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.