Progression in Mathematics and Numeracy
“…progression must be embedded in learning and teaching and should form the basis of thinking in schools when designing and planning the school curriculum.”
Curriculum for Wales
Curriculum for Wales places the learner at the heart of curriculum design. This quote from the guidance emphasises the importance of understanding how learners progress and how this understanding will provide the starting point as schools design a curriculum for their learners. In order to support learners to progress along the learning continuum, practitioners need to develop a shared understanding of progression.
This workshop will start by addressing the four questions below, which come directly from the 'Supporting learner progression' guidance and which have been set by WG to guide you as you design your school curriculum:
What is progression?
What is a shared understanding of progression?
Why is a shared understanding of progression important?
How should schools and settings develop this shared understanding?
Curriculum for Wales requires an understanding of progression in learning before moving on to discussions on how the learning will be assessed.
This workshop aims to support you to develop a shared understanding of progression in Mathematics and Numeracy, offering activities for you to undertake with your teams in school.
"This is a big cultural shift from the way things have been done before. We know that assessment must build on progression: being clear on what needs to be learnt, why and how learners progress, determines how that should be assessed."
Jeremy Miles
What is progression?
Response and reflection
Share thoughts with your colleagues about what is meant by progression.
Here are a few thoughts which have previously been shared by practitioners:
moving forward
improving
developing
building on prior knowledge
transferring and applying.
Now that we have shared thoughts on what is meant by progression, it's important to agree on what we want our learners to make progress in.
If we want our learners to make progress in their geometrical reasoning, that would mean increasing what they know about shapes, and that is knowledge. If we want them to become aware of the relationships within the number system that would mean them showing progress in their understanding. If we want them to improve the strategies they use when they are problem solving using the four operations that would involve improving what they can do, which means making progress in skills, in their calculation skills. So, when planning for learning we will support our learners to increase their knowledge, to deepen understanding and to improve their skills, but we can also help them to develop positive attitudes and to establish sound values as they progress through their education.
Learners will therefore make progress in:
knowledge
understanding
skills
attitudes
values
"Progression in learning is a process of developing and improving in skills, knowledge and understanding over time. This focuses on understanding what it means to make progress in a given area or discipline as learners increase the depth, breadth and sophistication of their knowledge and understanding, skills and capacities, and attributes and dispositions."
Curriculum for Wales
What is a shared understanding of progression, and why is it important?
What?
Exploring, discussing and understanding together …
Expectations for how learners should progress
Coherent progression
How expectations compare with other schools/settings
Why?
To ensure…
Coherence and equity across the education system
Smooth transitions to support education and well-being
Sufficient pace and challenge
Having a shared understanding means having conversations in schools, clusters and beyond to ensure that we share the same high expectations of our learners so that they experience a smooth seamless learning journey from 3 to 16.
These conversations are of the utmost importance if we are to offer an equitable education at a pace and level of challenge which responds to the needs of all our learners.
How do we develop a shared understanding of progression?
The guidance states that ongoing professional dialogue around progression should happen within schools, clusters and with other schools or settings beyond the cluster. This will provide opportunities for practitioners to reflect on their understanding, compare their thinking and understand different approaches and practice.
In order to develop a shared understanding of progression, we start with the Principles of Progression. In Mathematics and Numeracy these are the five interdependent proficiencies and they are mandatory. They are:
Conceptual understanding
Communication using symbols
Fluency
Logical reasoning
Strategic competence
To ensure progress in any mathematics and numeracy learning, the proficiencies should be developed and connected in time and should also develop as learners progress in their education.
There is no hierarchy, they are each of equal importance and interdependent – one proficiency is closely linked to, and relies on, the development of the other – as shown in this image.
Numeracy is the capacity, confidence and disposition to use mathematics in daily life and so, progression in numeracy will require learners to apply and connect these proficiencies in a range of real-life contexts, across the curriculum.
Click on the dropdown bars below to see the narrative for each principle in Mathematics and Numeracy.
Conceptual understanding
Mathematical concepts and ideas should be built on, deepened and connected as learners experience increasingly complex mathematical ideas. Learners demonstrate conceptual understanding through being able to explain and express concepts, find examples (or non-examples) and by being able to represent a concept in different ways, flowing between different representations including verbal, concrete, visual, digital and abstract.
An increasing breadth of knowledge is achieved through the learners being introduced to new mathematical concepts. Depth of knowledge is achieved through learners being able to represent, connect and apply a concept in different ways and in different situations. The concepts that learners are introduced to will become increasingly complex, and understanding the way in which concepts connect will contribute to a growing understanding of the ideas within this Area. An understanding of how mathematical concepts underpin learning help learners make connections and transfer learning into new contexts.
Communication using symbols
Learners should understand that the symbols they are using are abstract representations and should develop greater flexibility with the application and manipulation of an increasing range of symbols, understanding the conventions of the symbols they are using.
The introduction and application of a new concept will involve developing an understanding of how symbols or expressions are abstract representations that succinctly describe a range of situations, thus contributing to a growing understanding of the nature of mathematics. The introduction of new symbols will add to the breadth of knowledge and the communication with symbols will contribute to refinement and growing sophistication in the use and application of skills
Fluency
As learners experience, understand and effectively apply increasingly complex concepts and relationships, fluency in remembering facts, relationships and techniques should grow, meaning that facts, relationships and techniques learned previously should become firmly established, memorable and usable.
Development of fluency and accuracy reflects the refinement and a growing sophistication in the use and application of skills.
Logical reasoning
As learners experience increasingly complex concepts, they should also develop an understanding of the relationships between and within these concepts. They should apply logical reasoning about these relationships and be able to justify and prove them. Justifications and proof should become increasingly abstract, moving from verbal explanations, visual or concrete representations to abstract representations involving symbols and conventions.
Refinement and growing sophistication in the use and application of skills will be demonstrated through the application of increasingly sophisticated logical reasoning. The development of an understanding of relationships between mathematical concepts and the development of justifications and proofs, leads to a growing understanding of the nature of mathematics and helps learners make connections and transfer learning into new contexts. The development of justifications and proof help support the increasing effectiveness of learners.
Strategic competence
Learners should become increasingly independent in recognising and applying the underlying mathematical structures and ideas within a problem, in order to develop strategies to be able to solve them.
Recognising mathematical structure within a problem and formulating problems mathematically in order to be able to solve them relies on an understanding of the ideas and disciplines within areas of learning and experience alongside a depth of knowledge. It also supports making connections and transferring learning into new contexts and developing increasing effectiveness as a learner. The recognition of the power of mathematics in enabling the representation of situations should lead to a growing appreciation of the usefulness of mathematics.
Response and reflection
Read and discuss the narrative for each principle of progression. Summarise each paragraph into bullet points, asking the question 'What do learners need to do to make progress within this principle?'
e.g. Conceptual understanding
increasing depth of understanding by explaining and expressing concepts
The process of undertaking this activity together will enable you to gain a shared understanding of what learners need to do to make progress.
Click on the image to see an example.
It can be refined and improved each time the discussions are revisited.
Response and reflection
This second activity may support you to develop a deeper understanding of each of the proficiencies.
On the google slide, you will see sentences on yellow post-its. Discuss the learning in each of the sentences and match it to one of the proficiencies.
Now that you have completed the activity, you may wish to see how the sentences are matched to each of the proficiencies.
These have been taken from the revised numeracy framework, which as you will know by now, is no longer statutory but it is a useful supplementary resource to assist with progression. It has been revised to support progression in Curriculum for Wales.
In the framework, you may have noticed that, for some of the threads of learning in the proficiencies, the statements are the same for each progression step, this is because the progression will come from the sophistication of the concept being learned. In other cases, progression in each proficiency builds as learners progress along the learning continuum.
What is the role of the Statements of what matters, the principles of progression and the descriptions of learning?
"The statements of what matters, principles of progression and the descriptions of learning articulate the essence of what should underpin learning and provide the same high expectations for all learners."
Curriculum for Wales
All three underpin learning and provide the same high expectations for all learners across Wales.
When designing your curriculum, they should be used in the following order, as they appear in the guidance:
statements of what matters (mandatory)
principles of progression (mandatory)
descriptions of learning (not mandatory)
When we decide what our learners need to learn, we go to the statements of what matters. They are mandatory and encapsulate the learning required to realise the four purposes.
When we want to understand progression in this learning , we go to the principles of progression.
The descriptions of learning serve as signposts to show how learners should progress in different threads of learning within each statements of what matters.
Selecting learning from the statements of what matters
In our last workshop, we suggested that you undertake an activity to select learning from the statements of what matters. The example below shows how one group of practitioners selected the learning from the statement Geometry focuses on relationships involving shape, space and position, and measurement focuses on quantifying phenomena in the physical world.
They have placed the selected learning on Post-its. The learning on the green Post-its refer to the four main ideas they want their learners to learn.
They went on to unpick the learning required to enable progress within each of the green Post- its.
This is the selected learning for 'How can I measure (estimate, measure, record and communicate) anything to compare and make decisions?'
The Post-its show the learning that will need to happen to enable progress within measuring:
In pink - the quantities being measured.
In blue - the aspects of learning for measuring position
In yellow - the threads of learning which run through all aspects of measuring, for example estimating and using approximations.
In orange - the higher order concepts which require multiple measures so a learner will progress in these when they have developed a deep understanding through all the proficiencies to be able to solve problems with measurements of any size.
The next activity will focus on learning to measure with length.
Response and reflection
In this activity, consider how learners will make progress when learning to compare and make decisions with 'length'. Discuss what progress in each of the proficiencies will look like from early to later learning.
On the slides, you will find suggestions for each of the proficiencies which you may find useful as a starting point as you discuss what, generally, does learning to measure with length look like for learners in the foundation phase, end of primary, and as they continue their journey to secondary school?
You may find these examples helpful in your discussions.
How the descriptions of learning help us to understand progression in learning
Although the descriptions of learning are not mandatory, they are there to support us to plan for progression in different threads of learning within each statement of what matters. Below you can see the descriptions for progression in measuring length. In blue are the descriptions for the statement of what matters for geometry and measures and in green are the descriptions from number. We can see that learning to measure is underpinned by learning in number. We can see that on this continuum - which encompasses 12 years of learning – learners progress from measuring the length of the potato grown in the school garden to the distance from here to the sun.
We must remember that the descriptions of learning should not be used as a starting point for curriculum design, neither should they be treated as boxes to tick. They are rather points of reference or signposts to help us as practitioners support our learners to move along the mathematics and numeracy continuum.
Another resource we have available as a reference point in Mathematics and Numeracy is the revised Numeracy Framework. Even though it is no longer statutory, it is a useful supplementary resource to assist with progression.
These activities have been designed to show how the principles of progression guide schools to design a curriculum which allows learners to progress in the learning required in the statements of what matters, the learning which enables learners to realise the four purposes.
What are the next steps for you and your teams?
As the importance of a shared understanding of progression to ensure coherence and continuity for your learners has been emphasised above, we suggest you undertake the following steps:
Arrange to meet with colleagues in your school, cluster and beyond to develop a shared understanding of progression
Create together a bullet-pointed summary of the principles of progression for your use when planning for learning
Select a thread of learning from the Statements of what matters and discuss how progress could look from early to later learning along the continuum
Continue these discussions until you have a shared understanding of progress in learning within each of the Statements of what matters.
Here's a video that consolidates this page that you may want to use for training purposes.
