and a 2D representation explicit, something that as science teachers I think we skip over too quickly — another example of the ‘curse of knowledge’, I believe.

Screenshot 2020-02-21 at 11.18.21.png

Magnetic Field on Magnetic Field

If we place the current carrying conductor inside the magnetic field produced by the permanent magnets, we can show the magnetic fields like this:

Screenshot 2020-02-21 at 11.32.31.png

Note that, in the area shaded green, the both sets of magnetic field lines are in the same direction. This leads a to stronger magnetic field here. However, the opposite is true in the region shaded pink, which leads to a weaker magnetic field in this region.

Screenshot 2020-02-21 at 11.33.27.png

The resultant magnetic field produced by the interaction between the two magnetic fields shown above looks like this.

Screenshot 2020-02-21 at 11.37.19.png

Note that the regions where the magnetic field is strong have the magnetic field lines close together, and the regions where it is weak have the field lines far apart.

The Catapult Field

This arrangement of magnetic field lines shown above is unstable and is called a catapult field.

Essentially, the bunched up field lines will push the conductor out of the permanent magnetic field.

If I may wax poetic for a moment: as an oyster will form a opalescent pearl around an irritant, the permanent magnets form a catapult field to expel the symmetry-destroying current-carrying conductor.

Screenshot 2020-02-21 at 11.50.55.png

The conductor is pushed in the direction of the weakened magnetic field. In a highly non-rigorous sense, we can think of the conductor being pushed out of the enfeebled ‘crack’ produced in the magnetic field of the permanent magnets by the magnetic field of the current carrying conductor…

Also, the force shown by the green arrow above is in exactly the same direction as the force predicted by Fleming’s Left Hand Rule, but we have established its direction using only the right hand grip rule and a consideration of the interaction between two magnetic field.

The Catapult Field for an electric motor

First, let’s make sure that students can relate the 3D arrangement for an electric motor to a 2D diagram.

Screenshot 2020-02-21 at 13.42.51.png

The pink highlighted regions show where the field lines due to the current in the conductor (red) are in the opposite direction to the field line produced by the permanent magnet (purple). These regions are where the purple field lines will be weakened, and the clear inference is that the left hand side of the coil will experience an upward force and the right hand side of the coil will experience a downward force. As suggested (perhaps a little fancifully) above, the conductors are being forced into the weakened ‘cracks’ produced in the purple field lines.

The catapult field for the electric motor would look, perhaps, like this:

Screenshot 2020-02-21 at 14.39.36.png

And finally…

On a practical teaching note, I wouldn’t advise dispensing with Fleming’s Left Hand Rule altogether, but hopefully the idea of a catapult field adds another string to your pedagogical bow as far as teaching electric motors is concerned (!)

I have certainly found it useful when teaching students who struggle with applying Fleming’s Left Hand Rule, and it is also useful when introducing the Rule to supply an understandable justification why a force is generated by a current in a magnetic field in the first place.

The catapult field is a ‘road less travelled’ in terms of teaching electromagnetism, but I would urge you to try it nonetheless. It may — just may — make all the difference.

Why does kinetic energy = 1/2mv^2?

Why does kinetic energy Ek=½mv2?

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

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

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

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

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

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

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

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

Step 2:    W= F x s

But remember s = v x t so:

Step 3:    W= F x vt

And also remember that F = m x a so:

Step 4:    W= ma x vt

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

Step 5:        Wd = (v / t) x vt

The ts cancel so:

Step 6:    W= mv2

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

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

Screenshot 2019-03-09 at 15.14.17

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

But wait: Ek=mv2!?!?

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

Houston, we have a problem.

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

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

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

So Step 3 should read:

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

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

Step 3:    Wd= F x (v / 2) t

And also remember that F = m x a so:

Step 4:    Wd= ma x (v / 2) t

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

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

The ts cancel so:

Step 6:    Wd= ½mv2

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

Screenshot 2019-03-09 at 17.58.45.png

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

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

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

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

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.

Crossing Cognitive Chasms With P-prims

Crossing a cognitive chasm . . .

Apparently, roughly 10% of humans still believe that the Earth is larger than the Sun. Do they believe this because they haven’t been properly educated? Possibly. Do they believe this because they’re stupid? Probably not.

In fact, the most likely explanation is that the individuals concerned just haven’t thought that much about it. The Earth looks big; the Sun looks like a small disc in the sky; ergo, the Sun is smaller than the Earth.

The individuals are relying on what Andrea diSessa (1988) would call a phenomenological primitive or p-prim: “These are simple abstractions from common experiences that are taken as relatively primitive in the sense that they generally need no explanation; they simply happen.”

What is a p-prim (phenomenological primitive)?

A p-prim is a pattern of thought that is applied across a range of contexts. For example, the “Ohm’s Law” p-prim — the idea that increased “effort” invariably leads to a larger “outcome” and that increased “resistance” always yields a smaller “outcome” — is routinely applied not just to the domain of electrical circuits, but to the physical world in terms of pushing and pulling objects, and not least to the domain of psychology in the context (say) of persuading a reluctant person to perform an action.

Examples of other p-prims would include:

  • The “Father Dougal” p-prim: things that look small really are small; large things always look bigger than small things.
  • The “More Is Better” p-prim: that more of any quantity is invariably better than a smaller amount.
  • The “Dynamic Balance” p-prim: equal and opposite competing “forces” or “influences” can produce an equilibrium or “static outcome”.

P-prims are not acquired by formal teaching. They are abstractions or patterns extracted from commonplace experiences. They are also, for the most part, primarily unspoken concepts: ask a person to justify a p-prim and the most likely answer is “because”!

Also, p-prims exist in isolation: people can easily hold two or more contradictory p-prims. The p-prim that is applied depends on context: in one situation the “Ohm’s Law” p-prim might be cued; in another the “Dynamic Balance” p-prim would be cued. Which p-prim is cued depends on the previous experience of the individual and the aspects of the situation that appear most significant to that individual at that particular time.

The KIP (Knowledge in Pieces) Model

diSessa integrates these p-prims (and many others) into a “Knowledge in Pieces” model:

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

The model is summarised more poetically by Dashiell Hammett (quoted by diSessa):

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

So, for example, a person might respond to the (to them) out-of-left-field question of “Which is bigger: the Earth or the Sun?” by simply selecting what seems to them a perfectly appropriate p-prim such as the “Father Dougal” p-prim: the Sun looks like a small disc in the sky therefore it is smaller than the Earth. It is important to note that this process often happens without a great deal of thought. The person reaches into a grab-bag of these small units of thought and takes hold of one that, at least at first glance, seems applicable to the circumstances. The person is simply applying their past experience to a novel situation.

Picking Your P-prim

However, as Anne Nelmes (2004) points out, the problem is that often the wrong p-prim is cued and applied to the wrong situation. As science teachers, is there a way that we can encourage the selection of more suitable p-prims?

Nelmes believes that there is:

Analogy has long been used to aid understanding of scientific concepts, both in and out of the classroom. Rather than trying to overtly change the misconception into the scientific conception, it may be as, or more, effective and certainly less time consuming to cue the right idea using analogy on a very low key level, without the pupils even realising that an analogy has been used. The idea of cueing correct ideas comes from work done by diSessa and others on p-prims (phenomenological primitives). These are small knowledge units which are cued to an active state to explain phenomena.

It is hoped the correct p-prim will be cued by use of the analogy and, if cued repeatedly, will strengthen.

One example presented by Nelmes that I find quite persuasive is in the context of students’ difficulty in accepting that good absorbers of heat radiation are also good emitters of heat radiation. A matt black surface will absorb a substantial fraction of the infrared radiation falling on it; however, matt black surfaces are also the most effective emitters of infrared radiation.

aborbers emitters

This seems a concept-change-too-far for many students; particularly as it often follows hard on the heels of good conductor = poor insulator and good insulator = poor conductor. Students find it hard to accept that a substance that is good at one thing can also be good at its opposite.

Nelmes suggests cueing a more appropriate p-prim for this context by the use of low key analogies such as:

  • Effective communicators are good at taking in information and good at giving out information.
  • Effective netball players are good at throwing the ball and catching the ball.

Nelmes’ research suggests that the results from such strategies may be modest but are generally positive. One telling example is the fact that many student answers featured “you” as in “I think this because when you are good at something, radiating, you are usually good at the other, absorbing heat.”

As Nelmes notes, the use of the personal pronoun in such answers suggests that students had, perhaps, absorbed the bridging analogy unconsciously.

Be that as it may, I think the p-prim and bridging analogy strategy is one I will be attempting to add to my teaching repertoire.

References

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.

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

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.

You can read more on the CTM here.