Building relational understanding with the Core Competencies and NCETM’s Big Ideas
What do the Maths — No Problem! Core Competencies and NCETM’s Big Ideas have in common? They’re both important maths mastery principles that work together to build relational understanding.
The NCETM’s Five Big Ideas are lesson design principles for teaching maths for mastery and the Five Core Competencies are attributes that help learners develop deeper thinking.
Seems fairly straightforward. But how do the Big Ideas and the Core Competencies align? Do they serve different purposes? What are the implications for teaching?
To save yourself hours of digging through the internet, keep reading.
What are the NCETM’s Five Big Ideas?
Let’s start by looking at the Five Big Ideas in a bit more detail.
The NCETM Five Big Ideas were created to enhance teaching for mastery. These research-based principles frame the lesson studies, professional dialogue and lesson design process within Teacher Research Groups (TRGs).
Mastery specialists from regional maths hubs have helped spread the Five Big Ideas through TRGs.
So, what are the Five Big Ideas?
- Representation and structure: representations used in lessons expose the mathematical structure so that students can do the maths without needing the representation.
- Mathematical thinking: if taught ideas are to be understood deeply, they musn’t be passively received but must be worked on by the student: thought about, reasoned with and discussed with others.
- Fluency: quick and efficient recall of facts and procedures and the flexibility to move between different contexts and representations of mathematics.
- Variation: Variation is twofold. It is firstly about how the teacher represents the concept, often in more than one way, to draw attention to critical aspects, and to develop deep and holistic understanding. It is also about the sequencing of the episodes, activities and exercises used within a lesson and follow-up practice, paying attention to what stays the same and what changes, to connect the mathematics and draw attention to mathematical relationships and structure.
- Coherence: breaking lessons down into small connected steps that gradually unfold the concept, providing access for all children and leading to a generalisation of the concept and the ability to apply the concept to a range of contexts.
What are the Maths — No Problem! Five Core Competencies?
Now, let’s take a look at what MNP brings to the table. The Maths — No Problem! Five Core Competencies are the attributes you want to see in your learners as you teach for mastery.
Learners who show these core competencies are set up for maths success.
The Five Core Competencies are:
- Visualisation: ask learners to show ‘how they know’ at every stage of solving the problem.
- Generalisation: challenge learners to dig deeper by finding proof.
- Communication: encourage learners to answer in full sentences. Try asking learners to talk about the work they’re doing or use structured tasks centred around a class discussion.
- Number sense: a learner’s ability to work fluidly and flexibly with numbers.
- Metacognition: teach learners to think about how they are thinking. This helps learners solve multi-step tasks and promotes the ability to keep complex information in mind.
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.
Developing relational understanding relies on lessons that encourage learners to make connections and delve deeper. Teaching relational understanding is demanding, but it’s worth it! By building a connected body of knowledge and skills, your learners can become true mathematical thinkers.
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