Units, you nit!

The S.I. System of Units is a thing of beauty: a lean, sinewy and utilitarian beauty that is the work of many committees, true; but in spite of that common saw about ‘a camel being a horse designed by a committee’, the S.I. System is truly a thing of rigorous beauty nonetheless.

Even the pedestrian Wikipedia entry on the 2019 Redefinition of the S.I. System reads like a lost episode from Homer’s Odyssey. As Odysseus tied himself to the mast of his ship to avoid the irresistible lure of the Sirens, so in 2019 the S.I, System tied itself to the values of a select number of universal physical constants to remove the last vestiges of merely human artifacts such as the now obsolete International Prototype Kilogram.

Meet the new (2019) SI, NOT the same as the old SI

However, the austere beauty of the S.I. System is not always recognised by our students at GCSE or A-level. ‘Units, you nit!!!’ is a comment that physics teachers have scrawled on student work from time immemorial with varying degrees of disbelief, rage or despair at errors of omission (e.g. not including the unit with a final answer); errors of imprecision (e.g. writing ‘j’ instead of ‘J’ for ‘joule — unforgivable!); or errors of commission (e.g. changing kilograms into grams when the kilogram is the base unit, not the gram — barbarous!).

The saddest occasion for writing ‘Units, you nit!’ at least in my opinion, is when a student has incorrectly converted a prefix: for example, changing millijoules into joules by multiplying by one thousand rather than dividing by one thousand so that a student writes that 5.6 mJ = 5600 J.

This odd little issue can affect students from across the attainment range, so I have developed a procedure to deal with it which is loosely based on the Singapore Bar Model.

A procedure for illustrating S.I. unit conversions

One millijoule is a teeny tiny amount of energy, so when we convert it joules it is only a small portion of one whole joule. So to convert mJ to J we divide by 1000.

One joule is a much larger quantity of energy than one millijoule, so when we convert joules to millijoules we multiply by one thousand because we need one thousand millijoules for each single joule.

In time, and if needed, you can move to a simplified version to remind students.

A simplified procedure for converting units

Strangely, one of the unit conversions that some students find most difficult in the context of calculations is time: for example, hours into seconds. A diagram similar to the one below can help students over this ‘hump’.

Helping students with time conversions

These diagrams may seem trivial, but we must beware of ‘the Curse of Knowledge’: just because we find these conversions easy (and, to be fair, so do many students) that does not mean that all students find them so.

The conversions that students may be asked to do from memory are listed below (in the context of amperes).

A table showing all the SI prefixes that GCSE students need to know

The Burnéd Hand Teaches Best

The burned hand teaches best. After that, advice about fire goes to the heart.

J. R. R. Tolkein, The Two Towers (1954)

As is often the case in an educational context, and with all due respect to Tolkein, I think Siegfried Engelman actually said it best.

The physical environment provides continuous and usually unambiguous feedback to the learner who is trying to learn physical operations . . .

Siegfried Engelmann and Douglas Carnine, Theory of Instruction (1982)

I am going to outline a practical approach that will help students understand that black objects are good emitters and good absorbers of infrared radiation.

What I propose is a simple, inexpensive and low risk procedure (similar to this one from the IoP) that won’t actually inflict any actual “burned hands” but will, hopefully, through a clever (imho) manipulation of the physical environment, speak directly to the heart — or at least to students’ “sense of mechanism” about how the world works.

Half human and half infrared detector

Obtain tubes of matt black and white facepaint. (These are typically £5 or less.) Choose a brand that is water based for easy removal and is compliant with EU and UK regulations.

We also need a good source of infrared radiation. Some suppliers such as Nicholl and Timstar can supply a radiant heat source that is safe to use in schools. Although these can be expensive to purchase, there may already be one hiding in a cupboard in your school. If you don’t have one, use a 60W filament light bulb mounted in desk lamp (do not use a fluorescent or LED lamp — they don’t produce enough IR!). Failing that, you could use a raybox with a 24W, 12V filament lamp to act as the infrared source. [UPDATE: Paul Bushen also recommends a more economical option — an infrared heat lamp.)

Use the facepaint to make 2 cm by 2 cm squares on the back of one hand in black and in white on the other. Hold each square up to the infrared source so they are a similar distance from it.

Hold the hands still in front of the source for a set time. This could be anywhere between five seconds and a few tens of seconds, depending on the intensity of the source. You should run through this experiment ahead of time to make sure that there is minimal risk of any serious burns for the time you intend to allocate. If you are using rayboxes then you might need a separate one for each hand.

Schematic representation showing two hands with white and black paint on the back being held up to an infrared source.
The human infrared detector

The hand with the black paint becomes noticeably warmer when exposed to infrared radiation. We can deduce that this is because the colour black is better and absorbing the infrared than the white colour.

Energy is being transferred via light into the thermal energy store of the hand.

Schematic representation of energy being transferred into the thermal energy store of the hand via light.

We can use a black painted hand as a rudimentary detector for infrared. The hotter it gets, the more infrared is being emitted.

Enter Leslie’s cube . . .

Direct perception of the infrared output from a Leslie’s cube

Fill a Leslie’s cube with hot water from a kettle and then get students to place the hand with the black square a couple of centimetres away from the black face of the cube. After a few seconds, ask them to place the same hand by the white face of the cube. (Although, for the best contrast, you should maybe try the polished silver side). Make sure the student’s hand does not actually touch the face of the Leslie’s cube, otherwise they may end up with an actual burned hand!

The fact that the black face emits more infrared radiation is immediately directly perceivable by the “infrared detector” hand which feels distinctly warmer than when it’s placed next to the black coloured face rather than the white face.

This procedure is, I think, more convincing to many students as opposed to merely using (say) a digital infrared detector and reading off a larger number from the dark side compared to the white side.

Understanded of the pupils

It is a thing plainly repugnant . . . to Minister the Sacraments in a Tongue not understanded of the People.

Gilbert, Bishop of Sarum. An exposition of the Thirty-nine articles of the Church of England (1700)

How can we help our students understand physics better? Or, in more poetic language, how can we make physics a thing that is more ‘understanded of the pupils’?

Redish and Kuo (2015: 573) suggest that the Resources Framework being developed by a number of physics education researchers can be immensely helpful.

In summary, the Resources Framework models a student’s reasoning as based on the activation of a subset of cognitive resources. These ‘thinking resources’ can be classified broadly as:

  • Embodied cognition: these are simple, irreducible cognitive resources sometimes referred to as ‘phenomenological primitives’ or p-prims such as ‘if-resistance-increases-then-the-output-decreases‘ and ‘two-opposing-effects-can-result-in-a-state-of-dynamic-balance‘. They are typically straightforward and ‘obvious’ generalisations of our lived, everyday experience as we move through the physical world. Embodied cognition is perhaps summarised as our ‘sense of mechanism’.
  • Encyclopedic (ancillary) knowledge: this is a complex cognitive resource made of a large number of highly interconnected elements: for example, the concept of ‘banana’ is linked dynamically with the concept of ‘fruit’, ‘yellow’, ‘curved’ and ‘banana-flavoured’ (Redish and Gupta 2009: 7). Encyclopedic knowledge can be thought of as the product of both informal and formal learning.
  • Contextualisation: meaning is constructed dynamically from contextual and other clues. For example, the phrase ‘the child is safe‘ cues the meaning of ‘safe‘ = ‘free from the risk of harm‘ whereas ‘the park is safe‘ cues an alternative meaning of ‘safe‘ = ‘unlikely to cause harm‘. However, a contextual clue such as the knowledge that a developer had wanted to but failed to purchase the park would make the statement ‘the park is safe‘ activate the ‘free from harm‘ meaning for ‘safe‘. Contextualisation is the process by which cognitive resources are selected and activated to engage with the issue.

Using the Resources Framework for teaching

I have previously used aspects of the Resources Framework in my teaching and have found it thought provoking and helpful to my practice. However, the ideas are novel and complex — at least to me — so I have been trying to think of a way of conveniently organising them.

What follows in my ‘first draft’ . . . comments and suggestions are welcome!

The RGB Model of the Resources Framework

The RGB Model of the Resources Framework

The red circle (the longest wavelength of visible light) represents Embodied Cognition: the foundation of all understanding. As Kuo and Redish (2015: 569) put it:

The idea is that (a) our close sensorimotor interactions with the external world strongly influence the structure and development of higher cognitive facilities, and (b) the cognitive routines involved in performing basic physical actions are involved in even in higher-order abstract reasoning.

The green circle (shorter wavelength than red, of course) represents the finer-grained and highly-interconnected Encyclopedic Knowledge cognitive structures.

At any given moment, only part of the [Encyclopedic Knowledge] network is active, depending on the present context and the history of that particular network

Redish and Kuo (2015: 571)

The blue circle (shortest wavelength) represents the subset of cognitive resources that are (or should be) activated for productive understanding of the context under consideration.

A human mind contains a vast amount of knowledge about many things but has limited ability to access that knowledge at any given time. As cognitive semanticists point out, context matters significantly in how stimuli are interpreted and this is as true in a physics class as in everyday life.

Redish and Kuo (2015: 577)

Suboptimal Understanding Zone 1

A common preconception held by students is that the summer months are warmer because the Earth is closer to the Sun during this time of year.

The combination of cognitive resources that lead students to this conclusion could be summarised as follows:

  • Encyclopedic knowledge: the Earth’s orbit is elliptical
  • Embodied cognition: The closer to a heat source you are the warmer it is.

Both of these cognitive resources, considered individually, are true. It is their inappropriate selection and combination that leads to the incorrect or ‘Suboptimal Understanding Zone 1’.

To address this, the RF(RGB) suggests a two pronged approach to refine the contextualisation process.

Firstly, we should address the incorrect selection of encyclopedic knowledge. The Earth’s orbit is elliptical but the changing Earth-Sun distance cannot explain the seasons because (1) the point of closest approach is around Jan 4th (perihelion) which is winter in the northern hemisphere; (2) seasons in the northern and southern hemispheres do not match; and (3) the Earth orbit is very nearly circular with an eccentricity e of 0.0167 where a perfect circle has e = 0.

Secondly, the closer-is-warmer p-prim is not the best embodied cognition resource to activate. Rather, we should seek to activate the spread-out-is-less-intense ‘sense of mechanism’ as far as we are able to (for example by using this suggestion from the IoP).

Suboptimal Understanding Zone 2

Another common preconception held by students is all waves have similar properties to the ‘breaking’ waves on a beach and this means that the water moves with the wave.

The structure of this preconception could be broken down into:

  • Embodied cognition: if I stand close to the water on a beach, then the waves move forward to wash over my feet.
  • Encyclopaedic knowledge: the waves observed on a beach are water waves

Considered in isolation, both of these cognitive resources are unproblematic: they accurately describes our everyday, lived experience. It is the contextualisation process that leads us to apply the resources inappropriately and places us squarely in Suboptimal Understanding Zone 2.

The RF(RGB) Model suggests that we can address this issue in two ways.

Firstly, we could seek to activate a more useful embodied cognition resource by re-contextualising. For example, we could ask students to imagine themselves floating in deep water far from the shore: do the waves carry them in any particular direction or simply move them up or down as they pass by?

Secondly, we could seek to augment their encyclopaedic knowledge: yes, the waves on a beach are water waves but they are not typical water waves. The slope of the beach slows down the bottom part of the wave so the top part moves faster and ‘topples over’ — in other words, the water waves ‘break’ leading to what appears to be a rhythmic back-and-forth flow of the waves rather than a wave train of crests and troughs arriving a constant wave speed. (This analysis is over a short period of time where the effect of any tidal effects is negligible.)

Both processes try to ‘tug’ student understanding into the central, optimal zone.

Suboptimal Understanding Zone 3

Redish and Kuo (2015: 585) recount trying to help a student understand the varying brightness of bulbs in the circuit shown.

4 bulbs in a circuit: Bulbs A, B and D are in series with the cell but bulb C is in parallel across bulb B.
All bulbs are identical. Bulbs A and D are bright; bulbs B and C are dim.

The student said that they had spent nearly an hour trying to set up and solve the Kirchoff’s Law loop equations to address this problem but had been unsuccessful in accounting for the varying brightnesses.

Redish suggested to the student that they try an analysis ‘without the equations’ and just look at the problems in simpler physical terms using just the concept of electric current. Since current is conserved it must split up to pass through bulbs B and C. Since the brightness is dependent on the current, the smaller currents in B and C compared with A and D accounts for their reduced brightness.

When he was introduced to [this] approach to using the basic principles, he lit up and was able to solve the problem quickly and easily, saying, ‘‘Why weren’t we shown this way to do it?’’ He would still need to bring his conceptual understanding into line with the mathematical reasoning needed to set up more complex problems, but the conceptual base made sense to him as a starting point in a way that the algorithmic math did not.

Analysing this issue using the RF(RGB) it is plausible to suppose that the student was trapped in Suboptimal Understanding Zone 3. They had correctly selected the Kirchoff’s Law resources from their encyclopedic knowledge base, but lacked a ‘sense of mechanism’ to correctly apply them.

What Redish did was suggest using an embodied cognition resource (the idea of a ‘material flow’) to analyse the problem more productively. As Redish notes, this wouldn’t necessarily be helpful for more advanced and complex problems, but is probably pedagogically indispensable for developing a secure understanding of Kirchoff’s Laws in the first place.

Conclusion

The RGB Model is not a necessary part of the Resources Framework and is simply my own contrivance for applying the RF in the context of physics education at the high school level. However, I do think the RF(RGB) has the potential to be useful for both physics and science teachers.

Hopefully, it will help us to make all of our subject content more ‘understanded of the pupils’.


References

Redish, E. F., & Gupta, A. (2009). Making meaning with math in physics: A semantic analysis. GIREP-EPEC & PHEC 2009, 244.

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

Series and Parallel Circuits — an unhelpful dichotomy?

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

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

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

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

The actual situation is more like this:

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

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

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

How to avoid the false dichotomy

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Conclusion

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

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

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

2021 Retrospective

A huge thank you to everyone who has viewed, read or commented on one of my posts in 2021: whether you agreed or disagreed with my point of view, you are the people that make the work of writing this blog so enjoyable and rewarding.

The top 3 most-read posts of 2021 were:

1. What to do if your school has a batshit crazy marking policy

Extract from blog post "What to do if your school has a batshit crazy marking policy"

This particular post was written back in 2019 and it’s sobering to realise that it is still relevant enough that it was featured by @TeacherTapp in November 2021. As edu-blog writers will know, this unlooked for honour generates thousands of views — thanks, @TeacherTapp!

Note to schools: please, if you haven’t done so already, please please please sort out your marking policy and make sure it is workable and fit for purpose. It would seem that, even now, teachers are being made to mark for the sake of marking, rather than for any tangible educational benefit.

2. FIFA for the GCSE Physics Calculation Win!

Extract from "FIFA for the GCSE Physics Calculation Win" blog.

The next post is one I am very proud of, even though FIFA is just a silly mnemonic to help students follow the “substitute-first-and-then-rearrange” method favoured by AQA mark schemes. Yes, FIFA did start life as a “mark-grubbing” dodge; however, somewhat to my own surprise, I found that the vast majority of students (LPAs included), can rearrange successfully if they substitute the numbers in first. Many other teachers have found the same thing as well — search #FIFAcalc on Twitter for some illustrative tweets from FIFAcalc’s biggest fans.

However, it is clear that the formula triangle method still has many adherents. I think this is unfortunate because: (a) they only work for a limited subset of formulas with the format y=mx; (b) they are a cognitive dead end that actively block students from accessing higher level STEM courses; and (c) as Ed Southall argues effectively, they are a form of procedural teaching rather than conceptual teaching.

3. Why does kinetic energy = 1/2mv^2?

Extract from blog explaining how to derive the kinetic energy equation

This post is a surprise “sleeper” hit also dating from 2019. It outlines an accessible method for deriving the kinetic energy formula. From getting a respectable but niche 200 views per year in 2019 and 2020, in 2021 it shot up to over 3K views. What is very encouraging for me is that most of these views come from internet searches by individuals from a wide range of backgrounds and not just my fellow denizens of the online edu-Bubble!

Here’s to 2022, folks!

Waves: Introduce With Trolleys, Not Slinkys

A number of interesting questions came up in the recent author Q&A run by @Chatphysics about my contribution to Cracking Key Concepts in Secondary Science: many thanks to Jinny Bell (@MissB0107) for hosting!

In this post, the question I want to address in more detail is: why do I recommend introducing waves using dynamics trolleys rather than a slinky spring or elastic rope?

I think I have a couple of persuasive reasons; but first a digression.

A thing doing a thing…

Robert Graves wrote his charming comedic poem Welsh Incident in 1929. It describes (at least as I read it) the visitation of disparate group of space aliens to a small town on the North Wales coast in the 1920s.

I will paraphrase one line where the Narrator says:

Then one of things did a thing that was finally recognisable as a thing.

The first example of a thing doing a thing…

Now don’t get me wrong: slinky springs and elastic ropes are brilliant examples of mechanical waves and I do use them (a lot!) — I just don’t use them as the first example.

The problem as I see it is that they could lead students to file wave mechanics in an entirely separate and independent category from mechanics.

I want students to see that wave behaviour isn’t distinct from forces and particles but rather is a direct (and fairly straightforward) consequence of a particular arrangement of particles with a specific pattern of forces between them.

Since the first example, is often the ‘loudest’ (metaphorically speaking), it’s not a bad idea to start with longitudinal waves.

I use standard wooden dynamics trolleys. Dowel rods or metal posts can be used to link the trolleys together. The system is more stable if a pair of springs is used at the front and back of each trolley. The springs used are the ones we typically use for the Hooke’s Law experiment.

A compression carrying energy along a line of trolleys linked by springs can be easily modelled:

Modelling energy transfer by compression of a longitudinal wave using a line of dynamics trolleys and springs

So can a rarefaction:

Using a line of dynamics trolley linked together with spring to demonstrate the transmission of a rarefaction pulse

Transverse waves can be modelled like this:

A transverse wave modelled using a line of dynamics trolleys linked with springs

Amongst the advantages of this approach are:

  • Students are introduced to an unknown thing (wave behaviour) by means of more familiar things (trolleys and springs)
  • The idea that there is no net movement of the ‘particles’ as energy is transferred is much more directly observable using this arrangement rather than the slinky or elastic rope.
  • The frequency of a wave (which in some ways is a more fundamental measurement than wavelength) can be associated with the repeating motion of a single ‘particle’ and extended outwards to the whole system, rather than vice versa.

You can read more in Chapter 25 of Cracking Key Concepts in Secondary Science.

Conclusion

I hope readers will try this demonstation: hopefully introducing students to a thing which is already recognisable as a thing will make wave behaviour more comprehensible and less like an unwelcome diversion into terra incognita.

Readers who are ‘rich in years’ like myself will recognise this demonstration as being adapted from the old Nuffield linear A-level Physics course.

You can listen to Richard Burton’s great reading of Robert Graves’ Welsh Incident here.

Dual Coding and Equations of Motion for GCSE

Necessity may well be the mother of invention, but teacher desperation is often the mother of new pedagogy.

For an unconscionably long time, I think that I failed to adequately help students understand the so called ‘equations of motion’ (the mathematical descriptions of uniformly accelerated motion using the standard v, u, a, s and t notation) because I suffered from the ‘curse of knowledge’: I did not find the topic hard, so I naturally assumed that students wouldn’t either. This, sadly, proved not to be the case, even when the equation was printed on the equation sheet!

What follows is a summary of a dual coding technique that I have found really helpful in helping students become confident with problems involving the ‘equations of motion’. This is especially true at GCSE, where students encounter formulas such as

Equation of motion (v^2 - u^2 = 2as)

for the first time.

A dual coding convention for representing motion

Table to show dual coding convention

Applying dual coding to an equation of motion problem

EXAMPLE: A car is travelling at 6.0 m/s. As the car passes a lamp-post it accelerates up to a velocity of 14.2 m/s over a distance of 250 m. Calculate a) the acceleration; and b) the time taken for this change.

The problem can be represented using the dual coding convention as shown below.

Diagram showing car accelerating past a lamp-post using the dual coding convention outlined in the blog

Note that the arrow for v is longer than the arrow for u since the car has a positive acceleration; that is to say, the car in this example is speeding up. Also, the convention has a different style of arrow for acceleration, emphasising that it is an entirely different type of quantity from velocity.

We can now answer part (a) using the FIFA calculation system.

Working showing solution for part (a)

Part (b) can be answered as shown below.

Working for part b of the problem

Visualising Stopping Distance Questions

These can be challenging for many students, as we often seem to grabbing numbers and manipulating them without rhyme or reason. Dual coding helps make our thinking explicit,

EXAMPLE: A driver has a reaction time of 0.7 seconds. The maximum deceleration of the car is 2.6 metres per second squared. Calculate the overall stopping distance when the car is travelling at 13 m/s.

We need to calculate both the thinking distance and the braking distance to solve this problem

The acceleration of the car is zero during the driver’s reaction time, since the brakes have not been applied yet.

We visualise the first part of the problem like this.

Using s=vt we find that the thinking distance is 9.1 m.

Now let’s look at the second part of the question.

There are three things to note:

  • Since the car comes to a complete stop, the final velocity v is zero.
  • The acceleration is negative (shown as a backward pointing arrow) since we are talking about a deceleration: in other words, the velocity gradually decreases in size from the initial velocity value of u as the car traverses the distance s.
  • Since the second part of the question does not involve any consideration of time we have omitted the ‘clock’ symbols for the second part of the journey.

We can now apply FIFA:

Working for braking distance part of problem

Since the acceleration arrow points in the opposite direction to the positive arrow, we enter it as a negative value. When we get to the third line of the Fine Tune stage, we see that a negative 169 divided by negative 4.4 gives positive 38.408 — in other words, the dual coding convention does the hard work of assigning positive and negative values!

And finally, we can see that the overall stopping distance is 9.1 + 38.4 = 47.5 metres,

Conclusion

I have found this form of dual coding extremely useful in helping students understand the easily-missed subtleties of motion problems, and hope other science teachers will give it a try.


If you have enjoyed this post, you might enjoy these also:

Circuit Diagrams: Lost in Rotation…?

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

I think that, possibly, there may be.

(Note: circuit diagrams produced using the excellent — and free! — web editor at https://www.circuit-diagram.org/.)

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

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

A simple circuit as usually presented

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

Measuring the potential difference across the cell

…and across the resistor.

Measuring the potential difference across the resistor

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

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

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

Try another way…

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

The same circuit drawn ‘all-in-a-row’

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

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

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

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

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

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

Crossing the leads on a voltmeter

A Hidden Rotation?

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

A hidden rotation…?

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

Conclusion

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

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

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

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

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

Signposting Fleming’s Left Hand and Right Hand Rules

Students undoubtedly find electromagnetism tricky, especially at GCSE.

I have found it helpful to start with the F = BIl formula.

Introducing the “magnetic Bill” formula

This means that we can use the streamlined F-B-I mnemonic — developed by no less a personage than Robert J. Van De Graaff (1901-1967) of Van De Graaff generator fame — instead of the cumbersome “First finger = Field, seCond finger = Current, thuMb = Motion” convention.

Students find the beginning steps of applying Fleming’s Left Hand Rule (FLHR) quite hard to apply, so I print out little 3D “signposts” to help them. You can download file by clicking the link below.

You can see how they are used in the photo below.

Photograph to show how to use the FLHR signpost

Important tip: it can be really helpful if students label the arrows on both sides, such as in the example shown below. (The required precision in double sided printing defeated me!)

Another example of the FLHR signpost in use

Signposts for Fleming’s Right Hand Rule are included on the template.


Other posts I have written on magnetism include:

Feel free to dip into these 🙂

Misconceptions and p-prims at ResearchED 2021

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

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

The PowerPoint can be downloaded below.

You can watch a short summary of the talk here.

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

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


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

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

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

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

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