DR TAREK EL BABA, Mathematics Educator, Dar Al Fikr Schools, Saudi Arabia; Researcher, Makassed University of Beirut, Lebanon
Introduction
During my career as a maths teacher, I have repeatedly encountered learners who believed that maths was something they simply ‘could not do’. In many cases, they were not lacking effort or ability, but had developed gaps in conceptual understanding that made increasingly abstract maths seem inaccessible. Over time, I became aware that mathematical achievement is closely connected to mathematical identity. When learners experience repeated difficulty, they begin to see themselves as unsuccessful mathematicians, which can affect their willingness to participate, take risks and engage with challenging problems.
Through my work in international school contexts, I have supported learners from diverse backgrounds and with different learning needs, observing that some learners require a different pathway into mathematical thinking. They do not necessarily need maths to be simplified; they need it to become visible.
One group who often experience these barriers are those with dyscalculia, a learning difficulty associated with persistent challenges in numerical processing, understanding quantities, recognising mathematical relationships and developing efficient calculation strategies (Butterworth et al., 2011). Research suggests that developmental dyscalculia affects three to six per cent of learners, although identification rates vary depending on assessment methods and educational contexts (Landerl et al., 2004).
Understandings of dyscalculia have moved beyond earlier deficit-based descriptions. Mathematical difficulties are not a reflection of intelligence, motivation or effort. Instead, learners may require carefully designed instruction that provides alternative routes towards conceptual understanding. In mainstream classrooms, learners with dyscalculia often experience difficulty when instruction moves rapidly from concrete experiences to abstract symbols without sufficient opportunities to connect these representations.
The Concrete–Pictorial–Abstract (CPA) approach provides a framework for addressing this challenge. Based on Bruner’s (1966) theory of cognitive development, CPA describes a progression from physical manipulation of objects to visual representation and, finally, abstract mathematical notation. It is sometimes called Concrete–Representational–Abstract, particularly in North American contexts. While terminology differs, the principle is the same: learners develop mathematical understanding by connecting experiences, images and symbols.
This case study documents a 10-week CPA-based intervention that explored:
- how CPA-based teaching can support conceptual mathematical development for a learner with dyscalculia
- how reflective practices, particularly journalling, can support metacognitive development and confidence.
Rather than presenting CPA as a universal solution, this article offers a practitioner reflection on how one learner’s experience changed when maths was approached via structured representation, dialogue and reflection.
Practitioner context and learner profile
The intervention took place in an international school in Saudi Arabia that followed a US curriculum. Working in an internationally diverse educational environment has strengthened my belief that inclusive maths teaching requires flexibility and responsiveness.
Student A was a middle school (aged 11–14) student who showed difficulties consistent with dyscalculia. Classroom observations and assessment information indicated challenges in several areas:
- understanding place-value relationships
- recalling multiplication and division facts efficiently
- interpreting fractions conceptually
- transferring knowledge to unfamiliar problems
- explaining mathematical reasoning.
At the start, Student A approached maths with uncertainty, sometimes completing familiar procedures but struggling to explain why methods worked. For example, when working with fractions, they could follow a demonstrated algorithm but found it hard to represent fractions visually or explain the relationship between parts and wholes.
My initial response could have been to provide extra practice exercises. However, experience had shown me that repetition alone does not always address conceptual gaps. The challenge was that mathematical ideas were not yet connected in a meaningful way. The intervention was thus designed around the question: How can we build understanding from the foundations up?
The CPA intervention
The intervention was delivered in individual sessions over 10 weeks. Each followed the CPA progression, beginning with concrete experiences, moving towards representations, and finally connecting understanding to abstract mathematical notation.
The intervention focused on three key areas:
- number sense and place value
- fractions and proportional reasoning
- calculation strategies and mathematical problem-solving.
Concrete phase: Creating mathematical meaning
The first stage focused on developing understanding via physical experiences. Manipulatives were selected according to the learner’s specific difficulties, rather than used as additional resources.
For place value, base-10 blocks were used to allow Student A to physically explore the relationship between units, 10s and 100s. Instead of memorising the fact that 10 units equal one 10, they could exchange 10 individual blocks for a larger representation and experience the relationship directly.
For fractions, fraction tiles and visual models helped Student A to understand that fractions represent relationships between quantities. This was particularly important because they had previously seen fractions as procedures to memorise.
Where specialist resources were unavailable, simple alternatives were used, including paper fraction models, counters, drawings and everyday classroom materials.
Representational phase: Connecting thinking and visual models
Once Student A developed understanding through concrete experiences, the intervention moved towards visual representations. The learner used number lines, bar models, diagrams and drawings to represent mathematical ideas.
During this phase, questioning became central. Rather than asking only whether an answer was correct, discussions focused on mathematical reasoning:
- How does your model show your thinking?
- What does this part represent?
- How do you know that your answer makes sense?
This encouraged the learner to move from performing to explaining maths.
Abstract phase: Moving towards mathematical independence
The final stage involved connecting the learner’s developed understanding to abstract notation. A key principle was that symbols should represent ideas already understood rather than replace understanding.
For example, when introducing fraction operations, Student A was not immediately given procedural rules. Instead, they first explored fractions using models, drawings and explanations. Once relationships between quantities became clearer, formal mathematical notation was introduced as a more efficient way to record thinking.
This approach changed the learner’s relationship with mathematical symbols. Previously, equations appeared as unfamiliar instructions to follow. Through CPA, symbols became meaningful representations of ideas that the learner already understood.
Evidence of learning: Mathematical development and confidence
As this was a single-learner case study, the findings should not be interpreted as evidence that CPA alone can overcome all difficulties associated with dyscalculia.
Progress was monitored through diagnostic activities before and after the intervention, classroom observations, mathematical discussions and learner reflections.
While mathematical progress was important, the most meaningful change was the learner’s developing confidence. Errors were once interpreted as evidence of inability rather than opportunities for learning. Over time, the learner became more willing to explore, explain and revise mathematical thinking. This reflects an important consideration: achievement and confidence are interconnected. Learners who believe that they can improve are more likely to engage with challenging tasks and persist through difficulty.
The Mathematics journal: Making learning visible
One of the intervention’s strongest aspects was a maths journal. Initially, this was a simple reflection tool, but it became an important evidence source of the learner’s developing mathematical identity.
At the beginning of the intervention, Student A’s reflections were brief and focused mainly on task completion:
I finished the questions today.
However, later reflections demonstrated deeper awareness:
The blocks helped me understand because I could see what was happening.
Maths was no longer viewed as a fixed ability that the learner lacked.
The journal has potential for mainstream classrooms across subjects and phases because it is accessible, inexpensive and adaptable. Teachers can incorporate short reflections through questions such as:
- What strategy helped you today?
- What representation helped you to understand?
- What mistake helped you to learn?
- How would you explain this idea to another student?
These prompts encourage metacognition and allow teachers to understand learners’ thinking processes.
Implications for maths teachers
Although the intervention was conducted individually, these principles have practical relevance for mainstream maths classrooms, and teachers do not need to create separate programmes for every learner experiencing difficulty.
1. Build understanding before introducing procedures
Allow learners to experience mathematical ideas before expecting them to manipulate symbols. For example, before introducing fraction algorithms, learners can explore fractions through models, diagrams and real-life contexts. This approach benefits learners with dyscalculia but also supports all learners by strengthening conceptual understanding.
2. Make manipulatives accessible
The purpose of manipulatives is to help learners to connect physical experiences with mathematical ideas. Effective implementation can begin with simple materials:
- paper fraction strips
- classroom objects
- counters
- drawings
- number lines
- digital manipulatives.
3. Adapt CPA for whole-class teaching
While this intervention was delivered via individual support, CPA principles can be adapted for classrooms. Teachers can:
- begin lessons with concrete exploration
- encourage multiple representations
- provide opportunities for mathematical discussion
- allow learners to explain strategies before recording answers.
Small-group instruction can also provide targeted support for learners who require additional scaffolding.
4. Develop mathematical communication
Learners may know procedures but struggle to explain reasoning. Questioning can support deeper understanding:
- How do you know?
- Can you show another way?
- What changed when we represented it differently?
5. Recognise confidence as part of progress
A learner who feels incapable may avoid opportunities to develop understanding. Teachers can support confidence by:
- valuing strategies rather than only answers
- celebrating improvement
- normalising mistakes
- providing opportunities for success
- encouraging reflection.
Limitations and reflection
This case study has several limitations. First, it focuses on one learner in a specific international school context, so its findings should not be generalised. Further research involving larger groups and different settings would provide more evidence on CPA implementation.
Second, it involved individual support sessions. Many classroom teachers work with large groups and limited resources. Future exploration should examine how CPA approaches can be embedded within whole-class instruction.
CPA provided a pathway from objects to representations to symbols. For teachers who are supporting learners with SEND, this represents the heart of inclusive maths education: creating environments where every learner has a pathway towards understanding, participation and confidence.
Read more from In Praxis: SEND and Inclusion
