The Maths Guarantee: How to boost students’ learning in primary schools

2.3 The way the brain works has important implications for how to teach maths

The research on how the human brain works has important implications for how schools plan their approach to teaching primary maths.

The concept of the instructional hierarchy sets out the predictable stages of skill development and maps these against the teaching approach that works best for each stage (see Figure 2.1).⁵⁵Haring and Eaton (1978) originated the concept. See also Burns et al (2010), Codding et al (2024), and VanDerHeyden and Burns (2005).

Figure 2.1: The instructional hierarchy shows that, for new material, students should be explicitly taught before doing application tasks

The instructional hierarchy suggests a different approach is needed to teach a new concept or skill than to extend students who have already achieved mastery. Teachers need to break new learning down into small chunks that are taught explicitly. This may seem counterintuitive to some. The goal of maths teaching is for students to apply the maths they have learnt in class to novel problems that require them to adapt what they know without teacher assistance (the final stage of the instructional hierarchy). Given this, the temptation can be to start teaching a new concept or skill by first focusing on meaty application tasks.

But starting at this stage is a mistake. Starting with activities that involve students applying a skill with minimal teacher support before they are proficient is a recipe for student disengagement and teacher frustration. Instead, teachers need to first support students to become accurate and fluent. The sections below explain how.

2.3.1 Acquisition: Teaching new maths concepts and skills

High-quality research shows that the most effective way to help students learn new maths material is to break the concepts down into small chunks, explain them clearly with examples, model what students need to know step by step, and provide students with plenty of opportunities to practise.⁵⁶See, for example, AERO (2024a), Ashman et al (2020), EEF (2022), Evans and Martin (2023), Fuchs et al (2021), Gersten et al (2009a), Merlo (2024), Ofsted (2021), S. R. Powell et al (2022), S. R. Powell et al (2024), and Rosenshine (2012). This form of teaching gives students what they need to succeed. It avoids overloading working memory and helps transfer information into long-term memory. And, because teachers constantly check for understanding and provide immediate feedback, students know whether they are succeeding. As a result, students experience a high rate of success, which can breed confidence and motivation, and help to combat anxiety about maths (see Box 7).⁵⁷Gunderson et al (2018), and Ma and Xu (2004). This approach benefits all novice learners but is particularly important for students who are behind in maths or have learning difficulties.⁵⁸Chodura et al (2015), Fuchs et al (2021), Gersten et al (2009b), Kroesbergen (2004), and Morgan et al (2015).

Box 7: Effective teaching can combat maths anxiety

‘Maths anxiety’ is the worry and tension that students can feel when doing maths.^aBuckley (2013).

By high school, many Australian students appear to experience some discomfort when doing maths. Forty per cent of Australian 15-year olds agreed with the statement ‘I get very tense when I have to do maths homework’.^bThe OECD average was 39 per cent. See De Bortoli et al (2024) Thirty-five per cent said they feel helpless when doing a maths problem.^cThe OECD average was 41 per cent. De Bortoli et al (ibid).

There is a well-evidenced connection between academic achievement and maths anxiety.^dBarroso et al (2021), Gabriel et al (2020), Ma (1999), and Szczygieł et al (2024). Very intense feelings of nervousness may impede working memory, making it hard for students to solve maths problems.^eAshcraft and Kirk (2001).

Research suggests this is mostly a learnt fear linked to poor prior maths achievement.^fSee, for example, Geary et al (2023), Gunderson et al (2018), Ma and Xu (2004), and Wang et al (2020). Note that different studies have produced different results on the nature of the relationship between prior achievement and maths, and the effect of potential moderators (such as students’ age and gender). See Barroso et al (2021). Therefore, helping students develop strong mathematical foundations can build confidence in maths. Simply avoiding maths for fear it will cause students anxiety is not a productive response.

Each lesson should have a clear purpose and measurable learning intention against which its success can be judged. That learning intention should be achievable within the lesson, and focus squarely on a particular maths skill or concept. Well-focused learning intentions distinguish this teaching approach from approaches designed around activities (see Box 8).

Box 8: The dangers of activity-based planning

Primary school teachers are often encouraged to structure lessons around games and activities. But if activities become the driving focus, there is real risk that the maths in the lesson is lost.

One suggested Year 3 lesson sequence for Australian teachers invites students to learn about fractions by exploring the optimal fraction to fill a bottle so it lands upright when flipped.^aSee reSolve (2018). This activity requires knowledge of fractions (e.g. 1/3, 1/5, etc.), volume (including millilitres), and data analysis.

If this activity is used before students are proficient with these aspects of maths, there is a real risk that students will spend a lot of time flipping bottles without attending to the underlying maths.

The more students think about something, the more they are likely to learn and remember it.^bWillingham (2009) Students may spend the lesson competing with friends at bottle flipping, and not thinking about how a 600 mL bottle filled to 200 mL is one-third full.

Activities like this can be fun for both students and teachers. While they have their place – such as at the end of term – they take up a lot of time, and are not a substitute for teaching the foundational knowledge and skills students need to master in order to progress with their learning.

A teaching approach based mainly on these activities also risks exacerbating existing inequities. Students who have the requisite knowledge and self-regulation skills find it easier to engage and consolidate their learning, while their peers who don’t sit passively (or disrupt the class), take less from the activity, and fall further behind.

Teacher explanation

To avoid overloading working memory, teachers should break new content into small chunks and ensure explanations are unambiguous and to the point. Ideally, the explanations should be short too, so that the bulk of class time is dedicated to practice.

During explanations, teachers should clearly explain what students need to know, and fully model mathematical procedures. It can be helpful if teachers also show ‘non-standard examples’ and ‘non-examples’ to clarify the boundaries of a maths concept.⁵⁹For example, students in a Foundation class might learn that a triangle is a three-sided shape. To show the boundaries of what counts as a triangle, the teacher could present non-standard examples, such as triangles rotated 45 degrees, or very skinny triangles. They could also show non-examples such as a pyramid (which is not a triangle because it is three-dimensional), a diagram of a pizza slice (not a triangle because it has one rounded edge), or a paperclip folded into a triangle-like shape (not a triangle because it is not enclosed). These non-examples help refine the students’ understanding from ‘a triangle is a three-sided shape’ to ‘a triangle is a closed, two-dimensional shape with three straight sides’. This example is adapted from White Rose Education (2024). Teachers should also introduce straightforward worded problems, to show a maths concept or skill in context.⁶⁰A frequent misconception is that worded problems are best used for extension, or only once students are fluent with a skill. But teachers should model and guide practice with simple worded problems sooner, because these ‘give meaning to mathematical operations such as subtraction or multiplication’: see discussion in Gersten et al (2009b, p. 26).

Teachers’ explanations may be supported by concrete or pictorial representations of maths concepts, such as base 10 blocks and diagrams.⁶¹Base 10 blocks – also known as dienes – are three-dimension blocks typically made out of wood or plastic that represent the base-10 place value system. A set normally consists of units (ones place), longs (tens place), flats (hundreds place) and blocks (thousands place). See discussion in Fuson and Briars (1990). These representations help students visualise and understand abstract maths concepts.⁶²Primary maths requires students to understand abstract concepts, such as that the symbol 5 represents five of a thing and that the percentage sign – % – denotes a number expressed as a fraction of 100. See Bouck et al (2018) and Carbonneau et al (2013). It is important that concrete representations are removed as soon as possible but as late as necessary (this is called ‘concreteness fading’) so that students have ample opportunities to practise working abstractly – a skill that is essential for later success as maths becomes more abstract.⁶³Working with maths symbols is faster than using concrete representations (i.e. it is quicker to compute 6 + 7 mentally than to arrange six counters, add seven more, and then count the total). Additionally, students need to be confident without concrete materials as they move into secondary school maths and tackle increasingly abstract topics, such as algebra.

In comparison to activity-focused teaching approaches (see Box 8), instruction that involves clearly explaining concepts requires teachers to have a higher degree of confidence with maths as they model problems in front of the class. For this reason, primary teachers’ knowledge of maths and how to teach it really matters (we discuss this further in Chapter 3).

Guided practice

After a whole-class explanation of a new concept, students benefit from applying the new knowledge through guided practice.

Guided practice in a whole-class setting involves students having the opportunity to work through problems step by step, with the teacher checking and providing immediate corrective feedback throughout.

The questions posed during guided practice should have the same underlying structure as questions the teacher modelled during their explanation. During guided practice, teachers provide scaffolds (or supports) and gradually reduce the amount of guidance as students master a concept or skill.⁶⁴One way to do this is through ‘faded worked examples’, in which students attempt partially solved maths problems. The number of pre-completed steps for each problem steadily decreases as students transition to solving the problems independently. See Renkl and Atkinson (2016) and Retnowati (2017).

To guide practice effectively, it is critical that teachers check students’ work and responses, so they know whether each student has understood their explanation, can follow the steps modelled, and is ready to apply their learning independently. To quickly check for understanding in a whole-class setting, teachers can use methods that elicit responses from all students at once (for instance by having students answer and display questions on mini whiteboards).

Independent practice

After guided practice as a whole class, students still need opportunities to practise new skills and knowledge independently, so that what they have learnt transfers to their long-term memory.

The focus of practice should initially be on accuracy.⁶⁵Haring and Eaton (1978).

 It helps if students are given questions structured similarly to those modelled during the explanation and completed during guided practice. For example, if a lesson is on calculating perimeter, the questions students attempt during independent practice should also be on calculating perimeter. This is called ‘blocked practice’.⁶⁶H.-B. Hwang (2025).

It is also best to start with questions that involve simple arithmetic, so students can focus their attention on the relevant maths processes without being overwhelmed by the numbers.⁶⁷For example, imagine a Year 5 lesson in which students are learning the standard method for adding fractions. This is a three-step process that involves finding the lowest common denominators, adding the numerators, and simplifying (where possible). It would be unwise to set, as the first practice question, 7/13 + 8/9 because the lowest common denominator of these fractions is 117 and the answer involves improper fractions. Such questions are best reserved for when students are fluent with the procedure of adding fractions.

2.3.2 Fluency: Improving the speed and ease with which students respond

Once students are consistently giving accurate responses, but still do so haltingly, teachers should focus on building fluency: the speed with which students can respond accurately.⁶⁸See Haring and Eaton (1978). See also Burns et al (2010, p. 71) for a discussion of different teaching strategies to build fluency and how these differ from strategies to build accuracy. Fluency is vital for maths.⁶⁹Kilpatrick et al (2001).

It means students can do more practice questions in a lesson, and a lot of practice is key for maths success. It also means they can tackle more complex problems.

To support students’ fluency, teachers should do some timed practice.⁷⁰Fuchs et al (2021). This is because once students achieve a high degree of accuracy (nearly 100 per cent correct), timed practice gives teachers more information about what students have mastered and stored in their long-term memory.⁷¹See VanDerHeyden and Solomon (2023). Timed practice results in rate-based data such as ‘digits correct per minute’.

Achieving fluency requires at least some repetitive practice.⁷²One study found that to master small sets of multiplication facts, students needed many practice sessions on average – sometimes as many as eight. In each practice session, students might rehearse a maths fact multiple times. Younger students and students with poorer maths skills tended to need even more repetitions. See Burns et al (2015, Table 1). Repetitive practice should not be dismissed as pointless ‘rote learning’ or ‘drill and kill’ (see Box 9). Mastering most hard things requires practice, as any budding student athlete or musical performer can attest. Maths is no different. To that end, extra practice outside the class can make a difference, but it should not be relied upon as the primary source of practice (see Box 10).

Box 9: Myth busting: Fluency practice is not ‘drill and kill’

Fluency practice is sometimes seen as being at odds with the goal of students applying maths to real-life scenarios. Some maths academics have gone so far as labelling the memorisation of times tables through repetition as ‘unnecessary and damaging’.^aBoaler and Confer (2015, p. 1). But as other academics have pointed out, this misunderstands the importance of having automatic recall of maths facts, and underestimates how engaging fluency practice can be.^bSee, for example, J. D. Stocker et al (2022).

Interventions that focus on improving lower-performing students’ maths fact fluency can have a positive impact on maths test scores, including students’ ability to complete worded application questions.^cIbid. Practice focused on procedural fluency can also reinforce underlying conceptual understanding.^dFor example, column addition emphasises place value: when adding the ones column in 17 + 14 the 11 ones can be regrouped into 1 ten and 1 one. See Fan and Bokhove (2014) for further discussion.

For fact fluency, only small doses of practice are needed if done daily.^eHernandez-Nuhfer et al (2020), and Duhon et al (2022).

Fluency practice can also be fun. To develop maths facts, students can race against the clock to complete as many questions as they can, or work in partners with flashcards, celebrating each new personal best.^fSee, for example, Fontenelle IV et al (2022) and Payne et al (2024). See also Stokke (2024). Fluency practice is also a great way for young children to achieve success in maths, which is highly motivating and helps develop positive attitudes towards the subject.

Box 10: Homework and primary school maths

Meaningful, focused practice helps students master maths. It follows that students benefit from additional practice at home, provided it is high-quality and purposeful.^aEEF (2021a). Homework may particularly benefit students who are behind.^bBartelet et al (2016).

Homework in maths should involve practising things learnt in class.^cEEF (2021a). Short, frequent homework may be more effective than longer bouts.^dEEF (2021a), and McJames et al (2024). There is promising evidence that digital apps can be an effective means of at-home practice for primary school students.^eBartelet et al (2016), and Roschelle et al (2016).

Whether to set homework will depend on students’ ages and the extent to which they are mastering grade-level content in class.

But primary schools should not operate on the expectation that practice will happen at home for all students. Some students may not have a quiet place to work.^fEEF (2021a). And, in the younger years, students may struggle without an adult to guide them. In such cases, students might need a place to get support at school, such as an after-school homework club.

Because individual circumstances impact students’ capacity to practise out of class, homework is out of scope for this report. Our focus is instead on maximising the impact of time in class.

Fluency with maths facts is essential

In addition to building students’ fluency with other parts of maths, primary schools play a vital role in ensuring students build fluency in maths facts in the early years. The goal should be for students to recall a maths fact with automaticity – without hesitation (within approximately two-to-three seconds) – because this indicates they probably have automatic recall of that knowledge (i.e. it is in their long-term memory and readily retrievable).⁷³Stickney et al (2012). Students should have automaticity with addition and related subtraction facts up to 10 + 10 by the end of Year 2 (at the latest) and with multiplication and division facts up to 12 × 12 by the end of Year 4 (at the latest).⁷⁴Year levels based on ACARA (2022b). See English Department for Education (2021, p. 331) for the fact fluency progression recommended in England.

Evidence suggests a minimum of four minutes of practice with maths facts per day, and that small daily doses of practice are best.⁷⁵Two minutes per day was the minimum dose required to see gains in Duhon et al (2022). We have suggested four minutes based on Stokke (2024), which allows for some set up time. There is evidence that two minutes of practice in the morning and the afternoon is better than four minutes of back-to-back practice in the morning. See Schutte et al (2015). For further research on optimising fluency practice, see Codding et al (2016), Duhon et al (2009), Hernandez-Nuhfer et al (2020), Payne et al (2024), and S. L. Powell et al (2022). This might be scheduled as part of the lesson warm-up. Schools should plan practice to help students focus on a manageable number of maths facts that they haven’t yet committed to memory.⁷⁶One proven method is incremental rehearsal. Rather than trying to memorise all the seven times tables at once, for example, students instead tackle a practice set that introduces just one unknown fact at a time until that fact has been mastered. See Burns (2005) for further details on this method. Adaptive technology can be helpful here.⁷⁷See, for example, Burns et al (2012), Hassler Hallstedt et al (2018), and Stickney et al (2012). See Ran et al (2021) for a meta-analysis of the benefits of computer technology for students struggling in maths more generally. It can personalise practice, pinpointing the facts individual students are yet to master. Technology can also gamify fluency practice, making it fun.

2.3.3 Generalisation: Applying learnt material in new contexts

Once students are accurate and fluent with a maths topic, students are properly prepared for, and benefit from, applying this knowledge flexibly.

At this stage of the instructional hierarchy, students benefit from questions and tasks that require them to figure out when it is appropriate to apply the skill they are now accurate and fluent in. Teachers can achieve this by designing sets of questions that involve a mix of question types (this is called ‘interleaved practice’).⁷⁸Taylor and Rohrer (2010), Rohrer et al (2015), and Rohrer et al (2020). This could be through cumulative tests or quizzes, or daily reviews that involve a mix of question types. Questions should be designed to require students to discriminate between problems and choose appropriate strategies. In the example below, students must choose whether to use addition or subtraction:

Ed ate 5 cookies, and he now has 9. How many did he start with?

Ed ate 5 cookies, and he began with 9. How many does he have?⁷⁹Rohrer et al (2017, p. 4).

Interleaved practice should not be introduced when students are still developing accuracy in a skill, as it might confuse them.⁸⁰See discussion in Fischer (2023).

During the generalisation phase, students also benefit from applying previously learnt knowledge to non-routine problems, which are problems where the path to the solution is not immediately clear.⁸¹Ashman et al (2020), Evans and Martin (2023), Foster (2018), and Sweller et al (2024). Non-routine problems might include application tasks with multiple possible solutions (see Box 11), problems that present the maths skill in a different context, and problems that involve multiple components, requiring students to draw together various maths skills they have mastered. Non-routine worded problems can also be effective (see Box 12).

Box 11: Example of a rich maths application task

The end of Year 6 unit on calculating discounts might include a lesson with the following application task:
• Each student is to plan an action-packed weekend of entertainment.
• Students receive a fictional budget of $100 to spend across the weekend on as many activities as they can afford.
• Activities are listed on a flyer: $22 for bowling, $35 for indoor rockclimbing, $34 for an arcade games pass, and so forth.
• Students also receive a pack of six coupons – such as a 30% discount or ‘Only pay 1/3’. They can apply these to any activity, but can only use each coupon once.
• Students’ aim is to pack in as many activities as possible without going over budget.
• Students are to record the activities purchased, their working out to apply discounts, and the final price of each activity.^aTask sourced from Ochre (2024a).

This task is reserved for when students are confident with calculating discounts using fractions, percentages, and decimals. Before introducing the task, the teacher might recap necessary prior knowledge, and check students have mastered it.

They would then work through an example of applying a coupon to an expense. If students are struggling, they might choose to do more of the task as a whole class.

Students could be extended or supported with different activity prices and coupon discounts that are more or less challenging.

Box 12: Worded problems are an essential part of a healthy maths diet

Worded problems are sometimes conflated with extension, reserved only for high-achievers. But not all extension questions are worded (see Figure 2.2), and vice versa.

Worded problems help students apply the maths they know to real-life contexts – a core goal of learning maths. They require students to decipher a scenario and choose a suitable mathematical method to solve the underlying problem.

Primary school students often struggle to unpack worded problems.^aJitendra et al (2007), and Dela Cruz and Lapinid (2014). Teachers can help by equipping students with ‘attack strategies’ (often in the form of mnemonics), that give students a series of steps to follow to make sense of worded problems.^bSee, for example, S. R. Powell et al (2024).

Teaching students to distinguish between the underlying types of worded problems is also effective. But teachers need to be careful. A common mistake is to give students key words and associated math- ematical operations (e.g. ‘In a worded problem, ‘altogether’ tells you to add’). But this type of ‘key word’ strategy falls apart when students encounter questions like:

Ali bought 3 packs of textas, with 4 textas in each pack. How many textas does Ali have altogether?^cExample adapted from S. R. Powell and Fuchs (2018).

Instead, teachers should help students distinguish between underlying problem types and associated solutions.^dSee Fuchs et al (2010). This teaching strategy is termed schema-based instruction, since teachers help students detect the problem type – or ‘schema’ – of word problems. One example is the ‘combine’ problem type, which involves putting together two or more separate parts.^eSee S. R. Powell and Fuchs (2018) for a detailed explanation of common schemata.

Students also benefit from teachers modelling how to use diagrams to visualise worded problems, and to clarify the information they have and the information they need to find out.^fSee Kaur (2019), Mahoney (2012), and Morin et al (2017). See Booker et al (2020, pp. 58–59) for a practical example. Bar models are one example.^gNCTEM (2020) provides an overview of using the bar model for addition, subtraction, multiplication, division, ratio, and fraction questions. See Kaur (2019) for further examples and a summary of the model’s evidence base. Ochre (2024b) has free lesson resources modelling the use of the Bar Model. See Year 3, Unit 2, Lessons 13 and 14.

Worded problems also rely on students having a strong mathematical vocabulary.^hWanjiru and O’Connor (2015), Riccomini et al (2015), Ufer and Bochnik (2020), Vista (2013), and Carter (2011). A whole-school approach to explicitly and systematically teaching mathematical vocabulary is therefore necessary.

Figure 2.2: Not all extension problems involve words

Make your own magic square with 1/8 in the centre

More open-ended tasks should be carefully planned so the focus is on the maths. Tasks which involve a lot of extraneous elements – such as excessive cutting and pasting – may divert students’ attention from the mathematical thinking the task is intended to elicit. For example, asking Year 3 students to design animals out of fraction pieces (circles split in halves, quarters, eighths, and so on) risks students spending more cognitive energy thinking about which animal to create, and arranging the fraction pieces to form it, than about the maths involved in the fractions themselves.⁸²This idea comes from Lesson 2 in the instructional sequence outlined by Hubbard and Marino (2023). For an alternative fractions task aimed at Year 5-to-7s that foregrounds maths and has multiple possible solutions, see AMSI (2018, p. 3).

Non-routine problems, which require students to use maths flexibly, mark the final stage and desired destination of maths teaching. But starting an instructional sequence at this final stage is a mistake, based on a misunderstanding of how students learn. It risks overloading students’ working memory, thwarting their efforts to complete the task or learn the new skill, which can in turn fuel students’ feelings of failure and anxiety.⁸³These application tasks have higher ‘element interactivity’ – a term which refers to the number of task elements that must be processed in working memory. More complex tasks have a higher base level of element interactivity. But as learners gain expertise, they can retrieve more from long-term memory, decreasing the effective element interactivity of a complex task. See Ashman et al (2020). Starting at this final stage also creates equity issues: students who are lucky enough to have the required knowledge for the tasks (because, for example, they have been tutored at home) may engage more. And if the activities are designed as group tasks, students who struggle might sit back while more able peers do the work.

2.3.4 Maintenance: Frequently revisiting previously learnt maths concepts

Knowledge doesn’t stick in long-term memory without frequent retrieval. Retrieval practice involves students recalling information that they have already learnt. By trying to recall information, students strengthen their long-term memory and identify gaps in their learning.⁸⁴Roediger and Butler (2011), AERO (2024a), Breneman-Smith (2024), and Lyle et al (2020). Retrieval practice also helps students transfer information they know to novel situations: one goal of maths teaching.⁸⁵See Pan and Agarwal (2020) for a summary of the research.

Practice questions should be chosen so students periodically revisit learnt maths topics. This matters because information in long-term memory may fade when not frequently retrieved.⁸⁶See discussion in Baker (2018), May (2022), and Radvansky et al (2022). See Murre and Dros (2015) for estimates of retention length.

Retrieval practice should be designed so that questions are appropriately challenging, all students respond (for example, via mini whiteboards or cumulative quizzes), and misconceptions and wrong answers are readily identified and corrected.⁸⁷AERO (2022). Greater effort at recall during such sessions leads to better memory retention, so questions should be designed to require effortful recall.⁸⁸Agarwal et al (2020).

There are no hard-and-fast rules for the optimal interval of time before revisiting previously learnt topics. What matters most is that teachers space out retrieval practice, rather than concentrating it all in one lesson.⁸⁹AERO (2022), Agarwal et al (2020), Agarwal et al (2021), and Carpenter and Agarwal (2020). Schools might choose to implement retrieval practice through 10-minute lesson warm-ups that also include some fluency practice.

Designing effective retrieval practice requires careful whole-school planning to identify what content to revisit and when to revisit it.

Table 2.1: Practice snapshot: Selected indicators of effective maths teaching

Indicators of more effective practice	Indicators of less effective practice
Each lesson has a clear and measurable learning intention which serves as a tight focus for the lesson – activities and games are used intentionally.	Lessons are primarily focused on different activities, and lack a clear, maths-focused learning intention.
Teachers regularly check that students have understood, using methods such as mini whiteboards to get frequent, whole-class data on students’ success against the learning intention.	Students’ success rate is not clear at every step, and may only be apparent at the end of the class.
Before students apply learning independently, teachers spend time explaining and modelling concepts and procedures, and guiding whole-class practice (during which students get immediate feedback so they know if they are right).	When students are learning new content, teaching is not explicit and systematic: students independently explore or discover concepts before teacher explanations, and there is little whole-class, guided practice.
Teachers use whole-class participation tactics – such as mini whiteboards and think–pair–share – to maximise engagement.	Participation is uneven: a few students are doing all the work, with many students passive or off-task.
Students develop fluency with one strategy before new strategies are introduced, and are taught which strategy is most useful when.	Students are often expected to invent strategies – or use multiple strategies – to tackle a new problem, without regard to the efficiency of different strategies.
Teachers emphasise both procedural fluency and conceptual understanding, because they see them as mutually reinforcing.	Teachers prioritise only conceptual understanding and spend limited time on procedures, or vice versa.
Students periodically revisit taught concepts, with regular cumulative reviews.	Students do not revisit topics until the subsequent year.
Students all work on the same overarching learning intention, and the range of students’ needs is met through different scaffolds (e.g. some students spend longer in guided practice with the teacher while others do independent practice).	To meet a range of abilities, students are split into table groups’ and teachers attempt to split their attention by running several concurrent mini lessons on different curriculum content.