Research Review series: Mathematics (OFSTED, 25 mei 2021)
In this section, we discuss the instructional needs of pupils as they progress through the curriculum. When pupils are at the start of school life or starting a new sequence of learning, they need more instruction than pupils who are already competent in that topic area. Throughout sequences of learning, pupils benefit from teaching that is systematic and clear. Pupils can also develop further understanding when sets of exercises are curated to present new and useful number connections, as pupils rehearse recently taught content.
The novice needs more instruction, not less
Novice learners of new mathematics content need systematic instructional approaches similar to those used to teach early reading and writing. Teachers need to ensure daily dedicated time for teaching and practising component parts.[footnote 120]Like the code for language, it is useful to think about early mathematical content as also being a code. Not all pupils will crack, discover or invent this code for themselves. An approach that comes closest to guaranteeing foundational success in mathematics is one that acknowledges that:
To most effectively develop more comprehensive and abstract thinking about mathematics, children often need more than their natural, spontaneous learning.[footnote 121]
An approach like this should incorporate extra elements of explicit, systematic instruction. This will help to close the school entry gap in knowledge. It will also give more pupils the foundations for mathematical success,[footnote 122]as well as greater self-esteem.[footnote 123]
Use of intelligent variation in sets of exercises
There is a difference between content that pupils have recently learned and content that develops further in their minds through practice. Both can be planned for. Variation within sets of exercises can help pupils to learn:
the ranges and boundaries of strategy applicability
important patterns and rules
connections between varying problems[footnote 124]
how to focus
logical and systematic approaches to solving problems[footnote 125]
Leaders need to make sure they curate and control this approach. This is evident in the systematic use of variation in collections of tasks given to pupils in China, Hong Kong and Taiwan.[footnote 126]
Systematic instructional approaches also work well for all ages and stages
Proficient mathematicians are able to demonstrate success in problem-solving lessons.[footnote 127]However, it is easy then to assume that the activity that demonstrates a proficient mathematicians ability to problem-solve is the ideal means of acquiring proficiency.[footnote 128]Learning through participating in similarly open-ended problem-solving activities might be enjoyable for both teachers and pupils,[footnote 129]but it does not necessarily lead to improved results.[footnote 130]The adult in the room is an important mediator of pupils success.
Without the adult, even where content and sequence of content might be ideal, the learning of at-risk groups of pupils is compromised.[footnote 131]Evidence shows that pupils can learn from worked examples,[footnote 132]particularly if teachers help pupils to make sense of worked examples.[footnote 133]Questioning, as long as teachers take care with language and timing, can also aid instruction.[footnote 134]Teaching pupils how to construct and use visual representations can help pupils to convert information presented in a problem into symbolic equations.[footnote 135]
Systematic instruction might also offer up benefits beyond enhanced learning of facts, methods and strategies because pupils who are more successful develop better learning behaviours.[footnote 136]
Based on the above, high-quality maths education may have the following features
Teachers remember that it is not possible for pupils to develop proficiency by emulating expertise, but by emulating the journey to expertise.
Systematic instructional approaches to engineer success in learning are incorporated into all stages and phases.
Teachers aim to impart core content in alignment with the detail and sequence of the planned curriculum.
Teachers help pupils to avoid relying on guesswork or unstructured trial and error.
Pedagogy: consolidation of learning
As pupils progress through the curriculum, they need regular opportunities to rehearse and apply the important facts, concepts, methods and strategies that they have learned. When designing sequences of rehearsal, teachers need to consider both the quality and the quantity of practice that pupils need to develop their understanding and to make core content firm and precise in the mind. Practice needs to go beyond immediate accuracy and understanding. Sequences of rehearsal should help to prevent pupils forgetting content over time.
Categories of rehearsal/consolidation of learning
||Type 1 practice
||Type 2 practice
||Fact retrieval (recall)
||Explaining relationships between facts (derivation and parsing of number)
||Method rehearsal (exercises)
||Explaining principles, proving conceptual understanding (such as, use of informal methods, creating bar models and interpreting context)
||Strategies rehearsal (collections of problems with the same deep structure)
||Describing relationships between the problem and choices of strategy (proof/reasoning)
Some pupils are quick to grasp new content, while others might need more time to think, practise, recall and apply. Given that proficiency in mathematics requires pupils to attain a level of procedural fluency,[footnote 137]teachers should ensure that they give pupils adequate opportunities to practise. This is more likely to increase pupils levels of procedural fluency.
Consolidation of learning transforms pupils initial moments of success, realisation and understanding into long-term memories.[footnote 138]The younger the pupil and the lower the level of overall mathematical skills, the more time and the greater the number of repetitions needed to attain automaticity in facts and methods.[footnote 139]If a pupils recall fails, therefore, it might be that they need more practice[footnote 140]rather than just repeated teaching.[footnote 141]
In the most successful systems of mathematics education, systematic rehearsal is given more time and focus than in England. Powerful teaching and learning in classrooms where pupils do well are supported by regular homework assignments that require pupils to systematically rehearse content at home.[footnote 142]Teachers in these systems can plan future sequences of learning confident that pupils foundational knowledge is secure. Under these conditions, consolidation of learning is personalised by time taken to complete assignments where:
every child will have had to attend to every word, every problem, and every exercise included in every textbook.[footnote 143]
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]
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]
Textbook analysis can provide useful information about quality of rehearsal. In contrast to the ideal of systematic rehearsal aligned to sequences of learning:
English primary textbooks tend instead to move around rapidly and to constantly recapitulate.[footnote 151]
Lack of coherence and evidence-informed features in textbooks given to the youngest pupils could potentially influence the likelihood of laterSEND.[footnote 152]This may be compounded by the fact that textbooks are generally seen by teachers in England as a supplemental resource rather than a potential and valuable system of rehearsal.[footnote 153]This is different to how textbooks are used in countries where pupils do best, not just in terms of volume of questions, but also in terms of sequencing and alignment with the curriculum sequence.[footnote 154]
Textbooks are particularly important for low attainers[footnote 155]and they might also be useful for pupil and parent buy-in. Pupils know they need to concentrate in the lesson to be able to complete the homework and they know they need to complete the homework to understand the next lesson.[footnote 156]Parents can also easily check their childs progress.[footnote 157]
Systematic rehearsal does not always require textbooks, pencil and paper. For younger pupils, rehearsal of number bonds and sequences can draw on a canon of games and songs involving dice, dominoes and counting sequences.[footnote 158]Computing technology can also help pupils acquire number facts by providing them with enough repetitions and direct feedback in ways that they enjoy.[footnote 159]Pupils also experience more progress and enjoyment of computer maths games when core content is introduced as separate learning components that are systematically followed by mini-games than if content is entirely subsumed into the gaming conditions.[footnote 160]This approach can also be a useful intervention for pupils withSEND.[footnote 161]However, teachers should take care with this approach because not all children make the same progress when learning with computers.[footnote 162]
Tasks that are content-focused and achievable
In mathematics, studies suggest that long-term retrieval of core content should be a focus of teachers and leaders planning.[footnote 163]This means teachers should set pupils tasks that focus on rehearsal of facts, methods and strategies in addition to tasks that develop pupils understanding.
Activity observation might show that pupils are engaged with and enjoying an activity, but if pupils are spending large amounts of time making choices, working out what to do or setting out, such as when physical apparatus is involved, their attention and learning can be compromised.[footnote 164]For example, drawing, measuring and comparing the angles in a polygon to find out and then learn the formula for the sum of angles means that pupils think about the formula in the last few minutes of the lesson. Even imagery can be distracting: textbooks in countries where pupils do well have fewer non-content related and distracting illustrations, pictures and cartoons.[footnote 165]
Pupils are more likely to engage in disruptive behaviours if they are expected to complete tasks that they have not mastered the component parts of yet. They are more likely to stay on task and be motivated if tasks are achievable.[footnote 166]In turn, sustained completion of tasks helps pupils to improve their ability to focus.[footnote 167]It is better for pupils to initially learn and rehearse content as component parts before learning the conditions for its use within a composite skill.[footnote 168]This has implications for challenge because pupils tend to resort to using the methods they have most facility with, rather than those that are most valid and that have been recently taught, when faced with unfamiliar or demanding tasks. For example, pupils tackling tricky arithmetic problems will default to addition,[footnote 169]pupils who are new to working with algebra will default to arithmetic methods or trial and error,[footnote 170]and new learners of calculus will fall back on familiar concepts that are visually similar, but unrelated to the question.[footnote 171]
When pupils are ready to solve problems, they need to be able to hold a line of thought and to concentrate.[footnote 172]Background noise and general chit-chat have been shown to negatively affect pupils ability to understand what the teacher is saying, to maintain appropriate behaviours and to concentrate. The children who are most affected are those who are under 13 and children withSEND.[footnote 173]Studies have also shown that the ideal environment for periods of independent work is one that is not just quiet but is in fact near silent.[footnote 174]That is not to say that all rehearsal experiences should be silent. Group work can aid pupils development of explanations, providing it is tightly managed.[footnote 175]However, there are limits to its impact on learning as it does not always improve attainment and is difficult to implement.[footnote 176]Teachers should balance opportunities for discussion with pupils needs for quiet periods of time to think.
Scaffolds as aids, not crutches
Teachers need to give careful consideration to how they use scaffolds, frames, physical apparatus and alternative information sources for pupils identified as needing extra support. There is a distinction to be made between using physical apparatus to reveal useful information[footnote 177]and its habitual use as an outsourced memory.
Reliance and subsequent dependence on manipulatives and associated aids can hinder progression through the curriculum.[footnote 178]The implications are that teachers need to give pupils enough time to consolidate learning and they need to plan for how pupils will move away from using the manipulative. This will help to avoid pupils relying on manipulatives to work around gaps in core knowledge that might become barriers to learning later.
Balancing rehearsal of proof and explanations with rehearsal of facts, methods and strategies
There are 2 types of practice:
type 1 involves the rehearsal of core facts, methods and strategies that can be used to complete exercises and solve problems now and in the next stage of education
type 2 includes explaining, justifying and proving concepts using informal and diagrammatic methods, parsing and derivation of number
Teachers need to create balance between these 2. It is helpful for pupils to replicate explanations and proof as a way of improving their own conceptual understanding of the why, but when it comes to learning how to find solutions to problems, practice of the methods of calculation themselves so that they can be recalled in the long term is likely to be a key to proficiency,[footnote 179]particularly for pupils who are identified as being more likely to struggle.[footnote 180]This gives greater assurance that pupils can use core knowledge of facts, efficient methods and useful strategies in the next stage of their education. Close inspection of curriculums in countries where pupils do well shows that systematic rehearsal emphasises learning and applying core facts and methods alongside, rather than after, the development of conceptual understanding.[footnote 181]
Based on the above, high-quality maths education may have the following features
Educators plan to give pupils opportunities to consolidate learning that:
go beyond immediately answering questions correctly
align with the detail and sequence of the curriculum
are free of distraction and disruption
strike a balance between type 1 and type 2 practices
avoid creating a reliance on outsourced memory aids or physical resources
help pupils to avoid relying on guesswork, casting around for clues or the use of unstructured trial and error
Assessment during the learning journey is most useful when it focuses on the component knowledge that pupils have learned. This approach aids pupils confidence and makes it easier to analyse and respond to gaps in learning. In mathematics, pupils benefit from timed practice of knowledge that should be easily recalled, such as maths facts. The timing element gives assurance that pupils are not reliant on derivation.
Frequent low-stakes testing helps to prepare pupils for the final performance
Summative assessments of learning need to provide easily comparable information to all stakeholders, including parents and the pupils themselves, on a regular basis.[footnote 182]Module exams provide short-term goals and a sense of achievement, but they can promote a just in time approach to learning that means that knowledge is jettisoned soon after tests are taken. End-of-course examinations give greater assurance that the learning of content is long term.[footnote 183]This suggests that a mixture of approaches is best: regular tests of content recently taught and learned and an objective, fair and accurate summative assessment at the end of the year or course.
Leaders should, however, avoid conflating the 2 concepts with frequent use of summative tests, such as past papers. These can cause lower attaining pupils and pupils withSENDto be regularly reminded about what they do not know and cannot do (which may inculcate guesswork, misconception rehearsal or avoidance tactics). Over time, pupils who do not experience success can then become demotivated, which may negatively impact on their chances of attaining a pass when re-taking courses between 16 and 18.[footnote 184]
However, it is not the tests but lack of proficiency that causes this performance anxiety. Lack of proficiency can also be compounded with the use of realistic settings of story problems that present a language barrier for disadvantaged children.[footnote 185]Teachers can ensure that pupils come to see tests and testing as moments to shine by adopting the principles underpinning the causal pathway to motivation and enjoyment in the subject. This can be achieved by seeking to engineer proficiency and initial success in the subject.
When pupils obtain levels of proficiency, they look forward to and enjoy tests.[footnote 186]Competitive maths games are, for example, more effective for learning and retention than non-competitive games.[footnote 187]The goals of trying to achieve a personal best and doing well compared to the average mediate later attainment.[footnote 188]Therefore, in addition to ensuring pupils are well prepared for tests, leaders should ensure that benchmarks for success are understandable.
Frequent, low-stakes testing of taught content can help prepare pupils for summative tests by providing memory-enhancing opportunities to recall and apply taught content.[footnote 189]Low-stakes testing also works well when tests of component parts, such as mathematics facts, are timed.[footnote 190]If teachers give honest feedback, pupils interest and sense of self-efficacy also increases.[footnote 191]Teachers can also set benchmarks for mastery of facts and methods so that they can be assured that pupils are recalling rather than guessing or deriving.[footnote 192]Tests should therefore be aligned closely with curriculum sequences because generic tests are not able to give this feedback.[footnote 193]
Based on the above, high-quality maths education may have the following features
Pupils are well prepared for assessments through having learned all the facts, methods and strategies that are likely to be tested.
Teachers plan frequent, low-stakes testing to help pupils to remember content.
Lessons incorporate timed testing to help pupils learn maths facts to automaticity.
Systems at the school level
Leaders can support pupils progression through the mathematics curriculum by ensuring that pupils bookwork is of a high quality. This is important because when pupils calculations are systematic and orderly, they are better able to see the connections of number and to spot errors that can be corrected. Leaders can also plan to develop teachers subject and subject-pedagogic knowledge through giving teachers opportunities to work with and learn from each other. This, for example, helps new teachers to see and adopt useful ways of explaining core concepts, methods and strategies to the pupils they teach.
Calculation and presentation
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.
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
The need for firm foundations applies to all novices, including novice mathematics teachers. Regular observations are often viewed as a main driver of professional development, where teachers are given feedback on aspects to improve. Use of the teacher standards for these reactive approaches prioritises on-the-ground development of pedagogical and subject-pedagogical expertise, highlighting features that are absent or in need of correcting along the way.
It may be tempting for leaders to focus on teacher-to-student relationships as an indicator of high-quality teaching and learning. However, analysis of pupils attainment and attitudes suggests that a focus on pupils effort and interest in the subject may matter more.[footnote 194]Given that initial teacher training can be variable in terms of pedagogical, subject-pedagogical and subject-specific knowledge,[footnote 195]we cannot assume that all novice mathematics teachers will possess all the tools they need to make the most successful start.
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:
regular opportunities to observe and be mentored by experienced and successful teachers of mathematics
provision of sequenced schemes of learning, matching textbooks and teacher notes to aid explanations and help the novice teacher to bring the subject to life[footnote 197]
systematic plans to build these models of instruction and rehearsal over time so that future generations of teachers can benefit
collaborative planning with more experienced and successful teachers of mathematics
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]
Teachers should also seek to renew and improve their subject knowledge, even if they are teaching foundational concepts.[footnote 200]For example, teachers of primary age groups gain when they know the foundational principles that pupils can learn to help them with later algebra lessons.[footnote 201]
Based on the above, high-quality maths education may have the following features
School-wide approaches to calculation and presentation in pupils books.
School-wide approaches to providing time and resources for teachers to develop subject knowledge and to learn valuable ways of teaching from each other.
Throughout the review, the theme of engineering success, underpinned by systems thinking, predominates. These approaches seek to transform an offer of content into more of a guarantee that content can and will be learned. The outcomes of this systems thinking are the observed features and approaches of successful mathematics education:
detailed codification and sequencing of the facts, methods and strategies that pupils will acquire
instructional coherence and aligned rehearsal that increase the chances of understanding and remembering while minimising the need for guesswork or trial and error
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.
Quality and quantity of practice is a vital key that unlocks the development of dual tracks of conceptual understanding and procedural fluency. Further, in observing pupils relative expertise and proficiency, such as in a problem-solving lesson, teachers and leaders should be mindful of the journey that pupils took to achieve problem-solving proficiency. This journey will have involved more than the features and activities of the lessons that proficient mathematicians are taking part in at the time. Variation in the quality of mathematics education in England is likely to be the result of the absence of systems and systems thinking, as well as possible gaps in content, instruction, rehearsal, assessment and the plans for their evolution over time.