In algebra, pupils benefit from fewer but powerful representations and an iterative approach to sequencing the facts and procedures for working algebraically. [footnote 87] Abstract representations can be just as effective as contextualised representations. [footnote 88] The bar modelling method can be used as a bridge from arithmetic to early algebra. It is a useful interim method for abstracting arithmetic and algebraic expressions from word problems. [footnote 89] Teachers can even teach methods of evaluation of algebraic expressions and ways to set these out as a series of steps for pupils to learn by heart. [footnote 90] This contrasts with an approach of encouraging more informal, self-generated ways for pupils to solve linear equations. This may be self-limiting when pupils are faced with unconventional presentations of linear equations. Accurate calculations and careful presentation give pupils the ability to spot important and interesting patterns of number, as well as errors that need to be corrected. Calculation methods and presentation rules are procedural knowledge that need to be taught and rehearsed to automaticity. Some pupils might naturally develop ‘neatness’ and subsequent accuracy, but teaching and rehearsing this procedural knowledge gives greater assurance that more pupils will be able to see errors and spot patterns of number, as well experience a sense of accomplishment. Within these powerful mathematics education systems, the textbooks, teacher guides and workbooks are seen as a vital part of the infrastructure for efficiently transmitting subject knowledge and subject-pedagogical knowledge to new generations of pupils and teachers. This signals a need for teachers and leaders to avoid installing features and approaches in the absence of the ‘infrastructure’ underpinning their efficacy. It is also likely that the features that tend not to be observed or selected, such as the less glamorous quality and quantity of practice, are also integral to the overall success of novice mathematicians. Pupils can be helped with simple everyday objects and semi-concrete representations, such as Numicon, but the aim should be that pupils move to working with symbols and abstract representations. [footnote 72] The use of manipulatives, for example, does not always guarantee that a pupil will understand [footnote 73] and their use may distract pupils from thinking about content to be learned. [footnote 74] A method of calculation that relies on derivation may be useful in the short term and as a bridge to formal methods of written calculation that require pupils to accurately recall number bonds. [footnote 75] In the absence of learning this core knowledge, pupils may rely too much on estimation and looking around for clues, or they may develop the habits of guessing and copying. [footnote 76]

Declarative knowledge is static in nature and consists of facts, formulae, concepts, principles and rules. It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University) Childrenoftenthinktheyhave to find an answerand become caught up in this rather than the ‘what’ and ‘how’. Help them overcome this by starting withsomething to think and talk about,rather thansomething to calculate. Give them the answer to a calculation and askthemwhy, and what this tellsus.It will help themthink about what they are going to talk about, notsimplywait to be told how to do the calculation. 4. Vary representations There are a variety of ways that schools can construct and teach a good maths curriculum, and Ofsted recognises that there is no singular way of achieving high-quality education in the subject. However, the review identifies some common features of successful, high-quality curriculum approaches: When pupils learn and use declarative, procedural and conditional knowledge, their knowledge of relationships between concepts develops over time. [footnote 25] This knowledge is classified within the ‘type 2’ sub-category of content (see table below). For example, recognition of the deep mathematical structures of problems and their connection to core strategies is the type 2 form of conditional knowledge.The unexpected finding from cognitive science is that practice does not make perfect… Practice until you are perfect and you will be perfect only briefly… What's necessary is sustained practice.” The teaching of maths facts and methods is sequenced to take advantage of the way that knowing those facts helps pupils to learn methods, and vice versa.

This really does not matter. The practice quizzes are designed to enable learning to take place. That is why pupils have two opportunities to respond. So long as the person supervising the session has a good knowledge of the content being covered – and knowledge of effective teaching strategies to help pupils overcome barriers – there is strong potential for very good pupil progress within sessions. Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics. Pupils need to be fluent with the relevant facts and methods before being expected to learn how to apply them to problem-solving conditions. [footnote 94] Giving younger pupils the ability to understand word problems Ideally, pupils gradually cease to depend on some methods of counting and calculating, and associated resources, that they were taught earlier on. This is because reliance on some early counting and calculation methods, in the absence of learning valuable number facts, can hinder later progress. [footnote 71] Comparison of textbooks also reveals that the expected volume of calculations, exercises and collections of problems to be completed is higher in countries where pupils tend to do well. [footnote 148] The evidence points to the need for teachers to provide enough opportunities to practise taught facts, methods and strategies, as well as additional opportunities for overlearning. [footnote 149] Efficient pedagogies such as choral response, explicit timing and goal setting may help to increase the ‘rate’ of practice in lessons, if it is difficult to provide additional opportunities for overlearning. [footnote 150] Qualitysometimes it refers to ease of recall and computation (which the review refers to as ‘automaticity’) That is not to say that more ‘messy’ experimental workings should never be allowed. However, teachers can help to engineer calculation and presentation success by balancing experimental approaches with opportunities to learn how to be systematic, logical and accurate when applying taught facts, methods and strategies. Proactive professional development: the planned and purposeful pathway to expertise You can use this guidance to help plan teaching the statutory mathematics curriculum in primary schools in England. This guidance:

This is easier if the mathematics curriculum focuses on core content early and leaders prioritise and value consolidation. Minimising off-task behaviour may also help to maximise the amount of time available for retrieval, rehearsal and consolidation of learning. Pupils who do well tend to have spent more time on the subject. [footnote 108] EquityIn the English mathematics education system, emphases on reactive approaches are associated with a wide attainment spread and a long tail of under-achievement. Almost 180,000 students had to re-sit GCSE mathematics in 2019. Of these, only 22.3% achieved a standard pass (grade 4) or above. [footnote 109]