Modeling is a process for which you have to choose tools

The process

Modelling is all about creating representations with which one can reason. At their best these are responsive: dynamic if accounting for changes over time, or explorable if representing a process. It has for some years been possible to create computational representations of facets of the physics that we teach. Here I'll explore what might be done to support current curricular modelling statements and see how to evolve the current practice.

Comparing these models with the real world decides their value. A good match and you have done a good piece of science. This can be hugely satisfying. Activities with modelling software are designed to allow children this pleasure. So they should be writing the rules to define what the system does.

The tools

Modelling tools are, in their essence, exercises in intelligible design. That is, they carry messages about what they represent in their structure and affordances. Since modelling is about doing the mostest with the leastest — at least in physics, it follows that the tool itself can usefully proscribe some possibilities.

In essence, modelling tools should be lean and focussed on making available (and salient) the pre-eminent modes of reasoning. Furthermore, the design of the tool should minimise the chances of syntactic errors.

Some simple ideas about an evolving system

• when representing the evolution over time, something obviously happens during that time
• the imagined, created world needs to evolve by itself, without interventions
• in elementary physics, there are a limited set of evolutions over time, and these are mostly kinematics
  • a accumulates v
  • v accumulates x
• Do...while a certain condition holds (doWhile functions) are a fundamental iterating structure, because:
  • iterating forever to an infinite future is very risky (so, no-keeping-on-going forever)
  • processes often also need terminating conditions other than a certain duration

Some simple ideas about a constrained system

• no evolution in time, only an exploration of constraint relationships, so of possibilities and of impossibilities
• the properties of geometric figures can be co-opted to reason about some such relationships

How to choose a particular tool

Criteria for being interesting

• the range of processes or situations that you and the children can express with the tool
• the range of partial and completed models that are intelligible enough to share
• similar patterns of thinking should be visibly rendered as such
• it should be possible to view the big picture, rather than getting mired in technicalities

Questions about pedagogy

• what exactly are the milestones and possible paths on the learning journey?
• how can you manage the learning journey?
• affordances and resistances of the chosen system
• there should be an element of mild surprise or pleasure obtained when the model runs
• children's ideas should be implementable in different ways: modelling should be an open activity

Selecting and managing modelling tools

Just what are modelling tools for?

You might start by looking for either

• pedagogic utility or
• and well-established teaching models as these are a primary concern if trying to change practice, whereas philosophical verisimilitude is probably less of a concern.

Here is fruitful thought: what we think of as real is what we use to act on the world. That is interventions are central to our idea of what is real. The representational facilities of modelling tools can be exploited to make a wider range of entities manipulable and therefore real.

As far as learning is concerned you'll be looking for something that'll be:

• plausible
• fruitful
• intelligible

but you also need to bring in the importance of immediately accessible affordances as an encouragement to improvise and explore further.

In this respect what does modelling have to learn from coding?

Essentials for considering a modelling system

• there should be a range of situations or processes that can be expressed with the tool, in ways that support the learning of physics
• a range of partial and completed models can be shared with pupils that are intelligible, can be explored and developed
• an element of mild surprise or pleasure should be obtained when the model runs—not only 'just what was expected'
• there should be a successful attempt to minimise the syntactic errors by the cunning design of the expressive medium

Strategic decisions to do with managing the modelling process

• how to show the big picture whilst modelling—the zoomed-out view that keeps the aim of the whole in sight whilst we're wrestling with the minutiae
• how to make the same pattern of thinking visible, across many contexts (search and replace in code is a valuable tool in this regard)
• how to implement the same model in different expressive media to help with abstraction
• how to foreground the re-useable parts of the learning journey as the model is developed?
• plan to have to hand a range of techniques for managing the learning journey - placing of milestones and signposting of (multiple) pathways?
• how to exploit the affordances of the system in which implement the modelling

Reflections on two kinds of variables based modelling system

There are two kinds of models, or at the very least two modes of thought, that relate physical quantities, and it's important to distinguish between them, for rather fundamental reasons.

These are models that relate quantities that accumulate and models that relate quantities that are constrained. In the former case, the physical quantities evolve with time, and in the latter, they do not.

Physics is replete with atemporal relationships between physical quantities:

FractionCBA{a}{F}{m}

I = VR

FractionCBA{density}{mass}{volume}

F = GMmSymbolSuper{r{2}}

These are all true at an instant (any instant). There's no essential temporal variation. You might think of them as 'snapshot relationships: describing possibilities; proscribing impossibilities. With such a view, other relationships are also of this kind:

v = a × t

and

s = u MultipliedBy t + WordFraction{1{2} MultipliedBy a MultipliedBy SymbolSuper{t}{2}}

Here t stands for the time on the clock. (There's rather a lot of advice about being careful about time in the literature, usually focussed on the teaching of kinematics, but you ought to keep it in mind in all areas of teaching so that you can represent the coherence of physics well). Such a time-on-clock is also known as an instant.

Laurence Viennot has written extensively and convincingly about a tendency of children to interpret these kinds of relationships in terms of stories with a timeline. A favourite example is:

ProductABCD{P}{V}{n MultipliedBy R}{T}

This is a relationship between macroscopic variable, true at every instant, so a canonical snapshot relationship, where the physical quantities are constrained. Perhaps because humans seek causes, and tell ourselves explanatory stories in terms of these causes, we tend to reason with this kind of relationship in terms of first one quantity changes, then this changes the next, so the final quantity remains constant. The important point here is that there is no then as a consequent action: it's a snapshot relationship. Much discussion of how to present Newton's third law would be clarified by directing attention to this atemporal or instantaneous constraining.

So in modelling, we ought to respect the difference between constraining and accumulating models. Accumulations ought to be about evolution over time, and so you'd be able to iterate over time. Other models that do not involve such accumulations ought to steer clear of iterations altogether. Here's an example where you could iterate (that is the modelling system permits it – and it's slick) but one ought not to, and if your aim is present the tool as something which exposes the thinking in a rather fundamental and helpful way. The example is the maximum power dissipated in a load resistor(R)in a circuit where the cell has an internal resistance(r). This might not be importantly all if using the modelling tool as a construction kit for building simulations, where the engine is probably hidden. But here I'm discussing a modelling tool as an expressive medium, so the maximum power theorem and similar relationships are better not expressed as accumulation relationships.

Linking the two kinds of thinking and expression

• In practice both kinds of relationships turn up in physics models, sometimes in the same model.
• Models with evolution gives time traces of trajectories through phase space and, when frozen, give geometric figures (e.g. a graph - this is could then be explored by a constraint model). There is some thinking and research to be done on the extent to which it is, in general, possible to exploit the properties of geometric figures to review relationships.
• So you could run an accumulating model, then freeze the output: then constraints modelling can be used to look back at frozen stream (This looks like a two step process of generating and then inspecting). Typically this is what data analysis programs do: represent a stream of data as a geometrical object, and then use tools to analyse it. I'm not aware of any that use any geometrical object other than a scatterplot. To restrict the output of accumulating models to this kind of inspection of output makes them seem less like live evolving systems.
• There remains an issue of how to represent the two kinds of relationships in identifiably different ways, particularly if within the same model.

An open question

There are two different kinds of programming languages:

• functional programming, of which Mathematica and the Wolfram Computable Language might stand as examples
• and procedural programming, of which Python and Scratch might stand as examples

Much of the talk of coding in schools focusses on procedural thinking, and this maps well onto evolutionary models, that represent processes, but perhaps do not do so well with those that represent situations. I think that there is space for further development by thinking about the affordances of these two.