Teaching kinematics – in 2 dimensions & 8 steps

An easy start…

You can connect children's lived-in world with how you'd like them to represent and reason about kinematics by starting in two dimensions. Such a starting point also makes it easier for you to introduce the kinematic quantities. Learn to love reasoning with arrows to reduce your worries about making such a change in your teaching. This assumes that you currently follow a more-or-less conventional approach in starting with relationships between speed, distance and time and then relying on graphical representations to support much of the discussion.

There are five supporting lines of argument for this approach, each of which is, in effect a pedagogic suggestion, being an interpretation of current research evidence and successful practice:

An outline of a teaching sequence, here cast as an imagined learning journey (so with a focus on what the children will be reasoning about and with as they experience the learning journey).

I think there is a reasonable chance that such a narrative line in kinematics could work for a good many children and their teachers in perfectly ordinary classrooms, while supporting the development of fruitful and intelligible styles and patterns of reasoning in physics for children 8-16 years old. There is certainly a lot of evidence that the current widespread approach based on graphs and algebra all in one dimension leaves room for improvement ("Educational research tells you what not to do rather than what to do"). This approach may also throw up issues, but I think there is a good chance that many of the tripwires that afflict the current scheme are removed, and no (apparent) new tripwires laid.

**Narrative Step 1: real stories of motion**

Start with lots of real stories of motion in a horizontal plane, mostly qualitative and semi-quantitative. These could be a 'drone's-eye' (perhaps even live) or 'bird's eye' view: a two-dimensional view of a tracked motion. These have exploitable resonances in lived-in world representations, such as GPS-enabled mappings of personal journeys (Strava, MapmyRide), maybe aerial traffic cams or city mapping of traffic flows. Start with stories of journeys told in words and diagrams with very young children, so rehearsing the need for taking a point of view. Try switching between first-person and third-person perspectives, and maybe exploiting links to local approaches to geometry, probably most often met in varieties of turtle geometry. Encourage framing of first and third-person descriptions using 'how fast, for how long and in what direction' or 'how far, and in what direction'. This lays in a store of resources to draw on when predicting displacements from velocities later on.

**Narrative Step 2: recording and generating journeys**

In the school laboratory this could be coupled with strobed long exposure images of polished steel ball bearing: records of 'natural motions' on different sculpted surfaces. Or put a suitably mounted 'flasher' to work in similar environments. (Such a flasher can be rather cheaply fettled using a flashing bicycling light as a starting point).

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An interactive multiple exposure 'image'

This line of development could be technologically supported by using toy robots that can be programmed to traverse two-dimensional paths. Introduce challenges to explore or reinforce an understanding of sequences of track segments, beginning to formalise the idea of a motion as a sequence of track segments. I'd suggest restricting these to robots that traverse a plane, as two dimensions are pedagogically optimal.

On the screen, sequences of displacements could render a track visible, track segment by track segment. Or you could simply draw on a map: idealising as you go, by restricting the track segments to straight lines(so mimicking the light streaks which appear in the long-exposure photographs as a result of fast-moving objects). You are providing a low-resistance route to predicting or retrodicting journeys as a sequence of arrows.

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On paper, now is the time to introduce motion diagrams, with the separation of the images or dots representing a track segment. I'd suggest moving from qualitative to semi-quantitative and then quantitative interpretations often, to encourage physical understanding.

The first formalisation is a plan view of a motion diagram which is intrinsically two-dimensional and composed of a sequence of track segments. Again it might be useful to make strong links with past experiences, connecting this representation with the descriptions of journeys as sketch maps, so a third-person point of view, and with reports of such journeys from the first-person point of view.

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From these formal diagrams draw out the idea of taking a point of view and recording when something was where, so relating a sequence of displacements to the motion diagram. This sequence of displacements can then be simply represented by a sequence of stylised arrows, from the chosen point of view.

**Narrative Step 3: accumulations change displacement **

Then introduce accumulations, relating the velocity ( just shown by an arrow, perhaps on even a map) to predictions about where the actor will end up. At this stage, 'velocity' is simply a label for the arrow, which shows 'how fast and in what direction', referring back to the groundwork you laid earlier on.

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watch Rana go: displacement is accumulated

Then segue into a series of predictive stories, starting with a sequence of velocities and predicting the displacements. For each step of the story, you can focus on the detail of the accumulation, varying the velocity and duration of the step, to vary the accumulated displacement. Iterating across the steps is the crux of 'reasoning with arrows': accumulation, which is just addition, is done by placing arrows tip to tail. ￼

displacement is accumulated by velocity over a duration

You could look at successive intervals, seeing how changing velocities changed displacement.

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displacement is accumulated by velocity over some intervals

As these stories of motion can be described as track segments, as represented in the laboratory records, you can bring together the experimental records of motion with a predictive model:

**Narrative Step 4: accumulations and tracks**

Manipulating the representations by adding arrows, as here, enables children to reason about what will happen. This set of arrows predict a track, made of track segments, as before.

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from accumulations to track segment

These predictive stories rely on 'velocity telling displacement how to change' (less demotically: velocity accumulates displacement). This is the first tautology of the pair that comprise kinematics. Here you are also re-creating the 'stories of journeys', so linking back to the lived-world. These predictions have demonstrable utility.

**Narrative Step 5: changing velocity**

Now extend the idea of accumulations to acceleration accumulating velocity: it may be better to resist the temptation to place the velocity vectors tip to tail at the this stage to avoid any suggestion of representing a track that is followed.

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velocity is accumulated by acceleration over some intervals

Again, you can look into the detail of the reasoning with arrows here.

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velocity is accumulated by acceleration over a duration

Then, as a second step, link both accumulations, one after another, to predict the track of a journey.

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two accumulations, from acceleration to velocity to displacement

This introduces the second tautology of kinematics ('acceleration tells velocity how to change', again in its natural two-dimensional environment).

**Narrative Step 6: to one dimension**

After completing the two-dimensional sequence, the one-dimensional case might be introduced as useful in some special cases. With the full understanding of the relationships between acceleration, velocity, displacement developed in the two-dimensional work, and the consequent ability to reason with arrows, the greater reliance on diagrams with which to reason, there should be less reason to trip ourselves in complex circumlocutions, or in what might appear to the children as somewhat ad-hoc sign conventions.

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One-dimension, velocity accumulates displacement

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One-dimension, acceleration accumulates velocity

**Narrative Step 7: algebra, at last **

What now? Well, in one sense, all the hard work is done – the connections between acceleration, velocity and displacement are made, and with due practice, can be secure. The scalar quantities of distance and speed are rather straightforward to introduce with the resources to hand. However, there are bridges to other representations to build: to algebra, to graphical representations.

It is highly likely that you'll only be dealing with scalars or the magnitudes of the vector quantities, which is why this is perhaps best delayed until after the introduction to kinematics suggested here.

**Narrative Step 8: and finally, graphs**

I do think that graphs, perhaps best seen as more-or-less as frozen accumulations, might well be introduced rather late on, with some simple algebraic relationships preceding those.

These can be constructed from the available materials. Still, you’ll have to decide rather carefully how to map the components onto the vertical axes of the graphs, assuming the horizontal axis, as is conventional, remains duration. This is particularly true if you want to build this bridge to graphs early on in the sequence: here you might restrict the quantities represented on the vertical axes to scalar quantities, so avoiding the formal complexities of vectors, once you stop thinking of them as arrow-like objects.

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For some, now might be the time to formalise the vector representation as an extension to their understanding of kinematics, as an ordered set of numbers. Or you might relate the predicted movements and vector representations to graphs of the components of vectors. The next step depends on the direction of travel and the desired destination.

Let's start with perhaps the most controversial suggestion, which is that you begin by reasoning about motions in two dimensions. Note that almost all representational surfaces on which we inscribe records of journeys, or which we use to make plans for journeys, are two-dimensional: from formal paper maps, through GPS trails, to sketch maps of how to get from home to the local shops. This is not accidental. Stories about journeys involve relating ourselves to the local geometry and often involve talking about, for example: 'going left', 'going right', 'going straight on', and 'turning half-left' by some landmark or after travelling for some distance or duration(so with some implied speed). You could co-opt these patterns of thinking and talking in the students' experiences: not something that can so readily be done if we start along the conventional route of teaching about kinematics in one dimension. The phrases that naturally involve two dimensions are a widely-used way to give directions and to talk about paths that are followed. It's possible to see connections here with the revolution in geometrical understanding derived with computational approaches embedded in Logo-like languages. These often appear in many early introductions to computational activity, in languages such as Logo itself or Scratch. The widespread opportunities to 'program' in a set of instructions, much like offering guidance in getting from A to B when 'programming' a human, provides additional support for thinking in two dimensions, often using relative positioning, (implicitly)relative velocities and durations. The availability of more-or-less educational computational toys such as Sphero™ further widens access to this 'two-dimensions first' approach to kinematics.

'Taking a point of view' is neither more nor less than establishing a frame of reference. Still, it is perhaps a more accessible formulation as a starting point (for development of the idea see, for example, the web resources for Supporting Physics Teaching). It is essential, but often not explicit, to take a point of view in order to record a velocity or a displacement. Starting with "what Charlie notices" or "what Alice records", rather than with somewhat formal arrays of measuring rods and clocks is a better way to make it more likely that this essential step of taking a point of view and incorporating that into the records of motion will become more widespread. Such an approach can make an approach to relative velocities, which often appears as a later, somewhat optional complication in current curricula, appear both more natural and more essential. You can start somewhat informally by exploring what Charlie or Alice (in an example where this pair are not co-moving) notice, then what they record, and how that move to formalisation can be handled by either character. Perhaps, if you wanted to lay the real foundations for thinking about relativity, you'd compare what they do and do not agree on in describing precisely the same process.

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This actor, or observer-based, approach also aligns well with the stories-of-journeys strategy promoted in the discussion about starting with two dimensions.

Velocity is often defined as the rate of change of displacement, and acceleration as the rate of change of velocity, and there comes a time along the learning pathway when this conceptually charged formalisation, with its infinitesimals, is an essential part of the physicist's toolkit. But perhaps not when starting out. Here integration, as simple repetitive addition, seems much more straightforward than subtraction and division, so differentiation. 'Acceleration accumulates velocity' ( and the more demotic version, 'acceleration tells velocity how to change' is more approachable still, but remains operationally precise. 'Acceleration accumulates velocity' is intrinsically a more straightforward formulation than 'acceleration is the rate of change of velocity', as assayed by the number and subtlety of mathematical operations required.

Indeed, numerical methods for solving sets of equations use just such repetitive addition to predict the velocity from a given acceleration, and the displacement from a given velocity. Of course, such predictions are risky: here the velocity as reported by the GPS-enabled device does not guarantee that I'll end up in the sea and miss Edinburgh, or that I'll travel directly from the current location, shown by the cross-hairs, to Edinburgh.

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Either approach must be sensitive to the real situation: in the case of modelling with either rates or with accumulations, the situation determines the interval used for calculations.

This consideration becomes much more persuasive when considering vectors as a part of the didactic mix, as adding vectors can be made rather straightforward by treating them as arrows and doing the addition graphically.

Velocity, acceleration and displacement are of course vector quantities, and therefore both magnitude and direction are essential to their nature. 'Simplifying' such quantities by restricting them to a single dimension does entangle teachers in verbiage, as evidenced by this series of quotations from experienced teachers: 'signed quantities, but not like power', 'the deceleration is negative', 'these are one-dimensional vectors, but not just like numbers on the number line', 'acceleration happens when the speed is not in same direction', 'directed speed'. All of these, which cause significant cognitive loading for teachers and children, could be sidestepped, I think, by using vectors in their natural introductory environment of two dimensions. This may be one place where the affordances of the computer as a (two-dimensional!) representation machine can kickstart a reformulation of our customs and rituals. Adding vectors is now easy: 'just place the arrows tip to tail'. With computational assistance, the arrows become objects-to-reason with, tools for thought. Later, perhaps much later, one can burrow down into components and algebraic representations. But in introductory teaching, we might try and get the physics established with the most approachable representations that we share with mathematics: just arrows. Arrows such as those which appear on sketch maps, on GPS displays and in diagrams always show velocity from a point of view.

Didactically, there is no need to introduce acceleration or velocity as be a derived quantity, whatever the status as an SI unit, or the desired final definition. Tools to measure acceleration more or less directly are widely available, and not so expensive, even if the ability to represent it as a vector is often missed. (There is a gap in the market for a good Doppler effect based velocity measuring tool). Alternatively, at a larger scale, that GPS technology is widely embedded now presents many possibilities for representing the quantities, in didactically appropriate ways. Melding such sensors effectively to computational power and portable high-resolution screens offers genuine choices about how to present the data. Dials, gauges, data tables and cartesian graphs may be comfortingly familiar to those inducted into physics when the current representational flexibility was not available, but it does not follow that this is the best or only pedagogic path. I think we could make more of vectors, particularly in teaching about kinematics.

These are important, but you should get to them at the right moment. These are highly abstract representations of motion, embedding many conventions far removed from our starting point of describing journeys. (There is little less removed from the lived-in world than graphing my bicycle journey to the shops in one dimension. Yes, it can have value, but the highest value may lie some way down the didactical path.) It seems at least likely that the best place to learn about the intricacies of constructing and interpreting graphs is not (as so often happens at the moment) in studies of motion, for however many carefully engineered sensors are employed in automatically generating the graph, time remains spatialised. Two axes extended in space (measured in metres) must represent seconds, metres, metres^{second-1} and metres^{second-2}: this is not obviously or immediately a well-formed tactic if clarity of representation is the aim. It is also the case that algebraic representations, manipulations and conventions can serve physics well, but cannot replace a physical understanding. There is more to understanding kinematics than achieving algebraic and arithmetic fluency, and the approach suggested here ensures that these fluencies are no longer a necessary precursor.

The difference between vectors and scalar quantities is important in physics, and so vectors deserve to be introduced so that they do real enabling work, allowing for learners to see how they are a powerful tool with which to think. The motivation behind this stricture is that an idea only becomes meaningful when we reason with it: it is incorporated into a mental model.

Connections between kinematics and the children's lived-in world can be strengthened by spending much more time developing qualitative descriptions before formalising kinematics. This can focus on stories of motion, to lay the groundwork for seeing motion diagrams as depictions and even records of movement, with the potential for insightful interpretation.

Working with local geometries has supported significant gains in mathematical understanding, and such gains should also help us here. Focusing on motion in a plane exploits the sweet spot, where manipulated simplicity meets representational adequacy. There is no easy access to three-dimensional spaces in which to draw, manipulate and represent. And one-dimensional spaces do not contain enough information to easily develop the idea of vector quantities, which will be essential to kinematics.

Such two-dimensional motion diagrams serve to connect the vector accumulations, which are the core manipulated representation, to the lived-in world. As such, they provide a transitional tool for both noticing and recording: two-dimensional representations are what one might expect children to reason with. Reasoning fluently with tautologies is essential: kinematics is not a set of empirical relationships (the relationships between acceleration, velocity and displacement are essential to the meaning of the terms, and could not be otherwise: they are a toolkit with which to apprehend the world, not something we learn about the world). Therefore introducing the quantities simply as empirically calculated is a misstep: a misrepresentation of 'thinking in physics'. As a consequence, it is probably better to start with some predictive narratives as these mimic the predictions that explore understandings, in exemplifying mental models in action, rather than a set of calculations from data. The central resources from which these mental models are constructed are accumulations and vector quantities.

One can accumulate vectors graphically because the grouping of magnitude and direction into a single graphical object allows you to accumulate unified entities while working in more than one dimension, which turns out to be crucial to avoiding difficulties in constructing an idealised learning journey for the topic. In constructing such a journey, making early use of these operationalised quantities is crucial: allowing time for these to become tools to think with.

It seems to me more than likely that kinematics is a really bad place to learn about graphs, perhaps because it relies on relating changes in space to changes in time, whereas scatter graphs spatialise both variables. This is probably an issue to be concerned about in any proposal to learn about graphs with all data based on a time series.

Above all in implementing a sequence, avoid a dash to formalism, especially one well-abstracted from the lived-in world.

Visitors to 'physics land' have wondered, with some justification, about our current practices:

"While it is easy for pupils to see that ‘speed’ is a very useful concept with everyday applications, it is not so easy to see what the ‘motivation’ is for introducing one-dimensional velocities...I found it difficult to get a sense of how pupils were persuaded that this opened up new areas of understanding of motion."

I'd hope that this observation would not apply to the current scheme, where the vectors do real work in the mental model that the scheme promotes, and form a necessary part of a coherent explanatory narrative. There will, of course, as with any new approach, be some transitional pain, but I think that it's at least plausible that such an approach could be fruitful and cause significantly fewer difficulties for both teachers and learners in being able to reason with the tautologies of kinematics.

For thinking about time and motion:

Arons, A.B.(1997): Teaching introductory physics, Wiley

For thinking about reasoning:

Johnson-Laird, P.N.(1983): Mental Models, Cambridge University Press.

For thinking about accumulations:

Lawrence, I. (2004). Modelling simply, without algebra: beyond the spreadsheet. Physics Education, 39(3), 281-288.

Lawrence, I(2007). Re-ordering kinematics through simple computer-mediated tool. in GIREP EPEC Conference Frontiers of Physics Education (2007; Opatija) Selected contributions, edited by R. Jurdana-Šepić, V. Labinac, M. Žuvić-Butorac, A. Sušac (Rijeka: Zlatni rez), ISBN 978-953-55066-1-4. p. 361–367.

For thinking about explanatory sequences:

Ogborn, J.(1996): Explaining Science in the Classroom, Open University Press

A sample of research into the success of conventional approaches:

Shaffer, P., McDermott, L.(2005). A research-based approach to improving student understanding of the vector nature of kinematical concepts. American Journal of Physics 73, 921.

For how to make aN LED flasher:

T. Terzella, J. Sundermier, J. Sinacore, C. Owen, H. Takai(2008): Measurement of g Using a Flashing LED. The Physics Teacher, Vol. 46, October 2008, p.395-397

To help you reflect on the value of critical thinking and teacher rituals:

Viennot, L. (2001). Reasoning in Physics. Dordrecht: Kluwer Academic Publishers.

Viennot, L(2014): Thinking in Physics: The pleasure of reasoning and understanding, Springer

As starters for sources of stories of journeys :

MapmyRide is at https://www.mapmyride.com.

Strava is at https://www.strava.com.

A (cheapish) programmable and steerable robot is Sphero™, at https://www.sphero.com.