ADVANCED MATHEMATICS FOR FORM FIVE – ALL TOPICS

GET BETTER AT MATHEMATICS – PART 1

1. Focus on Understanding Instead of Just Memorizing

Instead
of relying solely on memorization, prioritize understanding the core principles
of math. This is what distinguishes true mastery from rote learning. Understanding
the ‘why’ behind a method equips you to apply it across various situations. For
instance, grasping the concept of variables in algebra enables more effective
problem-solving. To reinforce new concepts, connect them to real-life
scenarios. For example, relate percentage calculations to shopping discounts or
consider ratios in the context of recipe ingredients

2.
Understand the Math Before Moving on To Another

Math
is similar to reading because if you don’t know how your letter sounds, you
have no chance of uttering words, so you’d be unable to read a phrase or a
sentence. All math courses follow a prescribed order since each topic builds on
the preceding one. If you’re experiencing difficulty with a particular subject,
work on it until you comprehend it and solve it. Don’t skip various topics
because it might hamper your progress as you go on.

3.
Implement daily practice

Math
practice is important. Once you understand the concept, you have to nail down
the mechanics. And often, it’s the practice that finally helps the concept
click. Either way, math requires more than just reading formulas on a
page. Daily practice can be tough to implement, especially with a math-averse
child. This is a great time to bring out the game-based learning mentioned
above. Or find an activity that lines up with their current lesson.

………….

Most other schools took ad-hoc approaches to supporting pupils who had moved set. These had varying degrees of success.

Example of stronger practice:

One school employed a subject specialist whose role included providing extra support for pupils who needed it. When a pupil moved teaching group, they received additional small-group or one-to-one teaching from the intervention tutor to make sure that they did not have any gaps in necessary prerequisite knowledge. This made it more likely that the pupil would integrate successfully into the new class.

98. In some schools visited, lower-attaining pupils completed a key stage 4 entry-level qualification. This worked most effectively when leaders designed the mathematics curriculum so that pupils could also be entered for GCSE mathematics, where appropriate. Using the entry-level qualification in this way made sure that the curriculum for these pupils was not narrowed and did not limit the qualification that they could achieve.
99. In a small number of schools, high-attaining pupils were given the opportunity to study for additional mathematics qualifications beyond GCSE, including level 2 additional mathematics and level 3 free-standing mathematics qualification (FSMQ) additional mathematics. Typically, pupils took the examinations for these qualifications at the end of Year 11, alongside their GCSE mathematics, rather than taking their GCSE mathematics early. This approach benefited pupils, as it allowed the curriculum to be structured in a coherent manner. It made the links between GCSE mathematics and the additional qualification clear, rather than artificially dividing the mathematical knowledge between GCSE and the additional qualification.

Pedagogy: new learning

Summary from the research review

The novice, whether they are starting school or starting a new topic, needs more instruction rather than less. Teaching should help them on the journey to expertise. Ideally, teaching should be systematic, follow the curriculum sequence and help pupils to understand. A systematic approach works well for pupils of all ages and stages.

100. In many schools where mathematics provision had historically not been strong, leaders discussed the challenges of staff recruitment and having less-experienced mathematics teams, often with a number of non-specialist teachers. In these schools, it was common to see teachers using plans and resources developed by others – either from commercial schemes, from multi-academy trust subject teams or developed ‘in house’ by more experienced staff. Textbooks were used in very few schools. When they were, they were usually associated with the specific commercial mathematics scheme that the school was using. Requiring teachers to use provided lesson plans and resources gave leaders a level of quality assurance of the mathematics provision that pupils would receive from teachers who lacked some subject knowledge or subject specific pedagogical knowledge.
101. In almost all schools, leaders said that teachers were expected to adapt and adjust provided resources to meet the specific needs of pupils in their classes. It was, however, relatively rare to see adaptations take place to resources usually being used without any adjustments. On occasions, this led to pupils studying new content without being secure with necessary pre-requisite learning or pupils moving onto new content before they were ready. Some leaders identified that pre-prepared lesson plans and resources were a ‘necessary but insufficient’ step on the department’s journey of improvement. They had correctly identified that some teachers’ subject knowledge and subject teaching knowledge needed to improve to allow them to make informed decisions about appropriate necessary adjustments to meet the needs of pupils in their classes.
102. In schools where mathematics provision had historically been strong, teachers were often given more flexibility to write or source the resources to use with their classes. This was a successful approach in this context. Often, these departments would have physical and electronic stores of shared resources to reduce teachers’ workload. The departments typically had settled staffing, and new staff were inducted relatively informally into the department’s agreed approaches.
103. In some schools, we saw misconceptions being introduced by teachers. These errors usually occurred in lessons taught by less experienced or non-specialist teachers, who lacked the subject knowledge or subject teaching knowledge to go ‘off script’ when responding to pupils’ questions or when they observed misconceptions.

Example of stronger practice:

The mathematics leader in one school had identified that the use of pre-prepared resources and lesson plans was necessary, in their context, to support the number of non-specialist staff in their team but was not sufficient to ensure high-quality provision. They had identified that ‘off-script’ provision (when teachers needed to respond to pupil questions, misconceptions or errors) was sometimes poor. Senior leaders had dedicated significant subject meeting time to providing CPD to teachers in the department. This aimed to improve subject knowledge and subject teaching knowledge.

Example of stronger practice:

One small trust had developed a central ‘resource hub’ of high-quality resources that teachers could use with their classes. These resources, why they were viewed as being high quality and how they aligned with the curriculum were discussed in department meetings. This approach had been introduced because leaders had noticed that the resources being selected by teachers did not always align with the planned curriculum and this was limiting pupils’ learning.

The ‘resource hub’ was constantly evolving as teachers from across the trust developed their understanding of the curriculum and effective resource design, and added new resources that were quality assured by mathematics leaders. Leaders and teachers identified that this approach:

provided teachers with the professional autonomy to select resources that met their pupils’ needs

ensured that resources selected aligned with the curriculum

developed teachers’ expertise

helped teachers to manage their workload while meeting the needs of pupils in their classes

Pedagogy: consolidation of learning

Summary from the research review

Practice helps pupils to understand and remember mathematical knowledge. There are broadly 2 types of practice. Type 1 involves retrieving and rehearsing facts, methods and strategies to the point of familiarity, speed and accuracy. Type 2 practice is more exploratory. It requires pupils to explain relationships, prove that they understand them and describe their reasoning. Both types are important. Pupils need quantity and quality of practice to help them understand and commit knowledge to long-term memory. This practice does not always involve textbooks and worksheets. It can include songs, games and rhymes.

Tasks should help pupils to focus on the mathematics to be learned. They should provide for overlearning and, ideally, include variation. Support for learning and understanding should be gradually withdrawn over time. Tasks should give pupils opportunities to be successful, rather than having to rely on guesswork or unstructured trial and error.

104. In all schools, pupils were given the opportunity to practise new content in lessons. In most schools, teachers and leaders carefully selected exercises that became more complex over time. This ensured that pupils’ knowledge deepened as they worked through the exercises set over a sequence of lessons. In a minority of schools, however, teachers had not planned for exercises to become more complex over time. As a result, pupils remained unable to apply their knowledge to more complex problems.
105. Most schools gave pupils opportunities to practise mathematics after they had first learned it. For example, homework was routinely used to practise prior learning, rather than focusing on current learning. Teachers also provided frequent low-stakes quizzes on prior learning and lessons in which pupils routinely spent time answering questions on topics they had studied previously. These approaches helped pupils to develop accuracy and speed in their mathematics.
106. There was much planned practice in the schools visited. However, in only a few schools had teachers agreed on how much practice was sufficient. Often, the amount of practice was determined by how long was left in the lesson once the teacher had modelled new content. It was not based on judgements about whether pupils had successfully engaged with sufficient, well-designed practice to learn the intended mathematics to automaticity. Some pupils were moved on to new content before they had had a chance to get to grips with what they had learned. This was most typically an issue for pupils who were grasping new concepts more slowly. These pupils’ workbooks showed that, in some cases, they would complete just a fraction of the practice of their classmates. They were not benefiting from the carefully developed exercises, and their practice was restricted to a few routine applications that were not practised enough for security. This left many pupils unable to build successfully on these concepts later.
107. In most schools visited, pupils did not practise problems enough that required them to explain, prove, justify or describe relationships. This runs the risk that pupils have a shallower understanding of mathematics, with their knowledge restricted to routine application of learned algorithms.

Assessment

Summary from the research review

Frequent, well-timed, low-stakes testing is useful for checking pupils’ knowledge of key facts and methods. This helps pupils to remember and gives leaders an insight into the gaps in pupils’ knowledge. Summative assessment should assess what pupils have learned and rehearsed, rather than what they do not know and cannot do.

Assessment for learning

108. In all schools, leaders said that teachers would routinely make sure that pupils had securely learned conceptual knowledge before starting new learning that built on it. However, this assertion was consistently accurate in very few of the schools visited.
109. Teachers used a range of approaches to assess pupils’ pre-existing knowledge/understanding. These included quick low-stakes quizzes, with pupils’ answers displayed on mini-whiteboards or electronic devices. When used with well-considered questions, these approaches gave teachers appropriate assurance that pupils had the knowledge and understanding necessary to move on to the next stage of the curriculum.
110. However, there were weaknesses in how effectively teachers struck a balance between assumption and assessment when checking pupils’ pre-existing knowledge. In a minority of lessons, teachers assumed that the majority of pupils had a firm grasp of pre-existing knowledge based on the answers of only 1 or 2 pupils. In others, teachers knew that pupils had been taught prerequisite knowledge and assumed that they had learned it. In these lessons, teachers moved on to new content while significant numbers of pupils had gaps in their knowledge or a lack of automaticity that would limit their chances of successfully learning new mathematics.
111. The mathematics national curriculum states, ‘Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on’. Despite this, in just under half the schools visited, teachers’ implicit, and occasionally explicit, focus was on ‘covering the curriculum’ rather than on securing learning. In these schools, teachers often said that they felt pressure to ‘stick to the timings in the scheme of work’ for fear of not completing the scheme by the end of the academic year.
112. In the other schools, teachers’ focus was more clearly on securing learning. In these schools, teachers moved on to the next conceptual step only when pupils were ready. Some schools achieved this by making sure that there was sufficient slack in the curriculum’s suggested timings. Others did so by ensuring that teachers worked through the curriculum at the pace suitable for pupils’ needs, and had secure methods for passing information on to the next teacher at the end of each academic year.

Assessment as learning

113. The majority of schools visited provided frequent opportunities for pupils to recall and apply mathematics they had learned previously. These were usually brief activities carried out in most, or sometimes all, mathematics lessons.
114. In most of these schools, teachers monitored pupils’ success in these recall activities. In the most successful lessons, when teachers identified a lack of secure knowledge, they quickly decided whether the best course of action was to spend a short time addressing the difficulties as a class or to revisit the topic in more detail in a future lesson. In less successful lessons, teachers sometimes tried to address the difficulties in the same lesson, but without careful planning or appropriate resources. This tended to be more time-consuming and was usually less successful.

Example of weaker practice:

In one lesson visited, pupils were adding fractions with different denominators as part of a ‘retrieval practice’ activity. A significant number of pupils in the class simply added the numerators and the denominators together. The teacher tried to address this common misunderstanding by re-teaching the concept to the class. The teacher selected 5/12 + 3/8 as the example to use to explain it. The teacher changed both fractions into 96ths, rather than 24ths, while talking about ‘lowest common multiple’. The lack of planning led to a poor choice of example to re-explain the method and limited the impact of the intervention.

115. In the majority of schools, teachers had a common understanding about the purpose of these opportunities: to practise the most crucial knowledge and skills that pupils had learned previously, but that was not yet learned to automaticity. When pupils practised retrieval for this purpose, they generally had a high success rate, as they were embedding content they already knew. In some schools, teachers did not have a common understanding of the purpose. Some teachers used ‘retrieval practice’ for practising exam-style questions or for assessing whether pupils were ready to move on to new learning. This lack of common purpose meant that retrieval practice was less successful in achieving its primary aim: to embed previously learned knowledge more deeply in long-term memory.

Example of stronger practice:

In one school, leaders had identified what important mathematics content needed to be learned to automaticity from each stage of their curriculum. They had collectively developed resources to be used for these brief retrieval activities that took place in lessons, and pupils recorded their answers on mini-whiteboards. Teachers had a common understanding of how to follow up these activities. They expected a high success rate. Any minor errors dealt with ‘in the moment’. More often, the teacher noted the errors, which fed into their planning for future lessons.

116. In some schools, pupils were practising mathematics that they had not fully understood at the time of first learning. Pupils were repeating this failure including misconceptions. These misconceptions were becoming embedded.

Example of weaker practice:

One department had a policy that every mathematics lesson began with pupils being asked to complete ‘fast 5’ questions drawn from prior learning. A significant number of pupils in the class incorrectly found 5% by dividing by 5. Following the department policy, the teacher displayed the correct answers to the ‘fast 5’ questions on the board, and pupils self-corrected any wrong answers. Pupils’ workbooks showed that many of them were regularly unable to recall prior learning. They had made this error repeatedly, and dutifully written the correct answer, during spaced practice activities over a number of months. Further investigation of pupils’ workbooks showed that many had consistently made this error when they were first taught how to find percentages without a calculator earlier in the year. At no stage had the teacher been concerned that pupils were regularly unable to recall the intended curriculum content or identified that pupils had this particular misconception and taken steps to address it.

117. These approaches were, usually, limited to the recall of a restricted repertoire of mathematical facts and procedures. Where used consistently well, these approaches were highly effective in supporting pupils to develop automaticity in recalling and accurately applying whatever mathematics needed to be learned.
118. In very few schools were pupils required to practise solving problems using the facts and methods they had previously learned. The exception was when they completed exercises when first applying new mathematical techniques to problems. This lack of practice, except at the point of first learning, prevented pupils from becoming fluent at identifying likely appropriate mathematical approaches to use when faced with unfamiliar problems.

Example of stronger practice:

In one school, leaders had identified that, as well as initially completing exercises with structurally similar problems in class, pupils needed to decide on the appropriate approaches to use when faced with an unfamiliar problem. Leaders had designed their curriculum to ensure, through spaced practice activities, that pupils were solving problems after the point of first teaching. These spaced practice activities required pupils to recall prior learning and choose the most appropriate approach to solve familiar and superficially unfamiliar problems. The problems that pupils had to solve varied from short word problems to longer problems, such as the ‘Frogs’ investigation. These were aligned with pupils’ prior learning.

Assessment of learning – have curriculum goals been achieved?

119. Most schools used termly or half-yearly tests to find out whether pupils had successfully learned the taught curriculum. In most of these schools, the tests were carefully designed to assess what pupils had been taught. In a minority of schools visited, pupils were asked to take tests that included topics that they had not studied and questions that were therefore impossible for them to answer. This was particularly the case when the school was using commercial tests or past GCSE examination papers. This is an inefficient use of pupils’ time, which could be better spent learning new mathematics. It could also harm pupils’ perceptions of their mathematical capability.
120. Teachers routinely followed summative assessments with activities designed to review pupils’ performance in the tests. While summative assessments do not lend themselves to formative purposes, they were most helpful when teachers used them to:

isolate the specific gaps in the building blocks of knowledge that pupils needed to answer each question correctly

plan new teaching and practice sequences to address those gaps

121. Going over the test by modelling correct answers was not so successful. It did not identify the specific gaps in pupils’ knowledge or provide pupils with additional teaching or opportunities to practise the areas of mathematics they had struggled with.
122. In the majority of schools, test outcomes were converted into some form of grade or attainment descriptor. The thresholds for expected performance were most often set in line with those for GCSE exams. In many schools, little attention was paid to identifying gaps in pupil knowledge and effectively addressing them. Accepting this low level of pupil success as ‘good enough’ and moving on results in many pupils moving to the next stage of their learning with significant gaps in their mathematical knowledge.
123. In some schools, information from internal and external summative assessments was used to systematically provide feedback on the effectiveness of the way the curriculum was designed or implemented. In these schools, leaders used the assessment information to inform the ongoing development of the curriculum. As a result, the design and implementation were increasingly effective in supporting pupils to learn.
124. In a small number of schools, assessments served little purpose other than determining a current or forecast grade to report to parents and deciding whether a pupil should change teaching class.
125. Many schools visited use the outcomes of tests to track pupils’ progress against GCSE, or similar, targets. Teachers were sometimes expected to make intervention plans for pupils who were ‘behind target’. This resulted in some pupils with identical mathematical needs not receiving the support given to others. Allocating intervention resources in this way inappropriately limits pupil access to support based on the mathematics they knew potentially 5 years previously.

Systems at the school level

Summary from the research review

School-level systems strengthen the consistency of a pupil’s journey to proficiency. They include monitoring approaches, staff training, resource allocation, teaching and learning expectations, and ways of raising the subject’s status, and ways of sharing information between stakeholders. Professional development should be a planned and purposeful pathway to expertise in teaching and subject leadership.

126. Schools visited had taken differing approaches to timetabling mathematics lessons. Averaging over 5 years, the majority of schools ensured that pupils had between 3.5 and 4 hours of mathematics lessons per week. In a small number of schools, pupils had only 3 hours.
127. Schools varied in how they split mathematics curriculum time across the week. In most schools, pupils had mathematics lessons on 3 or 4 days per week. However, in some schools, pupils had 2 much longer lessons per week instead. In a small minority of schools, pupils had a mathematics lesson every day of the week.
128. In schools that varied the number of mathematics lessons per week between year groups, it was usual for pupils in Year 11 to have slightly more mathematics lessons per week than pupils in key stage 3. Often, this additional teaching time in Year 11 was at the expense of non-examined subjects such as personal, social, health and citizenship education (PSHCE), religious education or physical education. In a small number of schools, Year 11 pupils received an extra lesson of mathematics teaching per week in compulsory ‘after-school’ lessons.
129. At some stage, all schools visited arranged teaching groups based on pupils’ current levels of attainment. Usually, this was based on current levels of mathematics attainment (sets), although occasionally grouping was based on attainment across a wider range of subjects (streams). Almost all schools arranged the school timetable so that pupils could change teaching group midway through the school year if this was judged to be necessary to support their mathematical development.
130. When pupils were placed into classes based on current attainment, it was often on the basis of key stage 2 national curriculum test scores, or baseline assessments taken as pupils started Year 7. It was rare for leaders to include information from Year 6 teachers when deciding classes. This sometimes led to pupils being placed in a teaching class that did not meet their needs, when their key stage 2 assessment did not accurately reflect their level of mathematical knowledge.
131. Where mathematics groupings were not based on current attainment, this was usually in Year 7. A number of leaders said that they had introduced mixed-attainment teaching in Year 7 as a temporary response to the disrupted education and lack of key stage 2 national curriculum test results due to COVID-19. Many of these schools were intending to return to ‘sets’ once they were confident that they had sufficient and accurate assessment information to do so.
132. A small number of schools timetabled an additional mathematics lesson per fortnight for Year 7 pupils who had arrived with significant gaps in their mathematical knowledge. These pupils were typically identified through discussions with primary schools and through their key stage 2 scores. These groups tended to study a curriculum with an assumed earlier starting point rather than identifying and addressing specific gaps in knowledge.
133. Several schools identified that external tutors had had limited success with their pupils. This was because the tutors’ work was not always aligned with the school’s curriculum thinking. These schools had had more success through employing tutors directly or by spending time making sure that external tutors understood and applied the school’s mathematics curriculum.
134. Most schools used departmental meeting time to improve the curriculum and the way it was put into practice. Many subject leaders said that, over the last few years, there had been a significant move from using subject meeting time for administration activities to using it as an opportunity to think about effective teaching.
135. There were significant differences in how often subject teams met. In some schools, subject teams met almost weekly. In other schools, meetings were limited to once or twice a half term. Providing limited opportunities for members of the mathematics department to work together increases the risk of incoherence in implementing the mathematics curriculum.
136. There was a noticeable difference in teachers’ CPD between schools with historically stronger provision and those where it had been weaker. In schools that had historically stronger provision, CPD was most often department-led, and often focused on effective teaching of specific parts of the mathematics curriculum. In schools with historically weaker provision, CPD was often at whole-school level on more generic themes, such as ‘retrieval’. In these schools, each department was sometimes given time to consider how this CPD could improve mathematics provision in the school.

Appendix

Methodological note

This thematic report draws on findings from 50 research visits to schools in England. These visits were carried out between September 2021 and November 2022.

Deep dives into mathematics took place as part of scheduled school inspections under the education inspection framework. They were carried out by inspectors with relevant expertise in mathematics education who had received training for this work. They carried out a deep dive as part of our methodology for evaluating the quality of education. Inspectors gathered a rich range of data from speaking to senior leaders, subject leaders and teachers, visiting mathematics lessons and speaking to pupils. They also reviewed pupils’ work in mathematics.

Various criteria were monitored consistently to identify characteristics for the sample that risked being underrepresented. These criteria were: region, inspection outcome, disadvantage quintile, size of school, and a rural or urban location. We made sure that the sample was broadly representative of the national picture and there was some representation from schools with different characteristics. The deep dive evidence collected was split evenly between primary and secondary schools.

Inspectors gathered qualitative evidence about mathematics education in schools they visited. The range of evidence gathered across these visits enabled us to identify common themes in mathematics education which are likely to be relevant in a wide range of schools.

Inspectors focused on gathering evidence which related to the following areas:

curriculum

pedagogy

assessment

school-level systems and their impact on mathematics education

When analysing this evidence, we drew on the conception of quality in mathematics education which we outlined in our mathematics research review. This enabled us to consider how mathematics education in English schools relates to our best evidence-based understanding of how schools can ensure a high-quality mathematics education for all pupils.

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Primary

Curriculum intent: identifying what pupils need to know and do

Summary of the research review

The curriculum should identify and sequence, in small steps, declarative, procedural and conditional knowledge, and plan for pupils to learn this in small steps. This will make sure pupils’ so that pupils’ knowledge builds steadily over time. Linked facts and methods should be sequenced to take advantage of the ways that knowing facts helps pupils to learn methods, and knowing methods helps them to learn facts. Declarative and procedural knowledge can be combined and taught as strategies for problem-solving. A well-sequenced curriculum, and systematic teaching and opportunities for practice help pupils to become proficient in mathematics. This leads to success and motivation in the subject.

1. Whether adopted or ‘home-grown’, most primary schools’ mathematics curriculums covered the national curriculum and were carefully sequenced. Leaders and teachers were aware of the importance of setting out a coherent path of progression for pupils. Often, planning included ‘small steps’ for each lesson, revisiting of previous learning and scheduled ‘buffer zones’ to allow teachers to respond to pupils’ needs.
2. Occasionally, subject areas such as geometry were moved to later in the academic year. This was so that pupils could focus on foundational number and arithmetic at the start of the academic year. A small proportion of pupils were relatively insecure in their knowledge of geometry facts. This was possibly because geometry topics had been allocated to the summer term, leading to long gaps before pupils revisited them. In contrast, some schools had organised their curriculums so that a day a week was devoted to geometry, to make sure that pupils were regularly revisiting this topic.
3. Leaders’ reasons for adopting commercial schemes included a desire to improve sequencing and resources. These schemes often included planned sequences of recapping and revisiting previously learned content. Leaders realised that this reduced the planning workload for teachers. This is a positive development.
4. ‘Home-grown’ curriculum plans were sometimes the result of collaboration across a multi-academy trust. In many schools, the NCTEM’s ‘ready-to-progress’ criteria provided a useful framework for content. This helped leaders and teachers to see how concepts were interconnected and built over time. Additional resources from commercial schemes, to provide much-needed extra practice for pupils, were common.
5. Leaders wanted staff to follow the curriculum plans closely. This made sure that pupils experienced a smoother path to proficiency, supported with consistent language and routines, rather than a series of disjointed lessons. However, leaders also encouraged teachers to adapt lessons in response to the needs of their classes. The NCETM’s ‘ready-to-progress’ criteria helped many teachers to prioritise key content.
6. In some schools, curriculum sequencing in the Reception Year was less detailed. Occasionally, leaders used early learning goals for curriculum planning. Potentially, this approach is problematic because the early learning goals are not frameworks to be used in this way. In contrast, the most effective Reception Year curriculum planning was as detailed as the planning that teachers of older pupils had access to. In many schools, staff had worked together to make sure that the curriculum prepared children for Year 1.
7. In some schools, leaders made sure that parents and pupils could find out what pupils would learn. One way of doing this was through sharing ‘knowledge organisers’. These were used as well as, rather than instead of, curriculum planning.

Declarative knowledge

Summary of the research review relevant to declarative knowledge

The curriculum should identify and sequence key facts, formulae, concepts and vocabulary. This helps pupils to avoid relying on derivation, guesswork or looking for clues.

8. In many schools, staff wanted pupils to learn key mathematics facts by heart. Pupils knew that this knowledge was important, too. The introduction of the times tables check in Year 4 has undoubtedly raised the profile of this type of knowledge. Curriculums generally emphasised mathematics facts, such as times tables, alongside helping pupils to understand the connections within ‘families’ of numbers. It is likely that these factors are the reason why it was rare to see pupils at key stage 2 having to rely on times tables grids.
9. Pupils in key stage 1 were often expected to develop flexibility and ‘deep understanding’ when thinking about number. However, there appeared to be less emphasis on learning addition and subtraction tables (number bonds) by heart. This is potentially problematic, as pupils need to be able to recall this type of knowledge quickly in order to access more complex mathematics in key stage 2.
10. In some schools, the curriculum in the Reception Year and key stage 1 emphasised both understanding and quick recall of addition facts. The NCETM’s Mastering Number programme was particularly helpful. In these schools, pupils were successful and received lots of praise. They were learning how to subitise (recognise a number of objects without having to count), understand numerical concepts and recall addition facts. Key features of this programme, in addition to a carefully sequenced curriculum, include:

whole-class teaching

use of a rekenrek (a type of counting frame)

low-distraction dice patterns for subitising

clear diagrams and representations

regular questioning

videos for staff training

11. Many schools’ curriculums identified and sequenced mathematical vocabulary, sentence stems and speaking frames. It was rare to see pupils relying on a ‘working wall’ for such vocabulary. Teachers explained and modelled important, age appropriate, mathematical vocabulary, such as ‘part – whole’, ‘lowest common multiples’, ‘array’ and ‘integer’. In Reception Year, songs and rhymes to helped with early language learning. Older pupils used technical vocabulary in their discussions and their writing. Occasionally, pupils had gaps in their vocabulary that affected their ability to reason and problem-solve.
12. In most schools, teachers quickly identified rare misconceptions in declarative knowledge. Older pupils were often accurate in their recall of declarative knowledge. However, in younger year groups, some pupils were unable to subitise or easily recall addition facts. Staff noted the impact of the pandemic restrictions on pupils’ ability to recall this knowledge. In response, leaders adapted planning to give them additional opportunities to revisit it. Occasionally, pupils’ books showed that they inverted numbers. This phenomenon may be linked to a lack of coherent sequencing, or insufficient modelling and practice in the Reception Year.
13. Younger pupils’ inability to subitise or easily recall addition facts hampers their progress. They may be able to understand a teachers’ instruction in, for example, Year 2 or 3, but they struggle to complete tasks with the speed and accuracy of their peers. They eventually obtain the correct answers (‘getting by’), thus demonstrating their understanding, but are less likely to remember this new knowledge. This cycle continues, but with pupils increasingly unable to understand, let alone apply, new knowledge. This is likely to be one of the reasons why interventions are so ubiquitous in Year 6: pupils’ internal struggles manifest after a significant amount of time has elapsed.

An example of stronger practice, in which pupils’ success with fractions was underpinned by strong knowledge of mathematics facts:

In one school, a joined-up approach to curriculum, teaching and pupil practice had helped older pupils to gain knowledge and confidence in working with fractions. Among many other positive features, the mathematics lead was helping to steer teachers towards consistency in carrying out the school’s chosen scheme of learning. Teachers were encouraged to slow the pace of learning, if necessary, to make sure that pupils mastered this knowledge before moving on. Learning key mathematics facts in class, for homework and through competitions between classes was part of a recently revised strategy to help pupils learn mathematics facts to automaticity (the point at which they could use them automatically). The mathematics lead ran a mathematics club for pupils who were less likely to be able to do homework at home. A Year 6 book scrutiny that focused on fractions showed that the sequence of lessons on fractions built in complexity in a logical order. Pupils successfully used different pictorial and concrete representations to help them to understand fractions. They were getting enough practice in calculating fractions. Pupils’ highly accurate work with equivalent fractions showed the impact of leaders’ work to make sure that pupils learned their mathematics facts to automaticity.

Procedural knowledge

Summary of the research review relevant to procedural knowledge

There is a difference between methods that help pupils to understand concepts and perform mental calculations and methods that are efficient and useful now and in the next stage of learning. The curriculum needs to carefully sequence the teaching of mathematical methods. It should allow for some early methods, such as one-to-one counting, parsing, derivation and complex diagrams, to fade over time (‘designed obsolescence’). Pupils should learn the most efficient, systematic and accurate mathematical methods, so that they can use them for more complex calculations and in their next stage of learning

14. Many primary schools’ policies for calculation set out how pupils will learn procedural knowledge in a logical way. This made sure that language and models were consistent. Sequences tended to move from expanded and informal methods to compact and formal methods. This approach prioritises pupils developing an understanding of mathematical ideas. For example, using the ‘grid method’ of multiplication helps pupils to understand place value and the concepts that underpin multiplication. However, this can be at the expense of developing automaticity in using efficient and formal methods. The expectation that pupils should develop significant automaticity in these procedures appeared to be less of a priority. This is problematic because efficient and formal methods, once learned, enable pupils to engage with problems using larger numbers, for example.
15. In some schools, leaders encouraged pupils to learn and then choose from a wide range of methods. The goal was to ensure that pupils had different methods at their disposal to solve a range of problems, and to practise selecting from them. However, such an approach could lead to pupils choosing ‘easier’ methods and not getting enough practice in using the methods they will need most in the future. This would make it more difficult for them to progress through the curriculum step by step
16. In some schools, younger pupils could use calculation tools quickly and accurately. For example, they used number lines or a rekenrek to help with addition and subtraction. This is likely to be the result of careful teaching of how to use these tools and of pupils’ secure knowledge of ‘one-to-one correspondence’ (the ability to match an object to the corresponding number and recognise that numbers represent a quantity), subitisation and ‘addition facts’.
17. Knowing how to set out written work is another form of procedural knowledge. In the best examples, leaders saw presentation as a part of the mathematics curriculum. Teachers carefully modelled and taught this form of knowledge. Textbooks and worksheets also helped to guide and support pupils’ presentation, which gave them a sense of pride. Careful presentation is also likely to help pupils spot patterns and identify their own mistakes.
18. When leaders talked to us about gaps in pupils’ knowledge, they often mentioned fractions. This was interesting, as this topic was generally sequenced well. Pupils grasped the conceptual knowledge underpinning this topic. Some leaders referred to the difficulties in teaching this topic online. However, working with fractions involves knowing and using procedural knowledge. Pupils can encounter difficulties when teachers have not prioritised procedural automaticity enough.
19. The bar modelhad been integrated into mathematics curriculums, which was a positive development. In addition to helping pupils understand proportion, for example, it also helped older pupils to solve word problems and algebraic equations.
20. In some schools, some older pupils could not remember methods of multiplication and division. Pupils need procedural fluency to be able to solve a range of problems and to then learn which types of problems a method is useful for. Both of these develop with practice. Lack of procedural fluency is likely to be one of the reasons why pupils eventually need interventions in Year 6. Their lack of procedural fluency may not be apparent until they encounter a sample test paper that requires them to choose which method to use.

Conditional knowledge

Summary of the research review relevant to conditional knowledge

Pupils should be able to recall facts and methods to some level of automaticity before using them for wider problem-solving. The curriculum should reflect this optimal ordering.

‘Problem-solving’ is not a generic skill, and pupils cannot become problem-solvers by imitating the activities of experts. Pupils need to learn strategies and the most useful combinations of facts and methods to solve types of problem. Since it is not possible for pupils to encounter every possible problem, a suitable curriculum identifies strategies to solve a range of problem types (topic-specific). For younger pupils, these include how to interpret word problems that are more common at their stage of learning. This approach helps pupils to know what to do, without relying on guesswork.

21. Leaders in all schools wanted pupils to be able to reason and solve a wide range of problems. However, curriculum approaches differed. In many schools, curriculum plans specified the models, explanations and sentence stems that would teach pupils strategies for wider problem-solving. Often, teachers dedicated a section of the lesson to this form of knowledge. This was a positive approach. One pupil said to us, ‘They do it in a structured way by looking at a problem and then we do a similar one.’
22. In some schools, however, reasoning and problem-solving were an activity or task, and something that pupils could choose. This approach may result in some pupils skipping ahead of vital practice of facts and methods, or sticking with repeated practice of already-secure knowledge. This is problematic because pupils are entitled to learn all types of knowledge.
23. Some leaders had identified lack of fluency in procedures and lack of language and comprehension as barriers to reasoning and wider problem-solving. This shows that leaders were increasingly aware of the forms of knowledge that pupils need, to be able to reason and problem-solve.
24. A lack of conditional knowledge ultimately leaves pupils unable to choose the best method when completing a mixed set of questions, for example during a test. Multiple factors contribute to this. For example, pupils may lack automaticity in using declarative knowledge and procedural fluency. If pupils’ declarative and procedural knowledge is secure, it is likely that they do not know the ranges and boundaries of the problems to which the methods apply. This comes from teaching and practising when to use methods, based on identifying underlying mathematical structures. In schools where pupils choose problem-solving challenges, some pupils will discover this form of knowledge, but many will not. This contributes to differences in pupils’ attainment and a need for more intervention in Year 6.

Meeting the needs of pupils

Summary of the research review relevant to meeting pupils’ needs

A well-sequenced path to proficiency, with the small steps identified, is important for all pupils and crucial for pupils with SEND. This helps pupils to keep up and reduces the need for catch-up support. Many pupils with SEND benefit from explicit, systematic instruction and from practice in using declarative and procedural knowledge. They may also need more time to complete tasks and opportunities to practise, rather than different tasks or curriculums. The value of knowing crucial facts and methods to automaticity is even more important for some pupils, such as those with autism spectrum disorder. This is because it frees up working memory for listening, learning and thinking about new knowledge.

25. The ‘keep up, not catch up’ approach, often directly referred to by leaders, made sure that pupils really understood and remembered what was being taught before moving on. This is an inclusive approach, particularly for pupils with SEND, provided there is enough support in place to help them keep up. Many pupils with SEND were following the same curriculum, with support and adaptations in class. For example, they received the same teaching as the main class, and then the teaching assistant supported them during their independent practice. However, this support may circumvent, rather than close gaps in knowledge.
26. Some of the more effective examples of additional help included pre-teaching and same-day interventions. This support gave pupils, including those with SEND, vital additional opportunities to review and practise core knowledge. Some SEND coordinators had introduced an additional ‘precision teaching’ programme for pupils who were working well below age-related expectations.
27. In some schools, leaders tried to maintain the ideal of all pupils moving on together when this was not a successful approach for pupils who were working at significantly below age-related expectations. Pupils in some schools, particularly those with SEND, were less likely to be ‘keeping up and catching up’ in lower year groups. Further, some pupils were less secure in the basic facts than their peers. Teachers’ instructions and explanations of relatively advanced mathematical concepts were beyond their comprehension, even if the teacher explained them well. In these situations, the ‘appearance’ of inclusivity may be taking the place of real inclusivity. These pupils’ needs might be better met if they were to learn different content practised using different tasks. This could be in groups of pupils with a similar level of attainment. This approach would be similar to how schools manage the teaching of early reading for older pupils who have yet to master basic reading skills.
28. Interventions were common. They were more likely to happen in Year 6 and, to a lesser extent, Year 2. In some schools, leaders had reduced class sizes and introduced setting in older year groups. As one school leader reported, the ‘gaps just got bigger and bigger’. These year groups are associated with external accountability measures, in the form of end of key stage tests. The focus on these year groups shows that leaders were having to balance the needs of individual pupils with the pressures of accountability. But leaders knew that some pupils needed to spend more time developing fluency and confidence in ‘the basics’ in earlier years. We observed differences in the quality and quantity of practice and handwriting proficiency in pupils’ books. These are likely to be contributing to holding back pupils with and without SEND.

Pedagogy: teaching the curriculum

Summary of the research review relevant to teaching

The novice, whether they are starting school or starting a new topic, needs more instruction rather than less. Teaching should help them on the journey to expertise. Ideally, teaching should be systematic, follow the curriculum sequence and help pupils to understand. A systematic approach works well for pupils of all ages and stages.

29. In many schools, a consistent approach to designing and implementing the curriculum, with an emphasis on content and ‘small steps’ sequencing, involved a shift of responsibility for the curriculum from the individual teacher to the school’s leadership. This approach assured leaders that pupils’ progression through the curriculum was joined up and balanced and that teachers were using mathematical language and representations consistently. Additional whiteboard resources, often associated with commercial schemes, included representations that were clear and consistent. This helped pupils to understand underlying mathematical structures better.
30. In the Reception Year, mathematics teaching sessions happened daily and tended to be quite short. This is to be expected, given the shorter attention spans of children in this age group. Many practitioners planned to use fun, interesting books, songs and rhymes to help children learn the language and concepts of early mathematics. This is positive, considering leaders’ observations that speech and language delays were increasingly common.
31. Teachers often showed strong subject teaching knowledge in the classroom. They used careful explanations and demonstrations. They were able to break learning down into small steps for pupils. This shows the effect of strong networks of support, such as from the Maths Hubs. Where available, accompanying progression documents were also useful. By setting out what pupils had learned and what they would learn in the future, teachers could understand how each lesson fitted into the bigger picture of mathematics progression. Teachers adapted plans to make sure that pupils were confident before moving on to new learning.
32. Where the curriculum was chosen well and sequenced, and adapted to meet pupils’ needs, this was reflected in the activities in pupils’ books. These showed how learning was built from small component tasks to more complex mathematical processes.
33. Leaders often viewed teaching and use of mathematical vocabulary as ‘non-negotiable’ in lessons. Teachers introduced new vocabulary at the start of lessons and used it throughout. They frequently gave pupils opportunities to revisit this form of knowledge. This approach helped pupils to understand and remember more.
34. Teachers often used the concrete-pictorial-abstract approach to teach new ideas and methods. Pupils were often able to use the objects used in demonstrations themselves. This is helpful for pupils, and leaders were keen for this to happen. However, occasionally, there was a tendency for teachers to use too many concrete and pictorial demonstrations. When this happened, pupils ended up confused from multiple representations.
35. In many schools, leaders expected teachers to use frequent questioning in lessons. They understood that a lack of back-and-forth interactions had been one of the limitations of online learning during the pandemic. Questioning tended to be used well. Familiar sets of questions were almost routine: ‘What do you notice?’ ‘What’s the same and what’s different?’ and ‘Convince me’. Many teachers used questioning deftly throughout lessons to check whether pupils were ready to learn the material, to check their understanding and to encourage their reasoning. In some schools, teachers targeted questioning at pupils who they knew needed extra practice. This made sure that opportunities to take part in the lesson were distributed fairly. However, in a minority of schools, questioning was less effective. Pupils were expected to second-guess what was going to be taught. More confident pupils would be able to volunteer, while others would stop concentrating. It is likely that, in these situations, pupils who need systematic, explicit instruction were missing out on opportunities to understand and learn new knowledge.
36. In many schools, lessons were efficient because of the use of routines. For example, in one school, pupils recited times tables during the transition time between instruction and going to their tables. Many teachers used routines and hand signals that younger pupils were familiar with from their phonics lessons. Pupils knew what was expected of them and what was going to happen next. This meant they could focus on new learning. The use of routines was associated with pupils being on task.
37. In some schools, pupils did not always listen or take part. This included, but was not limited to, pupils with SEND. Some pupils with challenging behaviour had gaps in their workbooks where there should have been evidence of practice. These pupils were practising much less than their peers. This may lead to a vicious circle where pupils with poor behaviour increasingly do not understand what is happening in the classroom, fall behind and are ultimately labelled as having SEND.
38. In some schools, the classroom layout affected pupils’ learning opportunities. In classrooms where pupils faced the teacher, pupils engaged more and were better able to listen and pay attention. This makes sense, as teachers can better gauge pupils’ reactions and know whether they need another explanation or worked example. In contrast, pupils found it difficult to concentrate in classrooms where they had been split into multiple groups for teaching and practice, mainly because of the noise.

Pedagogy: pupils’ practice

Summary of the research review relevant to pupils’ practice

Practice helps pupils to understand and remember mathematical knowledge. There are broadly 2 types of practice. Type 1 involves retrieving and rehearsing facts, methods and strategies to the point of familiarity, speed and accuracy. Type 2 is more exploratory. It requires pupils to explain relationships, prove that they understand them and describe their reasoning. Both types are important. Pupils need quantity and quality of practice to help them understand and commit knowledge to long-term memory. This practice does not always involve textbooks and worksheets. It can include songs, games and rhymes.

Tasks should help pupils to focus on the mathematics to be learned. They should provide for overlearning and, ideally, include variation. Support for learning and understanding should be gradually withdrawn over time. Tasks should give pupils opportunities to be successful, rather than having to rely on guesswork or unstructured trial and error.

39. In most schools, staff and pupils knew that practice, including the need for overlearning, was important. This is very positive. However, practice was not always accompanied by checks to ensure that all pupils were learning the intended declarative, procedural and conditional knowledge, to automaticity, before moving on.
40. Leaders had moved away from an assumption that correct answers showed that work was ‘too easy’. When choosing schemes of work, some leaders looked for opportunities for retrieval, including quizzes. Worksheets accompanying schemes of learning were generally well designed. They included worked examples to help pupils understand, and did not contain distracting pictures. This was very positive for pupils.
41. Teachers consistently built in opportunities for pupils to rehearse knowledge. For example, they would start lessons with a ‘fluent in 5’ approach. Schools were increasingly adding a short, discrete session of extra practice. This usually took place in the afternoon and included mental arithmetic. In one notable example, leaders had built ‘fast fractions’ into this extra session, and pupils had opportunities to practise finding fractions of amounts and counting in fractions. Some schools used the early morning registration period to provide extra opportunities for pupils to receive focused support or practise content they had been taught. These additional sessions helped pupils. However, the need for them may indicate deficiencies in the design and implementation of the curriculum, including opportunities for practice, in pupils’ main lessons.
42. Opportunities for practice sometimes skipped plainer, ‘type 1’ practice and moved too quickly to wider problem-solving. Teachers provided additional workbooks when they thought pupils needed more practice of the basics. However, if this happens frequently it may indicate that, despite being carefully presented, worksheets and tasks do not always provide enough practice for pupils.
43. Linked to the issue of limited practice on worksheets, there was often no consensus among leaders on the amount of quality and quantity of practice that gives assurance that pupils have learned what was intended. Leaders and teachers often needed a better understanding of what an adequate amount of practice is.
44. Staff in the Reception Year frequently made sure that children had repeated exposure to mathematical language and concepts. They planned mathematics activities carefully. Simple, rather than complex, activities helped children to think about mathematical concepts. However, when they were not working in a group with an adult, children tended to choose opportunities for practice from what was laid out in their classroom. Schools need to have a system for monitoring children’s access to mathematics-related tasks from these resources. Otherwise, some children will get more practice than others, and some children will make less progress.
45. Most pupils’ workbooks showed some quality and quantity of practice. However, there was some variation in pupils’ completion rates and access to wider problem-solving. Older pupils’ confidence with recall was associated with high-quality book work. In addition to pupils’ relative proficiency, quality and quantity of practice is likely to be linked to how well they focus on the work, and the amount of effort they make. Pupils who were confident working with fractions had also experienced plenty of practice in calculating with fractions and knew important mathematics facts to automaticity. However, where pupils’ work in key stage 1 was less thorough, they struggled to recall number bonds, for example.
46. In many schools, songs and rhymes gave pupils low-stakes opportunities (where pupils can make mistakes without penalty) to practise counting, shapes and vocabulary. Times table starters (such as ‘rolling your numbers’) and songs in lessons helped pupils to remember important knowledge. This is positive, as it helps all pupils and creates a culture in which mathematics is celebrated. Pupils, too, when asked about what helped them remember, talked about the usefulness of songs.
47. Frequent use of choral response (responding in unison) for low-stakes practice of concepts, vocabulary and mathematical sentences was a positive theme. In one example, leaders and teachers informed us that this was similar to their phonics approach. They noted that the approach was more inclusive, helping pupils with SEND and reducing anxiety about mathematics. This ‘choral response’ is a key feature of the NCETM’s Mastering Number programme for younger pupils. A ‘my turn your turn’ approach to modelling and reciting stem sentences helps pupils to learn new mathematical language and understand important concepts.

Example of stronger practice, incorporating choral response:

In one school, leaders and teachers informed us how the use of choral response was similar to their phonics approach. The approach was more inclusive, helping pupils with SEND and reducing mathematics anxiety:

‘Lessons [include] lots and lots of practice, a “ping-pong” approach between teacher and children, partner work, small steps of progress with teachers circulating… There is a lot of whole-class choral response with the teachers… You can hear who has got it slightly wrong without having to draw attention to the child.’

‘It does feel “repetitive”, but this is necessary.’

48. In many schools, homework consisted of arithmetic practice and access to online platforms to rehearse key mathematics facts. The latter was very popular with pupils. Homework tended to require pupils to practise something they had recently learned. This was because leaders wanted homework to require minimal parental support. In some schools, Year 6 pupils were set questions similar to those in national curriculum tests as homework. This is potentially problematic, as some questions require pupils to know relatively complex strategies and have high levels of understanding. Pupils who are working below age-related expectations become confused or resort to guessing. Leaders were often careful to make sure that test questions were related to content that pupils had already been taught. Some leaders laid on an extra homework club for pupils who were unable to complete homework at home.
49. There has been a cultural shift away from the previously ubiquitous ‘3 levels of differentiation’. This was where pupils were assigned or chose tasks set at different levels. In practice, this reinforced different declarative, procedural and conditional knowledge. In schools where pupils chose their own level of task, some pupils found this choice motivational, as it was a source of pride to take on the difficult ‘challenge’. However, only a subset of pupils would access the reasoning or problem-solving version of whatever was being learned. Further, some pupils would proceed to the ‘extra challenge’ that teachers had prepared for early finishers, while others might not finish basic tasks. This approach may cater for a wider range of proficiency in the moment, but it does not allow each pupil to reinforce the knowledge they need to learn. This reinforces diverging rates of progress and attainment. In some schools, leaders and teachers overcame this by setting a minimum expectation that all pupils should at least do the basic tasks to consolidate their knowledge of facts and methods. Pupils experienced independent practice that was largely similar, at least at the beginning of sequences of learning. Leaders were aware that doing the same thing can, in some circumstances, be ‘OK’, and that every pupil needed to be familiar with the basics.

Assessment

Summary of the research review relevant to assessment

Frequent low-stakes testing (that is, without risk of failure), with an element of timing, is useful for checking pupils’ knowledge of key facts and methods. This helps pupils to remember and gives leaders an insight into gaps in pupils’ knowledge. Assessment at the end of a year or phase should assess pupils on what they have learned and rehearsed, rather than on what they do not know and cannot do.

50. Many teachers used live marking in lessons, as well as whole-class feedback. They swiftly noticed pupils’ successes, misconceptions and errors. They could direct pupils to revisit knowledge at the start of the lesson, adjust the next lesson for all or focus on pupils who needed additional support. In some schools, same-day marking fed into short interventions in the afternoon. The speed and responsiveness of this approach minimised workload for teachers, as it was less likely that pupils would carry forward their errors and misconceptions.
51. Most leaders checked children’s knowledge on entry to Reception Year. Practitioners were responsive to children’s needs, but there was little evidence that they were systematically addressing gaps in the children’s mathematical knowledge. It is possible that, at this stage of learning, gaps in learning slip through the net. More positively, in some schools, leaders were assessing pupils’ ‘addition facts’. This reflected leaders’ awareness of the importance of declarative knowledge.
52. Most schools used regular end-of-unit tests aligned with the school’s curriculum. Pupils would be tested on what they had learned and practised. This highlighted to leaders which forms of knowledge pupils needed to revisit. Intervention plans and resources were available for teachers to use with some commercial test schemes. Occasionally, assessments did not include geometry, data and work with coordinates. Some leaders had developed their own assessments to test and identify gaps in knowledge. Some used multiple-choice questions to expose misconceptions. Both approaches are positive, because they increase the level of detail of diagnostics. In some cases, benchmarks for proficiency had been set at an expectation of 80% accuracy. This is a more appropriate benchmark than the benchmark for ‘meeting age-related expectations’ at the end-of-key-stage tests (currently only around 50% accuracy). However, there was less testing for, or understanding of, benchmarks for ‘procedural fluency’. This indicated a lack of knowledge in this area.

9 Ways To Improve Math Skills Quickly | Prodigy

Math class can move pretty fast. There’s so much to cover in the course of a school year. And if your child doesn’t get a new math idea right away, they can quickly get left behind.

If your child is struggling with basic math problems every day, it doesn’t mean they’re destined to be bad at math. Some students need more time to develop the problem-solving skills that math requires. Others may need to revisit past concepts before moving on. Because of how math is structured, it’s best to take each year step-by-step, lesson by lesson.

This article has tips and tricks to improve your child’s math skills while minimizing frustrations and struggles. If your child is growing to hate math, read on for ways to improve their skills and confidence, and maybe even make math fun!

But first, the basics.

The importance of understanding basic math skills

Math is a subject that builds on itself. It takes a solid understanding of past concepts to prepare for the next lesson.

That’s why math can become frustrating when you’re forced to move on before you’re ready. You’re either stuck trying to catch up or you end up falling further behind.

But with a strong understanding of basic math skills, your child can be set up for school success. If you’re unfamiliar with the idea of sets or whole numbers, this is a great place to start.

What are considered basic math skills?

The basic math skills required to move on to higher levels of math learning are:

Addition — Adding to a set.

Subtraction — Taking away from a set.

Multiplication — Adding equal sets together in groups (2 sets of 3 is the same as 2×3, or 6).

Division — How many equal sets can be found in a number (12 has how many sets of two in it? 6 sets of 2).

Percentages — A specific amount in relation to 100.

Fractions & Decimals — Fractions are equal parts of a whole set. Decimals represent a number of parts of a whole in relation to 10. These both contrast with whole numbers.

Spatial Reasoning — How numbers and shapes fit together.

How to improve math skills

People aren’t bad at math — many just need more time and practice to gain a thorough understanding.

How can you help your child improve their math abilities? Use our top 9 tips for quickly and effectively improving math skills.

1. Wrap your head around the concepts

Repetition and practice are great, but if you don’t understand the concept, it will be difficult to move forward.

Luckily, there are many great ways to break down math concepts. The trick is finding the one that works best for your child.

Math manipulatives can be a game-changer for children who are struggling with big math ideas. Taking math off the page and putting it into their hands can bring ideas to life. Numbers become less abstract and more concrete when you’re counting toy cars or playing with blocks. Creating these “sets” of objects can bring clarity to basic math learning.

2. Try game-based learning

During math practice, repetition is important — but it can get old in a hurry. No one enjoys copying their times tables over and over and over again. If learning math has become a chore, it’s time to bring back the fun!

Game-based learning is a great way to practice new concepts and solidify past lessons. It can even make repetition fun and engaging.

Game-based learning can look like a family board game on Friday night or an educational app, like Prodigy Math.

Take math from frustrating to fun with the right game, then watch the learning happen easily!

3. Bring math into daily life

You use basic math every day.

As you go about your day, help your child see the math that’s all around them:

Tell them how fast you’re driving on the way to school

Calculate the discount you’ll receive on your next Target trip

Count out the number of apples you need to buy at the grocery store

While baking, explain how 6 quarter cups is the same amount of flour as a cup and a half — then enjoy some cookies!

Relate math back to what your child loves and show them how it’s used every day. Math doesn’t have to be mysterious or abstract. Instead, use math to race monster trucks or arrange tea parties. Break it down, take away the fear, and watch their interest in math grow.

4. Implement daily practice

Math practice is important. Once you understand the concept, you have to nail down the mechanics. And often, it’s the practice that finally helps the concept click. Either way, math requires more than just reading formulas on a page.

Daily practice can be tough to implement, especially with a math-averse child. This is a great time to bring out the game-based learning mentioned above. Or find an activity that lines up with their current lesson. Are they learning about squares? Break out the math link cubes and create them. Whenever possible, step away from the worksheets and flashcards and find practice elsewhere.

5. Sketch word problems

Nothing causes a panic quite like an unexpected word problem. Something about the combination of numbers and words can cause the brain of a struggling math learner to shut down. But it doesn’t have to be that way.

Many word problems just need to be broken down, step by step. One great way to do this is to sketch it out. If Doug has five apples and four oranges, then eats two of each, how many does he have left? Draw it, talk it out, cross them off, then count.

If you’ve been talking your child through the various math challenges you encounter every day, many word problems will start to feel familiar.

6. Set realistic goals

If your child has fallen behind in math, then more study time is the answer. But forcing them to cram an extra hour of math in their day is not likely to produce better results. To see a positive change, first identify their biggest struggles. Then set realistic goals addressing these issues.

Two more hours of practicing a concept they don’t understand is only going to cause more frustration. Even if they can work through the mechanics of a problem.