One of the great challenges in teaching is working out how far you need to go in any given unit of work to secure mastery of the content here and now – versus the extent to which you can move on, trusting that, over time, a deeper understanding will emerge. I’ve heard Mark McCourt explore this brilliantly in relation to understanding mathematics. Over time, as we learn more about how numbers, geometry and algebra work, we look back at what we learned before with fresh insights; we see links that previously would have made no sense; we see specific situations as merely specific examples of a general case – e.g. squares are now specific kinds of quadrilaterals with equal sides and internal angles; angles in a triangle adding up to 180 is just one of a series of angles in polygons, fitting the wider pattern 180(n-2) with n = 3. But also the way fractions add and negative numbers behave make more and more sense – our Age 15 understanding of operations is deeper than our Age 7 understanding.
As we understand more, the granularity of our understanding is sharper within the same content area:
I think this is an important lens through which to review curriculum design and progress monitoring; it’s also something that more experienced teachers should support newer teachers with – sharing that tacit knowledge around how worried/relaxed to be about students totally nailing a subject at any moment. Sometimes, you need to get it absolutely right at this point in time; other times it’s ok to leave students to cogitate for a while… not yet fully comprehending in detail.. because you know it will all fall into place later. But, of course, there has to be a later….the curriculum has to be designed with this in mind.
In this image, the curriculum is set of units that overlap in various ways.. but some parts are not revisited; they don’t add up to anything beyond; it’s a set of mainly separate challenges in sequence. Learning more would mean extending the diagram to the left or right… more smaller peaks.
However, some areas of the curriculum not made up of separate units; they require us to think about a core set of ideas that gradually increases over time. All the previous knowledge is subsumed within the new knowledge as children’s understanding matures. We’re never leaving anything behind so there is always scope to continue exploring all the older material in amongst the new:
A typical example is with writing.. writing has many sub-elements of knowledge but ultimately it continues to add up to something coherent. We might introduce new elements over time, but we’re always accumulating and practising previously learned elements. This maturation process takes time to understand and trust – because students mature in their writing at different rates. There’s a bit of ‘fake it to make it’ at times – the writing doesn’t seem authentic at first; but then the ideas get assimilated and students learn to craft writing in a more natural style, appropriate for the context.
Here’s another diagram. Here, the final triangle spans all the topics that came before. There’s a path forward that takes in an ever wide view, linking, building, connecting, widening, deepening..
In subjects like science, we might need to create a sense the knowledge coming together over time more deliberately. When you’ve learned all the component areas of knowledge, you can look at, say, a piece of ice floating in water – or an iceberg in the ocean – and all kinds of ideas coalesce – the kinetic model, intermolecular forces, the chemistry of water – hydrogen bonds and all- temperature/ heat/specific and latent heat, density, Archimedes and buoyancy… all these ideas connect and when we stand back, what were once separate details of knowledge, all make a lot more sense as they link to each other.
It could apply to literature and history where you cover this text here, this one there; this period and battle here; this period and empire story there. But, now look – scanning backwards – we see the connections, the themes, the concepts, the big ideas all threading through. They may not have been obvious at first, they have been impossible to notice as general ideas within any given example – but if spend time taking the wider view, deliberately and intentionally, with a more mature view of the subject, the richness of all we’ve covered can reveal itself anew.. and that’s exciting.
The point here is that it really helps, as you teach, to know how and when ideas will link up in the future. It helps to have walked the path so that you can signpost the way ahead, reassuring your students that, whilst things might not be clear right now, they will in the end. That helps you to pace the curriculum, to move through it allowing certain gaps to persist because you know when you’ll be reviewing them, looking back over all you’ve covered in a more holistic fashion.
To make this abstract idea more literal.. here’s the view from the top of the mountain. You can survey the scene – all that terrain you’ve covered, each lower peak you conquered, each valley you entered – it all makes more sense now as you scan backwards, looking back down the mountain.