Last week, Ofsted released their research report into effective and high-quality teaching of maths. They intend to use the findings to inform their approach to inspecting formal maths education from Reception upwards. In their report (available here), they make a difference between curriculum and pedagogy, and state that there is no single route to providing a high-quality maths education. In this blog, I've tried to draw out the themes and main findings and implications that will be useful for teachers and leaders. Any errors are on my part in understanding the report, though I hope I've done it justice, as I think there are some brilliant arguments drawn from the research, particularly for how maths curriculums can enable all to succeed. It took me quite some time to digest the report, and so I hope this saves you some time and helps you in some way, shape or form. (I've also organised the research into some key themes to make this blog post easier to digest than I found the report.)
Defining mathematical knowledge
The report recognises the lack of precise definition for what 'fluency' is, and so provides examples of both fluency of facts, methods and strategies, alongside their conceptual counterparts of understanding relationships, principles and reasoning. Personally, I find this a very useful way of considering the knowledge which is to be taught to pupils, although it jars with some of my previous understanding around the different elements of the curriculum (not necessarily a bad thing!). Ever the pragmatist, I think that knowing which knowledge we need to define from a curriculum design perspective, is a large step in the right direction - particularly in creating equity for children who would not be considered 'natural' mathematicians.
Ofsted Definitions of Knowledge
Declarative knowledge: This can be introduced with "I know that" and refers to facts and formalae, and the relationship between facts; Ofsted refer to the latter as conceptual understanding.
Procedural knowledge: This can be introduced with "I know how" and refers to methods, and the principles underpinning them. An example of this is a missing number problem.
Conditional knowledge: This can be introduced with "I know when" and refers to knowledge and understanding of strategies which can be used to reason and solve problems. This extends to combinations of declarative and procedural knowledge which then become strategies for particular types of problems.
Organising the curriculum
☝️The curriculum should be ambitious for all pupils. Maths should not be viewed as a subject in which only 'naturals' can succeed. Pupils should move through the curriculum at broadly the same pace, and intervention should only be used where necessary.
📈The curriculum should guarantee long term learning. It should be sequenced carefully, starting with a focus on the core content in primary school. The sequencing of the curriculum can expose pupils to new and consistent patterns of information which teachers should make visible and explicit to pupils. Declarative and procedural knowledge should be sequenced together so that pupils become familiar with their reciprocal relationships.
⏳Sufficient time should be dedicated for children to acquire foundational knowledge. They need considerable time and effort to learn these early on. Children should acquire knowledge thoroughly to reduce the need for it to be relearned and reduce the chance of it being forgotten. The curriculum should also provide time for rehearsal and consolidation.
🌿 Low-differentiating systems have higher levels of proficiency.
🧩 Pupils should become proficient in core knowledge which can be recalled and deployed with speed and accuracy. Once this knowledge is automatic, problem solving should be taught and learned. A successful curricular approach to teaching problem solving includes teaching the useful combinations of facts and methods, how to recognise the different types of problems, and how to pair the strategies with the deeper structures of the problems. Proficiency in reading is a key component of children being able to access word problems.
⛓The timely link between procedural knowledge and problem-solving practice enables some children to intuit the connections between the strategy and the conditions for their use. All children need experience of this, however, as this will be a greater challenge when confronted with problem types in test papers.
🗒 It is important for children to have efficient, accurate and clear methods for the four operations. Equally as important is the neatness and logical approaches behind them. This is because they reduce the risk of accidental errors. Basically, teach the methods in the National Curriculum. Pupils should become automatic in their procedural knowledge so that they do not rely on previously taught methods that are inefficient.
∑ The abstract level of understanding should always be the aim.
Setting the foundations in Reception
🧒 What children know as they enter Reception is dependent on their parental input and early exposure to the basics of maths. This early acquisition is a significant predictor of later success. EYFS providers need to know which children have not had exposure to early maths at home, so that they can have systems in place to ensure that these children benefit from any 'maths rich' environments.
🎭 There is some EFYS research which shows that those children who are already proficient in everyday language used to describe quantity, shape and time, can be negatively impacted when they are expected to go over old content, that might be new to disadvantaged pupils. This tension needs to be considered and a balance sought.
💠 In the Early Years, children need to learn the 'code' of maths - much like children need to learn the 'alphabetic' code of English e.g. that a numeral represents a quantity. It is vitally important for children to quickly and easily recall maths facts. There are over 100 basic addition and subtraction facts which must learned to automaticity before children can approach different types of problems.
⏭ The EYFS is where key conceptual relationships which underpin the curriculums of KS3 and beyond are first taught. Children will rely upon these for later success. Even though children are likely to pick up relationships over time, they all benefit from having structures and patterns pointed out to them.
Ambition for all
The report has at its heart the ambition for all pupils to enjoy and be successful in mathematics. Some of the implications therefore tend to focus on children who might ordinarily find school maths more of a challenge. All children have an entitlement to the core curriculum. This includes problem solving, which should be taught to all children.
Leading all pupils to success
👩🏼🏫 The report also mentions appropriate practices which are useful and beneficial for groups of children such as those with SEND and disadvantaged children. It provides, in many ways, ways of enabling ambition and high expectations for pupils to be effected and enacted.
⏳ The long-term retrieval of core content should be a focus of teachers' and leaders' planning.
👨🏻🏫 Novice learners require more instruction (I've interpreted 'novice' to mean pupils in general). They benefit from being given the ability to recognise the deep structure of problem and being able to swiftly deploy a suitable strategy. Instructional methods which mirror those used to teach early reading and writing can be effective when teaching novice learners of maths.
🌟 Use variation to plan for children to develop content inter minds. This applies to all children, but those who grasp things more rapidly can be extended by providing additional purposeful, intelligent practice.
☝️Explicit instruction is a highly effective way of teaching disadvantaged children and those with SEND. These children also benefit hugely from systematic rehearsal of declarative and procedural knowledge.
🧠 ASD children can benefit by using any propensity for memorisation to memorise core facts and methods so that they can dedicate more thinking time to learning problem-solving strategies
🏅 Children will develop more positive attitudes to maths when they experience more success. Games can also foster enjoyment but do not always optimise learning.
🥺 Children develop anxiety when they fail to acquire knowledge: it is not caused by the nature of the subject. By focusing on core knowledge acquisition and having determination to stimulate success, we are more able to increase children's motivations and enjoyment.
👎 Infrequent mistakes can be learned from but consistent mistakes can lead novices to anxiety. They represent weak foundational knowledge.
✅ Take a proficiency-first approach rather than expecting children to learn through mistakes. This is not to say mistakes should be a negative, rather they should not be the preferred route of children learning about procedures and concepts.
Teaching problem solving
🧬 Problem solving is not a generic skill which pupils can transfer from to multiple topics or domains. Problem solving strategies are topic-specific and can be planned into the sequence for that topic. This can help prevent gaps from emerging and creates a more inclusive curriculum. Pupils can then develop further conceptual understanding through applying procedures to classes of problem.
💬 Teach strategies to convert the deep structure of word problems into simple equations.
📆 Use sequences of rehearsal to prevent children from forgetting what they have been learned. Use your knowledge of the children to balance the quantity and quality of the practice they need. There is a direct link between the amount of practice and the pupils' levels of procedural fluency. Plan to rehearse key content in terms of number facts and procedures, as well as the conceptual understanding which sit alongside these, such as relationships.
🎶 Textbooks, games and songs can be used rehearse key content.
🏠 Use homework to systematically rehearse content at home.
🔁 Set tasks that focus on pupils' rehearsal of content as well as tasks that develop understanding.
Reducing Cognitive Load
↘️ Reduce the need for pupils to spend lots of time making choices as this splits their attention.
🖼 Consider any imagery you use during lessons as non-content related imagery can be a distraction.
🔬 Break down composite skills into their component parts and provide opportunities for children to practise these before practising the composite skill.
🔇 The classroom environment should support periods of sustained concentration. The ideal level of noise is almost silence, particularly for children under 13, and those with SEND.
🤔 Scaffolding is essential but should be carefully considered so that children do not become dependent on them.
🧮 Manipulatives should be used to reveal useful information rather than be used as an external memory device. Using resources is more valuable when relationships are focused upon. Don't forget to plan for how children will stop using manipulatives.
🚨 Manipulatives can also be a distraction and do not always ensure that a pupil will understand the concept they aim to represent.
🧮 The most useful resources for early methods are those associated with efficiency, accuracy and simplicity. There are also some equipment, such as a sorobon, which can inherently support the acquisition of key facts as well as displaying the connections that lead to stronger recall and understanding e.g. bridging through five or ten.
🗓 Frequent low-stakes testing can prepare children well for their final performance.
🤯 Overusing past papers should be avoided as this can remind lower-attaining pupils about what they do not yet know and enforce poorer approaches to problem solving, such as trial and error. A lack of proficiency can cause performance anxiety. Tests should be closely aligned to the curriculum sequences because generic tests do not give pupils the type of feedback they need to increase their interest or sense of self-efficacy. Presumably this is because the sequencing of the curriculum is out of the pupils' control.
Implications for Leaders
⏲ Ensure there is sufficient, dedicated time for maths.
📖 Ensure that bookwork is of a high standard, and focus on presentation. Excellent presentation allows pupils to see connections and spot errors.
✅ Methods and presentation rules are types of procedural knowledge that need to be taught and rehearsed to automaticity.
📝 Pupils should be taught how to use jottings but this should not be done at the expense of accurate presentation, which should also be the usual practice.
🎗Support novice teachers to teach maths effectively by providing robust support: do not leave it to them to develop their own methods from scratch. Novice teachers also benefit from collaborative planning with more experienced and successful teachers of maths.
📚Provide sequenced schemes of learning and supplementary materials such as teacher guides and CPD to help novice teachers bring the subject to life.
📆 Have systematic plans to build models of instruction and rehearsal over time.
🤔 Teachers across phases benefit from renewing and improving their subject knowledge, even if they are teaching foundational concepts. The relationships that underpin number bonds, for example, are directly linked with those of KS3 algebra.
My Own Take
I'm really quite pleased by these findings and I think that they offer schools a clear route forward in considering how to build on the successes of mastery approaches that most schools have introduced since the beginning of the new curriculum. The parts which stand out the most for me are those which demonstrate how all children can be successful at maths, how success breeds enjoyment, and how teachers should break down the complexity of maths to enable children to be as successful as possible.
What really grabbed me was the implications of Cognitive Load Theory in respect of mathematics learning; breaking down composite skills into components of knowledge, and providing adequate practice before bringing these together to solve problems and more challenging question makes so much sense. Although the report does not overtly refer to CLT, it does refer to overloading working memory, which I think is such a barrier for so many children - particularly if they are not fluent in the core content from previous year groups.
Another really useful implication from the report surrounds the use of manipulatives. I think it's great that the use of manipulatives has become so widespread, but the findings from the report - that they should be used purposefully and not as a way of outsourcing memory - make a lot of sense and show that schools need to be intentional and purposeful about the role manipulatives play in children's learning journeys.
I hope you have found this blog post useful and here's to ensuring that all children benefit from a high-quality maths education!
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