On teaching with waves

Immateriality and the state of practice

"The world is made of kisses and not stones" (Rovelli)

Waves are in some sense the prototypes of transmission of influence without stuff also being transmitted, so could or should play a leading role in reflecting this insight (generally, and for good reason, any elaboration of the idea of a field comes after the idea of a wave). So it is rather important that the theoretical idea of a wave gets well-established in a way that the power and applicability of the idea across different contexts can be realised. So if we’re teaching about waves, then we’d like children be able to represent and reason about disparate phenomena using ideas that are distinctively wave-like. Such a demand requires that we have first in the front-of-mind a clear idea of a wave transposed from its natural rather algebraic realm into a form that is not diluted to the point of being a classificatory label for certain kinds of phenomena. To be teaching the idea of a wave is to capture the essence inherent in y = Asin(kx -ωt).

This is noticeably absent from the pre-16 physics criteria in the UK 2018(for example, it is one of 7 headings in the IGSE, yet the detail does not express the idea of delayed mimicry, or the idea of superposition). The idea of a wave is also not salient in the professional discussions amongst teachers of physics, for example on TalkPhysics or PTNC. Nor is it a feature of pupil textbooks, or teacher guidance associated with purchased materials. So there remains a largely unexplored question of how an immaterial approach to waves can support these interaction models rather than tending to reify the waves, or to elide without comment some of the essential features of the wave.

To further illuminate the issue, here is a piece of advice from a coach of teachers of physics, appearing high on search engine rankings: "Get the students to wave to you and they all can. So they understand the concepts of waves. Have a discussion of what makes a wave, a wave." [There is no indication of how to achieve the discrimination, or what such a distinction would look like on the page, or perhaps the site].

It may be somewhat harsh but the majority of highly rated teacher-advice sites, as located by search engine, are a series of declarative statements, often without any discernible cognitive glue, which might serve either to provide coherence, or to illuminate a pattern of thinking which the learner could apprehend.

This gap seems to me to be of a historical habit of mind – perhaps even a 'teacher ritual', as elsewhere there remain many other buttressing resources available, which do not follow this 'inessentials' approach. For example swell and surf forecasts focus on amplitudes, frequencies and expected arrival times, depending on the paths followed from the source.

Two lines of approach

If the teaching approach is phenomenon-led then this implies an almost zen-like approach to waves: ‘see this here – that's not the wave’. At the moment this seems to be more-or-less the approach adopted, but with not enough zen, leading to conflation of the phenomena and the theoretical construct. In other work supporting the development of teaching approaches, for example in electric circuits, in introducing descriptions with forces, and pre-eminiently in refocussing the teaching of energy, the preservation of this separation between the lived-in world (particularly tangible experiences and noticed phenomena) and elaborated theoretical descriptions has proved crucial: I suspect that the introductory teaching of waves (remaining, as is usually done pre-16, where algebraic fluency is limited).

A second approach, developed here, is more theoretically-grounded. I think this approach could reap dividends in that teachers will be able to be clearer about the intended outcomes and how these relate to the depiction of a wave to support children’s reasoning. Central to this is to maintain the separation of influence and stuff.

Resources to think with

Two essential targets a fully-realised idea of a wave are superposition and the transmission of information. What might serve as experiential resources with affordances that provide credible and fruitful learning pathways to these two ideas?

Steps might be a useful prototype for the superposition facet of waves as they necessarily entangle time and space with the idea of the speed of progression. And one can be more-or-less in-or-out of step, which serves as a fruitful precursor to the idea of phase. Add the idea of varying amplitude, and so of varying contribution from a path or source and a learning pathway to superposition becomes conceivable.

A second building block is the idea of a mimic, or more demotically, a ’copy-cat’. Just as I can mimic the action of another, so I can imagine a detector mimicking the action of a source after a delay, providing the resources to build the idea of a wave on top of well-established source-medium-detector model used in introducing seeing and hearing.

Beside this pair the traditional distinction between longitudinal and transverse remains much less important: only distinguishing two sub-genre, and not serving to distinguish the genre from others.

A didactical path: Imitation and delay

First I’d suggest developing the idea of a wave slowly and carefully, with the building blocks as:

  1. imitation of what is happening at the source,
  2. after a delay,
  3. depending on distance from the source.

Such an episode moves from a single imitator (mimic, copy-cat) to an array of imitators at different distances. These represent the medium, and arranging them regularly gives a recognisable conventional representation of a wave. The gain is that now children know what the component parts are and how they work together to give this overall pattern.

A guided graphic narrative to the component ideas of a wave


Then launch into an episode exploring different patterns of motion in the source, still imposing a delay on the imitators depending on their distance from the source. This is the imperative, expressed as a memorable slogan: ”do like me, but later”. The imitators now show a variety of waveforms, and the sequence reinforces the essential component facets of what it is to be a wave.

A guided graphic narrative to different wave patterns


Not all sources vibrate continuously, so launch into the a third episode moving from a single delayed imitation, to pulses, to wave packets, and back to continuous waves.

A guided graphic narrative to pulses and waves


A didactical path: Steps and superposition

Next introduce an episode that builds from the idea of delays to contributions in and out of step, and so to the idea of superposition.

By altering the delay between a source and its imitator, you can make them out of step, by altering the trip time from source to imitator (set by speed at which the path is covered and the length of the path).

The imitator could be at a detector. If there is more than one path from source to detector, then contributions from those paths might end up in or out of step, as a result of the trip times for those paths. Add the contributions to find the resultant displacement at the detector. The amplitude at the detector could be large, zero, or anything in between. It all depends on whether the contributions are in or out of step.

A guided graphic narrative building from steps to superposition


Teaching and learning issues

The central new issue here is the assertions of identity across the physical/ conceptual divide – so conflating phenomena with representations: e.g.:‘sound is a wave’. The lived-in world should remain correlated with to physics-world rather than merged with it: to do physics is to reimagine the world, not to recite facts.

There also appears to be too much hand-waving and the deployment of vague words, leading to the suspicion that children are being short-changed, with the idea of a wave being used as simply an act of classification, rather than as a mental tool with which one can reason about influences propagating so as to bridge distances without any particles having to make the trip. In which case the rigorous idea of wave is not being developed or deployed with the consequence that the teaching and learning is not as intelligible, fruitful or plausible as it could be.

There are particular difficulties associated with teaching about light or sound, or radiating, which have already been identified in supporting physics teaching( SPT).

Winding up

The concern is to get away from content-free statements, such as ‘light is a wave’, when the idea of a wave is not spelt out. Waves are theoretical constructs, a whole intellectual mechanism or apparatus, useful for describing some phenomena. The key question is whether the idea of delayed mimicry of periodic motion is an essential piece of reasoning, or not. If not, then I suspect we continue to do little more than categorise – useful, but not defensible as a central occupation in an education in physics. Any trace of framing the descriptions as if one is discussing particles or other stuff moving from source to detector needs to be lost. This has implications for any courteous introduction to the idea of photons, not dealt with here (it is possible to construct an intelligible learning journey, without discontinuous jumps).

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References

Pearson Edexcel International GCSE in Physics – Specification – Issue 2 – April 2018 © Pearson Education Limited 2018