Images, Resonances, Echoes, take 6

Well, here in the UK the gloom continues, alleviated by birdsong and unseasonably predictable blue skies. I hope you're coping well, however well your country is dealing with the challenges of the day, and that maybe there will be some consolations in losing yourself in energy for a while. Because yes, the energy blockbuster has arrived, and I apologise for the unusual length. You might prefer to read on the web, so making the links more comfortable to follow. A click on the title links you through to a web version.

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Images of plausibility and intelligibility - and energy?

The image of energy is just a little tarnished. We've drifted away from the style and interests the pragmatic thermodynamicists who created the idea of energy (too much "this is definitely right"; "how could you suggest that"). I think we need to recapture some of their intent by considering first to what uses we might put a completed description in terms of energy. In working towards that, there is a kind of choice tree:

Choosing a kind of description

First off, that energy is a calculated quantity remains a touchstone. Tripwires laid around to deflect from a journey heading towards that idea should be avoided. But it's not the only calculated quantity, so what's so special about the idea?

Mechanism and narrative stories are comforting, and a standard explanatory structure. However, energy is a constraint on what's possible, so stands aside from this narrative structure. However, the lived-in world is, the world of scientific reasoning that we're trying to account for is in this part, not made from or built of this kind of explanatory structure. (Energy descriptions are not of the form: 'First this, then that', as a causal consequence). The teaching of energy should respect that reality: energy is not the 'go of things'. Energy is instead the ultimate 'limiting factor', determining what is and is not possible, so we should show it as such.

"How?" unpacks to three questions: "How much?"; "At what rate?"; "By what mechanism?".

The first question is about energy, and so about augmenting and depleting a stored quantity (find energy in stores). The second is about power, therefore a quantity that's defined when a process is happening(find power in pathways). Both these questions require calculated quantities, so depend on a clear description of a physical change or process.

There are three levels of description.

1. Describe a physical process

2. Quantify, then perhaps develop an explanatory mechanism

3. Generate an energy account

Three levels of description

The third question is not an energy description - it belongs to a different semiotic register - although it may be useful in identifying calculations that might be insightful, and so denote fruitful qualitative descriptions in identifying stores and pathways. That's the last you'll see of mechanisms here.

(Because power is about what's happening, it's easier to connect to the lived-in world, and perhaps move from that into discussions of mechanisms. Power and energy might be co-developed, as opportunities for their natural use occur, rather than developing at first one, and then the other.)

The critical test for any proposed description is:

'How are children expected to represent the core ideas to themselves and how can they reason with these representations to form predictive mental models, as simulacra of scientific descriptions?'

That's been a good deal of essential theoretical throat-clearing. Now for some suggestions.

Firstly, introduce the idea that energy is a tool to think with…suitable for some tasks but not for others (see the choice tree above). As a tool to think with, we have to exemplify the thinking to explain what the tool can and cannot do. So any attempt to teach 'the concept of energy', then to put the idea to work, results in understandable difficulties. (Always intimidate with Feynman: his introductory explanation is about a tool for thinking put to work, in a metaphorical story).

So, teach energy by choosing examples which illuminate by developing plausible, intelligible and fruitful insightful descriptions. Then children can learn to reason with the idea of energy in ways that do not distort its usage in the scientific community, by engaging with meaningful and purposeful descriptions. So what's a purposeful description? Back to the thermodynamicists for insight: energy calculations, first of all, tell us what can or cannot happen – which changes are or are not possible. (In echoes below you'll see how to reinterpret this core to develop other insights). To create an energy description is to seek and insight – it's an inherently purposeful activity.

So here are four purposeful steps in teaching, matched to the three levels:

1. Describe the physical process in everyday terms

2. Notice and record physical quantities

3. Generate an energy description

4. Relate the insights gained back to the first two levels

Four steps

Principles for developing an energy description:

1. Any such description depends on a clear physical description, using discipline-specific quantities, of a phenomenon in the lived-in world. Such physical descriptions may include mechanisms but should not invoke "energy".

"Energy: not a mechanism; not tangible."

2. Any such description must be purposeful: the description should be capable of yielding insight.

"Only describe with energy where this yields insight."

3. Energy is conserved and dissipated, and changes calculated in various ways but is just energy: qualifying adjectives are unhelpful.

"Energy is just energy."

4. A description constrains possible processes but does not predict whether a process happens, or explain how a process happens.

"Energy insights reveal impossibilities: what could happen; not what does happen."

5. An energy description represents a radical simplification, well-matched for describing some, but not all processes.

"Where energy helps, the description is simple."

Avoiding this advice might lead to several missteps.

Tripwires to avoid

More reading

Coopersmith, J. (2015) Energy, the Subtle Concept, OUP

Elkana, Y. (1974) The discovery of the conservation of energy, Hutchinson Educational

Feynman, R., Leighton, R.B., Sands, M. (1964) The Feynman Lectures on Physics. Addison–Wesley. (Section 4.1)

Resonances - preparing the ground for energy

Well, we now have an image of a description; how can we prepare the ground?

1. Foregrounding constraint relationships – those which show possibilities and impossibilities – for changes where the quantities are less abstract(so level two).

2. Developing the idea of a trade-off, thus establishing that there may be choices in getting a job done, but there are hard limits to those choices.

3. Developing a semi-quantitative description for the calculated stored quantity that is energy (and maybe also a linked version for power).

This last one is reasonably well-rehearsed and tested, mainly through the influence of the supporting physics teaching initiative.

Eight stores

(There is a bonus section at the end, explaining the thinking behind the eight stores.)

Four pathways

Energy is not the 'go of things': descriptions which exploit the idea of energy do not provide causal mechanisms. Instead, descriptions about energy are constraints on what may or may not happen.

Here the trouble with cause and effect is that it is a story-telling mechanism, and the telling of a story depends on taking a personal point of view: it depends on a narrative structure. Such structures are great for explaining 'how something goes' to ourselves but have their limitations, and different story-tellers may choose a different cause. Philosophers have explored such problems down the ages, with Hume being a very significant inflexion point. You could borrow a metaphor from the internet, – often accounts in physics are not best structured as a set of 'if this then that' because this relies on many un-explicated necessary conditions. As anyone who codes will realise, there are auxiliary assumptions behind the correct functioning of a piece of code.

To put it another way, any assertion of cause relies on a host of ceteris paribus conditions, which are often unexamined. Physics has developed to be more concerned with the "how" and not with the "why", so with stating correlations. The relationship a=F/m is of this kind: there is no arrow of time, no before and after, hidden in the assertion of equality. It is not, in short, a causal relationship, at least in the way that story-tellers, and perhaps, therefore, any natural language-users want to use cause and effect. Instead, it is a constraint relationship, describing a possibility space. It and many other relationships in physics have this in common with energy descriptions.

Constraining: no arrow of time; no cause

So to try to decide whether acceleration causes force or force causes acceleration is to conflate two different story-telling traditions( and why does no one speak up for mass?).

Acceleration, force and mass constrained

Similarly, for I=V/R. Maybe it's worth thinking about why no one even thinks about launching a discussion about cause and effect for the relationship density = mass/volume.

Most relationships in elementary physics are like this, comparing the values of physical quantities before a change with those afterwards and in other words, constraining the possible values between a pair of snapshots.

t,x,v constrained

(These are different from accumulation relationships: the argument for preferring to focus on accumulation relationships in introductory kinematics appears in IRE05.)

These constraint relationships describe possibilities and so also impossibilities. Given one set of physical values, many other combinations of values are ruled out. It is a way of capturing and expressing the regularities in the world. Counterfactual arguments provide ways of imagining a world in which such regularities either did not exist or had yet to be discovered.

showing some of what's known

knowing nothing

These kinds of accounts in the sciences are expressed with constraint relationships. They are not linear path narratives, a series of 'and next' statements: like a novelist's arrangement of events. These are a special kind of story, one that uses a particular type of thinking, not closely related to the causal. This thinking is wholistic and systemic, rather than atomistic and mechanical. The constraint shows possibilities and impossibilities and does not effect a particular outcome.

Energy is perhaps the apogee of this kind of thinking. Notice it first in less abstract contexts (level two descriptions) and practise there, preparing for working at level three.

A comfortable and accessible place to start introducing the idea of limits to possibilities, without worrying about the detail of mechanisms, is in getting simple lifting jobs done. Here choices are made, but there are hard limits to the options, and necessary trade-offs. A trade-off is a useful handle for a linked pair of quantities, say force and distance. Piaget got there first with compensated quantities, so I'm suggesting a move away from labelling them compensated quantities, which I have indicated in the past. (Not only is it probably unwise to step on his turf, but there are Piagetian studies of compensation for flasks of different cross-sectional areas, and of the relationship between mass, volume and density, which are not the focus here). I suggest that we can fruitfully use 'trade-off' when choosing values of a pair of physical quantities whose product is either power or energy, to achieve some required value in joule or watt.

You might start with pulleys, use to support a load that is too heavy for you alone. You hold only one rope of the number that leave the load, so only have to support a fraction of the load, the remaining ropes account for the remainder. However, the trade-off is now that you have to pull through lots of rope.

Load-sharing with pulleys

You could extend this sharing idea to a hydraulic machine, where increasing the area under the load again allows you to choose to support less, but still find that you have chosen to push further.

A hydraulic load-sharing helper

Levers provide another way of exercising your choices in a trade-off.

The lever as a load-sharing helper

You could represent the idea of a trade-off as the area of a box, one side represented by each of the quantities involved in the trade-off:

Trade-offs for hydraulics

Trade-offs for levers

The orange bar, as ever, is the quantity of energy needed to perform the task. Not enough energy: the task is impossible. Choose how to

Force, distance, trade-off

This pattern of trade-offs applies to filling any store or calculating the power in any pathway, for example.

Trade-offs in filling the gravity store

Trade-offs in filling the kinetic store

two circuits, power, trade-offs

Three preparatory steps for a gentle path: a semi-quantitative representations that can be reasoned with; thinking with constraint relationships; choosing trade-offs.

More reading

Viennot, L.(2007): Reasoning in Physics: The Part of Common Sense. Springer.

Echoes - how this might play out in classrooms

The four steps

Here four steps in developing an energy description map to three levels of description.

The structure of the energy descriptions, which is part of a passage of teaching consists of three levels:

Level one is located in the lived-in world and consists of a description of changes in everyday objects using common terms.

Level two has two complementary halves. The first half is an operationalisation of measures by a process of noticing and recording elements of the lived-in world, bring them into focus so that they can be a part of a physical description. The second half is complementary to this and provides imagined mechanisms which can be a mental model to enable you to get a grip on what is happening during a process( and so provide an explanatory mechanism).

Level three is where the idea of energy gets a grip: it depends on the operationalised measures established in the second level. Still, it does not depend on (a grasp of) – and, complements – the physical mechanisms described in the second level.

Any proposed passage of teaching, which invokes the idea of energy should develop this idea through the levels. Developing an energy description involves working through the levels with your class: don't jump straight to an account in terms of energy. This structure should reduce difficulties by providing a framework in which talk about energy is much more likely to consist of sensible talk, and not a word game. Tyr using non-verbal resources to complement the words, as words by themselves are somewhat slippery, and not well suited to introducing children to precise yet abstract ideas.

A pair of examples:

freewheeling downhill

a solar light

The development of the energy description here sets four common patterns of reasoning, often required in assessment items, in a common framework.

to calculate efficiencies

to proscribe possibilities

to calculate a physical quantity

to explore trade-offs

The idea of energy is a tool to think with–suitable for some tasks but not for others.

You could knock a nail in with an adjustable spanner, or maybe even tighten a screw. But for some tasks, the spanner is particularly well-suited – by design. The idea of energy has evolved in precisely this way–whereas it is possible to try and describe every process with energy, it is not fruitful to do so.

Teach energy by choosing examples which illuminate by developing plausible, intelligible and fruitful insightful descriptions. Then children can learn to reason with the idea of energy in ways that do not distort its usage in the scientific community, by engaging with meaningful and purposeful descriptions.

More reading

Supporting Physics Teaching, now partially available on IoP Spark, had two topics on energy, for teaching 11-14 and 14-16 year old children, including more teacher background, and more examples (But please note that the thinking here goes beyond what was there).

Bonus Reading, in-line, right here:

On choosing a selection of stores

Here is the physics: "Energy is energy and a calculated quantity which is conserved, but which can be dissipated".

On the way to being able to do such calculations, and appreciate the implications of conservation and dissipation, it is possible to develop qualitative descriptions to assist the transition from the physical descriptions to the energy calculations. These descriptions help focus on noticing changes in certain facets of the physical world and using them as clues to suspect that there might be calculable changes which might yield insightful energy descriptions.

Developing a description in terms of stores is a suggestion for a milestone on an imagined learning journey[1] culminating in such calculations. That some might not complete the journey, and that many of these will be functioning voting citizens adds implications about the consequences of wise choices.

So the slogan "find energy in stores" is only a start, albeit a useful one because it lays a foundational principle for the didactical transposition of the idea of energy. It would help if you then considered the uses to which your representations of energy will be put and design them accordingly. In this, discussion about depleting energy resources to power our warm houses, cook our food and satiate our predilection for rushing about is a non-trivial consideration. That provides a framework for making choices. Ally this to a concern to develop a parsimonious and elegant description, that segue smoothly into later formal as an additional consideration to this essential citizenship role. In addition to this pair, other significant constraints include:

an unwillingness to incorporate too many spelling challenges

-a similar reluctance to rely on special words ('system' is a prime candidate here);

laying too many tripwires for other topic areas (as energy is a unifying idea, poor choices here will have effects elsewhere);

how current practitioners will interpret suggestions.

You'll have to make choices, so base decisions on explicit reasons.

SPT settled on eight stores and four pathways. Here I'll discuss some reasoning behind the selection and naming of the eight stores. Neither the names or the graphical representations are 'to-be-learnt'. Instead, they are equal partners in enabling the constructing insightful energy descriptions: both words and diagrams stand for the idea that there is a recognisable physical change which could result in a calculation of a change in energy.

While writing SPT the terms "sound energy", "light energy" and "electrical energy" were in everyday use, and often led to descriptions which were not a didactical transposition of how the idea of energy is used in the sciences. Similarly, chains of energy 'forms' were commonplace, again not representing practice in the sciences at all well. And there were a lot of word games in play, not least in mark schemes for public exams.

So this suggestion focusses on eight stores, each of which is mapped onto one or more calculations.

The physical changes you look for and the calculations you can do are here.[link]

The elastic store functions as a prototype, because it is physically manipulable, and its evident stability as a store. Simply put, stretch a rubber band, hook it up at that stretched length, and leave it there (as you cannot do with the unhelpful trio of light/sound and electric, met above). Later relax the band a bit, and you can get a job done. Not any job, but one constrained by how much relaxing you have allowed in the band. The energy shifted from a store is, therefore, a constraint on what can happen: some things are impossible if depleting the energy in the store does not shift enough energy. Plus energy = 1/2 kx^2 is a calculation that lies not too far in the future, in a learning journey. I avoided elastic potential energy because any qualifying adjective for energy diminishes the force of the unity of energy and on the grounds of eliminating unnecessary words (bye-bye "potential").

You could choose to stretch the 'spring of the earth' otherwise known as gravity and make similar points about being able to hook it up (maybe by storing water in a reservoir and leave it there). You might choose to call this a gravistatic store. Still, simplicity won out over pedantry here (allocating SPAG marks never was a fav activity, and was never much about the ideas). Being able to identify a change in the separation of two massive objects, and associate this change with a difference in the energy in the store, linking to the separations of the ends of the spring in the comparable elastic store, is a crucial teaching move.

Since the energy stored in the band is now the energy stored in the field, your thoughts might move on towards electrostatic stores and magnetostatic stores. Or even further, to field potential stores, covering all three. Or it might not. The idea of a field potential store was considered briefly, then rejected mostly on the grounds of too much abstraction, too soon.

So why not electrostatic and magnetostatic- after all in principle you could think of separate calculations corresponding to filling or emptying each store? Well, for starters, the word 'electrostatic' is just too likely to be simplified, it's just temptingly close to 'electric', which had an unfortunate history of practice in the topic(as alluded to above). And, if you use 'electrostatic', then you really ought to go for 'gravistatic' to match, which is just ugly. Then there is also the issue of potential collateral damage. You might lead others astray in representing electric circuits, mainly because of how a gravistatic analogy is often used to introduce a potential difference. But this analogy is not with an electrostatic situation, so again the electrostatic is deprecated because of the possible conflation of ideas. Finally, one cannot feel electrostatic forces as a result of separated charges directly, as you can with separated masses and gravity, leading to the need for multi-step explanations to link changes in the energy in that store to the physical changes.

'Magnetostatic' has the benefit of being able to associate physical experiences of forces when separating magnets with changes in the energy store, but this does not map onto calculations that will be done any time soon. So this could be a contender but is hobbled by its necessary association with electrostatic, where the multiple reasons why this might not be sound suggestion are not enough to offset the advantages of the physical experiences associated with magnetostatic.

But that we can enable children to feel 'magnetic springs' in action, and that there is a close connection between the electric and the magnetic, and to be made in short order in conventional curricula, suggests that there is value in describing a compound store, the electromagnetic. This decision is further bolstered by the unification of the electric and the magnetic fields, not something we have yet managed to do with the gravitational, which is a further point against the "field potential" store.

Again, using the elastic store as a metaphor, one can think of chemical springs, where atoms are re-arranged to store more of less energy, and nuclear springs, where it is the nucleons that are similarly re-arranged.

Which leaves the vibrational store, standing in for calculations of 1/2 kA^2. The interplay of energy changes associated with changes in movement and with changes in position are essential to the energy calculation and the nature of a vibrating object: taking a snapshot and calculating wreaks a fatal distortion of what it is to be a vibrating object. It is not something that occasionally happens to be in such a physical situation that you can calculate the energy that's been shifted to a gravity or elastic or electricmagnetic store. At other times fortuitously you can calculate a change in an energy store. That's also true, after all, of a large number of processes which are not vibrators. Apart from the utility in the discussion of the depletion of resources, such a store provides a useful starting point for a discussion of waves, and how they might fill or empty stores.

These stores represent the calculations to come, so laying the groundwork for thinking about constraints – about what cannot happen. But there is also some semi-quantitative reasoning that is of immense value in thinking about depleting resources and in planning for sustainability which typically appears too early on in curricula for calculations to be possible. Whatever the choice of stores, this should encourage and enable such discussion, linking the real decisions to be made to the energy descriptions.

[1] I like this description of 'imagined learning sequences', although others would call this a didactical pathway. In any case, it is an invitation to try a 'teaching experiment', to persevere with implementing the line of thinking in your classroom, because it is well-enough supported by plausible reasons that you can see a local adaption. There are also explicit principles, giving you and your children a better chance of making sense of a connected set of ideas in the sciences.