Bringing together two sets of constraints

Focusing on the learners:

Distinguishing–eliciting–connecting. How to:

• idealise from everyday experiences the lived-in world to develop the idea of a natural motion
• distinguish between everyday motions of the lived-in world and the idealised world of Newtonian mechanics
• draw out children's ideas of force and motion
• reactivate, and then use children's prior learning

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:

• represent idealised motions as natural motions in a frictionless world
• deal with remote causes—action at a distance forces
• introduce mass, and not conflated with quantity of matter

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.

A common mistake: force  →  motion and motion  →  force

Wrong Track: The car has a force which keeps it moving along the motorway. If it's moving it must carry a force.

Right Lines: A force acting on an object changes its motion.

Giving forces to things

Thinking about the learning

It's a very common belief that objects that are moving must possess a force to keep them going. When questioned, a student might typically argue:

Stephen: The ball is carrying the force you gave it when you threw it to start it off.

Students often label moving objects with force arrows in the direction of motion. You're also likely to meet:

Nathan: The ball will stop when its force is used up.

The idea that a moving object carries a force, usually traced back to the force that was originally exerted, is common sense and it is almost correct. A moving object might indeed have been forced to start its motion. But, and it is a big but, this force changed the motion, in getting it going – it was the starting push. Once the agent doing the pushing stops exerting the force (so when the object no longer interacts with that aspect of its environment) the force is no longer there. The object continues to move but does not carry a force. The force changed the motion.

But both the energy in a kinetic store and the momentum do increase (see episode 03 for more on measures of motion) as a result of the action of the force.

Thinking about the teaching

The big challenge here is to get a visceral belief about the truth of Newton's first law embedded. In the lived-in world we can't escape the effects of frictional interactions. Accurate descriptions of everyday motions always involve frictional forces (grip, slip, drag). So the task is to imagine the natural motion of the objects in a world far removed from students' everyday world, and then – and only then – to add complexity back into the description.

Everyday experience is not on our side – to keep something moving we know we need to keep pushing it. This continued application of a force is required simply to oppose the retarding effect of friction. In a world without friction there would be no need to keep pushing. Objects, once in motion, would carry on moving. Pupils don't live in a world without friction, but there would be other interesting side effects.

You need to engineer the learning environment so that students' imagination can trump their (prior) experience.

Empirical discoveries, computing, and defining

The equals sign appears in many relationships, but also in computer programs. In these two contexts the meanings are different, but the symbol is the same.

When we use it for doing calculations there is a danger that we conflate the two meanings, tending to undermine both.

There are three separate use cases:

• Assignment – used in computations.
• Equality – used in empirical assertions.
• Identity – used in definitions.

Taking care with symbols, one might use different symbols for different meanings. The difficulty is in computation, both when using pencil and paper, and when using software.

In many cases in physics, an = is often making an assertion about the world. It's a claim that what appears on its left hand side is identical in value (so physical quantity – number plus unit) to what appears on its right hand side. You test the validity of this assertion of the relationship between the two sets of values by empirical test. The use of the = sign is the final step in explaining how you think the world works.

In some other cases, the = sign plays a part in creating a framework that allows you to describe the world. In this case you're defining new entities in terms of existing ones. These are not empirical statements, and so could not be wrong (they're stipulative definitions).

Here is a simple example:durationspeed = distance

There is no experiment you could do to show that this is wrong.

(Some choose to write these kinds of relationships as:distanceduration  ≡  speed. We've not chosen that representation here, because it's not widespread.)

An assertion of equality

Good mathematical habits can help you to think tidily, and not lead your students off down the wrong tracks.

As we've suggested you present it (following the development through the SPT: Forces topic, and the SPT: Motion topic), this kind of statement represents an empirical statement about the world.

forcemass = acceleration

The quantities on the left hand side are found (by experiment) to be equal to the quantities on the right hand side. There's nothing voluntary about it: this is the way the world is.

You might want to (and perhaps should) debate the empirical status of Newton's second law with colleagues (what experiments could you do to show that it is false?), but you do need to present a coherent line of reasoning to students, and we think the position taken here is plausible, intelligible and fruitful.

Some would argue for:

force = mass × acceleration. We don't think that's all that helpful in learning to use the relationship.

Teacher Tip: Use the = to mean that the left hand and right hand sides of a relationship are equal, rather than as an act of assignment (let's make this equal to that). So always, implicitly (by aligning the = in multiple line calculations) or explicitly (by always writing things out in full), ensure this balance by having a left and right hand side.

Forces on objects moving at constant speeds

Wrong Track: It's just stood there: no forces are acting on it by the environment.

Wrong Track: The speed is steady: no forces are acting.

Right Lines: Objects that appear to be affected by no force might have several forces acting on them which all add to zero.

Talking about forces acting on objects in equilibrium

Thinking about the learning

Many situations used to discuss forces, particularly when first introducing the action of adding force arrows acting on objects to replace the interactions with their environment, are of objects in equilibrium. Very often these objects are stationary.

Such forces are often said to cancel out, reinforcing the idea that the forces were never really there in the first place.

Thinking about the teaching

We'd suggest explicitly going through the process of:

• Identifying and isolating the object.
• Noticing interactions of the object with its surroundings and replacing these with forces acting on the object, exerted by its environment.
• Drawing a diagram showing all these forces.
• Adding these forces as vectors, tip to tail, to show the resultant force.
• If the resultant is zero, saying so, and adding the explanatory phrase: because all the forces add to zero.

It's particularly important to keep forces and velocities separate. The resultant force changes the velocity. Even though both are represented by arrows (because they're both best represented using vectors), they're very different things. Here we suggest that you show only force arrows on the diagrams. There's much more on representing different quantities with arrows throughout this topic.

A non-local piece of the environment interacts with our object

Wrong Track: How can the Earth pull on the Moon to make it fall? It's not even touching it!

Right Lines: Some forces can act on objects without touching them. Gravity is one example of these non-contact forces.

Explaining action-at-a-distance

Thinking about the learning

If a student argues in this way, you might encourage them by recounting something of the history of the development of the idea of a field. It's possible to argue that this (very powerful) idea came into existence precisely to help scientists with their unease about this somewhat spooky action-at-a-distance. Certainly acting on something physical without touching it seems a very alien idea.

Thinking about the teaching

Gravitational, magnetic and electric effects happen at a distance: they are in a special class of non-contact forces. If students think that the action-at-a-distance effects of gravity are obvious, then you might well be worried. It's a strange way of thinking, rather different from the thinking in identifying all the other kinds of forces (normal and retarding).

However, the idea of a field is even more pervasive than the idea of a force in more recent physics, so it's something we may just gradually get used to. It's certainly an idea that's proved to have enduring cultural worth, and therefore worth investing some time in.

It's probable that it grew from worrying about how atoms interact – a line of thinking attributed to Boscovich (pictured here) – and was then picked up by Faraday to help him understand the non-local interactions of magnets and electromagnets. However, take care to keep magnetism and gravity separate as conflating the two is a well-known difficulty (see the SPT: Earth in space topic for more details).

How to show something is a vector

Teacher Tip: If some quantity is a vector, we'd suggest using an explicit notation to make that very clear. So we'd suggest →x instead of just writing the vector in a bold or italic face.

So acceleration is shown to be a vector by writing →a.

And velocity is shown to be a vector by writing →v.

In this way you'll help to keep separate scalars and vectors in students' minds.

You'll also be using something that they can mimic, and so help clarify their own thinking, when expressing ideas for themselves.

Teacher Tip: Vectors: acceleration, →a; velocity, →v; displacement, →d.

Scalars: speed; distance; mass.

Adding skins to functional models

Depending on the background and interests of your students, the following phrases may be useful.

Teacher Tip: It's not what it looks like, but what it does.

In physics, models don't have to resemble what they are models of. So the term model is used in a rather particular way. You'd be a bit surprised to buy a model car in a shop that did not have four wheels and windows. But models of cars used in physics may not have either.

The target in physics isn't representational models, but functional models. This is the origin of the impact of the phrase: Consider a spherical cow. It's not that physicists think that cows are spherical, rather that a sphere is simple enough that they can use it to capture some essential feature of how a cow works, which enables them to make predictions, and so understand what a cow does.

The idea of skinning – common in software and also in buying new covers for your mobile – may be useful here. Neither of these alter the functionality in any way: they only change the decorative veneer. Underneath is just the same reality: it's still just a browser, or just a Nokia.

Teacher Tip: A mass acted on by one or more forces can be skinned in many different ways. It could be skinned as a cat, a space rocket, or a thrown tennis ball.

Imagining the world in terms of forces and masses is seeing beyond the skin, down into the animating mechanisms. Since such a view makes many apparently different things appear identical, you might choose to call it a deeper reality. It's certainly a much more parsimonious and coherent view of the world – and those are key markers of the output of thinking like a physicist. You keep chipping away until the ultimate essence of a thing is revealed.

Einstein: A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability.

Said, and best left unsaid

Teacher Tip: We'd suggest using the phrase the forces add to zero as a step on the way to saying the resultant force is zero, and definitely avoiding the forces cancel out.

There are two main reasons for suggesting this:

• Adding the forces and finding that the resultant is zero is exactly what you do.
• The process of cancelling might evoke memories of dealing with ratios or fractions. That's not appropriate for vectors – there are no good mathematical rules for performing these operations with vectors.

Some parts of you are harder to shift than others – and that's not all in the mind

Remember that, in getting parts of your body moving, it's not only the mass – there's bound to be something about levers in there somewhere as well. The complexity of the situation suggests that using body parts may not be a good idea when introducing inertial effects.

Teacher Tip: Don't use parts of the body, or really anything that doesn't act as a point mass (so where you need to think about the internal structure), to introduce inertial mass and its effects. Concentrate on simple objects.

Choosing and representing

Teacher Tip: There's a very important connection between mass, force and acceleration: I choose the mass and the force. The acceleration is a result of these choices. Remind yourself of that by writing: forcemass = acceleration each time.

Teacher Tip: To be even more precise, in this relationship there is one thing which is shown by just a number (mass is a scalar), and two things are shown by a representation where direction is important (force and acceleration), so we use ordered sets of numbers for this pair. Now you'd remind yourself of that by writing: →a = →Fm.

The effects of several forces

Wrong Track: This force tries to push the box in this direction and these forces try to push the box in that direction. So the box is trying to move in two directions at once.

Right Lines: Things like boxes are not like you or me. They don't try to move. Here's a better way to describe it, less likely to lead us into unhelpful thinking. Both forces act on the box. The resultant force is the sum of these two forces. The resultant force changes the motion.

Find the resultant force first

Thinking about the learning

Expressions such as the surfaces are trying to move past each other are rather common when talking about frictional forces. That's not the only place: the danger arises every time there is more than one force acting on an object.

Thinking about the teaching

Avoid these difficulties by relating the changes in motion to the resultant force acting. Particular difficulties with frictional forces can be mitigated by using the distinction between grip, slip and drag (introduced in the SPT: Forces topic and again in the Physics Narrative on this episode).

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

Try these

• treating empirical relationships with respect
• treating the reimagining of natural motion as if it's problematic
• dealing with action at a distance respectfully
• using resultant force to predict motion

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

• speaking as if force is the kind of thing that gets carried
• being careless about scalars and vectors
• speaking of forces trying to do things

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.