New consultation on accountability looks to shake up Progress 8 - but will it incentivise what it hopes to?

The long-awaited schools white paper, 'Every child achieving and thriving', has been published today. Leading the way are the reforms to the SEND system, as well as the consultation on those reforms, which I know many have been anticipating.

However, as a former assistant headteacher in charge of data, it was the consultation on secondary school accountability measures announced, that really caught my eye. The consultation proposes four major changes to the secondary school accountability measures.

Changes to Progress 8

Two of the changes relate to the Progress 8 measure.

Replacing the three EBacc and three open bucket slots in the current progress 8 measure with two science slots, two 'breadth' slots and two 'choice' slots

This is probably the biggest change announced in the consultation, as we see the final draft of what was originally proposed in the government's response to the Curriculum and Assessment Review in November, with some tweaks and further information.

Ostensibly this is to attempt to reverse the 'decline' in the take up of arts subjects since the introduction of the EBacc back in 2010. However, opinion remains divided as to how big an impact the EBacc has actually had on arts take ups.

As can be seen in these graphs (which I generated with the help of Google's Gemini AI tool), the only arts subject to experience a significant decline in the last 15 years is design and Technology. However the rate of this decline is similar in Wales (which does not have the EBacc performance measure) when compared to England. This is much more likely to be attributable to the increased costs to schools in offering DT at GCSE, and the significant fall in the recruitment of design and technology teachers meaning that some schools will simply not be able to recruit DT teachers to be able to offer DT as a GCSE option. This is not to mention the changes to the design and technology GCSE, the removal of the Food GCSE from the DT umbrella and the rise in vocational qualifications that mirror different aspects of the design and Technology GCSE, which will all have some impact on the reported take up of design and technology at GCSE. The other subjects in these graphs all show similar rates of decline across England, Scotland and Wales (with the exception of drama in Scotland), suggesting wider societal factors are at play here than simply the introduction of the EBacc.

Even with the removal of the EBacc performance measure, it is hard to see how this can do much to improve the take up of arts subjects. I am sure there are some schools out there that will force pupils down an EBacc pathway simply to try and boost their EBacc take up figures, however I would suggest the majority of schools will be ensuring as many pupils as possible take an EBacc option because either:

1. They believe in the messaging from the previous government that these qualifications are truly the gateway qualifications to further academic study, or
2. Their curriculum and staffing is set up for offering more of the EBacc subjects through KS3 and KS4 than arts subjects.

This second point is not to be underestimated. To offer more creative subjects at GCSE, or to increase take up, schools need to spend more time at KS3 preparing pupils for GCSEs in these subjects. This means diverting time at KS3 away from other subjects (most likely the humanities), towards these subjects. This requires more teachers, more specialist equipment or larger spaces (in the case of drama and dance), that many schools will not be set up to provide. Smaller schools especially would struggle with the financial burden of these subjects compared to predominantly classroom based subjects such as history, geography and RS if take up of the arts were to significantly increase, and would almost certainly have to reduce their humanities staffing. These smaller schools are already likely to be reviewing their staffing following the government pledge during the aforementioned curriculum and assessment review to ensure that the three separate science GCSEs are available in every school - if these schools have to find extra money for science teachers and science equipment they are even more unlikely to be able to fund increases in arts teaching and equipment.

Simplified banding processes

Instead of the current banding process, which sees schools grouped into five groups based on the confidence intervals of their P8 figure, the government is proposing to simply chop schools into five quintiles based on their P8 figure, so the bottom 20% would be 'well below average', the next 20% labelled 'below average' and so on. This compares to the distribution of scores in 2024 shown below (note the image was actually produced in 2019, but the figures remained the same until 2024).

The government say this is to address issues created by confidence intervals, such as smaller schools having such wide confidence intervals that they can never be anything other than average. 

Whilst I appreciate that the current system is more convoluted, I can't help but feel that the replacement is too simplistic. The government have said that they will mitigate against the loss of confidence intervals by publishing three years worth of data alongside each other, as well as cohort sizes and an explanation about the inherent uncertainty due to cohort sizes, however it still feels off to me to have all of these categories be the same size. The figures above suggest an almost normal distribution of schools - in a normal distribution approximately 38% of the data is within 0.5 standard deviations of the mean, with about 15% then between 0.5 and 1 standard deviations on each side, and a similar proportion above or below 1 standard deviation from the mean.

Whether arrived at using the current methodology or using percentile (as opposed to quintile) measures, this distribution of schools feels right to me.

New measures introduced

Alongside these changes, the government is suggesting introducing two new measures for school accountability.

New measure for those that didn't meet the expected standard

It has long been recognised that a small number of pupils performing poorly can drastically alter a school's P8 score. The previous government went some way to address this by introducing a cap for how negative a pupil's P8 score can be, however this government is looking to go further by including a new measure of progress alongside P8 for those pupils that come to secondary school without having met the expected standard in English and maths.

The proposal is to calculate a best-fit progress score across all the subjects that a pupil sits individually - basically calculating a P3, P4, P5 etc. score and allowing the school to take the highest of these. It is hoped that this will allow schools to continue to encourage lower prior attaining pupils to attempt a broad curriculum, whilst allowing schools to highlight the progress pupils make in areas even if those pupils don't as well in other areas, or don't fill all eight of the P8 buckets.

While a laudable attempt, I can see this process being very open to 'gaming' by less scrupulous schools. It would be very easy for a school to decide that a pupil isn't on track to achieve a grade, or only on track to achieve a grade 1, in a particular subject and so redirect time for the pupil away from this subject into another subject to try and secure improvement there - particularly the double-weighted subjects of English and maths. I guess it will depend on how this measure ends up being used in accountability as to how much effort schools might put in to maximising it (this measure won't be made public in the first year), but I can certainly see the potential for this to provide perverse incentives for schools to possibly narrow the curriculum for a child rather than broadening it.

New additional achievement measure for high attainers

Alongside the current measures of percentage of pupils achieving grade 5+ and grade 4+ in English and maths, there is a proposal to include a new measure for the proportion of pupils achieving grade 7+ in English and maths. The government says that this should reinforce 'the incentives for schools to provide a rich and stretching education for all children'.

This seems to suggest that the government believes that there are schools out there that that don't try and push as many pupils as possible into those top grades - which I honestly don't think is the case. The Progress 8 measure already ensures that schools try and maximise the attainment of all pupils - and it is generally well recognised that moving a child from grade 6 to grade 7 is much easier than moving them from 2 to 3 or 3 to 4 (certainly in maths). For maths specifically it is also the case that many post-16 providers don't accept pupils for A-Level maths with less than grade 7, so schools that are able to often provide support for pupils to achieve this benchmark.

Like the previous new measure, this one seems open to gaming - possibly even more so - and could easily incentivise schools to narrow the curriculum rather than make it more 'rich and stretching'. I can see the scenario where pupils might be pushed into extra English and/or maths if they are deemed able to reach grade 7, even if this costs them a grade in another subject. From a schools point of view, the Progress 8 contribution is greater if a pupil gets, say, a grade 7 in maths and a 4 in history, than if they get a grade 6 in maths and a 5 in history (due to the maths being double weighted). The fact that this will now also improve another accountability measure if the pupil also gets a grade 7 in English as well as maths can, in my opinion, only serve to push more desperate schools to prioritise English and maths grades over the wider curriculum.

The missing piece of the puzzle here is how these new measures will feature in Ofsted's process for holding schools to account. Given the focus throughout the latest framework on pupils with SEND or disadvantage, I would expect the measure for those that didn't meet the expected standard (which is over-represented by pupils with SEND or disadvantage) to feature prominently in their thinking.

An interesting property of linear sequences - inspired by the 1% club.

The 1% club is one of my favourite quiz shows. It is the only quiz show I have actually applied to be on (no success unfortunately) but I play along on the app all the time, and also regularly complete the daily question that comes through the app. Yesterday (27th January 2026) had a very interesting question (from a maths point of view) that sparked a little dive into linear sequences. I resisted posting it yesterday as I didn't want to provide spoilers for any readers that also play along.

So, the 1% club daily question yesterday was this: 

What two digit number replaces the question marks in this sequence of numbers:

92, 23, 53, 83, 14, 44, ??

What made this interesting was the way I achieved the correct answer was very different to the way the app explained how to arrive at the answer (if you want to try and answer before I reveal the solution then don't scroll down too far!)

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The correct answer was 74. The reasoning the app gave was that if you reverse the digits of the list you get the sequence 29, 32, 35, 38, 41, and 44 and so the next value would be 47 which, when reversed gives 74. Which makes perfect sense. But it isn't how I arrived at 74.

I (as I am sure many other readers also) noticed that a lot of the jumps were +30 and that those that weren't were -69. There also seemed to be a regularity to when these jumps appeared; a jump of -69 followed by two jumps of +30. Given the jump of -69 from 83 to 14, I reasoned there would be a jump of +30 (although I was wrong about the regularity of the pattern of jumps as the next would actually be another -69).

Of course, once I realised that these two approaches both gave the same answer, I absolutely had to try and decide whether this was a property of this particular set of numbers, or whether it would be true for the reverse digits of all linear sequences made of two digit numbers.

Rather than diving in with the algebra straight the way (that is coming, don't worry), I decided to play with a few more sequences first to create further examples and see if this sequence was obviously a unique case (a very good problem solving strategy in general I find to allow for pattern spotting).

So I tried 30, 34, 38, 42, 46, 50 becoming 03, 43, 83, 24, 64, 05 - which quickly disabused me that there was any regularity to when a sequence went up or down, and then I tried 17, 24, 31, 38, 45, 52 becoming 71, 42, 13, 83, 54, 25.

It was at this point that I realised that the value of the differences were always 99 apart in the reversed sequences, in the first 30 and 69, in the second 40 and 59, in the third 70 and 29. It took me an embarrassingly long time to recognise that the subtractions were happening when the original linear sequence bridged a 10, or that if the linear sequence was going up in 3 (say) that the reversed sequence should be going up in 30.

I started to explore the algebra at this point a little, but quickly realised that I was getting confounded by the fact that I had only tried differences in the original linear sequences that were less than 10, so I tried 26, 39, 52, 65, 78, 91 becoming 62, 93, 25, 56, 87, 19 (which showed me it wasn't so simple as subtractions occurring when the original sequence bridged a 10, but was more about the units digit becoming smaller - which should have been obvious really) and also 12, 35, 58, 81 becoming 21, 53, 85, 18. This confirmed that the sum to 99 was still a thing - or more precisely that the subtractions were the positive differences subtract 99.

At this point I dived properly into the algebra, which I did as follows (again, if you want to try it first then don't scroll down):

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(I added some text to show clearly what the algebra implied that I didn't write in my own scribblings).

In terms of this as a task for pupils, I think there would be something interesting in offering KS3 pupils a chance to explore 'reverse linear' sequences - probably at a distance from linear sequences themselves. I think it might reinforce some properties of linear sequences and it would be very interesting to see if they spot the 99 link and how they try and justify it.

I definitely think there would be something about using the proof with a GCSE/Further GCSE/A-Level class, either as an example of constructing a logical proof or as an exercise for them as part of their practise in creating a deductive proof.

Of course, the question remains about what happens with linear sequences that stray into 3 digit numbers (single digits are trivial as we can just treat them as two digit numbers with first digit 0). I have answered this question to my own satisfaction and so will leave it as an exercise for the interested reader with one hint, which comes from when I shared the initial problem with other maths teachers at Twinkl and one of them came up with a third approach to the original problem (which is equivalent to what I have outlined and also leads to the correct answer):

"Add 30 each time but if the answer goes over 100 add the 100s digit to the ones digit".

A mathematical curiosity?

 In writing my new book 'Practising Maths' I referenced a lovely result (you will have to buy it to see how) that sums such as 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1, etc. all produce square numbers.

If you haven't come across this result before then feel free to have a look at it for a minute (even try and prove it) - if you are familiar with consecutive triangular numbers summing to square numbers, it is closely related.

The curiosity I noticed was that I knew 121 was also square. So I became interested in the fact that 1 + 2 + 1 is square, and 121 is square. I decided to look into the others, and it turns out they are also square! Well, the ones up to 12345678987654321 are square anyway.

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This of course raised a question - is this a reflection of something deeper? You might like to spend some time exploring and coming to your own conclusion before you read on.

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I guess the truth is a little of both.

If we consider squaring polynomials of increasing order with unit coefficients we get the following:

These are, of course, the same expressions as above, but in base x rather than base 10. So, if we substitute x = 1 into the expressions we get the sums on the left of the above table. However, if we substitute x = 10 into the same expressions, we get the numbers on the right of the table.

In terms of a task, we could offer the first few rows of the table to pupils and ask them what they notice/wonder. They might explore when the pattern breaks and why. We might encourage them to write out the numbers using explicit base 10 notation, such as 1 × 100 + 2 × 10 + 1 and see what insights this brings out. Pupils with the necessary algebra skills might even explore the expansions given above. Or we might just show it to pupils as an example of a mathematical curiosity.

New Maths and Dyscalculia Assessment!

A new assessment for identifying and supporting difficulties in mathematics learning was launched in late July that has the potential to be of significant help in strengthening school’s and parent’s ability to appropriately plan for learners struggling with key aspects of maths study.

The assessment has been designed by the co-founder of the Dyscalculia Network and experienced specialist teacher, Rob Jennings, alongside Jane Emerson, the Director of Emerson House – a centre for dyscalculia, dyslexia and dyspraxia.

Rob Jennings and Jane Emerson (used with permission)

Split into 19 sections, the assessment provides for a comprehensive examination of learner’s abilities regarding early number concepts such as number sense and counting, different mental and written calculation strategies, interpreting word problems, working with and converting between fractions and decimals, as well as basic length measure. The sections have a mix of verbal and written questions, with some of the earlier ones requiring the use of counters. The authors suggest that the assessment should take roughly one hour (based on trials that have taken place to help refine the test questions and assessment approach), however are at pains to point out that the assessment should not be limited by time, either overall or for any one of the sections, as this could lead to anxiety for the pupil that could skew the results.

What sets this assessment apart, for me, from other assessments and on-line screeners for dyscalculia and/or other maths difficulties, is the level of detail that the assessor (which could be a parent, teacher, or TA – not necessarily a qualified assessor) is encouraged to capture about the pupil. As well as simply getting an answer correct or incorrect, the assessor is encouraged to note down (through the use of a provided assessor’s booklet) how long each section took, the strategies that pupils used (to help capture whether these are efficient or immature strategies), and any comments or questions that the pupil makes – either to themselves or to the assessor. This sort of information is potentially crucial in formulating a proper plan for addressing the difficulties that the assessed pupil is facing with mathematics. In addition, there are actually two assessments, an A assessment and a B assessment, which means that they could be used as a pre- and post-test for an intervention specifically designed to support the pupil.

Access to the assessment, assessor’s booklet, and answers comes through the purchase of the companion guide called (straightforwardly), “The Maths and Dyscalculia Assessment”, with a link and redemption code for the online materials included in the guide.

The companion guide comes with much more than simply a step-by-step guide on administering the test itself. Included is also guidance for the assessor in preparing for the test, including how to make sure the necessary things are organised in advance, how to create a good environment for conducting the test, what to be on the lookout for and to record during the assessment, and even making sure that the learner is at ease during the assessment. There is also a chapter on interpreting  the results of the assessment, including what issues may have been highlighted by the assessment, what might then be included in a teaching plan if these issues have arisen, and suitable specific interventions that might be required, which I would definitely recommend reading this before  administering the assessment – I feel like it would sharpen my focus on certain aspects of the test and approaches that a child takes to the test beyond what is given in the step-by-step guide.

In addition to the guidance provided for before, during and after the assessment, the companion guide also contains a host of other information and support for working with pupils that have difficulties with mathematics, including different checklists or screeners that could be used in advance of the assessment or to support its findings, a template summary report along with a completed exemplar to help capture the results of the assessment and plan for future teaching with the pupil, and an in-depth family questionnaire that can be used to provide extra detail and context to be used in the summary report. Both the templates for the summery report and the family questionnaire are both included in the online materials accessed through the guide, which means that the complete assessment package can be used with as many pupils as is needed. The guide also contains lots of information about how dyscalculia and other maths specific and non-specific learning difficulties (such as maths anxiety, dyslexia and the like) might impact maths learning and attainment, as well as some interesting statistics about occurrence of maths difficulties and co-occurrence with other difficulties, and a host of sources of further information and resources that could be useful for parents and educators.

As I went through the book and the online assessments, I reflected heavily about the numerous pupils I had encountered that displayed some or many of these difficulties. In the last department that I led we benefited from the assistance of a part-time numeracy intervention tutor and I can see how this sort of assessment would be invaluable in supporting her work alongside those of the main class teachers, as well as contributing hugely to the work of our SENDCo and inclusion team in pinpointing the difficulties that pupils were having and initiating conversations with parents and other agencies about the diagnoses and support that these pupils might benefit from. Despite my limited experience in the field, I have never seen anything that is designed to capture how pupils approach maths problems alongside their accuracy and time taken in completing them, or the depth of insight to guide future planning that this test provides and for that reason alone I think this book/assessment is well worth a look for any school or parent that wants to get a real handle on the maths difficulties that their children are facing.

From GCSE Sequences to Calculus

As part of the recent mathsconf39 I ran a session on using structure to promote algebraic thinking, and as part of that I looked at using a couple of different structures that can be used to help with the teaching of sequences. This got me reflecting on a link that I taught and highlighted to my Level 2 Further Maths and A-Level students about the links between the sequences they learn at GCSE/pre-GCSE and the calculus with polynomial functions that they first learned at Level 2 Further Maths or A-Level.

Part of the first work that pupils do with sequences is learn about linear sequences, and their defining characteristic being that of constant difference. Later they learn about quadratic sequences, and their defining characteristic being constant second difference. I always highlight that this means that the differences in the terms for a quadratic sequence form their own linear sequence. This establishes a progression from quadratic to linear to constant, or, in reverse, constant to linear to quadratic. I then ask them to predict what type of sequence would have a quadratic difference – which pupils quickly identify as cubic and can then extrapolate further from there.

This of course perfectly mirrors the progression that different polynomials go through when they are differentiated or integrated. Cubics differentiate to quadratics, which differentiate to linears, which differentiate to constants. So, when I first introduce differentiation to pupils, I ask them about where they have seen this progression before. Someone almost invariably mentions sequences, and if they don’t, I might start by writing out a quadratic sequence to prompt them.

A great question to ask students then is, of course, why? Why this connection between sequences and differentiation? This allows me to reinforce the idea of differentiation as about the rate of change, as when we are identifying the type of sequence we are examining the change between the terms.

It is worth then examining a particular sequence and its nth term, and the links to differentiation. I will typically pick one that has a coefficient of n², such as the sequence 3n² + 4n – 1:

n = 1		= 2		= 3		= 4
6		19		38		63
	+13		+19		+25
		+6		+6

 

If we differentiate the nth term of the sequence, we of course get 6n + 4, whereas the nth term of the linear 1^st differences is 6n + 7. At first this appears to be a discrepancy, until you see that the 1^st differences are in the 1.5^th, 2.5^th and 3.5^th position. We can account for this by shifting each value back by 3 (half of 6):

n = 1		= 2		= 3		= 4
6		19		38		63
10		16		22		28
	+6		+6		+6

 

Or by substituting n = 1.5 and T_n = 13 into 6n + a = T_n:

6 × 1.5 + a = 13

9 + a = 13

a = 4

Either way the nth term of the first differences can be shown to actually be 6n + 4, which is precisely what we get when we differentiate the nth term.

This provides a nice way of interleaving other algebraic manipulation and structure into the study of calculus, as well as reinforcing the idea that differentiation and sequences are two sides of the same mathematical idea – sequences being an introduction the study of changes and growth of discrete functions whilst differentiation (and integration) being the study of changes and growth of continuous functions. It also just adds to the impression for pupils that the landscape of mathematics is connected in unexpected but beautiful ways.