Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

In case you missed it, we’re spending this year recruiting 60 schools with an ambition to transform maths outcomes for their students, and make failure obsolete.

If you would like your school to join them:

School network leaders, use this form to learn more.

School leaders, including subject leaders, use this form to learn more.

We will build out programmes for individual teachers in the future, and if you would like to be kept informed, you can express interesting using this form.

In the UK, the spiral curriculum is ubiquitous in schools. What is a spiral curriculum?

A spiral curriculum is one type of curriculum model, and it sequences topics which are revisited again at a later stage where more of the topic is taught. A successful spiral curriculum is considered an upward spiral, where a concept’s foundation is taught at one learning stage, and then added to at a later stage.

The spiral curriculum was developed by Jerome Bruner in the 1960s after his publication “The process of learning”.

What if I told you that the spiral curriculum wasn’t effective?

Here are two of the reasons:

1. It does not promote sufficient review of a unit once it’s completed because you have to move onto the next topic, which is different. Yes, there may be some time allocated to review the topic, but then moving onto a new topic increases the likelihood of the review work not being seen for a delayed period of time.

2. Another disadvantage is that a spiral curriculum limits a student’s opportunity to master mathematical concepts. If a topic, such as fractions, is broken up over a 5 year period with different topics taught in between each fraction unit, that risks students developing a superficial understanding of a mathematical concept.

What can we replace a spiral curriculum with?

Answer: A strand curriculum

What is a strand curriculum?

A strand curriculum organises each lesson around multiple topics, each addressed for 5-10 minutes, and all topics are revisited day after day over dozens of lessons.

Connecting Maths Concepts is one, if not the only, mathematics curriculum which is organised as a strand curriculum.

In lesson 1, of a programme, multiple skills or topics are introduced, and new knowledge is added to the same topics over 100+ lessons. Each lesson spends around 85% of its time on knowledge consolidation, and only 15% introducing new knowledge.

This is visible from CMC’s Scope and Sequence Chart.

In Level E, the topics introduced in lesson 1 are:

1. Addition number families: Number family foundations (until lesson 6)

2. Fractions (until lesson 30)

3. Geometry: shapes (until lesson 5)

4. Column subtraction (until lesson 30)

The next topics are introduced:

1. Addition number families: word problems (Lesson 6 until 130)

2. Whole numbers (Lesson 5 to 120)

3. Multiplication number families: word problems (Lesson 9 to 130)

4. Facts: Multiplication/Division facts with number family (Lesson 11 to 130)

5. …

A topic is introduced and then appears in almost every lesson until the concept is mastered. And in every lesson, several ongoing topics - ongoing strands - are always being taught.

Here is a visual below:

In addition to this, as a concept is mastered,

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Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

In case you missed it, we’re spending this year recruiting 60 schools with an ambition to transform maths outcomes for their students, and make failure obsolete.

If you would like your school to join them:

School network leaders, use this form to learn more.

School leaders, including subject leaders, use this form to learn more.

We will build out programmes for individual teachers in the future, and if you would like to be kept informed, you can express interesting using this form.

A version of this article originally appeared in the ResearchED Magazine. Here, I’ve edited and expanded upon the original.

Connecting Maths Concepts (CMC) is a Mathematics Direct Instruction (DI) programme. It is a packaged resource which includes a teacher scripted presentation book, additional teacher guides for instructional strategies, pupil textbooks and workbooks with an answer key, and placement tests.

I used this textbook series as a remedial programme for intervention with Year 7 and Year 8 whilst at Michaela Community School. The CMC textbooks changed my understanding of mathematics and made me appreciate the intricate and expertly designed structure of Direct Instruction. More importantly, it closed the most fundamental knowledge gaps the weakest pupils had, and accelerated their learning in their mainstream lessons. Using these materials was the best teacher training I ever received.

CMC has been shaped through extensive field testing, and it is different to traditional study programmes because their philosophy is, “if teachers or students have trouble with material presented, the program is at fault.”^[1] To ensure that there is no fault with the programme, DI pays great attention to all aspects of the teaching process.^[2] The programme strives to be logically faultless and it is accepted that “if any one element of instruction is not done well, high quality instruction in other areas may not compensate for it.”^[3]

CMC provides resources for the teacher and pupils which have been designed so all aspects of the teaching process have been catered for.^[4] These aspects consist of three main components of DI which allow all children to learn effectively and efficiently:^[5]

1. Programme design

2. Organisation of instruction

3. Student-teacher interaction techniques

Here, I’m just going to focus on programme design, and its five elements.

1) Analysing the content matter

DI’s goal “is to teach generalised skills.”^[6] For this to be possible, the concepts, rules and teaching strategies must be identified.

For example, a task at Level D of the programme is to state a fraction given a diagram. A variety of shapes is provided, along with a variety of number line representations. This way, pupils can generalise their concept of fraction and apply it in future to the widest possible range of examples.

Tasks below include expressing a mixed number on a number line, showing that two fractions are equivalent using a diagram, and finding the fraction needed to make one.

Image 1: Tasks from the Level D CMC textbook series

Identifying the content matter is the first step of programme design.

2) Clear Communication

Given that the content matter has been identified, the second aspect of programme design is clear communication. This means creating an instructional sequence such that pupils can apply a generalised strategy in a wide range of examples.^[7] One part of this is called “general case programming”, where instruction is designed to communicate one and only one meaning, for all situations.^[8]

For example, to say how to state a fraction, given a diagram, a poor communication might look like this:

“The numerator is the total number of shaded pieces. The denominator is the total number of pieces”

Instead, the Level D programme communicates how to state a fraction, given a diagram, like this:

The top number is the total number of shaded pieces. The bottom number is the total number of pieces in one unit.

This instruction didn’t change at any point throughout the textbook, it does two things better:

1. it simplifies the language so communication is more likely to be successful

2. in the first example you would have to change the language as soon as more than one diagram was present, since the total number of pieces visible no longer tells you how many pieces are in ‘one whole’ or ‘one unit’

It also aligns with all situations involving fractions: a proper fraction, an improper fraction, or a fraction which simplifies to 1.

This is the same language used when stating a fraction from a shape or a number line.

The instruction deliberately uses the language ‘top’ and ‘bottom’ rather than denominator because it is “learner friendly” instruction. The same wording is used throughout.^[9]

3) Instructional Formats

Next, instructional formats are created. ‘Format’ here refers to the way a teacher presents each question or explanation. The scripted presentation book is very helpful in providing each explanation for a concept which allows them to use “effective, well designed and precise language to communicate clearly with all students.”^[10] In terms of the questions, the initial format of a set of questions will be structured to support pupils but then the format changes so pupils can apply their understanding independently.

For example, here is the transition between a sequence of exercises over 4 lessons where pupils learn how to state a fraction from a diagram.

Practice Set 1

Practice Set 2

Practice Set 3

Practice Set 4

Image 2: Practice exercises from Level D CMC textbook series

The format of the exercise changes each day:

1. The first set is focused on the use of shape.

2. The second set includes tasks where the fraction can simplify to an integer.

3. The third set is a mixture of number lines and shapes.

4. The fourth set shows fractions that are all integers, and includes representations with only ‘one part in the unit’, for the first time.

The initial support is vitally important because it ensures a high level of initial success. Then, with each new practice set, the process of “fading” the format comes into play; the format goes from “highly supportive to highly independent.”^[11]

4) Sequence of Skills

The sequence in which skills are taught impacts the success of the learning process, starting with daily practice. ^[12] For example, the skill of stating a fraction from a diagram is included in 40 consecutive lessons in one form or another. The continual review of one skill allows pupils to develop automaticity, and so ‘re-teaching’ is unnecessary.^[13] The alternative is teaching a skill which isn’t reviewed in the future, which means a pupil’s understanding of that skill deteriorates and re-teaching is required. ^[14] The sequence builds up over the days until, eventually, pupils apply the generalised strategy to more exceptional situations (examples below).

5) Track Organisation

A track is an organisational framework where one skill is developed over multiple lessons. For each skill (which might be what we’d call ‘an atom’, but is usually less granular), there is a track. This means that, in each lesson, arout 4-5 skills are included, replacing the narrow focus of a single new ‘learning objective’, occupying the entire lesson. ^[15]

Image 3: Scope and sequence chart from CMC Level C^[16]

This way DI “can extend the teaching and practice of a skill across many lessons and weave pre-requisite skill tracks into the tracks that integrate these skills into more complex strategies.”^[17] Each skill is developed with only one small change at a time to avoid pupils becoming overwhelmed with a large quantity of new information.^[18] This allows pupils to learn new concepts effectively and efficiently.

CMC is an extraordinary resource which has helped pupils learn more in less time. CMC demonstrates that “higher order thinking depends on the mastery of more basic skills and involves the integration of concepts, rules, and strategies.”^[19]

So, where does all of this lead?

Above, I mentioned ‘more exceptional’ situations.

Below, I’ll show you where Level D starts the Fractions track, and then where it ends.

The ending is important, since teachers who pick up a copy of the programme often look at the start, and its simple tasks, and write it off as too easy; they don’t recognise where it will eventually lead, and - importantly - that it will take every child there with almost complete certainty.

I’ll also show how many lessons it takes to get from start to end.

Read more

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

Welcome to our new section, Unstoppable Resources!

Kris will continue his deep dive into Atomisation, which underpins much of the analytical work we do. It’s intended for high-performing school and subject leaders.

But what about the operational?

Let’s face it, when you’re a busy teacher and you have five lessons to teach today, analysis only gets you so far. That’s before we discuss the rest of your team. Ultimately, you need resources, lots of resources. You need to know what to present, what examples to use, and you need plenty of exercises for your students to work through. Concrete examples go a long way here.

With our Unstoppable Schools this is an integral part of what we provide. Here, we’re going to offer some insight into how it all works. How can you produce what you need at scale? What format should it take? What’s out there already? How do you determine high-quality instructional materials from profitable pulp? If you centralise or purchase resources for your department, does that deskill teachers?

The good news is if you’re subscribed to Unstoppable Learning, then you’re already subscribed to these posts, but if you’re only here for the pure analysis, you can edit the sections you’re subscribed to in your profile.

To kick off, a quick introduction to Naveen, for anyone who hasn’t been following her work this last decade: as part of the foundational team at Michaela Community School, Naveen taught a programme called Connecting Math Concepts (CMC). She used it to teach the very weakest children in Year 7. Naveen has described this experience - simply using the programme to teach - as the best training and professional development of her career.

It revolutionised her practice.

It informs much of her resource design to date.

And Naveen has created a lot of resources. Entire curricula, including supporting lessons and materials. Start to finish. Year 7-11 for more than one multi-academy trust; this includes the nation’s largest, United Learning, for its 55 secondary schools.

It has even informed her work in high-level curriculum design for many other MATs and governments around the world.

Why has Naveen taken so much from it?

Because of the impact she witnessed.

In one class were eleven-year-old children who couldn’t count. Oh, they could say ‘One, two, three…’ and so forth, up to ‘twenty,’ but they couldn’t count. They didn’t know what they were saying. Put seven objects in front of them and they couldn’t identify that as ‘seven.’ They had learnt to ‘say the words’ up to twenty the way you might memorise the lyrics of a song in foreign language: the sounds and music have stuck, but you have no idea what you’re saying.

By Year 9, this cohort of children were back in mainstream lessons, part of a cohort of children where 99.1% achieved at least Grade 4 in their GCSE exam, and 98.3% achieved a Grade 5.

Since then, Naveen has carried out trust-wide implementation of the CMC programme in the primary and secondary schools of three English academy trusts, Astrea, Athena, and City of London, and supported the implementation in one Australian chain of schools, Mastery Schools Australia (MSA).

At Astrea Academy Trust the weakest pupils, on average, gained 1.7 years on their numeracy age in twelve weeks, and were on track to gain at least 5.1 years in one academic year.

To understand the impact at Athena, you need to understand a little about English SATs: children in English primary schools all sit national exams in maths, reading and writing in Year 6 (fifth grade). Performance on these exams is reported as a scaled score out of 120, with 100 being needed to meet the expected standard (the only grades are ‘met expected standard’ and ‘not met….’)

The national average for meeting expected standard at the time was 73%.

At Athena Learning Trust, Naveen implemented the CMC programme in two primary schools. In 2024, before CMC, their performance on these exams was 54% and 45% meeting the expected standard.

In 2025, following CMC implementation, their results were 74% and 91% respectively, a 20 percentage point and 46 percentage point increase in just one year.

It’s results like this that led to the Athena Trust and Coolangatta (MSA) earning the Becker Excellent School Award, and extraordinary teachers like Bryce, Brooke and Jamie at Southport (MSA) earning the Engelmann Excellence in Education Awards.

So, why not get CMC in your school or trust right this minute?

Naveen might have instigated and led these initiatives, but she certainly didn’t achieve those results on her own. Network leaders, school leaders, middle leaders, individual teachers, instructional coaches, they were all responsible and deserve the credit for these outstanding outcomes. People like Jon Owen, Emma Kerr, Kim Flynn, Tamara Bressi, Rebecca Sanders, Kevin Surrey, Suzy Wybrow, Avin Bissoo, Claire Ankers, Sarah Green, Toni Hatten-Roberts, Michael Roberts, Joe Kirby, Tehrim Valibhai, Mouhssin Ismail, Hywel Jones, Bryce Phillips, Brooke Schembri, Jamie Phillips… this long list isn’t even comprehensive. In short, unless you have a real team of exceptional individuals, people prepared to put ego to one side and learn from these extraordinary programmes and their wrap around support, it cannot work. You might have heard that these programmes include scripts for teachers. It’s true. Sounds simple. Read the script. You would be astounded by how much there is to learn to deliver scripted lessons effectively.

Maybe you’re ready for this; perhaps you have the team. If so, reach out. We have nothing to offer here, but we can put you in touch with the right people.

Or maybe this isn’t what you’re looking for; perhaps you’re working in the secondary setting, where CMC doesn’t apply.

CMC doesn’t apply, but its theoretical underpinnings do. The work that Engelmann and Carnine outlined in their Theory of Instruction underlies so much of what we do more broadly at Unstoppable Learning, and Kris’ writing in particular.

But Theory is very analytical. CMC is operational. It takes the analysis and shows you one way it can be concretely applied.

So, Naveen will kick off this section with a series exclusively on CMC, and my, is there a lot to discuss. It’s a programme so rigorous, so detailed, and so unusual in its design - it would have to be, to get these kinds of outcomes - that it will surprise and inspire

Stay tuned as we explore this magnum opus in the coming weeks; learn how it applies the principles in Theory, and how you can lift and apply those same principles in your own schools.

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Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.