Educational theory & research

Research underscores the benefits of spaced practice, in which study sessions are distributed over time to improve long-term retention. Studies by Emeny, Hartwig, and Rohrer (2021) found that spaced practice in mathematics improves test scores and reduces overconfidence, helping students build accurate and enduring mathematical knowledge. Kang (2016) highlights the efficiency of spaced repetition for enhancing learning retention, which has policy implications for instructional design. Dunlosky and Rawson (2015) further support spaced practice with practical recommendations for the classroom. The EMP’s Daily Review and Quick Teach (DRQT) sessions, in addition to the overall structure of the program, which strategically spaces and revisits core concepts over time, integrate these findings, thus promoting long-term retention.

Retrieval practice, or the act of recalling information from memory, has been shown to improve memory retention and reinforce learning pathways. Roediger and Karpicke (2006) describe the power of testing as an active recall strategy that strengthens memory and aids long-term learning in educational contexts. Additionally, Roediger and Butler (2011) emphasise the critical role of retrieval practice in enhancing long-term retention, showing that repeated recall strengthens the encoding of knowledge more effectively than passive review. Within EMP, retrieval practice is embedded across DRQT sessions and explicit lessons, allowing students repeated opportunities to recall and apply previously learned content, thereby solidifying their understanding over time.

Interleaving is the practice of mixing different problem types within a single practice session, which has been shown to promote the differentiation and understanding of similar concepts. Rohrer, Dedrick, and Agarwal (2017) demonstrated that interleaved practice in mathematics allows students to learn and contrast different types of problems more effectively, improving concept mastery. A systematic review by Firth, Rivers, and Boyle (2021) supports this, indicating that interleaving fosters concept learning by requiring students to discriminate among problem types. Further research by Rohrer, Dedrick, and Burgess (2014) suggests that interleaved mathematics practice benefits students beyond superficial problem similarities, fostering deeper conceptual understanding. EMP integrates interleaving both on a micro level, within DRQT segments that mix various mathematical topics, and on a macro level, interleaving content from multiple curriculum strands over time, promoting cross-topic comprehension and retention.

Cognitive Load Theory, developed by Sweller, emphasises the importance of instructional design that minimises unnecessary cognitive demands. Sweller, Ayres, and Kalyuga’s (2011) foundational work on CLT demonstrates that reducing extraneous cognitive load, such as through clear, straightforward problem setups, enhances learning by focusing cognitive resources on intrinsic learning tasks. Additionally, Lovell’s Cognitive Load Theory in Action (2020) elaborates on practical methods for reducing load in classroom materials, highlighting that removing redundancy, split-attention effects, and transient information improves learning outcomes. The EMP integrates these CLT strategies comprehensively across its materials. Given the influence of Oliver Lovell as one of the program’s authors, specific attention is paid to minimizing extraneous load, thereby allowing students to focus on fundamental mathematics skills and concepts without cognitive distractions.

By embedding these research-based principles into its structure, EMP provides an evidence-informed approach to mathematics instruction that fosters students’ ability to retain and apply key concepts effectively.

The Explicit Mathematics Program (EMP) is grounded in robust mathematics education research, adopting proven instructional methods to foster fluency, problem-solving skills, and conceptual understanding. Foundational sources include Teaching Primary Mathematics by George Booker (2020), Steven Norton (2023), Sarah Powell (2011), and Elementary and Middle School Mathematics: Teaching Developmentally by John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams (2020).

George Booker’s Teaching Primary Mathematics (2020) serves as a core foundation for EMP, particularly in developing number sense and structuring the gradual shift from concrete to abstract understanding. For example, Booker advocates for introducing comparison tasks using “concrete concrete” strategies, followed by “concrete number” and finally “number number” strategies, as seen on pages 134-135. Another instructional sequence from Booker used in the EMP involves teaching two-digit numbers in stages: starting with linguistically familiar groups such as the 40s, 60s, 70s, 80s, and 90s, before introducing more complex sequences like the 30s, 50s, 20s, and finally the teens. This sequence supports students in mastering place value and number sense gradually, reinforcing their understanding of base-10 structure in mathematics.

Steven Norton’s work emphasises using concrete tools alongside abstract representations, supporting the scaffolded development of conceptual understanding and connections between both concrete and pictorial representations, and their abstract counterparts. One example of how the EMP adopts his approach is by integrating bundling sticks and similar concrete tools when introducing algorithms. Norton’s work in this space is further built upon in the EMP through refined scripts, particularly influenced by Toni Hatten-Roberts’ work within Mastery Schools Australia.

Powell’s (2011) research on schema-based instruction for solving word problems is fundamental to the EMP’s problem-solving approach. Powell advocates for the use of schemas to organize and approach different types of word problems, typically through equations. The EMP adapts this strategy by incorporating schemas using number bonds and the bar model instead, offering students more visual and flexible tools for problem-solving whilst still basing problem identification on robust schema-based approaches, and avoiding unreliable word cuing methods. By presenting problems through number bonds and the bar model, students gain a clearer understanding of part-whole relationships and structure, which enhances their independence in approaching complex problems.

The developmental progression strategies outlined in Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, and Bay-Williams (2020) were particularly influential during initial scope and sequence work within the EMP.

Reports from the Centre for Independent Studies (CIS) and the Australian Education Research Organisation (AERO) reinforce EMP’s balanced focus on procedural fluency and conceptual understanding. CIS publications such as The Need for Speed: Why Fluency Counts for Maths Learning (CIS, 2023) underscore the importance of fluency, a principle that EMP incorporates through dedicated fluency-building exercises across lessons. AERO’s report Developing Maths Proficiency (AERO, 2024) also supports EMP’s dual emphasis, supporting the program’s efforts to help students achieve a balanced mastery of skills and concepts.

The Explicit Mathematics Program has undergone a rigorous and multi-layered quality assurance process to ensure accuracy, clarity, and efficacy. Each component of the program has been meticulously reviewed through at least five rounds of proofing, reflecting a strong commitment to providing a reliable, well-structured resource for mathematics instruction in primary schools.

Each instructional script within the EMP has been carefully reviewed by a team of four prominent experts in the field of mathematics education: David Morkunas, Oliver Lovell, Toni Hatten-Roberts, and Dr. Wendy Taylor. This expert team’s collaborative oversight has been pivotal in refining both the pedagogical accuracy and practical implementation of each lesson, ensuring that content is both rigorous and accessible for teachers and students alike. These experts bring a wealth of experience in educational best practices, which has directly informed the structure and instructional approach of the EMP.

In addition to expert reviews, the EMP has been proofread by two independent copy editors, one of whom has a strong background in educational publishing, with over 2 decades of experience at The Education Center (United States), and a history of writing and editing educational materials tailored for classroom use, supporting teachers with ready-to-use scripts and age-appropriate content. Their expertise has contributed to a polished and teacher-friendly final product, ensuring that EMP materials meet high standards of readability and accessibility.

The EMP has been rigorously trialled in diverse school settings across Australia, representing a variety of socioeconomic contexts. These trials have taken place in schools from low and high SES communities, spanning Western Melbourne, the Goldfields, metropolitan areas, and rural regions of Western Australia. Feedback from these trial schools indicates that the EMP is not only straightforward to implement but has also significantly accelerated student progress in mathematics. Teachers and leadership across these contexts have reported positive outcomes, with students showing marked improvements in foundational skills and increased engagement in mathematics lessons. Testimonials from some of these trial schools are included below:

The EMP is mapped to all state and curricular strands across the country, and is therefore appropriate for use across all states and territories within Australia. This has been completed internally within the EMP project team. More recently, explicit mapping documents for public use have been created for both the Australian and NSW curriculums, available on our programs page. Further mapping documents will be available shortly.

There are two types of assessments within the Explicit Mathematics program, Formative Assessments and Summative Assessments. Formative Assessments occur every 5 blocks (approximately every 3 weeks), and act as a rapid indicator of student progress which can be used to tailor pre-scheduled Catch-up or Top-up sessions to specific skills.

To support reporting requirements, the EMP also provides a Summative Assessment each semester (two in total). These assessments cover all 6 strands and, using our supporting spreadsheet, student data is automatically converted to a progression point, dependent upon the selected state/territory/national curriculum.

Assessment samples, scopes and sequences, and more can be accessed on our programs page: https://explicitmathematicsprogram.com/programs/

The EMP is designed to be easy to use and implement in any school across Australia. All trial schools have used the program without any external training, simply working through the in-built implementation guide within the Teacher Lesson Book (teacher manual).

That being said, the following would likely be helpful to schools considering the EMP:

We hope that you have found this overview of the EMP program and please reach out any time to admin@explicitmathematicsprogram.com with questions or for further information.