Our pedagogy resources permitted Key Stage two children to be active learners, deepen their understanding of maths and confidently use appropriate vocabulary in context.
In this blog, Doug Harmer, with input from Samantha Hodges and Amy Bond, will explore specific ‘well known’ principles in the teaching and learning of mathematics through the eyes of both teachers and learners. Within that scenario, a recipe for success is clearly formulated through a delightfully simple menu of applying theory into practice that is analysed through the rationale of its intent, implementation, and subsequent impact.
To heighten the taste buds from the outset, an aroma of mathematics education theory needs to be concocted through the mixture of familiar ‘household’ ingredients. The Head Chef in this mathematical kitchen, who, for the purposes here, is responsible for the most significant input into the menu, is Zoltan Dienes (1916 to 2014). He is well known and adored for his creation of the widely used ‘Dienes Apparatus’, otherwise known as base-10 equipment.
This concrete material facilitates in-depth understanding of place value, linking representations, and provides opportunities to reason and expand mathematical language. But the man was not merely a manufacturer, far from it; he understood how learning in maths could be effectively initiated, devised a set of principles to enable that end; and stuck to them.
At the top of the ‘carte du jour’ is his Dynamic Principle that establishes a general framework within which the learning of mathematical concepts can occur. Thomas Post (1988) explains that this,
‘…suggests that true understanding of a new concept is an evolutionary process involving the learner in three temporally ordered stages. The first stage is the preliminary or play stage. The learner here experiences the concept in a relatively unstructured but not random manner. For example, when children are exposed to a new type of manipulative material, they characteristically ‘play’ with their newfound ‘toy’.
Dienes suggests that such informal activity is a natural and important part of the learning process and should therefore be provided by the classroom teacher.
Following the informal exposure afforded by the play stage, more structured activities are then appropriate. This is the second stage. It is here that the child is given experiences that are structurally similar (isomorphic) to the concepts to be learned.
The third stage is characterized by the emergence of the mathematical concept with ample provision for reapplication to the real world’.
The next culinary delight in this pedagogical menu is provided by The Characteristics of Effective Learning in the EYFS framework entitled ‘Development Matters’. This went ‘through the mixer’ in July 2021 and became the Characteristics of Effective Teaching and Learning. The general (adapted) principles were retained however and related to:
Playing and exploring – engagement
- Investigating and experiencing things
- Playing with what they know
- Being willing to ‘have a go’
Active learning – motivation
- Being involved and concentrating
- Keeping on trying
- Enjoying achieving what they set out to do
Creating and thinking critically – thinking
- Having and developing their own ideas
- Making links between ideas
- Developing strategies for doing things
Downing (2021) highlights that:
‘The purpose of the ‘Characteristics of Effective Teaching and Learning’ (CETL) section of Early Years Foundation Stage (EYFS) guidance is to signpost teachers towards prioritising not just what students learn, but how they learn.’
Effective learning is, of course, an ongoing life skill. Bandura (1977) links this to ‘self-efficacy’ (a person’s belief in their ability to succeed in a particular situation) and describes those who view challenging problems as tasks to be mastered, develop deeper interest in the activities in which they participate, form a stronger sense of commitment to their interests and activities, and recover quickly from setbacks and disappointments (as signified in the CETL).
Bandura, (1992) also explains that ‘beliefs and attitudes supporting self-efficacy form in early childhood, while the growth of self-efficacy continues to evolve throughout life as people acquire new skills, experiences, and understanding’.
This ongoing life skill can be nurtured in the teaching of maths from the outset and a key ingredient in enabling this scenario, within the evolving recipe described here, is provided by Master Chef Jerome Bruner, of ‘Concrete – Pictorial – Abstract’ (CPA) approach fame.
He asserted that children show their mathematical understanding through the ‘cocktail’ of making connections via multi-representation, namely:
His rationale centred on the idea of ‘blending’ where: ‘It is not the type of representation that is indicative of the child’s understanding, but the ability to translate between these models.’
This translation needs to include Enactive representation (action-based) - Iconic representation (image-based) - Symbolic representation (language-based) and includes children’s abilities to identify which approach is the most effective in different situations, combine different concepts to solve problems and be able to apply knowledge in real life situations. (Bruner, 1966)
All these ingredients need to be carefully bound together and are held securely in place using questioning scaffolds, prompts and clues. In the ESSENTIALmaths materials produced by the Primary Maths Team at Herts for Learning, this is achieved through a consistent mix of the (non-exhaustive) following examples:
I shared this menu, with its recipe for success, with two colleagues, Samantha Hodges and Amy Bond, from Breachwood Green JMI School, who are both Leading Teachers in the NCETM Mastery Readiness Programme, led in Hertfordshire by HfL in conjunction with Matrix Maths Hub.
They both recognised the potential impact and were excited about initially trialling the processes in their own classrooms:
‘After initial discussions with Doug regarding the implementation of maths mastery within the school and carrying out pupil voice, it was clear that the ‘Characteristics of Effective Teaching and Learning’ were not embedded throughout our mathematical curriculum. Doug introduced us to the mastery approach using the techniques of exploring, active learning and questioning across topics.
Beginning topics with exploring meant that the children initiated the learning and it allowed teachers to assess the secure aspects within the topic and identify the gaps that needed to be addressed. Using the concrete resources permitted the children to be active learners, deepen their understanding and confidently use the appropriate vocabulary in context. The flexibility within exploring gave children the opportunity to use their critical thinking independently without being scaffolded by the teacher, leading to us understanding the way in which they processed the strategies being taught. Following on from this, it allowed us to plan around their misconceptions and gave a greater understanding of the children's areas of strength and areas for development to plan towards.
Originally, we had concerns over more planning and having to reshape the way we teach and deliver lessons, however, embedding this approach has made us aware that it was always in the planning and now we prioritise the mastery approach across our maths lessons. As well as improving our own pedagogy, it opened our eyes to the mathematical understanding of our whole class – including children that had previously missed the opportunity to show their full potential in the subject.
It was clear that a strand of mathematics that was not as secure within our school was the children’s ability to draw conclusions from their work and justify their answers. We have found that since using the mastery approach of questioning, this has developed the way the children in Key Stage Two draw conclusions from their own learning.
Even simple questions such as, ‘What do you know?’ sparked a whole new learning opportunity.
The other question that we have found that allowed the children to lead their learning and draw their own conclusions was, ‘How else could you show that?’ This gave the children the opportunity to think outside the box and explore the learning in their own way rather than being led down a particular path by the teacher. The children found that there could be fifteen different ways represented and discussed in the classroom rather than just one way as a whole class. This lent itself to in-depth discussions with the use of the correct vocabulary for children to justify their answers.
Before implementing the mastery approach, links between concepts across the maths curriculum were not embedded. Now, lessons are more flexible and not as scripted to allow the learning to follow the children’s needs and curiosity. Sometimes, it means the lesson does not follow the structure you would expect, however, the children’s understanding at the end and the links that they can make between topics is stronger. This has shown us that for the approach to be successful, the subject knowledge of the staff must be secure as who knows where the lesson might end up and what topics you might cover unexpectedly!
A Year Three lesson proved how inclusive the mastery approach is for all children. The learning objective of the lesson was for the children to group regular and irregular polygons.
In this lesson, the teacher became an observer and a guide to the children’s learning by offering questioning opportunities rather than directing and deciding on their learning path.
The lesson began with the children being given a set of shapes and told to independently put them into two groups. Children who teachers would usually class as previously higher attaining were the ones who struggled with the lack of input as it was more about them taking control. At first, the children grouped these into warm and cold colours and although this was not expected, they had used classification and showed that they understood different groups. This was then extended into asking the children to group them using the properties of shapes, such as the number of sides. Without understanding the meaning of the words ‘regular’ and ‘irregular’ explicitly, the children independently managed to group the shapes correctly and by the end of the lesson, they could articulate why each shape was in the group without the teacher directly explaining. This allowed for the children to have a much deeper understanding of the knowledge as well as a sense of pride and accomplishment from being given the opportunity to form conclusions from the concrete materials.’
Samantha and Amy’s delectable outcomes were shared during a whole school INSET session that linked the menu, with its recipe for success, to the term ‘Mastery’. This was jointly delivered by the Leading Teachers and myself to a variety of staff from the school. To enable them to gain both an overview and in-depth understanding, the session began by asking the staff to solve a multi-step problem.
This was carried out with engagement and the desire to be correct by all staff which was then shared collaboratively. I then suggested that this delightful scenario had occurred because they all had both the skills and the inclination to succeed and were in a situation that allowed them the time to explore with the support of their colleagues, which was then linked explicitly to the overarching rationale of the ‘Dynamic Principle’.
The skills needed were also analysed with the outcome that as adults, these had been acquired over time with constant revisiting in a wide variety of scenarios and the necessary practice to allow them to be developed deeply. This was then linked proportionally to the idea of mastery with a focus on the small steps, and further exploration needed for children to acquire the skills.
The staff were given a wide range of concrete materials and paper and coloured pens and asked to use multi-representation to prove that their answers were correct.
This was also carried out with engagement and enthusiasm and shared collaboratively.
The use of the CPA approach by the staff was analysed and explored.
As seen above, the correlation of the beadstring and the tens frames (scaffolding) and the sweets (concrete manipulatives) demonstrated clear understanding which was also connected through pictorial representations and in the abstract. I made the point that the use of the associated mathematical language was prevalent as the staff carried out the activity.
This included statements such as, ‘It must be because…’ and ‘It cannot be because…’. These were linked to the necessary skills that allowed that language to be utilised. The staff agreed that these skills had been developed over time and were not merely taught ‘in the moment’.
If the activity was carried out by children, then they would need to use and apply their previous learning. This was then expanded to reveal the multitude of concepts involved in solving the ‘sweet’ problem, including:
- more than
- number magnitude
- odd and even
- ten and some more
- teens number names
- numbers to twenty
- division and sharing and grouping
- factors and multiples
- times tables
- balance and equivalence
- potentially unfamiliar language, including: ‘No more than’ in context, ‘left over’, ‘could’ and ‘more than one answer’
This was then linked to the children mastering deep understanding of concepts. If the ‘menu’ and the associated effective pedagogy is utilised, then children will be able to make networks of connections that can be used and applied across different domains.
Within number and calculation, for example, relationships can be recognised between odd and even numbers, multiples and factors, times tables and division and ‘left over’ can be linked to remainders where, in this context, it does not need to be rounded up or down.
It may also be the case, as another example, that children instantly recognise that subitising is more efficient than counting as an efficient starting strategy.
The clear points made here were that the children should not be given the initial question regarding the sweets in its entirety if they do not have deep understanding of the key concepts involved that have been developed over time. These key concepts need to be taught in a way that allows the children to master not only their deep understanding but also to make the connections to other ideas and concepts. The teaching approaches discussed and analysed here are particularly useful when children are encountering new concepts and they may not be appropriate for all lessons, where, of course, we want the children to have opportunities to revisit, rehearse and practice learning, although they can still carry that out through reasoning and solving problems.
It may be the case that a previously taught concept needs to be revisited and explored further in another context. I shared an example of where I had worked with a Year Two teacher in another school who had previously taught ‘more than’ and ‘less than’ but when the children were asked to solve a similar sweet problem, they had a clear misunderstanding of ‘no more than’. This was then explicitly taught through the exploratory approach in a focused session that was then augmented in separate fluency sessions. The teacher and I created our own slides and the teacher used them as part of their daily 10-15 minute fluency sessions that focus on embedding previously taught skills:
This was instigated using the displayed mathematical language alongside sentence scaffolds where the children were asked to consider the starting points of:
- I know that …
- I think that …
- I notice that …
- If … then …
- It reminds me of …
Fluency and reasoning mathematically were then explored and analysed with regard to the aims of the National Curriculum for Mathematics (2014) where children need to:
These were also linked and associated to the overarching rationale of the menu already discussed here and were then aligned with this statement from the aims of the NC:
This interconnectivity, that can be facilitated through the contents of the overall menu described here, was explicitly linked to the lessons shared by Samantha and Amy, and then analysed with regard to the impact on both teacher and learner. This was formulated into a Venn Diagram that produced the following outcomes, impact, and guidelines.
- exploration provides consistent ongoing formative assessment for learning
- the utilisation of small steps provides deep understanding that is clear through multi-representation.
- effective pedagogy includes differentiation by scaffolding support using the enactive, iconic and symbolic approach linked to explicit mathematical language
- the use of existing planning, but in a far less complex way which is not prescriptive, meets the needs of all children
- the approach initially appears to be ‘risky’ as the route that the lesson is taking is not specifically prescribed but it provides the children with the confidence needed to gain understanding
- learning can be fun with constant involvement for all from the outset during both the initial and subsequent lessons
- the use of the CPA approach is an effective learning device for all children as it includes the facilitation of confirmation and proof
- active learning allows for engagement for all children and provides ownership of learning that is personalised.
- working in this way allows the children to have confidence in understanding key concepts and their efficacy is both attained and realised which also enhances their mindsets
The overarching connection (as can be seen in the Venn Diagram) between teacher and learner is ‘confidence’. This was discussed at length, especially regarding the teacher attaining the confidence to change their pedagogy and thus providing the children with not only the confidence to learn more effectively but to be able to articulate their clear understanding in a variety of ways.
As the title of this blog suggests, the development and outcomes of this whole procedure have been ‘Dynamically Delicious’ and to that end I have created a menu as a framework:
Samantha and Amy sum this up by saying:
‘We are still at the beginning of our mastery journey as a school; however, we have already seen the positive impact this pedagogy has had on all children’s mindset, quality of work, enjoyment, and engagement with maths. We look forward to seeing where it takes us next.’
Professional development opportunities:
The Concrete, Pictorial, Abstract (CPA) approach is an essential pedagogical tool in developing deep and sustained understanding of maths.
These one-day, face to face, hands-on courses will focus on how concrete resources can support developing deep conceptual understanding of key areas of the mathematics curriculum.
Develop practice further in:
- purposeful use of concrete materials
- exploring maths in a hands-on way in class
- exposing and deepening learning for all pupils using manipulatives
- teacher modelling and pupil exploration
- developing pupil talk and recording
- use of a working wall
November 2022 CPA approach to secure mathematics learning:
Bruner, J. S. (1966). Toward a Theory of Instruction. Cambridge: Harvard University Press.
Post, T. (1988). Some notes on the nature of mathematics learning. In T. Post (Ed.), Teaching
Mathematics in Grades K-8: Research Based Methods (pp. 1-19). Boston: Allyn & Bacon.
Bruner - Learning Theory in Education. McLeod, S (2019) [accessed 10/5/22]
Development Matters - Non-statutory curriculum guidance for the early years foundation stage DfE (First published September 2020 - Revised July 2021) [accessed 10/5/22]
Bandura, A. (1992). Exercise of personal agency through the self-efficacy mechanism. In R. Schwarzer (Ed.), Self-efficacy: Thought control of action (pp. 3–38). Hemisphere Publishing Corp.