Effective Maths Teaching: A Guide to Teaching Basic Mathematical Concepts

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Pupils that score 100% on any of the quizzes – timed or practice – can be challenged to look at their detailed report, which shows their time, and beat the time.) Interventions should include explicit and systematic teaching. Explicit instruction is absolutely necessary when teaching content that learners couldn’t discover for themselves or when discovery may be inaccurate, inadequate, incomplete, or inefficient. For example, achild may demonstrate confident recall of times tables having practised them during remote learning, but have gaps in their understanding of addition and subtraction of fractions. The teacher can support children to build on this fluency by planning in ways they can explicitly use their times tables, when reviewing addition and subtraction of fractions.

In contrast, a visually simple counting frame (such as a soroban, commonly used in countries where pupils experience early success) is a resource that represents an efficient and powerful early method of calculation. The method associated with this resource, once the pupil has been taught how to use it, consistently presents accurate connections of number that can be learned and then later recalled as number sequences, rules and bonds. Giving young pupils an efficient, less distracting method of calculation that is not associated with other familiar activities (such as toys used for social play) helps them to see past the methods and any associated resources to new connections of number. [footnote 77] In the case of a simple counting frame, children no longer need it once they have learned to recall the number bonds, sequences, patterns and rules automatically. [footnote 78] The maths facts that they have acquired because of familiarity with and use of a powerful method can then aid their ability in mental arithmetic. [footnote 79] It is not the resource itself but the fact that its use is associated with efficiency, accuracy and visual simplicity that is the most important feature of powerful early methods. Methods for more complex measurements and calculations 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] Equity Further, the option of problem-solving as part of task differentiation does not guarantee that all pupils will learn problem-solving strategies. Leaders and teachers should ideally view learning of all core content, including the links between content, as an entitlement and therefore a pre-planned pathway for all pupils. Balancing new learning and rehearsal of learning However, research also shows that the unique organisation and powerful declarative memory systems of many people with autism help them study, and develop proficiency in, the subject. [footnote 119] Potentially, a powerful declarative memory system can take on a compensatory role; thus many pupils with autism might benefit from a deliberate focus on memorisation of core facts and methods. Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education.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 Choice of examples: Rowland, T. (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational studies in mathematics, 69(2), 149-163.

The explicit teaching of cognitive and metacognitive strategies is integral to high-quality teaching and learning, and these strategies are best taught within asubject and phase specific context. Approachessuch asexplicit instruction, scaffolding and flexible grouping are all key components of high-quality teaching and learning for pupils. Leaders could consider incorporating more proactive approaches that close gaps and allow novice teachers to adopt and improve expert teaching methods, rather than develop their own aspects of effective mathematics teaching from scratch. [footnote 196] Such approaches could include: Close examination of lesson planning and teachers’ thoughts about lesson planning in education systems where pupils do well reveal an intense focus on underlying knowledge structures and connections rather than the surface coherence of activities and teaching. This means that teachers are planning for what pupils will be thinking about or with, not what they will be ‘doing’.The evidence presented here supports careful consideration of sequencing and content that makes a mathematics curriculum a guarantee of long-term learning. Useful facts and efficient and accurate methods are ideally paired within a topic sequence. Strategies for solving problem types are then best taught and learned once pupils can recall and deploy facts and methods with speed and accuracy. When planning curriculum content, teachers also need to prioritise ‘forward-facing’ knowledge. This goes beyond important facts of number. It includes the mathematical methods that pupils will take with them on their journey. The ideal aim is for pupils to attain proficiency, not just collective moments of understanding, familiarity or experience. This will help pupils to develop motivation in the subject. Selecting and sequencing core declarative, procedural and conditional knowledge The message of quality over quantity of procedural knowledge also applies throughout key stages 3 and 4. 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. In contrast, pupils in England spend less time on mathematics homework than pupils in high-performing countries. [footnote 144] The fact that extra rehearsal, particularly in core content, helps pupils attain automaticity in recall and use of facts and methods [footnote 145] may explain some of the increases in attainment following the introduction of the ‘numeracy hour’ into English primary schools. [footnote 146] Conversely, when lessons and therefore rehearsal opportunities are cut, attainment declines. [footnote 147]

Additional risks arise from mixing and matching a toolkit of informal and self-generated methods for working with larger numbers and more complex calculations as pupils progress through the curriculum. This increases the likelihood of pupils generating errors and structuring written records poorly, which may lead to confusion. [footnote 84]

Mathematics teaching practice

Furthermore, if this core content has been sequenced well and pupils have learned it thoroughly, they are less likely to forget and are therefore unlikely to need to ‘re-learn’ it later. [footnote 34] A focus on core knowledge in younger year groups can be achieved by focusing on depth over breadth, covering fewer core topics but in more detail. Amplifying the curriculum through instruction, rehearsal and assessment plans 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]

With some ‘creative’ timetabling, interventions won’t impact other areas of the curriculum. Many schools hold interventions during assembly time and after school. 8. Use your time wiselyCognitive strategies include subject-specific strategies or memorisation techniques such as methods to solve problems inmaths. sometimes it refers to ease of recall and computation (which the review refers to as ‘automaticity’) Japanese lesson study is an example of a systematic approach to sharing subject-pedagogical knowledge that builds and shares subject-pedagogical knowledge at organisational, local and national scales. [footnote 198] The fact that lesson study is a system should also alert teachers and leaders to the dangers of adopting ‘surface features’ and not systems. This may also explain why attempts to install (the surface features of) ‘lesson study’ as a curricular or pedagogical intervention leads to somewhat less convincing results. [footnote 199] These practices significantly improve proficiency in word problem solving and operations. 4. Use staff strategically