Physics

FIFA for the GCSE Physics calculation win

e=mc2andallthat41 Comments

Student: Did you know FIFA is also the name of a video game, Sir?

Me: Really?

Student: Yeah. It’s part of a series. I just got FIFA 20. It’s one of my favourite games ever.

Me: Goodness me. I had no idea. I just chose the letters ‘FIFA’ completely and utterly at random!

The FIFA method is an AQA mark scheme-friendly* way of approaching GCSE Physics calculation questions. (It is also useful for some Y12 Physics students.)

I mentioned it in a previous blog and @PedagogueSci was kind enough to give it a boost here, so I thought I’d explain the method in a separate blog post. (Update: you can also watch my talk at ChatPhysics Live 2021 here.)

The FIFA method:

  1. Avoids the use of formula triangles
  2. Minimises the cognitive load on students when approaching calculations.

Why we shouldn’t use formula triangles

Formula triangles are bad news. They are a cognitive dead end.

Screenshot 2019-10-27 at 15.34.54

During a university admissions interview for veterinary medicine, I asked a prospective student to explain how they would make up a solution for infusion into a dog. Part of the answer required them to work out the volume required for a given amount and concentration. The candidate started off by drawing a triangle, then hesitated, eventually giving up in despair. […]

They are a trick that hides the maths: students don’t apply the skills they have previously learned. This means students don’t realise how important maths is for science.

I’m also concerned that if students can’t rearrange simple equations like the one above, they really can’t manage when equations become more complex.

— Jenny Koenig, Why Are Formula Triangles Bad? [Emphases added]

[Update: this 2016 article from Ed Southall also makes a very persuasive case against formula triangle.]

I believe the use of formula triangle also increases (rather than decreases) the cognitive load on students when carrying out calculations. For example, if the concentration c is 0.5 mol dm-3 and the number of moles n required is 0.01 mol, then in order to calculate the volume V they need to:

  • recall the relevant equation and what each symbol means and hold it in working memory
  • recall the layout of symbols within the formula triangle and either (a) write it down or (b) hold it in working memory
  • recall that n and c are known values and that V is the unknown value and hold this information in working memory when applying the formula triangle to the problem

The FIFA method in use (part 1)

The FIFA acronym stands for:

  • FORMULA
  • INSERT VALUES
  • FINE TUNE (this often, but not always, equates to rearranging the formula)
  • ANSWER

Lets look at applying it for a typical higher level GCSE Physics calculation question

Screenshot 2019-10-27 at 16.04.29.png

We add the FIFA rubric:

Screenshot 2019-10-27 at 16.13.00.png

Students have to recall the relevant equation as it is not given on the Data and Formula Sheet. They write it down. This is an important step as once it is written down they no longer have to hold it in their working memory.

Screenshot 2019-10-27 at 16.18.15.png

Note that this is less cognitively demanding on the student’s working memory as they only have to recall the formula on its own; they do not have to recall the formula triangle associated with it.

Students find it encouraging that on many mark schemes, the selection of the correct equation may gain a mark, even if no further steps are taken.

Next, we insert the values. I find it useful to provide a framework for this such as:

Screenshot 2019-10-27 at 16.27.41.png

We can ask general questions such as: “What data are in the question?” or more focused questions such as “Yes or no: are we told what the kinetic energy store is?” and follow up questions such as “What is the kinetic energy? What units do we use for that?” and so on.

Screenshot 2019-10-27 at 16.35.54.png

Note that since we are considering each item of data individually and in a sequence determined by the written formula, this is much less cognitively demanding in terms of what needs to be held in the student’s working memory than the formula triangle method.

Note also that on many mark schemes, a mark is available for the correct substitution of values. Even if they were not able to proceed any further, they would still gain 2/5 marks. For many students, the notion of incremental gain in calculation questions needs to be pushed really hard otherwise they will not attempt these “scary” calculation questions.

Next we are going to “fine tune” what we have written down in order to calculate the final answer. In this instance, the “fine tuning” process equates to a simple algebraic rearrangement. However, it is useful to leave room for some “creative ambiguity” here as we can also use the “fine tuning” process to resolve difficulties with units. Tempting though it may seem, DON’T change FIFA to FIRA.

We fine tune in three distinct steps (see addendum):

Screenshot 2019-10-29 at 12.17.55.png

Finally, we input the values on a calculator to give a final answer. Note that since AQA have declined to provide a unit on the final answer line, a mark is available for writing “kg” in the relevant space — a fact which students find surprising but strangely encouraging.

Screenshot 2019-10-29 at 12.16.46.png

The key idea here is to be as positive and encouraging as possible. Even if all they can do is recall the formula and remember that mass is measured in kg, there is an incremental gain. A mark or two here is always better than zero marks.

The FIFA method in use (part 2)

In this example, we are using the creative ambiguity inherent in the term “fine tune” rather than “rearrange” to resolve a possible difficulty with unit conversion.

Screenshot 2019-10-27 at 17.20.42.png

In this example, we resolve another potential difficulty with unit conversion during the our creatively ambiguous “fine tune” stage:

Screenshot 2019-10-27 at 17.33.05.png

The emphasis, as always, is to resolve issues sequentially and individually in order to minimise cognitive overload.

The FIFA method and low demand Foundation tier calculation questions

I teach the FIFA method to all students, but it’s essential to show how the method can be adapted for low demand Foundation tier questions. (Note: improving student performance on these questions is probably a more significant and quicker and easier win than working on their “extended answer” skills).

For the treatment below, the assumption is that students have already been taught the FIFA method in a number of contexts and that we are teaching them how to apply it to the calculation questions on the foundation tier paper, perhaps as part of an examination skills session.

For the majority of low demand questions, the required formula will be supplied so students will not need to recall it. What they will need, however, is support in inserting the values correctly. Providing a framework as shown below can be very helpful:

Screenshot 2019-10-27 at 17.47.24.png

Also, clearly indicating where the data came from is useful.

Screenshot 2019-10-27 at 17.55.45.png

The fine tune stage is not needed, so we can move straight to the answer.

Screenshot 2019-10-27 at 18.01.07.png

Note also that the FIFA method can be applied to all calculation questions, not just the ones that could be answered using formula triangle methods, as in part (c) of the question above.

Screenshot 2019-10-27 at 18.06.16.png

And finally…

I believe that using FIFA helps to make our thinking and methods in Physics calculations more explicit and clearer for students.

My hope is that science teachers reading this will give it a go.

You can read about using the FIFA system for more challenging questions by clicking on these links: ‘Physics six mark calculation? Give it the old FIFA-one-two!‘ and ‘Using the FIFA system for really challenging GCSE calculations‘.

PS If you have enjoyed this, you might also enjoy Dual Coding SUVAT Problems and also Magnification using the Singapore Bar Model.

*Disclaimer: AQA has not endorsed the FIFA method. I describe it as “AQA mark scheme-friendly” using my professional own judgment and interpretation of published AQA mark schemes.

Addendum

I am embarrassed to admit that this was the original version published. Somehow I missed the more straightforward way of “fine tuning” by squaring the 30 and multiplying by 0.5 and somehow moved straight to the cross multiplication — D’oh!

My thanks to @BenyohaiPhysics and @AdamWteach for pointing it out to me.

Screenshot 2019-10-27 at 16.58.23.png

Potential Divider Circuits and the Coulomb Train Model

e=mc2andallthat11 Comments

A potential divider circuit is, essentially, a circuit where two or more components are arranged in series.

(a) Two resistors in series; (b) an ammeter (top) and an electric motor in series; (c) (L to R) a resistor, filament lamp and variable resistor in series

For non-physicists, these types of circuit can sometimes present problems, so in this post I am going to look in detail at the basic physics involved; and I am going to explain them using the CTM or Coulomb Train Model. (You can find the CTM model explained here.)

In the AQA GCSE Physics (and Combined Science) specifications, students are required to know that:

Extract from p.26 of the AQA spec

First, let’s look at the basics of describing electric circuits: current, potential difference and resistance.

1.0 Using the CTM to explain current, potential difference and resistance

Pupils tend to start with one concept for electricity in a direct current circuit: a concept labelled ‘current’, or ‘energy’ or ‘electricity’, all interchangeable and having the properties of movement, storability and consumption. Understanding an electrical circuit involves first differentiating the concepts of current, voltage and energy before relating them as a system, in which the energy transfer depends upon current, time and the potential difference of the battery.

The notion of current flowing in the circuit is one which pupils often meet in their introduction to a circuit and, because this relates well with their intuitive notions, this concept becomes the primary concept. (Driver 1994: 124 [italics added])

To my mind, the CTM is an excellent “bridging analogy” that helps students visualise the invisible. It is a stepping stone that provides some concrete representations of abstract quantities. In my opinion, it can help students

  1. move away from analysing circuits in terms of just current. (In my experience, even when students use terms like “potential difference”, in their eyes what they call “potential difference” behaves in a remarkably similar way to current e.g. it “flows through” components.)
  2. understand the difference between current, potential difference and resistance and how important each one is
  3. begin thinking of a circuit as a whole, interconnected system.

1.1 The CTM and electric current

Let’s begin by looking at a very simple circuit: a one ohm resistor connected across a 1 V cell.

A simple circuit
A very simple circuit

Note that it is a good teaching technique to include two ammeters on either side of the component, although the readings on both will be identical. This is to challenge the perennial misconception that electric current is “used up”. Electric charge, according to our current understanding of the universe, is a conserved quantity like energy in that it cannot be created or destroyed.

The Coulomb Train Model invites us to picture an electric circuit as a flow of positively charged coulombs carrying energy around the circuit in a clockwise fashion as shown below. The coulombs are linked together to form a continuous chain.

The CTM version of a simple circuit
The CTM applied to the very simple circuit shown above.

The name coulomb is not chosen at random: it is the SI unit of electric charge.

The current in this circuit will be given by I = V / R (equation 18 in the list on p.96 of the AQA spec, if you’re keeping track).

Using the AQA mark scheme-friendly FIFA protocol:

The otherwise inexplicable use of the letter “I” to represent electric current springs from the work André-Marie Ampère (1775–1836) and the French phrase intensité de courant (intensity of current).

From Q = I t (equation 17, p.96), current is a flow of electric charge, since I = Q / t. That is to say, if a charge of 2 coulombs passes (AQA call this a “charge flow”) in 2 seconds, the current will be …

A current of 1 amp is therefore represented on the CTM as 1 coulomb (or truck) passing by each second.

1.2 The CTM and Potential Difference

Potential difference or voltage is essentially the “energy difference” across any two parts of a circuit.

The equation used to define potential difference is not the familiar V = IR but rather the less familiar E = QV (equation 22 in the AQA list) where E is the energy transferred, Q is the charge flow (or the number of coulombs passing by in a certain time) and t is the time in seconds.

Let’s see what this would look like using the CTM:

(a) Circuit diagram showing how the measure the potential difference across a 1 V cell. (b) The same circuit represented using the CTM. (Note that the “white gloves” on the ends of the voltmeter connections are intended to be reminiscent of the white gloves of a snooker referee, indicating that the voltmeter does not disrupt the flow of the coulombs: in other words, the voltmeter has a high resistance.)

For the circuit shown, the voltmeter reading is 1 volt.

Note that on the CTM representation, one joule of energy is added to each coulomb as it passes through the cell.

If we had a 1.5 V cell then 1.5 joules would be transferred to each coulomb as it passed through, and so on.

(a) Circuit diagram showing potential difference measured across a connector with negligible resistance. (b) The same circuit represented using the CTM

If the voltmeter is moved to a different position as shown above, then the reading is 0 volts. This is because the coulombs at the points “sampled” by the voltmeter have the same amount of energy, so there is zero energy difference between them.

(a) Measuring the potential difference across a resistor. (b) The same circuit shown using the CTM.

In the position shown above, the voltmeter is measuring the potential difference across the resistor. For the circuit shown (assuming negligible resistance in all other parts of the circuit) the potential difference will be 1 V. In other words, each coulomb is losing one joule of energy as it passes through the resistance.

1.3 The CTM and Resistance

(a) Measuring the current through and the potential difference across a resistor. (b) The same circuit represented using the CTM.

In the circuit above, the potential difference across the resistor is 1 V and the current is 1 amp.

Resistance can therefore be thought of as the potential difference required to drive a current of 1 amp through that part of the circuit. It can also be thought of as the energy lost by each coulomb when a current of 1 amp flows through that part of the circuit; or, energy lost per coulomb per amp.

1.4 Summary

On the diagrams below, the coulombs are moving clockwise.

Summary of CTM

2.0 The CTM applied to a potential divider circuit

A potential divider circuit simply means that at least two resistors are in series so that the potential difference of the cell is shared across the resistors.

2.1 Two identical resistors

Because the two resistors are identical, the 3 V supply is shared equally across both resistors. That is to say, there is a potential difference of 1.5 V across each resistor. But let’s check this by applying V = IR (eq. 18). The total potential difference is 3 V and the total resistance is 1 ohm + 1 ohm = 2 ohms.

Now let’s use V = IR to check that the potential difference across each separate resistor is indeed half the total supply of 3 V. The resistance of one resistor is one ohm and the current through each one is 1.5 A. So V = 1.5 x 1 = 1.5 V.

But what would happen if we doubled the value of each resistor to 2 ohms?

Well, the current would be smaller: I = V/R = 3/4 = 0.75 amps.

The potential difference across each separate resistor would be V = I R = 0.75 x 2 = 1.5 V

So, the potential difference is always split equally when two identical resistors are placed in series (although, of course, the total resistance and the current will be different depending on the values of the resistors).

2.2a Two non-identical resistors

Let’s consider a circuit with a 2 ohm resistor in series with a 1 ohm resistor.

In this circuit, the total resistance is 1 ohm + 2 ohms = 3 ohms. The current flowing through the circuit is I = V / R = 3 / 3 = 1 amp.

So the potential difference across the 2 ohm resistor is V = IR = 1 x 2 = 2 V and the potential difference across the one ohm resistor is V = IR = 1 x 1 = 1 V.

Note that the resistor with the largest value gets the largest “share” of the potential difference.

2.2b Two non-identical resistors (different order)

Now let’s reverse the order of the resistors.

The current remains unchanged because the total resistance of the circuit is still the same.

Note that the largest resistor still gets the largest share of the potential difference, whichever way round the resistors are placed.

2.3 In Defence of the CTM and Donation Models

Many Physics teachers prefer “rope models” to so-called “donation models” like the CTM.

And it is perfectly true that rope models have some good points such as the ability to easily explain AC and a more accurate approximation of what happens when current starts to flow or stops flowing. The difficulty in their use, in my opinion, is that you are using concepts that many students barely understand (e.g. friction to model resistance) to explain how very unfamiliar concepts such as potential difference work. Also, the vagueness of some of the analogs is unhelpful: for example, when we compare potential difference to “push”, are we talking about the net resultant force on the rope or simply the force needed to balance the frictional force and keep it moving at a steady speed?

To my way of thinking, the CTM has the advantage of encouraging quantitative thinking about current, potential difference and resistance almost from the moment of first teaching. Admittedly, it cannot cope with AC — but then again, we model AC as a direct current when we use RMS values. Now admittedly, rope models are far better at picturing what happens in the initial fractions of a second when a current starts to flow after closing a switch. Be that as it may, the CTM comes into its own when we consider the “steady state” of current flow after the initial surge currents.

One of the frequent criticisms (which is usually considered quite damning) of this type of model is “How do the coulombs know how much energy to drop off at each resistor?”

For example, in the diagram above, how do the coulombs “know” to drop off 1 J at the first resistor and 2 J at the second resistor?

The answer is: they don’t. Rather, the energy loss is due to the nature of the resistor: think of a resistor as a tunnel lined with strip curtains. A coulomb loses only a small amount of its excess energy passing through a low value resistor, but a much larger amount passing through a higher value resistor, as modelled below.

Strip curtain model for CTM
A 1 ohm and 2 ohm resistor modelled as strip curtains

FWIW I therefore commend the use of the CTM to all interested parties. 

References

Driver, R., Squires, A., Rushworth, P., & Wood-Robinson, V. (1994). Making sense of secondary science: Research into children’s ideas. Routledge.

Thermal Energy and Internal Energy

e=mc2andallthat1 Comment

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

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

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

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

Thermal Energy Through the Macroscopic Lens

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

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

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

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

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

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

Internal Energy Through the Microscopic Lens

Screenshot 2019-04-14 at 15.27.25.png

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

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

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

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

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

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

Screenshot 2019-04-14 at 16.02.55.png

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

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

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

AQA: Oops-a-doodle!

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

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

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

Screenshot 2019-04-14 at 16.44.38.png

References

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

 

 

Why does kinetic energy = 1/2mv^2?

e=mc2andallthat3 Comments

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: Wd = F x s

But remember s = v x t so:

Step 3: Wd = F x vt

And also remember that F = m x a so:

Step 4: Wd = ma x vt

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

Step 5: Wd = m (v / t) x vt

The ts cancel so:

Step 6: Wd = 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: Ek = 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.png

It 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 = m (v / t) x (v / 2) t

The ts cancel so:

Step 6: Wd= ½mv2

Based on this, of course, Ek = 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.

Another way of representing the kinetic energy as a solid prism is shown below.

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, Ek = ½mv2 by 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

e=mc2andallthat15 Comments

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.

IoP Energy for busy teachers

e=mc2andallthat11 Comments

The first rule of IoP Energy Club is: you do not talk about energy . . .

. . . unless you’re gonna do a calculation.

— with apologies to Brad Pitt and Chuck Palahniuk

In the UK, the IoP (Institute of Physics) has developed a model of energy stores and energy pathways that has been adopted by all the exam boards. Although answers couched in terms of the old “forms of energy” model currently get full credit, this will almost certainly change over time (gradually or otherwise).

This post is intended to be a “one stop” resource for busy teachers, with suggestions for further reading.

Please note that I have no expertise or authority on the new model beyond that of a working teacher who has spent a fair amount of time researching, thinking about and discussing the issues. What follows is essentially my own take, “supplemented by the accounts of their friends and the learning of the Wise” (if I may borrow from Frodo Baggins!).

Screen Shot 2018-08-07 at 14.53.19.png

Part the First: “Why? For the love of God, why!?!”

The old forms of energy model was familiar and popular with students and teachers. It is still used by many textbooks and online resources. However, researchers have suggested that there are significant problems with this approach:

  1. Students just learn a set of labels which adds little to their understanding (see Millar 2014 p.6).
  2. The “forms of energy” approach focuses attention in the wrong place: it highlights the label, rather than the physical process. There is no difference between chemical energy and kinetic energy except the label, just as there is no difference between water stored in a cylindrical tank and a rectangular tank. (See Boohan 2014 p.12)

The new IoP Stores and Pathways model attempts to address these issues by limiting discussions of energy to situations where we might want to do calculations.

Essentially, the IoP wanted to simplify “energy-talk” and make it a better approximation of the way that professional scientists (especially physicists) actually use energy-concepts. The trick is to get away from the old and nebulous “naming of parts” approach to a newer, more streamlined version that is fit for purpose.

Part the Second: How many energy stores?

The second rule of IoP Energy Club is: you do not talk about energy . . .
. . . unless you’re gonna do a calculation.

— with apologies to Brad Pitt and Chuck Palahniuk

The IoP suggests eight named energy stores (listed below with the ones likely to be needed early in the teaching sequence listed first).

Screen Shot 2018-08-07 at 12.19.15.png
From: http://supportingphysicsteaching.net/SPTGraphics/EnEightEnergyStoresINCC.svg. Note: typographical errors in original (accessed 7/8/18)

Many will be surprised to see that electrical energy, light energy and sound energy are not on this list: more on that later.

There are, I think, two very important points:

  1. All of these energy stores represent quantities that are routinely measured in joules.
  2. All of the energy stores represent a system where energy can be stored for an appreciable period of time.

For example, a rattling washing machine is not a good example of a vibration energy store as it does not persist over an extended period of time: as soon as the motor stops, the machine stops rattling. On the other hand, a struck tuning fork, a plucked guitar string or a bell hit with a hammer are good examples of vibration energy stores.

Similarly, a hot object is not a vibration energy store: it is better described as a thermal energy store. Thermal energy stores are useful when there is a change in temperature or a change in state.

Likewise, a lit up filament bulb is not a good example of a thermal energy store because it does not persist over an extended period of time; switch off the current, and the bulb filament would rapidly cool.

Note also that the electric-magnetic energy store applies to situations involving magnets and static electric charges. It is not equivalent to the old “electrical energy”.

The thread linking all the above examples is we limit discussions of energy to situations where we could perform calculations.

Thermal energy store is an appropriate concept for (say) the water in a kettle because we can calculate the change in the thermal energy store of the water and the result is useful in a wide range of situations. However the same is not true of a hot bulb filament as the change in the thermal energy store of the filament is not a useful quantity to calculate (at least in most circumstances). For further discussion, see this blog post and also this section of the IoP Supporting Physics website.

Part the third: How many energy pathways?

The third rule of IoP Energy Club is: there ain’t no such thing as ‘light energy’ (or ‘sound energy’ or ‘electrical energy’).

— with apologies to Brad Pitt and Chuck Palahniuk

In the new IoP Energy model, there is no such thing as a “light energy store”. Instead, we talk about energy pathways.

Energy pathways describe dynamic quantities that are routinely measured in watts. That is to say, they are dynamic or temporal in the sense that their measurement depends on time (watts = joules per second); energy stores are static or atemporal over a given period of time.

It is not useful to talk about a “light energy store” because it does not persist over time: the visible light emitted by (say) a street lamp is not static — it is not helpful to think of it as a static “box of joules”. Instead it is a dynamic “flow” of joules which means its most convenient unit of measurement is the watt.

As an analogy, think of an energy store as a container or tank; in contrast, think of a pathway as a channel or tap that allows energy to move from one store to another. )

You can read more on the “tanks and taps” analogy here.

Screen Shot 2018-08-07 at 15.27.23

The cautious reader should note that the IoP describe slightly different pathways which you can read about here. (Mechanical and Electrical Working are in, but the IoP talk about “Heating by particles” and “Heating by radiation”; on this categorisation, sound would fit into the “Mechanical Working” category!)

The fourth rule of IoP Energy Club is: I don’t care what you call it, if it’s measured in watts, it’s a pathway not an energy store, OK?

— with apologies to Brad Pitt and Chuck Palahniuk

You can look forward to more ‘IoP Energy Club Rules’, as and when I make them up.

Important note: all of the above content is the personal opinion of a private individual. It has not been approved or endorsed by the IoP.

References

Boohan, R. (2014). Making Sense of Energy. School Science Review, 96(354), 33-43.

Millar, R. (2014). Teaching about energy: from everyday to scientific understandings. School Science Review, 96(354), 45-50.

The Life and Death of Stars

e=mc2andallthat3 Comments

Stars, so far as we understand them today, are not “alive”.

Now and again we saw a binary and a third star approach one another so closely that one or other of the group reached out a filament of its substance toward its partner. Straining our supernatural vision, we saw these filaments break and condense into planets. And we were awed by the infinitesimal size and the rarity of these seeds of life among the lifeless host of the stars. But the stars themselves gave an irresistible impression of vitality. Strange that the movements of these merely physical things, these mere fire-balls, whirling and traveling according to the geometrical laws of their minutest particles, should seem so vital, so questing.

Olaf Stapledon, Star Maker (1937)

Star Maker Cover

And yet, it still makes sense to speak of a star being “born”, “living” and even “dying”.

We have moved on from Stapledon’s poetic description of the formation of planets from a filament of star-stuff gravitationally teased-out by a near-miss between passing celestial orbs. This was known as the “Tidal Hypothesis” and was first put forward by Sir James Jeans in 1917. It implied that planets circling stars would be an incredibly rare occurrence.

Today, it would seem that the reverse is true: modern astronomy tells us that planets almost inevitably form as a nebula collapses to form a star. It appears that stars with planetary systems are the norm, rather than the exception.

Be that as it may, the purpose of this post is to share a way of teaching the “life cycle” of a star that I have found useful, and that many students seem to appreciate. It uses the old trick of using analogy to “couch abstract concepts in concrete terms” (Steven Pinker’s phrase).

Screen Shot 2018-06-24 at 16.49.15.png

I find it humbling to consider that currently there are no black dwarf stars anywhere in the observable universe, simply because the universe isn’t old enough. The universe is merely 13.7 billion years old. Not until the universe is some 70 000 times its current age (about 1015 years old) will enough time have elapsed for even our oldest white dwarfs to have cooled to become a black dwarf. If we take the entire current age of the universe to be one second past midnight on a single 24-hour day, then the first black dwarfs will come into existence at 8 pm in the evening…

And finally, although to the best of our knowledge, stars are in no meaningful sense “alive”, I cannot help but close with a few words from Stapledon’s riotous and romantic imaginative tour de force that is yet threaded through with the disciplined sinews of Stapledon’s understanding of the science of his day:

Stars are best regarded as living organisms, but organisms which are physiologically and psychologically of a very peculiar kind. The outer and middle layers of a mature star apparently consist of “tissues” woven of currents of incandescent gases. These gaseous tissues live and maintain the stellar consciousness by intercepting part of the immense flood of energy that wells from the congested and furiously active interior of the star. The innermost of the vital layers must be a kind of digestive apparatus which transmutes the crude radiation into forms required for the maintenance of the star’s life. Outside this digestive area lies some sort of coordinating layer, which may be thought of as the star’s brain. The outermost layers, including the corona, respond to the excessively faint stimuli of the star’s cosmical environment, to light from neighbouring stars, to cosmic rays, to the impact of meteors, to tidal stresses caused by the gravitational influence of planets or of other stars. These influences could not, of course, produce any clear impression but for a strange tissue of gaseous sense organs, which discriminate between them in respect of quality and direction, and transmit information to the correlating “brain” layer.

Olaf Stapledon, Star Maker (1937)

Crossing Cognitive Chasms With P-prims

e=mc2andallthat11 Comments

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.

Electrifying Engelmann

e=mc2andallthat6 Comments

It is a long-standing and melancholy truth that, despite the best efforts of many legions of Physics teachers, many students continue to not only dislike electricity, but to hate it with the white-hot intensity of a million suns.

What we have here, I think, is a classic failure to communicate.

A final fact is that samenesses and differences of examples are more obvious when the examples are juxtaposed. This fact implies that the continuous conversion of examples provides the clearest presentation of samenesses and, differences because it creates the changes that occur from one example to the next.

— Siegfried Engelmann and Douglas Carmine, Theory of Instruction (1982) p.46

Looking at my own teaching, I certainly attempt to juxtapose a number of circuits. I really want to highlight the similarities and differences between circuits in order to better develop my students’ understanding. But the problem is that both limited resources and other practical considerations mean that the juxtapositioning cannot happen by continuous conversion, except very rarely.

For example, I would set up (or ask students to set up) a circuit with a single bulb with an ammeter, then I (or we) would disassemble the circuit and rebuild it with the ammeter in a different position, or a second bulb added in series or in parallel . . .

It occurs to me that what we are relying on to thread these juxtapositions together in students’ minds is a sequence of circuit diagrams. I suppose it’s another case of the curse of knowledge writ large: experts and novices think differently.

As a beginning teacher, I remember being genuinely shocked that many students found it easier to interpret a photograph or a 3D drawing rather than the nice, clutter-free, minimalist lines of a circuit diagram.

Without a doubt, many students retain strong visual impressions of many of the circuit diagrams they encounter, but they do not parse and decode the diagrams in the same way as their teachers do.

And that, I think, is the major problem when we are introducing electric circuits.

But what to do?

— R. S. Thomas, The Cure

Can we introduce the important aspects of electrical circuits by continuous conversion of examples?

I think we can. And what is more, I think it will be more effective than the itty-bitty assembly and disassembly of circuits that I have practiced to date.

Conservation of electrical current (and current in parallel circuits) by continuous conversion

Parallel Circuit

This is introduced with a teacher demonstration of the above circuit. Students are invited to note the identical readings on both ammeters and asked to explain why they are identical. They are then asked to predict the effect of adding a second bulb in parallel. The teacher then adds the second bulb by connecting the flying lead. The process is repeated with the third and fourth bulbs, with the teacher testing students’ understanding by asking them to predict the change in current readings as bulbs are added and removed. The teacher also tests students’ understanding of the conservation of current by asking students to predict whether the reading on both ammeters will be the same or different as bulbs are added and removed.

I find it useful to include a bulb that is not identical to the other three. It should be noticeably brighter or dimmer than the other three with the same p.d. so that students do not make the incorrect inference that the current always increases or decreases in equal steps when the circuit is changed.

The teacher could also draw the original circuit on a student whiteboard and ask students to do likewise. The changes that are about to be made could be described and students could be asked could alter the picture/circuit diagram and write their prediction on their whiteboards. They could then compare their version with the teacher’s and their prediction could be quickly tested by making the proposed changes “live” in front of the students.

If resources and time permit, students could then, of course, go on to construct their own parallel circuits as a class practical. However, I think it is important that these vital, foundational ideas are introduced (or re-introduced!) via a teacher demonstration to avoid possible cognitive overload for students.

Series circuits by continuous conversion

Series Circuit

In this demonstration circuit, four of the three bulbs are short-circuited so that they are initially unlit. The teacher asks students to explain only one bulb in the circuit is lit: it is helpful if they have previously encountered parallel circuits and can explain this in terms of electrical current taking the “easier” route (assuming they have not yet encountered the concept of electrical resistance).

Again, the two ammeters allow the teacher to emphasise and test students understanding of the idea that current is conserved.

The teacher then asks students to predict the change in current reading when switch X is opened: will it increase or decrease? Why would it increase or decrease? The process is repeated with switches Y and Z and students’ understanding is tested by asking them to predict the effect on the current reading of opening or closing X, Y or Z.

As before, the teacher would amend her circuit diagram on her student whiteboard and students would do likewise. For example: “I am going to open switch Y. Change the circuit diagram. Show me. What will happen to the reading on the left hand ammeter? What will happen to the reading on the right hand ammeter? Explain why.”

Again, I recommend that at least one out of the four bulbs in not identical to the other three to help prevent students from drawing the incorrect inference that the current will always increase or decrease in identical steps.

When Harold Met William

e=mc2andallthat3 Comments

Legend has it that in 1988, U.S. Presidential candidate Michael Dukakis opened an election rally in front of a huge crowd in a red state with the ringing words: “This joke will appeal to the Latin scholars amongst you…” He went on to lose decisively to George H. W. Bush.

On that note, this joke will appeal to all the Physics teachers (and other aficionados of the dot-and-cross convention).

Harold

For the non-physicists amongst you, this is an illustration of the dot-and-cross convention, which allows us to represent 3D objects on a 2D diagram. The dot represents a vector emerging out of the plane of the paper (think of an arrow coming towards you) and the cross represents a vector directed into the plane of the paper (think of an arrow going away from you).

520px-VFPt_Solenoid_correct2.svg
A solenoid (electromagnet) represented using the dot-and-cross convention. From http://www.wikiwand.com/en/Solenoid

I’ll get my coat…