# Prior-learning dependencies flow between Maths GCSE subject domains

Just over two years ago I published the post, You’ve never seen the GCSE Maths curriculum like this before, summarising my analysis of the prior-learning dependencies within GCSE Maths learning objectives. The curriculum map I produced showed the importance of students having high fluency with specific number related topics, such that the topics do not consume working memory resources when students were later learning GCSE Maths concepts dependent upon these. The analysis suggested which topics we should prioritise our students developing automaticity with at KS3.

We prioritised our KS3 around the nodal topics with the most out-going links (dependencies) with the aim that these should no longer be working-memory-consuming barriers to students’ learning of GCSE topics. Numeracy Ninjas was also created specifically to develop fluency in the top nodal topics.

From my nodal diagram, I’ve always had a sense that fluency with number underpins the other domains, but I recently found a way to visualise the flow of the dependencies between domains more clearly using a Sankey Diagram:

The diagram shows how the prior-learning links in the node diagram flow between the different subject domains within GCSE Maths. As suggested by the node diagram, the importance of number topics being the required prior-learning links for all three other domains (and further number topics) is very clear. A lack of automaticity with number will inhibit students’ ability to learn algebra, shape or data topics as their working memories will be consumed by the lack of fluency with the underlying prerequisite number topics. Number is the ‘enabler’ for the other domains.

Number empowers the other three domains, but the next observation is how algebra then becomes an important prior-learning domain to learn shape, space and measures efficiently. Intuitively this makes sense, through the use of lots of formulae and associated algebraic manipulations within shape, space and measures.

Thirdly, prior-learning of the shape and data domains typically only form dependencies for their own domains later on.

In very general terms,:

- To learn GCSE Maths number topics requires prior-learning of number
- To learn GCSE Maths algebra topics requires prior-learning of number and algebra
- To learn GCSE Maths data topics requires prior-learning of number and data
- To learn GCSE Maths shape topics requires prior-learning of number, algebra and shape

In sequencing terms, this suggests that a KS3 built upon these principles would first start with number, then progress to algebra. Once fluency within these domains is reached, the necessary prior learning will be in place to then efficiently study shape, which has the most number of domain dependencies, 3. Data can be studied at any point once fluency with number has been established.

The above conclusions from the Sankey Diagram are of course very generalised and could easily be misinterpreted and misused. I believe they suggest a possible sensible order for the sequencing of a KS3 curriculum in terms of general themes. The flow of the mastery-based KS3 curriculum we developed at Wyvern College follows this order. However, I would caution against developing a ‘domain-silo-only’ approach as many of the interesting and deep learning opportunities within teaching maths comes from making connections between topics, both within and ** outside** their own domains. Whilst it would seem reasonable that developing fluency with basic skills would logically require a KS3 scheme of learning with the above sequencing, I also believe effective teaching would occasionally dip into/span across other domains to develop the transferability of learning and deep links required for a mastery-based teaching approach.

As ever, there are general principles and then required exceptions…

What is needed — perhaps it’s been done already — is some research on the optimal time periods — which may differ among individuals — between mastering a concept (say, simple ratios) and rehearsing it so that it is not forgotten.

Ideally earlier-learned concepts would be seamlessly extended into newly-learned ones, as when vector problems include ratios.

But if this can’t be done naturally, there ought to be some mechanism whereby it just can’t happen that, say, 6 months passes after a pupil has learned how to add fractions, before he or she has to do it again.