Pedagogy Postcard #20: The abstract-concrete connection

A series of short posts about specific elements of teaching practice that I think are effective and make life interesting. Some are based on my own lessons and others are borrowed from lessons I’ve observed.


There’s a tipping point in many subject areas when the learning moves beyond the territory of tangible ideas, objects and observable phenomena into the realm of the abstract.  Taking students across this divide is a key aspect of enabling them to access higher level learning – or the next step in the learning.  It starts early on in numeracy; the simplest ideas about number require making abstract-concrete connections.  Having an intuition for the relative size of 2/3, 0.7 and 3/4 requires a mental model to be constructed.  Number bonds  (knowing that 6.4 + 3.6 = 10 and 637 + 363 =1000) and lots of other patterns require learners to have a good sense of scale within each power of ten. I’ve said before that the number line is probably the most important diagram in the universe!  It’s the key to creating mental models for numbers.  17 – -6 = 23 makes perfect sense on a number line; without it, the ‘a minus of a minus is a plus’ rule is just an abstraction.

All maths teachers will know this.  It applies to algebra, geometry, number, shape and space – the whole thing.  But what about other subjects? In history, a basic abstract concept is the idea of historical time. I’ve written about the importance of chronology here.   To some younger students, your starting point is something like this:

When did it happen?
When did it happen?

It’s hard for historians; it’s doubly hard for geologists and evolutionary biologists.  Yes – there has been time for evolution to work!  It requires a deliberate process to enable students to make sense of the scale of time – in decades, centuries and millennia so that they can deal with the development of historical themes and the timescales of physical and biological processes.

Later on, bigger ideas like democracy, ideology, communism, revolution, fascism and colonialism can also seem extremely abstract; it’s a gradual process  – a slow awakening – that leads students to develop an appropriately nuanced understanding of these terms.  I’d suggest that it helps considerably to acknowledge that this takes time.   Learning a new term like ‘fascism’ in terms of a definition is just the start; but it only becomes a concept students can use with confidence after a great deal of further exposure, referencing to and fro between the general definition and the specific cases, each of which are slightly different.

This applies in science too. Here the main challenge is to link macro phenomena to explanations at an atomic or molecular level.  Looking at ice, water and steam, for example, is a common learning experience.  How do we account for their different properties?

Screen shot 2014-04-14 at 13.40.56

In order for students to make their mental models, it pays to make connections between the macro properties  – their state of matter, capacity to flow, melt, boil and so on – whilst considering the molecular properties side by side.   But even here you need to watch for misconceptions.  Dylan Wiliam reports a case of students looking at  a diagram of water like the one above and, when asked to point to the water, they pointed at the white space between the molecules; the idea that the molecules are the water hadn’t registered.  Even when students get the idea about molecules, its hard for them to fully appreciate the scale – that there are billions of billions of molecules in an ice cube.  We need to work on this constantly – the abstract-concrete connection needs a drip feed, reinforced continually because it won’t happen fast or uniformly across a class.

Here is that same idea taken to another level.  At A level, it’s possible to set up elaborate apparatus like this using sensors to measure pressure and temperature for a gas trapped in a metal sphere, plotting the results on a computer.

photo (64)

Students can engage with the macro objects and the data; the numbers make patterns as numbers.  But to link this at a conceptual level to molecular processes requires deeper abstract-concrete probing.  Pressure is all about molecules hitting the sides of the container and it pays to get students to visualise that happening.

At A level we explain these things as part of the course:

Kinetic theory linked to idea gas laws. One of my favourite bits of physics.
Kinetic theory linked to idea gas laws. One of my favourite bits of physics.

But lower down, it’s still important to bash away at those abstract concepts.  A puddle evaporating is all about individual molecules escaping from the surface.  A chemical reaction is about atoms forming bonds in a new arrangement.  But that’s not what we see. Chemistry teachers needs to reinforce this continually, linking what is seen (the green stuff turning into black stuff) to a specific reaction (thermal decomposition of copper carbonate) and to the chemical equation and associated molecular model.  This applies to photosynthesis and respiration in plants; however hard you try, it’s a massive leap from learning the words and the standard reactions as abstract ideas to appreciating their true scale and complexity as concrete processes in plant cells.

As a science teacher I find it helps to bang at away at this all the time: Macro – what do we see? Micro – what’s really going on? Model – what model helps us to link those things together? Can we draw it? What’s the scale? And so on….

Finally, there is an abstract-concrete element to language learning that is supremely important:  Nominative, dative and accusative; object and subject; reflexive pronoun; future perfect; the subjunctive.  This is all so very abstract and learners need to find a way through the conceptual maze to reach the point that eventually it becomes more or less intuitive.  That requires teachers to make some good judgements about where to underline the conceptual linguistic framework and when to ride over it.  Sometimes it just doesn’t matter why we say it like that; we just do! At other times, the concepts are key – or else it’s all just guessing and memorising.

What’s the message? For me, its that we should be explicit about this journey. We should talk about making mental models and about the natural process involved in linking the abstract with the concrete.   It’s often necessary to create a very much simplified version of reality to allow students to grasp it and then to make it more complicated.  As the models become more sophisticated we should talk about their limits. We should acknowledge that this is difficult at times  – and that it’s a slow process that requires persistence; students need to develop a tolerance for uncertainty an ambiguity, for being a bit fuzzy.   Visual images, models, written descriptions and practical experiences all have a role to play in making the connections – who really knows what the best method is for each person? It always seems that you need to come at these things from all angles so that the ideas take hold more and more securely.  Explaining the same thing a hundred ways – that’s the teacher’s craft.

See also Great Lessons 6: Explaining.



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