Maths Mastery: Evidence versus Spin

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On Friday 13 February, the Education Endowment Foundation (EEF) published the long-awaited evaluation reports of two randomised control trials (RCTs) of Mathematics Mastery, an Ark-sponsored programme and recipient of one of the EEF’s first tranche of awards back in 2011.

Inside-out_torus_(animated,_small)Inside-out_torus_(animated,_small)EEF, Ark and Mathematics Mastery each published a press release to mark the occasion but, given the timing, none of these attracted attention from journalists and were discussed only briefly on social media.

The main purpose of this post is to distinguish evidence from spin, to establish exactly what the evaluations tell us – and what provisos should be attached to those findings.

The post is organised into three main sections which deal respectively with:

  • Background to Mathematics Mastery
  • What the evaluation reports tell us and
  • What the press releases claim

The conclusion sets out my best effort at a balanced summary of the main findings. (There is a page jump here for those who prefer to cut to the chase.)

This post is written by a non-statistician for a lay audience. I look to specialist readers to set me straight if I have misinterpreted any statistical techniques or findings,

What was published?

On Friday 13 February the EEF published six different documents relevant to the evaluation:

  • A press release: ‘Low-cost internet-based programme found to considerably improve reading ability of year 7 pupils’.
  • A blog post: ‘Today’s findings: impact, no impact and inconclusive – a normal distribution of findings’.
  • An updated Maths Mastery home page (also published as a pdf Project Summary in a slightly different format).

The last three of these were written by the Independent Evaluators – Jerrim and Vignoles (et al) – employed through the UCL Institute of Education.

The Evaluators also refer to ‘a working paper documenting results from both trials’ available in early 2015 from http://ideas.repec.org/s/qss/dqsswp.html and www.johnjerrim.com. At the time of writing this is not yet available.

Press releases were issued on the same day by:

All of the materials published to date are included in the analysis below.

Background to Maths Mastery

What is Maths Mastery?

According to the NCETM (October 2014) the mastery approach in mathematics is characterised by certain common principles:

‘Teachers reinforce an expectation that all pupils are capable of achieving high standards in mathematics.

  • The large majority of pupils progress through the curriculum content at the same pace. Differentiation is achieved by emphasising deep knowledge and through individual support and intervention.
  • Teaching is underpinned by methodical curriculum design and supported by carefully crafted lessons and resources to foster deep conceptual and procedural knowledge.
  • Practice and consolidation play a central role. Carefully designed variation within this builds fluency and understanding of underlying mathematical concepts in tandem.
  • Teachers use precise questioning in class to test conceptual and procedural knowledge, and assess pupils regularly to identify those requiring intervention so that all pupils keep up.

The intention of these approaches is to provide all children with full access to the curriculum, enabling them to achieve confidence and competence – ‘mastery’ – in mathematics, rather than many failing to develop the maths skills they need for the future.’

The NCETM paper itemises six key features, which I paraphrase as:

  • Curriculum design: Relatively small, sequenced steps which must each be mastered before learners move to the next stage. Fundamental skills and knowledge are secured first and these often need extensive attention.
  • Teaching resources: A ‘coherent programme of high-quality teaching materials’ supports classroom teaching. There is particular emphasis on ‘developing deep structural knowledge and the ability to make connections’. The materials may include ‘high-quality textbooks’.
  • Lesson design: Often involves input from colleagues drawing on classroom observation. Plans set out in detail ‘well-tested methods’ of teaching the topic. They include teacher explanations and questions for learners.
  • Teaching methods: Learners work on the same tasks. Concepts are often explored together. Technical proficiency and conceptual understanding are developed in parallel.
  • Pupil support and differentiation: Is provided through support and intervention rather than through the topics taught, particularly at early stages. High attainers are ‘challenged through more demanding problems which deepen their knowledge of the same content’. Issues are addressed through ‘rapid intervention’ commonly undertaken the same day.
  • Productivity and practice: Fluency is developed from deep knowledge and ‘intelligent practice’. Early learning of multiplication tables is expected. The capacity to recall facts from long term memory is also important.

Its Director published a blog post (October 2014) arguing that our present approach to differentiation has ‘a very negative effect’ on mathematical attainment and that this is ‘one of the root causes’ of our performance in PISA and TIMSS.

This is because it negatively affects the ‘mindset’ of low attainers and high attainers alike. Additionally, low attainers are insufficiently challenged and get further behind because ‘they are missing out on some of the curriculum’. Meanwhile high attainers are racing ahead without developing fluency and deep understanding.

He claims that these problems can be avoided through a mastery approach:

‘Instead, countries employing a mastery approach expose almost all of the children to the same curriculum content at the same pace, allowing them all full access to the curriculum by focusing on developing deep understanding and secure fluency with facts and procedures, and providing differentiation by offering rapid support and intervention to address each individual pupil’s needs.’

But unfortunately he stops short of explaining how, for high attainers, exclusive focus on depth is preferable to a richer blend of breadth, depth and pace, combined according to each learner’s needs.

NCETM is careful not to suggest that mastery is primarily focused on improving the performance of low-attaining learners.

It has published separate guidance on High Attaining Pupils in Primary Schools (registration required), which advocates a more balanced approach, although that predates this newfound commitment to mastery.

NCETM is funded by the Department for Education. Some of the comments on the Director’s blog post complain that it is losing credibility by operating as a cheerleader for Government policy.

Ark’s involvement

Ark is an education charity and multi-academy trust with an enviable reputation.

It builds its approach on six key principles, one of which is ‘Depth before breadth’:

‘When pupils secure firm foundations in English and mathematics, they find the rest of the curriculum far easier to access. That’s why we prioritise depth in these subjects, giving pupils the best chance of academic success. To support fully our pupils’ achievement in maths, we have developed the TES Award winning Mathematics Mastery programme, a highly-effective curriculum and teaching approach inspired by pupil success in Singapore and endorsed by Ofsted. We teach Mathematics Mastery in all our primary schools and at Key Stage 3 in a selection of our secondary schools. It is also being implemented in over 170 schools beyond our network.’

Ark’s 2014 Annual Report identifies five priorities for 2014/15, one of which is:

‘…developing curricula to help ensure our pupils are well prepared as they go through school… codifying our approach to early years and, building on the success of Maths Mastery, piloting an English Mastery programme…’

Mathematics Mastery is a charity in its own right. Its website lists 15 staff, a high-powered advisory group and three partner organisations:  Ark, the EEF (presumably by virtue of the funded evaluation) and the ‘Department for Education and the Mayor of London’ (presumably by virtue of support from the London Schools Excellence Fund).

NCETM’s Director sits on Mathematics Mastery’s Advisory Board.

Ark’s Chief Executive is a member of the EEF’s Advisory Board.

Development of Ark’s Maths Mastery programme

According to this 2012 report from Reform, which features Maths Mastery as a case study, it originated in 2010:

‘The development of Mathematics Mastery stemmed from collaboration between six ARK primary academies in Greater London, and the mathematics departments in seven separate ARK secondary academies in Greater London, Portsmouth and Birmingham. Representatives from ARK visited Singapore to explore the country’s approach first-hand, and Dr Yeap Ban Har, Singapore’s leading expert in maths teaching, visited King Solomon Academy in June 2011.’

In October 2011, EEF awarded Ark a grant of £600,000 for Maths Mastery, one of its first four awards.

The EEF’s press release says:

‘The third grant will support an innovative and highly effective approach to teaching children maths called Mathematics Mastery, which originated in Singapore. The programme – run by ARK Schools, the Academies sponsor, which is also supporting the project – will receive £600,000 over the next four years to reach at least 50 disadvantaged primary and secondary schools.’

Ark’s press release adds:

‘ARK Schools has been awarded a major grant by the Education Endowment Foundation (EEF) to further develop and roll out its Mathematics Mastery programme, an innovative and highly effective approach to teaching children maths based on Singapore maths teaching. The £600,000 grant will enable ARK to launch the programme and related professional development training to improve maths teaching in at least 50 disadvantaged primary and secondary schools.

The funding will enable ARK Schools to write a UK mathematics mastery programme based on the experience of teaching the pilot programme in ARK’s academies. ARK intends to complete the development of its primary modules for use from Sept 2012 and its secondary modules for use from September 2013. In parallel ARK is developing professional training and implementation support for schools outside the ARK network.’

The project home page on EEF’s site now says the total project cost is £774,000. It may be that the balance of £174,000 is the fee paid to the independent evaluators.

This 2012 information sheet says all Ark primary schools would adopt Maths Mastery from September 2012, and that its secondary schools have also devised a KS3 programme.

It describes the launch of a Primary Pioneer Programme from September 2012 and a Secondary Pioneer Programme from September 2013. These will form the cohorts to be evaluated by the EEF.

In 2013, Ark was awarded a grant of £617,375 from the Mayor of London’s London Schools Excellence Fund for the London Primary Schools Mathematics Mastery Project.

This is to support the introduction of Mastery in 120 primary schools spread across 18 London boroughs. (Another source gives the grant as £595,000)

It will be interesting to see whether Maths Mastery (or English Mastery) features in the Excellence Fund’s latest project to increase primary attainment in literacy and numeracy. The outcomes of the EEF evaluations may be relevant to that impending decision.

Ark’s Mathematics Mastery today

The Mathematics Mastery website advertises a branded variant of the mastery model, derived from a tripartite ‘holistic vision’:

  • Deep understanding, through a curriculum that combines universal high expectations with spending more time on fewer topics and heavy emphasis on problem-solving.
  • Integrated professional development through workshops, visits, coaching and mentoring and ‘access to exclusive online teaching and learning materials, including lesson guides for each week’.
  • Teacher collaboration – primary schools are allocated a geographical cluster of 4-6 schools while secondary schools attend a ‘national collaboration event’. There is also an online dimension.

It offers primary and secondary programmes.

The primary programme has three particular features: use of objects and pictures prior to the introduction of symbols; a structured approach to the development of mathematical vocabulary; and heavy emphasis on problem-solving.

It involves one-day training sessions for school leaders, for the Maths Mastery lead and those new to teaching it, and for teachers undertaking the programme in each year group. Each school receives two support visits and attends three local cluster meetings.

Problem-solving is also one of three listed features of the secondary programme. The other two are fewer topics undertaken in greater depth, plus joint lesson planning and departmental workshops.

There are two full training days, one for the Maths Mastery lead and one for the maths department plus an evening session for senior leadership. Each school receives two support visits and attends three national collaborative meetings. They must hold an hour-long departmental workshop each week and commit to sharing resources online.

Both primary and secondary schools are encouraged to launch the programme across Year 1/7 and then roll it upwards ‘over several years’.

The website is not entirely clear but it appears that Maths Mastery itself is being rolled out a year at a time, so even the original primary early adopters will have provision only up to Year 3 and are scheduled to introduce provision for Year 4 next academic year. In the secondary sector, activity currently seems confined to KS3, and predominantly to Year 7.

The number of participating schools is increasing steadily but is still very small.

The most recent figures I could find are 192 (Maths Mastery, November 2014) or 193 – 142 primary and 51 secondary (Ark 2015).

One assumes that this total includes

  • An original tranche of 30 primary ‘early adopters’ including 21 not managed by Ark
  • 60 or so primary and secondary ‘Pioneer Schools’ within the EEF evaluations (ie the schools undertaking the intervention but not those forming the control group, unless they have subsequently opted to take up the programme)
  • The 120 primary schools in the London project
  • Primary and secondary schools recruited outwith the London and EEF projects, either alongside them or subsequently.

But the organisation does not provide a detailed breakdown, or show how these different subsets overlap.

They are particularly coy about the cost. There is nothing about this on the website.

The EEF evaluation reports say that 2FE primary schools and secondary schools will pay ‘an upfront cost of £6,000 for participating in the programme’.

With the addition of staff time for training, the per pupil cost for the initial year is estimated as £127 for primary schools and £50 for secondary schools.

The primary report adds:

‘In subsequent years schools are able to opt for different pathways depending on the amount of support and training they wish to choose; they also have ongoing access to the curriculum materials for additional year groups. The per pupil cost therefore reduces considerably, to below £30 per pupil for additional year groups.’

In EEF terms this is deemed a low cost intervention, although an outlay of such magnitude is a significant burden for primary schools, particularly when funding is under pressure, and might be expected to act as a brake on participation.

Further coyness is evident in respect of statutory assessment outcomes. Some details are provided for individual schools, but there is precious little about the whole cohort.

All I could find was this table in the Primary Yearbook 2014-15.

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EEF maths mastery performance

It suggests somewhat better achievement at KS1 L2b and L3c than the national average but, there is no information about other Levels and, of course, the sample is not representative, so the comparison is of limited value.

An absence of more sophisticated analysis – combined with the impression of limited transparency for those not yet inside the programme – is likely to act as a second brake on participation.

There is a reference to high attainers in the FAQ on the website:

‘The Mathematics Mastery curriculum emphasises stretching through depth of understanding rather than giving the top end of pupils [sic] new procedures to cover.

Problem solving is central to Mathematics Mastery. The great thing about the problems is that students can take them as far as they can, so those children who grasp the basics quickly can explore tasks further. There is also differentiation in the methods used, with top-end pupils typically moving to abstract numbers more quickly and spending less time with concrete manipulatives or bar models. There are extension ideas and support notes provided with the tasks to help you with this.

A range of schools are currently piloting the programme, which is working well in mixed-ability classes, as well as in schools that have set groups.’

The same unanswered questions arise as with the NCETM statement above. Is ‘Maths Mastery’ primarily focused on the ‘long tail’, potentially at the expense of high attainers?

The IoE evaluators think so. The primary evaluation report says that:

‘Mathematics Mastery intervention is particularly concerned with the ‘mastery’ of basic skills, and raising the attainment of low achievers.’

It would be helpful to have clarity on this point.

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How influential is Maths Mastery?

Extremely influential.

Much educational and political capital has already been invested in Maths Mastery, hence the peculiar significance of the results contained in the evaluation reports.

The National Curriculum Expert Panel espoused mastery in its ‘Framework for the National Curriculum‘ (December 2011), while ducking the consequences for ‘stretch and challenge’ for high attainers – so creating a tension that remains unresolved to this day.

Meanwhile, the mastery approach has already influenced the new maths programme of study, as the NCETM document makes clear:

‘The 2014 national curriculum for mathematics has been designed to raise standards in maths, with the aim that the large majority of pupils will achieve mastery of the subject…

… For many schools and teachers the shift to this ‘mastery curriculum’ will be a significant one. It will require new approaches to lesson design, teaching, use of resources and support for pupils.’

Maths Mastery confirms that its Director was on the drafting team.

Mastery is also embedded in the national collaborative projects being undertaken through the Maths Hubs. Maths Mastery is one of four national partners in the Hubs initiative.

Ministers have endorsed the Ark programme in their speeches. In April 2014, Truss said:

‘The mastery model of learning places the emphasis on understanding core concepts. It’s associated with countries like Singapore, who have very high-performing pupils.

And in this country, Ark, the academy chain, took it on and developed it.

Ark run training days for maths departments and heads of maths from other schools.

They organise support visits, and share plans and ideas online with other teachers, and share their learning with a cluster of other schools.

It’s a very practical model. We know not every school will have the time or inclination to develop its very own programmes – a small rural school, say, or single-class primary schools.

But in maths mastery, a big chain like Ark took the lead, and made it straightforward for other schools to adopt their model. They maintain an online community – which is a cheap, quick way of keeping up with the best teaching approaches.

That’s the sort of innovation that’s possible.

Of course the important thing is the results. The programme is being evaluated so that when the results come out headteachers will be able to look at it and see if it represents good value.’

In June 2014 she said:

‘This idea of mastery is starting to take hold in classrooms in England. Led by evidence of what works, teachers and schools have sought out these programmes and techniques that have been pioneered in China and East Asia….

…With the Ark Schools Maths Mastery programme, more than 100 primary and secondary schools have joined forces to transform their pupils’ experiences of maths – and more are joining all the time. It’s a whole school programme focused on setting high expectations for all pupils – not believing that some just can’t do it. The programme has already achieved excellent results in other countries.’

Several reputations are being built upon Maths Mastery, many jobs depend upon it and large sums have been invested.

It has the explicit support of one of the country’s foremost academy chains and is already impacting on national curriculum and assessment policy (including the recent consultation on performance indicators for statutory teacher assessment).

Negative or neutral evaluations could have significant consequences for all the key players and are unlikely to encourage new schools to join the Programme.

Hence there is pressure in the system for positive outcomes – hence the significance of spin.

What the EEF evaluations tell us

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Evaluation Protocols

EEF published separate Protocols for the primary and secondary evaluations in April 2013. These are broadly in line with the approach set out in the final evaluation reports, except that both refer much more explicitly to subsequent longitudinal evaluation:

‘In May/June 2017/18 children in treatment and control schools will sit key stage 2 maths exams. The IoE team will examine the long–run effectiveness of the Maths Mastery programme by investigating differences in school average maths test scores between treatment and control group. This information will be taken from the National Pupil Database, which the EEF will link to children’s Maths Mastery test scores (collected in 2012 and 2013)’.

‘In May/June 2018 children in treatment and control schools will sit national maths exams. The IoE team will examine the long – run effectiveness of the Maths Mastery programme by investigating differences in average maths test scores between treatment and control group. This information will be taken from the National Pupil Database, which the EEF will link to children’s Maths Mastery test scores (collected in 2013 and 2014) by NATCEN.’

It is not clear whether the intention is to preserve the integrity of the intervention and control groups until the former have rolled out Mastery to all year groups, or simply to evaluate the long-term effects of the initial one-year interventions, allowing intervention schools to drop Mastery and control schools to adopt it, entirely as they wish.

EEF Maths Mastery Project Homepage

The EEF’s updated Maths Mastery homepage has been revised to reflect the outcomes of the evaluations. It provides the most accessible summary of those outcomes.

It offers four key conclusions (my emphases):

  • ‘On average, pupils in schools adopting Mathematics Mastery made a small amount more progress than pupils in schools that did not. The effect detected was statistically significant, which means that it is likely that that improvement was caused by the programme.’
  • ‘It is unclear whether the programme had a different impact on pupils eligible for free school meals, or on pupils with higher or lower attainment.’
  • ‘Given the low per-pupil cost, Mathematics Mastery may represent a cost-effective change for schools to consider.’
  • ‘The evaluations assessed the impact of the programme in its first year of adoption. It would be worthwhile to track the medium and long-term impact of the approach.’

A table is supplied showing the effect sizes and confidence intervals for overall impact (primary and secondary together), and for the primary and secondary interventions separately.

EEF table 1 Capture

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The support materials for the EEF’s toolkit help to explain these judgements.

About the Toolkit tells us that:

‘Average impact is estimated in terms of the additional months’ progress you might expect pupils to make as a result of an approach being used in school, taking average pupil progress over a year as a benchmark.

For example, research summarised in the Toolkit shows that improving the quality of feedback provided to pupils has an average impact of eight months. This means that pupils in a class where high quality feedback is provided will make on average eight months more progress over the course of a year compared to another class of pupils who were performing at the same level at the start of the year. At the end of the year the average pupil in a class of 25 pupils in the feedback group would now be equivalent to the 6th best pupil in the control class having made 20 months progress over the year, compared to an average of 12 months in the other class.’

There is another table showing us how to interpret this scale

EEF table 2 Capture

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We can see from this that:

  • The overall Maths Mastery impact of +0.073 is towards the upper end of the ‘1 months progress’ category.
  • The ‘primary vs comparison’ impact of +0.10 just scrapes into the ‘2 months progress’ category.
  • The secondary vs comparison impact of +0.06 is towards the middle of the ‘1 months progress category’

All three are officially classed as ‘Low Effect’.

If we compare the effect size attributable to Maths Mastery with others in the Toolkit, it is evident that it ranks slightly above school uniform and slightly below learning styles.

A subsequent section explains that the overall impact rating is dependent on meta-analysis (again my emphases):

‘The findings from the individual trials have been combined using an approach called “meta-analysis”. Meta-analysis can lead to a more accurate estimate of an intervention’s effect. However, it is also important to note that care is needed in interpreting meta-analysed findings.’

But we are not told how, in light of this, we are to exercise care in interpreting this particular finding. There are no explicit ‘health warnings’ attached to it.

The homepage does tell us that:

‘Due to the ages of pupils who participated in the individual trials, the headline findings noted here are more likely to be predictive of programme’s impact on pupils in primary school than on pupils in secondary school.’

It also offers an explanation of why the effects generated from these trials are so small compared with those for earlier studies:

‘The findings were substantially lower than the average effects seen in the existing literature on of “mastery approaches”. A possible explanation for this is that many previous studies were conducted in the United States in the 1970s and 80s, so may overstate the possible impact in English schools today. An alternative explanation is that the Mathematics Mastery programme differed from some examples of mastery learning previously studied. For example classes following the Mathematics Mastery approach did not delay starting new topics until a high level of proficiency had been achieved by all students, which was a key feature in a number of many apparently effective programmes.’

 

There is clearly an issue with the 95% confidence intervals supplied in the first table above. 

The Technical Appendices to the Toolkit say:

‘For those concerned with statistical significance, it is still readily apparent in the confidence intervals surrounding an effect size. If the confidence interval includes zero, then the effect size would be considered not to have reached conventional statistical significance.’ (p6)

The table indicates that the lower confidence interval is zero or lower in all three cases, meaning that none of these findings may be statistically significant.

However, the homepage claims that the overall impact of both interventions, when combined through meta-analysis, is statistically significant.

And it fails entirely to mention that the impact of the both the primary and the secondary interventions separately are statistically insignificant.

The explanation of the attribution of statistical significance to the two evaluations combined is that, whereas the homepage gives confidence intervals to two decimal places, the reports calculate them to a third decimal place.

This gives a lower value of 0.004 (ie four thousandths above zero).

This can be seen from the table annexed to the primary and secondary reports and included in the ‘Overarching Summary Report’

EEF maths mastery 3 decimal places Capture

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The distinction is marginal, to say the least. Indeed, the Evaluation Reports say:

‘…the pooled effect size of 0.073 is just significantly different from zero at conventional thresholds’

Moreover, notice that the introduction of a third decimal place drags the primary effect size down to 0.099, officially consigning it to the ‘one month’s progress’ category rather than the two months quoted above.

This might appear to be dancing on the head of a statistical pin but, as we shall see later, the spin value of statistical significance is huge!

Overall there is a lack of clarity here that cannot be attributed entirely to the necessity for brevity. The attempt to conflate subtly different outcomes from the separate primary and secondary evaluations has masked these distinctions and distorted the overall assessment.

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The full reports add some further interesting details which are summarised in the sections below.

Primary Evaluation Report 

EEF maths mastery table 4

Key points:

  • In both the primary and secondary reports, additional reasons are given for why the effects from these evaluations are so much smaller than those from previous studies. These include the fact that:

‘…some studies included in the mastery section of the toolkit show small or no effects, suggesting that making mastery learning work effectively in all circumstances is challenging.’

The overall conclusion is an indirect criticism of the Toolkit, noting as it does that ‘the relevance of such evidence for contemporary education policy in England…may be limited’.

  • The RCT was undertaken across two academic years: In AY2012/13, 40 schools (Cohort A) were involved. Of these, 20 were randomly allocated the intervention and 20 the control. In AY2013/14, 50 schools (Cohort B) participated, 25 allocated the intervention and 25 the control. After the trial, control schools in Cohort A were free to pursue Maths Mastery. (The report does not mention whether this also applied to Cohort B.) It is not clear how subsequent longitudinal evaluation will be affected by such leakage from the control group.
  • The schools participating in the trial schools were recruited by Ark. They had to be state-funded and not already undertaking Maths Mastery:

‘Schools were therefore purposefully selected—they cannot be considered a randomly chosen sample from a well-defined population. The majority of schools participating in the trial were from London or the South East.’

  • Unlike the secondary evaluation, no process evaluation was conducted so it is not possible to determine the extent to which schools adhered to the prescribed programme. 
  • Baseline tests were administered after allocation between intervention and control, at the beginning of each academic year. Pupils were tested again in July. Evaluators used the Number Knowledge Test (NKT) for this purpose. The report discusses reasons why this might not be an accurate predictor of subsequent maths attainment and whether it is so closely related to the intervention as to be ‘a questionable measure of the success of the trial’. The discussion suggests that there were potential advantages to both the intervention and control groups but does not say whether one outweighed the other. 
  • The results of the post-test are summarised thus:

‘Children who received the Mathematics Mastery intervention scored, on average, +0.10 standard deviations higher on the post-test. This, however, only reached statistical significance at the 10% level (t = 1.82; p = 0.07), with the 95% confidence interval ranging from -0.01 to +0.21. Within Cohort A, children in the treatment group scored (on average) +0.09 standard deviations above those children in the control group (confidence interval -0.06 to +0.24). The analogous effect in Cohort B was +0.10 (confidence interval -0.05 to 0.26). Consequently, although the Mathematics Mastery intervention may have had a small positive effect on children’s test scores, it is not possible to rule out sampling variation as an explanation.’

  • The comparison of pre-test and post-test results provides any evidence of differential effects for those with lower or higher prior attainment:

‘Estimates are again presented in terms of effect sizes. The interaction effect is not significantly different from zero, with the 95% confidence interval ranging from -0.01 to +0.02. Thus there is little evidence that the effect of Mathematics Mastery differs between children with different levels of prior achievement.’

The Report adds:

‘Recall that the Mathematics Mastery intervention is particularly concerned with the ‘mastery’ of basic skills, and raising the attainment of low achievers. Thus one might anticipate the intervention to be particularly effective in the bottom half of the test score distribution. There is some, but relatively little, evidence that the intervention was less effective for the bottom half of the test distribution.

So, on this evidence, Maths Mastery is no more effective for the low achievers it is intended to help most. This is somewhat different to the suggestion on the homepage that the answer given to this question is ‘unclear’.

Several limitations are discussed, but it is important to note that they are phrased in hypothetical terms:

  • Pupils’ progress was evaluated after one academic year::

’This may be considered a relatively small ‘dose’ of the Mathematics Mastery programme’.

  • The intervention introduced a new approach to schools, so there was a learning curve which control schools did not experience:

‘With more experience teaching the programme it is possible that teachers would become more effective in implementing it.’

  • The test may favour either control schools or intervention schools.
  • Participating schools volunteered to take part, so it is not possible to say whether similar effects would be found in all schools.
  • It was not possible to control for balance – eg by ethnic background and FSM eligibility – between intervention and control. [This is now feasible so could potentially be undertaken retrospectively to check there was no imbalance.]

Under ‘Interpretation’, the report says:

‘Within the context of the wider educational literature, the effect size reported (0.10 standard deviations) would typically be considered ‘small’….

Yet, despite the modest and statistically insignificant effect, the Mathematics Mastery intervention has shown some promise.’

The phrase ‘some promise’ is justified by reference to the meta-analysis, the cost effectiveness (a small effect size for a low cost is preferable to the same outcome for a higher cost) and the fact that the impact of the entire programme has not yet been evaluated

‘Third, children are likely to follow the Mathematics Mastery programme for a number of years (perhaps throughout primary school), whereas this evaluation has considered the impact of just the first year of the programme. Long-run effects after sustained exposure to the programme could be significantly higher, and will be assessed in a follow-up study using Key Stage 2 data.’

This is the only reference to a follow-up study. It is less definite than the statement in the assessment protocol and there is no further explanation of how this will be managed, especially given potential ‘leakage’ from the control group.

Secondary Evaluation Report

EEF maths mastery table 5

Key points:

  • 50 schools were recruited to participate in the RCT during AY2013/14, with 25 randomly allocated to intervention and control. All Year 7 pupils within the former experienced the intervention.  As in the primary trial, control schools were eligible to access the programme after the end of the trial year. Interestingly, 3 of the 25 intervention schools (12%) dropped out before the end of the year – their reasons are not recorded. 
  • As in the primary trial, Ark recruited the participating schools – which had to be state-funded and new to Maths Mastery. Since schools were deliberately selected they could not be considered a random sample. The report notes:

‘Trial participants, on average, performed less well in their KS1 and KS2 examinations than the state school population as a whole. For instance, their KS1 average points scores (and KS2 maths test scores) were approximately 0.2 standard deviations (0.1 standard deviations) below the population mean. This seems to be driven, at least in part, by the fact that the trial particularly under-represented high achievers (relative to the population). For instance, just 12% of children participating in the trial were awarded Level 3 in their Key Stage 1 maths test, compared to 19% of all state school pupils in England.’

  • KS1 and KS2 tests were used to baseline. The Progress in Maths (PiM) test was used to assess pupils at the end of the year. But about 40% of the questions cover content not included in the Y7 maths mastery curriculum, which disadvantaged them relative to the control group. PiM also includes a calculator section although calculators are not used in Year 7 of Maths Mastery. It was agreed that breakdowns of results would be supplied to account for this.
  • On the basis of overall test results:

‘Children who received the Mathematics Mastery intervention scored, on average, +0.055 standard deviations higher on the PiM post-test. This did not reach statistical significance at conventional thresholds (t = 1.20; p = 0.24), with the 95% confidence interval ranging from –0.037 to +0.147. Turning to the FSM-only sample, the estimated effect size is +0.066 with the 95% confidence interval ranging from –0.037 to +0.169 (p = 0.21). Moreover, we also estimated a model including a FSM-by intervention interaction. Results suggested there was little evidence of heterogeneous intervention effects by FSM. Consequently, although the Mathematics Mastery intervention may have had a small positive effect on overall PiM test scores, one cannot rule out the possibility that this finding is due to sampling variation.

  • When the breakdowns were analysed:

‘As perhaps expected, the Mathematics Mastery intervention did not have any impact upon children’s performance on questions covering topics outside the Mathematics Mastery curriculum. Indeed, the estimated intervention effect is essentially zero (effect size = –0.003). In contrast, the intervention had a more pronounced effect upon material that was focused upon within the Mathematics Mastery curriculum (effect size = 0.100), just reaching statistical significance at the 5% level (t = 2.15; p = 0.04)

  • The only analysis of the comparative performance of high and low attainers is tied to the parts of the test not requiring use of a calculator. It suggests a noticeably smaller effect in the top half of the attainment distribution, with no statistical significance above the 55th This is substantively different to the finding in the primary evaluation, and it begs the question whether secondary Maths Mastery needs adjustment to make it more suitable for high attainers.
  • A process evaluation was focused principally on 5 schools from the intervention group. Focus group discussions were held before the intervention and again towards the end. Telephone interviews were conducted and lessons observed. The sample was selected to ensure different sizes of school, FSM intake and schools achieving both poor and good progress in maths according to their most recent inspection report. One of the recommendations is that:

The intervention should consider how it might give more advice and support with respect to differentiation.’

  • The process evaluation adds further detail about suitability for high attainers:

‘Another school [E] also commented that the materials were also not sufficiently challenging for the highest-attaining children, who were frustrated by revisiting at length the same topics they had already encountered at primary school. Although this observation was also made in other schools, it was generally felt that the children gradually began to realise that they were in fact enjoying the subject more by gaining extra understanding.’

It is not clear whether this latter comment also extends to the high attainers!

A similar set of limitations is explored in similar language to that used in the primary report.

Under ‘Interpretation’ the report says:

‘Although point estimates were consistent with a small, positive gain, the study did not have sufficient statistical power to rule out chance as an explanation. Within the context of the wider educational literature, the effect size reported (less than 0.10 standard deviations) would typically be considered ‘small’…

But, as in the primary report, it detects ‘some promise’ on the same grounds. There is a similar speculative reference to longitudinal evaluation.

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Press releases and blogs

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EEF press release

There is a certain irony in the fact that ‘unlucky’ Friday 13 February was the day selected by the EEF to release these rather disappointing reports.

But Friday is typically the day selected by communications people to release educational news that is most likely to generate negative media coverage – and a Friday immediately before a school holiday is a particularly favoured time to do so, presumably because fewer journalists and social media users are active.

Unfortunately, the practice is at risk of becoming self-defeating, since everyone now expects bad news on a Friday, whereas they might be rather less alert on a busier day earlier in the week.

On this occasion Thursday was an exceptionally busy day for education news, with reaction to Miliband’s speech and a raft of Coalition announcements designed to divert attention from it. With the benefit of hindsight, Thursday might have been a better choice.

The EEF’s press release dealt with evaluation reports on nine separate projects, so increasing the probability that attention would be diverted away from Maths Mastery.

It led on a different evaluation report which generated more positive findings – the EEF seems increasingly sensitive to concerns that too many of the RCTs it sponsors are showing negligible or no positive effect, presumably because the value-for-money police may be inclined to turn their beady eye upon the Foundation itself.

But perhaps it also did so because Maths Mastery’s relatively poor performance was otherwise the story most likely to attract the attention of more informed journalists and commentators.

On the other hand, Maths Mastery was given second billing:

‘Also published today are the results of Mathematics Mastery, a whole-school approach which aims to deepen pupils’ conceptual understanding of key mathematical ideas. Compared to traditional curricula, fewer topics are covered in more depth and greater emphasis is placed on problem solving and encouraging mathematical thinking. The EEF trials found that pupils following the Mathematics Mastery programme made an additional month’s progress over a period of a year.’

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EEF blog post

Later on 13 February EEF released a blog post written by a senior analyst which mentions Maths Mastery in the following terms:

Another finding of note is the small positive impact of teaching children fewer mathematical concepts, but covering them in greater depth to ensure ‘mastery’. The EEF’s evaluation of Mathematics Mastery will make fascinating reading for headteachers contemplating introducing this approach into their school. Of course, the true value of this method may only be evident in years to come as children are able to draw on their secure mathematical foundations to tackle more complex problems.’

EEF is consistently reporting a small positive impact but, as we have seen, this is rather economical with the truth. It deserves some qualification.

More interestingly though, the post adds (my emphases):

‘Our commitment as an organisation is not only to build the strength of the evidence base in education, across key stages, topics, approaches and techniques, but also ensure that the key messages emerging from the research are synthesised and communicated clearly to teachers and school leaders so that evidence can form a central pillar of how decisions are made in schools.

We have already begun this work, driven by the messages from our published trials as well as the existing evidence base. How teaching assistants can be used to best effect, important lessons in literacy at the transition from primary to secondary, and which principles should underpin approaches on encouraging children in reading for pleasure are all issues that have important implications for school leaders. Synthesising and disseminating these vital messages will form the backbone of a new phase of EEF work beginning later in the year.’

It will be interesting to monitor the impact of this work on the communication of outcomes from these particular evaluations.

It will be important to ensure that synthesis and dissemination is not at the expense of accuracy, particularly when ‘high stakes’ results are involved, otherwise there is a risk that users will lose faith in the independence of EEF and its willingness to ‘speak truth unto power’.

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Maths Mastery Press Release

By also releasing their own posts on 13 February, Mathematics Mastery and Ark made sure that they too would not be picked up by the media.

They must have concluded that, even if they placed the most positive interpretation on the outcomes, they would find it hard to create the kind of media coverage that would generate increased demand from schools.

The Mathematics Mastery release – ‘Mathematics Mastery speeds up pupils’ progress – and is value for money too’ – begins with a list of bullet points citing other evidence that the programme works, so implying that the EEF evaluations are relatively insignificant additions to this comprehensive evidence base:

  • ‘Headteachers say that the teaching of mathematics in their schools has improved
  • Headteachers are happy to recommend us to other schools
  • Numerous Ofsted inspections have praised the “new approach to mathematics” in partner schools
  • Extremely positive evaluations of our training and our school development visits
  • We have an exceptionally high retention rate – schools want to continue in the partnership
  • Great Key Stage 1 results in a large number of schools.’

Much of this is hearsay, or else vague reference to quantitative evidence that is not published openly.

The optimistic comment on the EEF evaluations is:

‘We’re pleased with the finding that, looking at both our primary and secondary programmes together, pupils in the Mathematics Mastery schools make one month’s extra progress on average compared to pupils in the other schools after a one year “dose” of the programme…

…This is a really pleasing outcome – trials of this kind are very rigorous.  Over 80 primary schools and 50 secondary schools were involved in the testing, with over 4000 pupils involved in each phase.  Studies like this often don’t show any progress at all, particularly in the early years of implementation and if, like ours, the programme is aimed at all pupils and not just particular groups.  What’s more, because of the large sample size, the difference in scores between the Mathematics Mastery and other schools is “statistically significant” which means the results are very unlikely to be due to chance.’

The section I have emboldened is in stark contrast to the EEF blog post above, which has the title:

‘Today’s findings; impact, no-impact and inconclusive – a normal distribution of findings’

And so suggests exactly the opposite.

I have already shown just how borderline the calculation of ‘statistical significance’ has been.

The release concludes:

‘Of course we’re pleased with the extra progress even after a limited time, but we’re interested in long term change and long term development and improvement.  We’re determined to work with our partner schools to show what’s possible over pupils’ whole school careers…but it’s nice to know we’ve already started to succeed!’

 .

There was a single retweet of the Tweet above, but from a particularly authoritative source (who also sits on Ark’s Advisory Group).

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Ark Press Release

Ark’s press release – ‘Independent evaluation shows Mathematics Mastery pupils doing better than their peers’ – is even more bullish.

The opening paragraph claims that:

‘A new independent report from the independent Education Endowment Foundation (EEF) demonstrates the success of the Mathematics Mastery programme. Carried out by academics from Cambridge University and the Institute of Education, the data indicates that the programme may have the potential to halve the attainment gap with high performing countries in the far East.

The second emboldened statement is particularly brazen since there is no evidence in either of the reports that would support such a claim. It is only true in the sense that any programme ‘may have the potential’ to achieve any particularly ambitious outcome.

Statistical significance is again celebrated, though it is important to give Ark credit for adding:

‘…but it is important to note that these individual studies did not reach the threshold for statistical significance. It is only at the combined level across 127 schools and 10,114 pupils that there are sufficient schools and statistical power to determine an effect size of 1 month overall.’

Even if this rather implies that the individual evaluations were somehow at fault for being too small and so not generating ‘sufficient statistical power’.

Then the release returns to its initial theme:

‘… According to the OECD, by age fifteen, pupils in Singapore, Japan, South Korea and China are three years ahead of pupils in England in mathematical achievement. Maths Mastery is inspired by the techniques and strategies used in these countries.

Because Maths Mastery is a whole school programme of training and a cumulative curriculum, rather than a catch-up intervention, this could be a sustained impact. A 2 month gain every primary year and 1 month gain every secondary year could see pupils more than one and a half years ahead by age 16 – halving the gap with higher performing jurisdictions.’

In other words, Ark extrapolates equivalent gains – eschewing all statistical hedging – for each year of study, adding them together to suggest a potential 18 month gain.

It also seems to apply the effect to all participants rather than to the average participant.

This must have been a step too far, even for Ark’s publicity machine.

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maths mastery ark release capture

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They subsequently changed the final paragraph above – which one can still find in the version within Google’s cache – to read:

‘…Because Maths Mastery is a whole school programme of training and a cumulative curriculum, rather than a catch-up intervention, we expect this to be a sustained impact.  A longer follow-up study will be needed to investigate this.’

Even in sacrificing the misleading quantification, they could not resist bumping up ‘this could be a sustained impact’ to ‘we expect this to be a sustained impact’

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[Postscript: On 25 February, Bank of America Merrill Lynch published a press release announcing a £750,000 donation to Maths Mastery.

The final paragraph ‘About Maths Mastery’ says:

‘Mathematics Mastery is an innovative maths teaching framework, supporting schools, students and teachers to be successful at maths. There are currently 192 Mathematics Mastery partner schools across England, reaching 34,800 pupils. Over the next five years the programme aims to expand to 500 schools, and reach 300,000 pupils. Maths Mastery was recently evaluated by the independent Education Endowment Foundation and pupils were found to be up to two months ahead of their peers in just the first year of the programme. Longer term, this could see pupils more than a year and a half ahead by age 16 – halving the gap with pupils in countries such as Japan, Singapore and China.’

This exemplifies perfectly how such questionable statements are repurposed and recycled with impunity. It is high time that the EEF published a code of practice to help ensure that the outcomes of its evaluations are not misrepresented.]  

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Conclusion

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 .

Representing the key findings

My best effort at a balanced presentation of these findings would include the key points below. I am happy to consider amendments, additions and improvements:

  • On average, pupils in primary schools adopting Mathematics Mastery made two months more progress than pupils in primary schools that did not. (This is a borderline result, in that it is only just above the score denoting one month’s progress. It falls to one month’s progress if the effect size is calculated to three decimal places.) The effect is classified as ‘Low’ and this outcome is not statistically significant. 
  • On average, pupils in secondary schools adopting Mathematics Mastery made one month more progress than pupils in secondary schools that did not. The effect is classified as ‘Low’ and this outcome is not statistically significant. 
  • When the results of the primary and secondary evaluations are combined through meta-analysis, pupils in schools adopting Maths Mastery made one month more progress than pupils in schools that did not. The effect is classified as ‘Low’. This outcome is marginally statistically significant, provided that the 95% confidence interval is calculated to three decimal places (but it is not statistically significant if calculated to two decimal places). Care is needed in analysing meta-analysed findings because… [add explanation]. 
  • There is relatively little evidence that the primary programme is more effective for learners with lower prior attainment, but there is such evidence for the secondary programme (in respect of non-calculator questions). There is no substantive evidence that the secondary programme has a different impact on pupils eligible for free schools meals. 
  • The per-pupil cost is relatively low, but the initial outlay of £6,000 for primary schools with 2FE and above is not inconsiderable. Mathematics Mastery may represent a cost-effective change for schools to consider. 
  • The evaluations assessed the impact of the programme in its first year of adoption. It is not appropriate to draw inferences from the findings above to attribute potential value to the whole programme. EEF will be evaluating the medium and long-term impact of the approach by [outline the methodology agreed].

In the meantime, it would be helpful for Ark and Maths Mastery to be much more transparent about KS1 assessment outcomes across their partner schools and possibly publish their own analysis based on comparison between schools undertaking the programme and matched control schools with similar intakes.

And it would be helpful for all partners to explain and evidence more fully the benefits to high attainers of the Maths Mastery approach – and to consider how it might be supplemented when it does not provide the blend of challenge and support that best meets their needs.

It is disappointing that, three years on, the failure of the National Curriculum Expert Panel to reconcile their advocacy for mastery with stretch and challenge for high attainers – in defiance of their remit to consider the latter as well as the former –  is being perpetuated across the system.

NCETM might usefully revisit their guidance on high attainers in primary schools to reflect their new-found commitment to mastery, while also incorporating additional material covering the point above.

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Postscript

A summary of this piece, published by Schools Week, prompted two comments – one from Stephen Gorard, the other from Dylan Wiliam. The Twitter embed below is the record of a subsequent debate between us and some others, about design of the Maths Mastery evaluations, what they tell us and how useful they are, especially to policy makers.

One of the tweets contains a commitment on the part of Anna Vignoles to set up a seminar to discuss these issues further.

The widget stores the tweets in reverse order (most recent first). Scroll down to the bottom to follow the discussion in chronological order.

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GP

February 2015

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8 thoughts on “Maths Mastery: Evidence versus Spin

  1. It strikes me that a model that is specifically designed to cover less content in the short-term is not really best evaluated after a short-term period.
    But then, I’m biased 🙂

  2. Both evaluation reports discuss whether the post-intervention test is advantageous to the intervention or the control group.

    In the case of the primary trial, the report identifies potential advantages to both groups, but does not quantify these.

    In the secondary trial, the results are broken down to account for differences in curriculum coverage, but the results from that aren’t significantly more impressive.

    Neither report suggests that their findings are compromised as a consequence

    …And of course the counter-argument is that one cannot pre-suppose a ‘tortoise and hare’ effect, or indeed an aggregation of 1/2 month effect sizes as Ark originally did.

    GP

  3. Thanks GP – a fascinating analysis and important message on the real world nature of ‘evidence’. As I read it, the secondary programme evaluation is fatally flawed, if for no other reason, than because the testing was done by the teachers themselves (ie. needs to be independently done as was the primary programme). The primary programme impact evaluation is much more convincing – although of course not statistically significant (in the conventional sense), but probably worth replicating the study, and then combining (or meta analysing) the two primary studies together, having of course pre-declared that intention. However, trying to combine the analyses from the two very different primary and secondary programmes doesn’t make sense (even if the secondary trial was well done). Personally I’m less bothered about the ‘spin’ – that’s just the way things are, but much more concerned about the methods and from that the integrity of the publicly funded and available findings. Also a bit surprising that there are so few comments being made here about these important issues.

  4. Thanks for taking the time and trouble to comment.

    I’m not really equipped to question the evaluation methodology for the separate interventions. (Until now I have taken it as read that anything commissioned by EEF and evaluated by IoE, one of its approved evaluators, would be ‘state of the art’, but your views suggest otherwise.)

    However, I do think that all concerned could have done a much better job of communicating the evaluation outcomes – and that includes the methodological health warnings. There’s not much point telling us that there are issues associated with combining the two sets of results in a meta-evaluation and then failing to spell out what those are!

    If, as you say, the combination makes no sense, surely there’s an onus on the EEF to explain exactly why they hold the opposite view?

    Apropos the silence (especially from the three key players), I’m worried on their behalves, as they’re giving unnecessary ammunition to conspiracy theorists. I should have thought it would have been less damaging reputationally to hold up one’s hand, plead mea culpa, and make the necessary corrections.

    GP

  5. Thanks for this analysis. As one of the 25 London control schools in 13/14 and now in 14/15 a mathsmastery adopter, I awaited the results of the EEF trial and was surprised by the small gain but even more surprised by the spin. I’m a fan of the approach and based on my very limited and entirely anecdotal experience of the programme it works. But the truth is the truth and I want the truth unadulterated by vested interests.
    For what it is worth, I think one potential problem with the RCT was contamination of the control group. When we applied to be part of the 13/14 programme, we really wanted to change the way we did maths. When we were assigned to the control group we looked around for something new to do in the interim ‘control’ year. So we adopted ‘primary advantage maths’ which covers a lot of the same ground ( concrete, pictorial, abstract approaches, bar models, the importance of number facts) while not being a mastery approach as such. So much so that the initial training once our ‘control’ period was over, was a bit of a let down (‘ but we do all this already’ said the yr1 teacher). However, 7 months on, we can see the difference. Without a doubt the biggest challenge is providing a suitable level of challenge for those who grasp maths quickly. However, really skilled and well trained teachers can do this well- not random investigations but activities with reasoning deeply embedded. But what usually happens when a teaching approach becomes trendy is that a superficial level of training and a huge dose of rhetoric paper of the cracks in lieu of the really deep training for our teachers this approach will need if the outliers in our classes are to benefit from it. Not saying this is what is happening with mathsmastery now – just what might happen if there is some diffusion- range mathsmastery-lite roll out.

    There will also be extreme outliers whose needs it can’t meet entitrely. The child still at p6 is still at p6 despite daily 1:1 catch ups, ore teaching etc.

  6. Thanks so much for engaging with my post. Your comments shed a fresh light on the implications for ‘control’ schools of being involved in a RCT.

    Apropos providing challenge for so-called ‘rapid graspers’ (I have some issues with that descriptor) is there in your opinion sufficient coverage of how to do that within the maths mastery training and support materials? If not, what is missing that would make the difference?

    Thanks

    GP

  7. Thank you so much for this detailed post, I have been waiting for some time for this report and it was kept rather quiet! You have managed to link a lot of important information in my head to give me a better overview-really appreciate this.

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