Acceleration
Forces and Motion

Accumulating changes - Teaching and learning issues

Teaching Guidance for 14-16

The Teaching and Learning Issues presented here explain the challenges faced in teaching a particular topic. The evidence for these challenges are based on: research carried out on the ways children think about the topic; analyses of thinking and learning research; research carried out into the teaching of the topics; and, good reflective practice.

The challenges are presented with suggested solutions. There are also teaching tips which seek to distil some of the accumulated wisdom.

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Things you'll need to decide on as you plan

Acceleration
Forces and Motion

Things you'll need to decide on as you plan: Accumulating Changes

Teaching Guidance for 14-16

Bringing together two sets of constraints

Focusing on the learners:

Distinguishing–eliciting–connecting. How to:

  • develop precise descriptions, building on everyday experiences
  • exploit the availability of technology such as videocameras to tell stories about motion and recording of motion

Teacher Tip: These are all related to findings about children's ideas from research. The teaching activities will provide some suggestions. So will colleagues, near and far.

Focusing on the physics:

Representing–noticing–recording. How to:

  • talk helpfully about clock time and duration
  • separate instantaneous from average
  • emphasise that all motion is relative motion
  • exploit common patterns in thinking
  • be clear about what is simply a convention, albeit a very useful one
  • exploit clear diagrammatic conventions

Teacher Tip: Connecting what is experienced with what is written and drawn is essential to making sense of the connections between the theoretical world of physics and the lived-in world of the children. Don't forget to exemplify this action.

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Being precise about measurable quantities

Acceleration
Forces and Motion

Being precise about measurable quantities

Teaching Guidance for 14-16

Reporting measurements

Wrong Track: The speed is 20. The acceleration is decreasing.

Right Lines: Charlie records Bob's velocity as +20 metre inverse second. The acceleration is negative, because the acceleration is reducing the changes in distance between Charlie and Bob in each second (the change in velocity is negative).

Taking care to make reports unambiguous

Thinking about the learning

The point of view to be taken is often not thought about, and the implicit decisions made in the initial description of the situation often impede deeper understanding.

This is particularly the case if such ambiguities are added to vague assertions about rates of change.

Thinking about the teaching

Choose an explicit recorder for the measurements (Alice, Bob or Charlie have served us well). Include this decision in the description of the motion. We'd also suggest taking the opportunity to spell out what's changing with time (e.g. velocity is rate of change of position).

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Choosing a point of view

Acceleration
Forces and Motion

Choosing a point of view

Teaching Guidance for 14-16

Recording a process and agreeing on the recorded measurements

Wrong Track: Everyone agrees on durations and locations, surely. Where things are and when they happen everyone has to agree on.

Right Lines: If you want to be accurate you have to be more careful: when something happens and where it happens depend on point of view. But your basic idea is right – there are some things that everyone will agree on.

Snippets and suggestions

Thinking about the learning

Here is a telling snippet of conversation:

Six year old: How long until the train gets in?

Dad: Five minutes.

Six year old: But how long is five minutes?

Looks like being a long five minutes for the dad.

Thinking about the teaching

Indicating where things are and when they happened turns out not to be so trivial. It took some time before Greenwich was adopted as the agreed zero of longitude, and even longer before the clocks all over the UK agreed with each other (this had to wait for the railways, which carried timepieces from one station to another).

The Airy Transit Circle at Greenwich (1851) was agreed as the zero of longitude for the world in 1884, largely because of the accumulation of existing practice. Many countries had their own origins, but the tonnage of shipping using the Greenwich meridian provided a significant factor in the debate.

Now it's been supplemented by a GPS-based system, which doesn't quite agree with the older standard. You can find out by how much by visiting Greenwich and turning on your own GPS unit.

But there's more:

Distances between different positions feel as if they ought not to depend on the point of view, assuming the journey from one to the other follows the same route.

Differences in clock times from the start to the end of a journey (the duration) also feel as if they should be invariant – that is, not depend on the point of view.

It took the simple, but deep, analysis of an Einstein to show that our common-sense reasoning here does let us down. Both the journey distance and the journey duration depend on the point of view – that's an important insight from the theory of relativity (even in his theory some things don't change – there are some invariants). These effects are only noticeable at very high speeds.

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A range of constrained speeds

Acceleration
Forces and Motion

A range of constrained speeds

Teaching Guidance for 14-16

A range of speeds and constraints

The speed of travel is often constrained by the physical environment of the object. The constraint may be technological, material, or even more fundamental. It's a good idea to have a number of these to hand, and perhaps to base some discussion around which category the example falls into, and whether anything we can do might change the speed.

Teacher: Work on the first Thames 1200-foot tunnel took 16 years and two months, an average rate of progress (allowing for the seven-year lay-off) of only 4 inches a day. This was a good measure of how sorely the project tested the technology of the day.

Teacher: Row a boat through water, and the waterline length of the boat will set your top speed. A longer rowing boat will always beat a shorter rowing boat in a race.

Teacher: There is an ultimate speed limit: nothing can exceed that speed. So the very structure of the universe itself seems to provide a kind of constraint. This speed also happens to be the speed of light in a vacuum. That makes light (and other electromagnetic radiations) rather special.

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Constant

Acceleration
Forces and Motion

Constant - use of this word in your classroom

Teaching Guidance for 14-16

Restrict the word constant to a single meaning

Teacher Tip: Take care how you use words, so that you don't say things that a moment's thought would reveal to be false.

For example, the owners of this gate meant regular or perhaps frequent.

It turns out that they certainly did not mean constant, unless the photograph lies. In the same way, we'd suggest that you don't use steady, or other near synonyms, to mean constant.

We'd suggest that you restrict the use of the word constant to mean having the same value at all times, and don't use synonyms for the word. So:

Teacher: The speed is constant.

Teacher: The acceleration is constant.

But not:

Teacher: The speed was steady.

Teacher: It's a steady force.

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Average and instantaneous values

Acceleration
Forces and Motion

Average and instantaneous values

Teaching Guidance for 14-16

Differences and similarities

Wrong Track: We covered 15 kilometre on our ride in 2 hours so we must have been travelling at 7.5 kilometre / hour.

Right Lines: The average speed is calculated from distanceduration. But it's possible that you weren't travelling at that speed for very much of the journey at all. If you made a movie clip of your cycle computer, showing your current speed, it's likely that for most frames it wouldn't be showing 7.5 kilometre / hour.

Making intelligent choices

Thinking about the learning

Dividing the distance covered during the journey (kilometre) by the duration of the journey (hour) will give an average speed:durationspeedaverage = distance

This global value can't be allowed to become confused with the values shown on the measuring instruments carried on the journey, which will show the values at a particular time, not over the whole duration of the journey.

Such right-now representations are true at an instant (so have no necessary connections to the history of the journey up to that instant). Here are some examples: resultant forces, velocities, accelerations, instantaneous speed, and positions.

Thinking about the teaching

You'll need to think about how to refer to measurements that represent an average for a whole journey, or for a significant interval during that journey.

You'll also need to think how to refer to values that represent a right-now value, for a particular clock time, at that instant.

Here are some suggestions, for thoughtful adaptation to your own practice and situation.

Resultant forces, forces, velocities, accelerations and positions are right-now representations, and so are true at an instant (no history). The clock reading is always a series of instants.

So time is always timeinstantaneous. t stands for this, and only this. Some instant, defined by a clock reading. A unique now-ness. So:

vinstantaneous is a longer way of writing v.

sinstantaneous is a longer way of writing s.

ainstantaneous is a longer way of writing a.

Finstantaneous is a longer way of writing F.

If we want to deal with them as averages (we'd suggest avoiding this until post-16 studies, if possible) we'd extend the conventions – for example – like this:

vaverage; saverage; aaverage.

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GPS and relative motion

Acceleration
Forces and Motion

GPS and relative motion

Teaching Guidance for 14-16

GPS – where does the US military choose its origins?

The Earth is often taken as a fixed point of view, without always taking the care that one should about making this explicit.

The GPS location system reinforces this, appearing to give an unequivocal location.

In both cases you ought to be careful, occasionally bringing to students' attention the implicit assumptions behind the rather definite sounding data. In some ways one needs to be even more careful when the GPS unit starts providing out speed and velocity data.

Even if you don't move, while sitting in your chair reading this sentence, the orbital velocity of the Earth will ensure that you're not at the same 'place' by the time that you get to the end. (The rotational velocity of the Earth at the equator is 465 metre inverse second, so from a certain point of view you'll have moved about 1000 metre in the 3 second it took to read.) Then there is the orbital velocity, of the Earth around the Sun.

That's all before considering that you might be reading the sentence on a train, or in a plane.

Teacher Tip: What's the point of view of the GPS system?

You may go further by going more into the system, showing that it relies only on timing and so differences in distance, from which everything else is computed.

This links rather well to the idea of making images by timing, in the SPT: Radiations and radiating topic.

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Locating events

Acceleration
Forces and Motion

Locating events

Teaching Guidance for 14-16

The world described in terms of events

Noticing where something is (its position) and when it's there (what the time on the clock is) are both right-now recordings of data.

So where something is and when it is there are both at an instant.

A record of a journey is a set of these pairs (d, clock time). These pairs are often referred to as events. A multiple exposure picture is a good depiction of such a sequence. Less valuable, but more accessible are successive frames on a movie clip of the motion.

You record a position and an instant to fix an event. A series of such recordings narrates the journey, as has become popular on websites where you're encouraged to share such stories by uploading your GPS tracks (which are just such a series of recordings of events).

Teacher: Let's look again at this motion on the movie clip. How long does each frame last?

Ed: A very short time – it depends on the video camera.

Teacher: What should we look for to get the best record?

Belinda: As short a time as possible.

Teacher: Good: as close to zero as possible.

Teacher: So when is the ball here [points]?

Carly: Just as the clock points to that time. Not before, not after.

Teacher: That's very clear. It's at that instant, and only that instant. At that clock time the positioninstantaneous is x. And it's only there right then, because the ball keeps moving.

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Noticing and recording

Acceleration
Forces and Motion

Noticing and recording

Teaching Guidance for 14-16

Two different words for two different processes

The first thing you need to do in developing physics is to bring a phenomenon into focus, separating it out from its background. So we need to notice a particular process or facet of the lived-in world. This typically involves non-specialist language, used in an everyday way, and will often involve pointing at things and other informal approaches.

Teacher: Notice the fringes here [points].

However, typically later, it might also involve some technical language, building on the students' understanding to bring some new facet to their attention.

Teacher: What do you notice about the velocity of the wind after an hour?

For both of these describing processes we'd suggest you reserve the verb notice.

Then you might decide to quantify the data:

Teacher: Let's record the displacement.

To do this you need to bring along the whole careful operationally defined apparatus of physical quantities.

We think that maintaining this distinction might help to make what's happening clearer to pupils as well as providing you with a platform to begin to make explicit how physicists seek reliable knowledge through measurement. (Making a measurement allows you to test the predictions of theory much more precisely, so that you have more chance of being wrong.)

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On taking a point of view

Acceleration
Forces and Motion

On taking a point of view

Teaching Guidance for 14-16

Choose images, diagrams or text to promote noticing what's happening from a different viewpoint

Teacher Tip: Try out this poem, or something very like it, to support seeing a motion from a non-obvious point of view.

'When Einstein was travelling to lecture in Spain,

He questioned a conductor time and again:

"It may be a while,"

He asked with a smile,

"but when does Madrid reach this train?" '

You might find this poem, or similar images or phrases, useful to help students re-imagine the world from different points of view.

Seeing the world from different points of view is something that will need practice – often lots of effortful practice.

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Supporting an appreciation of instantaneous events

Acceleration
Forces and Motion

Supporting an appreciation of instantaneous events

Teaching Guidance for 14-16

A sequence of more and more precisely defined events

Teacher Tip: It might help this discussion to have a number of movie clips, in the first of which the movement is blurred, because the exposure time for each image was too long. In other clips the image can be sharper, gradually approximating to an instant.

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On time

Acceleration
Forces and Motion

On time

Teaching Guidance for 14-16

Restrict time to time of day

We'd suggest that there's much to be gained from not using the word time in a cover-all way.

There are times when you'd like to refer to the period of time, during which, say, a force is applied, a journey takes place, or a process happens. This is to report something that can be recorded and measured. We suggest using the word duration for an elapsed period of time.

There are times when you'd like to accumulate a quantity over some short interval of time – to predict what might come to pass. We suggest using the word interval for this (typically short) period of time.

We'd suggest leaving the use of time to telling the time of day – that is, an instant.

Teacher Tip: Use duration for periods of time that are measured, or which are to be measured.

Teacher Tip: Use interval for periods of time to be used in computations.

Teacher Tip: Use time for recording the time of day, or clock reading – what the time is now (and not for how long we've been recording this process).

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Patterns of thinking

Acceleration
Forces and Motion

Patterns of thinking

Teaching Guidance for 14-16

Exploiting patterns whilst making them explicit

Teacher Tip: Three patterns of thinking have been widespread and explicit in the topic and throughout SPT. These are compensation, proportion, and accumulation. All three are used in this SPT: Force and motion topic. We suggest that you explicitly refer to the patterns in order to help show how common patterns of thinking are re-used in the topic.

Teacher Tip: Compensation: a trade-off between the physical quantities x and y, both of which contribute to the quantity z.

z = y × x

Teacher Tip: Proportion: the physical quantities x and y go up and down together, and when one is zero, so is the other.

x = constant × y

Teacher Tip: Accumulation: one physical quantity, y sets how another, x, grows over time.

Δ x = y ×  Δ t

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Separating the empirical from the stipulative

Acceleration
Forces and Motion

Separating the empirical from the stipulative

Teaching Guidance for 14-16

What's found and what's (hard-working) convention

Wrong Track: Surely we just decide that forcemass = acceleration is true, just as we decide Δ velocity Δ time = acceleration is true.

Right Lines: Some things are helpful conventions, things we invent to help us describe the world in a special way, so we all agree well enough to test the usefulness of those descriptions. Others are rules about the inventions – discoveries supported by our tests.

Thinking about the learning

Students tend to see all equations as a set of algorithms to be followed to get from a set of inputs to a set of outputs, rather than as relationships between quantities.

Bringing the relationships to life may need you to spend a little more time explaining the nature of the relationships.

Students often don't see any difference in the status of the relationships encoded in the equations, and this can trip them up.

Thinking about the teaching

Everything in kinematics is tautological – that is, true in terms of the definitions. That does not make it useless, empirically, but you should recognise it for what it is – a very sophisticated way of describing the world. This description is highly conventional, but not accidental, and well suited to purpose. In particular, it is an excellent prediction machine, that mimics reality with a high degree of precision, given appropriate inputs to the machine.

But to get started you need to have the accelerations, as well as the initial velocities and displacements. And getting these is a matter of recording the values, and often invoking dynamics.

Dynamics is not a convention: forcemass = acceleration is a statement that could be wrong. It is therefore an empirical statement, and can be, and has been, tested. The fit to the lived-in world is good.

There is also a third kind of relationship, also empirical. These relationships set the forces for this particular situation, so are not universal, like Newton's second law, nor conventional, like kinematics.

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Stepwise evolution and emphasising change

Acceleration
Forces and Motion

Stepwise evolution and emphasising change

Teaching Guidance for 14-16

Some values tell you how a system will evolve

Teacher Tip: Predicting the future, step by step, depends on knowing the values now and projecting forwards.

Expressed as a relationship, you'd get: quantitynew = quantityold + change

This business of change is so important that it's probably worth rearranging the relationship.

Teacher Tip: Make a point of emphasising that change = quantitynew − quantityold.

Knowing ainstantaneous and vinitial(1) allows you to guess what the vlater(1) will be after a short interval (the longer the interval, the riskier the guess), Δ tinterval(initial(1)_final(1)).

Knowing vinstantaneous and dinitial(2) allows you to guess what the dlater(2) will be will be after a short interval (the longer the interval, the riskier the guess), Δ tinterval(initial(2)_final(2)).

Using these instantaneous values to project forward in this way emphasises the fundamentals of kinematics, and uses the idea of accumulations.

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Use simple diagrams to probe understanding

Acceleration
Forces and Motion

Use simple diagrams to probe understanding

Teaching Guidance for 14-16

Use simple diagrams to explore students' ideas

Using these pairs of diagrams, together with probing questions, can reveal a lot about how the students think about motion.

Further discussions, and the resources to create simple diagrams of their own, perhaps seeing each others' challenges, can result in developing students' thinking effectively.

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Thinking about actions to take

Acceleration
Forces and Motion

Thinking about actions to take: Accumulating Changes

Teaching Guidance for 14-16

There's a good chance you could improve your teaching if you were to:

Try these

  • be clear about the role that mathematics is playing
  • separate convention from empirical discovery
  • Focus on fundamental relationships

Teacher Tip: Work through the Physics Narrative to find these lines of thinking worked out and then look in the Teaching Approaches for some examples of activities.

Avoid these

  • mistaking technical fluency for understanding
  • presenting the topic as a series of definitions

Teacher Tip: These difficulties are distilled from: the research findings; the practice of well-connected teachers with expertise; issues intrinsic to representing the physics well.

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