An Independent Review: An Evidence Based Analysis of the ‘Foundations of Maths’

Overview Document
An Evidence-Based Analysis of the “Foundations of Maths by Maths Australia” Training Course: Alignment with Contemporary Pedagogical Research and Recommendations for Enhanced Practice

Written by Kathryn Wylie
BHSHM., BEd., GradDipPsySc., BPsychSc(Hons. H1)

I. Executive Summary

• Course Overview: The “Foundations of Maths by Maths Australia” online training course offers a coherent, multi-sensory, and systematic approach to teaching mathematics, emphasising a language-based philosophy and the iCRAVE methodology (Intuitive, Concrete, Representational, Abstract, Verbalisation, Explicit) with consistent use of manipulatives.
• Key Strengths: The course’s strengths include its systematic progression, cumulative review, and ongoing assessment, providing a structured pathway for conceptual understanding.
• Alignment with Research: It aligns significantly with evidence-based practices such as explicit and systematic instruction, the Concrete-Representational-Abstract (CRA) approach, the importance of foundational skills, and strategic use of manipulatives, as supported by Hattie’s meta-analyses and AERO/AAMT reviews.
• Areas for Enhancement: Opportunities exist for more explicit integration with multi-tiered systems of support (MTSS) and more formalised teacher professional learning pathways.
• Overall Recommendation: Leverage the course’s strengths, integrate it into broader systemic support, enhance professional development, rigorously evaluate its impact, and foster consistency in pedagogical guidance across Australian education systems.

II. Introduction: Context of Mathematics Education in Australia

• Current Challenges: Australia faces a significant challenge in mathematics proficiency, with one in three students not achieving proficiency, and lower international rankings compared to countries like England and Singapore.   
• Cumulative Nature of Maths: Learning gaps emerge early and compound over time, making strong primary foundations essential. Many students enter high school lacking foundational skills.   
• Teacher Preparedness: Many primary teachers lack the necessary mathematics knowledge, confidence, and training to teach effectively.   
• Inconsistent Practices: The educational landscape is marked by a scarcity of quality-assured materials and inconsistencies in teaching practices.   
• Course’s Role: The “Foundations of Maths” course directly addresses these challenges by focussing on establishing early, systematic foundations to interrupt the cycle of compounding learning gaps.   

III. Pedagogical Foundations of “Foundations of Maths by Maths Australia”

• Mathematics as a Language:
  • Core philosophy: Mathematics is a language, and teaching it should mirror natural language acquisition, starting with concrete, multi-sensory experiences before abstract symbols.   
  • Implication: This approach aims to reduce confusion, anxiety, and lack of confidence caused by premature exposure to abstract concepts.   
• Multi-Sensory Learning and Developmental Stages:
  • Piaget’s Influence: Acknowledges that many students learn through concrete and tactile interactions until approximately age 12, advocating for continued use of manipulatives beyond early years to deepen understanding.   
  • Montessori & Mortensen: Inspired by Maria Montessori’s “sensorial foundations” and Gerry Mortensen’s development of blocks emphasising “counting and building rectangles” with “consistency of colour”.   
  • Cognitive Load Theory: The consistent colour and single tool approach minimises extraneous cognitive load, allowing students to focus on mathematical concepts.   
• Addressing Learning Differences:
  • Universal Benefit: The multi-sensory, concrete-first approach is presented as universally beneficial for all students, neurodiverse (e.g., dyscalculia, dyslexia) or neurotypical, by reducing cognitive overload and anxiety often associated with abstract teaching.   
  • Inclusive Practice: This approach aligns with AERO’s view of high-quality Tier 1 instruction as a general education initiative, proactively preventing learning gaps.   

IV. Core Methodologies and Instructional Design

• The CRA (Concrete-Representational-Abstract) Approach:
  • Concrete (Doing/Building): Students physically interact with 3D manipulatives (e.g., Integer Block Kit) to build concepts. Advocates consistent use of a single tool for a minimum of three years for whole numbers and an advanced kit for higher concepts.   
  • Representational (Pictorial/Drawing): Students draw their own accurate 2D representations of concrete constructions, serving as a essential bridge to abstract thought. This active drawing promotes deeper cognitive processing.   
  • Abstract (Symbolic): Students work with abstract symbols (numerals, operation signs), understanding their meaning is derived from prior concrete and representational experiences.   
• The iCRAVE Methodology: A comprehensive framework for every lesson.
  • I – Intuitive: Identify student’s current understanding and mastery level via placement tests to ensure appropriate starting point.   
  • C – Concrete: Hands-on interaction with manipulatives.   
  • R – Representational: Drawing accurate 2D representations.   
  • A – Abstract: Working with symbolic notation, connecting meaning to concrete experiences.   
  • V – Verbalisation: Students articulate understanding in their own words; clear, precise teacher verbal instruction is essential.   
  • E – Explicit: Clear, accurate, and well-prepared instruction by a knowledgeable teacher, setting students up for success. This aligns with Hattie’s high-impact factor of teacher clarity.   
• Role and Characteristics of Manipulatives:
  • Consistency: Consistent colour (e.g., green for 1, orange for 2) and design (interlocking blocks) minimises cognitive load and highlights interconnectedness.   
  • Accuracy & Reliability: “The blocks never lie,” providing immediate, accurate feedback, fostering confidence and self-efficacy. This aligns with Hattie’s research on effective feedback.   
  • Utility: Integer blocks for whole numbers, advanced kits for fractions, decimals, percentages, and algebra, using the same methodology consistently.   
• Systematic Instructional Sequence:
  • Sequential Progression: Counting (0-9) → Place Value → Addition → Subtraction → Multiplication → Division → Fractions → Decimals → Percentages → Algebra.   
  • Mastery Learning: Each concept builds logically on the previous, ensuring mastery before progression, counteracting compounding learning gaps (Matthew effect).   
• Emphasis on “Life Questions” and Problem-Solving:
  • Integrates real-world scenarios to make concepts relevant and promote transfer of knowledge.   
  • Students build problems with manipulatives, draw, write, and create their own “life questions” from abstract expressions. This aligns with Hattie’s “knowing-with” (transfer) and problem-solving teaching.

V. Alignment and Evaluation: “Foundations of Maths” in Light of Broader Educational Research

• Instructional Effectiveness:
  • Explicit and Systematic Instruction: Strong alignment with AERO and Hattie’s findings on explicit and systematic instruction for foundational skills and interventions. The course’s nuanced definition of “explicit” (teacher clarity, not rigid scripting) aligns with effective interpretations.
  • Manipulatives and CRA: Strongly supported by AERO and Hattie; CRA is beneficial for basic number facts, algebra, arithmetic, and overall maths skills. The consistent, single-tool approach optimises manipulative impact.
• Addressing Achievement Gaps and Student Needs:
  • Foundational Skills & Early Intervention: Course’s emphasis on early mastery aligns with Grattan Institute and AERO’s calls for strong primary foundations to prevent compounding gaps.   
  • Student Assessment & Data Utilisation: The “Intuitive” stage (placement tests) aligns with AERO’s recommendations for timely identification and CBM for progress monitoring. Hattie emphasises knowing prior knowledge and using assessment as feedback.   
  • Student Confidence & Maths Anxiety: The multi-sensory, concrete-first approach reduces anxiety and builds confidence, aligning with Hattie’s research on self-efficacy as a high-impact factor. Manipulatives provide self-correcting feedback, fostering autonomy.   
• Teacher Quality and Professional Learning:
  • Teacher Expertise: Course’s emphasis on knowledgeable, prepared teachers aligns with Hattie’s focus on teacher expertise and evaluative thinking.   
  • Professional Learning and Systemic Support: The course itself is PL. Broader research (Grattan, AERO, AAMT) highlights the need for quality, sustained PL, micro-credentials, and “Maths Hubs” to uplift teaching practice. The Australian Association of Mathematics Teachers (AAMT) also champions flexible pedagogy and building teacher expertise through ongoing professional learning and collegial collaboration. AERO specifically recommends consistent national guidance for Multi-Tiered Systems of Support (MTSS) implementation, drawing lessons from the US experience to avoid discrepancies and ensure system-wide scaling. The “Foundations of Maths” course’s systematic approach offers a structured framework that could be integrated into these broader professional learning initiatives, potentially serving as a core component of micro-credentials or as a model for consistent pedagogical guidance.   
  • Collaborative Professional Learning: While the “Foundations of Maths” course primarily focuses on individual teacher training, its systematic and explicit nature could facilitate collaborative professional learning. John Hattie emphasises that teachers working together to critique their impact leads to “collective teacher efficacy,” a powerful influence on student achievement. AERO also highlights the importance of collaborative professional learning, particularly in using data to individualise instruction. Integrating the “Foundations of Maths” methodology within professional learning communities could provide a common language and framework for teachers to discuss, analyse, and improve their collective impact on student learning.   
• Scalability and Implementation Challenges:
  • Implicit Scalability: The “Foundations of Maths” methodology demonstrates implicit scalability through its consistent use of a single manipulative tool and a systematic progression of instruction across all mathematical domains. The course’s design for “a teacher, whether it’s a parent, a tutor, or a school teacher” suggests its applicability in various teaching environments.   
  • Broader Challenges: Broader research identifies significant challenges to widespread implementation of effective pedagogical approaches. The Grattan Institute and AERO highlight concerns about inconsistent guidance across jurisdictions, a lack of quality-assured curriculum materials, and what Hattie refers to as the “implementation chasm” – the difficulty in translating research into consistent practice in schools. Hattie notes that many schools lacked a concept of “deep implementation,” often seeking quick fixes or superficial engagement with evidence-based programmes. A central barrier is the “pedagogy delusion,” a rejection of evidence in favour of fads and philosophical beliefs. AERO specifically recommends that Australia adopt a consistent national approach to MTSS to avoid the discrepancies observed in the US, emphasising clear and authoritative advice to schools. For the “Foundations of Maths” methodology to achieve broader systemic impact, it would need to navigate these implementation challenges, potentially through explicit integration with national policy frameworks, robust external evaluations, and targeted professional learning initiatives that address teacher readiness and foster deep implementation.

VI. Conclusions and Recommendations

The “Foundations of Maths by Maths Australia” training course presents a pedagogically sound and coherent approach to mathematics education, aligning significantly with contemporary evidence-based practices. Its core strengths lie in its foundational philosophy of teaching mathematics as a language, its comprehensive multi-sensory approach, the systematic application of the CRA/iCRAVE methodology, the consistent and optimised design of its manipulative tools, and its explicit, sequential instructional progression. These elements collectively foster deep conceptual understanding, build student confidence, and proactively address the compounding nature of learning gaps in mathematics. The course’s emphasis on “life questions” further enhances meaning-making and the transfer of mathematical knowledge to real-world contexts.

Despite its inherent strengths and alignment with research, opportunities exist for the “Foundations of Maths” methodology to achieve even greater impact through explicit integration with broader systemic educational initiatives.

Recommendations for “Foundations of Maths by Maths Australia” Course Developers:

1. Enhance Explicit Links to Multi-Tiered Systems of Support (MTSS): While the course’s pedagogical approach inherently supports Tier 1 instruction and intervention, explicitly framing its methodology within the MTSS framework would provide clearer guidance for schools seeking to implement tiered support. This could involve detailing how the iCRAVE methodology and systematic progression can serve as foundational Tier 1 practices and how its diagnostic elements (e.g., placement tests) integrate with CBM for Tier 2/3 identification and progress monitoring.
2. Provide Guidance on Data-Driven Decision-Making: Further elaborate on how teachers can systematically use data from placement tests and ongoing assessments to inform instructional adjustments and individualise learning pathways, beyond simply identifying a starting point. This could include guidance on interpreting student responses to inform specific pedagogical moves within the iCRAVE framework.
3. Explore Formal External Evaluation: Seek rigorous, independent external evaluations (e.g., randomised controlled trials or robust quasi-experimental designs) of the long-term impact and scalability of the methodology across diverse school settings in Australia. This would provide empirical evidence of its effectiveness at a broader scale, addressing existing research gaps in the Australian context.

Recommendations for Australian Educational Systems and Policymakers:

1. Adopt Consistent Pedagogical Guidance: Commission or endorse detailed, national guidance on effective mathematics teaching that aligns with evidence-based practices, including explicit and systematic instruction, the CRA approach, and the strategic use of manipulatives. This guidance should be consistent across jurisdictions and sectors to reduce teacher confusion and promote coherence.
2. Invest in Quality-Assured Curriculum Materials: Establish an independent quality assurance body to review and endorse high-quality, evidence-informed mathematics curriculum materials, potentially including resources like those developed by Maths Australia, to ensure schools have access to demonstrably effective tools.
3. Establish Robust Professional Learning Pathways: Develop and subsidise quality-assured micro-credentials and professional learning programmes focussed on primary mathematics pedagogy. These programmes should emphasise evidence-based practices, data literacy, and collaborative professional learning, potentially leveraging “Maths Hubs” to facilitate knowledge transfer and implementation fidelity.
4. Mandate Early Years Numeracy Screening: Implement a mandatory, research-validated early years numeracy screening tool to identify students at risk of falling behind, enabling timely and targeted interventions before learning gaps compound.
5. Implement Strategic, Data-Driven Assessment Systems: Utilise efficient and frequent curriculum-based measures (CBMs) for timely identification of student needs and ongoing progress monitoring. Ensure that assessment data is triangulated with teacher observations, student work, and student voice to inform individualised interventions and instructional adjustments, moving beyond mere ranking.
6. Strengthen Systemic Support and Technical Assistance: Establish national technical assistance centres, potentially through university partnerships, to provide ongoing support, resources, and professional learning for MTSS implementation and the broader adoption of evidence-based practices across the system.
7. Promote Collaboration and Shared Learning: Encourage and facilitate collaborative professional learning communities where teachers can critically examine their impact, share best practices, and collectively improve student outcomes, recognising that peer-to-peer learning is highly effective.
8. Re-evaluate and Abandon Ineffective Practices: Systematically review and phase out “fad” or unproven teaching methods and organisational structures (e.g., certain forms of grouping/tracking that consistently show low or negative impact) that do not demonstrably improve student achievement.

References

Australian Association of Mathematics Teachers Ltd. (2025). Pedagogy in Mathematics. Retrieved from https://aamt.edu.au/

Australian Education Research Organisation. (2023, May). Supporting students significantly behind in literacy and numeracy: A review of evidence-based approaches. Retrieved from https://edresearch.edu.au/

Hattie, J. (2023). Visible Learning: The Sequel: A synthesis of over 2,100 meta-analyses relating to achievement. Routledge. https://doi.org/10.4324/9781003380542

Hunter, J., Haywood, A., Parkinson, N., & Petrie, D. (2024, April). The Maths Guarantee: How to boost students’ learning in primary schools. Grattan Institute. Retrieved from https://grattan.edu.au/

Maths Australia. (n.d.). Multi-Sensory Maths Foundations Training. Retrieved from https://mathsaustralia.com.au/training/multisensory-maths-foundations/