Research School Network: Small steps to build Maths success Building coherence: the order of learning and making connections within the learning

Coherence is one of the 5 big ideas from the National Centre for Excellence in Teaching Mathematics. It is about seeing how the small steps fit into the bigger picture: the overarching interconnectedness of mathematical concepts. It includes two key elements:

- The order of the learning: how the whole is broken down into steps which are easier to take onboard and master in smaller chunks.
- Making connections within the learning: how the steps connect to one another and then beyond to other topics.

The second part of this is very important: without it, pupils can engage with the procedural steps but perhaps not understand the reasoning behind them. These ideas are also emphasised in the EEF’s Guidance Reports for Mathematics in Key Stage 2 and 3.

If we look at the topic of constructions, specifically constructing isosceles triangles, pupils might follow these steps:

- Draw a horizontal line segment as a baseline
- From one end of your line segment draw an arc above your line
- From the other end of your line segment draw an arc the same size as the previous one
- Join the point of intersection with the ends of your line segment

Here students have followed some procedural steps and whilst they have successfully constructed an isosceles triangle, they will unlikely have a thorough and deep understanding of why this works, the connection to the circumference of the circles they’ve drawn or if there’s any alternative sizes or methods. If the student then forgets these steps they can no longer successfully complete the construction.

However, a task, such as the one below, taken from the NCETM Checkpoints tasks allows students to construct many different isosceles triangles, and all of the other required triangle constructions but from logical reasoning and deep understanding rather than memorising procedures.

While all of this may sound like common sense, really engaging with the idea of coherence can also help us consider the success of our curriculum. The order of learning may be outlined in medium term plans and modelled in resources, but does that end up reflected in the delivery? It can be helpful to consider three different embodiments of a curriculum:

In this model we see that all schools are teaching the National Curriculum (intended curriculum). But the way in which schools structure schemes of learning differs (implemented curriculum); and the delivery of this differs by teacher affecting how students experience this (attained curriculum).

To reduce students’ cognitive load and ensure greater mirroring between the implemented and attained curriculum, departments can plan collaboratively. Departments can keep the concept of coherence in mind when they break down a particular topic and then annotate this plan with their considerations of language, representations, misconceptions and of course links to other mathematical topics. The collaborative nature of this work within a department is extremely powerful, allowing teachers to discuss how they deliver certain content, working out where the similarities and differences are, and thinking about the impact this might have on pupils who change teachers from year to year.

There is also something powerful about really thinking hard about our subject and remembering that Maths isn’t just following the steps: it’s a process of logical reasoning with marvellous interconnections to be explored.


Further reading:
Rosenshine’s principles of instruction: step 2

Emma McCrea ​‘Making every Maths lesson count’

NCETM Coherence in Primary and Secondary