How the Big Ideas and Core Competencies work together to build relational understanding
The last couple of Big Ideas don’t match up quite as neatly with the Core Competencies. But the ones that do align like this:
|Five Big IdeasLesson design principles for teachers
||Five Core CompetenciesDemonstrated by learners
|1. Representation and structure
|2. Mathematical thinking
||4. Number sense
When you apply the lesson design principles from Big Ideas to develop the Core Competencies, you help learners build relational understanding.
Relational understanding focuses on not just knowing a rule, but understanding why it works and establishing connections.
So, how do the Ideas and Competencies work together to develop relational understanding?
‘Visualisation’ and ‘Representation and structure’
Relational understanding is all about visualising and understanding the underlying structure behind problems. To build relational understanding, try using Ban Har-style questioning like:
“Can you see?”
“Can you imagine?”
It’s essential to allow learners space for visualisation before offering explanations. Also, try to avoid too much pencil on paper.
Using manipulatives helps learners to visualise and allows teachers to expose the structure of the mathematics at hand. But it’s vital to use manipulatives as tools — not toys.
How should you get started? I often build in time to allow learners to just play at first, especially if the resource is completely new to them. Manipulatives are a good way of promoting flexible thinking by asking those learners quick to arrive at an abstract solution to prove their thinking in a different way.
It’s worth noting that the Education Endowment Fund recommends removing manipulatives once understanding is secure to avoid over-reliance or procedural use of one particular model.
A critical component of scaffolding is making sure you carefully consider which representation to use. This helps provide access for all learners. When designing lessons, consider what to record on the board — even down to how colour-coding may aid understanding.
‘Generalisation’, and ‘Communication’
All three principles are about making connections, spotting links, noticing patterns and reasoning — which all help to build a connected body of knowledge.
Supporting learners’ generalisation skills can include getting them to explore whether statements are always, sometimes, or never true (or false — using the idea of negative variation).
Another good strategy is using peer discussion. Here, learners establish consensus around rules, examples and counterexamples (or non-examples). Encourage them to explain, describe and justify their methods and results, and reflect on their conclusions.
‘Fluency’ and ‘Number sense’
Fluency and number sense are closely related: partitioning facts, times tables facts and using connected facts like equivalent fractions. When learners are fluent, they can use the known to work out the unknown — an important component of relational understanding.
For me, number sense and fluency are all about noticing patterns, checking to see whether an answer is reasonable, and selecting efficient and appropriate methods of calculation. Sound number sense avoids emphasising procedural recall and rehearsal.