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?
Subitising
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
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.
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.
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.
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