What the Heck Is Structured Maths? What is the problem with the way we have been teaching maths?

There are numerous buzzwords in education today, but the term "structured maths" has left me somewhat puzzled. Structured Literacy is a term trademarked by the International Dyslexia Association (IDA), so is there any value in using a similar term for maths? If so, what on earth does it mean?

Let's start by addressing the problems with how we have been teaching maths:

Lack of Guidance

Firstly, if your experience has been anything like mine, we've had very little guidance on what to actually teach. We had numbers and a set of strands, but our teaching seemed to oscillate between these without any clear connection or deliberate intention.  I'd spend a week on this, a week on that, to be frank, I was aiming for coverage, not learning.

Ineffective Numeracy Approaches

Secondly, our numeracy approach has often prioritised counting as a means to calculate. This has led to many children struggling to develop effective strategies for achieving fluency in mathematics.  

Gaps in Understanding Math Acquisition

Thirdly, our understanding of how children actually acquire mathematical skills is lacking. Don't even get me started on how little we know about dyscalculia and supporting children with specific mathematical difficulties. We have been led to believe that by presenting maths in authentic situations and encouraging children to engage in 'productive struggle,' they would magically develop the necessary understanding to become fluent mathematicians.  There was little to no understanding around the vital importance of subitising and the role it plays in strong foundations.

Misconceptions About Conceptual Understanding

Lastly, we have been sold the idea that conceptual understanding must precede procedural knowledge. This means that children are expected to have a solid conceptual grasp before they are introduced to any procedural methods. Many of us have also fallen into the trap of thinking that learning basic facts would naturally come as part of a rich program and that timed testing, under all circumstances, causes math anxiety.

So what is it we actually need to be aiming for?

Like you, I am a learner on this journey.  I have a degree in teaching that taught me very little and everything I know now, is based on what I have learned on the job.  My interest in math started several years ago, when I was faced with children I didn't know how to help.  This journey led me to question everything I had ever been taught about math teaching.

Below are some areas I believe are crucial components of any math approach, needless to say, this is not an exhaustive list, but a good starting point.

Subitising

This is a blog post on its own really, but this is a blurb from a recent facebook post.

Subitising is the ability to instantly recognise the number of objects in a small group without the need to count them individually. This skill is fundamental in early mathematics development and plays a crucial role in building a strong numerical foundation. Subitising forms the basis of the scope and sequence and is the common thread throughout all sessions.

There are two types of subitising:
Perceptual Subitising: Recognising small quantities (usually 1-4 items) instantly.
Conceptual Subitising: Recognising larger quantities by seeing groups and patterns within the sets.
Subitising is vitally important for a range of reasons.
It is the foundation for number sense. Subitising helps children develop an intuitive understanding of numbers and their relationships. This foundational skill is critical for more advanced mathematical concepts.
It encourages pattern recognition and noticing. Through subitising, students learn to recognize patterns and groupings, which is essential for understanding more complex mathematical ideas like addition, subtraction, and multiplication.
Subitising allows string calculation skills. Learners who develop strong subitising skills can perform arithmetic operations more quickly and with greater accuracy because they can see the results of their calculations at a glance. Counting is not calculating.
It builds from what is innate. Identifying and supporting subitising skills in early learners can help prevent later difficulties with math. It provides a solid foundation upon which more complex mathematical skills can be built. Explicitly teaching subitizing is vital for all children, but crucial for those with specific math difficulties, where it might not come naturally.
We know from research that procedural understanding needs to go hand in hand with conceptual. Procedural understanding in mathematics involves knowing how to carry out a sequence of steps to solve a problem. Subitising has strong connections to procedural understandings.
Subitising helps learners understand the basic building blocks of mathematical operations. For example, recognising that two groups of three objects each make six objects supports the understanding of multiplication.
Subitising enables learners to visualise numbers and their relationships, making it easier to understand and remember mathematical procedures. This visualisation supports procedural fluency, which we know is so important.
When learners can subitize, they can more easily connect concrete experiences (like seeing groups of objects) to abstract procedures (like addition and subtraction algorithms). This connection solidifies their understanding of why procedures work.
Learners with strong subitising skills are less likely to make procedural errors because they have a more intuitive grasp of numbers and their relationships. They can quickly verify the reasonableness of their answers.
Subitising promotes flexible thinking, allowing learners to approach problems from multiple angles and choose the most efficient strategies for computation. For example, a student might quickly see that 7 + 3 is the same as 10, without needing to count each number.
Subitising is a critical skill that underpins many aspects of mathematical understanding and it is a crucial component of all mathematical teaching.  

Explicit Teaching using a Scope and Sequence

One of the key components for effective teaching is having a well-defined scope and sequence. This framework ensures that instruction is systematic, progressive, and comprehensive. Explicit teaching is crucial in this context. Contrary to the misconception that explicit teaching is monotonous, it is actually an interactive process that actively involves students. This method leaves no room for assumptions, as it requires teaching content in a clear, deliberate manner.

A highly effective model to use within explicit teaching is the gradual release model: "I do, we do, you do." This can also be adapted to fit various needs, such as "I do, I do, I do; we do, we do, we do; you do." The essence of this approach is to provide a structured path from teacher-led instruction to student independence.

The Importance of Visuals

Visual aids are fundamental in helping children understand and internalize numerical concepts. This is particularly relevant in the context of subitising, which involves recognizing the number of objects in a small group without counting. I am a strong advocate of using Woodin Patterns, as they provide a consistent visual reference. Other effective visual tools include tens frames, five frames, and Numicon. These visuals aid in reducing the dependency on counting for calculation, enabling children to develop more intuitive strategies for understanding numbers.

Teacher-Modelled Examples

Explicit teaching often involves the use of teacher-modelled examples, which I have come to appreciate significantly. When introducing a new concept or strategy, it is beneficial for the teacher to work through an example while verbalizing their thought process. This method, known as "thinking out loud," helps students understand the rationale behind each step. As students work on examples alongside the teacher, it remains crucial to continue modeling the approach and thought process. This strategy has proven to be highly effective, as my students have begun to emulate this method in their own problem-solving.

Spaced Practice

Spaced practice is a strategy grounded in the theory of the forgetting curve, which proposes that information is more likely to be retained in long-term memory when practice is distributed over time. In line with this theory, my scope and sequence intentionally revisits concepts regularly. Initially, new learning requires frequent repetition, which can be gradually spaced out over time as students begin to internalise the key understandings and concepts.

Repeated Practice and Interleaved Practice

Repeated practice is essential for embedding knowledge and skills in long-term memory. Whether it involves basic facts, numeral writing, reading numerals, or recognizing and using patterns, repetition is key. The consistent practice helps solidify these skills, making them more accessible for future use.  Interleaving of practice, refers to the combining of areas of practice and perhaps weaving the new through this as well.  It is something I try hard to do within the scope and sequence.

Retrieval Practice

Retrieval practice involves the active recall of information from long-term memory, which strengthens memory retention. I have incorporated retrieval practice into the beginning of my sessions. This can be done through various methods such as games, quizzes, or traditional tests. One effective technique I use is taped practice: I record a set of ten problems, state each problem, wait four seconds, and then provide the answer. Students try to write down the answer before my voice gives it away. This method allows me to observe which students have mastered the material and which ones need additional support.

Aiming for Fluency - Fluency Does Not Just Mean Fast

The ultimate goal of these teaching strategies is to achieve fluency, which involves more than just speed. Fluency is about building a deep understanding that is embedded in long-term memory. It consists of three components: working at an appropriate rate, maintaining accuracy, and demonstrating flexibility. We promote fluency by helping students develop strong schemas to connect new learning with existing knowledge. Subitising plays a crucial role in building these connections, laying a solid foundation for future learning.

CPA: Concrete-Pictorial-Abstract

The CPA approach involves using concrete materials, pictorial representations, and abstract symbols. It is a common misconception that these stages are linear. In reality, they should be integrated and used concurrently in teaching sessions. Ensuring that all three components are addressed in each session enriches the learning experience and aids in deeper understanding.
Conceptual and Procedural Interwoven

There is a misconception that conceptual understanding must precede procedural understanding. In truth, these two types of understanding develop together and reinforce each other. In some cases, procedural knowledge can enhance conceptual understanding. Integrating both ensures a more holistic approach to learning.

Use of Schema for Problem Solving

I recently discovered the work of Sarah Powell, which emphasises using schemas for problem-solving. This approach has revolutionised my teaching, enabling me to instruct the process more explicitly and, consequently, helping my students become more successful problem solvers.

Ensuring Children Understand Mathematical Language

In recent years, it has become increasingly evident that children are entering school with lower language levels compared to previous generations. This trend has significant implications for their overall academic development, particularly in mathematics, where understanding the specific language used is crucial. For children with language delays or Developmental Language Disorder (DLD), this challenge is even more pronounced. As educators, it is our responsibility to ensure that all children develop a strong grasp of mathematical language to succeed in their studies.

The Importance of Mathematical Language

Mathematics is not just about numbers and equations; it is a language in itself. Terms such as "sum," "difference," "product," and "quotient" are foundational to comprehending mathematical concepts. When children do not fully understand these terms, they are at a disadvantage. This can lead to confusion, frustration, and a lack of confidence in their mathematical abilities.

Challenges Faced by Children

1. Lower Language Levels: Many children are starting school with reduced exposure to rich vocabulary and complex sentence structures. This can hinder their understanding of mathematical instructions and problems.

2. Language Delays and DLD: Children with language delays or DLD face additional hurdles. These children may struggle to grasp and retain mathematical vocabulary, making it harder for them to follow along in class and take a full part in sessions.

3. Assumed Understanding: Teachers often assume that students understand the language of mathematics when, in reality, many do not. This assumption can lead to gaps in learning and comprehension.

So, is there such a thing as structured maths?  I am still not sure, but what I do know, is that there are components that do need to be in place to ensure ALL children can achieve in maths.  Some of the directions maths has taken over the past  fifteen to twenty years have led to many children struggling and believing themselves to be 'bad' at maths, we need this to change and the way we can change it, is by growing our understandings about how to best teach maths.  As always, we need to start with our own understandings first.

To Decodable Or Not To Decodable? That Is The Question

 This has been a hot topic recently.  With Tim Shanahan's blog post on the research behind decodable books and their use (of which there has not been a lot) Blog post here.

Firstly let me say, I love his work, his research and insights have shaped a lot of what I do now.   But let's put some perspective on the post.

Very basically, the research is limited and unclear, leading to his last comment.

"I think it’s okay to use decodable texts as part of phonics instruction, but such practice should be

 limited, and even beginning readers should be reading (not just listening to) more than decodable texts."

This issue is complex and, in my opinion, not as straightforward as "Decodable's are nice to have, but not needed," as some might believe based on this last comment.

It's crucial to consider both research and practice in this matter. Given the high cost and difficulty of conducting relevant research, we must rely on practical experiences to answer questions like what constitutes good practice, the role of decodable books, their intended use, and how they compare to other text types.  

One common misconception about decodable books needs to be addressed

"Decodable books are boring and will put children off reading." This claim is often repeated, but it overlooks the variety of well-written, engaging decodable books available for the intended audience—novice readers, not fluent adult readers. From an expert reader's perspective, these books might seem dull, but for children, the ability to read these books is exciting and empowering. In my experience, I've never encountered a novice reader who found their first decodable books boring; rather, they were thrilled by their reading capabilities and driven to read more.

Understanding what a decodable book is remains essential. A decodable book is any book that a child can read independently based on their knowledge of phonetic codes. While most books are decodable for adults, for beginners who might only know a few short vowels and consonants, decodable books primarily feature words that adhere to these simple phonetic patterns. For this reason, I prefer to refer to them as 'controlled texts,' which better describes how these books selectively introduce phonetic elements to prevent overwhelming young readers.

A significant issue with decodable books arises when they are introduced to teachers without adequate professional learning and development (PLD). Teachers need to understand the purpose of these texts, which brings me to a crucial point:

Understanding takes time

This is something I think we need to shout from the rooftops!

Reflecting on my own journey with decodable books illustrates the importance of being given time to grasp their role fully.  

Check these out, this first one is a post from December 2018, after a year inquiring into literacy development...YES 2018, six years ago.

December 2018

Then again in 2020, updating my journey.  This is two years in to finding out more...oh the glorious benefits of being given time.  

September 2020

These experiences show how my approach and appreciation for decodable books have evolved, significantly impacting my teaching practices.  Primarily, the point I want to make, is that it has taken me that time to get to where I am and very clearly, decodable books played a massive part in that journey!

If we fast forward to 2023, at the beginning of last year, you will also see how my attempt to replace levelled readers in my class, with decodable books and continue on with the way I'd always done things, simply through a more informed lens.  This led to frustration and a real sense of need to do things better for my Tier 1 learners (the whole class.)

The start of whole class reading

Then a reflection in July around my interpretation of whole class texts.

And the unintentional benefits of these texts

Along with my reflection on whole class reading (Tier 1) at the start of this year

So here we are, six years later, what does practice tell us about decodable books and their use?

So if there is very little reliable research on decodable books, what has my six years of work with decodable texts told me?  

Given we started using decodable texts for our Tier 3 learners and because of our journey into understanding dyslexia, I think I am well placed to comment on this.  Of course, this is as usual, just my opinion, I am 'just' a teacher, do not have any recognised credentials in this area and funnily enough, other than the odd course, have had no formal training.  

As you can see if you read the series of links I have posted above, my journey has been just that, a journey and the role of decodable books has changed slightly.  I did initially present decodable books as the dominant text children were reading.  Over time this has changed, with my class texts becoming less decodable and reaching outside of the 'code' that the majority of children know.  

All children still practice with decodable books and spend a lot of time reading them. I've posted a little about this 'just right for me' reading on Facebook if it is of any interest. For those children who are ready to move away from decodable texts during this practice stage, these books are provided. Again, you know your children, and this decision can be made on an individual basis. We all have children in our classes who appear to be more 'natural' readers; they are often not the norm, though, and beware, because this does not necessarily translate to spelling ability.

They all benefit from this practice, and through our whole class sessions, we are still working in a systematic, explicit way through a scope and sequence. This is very, very important, in my opinion, because this approach isn't just about reading.

I always come back to the point that when it comes to learning, children are more similar than they are different, and this assertion that all children learn differently is a bit of a misnomer, in my opinion. If you have not looked into learning styles and how they are not a thing at all, then you really need to. I love the book Why Don't Children Like School? written by Daniel Willingham.

This message is important...

My Tier 2 students exclusively use decodable texts or sentences tailored to the specific phonetic elements they are learning. This approach is crucial for their development. These students depend on these books, and it's essential for me to understand their importance.

Through my experience, I've realised that I can significantly impact these learners by using decodable texts rather than uncontrolled texts. Reflecting on my practice in 2017, I remember the struggle of supporting students classified as 'Tier 2' or 'Tier 3'—terms I was unfamiliar with at the time. Back then, all students received the same instruction, yet some remained stagnant at the early stages of reading, known as the pink and red levels. I was at a loss on how to assist them and they desperately needed me to recognise the value of decodable books.

Tier 2 students require decodable texts to develop specific skills and insights, affording them numerous opportunities to apply and practice these skills. They should not be overwhelmed by texts containing unfamiliar phonetic codes. These learners benefit from the focused practice that decodable books offer, and it's crucial for us to understand and support this method. Even minor changes in font can deter these students; they find texts with embellishments on letters like 'i' and 'f' perplexing. For this reason, I often type out the sections of texts we are working on, preferring to use Sunshine texts for these students.

It is equally important to provide children with dyslexia with decodable texts. I have witnessed the profound impact these texts have on such learners and would be dismayed to consider them optional.

Back to addressing the original comment this post is based on

All students benefit from the systematic, explicit instructional strategies we employ, which are not solely about reading but encompass spelling and writing as well. It's crucial to recognise that while students may differ in various ways, their fundamental learning needs are more alike than not. This perspective challenges the notion of distinct learning styles, a concept well critiqued in Daniel Willingham's "Why Don't Students Like School?"

In conclusion, while decodable texts are not a panacea, they are indispensable for certain learners, particularly those at Tier 2 and 3 levels, who require specific phonetic practice. These texts should not be seen as merely optional or supplementary but as essential tools supported by a deep understanding of their purpose and application. This approach ensures that all students, especially those facing the most significant challenges, have the best possible support to succeed in reading.

Before dismissing decodable books as 'nice to have' let's take time to build our understanding of why they are so important for many.