# On designing a sequence

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.