My morning routine and a nice maths prompt

I have a somewhat quirky morning routine. I think it started as a way to occupy my mind during my morning commute. I don’t drive a car and so for the last 11 years of my teaching career, all spent in the same school, I used to use a combination of roughly 40 minutes walking and an eight-minute train journey to get to school each day. I like to be up for quite some time before I have to be active – to ease myself into the morning – and so I used to get up at 6 am, leave the house around 7:15, and arrive at school shortly after 8 am. During that time before I left the house, and then during the journey to school, I run through a series of games on my phone with an almost religious regularity.

It starts with the daily Wordle, then the daily Quordle (both the Classic and the Sequence, as well as the Weekly when I solve my first Classic of the week). Then I move onto the maths games, starting with the Ooodle, the OoodleMax and the Time Square grid, followed by the Nerdle. Then it is onto geography, with the Worldle (and its different rounds), the Statele, and then the Globle. I finish off with the Daily Sudoku, and then have recently added the Countle at the end. Since leaving teaching I have continued to play these games as part of my morning routine before starting my work from home.

The reason I mention all of this is that my attempt at the OoodleMax today prompted me to consider a nice mathematical relationship, that would make an excellent prompt for learners that could be used at multiple levels.

This was today’s Ooodle Max:

The goal is to use the numbers from the keypad on the right-hand side, at most once each, to fill in the blanks and make the target. Like other Wordle type games, numbers go orange if they appear in the calculation, but are in the wrong place, green if they are in the correct place in the calculation, and grey if they are not used at all. This was my first guess:

Having used 12 and 15, and without really thinking about it, it occurred to me that to be close I should use 13 and 14 in the multiplication. I then knew the “2” would have to be the divisor of the division, so I then set about figuring about what the dividend would be. This was my second guess:

I was lucky in that I got the 13 and 14 the right way, but what struck me is the relationship between the value of 13 × 14 and 12 × 15. It reminded me of the result that I knew about, which is that the square of an integer is always one more than the product of its adjacent integers. This is, of course, easy to prove mathematically for learners that can expand binomials. Given three integers  then we have that . It got me thinking that there must be a wider pattern to this, which of course there is; given four numbers , the product  and  (so a difference of 2), and then given  we have  and  (a difference of 3) and so on. There is an obvious symmetry here, as we get further away from a central value the difference between the products increases by one and starting the list each time with  makes this relatively clear each time.

The algebraic exploration of this is clearly a nice activity for those that can access the algebra, particularly proving the overall general case. However, it also struck me that these sorts of related calculations, and the general structure underpinning the relationship, can be explored without the need for algebra, using concrete manipulatives:

These are representations of 2 × 3 and 1 × 4, and then 4 × 5 and 3 × 6 – following the same pattern as 13 × 14 and 12 × 15. The question then becomes how is the red array related to the yellow? It is a relatively simple matter to see that we can move counters from the bottom row of the yellow arrays and attach them to the side to create the red arrays, but we will always have two counters left over.

It should also be within the capabilities of many learners to reason why this must always be the case; the number of rows in the red array is two less than the number of columns in the (original) yellow array, and so when we move counters from the bottom row of the yellow array to the side to extend the size of each row, we will always leave two behind. This sort of argument can then be generalised further to any of the related calculations.

So, what would the task look like? I haven’t decided fully yet, but as an inquiry prompt it might be something like this:

Calculate the following:

        (a)   2 × 3 and 1 × 4   

        (b)   3 × 4 and 2 × 5

        (c)    4 × 5 and 3 × 6

        (d)   13 × 14 and 12 × 15

What do you notice about each pair of results?

Could you write down other pairs that would produce the same result?

Can you explain it?

Can you find other pairs of calculations that always follow a different rule that is like this rule?

Maths GCSE outcomes – simply a measure of how wealthy you are?

The latest GCSE outcomes have landed and, as always, the dissection by groups like Ofqual, and FFT has already begun. The headlines in maths are the slight drops in in the proportion of students achieving at or above the major headline grades (1, 4, 7) and above – a continuation of the downward trend that was started to address the grade inflation caused as a result of Centre/Teacher assessed grades awarded during the height of the COVID pandemic. This is lessened when we consider only those students aged 16, with 0.1 percentage points drop in the 1+ and 4+ figures, and a 0.3 percentage points rise at 7+; with the small overall drops attributed to the number of resit students rising, with the usual low proportion of these students achieving these thresholds.

In previous years, I would normally have either been buried in maths data, looking for the lessons we could learn for the following year and which of our students were close enough to a grade boundary to be worth entering for a review of marking or, more recently, analysing whole school data to look for successes or issues in subjects that we need to get ahead of and discuss with the head of department. Without these burdens this year, I enjoyed diving into the detail around the local and national pictures for GCSE Maths in particular, using the excellent Ofqual analytics visualisations, particularly the map of how GCSE results vary by county for each of the grade boundaries.

(Image taken from Ofqual Analytics site and shows the proportion of 16-year-old students achieving grade 7 or above in the different counties in England).

The visualisation is fairly similar for grades 4 and up, showing the much larger proportions of 16-year-olds achieving the top grades across the southern band of counties stretching from Gloucestershire in the West through to Kent in the East, passing through Oxfordshire, Berkshire, Buckinghamshire, Hertfordshire, Greater London and Surrey (the other counties that are at 22% or better are Warwickshire, Cambridgeshire, Rutland and Dorset).

I have used this map before, but typically just to get the regional averages for different grades and different subjects in my school, so that our outcomes can be compared to our region as well as the national picture. However, taking a broader view this year, I decided to look at how this maps to the differences in deprivation across the country.

(Image taken from the Office for National Statistics household deprivation maps based on the 2021 National census data)

This map shows local authority district rather than county, and there are typically two or more local authorities within each county, but the overall picture is remarkably similar to the outcomes map, with the lower deprivation areas spanning that line from Gloucestershire through to Kent. The outliers in this are actually around London (where per-pupil funding can be nearly double other areas in the country) and parts of Yorkshire, Lancashire and Cumbria – North Yorkshire’s outcomes for 7+ are in the 20-22% range (which is around the average for the country as a whole) and all three counties have areas (arguably the more populous areas) with the highest levels of deprivation (15% or higher).

Now, this is not a new phenomenon. In fact, if you roll the Ofqual map back through each year back to 2019 (as far as the map goes) the picture remains very similar – even through the inflated COVID years. But I had never had the regional divide in both top outcomes and household wealth driven home to me in the way that comparing these two maps brings. It serves as a glaring reminder that a school can be doing truly great things and still be limited in the impact they can have due to the wealth in the area that they serve. Of course, there will be anomalies in all of these areas, schools that buck the trend for their region, individuals that achieve top grades despite coming from a significantly deprived background. Like most statistically significant patterns, there are always anecdotes that can be found that do not conform to the narrative. However, data on this scale simply doesn’t lie – your chances of achieving top outcomes in maths are considerably higher if you live in a less deprived area, than if you live in one of the most deprived areas.

The government is, of course, aware of this connection and are taking steps to try and address it, reportedly planning to reallocate “around £2bn of funding to the places and communities that need it most”. How much impact this will have on education remains to be seen – more and more schools are no longer directly under local authority control; however, many still purchase support from local authorities for things like behaviour and SEND. Still, the fact seems to remain that, until the disparity in deprivation between the home counties and the rest of the country is addressed, top outcomes for pupils in these areas are likely to remain starkly divided.

Should GCSE maths resits be scrapped?

Yesterday evening the Telegraph reported that the compulsory resitting of English and maths GCSE for those that don’t achieve at least grade 4 might be scrapped as an outcome of the current Francis Curriculum and Assessment review, with a Whitehall source reportedly stating that “scrapping of mandatory resits for English and maths GCSEs was likely to be on the cards”. This comes at a time when entries for GCSE resits are at an all time high, whilst achievement at grade 4 or better has dropped slightly from last year in maths.

Several possibilities have been mooted for reform of either pre-or post-16, including the Royal Society’s “driving test” style qualification for post-16 students, a return to a more modular style of assessment post-16, or OCR’s proposal to introduce a GCSE short course in maths during key stage 4 that students would sit at the end of year 10, forming part of a larger full GCSE maths qualification that students would sit at the end of year 11. Other, existing possibilities include expanding the use of functional skills qualifications, which is already allowed for students that achieve grades 1 and 2 at GCSE and has only just over half the take up at level 2 compared to the number of GCSE resits, or including qualifications like Pearson Edexcel’s level 2 Award in Number and Measure in the conditions of funding for post-16 so that these could be offered as an alternative.

Whilst all of these are worth consideration, they are clearly littered with problems. The biggest, and most obvious of these is where these reformed qualifications would sit in the larger qualification landscape. Currently, grades at GCSE provide students with either a level 1 (grades 1 to 3) or level 2 (grades 4 to 9) qualification in mathematics. The level 2 qualification in maths, rightly or wrongly, currently serves as a gateway to many post-16 opportunities, with most level 3 (A-Level or equivalent) qualifications requiring students to either have, or at least continue to work towards achieving, the equivalent of a grade 4 or better at GCSE maths. In addition, grade 4 in maths is a requirement for many career or university pathways. It is hard to see how any alternative qualification could significantly increase pass rates at level 2 whilst remaining comparable to GCSE maths. If, for example, three-quarters of the roughly 30% of students who don’t achieve a grade 4 in GCSE maths by age 16 went on to achieve a pass in an alternative level 2 qualification post-16, this would raise huge questions over the comparability of the two qualifications and pose a real risk to the status of GCSE maths.

The knock-on effect of this in schools would be devastating, with considerable numbers of students likely to “down tools” on GCSE maths study and simply wait it out until they could access the (clearly easier) alternative qualification at college or sixth-form. Given that schools in deprived areas, generally speaking, have lower achievement at grade 4 or better at GCSE, this could mean as many as 50% of students entering key stage 4 in these schools effectively opting out of GCSE maths study, further exacerbating the divide that already exists between outcomes compared to students in more affluent areas, and playing havoc with school accountability and league tables (not to mention behaviour). At a time where pressures over accountability and Ofsted are already a leading contributor to teacher, and especially leader, stress and retention issues, putting already disadvantaged schools in a position where engaging students in GCSE maths study is even more difficult would drastically exacerbate this situation. Coming also at a time where the government is trying to boost the number of people entering careers in STEM fields, action that is likely to reduce the overall number of students gaining a GCSE grade 4 in maths in favour of an alternative, less demanding, qualification would seem to be counter to that aim.

So, what is the alternative? It would seem to me that a possible approach would be to de-couple an alternative qualification from our existing qualification structure, whilst working with employers to ensure that such a qualification provides the clear indications for skills in numeracy and data handling that they report as lacking in the current workforce. This stand-alone qualification would not, therefore, be able to replace GCSE maths as a gateway to future qualifications, meaning that GCSE maths would maintain its importance within the qualifications and accountability landscape, but might provide employers with the evidence they need that a person has shown the necessary skills to be a valuable part of their workforce. If such a qualification provided a more nuanced account of a candidate’s skills than simply a numerical grade or a pass/fail indicator, instead generating a skills map showing competencies demonstrated over the different domains of the qualification, this would allow employers to assess whether an applicant had the necessary knowledge and skills for the role that they were seeking.

The only other possibility I can see is to remove GCSE mathematics from its position as a gateway qualification. This would almost certainly mean also having to remove or reduce its status as a core subject, and its prominence in school accountability measures – lest it lead to the damaging issues that I described above. Whilst I know there are many educators out there, including some in the maths community, that question the standing of maths as a core subject that all students must study until at least 16, given the aim highlighted above regarding increasing participation in STEM careers, and the clear links between attainment in mathematics and life time earnings, any such move to reduce the importance of mathematics in the school curriculum would likely only serve to widen the economic gaps that already exist in our country.

The current approach is clearly far from perfect, but I think that the option and opportunity to resit GCSE maths remains an important part of the offer for those students that don’t secure a grade 4 in the subject at 16. There are changes that could be made to the current system that might make things slightly better. Removing the November resit or limiting the availability of it to students who only narrowly missed out on grade 4 or have already achieved grade 4 and are simply looking to improve would be a starting point. This would mean that other students who were further from grade 4 have time to benefit from more real teaching rather than just going through an endless cycle of exam preparation with post-16 teachers being forced to try and secure marks quickly at the expense of a proper learning experience. In the future online assessment world, we could also look at the possibility of students being able to “top-up” their current GCSE, with competencies at the first attempt being recorded and attributed to the student, with that student then being able to focus primarily on demonstrating improvement in their weaker areas (whilst also having to show they have retained the fundamental knowledge and skills that are important in everyday maths). Changes to enhance the current system would, it seems to me, be much more likely to improve the situation for all stakeholders than the unintended consequences that would almost certainly arise from huge wholesale changes in either the pre- or post-16 maths qualification landscape.

Mathematics for pleasure – as important as reading for pleasure?

The importance of reading for pleasure has been tested and proven time and time again. The benefits of starting young with this, not just to education in English, but to other subjects and to mental well-being, are well documented. In fact, just this month, the English Department for Education has announced a “National Year of Reading”, backed by nearly £30 million, in order to support and encourage reading at all levels.

What gets talked about a lot less is the idea of doing maths for pleasure. There is plenty of importance laid on doing mathematics outside of school, the role that homework plays (provided it is well set and followed up with), and involving kids in maths around the house (like weighing and measuring ingredients or estimating the weekly shop). This is despite the fact that young children seem to enjoy mathematical activity tremendously, as much as they do reading. Pre-school children typically love sorting objects based on colour or size, stacking blocks, learning to count and measure, or playing shop with toy money and items. All of these are mathematical in nature but, as children age, mathematics seems to be relegated purely to the classroom, whilst reading is actively encouraged (for the most part) by both home and school.

No doubt, societal attitudes and maths anxiety play a part. It is estimated the nearly half of working-age adults only have the equivalent numeracy of an 11-year-old, and that one in every five adults have felt anxious when faced with a problem that is mathematical in nature. However, one possible way to reverse this for the next generation is to normalise the idea of “recreational maths”, playing with maths and numbers as a hobby, or a form of entertainment, as much as escaping into a good book is normal. Furthermore, there is good evidence that engaging in recreational mathematics increases both attainment and engagement with mathematics.

So, how might schools and parents engage children with continuing with mathematics for pleasure? Well, here are a few ideas:

1. Schools - include maths puzzles and games as part of homework: The EEF Teaching and Learning Toolkit page on homework (mentioned above) suggests that shorter homework tasks set twice is a week leads to the highest impact. Puzzles like Kakuro puzzles or Futoshiki puzzles are wonderful ways to prompt retrieval and develop reasoning strategies that can then be harnessed as part of the curriculum. You can also then highlight online sites where children can continue to play them on their devices, at different levels to suit them.
2. Parents - play or compete with your children: There are some very simple games on sites like “Mathszone” that can be played together or form a competition (I play the Ooodle, OoodleMax, TimesSquare grid and Countle every day on this site just for fun). The first two are very much like the popular word game “Wordle™” and provide similar stats, so you can see who is using the least guesses over time, how big a streak you and your children have built up. It doesn’t even matter if they are beating you, it is all in good fun!
3. Schools - Highlight sites like Wild Maths to your pupils: Wild Maths is basically nrich but for pupils rather than teachers. It has lots of different challenges that pupils can get stuck into and, importantly, they can comment to each other and even submit solutions that can be showcased on the site. The site is part of Cambridge University’s “Millienium Maths Project”, which also includes the site Plus Maths. The articles here are perhaps a little heavy for all but the best A-Level students, but the puzzles (some shared with Wild Maths) are accessible to at least KS4 pupils.
4. Parents - Take a look at the National Numeracy Family Maths Toolkit: This wonderful pack contains fun, maths-based activities for primary age children, but also resources to support mindset development for children all the way up to 16 as well as further information for parents themselves about helping boost their own confidence with numbers and supporting children in develop positive maths attitudes.
5. Schools - Support parents with their own maths anxieties: Negative attitudes to maths often stem from home. Parent’s may well have had a bad experience of maths from their own childhood. Offering opportunities to support parents with maths, in-person, on-line and through written communication (guidance for support and the like) can go a long way to allaying those fears, and providing parents with the confidence to talk with their children at home about mathematics. I know there are schools out there that have run parent maths workshops to great effect, so it is well worth considering.
6. Parents - Look out for maths-themed events: There are many maths-themed events going on in local communities, many of them free. Check your local library for listings, or give search on Eventbrite for “maths” to check for opportunities like these free STEM Masterclasses by the Royal Institution. These can make a great day out for children, particularly as an activity for the school holidays.

Positive attitudes to maths (and school in general) are easiest to forge when parents and schools work together. Hopefully this gives both some guidance for both to help children develop positive attitudes to maths and start doing maths simply for the pleasure of it.

The “Forgotten Third” needs help – but the problem is bigger than that

 With the latest government curriculum review due to report later this year, coupled with the (seemingly) imminent white paper that promises reform of vocational education at Key Stage 4 as well as changes to post-16 English and maths provision, speculation is rife as to what our new National Curriculum in maths will look like. Although the curriculum review interim report seemed to point more towards evolution than revolution, praising many aspects of our current system, one area that has come under increased scrutiny are those students that do not achieve grade 4 in maths (and/or English) by age 16 – the so called “forgotten third” (as it accounts for roughly one third of the 16-year-old cohort each year).

The Oxford, Cambridge and RSA (OCR) Examinations board is leading the charge in this area, already publicly calling for a new “short course” GCSE maths that would focus on “the fundamental maths skills that you need for life and work”. The idea is that pupils would sit this short course at the end of year 10 before sitting the full GCSE maths in year 11. There are obvious issues in this approach around the comparability of any grades awarded from the short course and the full GCSE maths, whether colleges and employers would accept and value the short course, and whether pupils who achieved a standard pass in the short course would then lose motivation to continue on to the full GCSE. This last one in particular needs careful consideration, as it is likely that those schools with the most disadvantaged and disaffected cohorts are likely to be more acutely affected.

I applaud any work that aims to support those pupils that find learning maths most difficult. Those pupils who don’t achieve a standard pass or higher by 16 are highly unlikely to do post-16, with only 17.4% of 17+ students achieving the standard in 2024. Given that, in the years that these students would have sat their GCSE maths in secondary school the grade 4 could be achieved with around 50% on foundation tier, and less that 20% on the higher tier, it is a sad state of affairs that only about two-thirds of pupils meet this standard after 12 years of schooling, and that less than 1 in 5 of those remaining then go on to achieve the standard at all. The government is already taking action to deal with this, with conditions of funding changes for academic year 2025-26 meaning the colleges and sixth-forms have to provide at least 100 hours of maths teaching for pupils without a standard pass within each academic year that they are re-sitting, and strongly encouraged to provide a further 35 hours on top of this, but it is clear that more will be needed to radically shift these numbers.

However, whilst these pupils clearly need more support both pre- and post-16, there is a larger issue at play here in the mathematical progress of a much larger group of pupils. I am speaking of the more than three quarters of 16-year-olds that do not achieve a grade 7 in maths.

Now, I am aware that the awarding of grades at GCSE is influenced by the distribution of prior attainment, and so the percentage of pupils that can be awarded a grade 7 is somewhat limited. But I am also aware that the National Reference Test (NRT) is sat each year by a sample of pupils to determine if there are differences between the most recent cohort and those in 2017 (the baseline year for the NRT). The NRT is sat in February/March of year 11 for those schools and pupils selected, with the data used to modify the distribution model that is used in the grading of GCSE exams. Except that it has never thrown up a need to change that model in any significant way (beyond a couple of percentage points from one year to the next). What this means that, the COVID pandemic disruption notwithstanding, we (and I count myself in this) have not really improved the number of pupils who gain access to the upper echelons of school-level maths. And when I say, “upper echelons”, I mean pupils that can open a higher tier paper and achieve 55% of the marks. This is the average number of marks that were required in 2024 to secure a grade 7. This figure is somewhat skewed however – with two of the four boards that offer GCSE maths in English schools the figure was less than 50% of the marks required. Bear in mind that 2016-2017 was the first year that the NCETM and Maths hubs started training Secondary Mastery Specialists (I know, because I was in the first cohort). Not that I am blaming the NCETM, the Maths Hubs, or even myself. The issue is a systemic one, finding its roots in a curriculum, assessment and accountability system that forces teachers to progress at a pace that has no choice but to leave some behind at each stage in order to try and maximise those that might reflect well on the school. The culmination of this is that many of the foundational ideas that should be secured earlier in schooling are not cemented, and therefore provide no foundation for higher learning. In reality, it should be the case that a significant number of pupils should be able to breeze a foundation GCSE paper by the end of year 9 (10 years of schooling in total) if not before, and that many of them should then have little further difficulty in the higher tier content.

Please note here that I am not advocating for pupils to sit foundation tier GCSE in Year 9 as a gateway to higher tier – this would only lead to the same problems that the proposed OCR course may do but on a grander scale. What I am advocating for is a curriculum that prioritises mastering the fundamentals and provides time for this to happen, an assessment system that measures this properly, and an accountability system that doesn’t incentivise pupils and schools to aim for enough surface-level knowledge that they can scrape what might be considered an acceptable standard, but in reality, is anything but. This is what the curriculum review needs to be focusing on, and what the implementation of it needs to make happen. This is a big departure from our current system, and will require drastic shifts in our approaches to make it a reality. However, this has to be the aspiration for a world class education system.

So, colleagues up and down the country, I have just one more thing to say: Viva la revolution.