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.

Everything that is wrong with the teaching of order of operations (and how to fix it)

Order of operations is one of my biggest issues when it comes to mathematics teaching in this country. For me, it completely epitomises the incorrect prioritisation of speed and accuracy using “tricks” and “hacks” over developing true, deep understanding of mathematical concepts. I am thoroughly tired of learners thinking that 9 – 6 + 3 is equal to 0 because “we do the addition first, so this is 9 – 9”, whereas the result of this calculation (as any good calculator will tell you) is 6.

This, of course, starts with acronyms like BODMAS/BIDMAS/BEDMAS, PEMDAS or a more recent entrant into the market, GEMS (standing for “Grouping symbols”, “Exponents”, “Multiplication and Division”, “Subtraction and Addition”). Acronyms like these have been around for over 100 years, although grew to greater prominence in England around the time of the introduction of the National Curriculum in 1988. They were designed to help mathematics learners remember the order of precedence for operations in standard algebraic notation, i.e. in an expression such as:

The precedence is for the calculation in the brackets to be completed first, giving:

Followed by the exponent, giving:

This highlights the problem with strict acronyms like BODMAS; a learner might now be tempted to try and calculate 8 ÷ 36 first (which will lead to the correct result, but is a less useful path to the result for most learners), and then multiply the result by 9. This is the same problem as in the first paragraph where a learner felt the need to calculate the addition before the subtraction. In fact, like addition and subtraction, multiplication and division are of the same precedence, and so can be carried out left to right. This gives:

GEMS overcomes this difficulty by showing multiplication and division with the same precedence (as well as subtraction and addition), and in the last decade many visualisations of BODMAS/PEMDAS have been produced to try and highlight that there is no precedence between multiplication and division, or between addition and subtraction, such as this one from the Oak National Academy:

However, there are still issues with this. The first is that, even presented like this (or as GEMS), that order of precedence is not strictly true. Take the same starting calculation as before, I can evaluate this is a number of different ways, such as:

 

In the first of these I simply did the multiplication before the bracketed sum or applying the exponent, because I recognised that the timing of the multiplication wouldn’t matter provided that I completely evaluated my divisor (i.e. the ”) and my dividend (, before attempting to divide one by the other. In the second (which, granted, is not a technique to calculation I would employ normally except under very specific circumstances), once I had evaluated the bracketed sum, I “saw” the resulting division as the product of two fractions (i.e. ) which would both simplify and leave a relatively simple fraction multiplication (given that the 3s cancel and the 4 cancels with the 2). This, at least, relegates the precedence of operations from a “rule” which must be obeyed, to guidelines that can be applied flexibly when you are fluent enough to recognise the allowed flexibility. And, for me, this is entirely the fluency we should be aiming to develop in our learners, not creating tricks to bypass. A similar example of this comes in trying to calculate this result:

Without a calculator, and following the precedence of operations, this is a difficult pair of multiplications that many learners of mathematics would struggle to compute correctly, followed by a sum that is only marginally easier. Certainly, even for those that would be successful, it is a time-consuming endeavour. However, the calculation can be made much simpler, to the point where most learners with basic place value knowledge would be able to compute the result:

Here I am recognising that, because both multiplications involve 9.2, the calculation can be simplified to 9.2 × a single number, in this case 10. Because multiplication by 10 is a much easier calculation than by 6.3 or 3.7, this is a far superior strategy than actually following the precedence of operations. However, notice that, in every sense that matters, I have added before I multiplied. This becomes even more apparent when we introduce numbers like surds, where the focus is not on evaluating the “exponents” (in this case, expressed as roots) and the precedence seemingly disappears entirely:

 

It is worth noting that, like the previous example, the first line contains two multiplications to be added, which can be simplified as both multiplications involve , and then the resulting multiplication is carried out after (using index laws), with the exponents/roots never even evaluated.

At this point, these guidelines become less “can be applied flexibly” and become more of a “if absolutely necessary”. So then the question naturally arises, when are they definitely necessary. When must the precedence of operation be followed, and under what conditions can we employ some flexibility. And this links to a much deeper question, namely, why is there a precedence at all?

Now, on the surface, this is not a deeper question. The precedence is there to ensure no ambiguity in the algebraic notation we are employing. For example, if calculating 5 + 3 × 4, the precedence tells me that this should be interpreted as “the sum of 5 and the result of 3 × 4”, rather than “the result of 5 + 3 multiplied by 4”. This is a case where the only correct order is to calculate that 3 × 4 = 12, and so calculate the result of 5 + 12. There are other notations where this potential ambiguity simply does not arise. For example, in polish notation, the same calculation would appear as + × 4 3 5. In this notation, the binary operation always precedes the numbers on which the operation takes place, so this would be read as an instruction to “sum the product of 4 and 3 with 5” and would proceed as:

whereas the alternative of adding the 5 and 3 before multiplying by 4 (which in algebraic notation would appear as (5 + 3) × 4) would be written as × 4 + 3 5 (read as “multiply 4 with the sum of 3 and 5).

What makes the question deeper is when we start to probe why might the convention be what it is when it comes to our ordinary, algebraic notation. Why have we decided to take 5 + 3 × 4 as the sum of 5 and the product of 3 and 4, and (5 + 3) × 4 as the product of the sum of 5 and 3 with 4? I think part of the answer lies in understanding important models for addition and multiplication, along with a fundamental law of arithmetic – the distributive law.

When it comes to addition, one of the key principles that our learners need to understand is that addition operates on a consistent unit or “like terms” to borrow from language generally used with algebra. When learners first start studying addition, this is implicit as all the numbers they work with are automatically in ones, e.g. 3 + 4 means 3 ones + 4 ones. Where it should be made more explicit than it currently is when learners start adding larger numbers in columns, for example 32 + 25. In columns we add the ones (7), and then add the tens (5), but what is important here is that it is this property of addition that allows this is that the ones are a consistent unit, and the tens are a consistent unit – it is literally “2 ones plus 5 ones is equal to 7 ones, and 3 tens plus 2 tens is equal to 5 tens”. Since the loss of multi-base arithmetic from the curriculum, what we have lost at this point is the opportunity to carry out similar sums outside of our normal place value columns, for example in base 8 this same sum becomes “2 ones plus 5 ones is equal to 7 ones, and 3 eights plus 2 eights is equal to 5 eights”. This doesn’t mean of course, that we can’t offer learners opportunities to calculate in multiple bases, or simply to simplify sums like this without evaluation, for example:

 

Learners don’t need to be explicitly taught that this is working in base 8, just what it means to simplify a calculation like this. This approach has major benefits later on as it ties to other occurrences of the same property of addition that learners tend to struggle with more; namely:

-        Why  can be added, but  cannot (at least until converted to a common unit, probably 77^ths).

-      Why  can be added, but  cannot (and this time there is no equivalent common unit to convert either of these to, except approximately).

-      Why  can be added, but  cannot (unless you know something about the relationship between  and  which would allow you to convert them to a common unit).

Of course, it is this last pair of sums that really drives this behaviour, as algebra (at least at school level) is just generalised arithmetic and so it is the fact that this common unit (the like terms I mentioned earlier) is required for addition in algebra that makes it equally true for all types of numbers. However, this shouldn’t only be made explicit once learners start to study algebra – if they haven’t had this property highlighted to them through lots of different work with lots of different types of numbers, then this generalised arithmetic comes out of the blue. This is what leads to many learners’ early difficulties with algebra – they simply haven’t worked explicitly with specific examples of the structures that we are asking them to then generalise. The key message here is “what am I counting?”, because if what you are counting is not the same type of thing, then these can’t be added together unless they can be made the same type of thing.

Where multiplication then fits into this is that multiplication creates/changes units; or at least this is one model for making sense of multiplication that has many useful properties. For example, when we write the calculation 3 × 5, we can think of this as “5 things worth 3”, or, alternatively, “3 things worth 5”. Money is an excellent example of those as it comes in different denominations (and yes, that word links very closely to denominators). For example, you can ask something like, “What do we have here?”

And learners might say “£53”. I would say “No” at this point, and highlight that I didn’t ask “how much money do we have here?”, but rather, “what do we have here?”. Learners will probably then respond correctly with “Three £1 coins and five £10 notes”. As they do I write down 3 + 5 × 10. Then explore how this arrives at £53 (and certainly not £80). Learners can then picture what it would mean to have (3 + 5) × 10 in terms of money (three £10 notes in one hand and five £10 notes in the other). We can then go on to explore other calculations like the ones below:

Can we write each of the calculations below as a single multiplication?

a)     2 × 3 + 5 × 3

b)     2 × 7 + 5 × 7

c)      2 × 12 + 5 × 12

d)     2 × 1000 + 5 × 1000

e)      2 × 1 million + 5 × 1 million

f)       2 × 3 + 5 × 7

g)     2 × 3 + 5 × 9

h)     2 × 3 + 5 × 6

i)       3² + 5 × 6

j)       2 × 3 + 5 × 2

Can you come up with a calculation of this type that can be written in (i) no other ways? (ii) one other way? (iii) two other ways? (iv) more than two other ways? What can you generalise from your calculations?

The last four are particularly interesting here, as 2 × 3 + 5 × 9 can be rewritten as 2 × 3 + 15 × 3 = 17 × 3; 2 × 3 + 5 × 6 can be rewritten as 2 × 3 + 10 × 3 = 12 × 3 or as 6 + 5 × 6 = 6 × 6, and even when we introduce indices in (i) we can still write this as 13 × 3, with the last one there to show it doesn’t matter if the same unit (in this case 2) is the first number in the multiplication or the second. This is laying the groundwork for being able to add fractions by finding equivalent fractions (just like 2 × 3 + 5 × 9 can be written as 2 × 3 + 15 × 3, so  can be written with a common unit of ninths). In general, the model again stretches into all areas of number and algebra, including standard form, with  being seen as “6 of these things called 10⁵”, surds (6 of these things called and algebraic expressions:

a)      2 × 3 + 5 × 3

b)     2 × n + 5 × n

c)      2n + 5n

d)     2n + 5m

e)     2(n + 1) + 5(n + 1)

f)       2(x + 1) + 5(x + 1)

g)      2(x + 1) + 5(1 + x)

h)     2(x + 1) + 5(x - 1)

i)       x(x + 1) + 5(x + 1)

This model also helps with something like  as we can see this as “2 of these things called 3²”, or alternatively “3² of these things called 2”. If we want to consider the first case, 3² is the unit, so we need to know what this is worth in ones so we can see what we have two of. In the second, we have 3² of these things called 2, and we need to know actually how many 2s we have.

Whilst this doesn’t move the precedence of operations from convention to required, it is suggestive of why it makes sense that we would consider multiplication as a higher precedence than addition – multiplication creates/changes the units so we need to look at these first and make sure that the units are consistent before we can then add. If the units aren’t consistent (for example in 3 + 5 × 10) then we carry out the multiplication in order to change the units (3 + 50) so we can then add the values with consistent units (3 ones plus 0 ones = 3 ones, and then 5 tens, so 53). As indices are, at least initially, an extension of multiplication, then we need to look at these and make sure they are also unit consistent (in the case of something like ), or that we know how much the unit is actually worth (in the case of something like ).

So, what about the distributive law? What is this and where does it fit in?

The distributive law is a fundamental axiom of mathematics, that states (in our usual notation):

Learners normally see this when they are working with algebraic expressions, particularly in expanding and factorising. However, it is the same law that governs most of the properties that I have just talked about. For example, why does addition require consistent “units”; because I need a “factor” I can take out to enable the addition in ones. For example, why does 2 × 3 + 5 × 3 = 7 × 3? Because:

Or why does ? Because . The “unit” that multiplication creates or changes is the factor, and without that common factor across all of the terms to be added, the distributive law doesn’t apply. Our use of brackets becomes crucial, then, to show what is being distributed over what. And, what can be distributed over what also ties very closely to our precedence. For example, we have seen that multiplication (by definition) distributes over addition, but that the reverse is not true (e.g. 3 + (4 × 5) is not equal to 3 + 4 × 3 + 5). Indices will then distribute over multiplication, e.g. (2 × 3)² = 2² × 3² (which is easy to demonstrate with small, positive integer powers). However, indices do not distribute over addition, e.g. (2 + 3)² ≠ 2² + 3², because (2 + 3)² actually means (2 + 3) × (2 + 3). Similarly, multiplication does not distribute over indices, e.g. . This gives us that: 

 

 

This, of course, is our precedence of operations!

The distributive law is independent of notation – it is a fundamental law of arithmetic. In Polish notation,   equivalent to saying . Distributivity is true no matter how you write it, but it perfectly explains why we have a precedence in operations with indices and roots (which are just indices) at the top, multiplication and division (which is just multiplication) on the next layer, and then addition and subtraction (which is just addition) at the bottom. In this sense, although we have to define a precedence to ensure our standard algebraic notation can be interpreted incorrectly, the choice of this precedence is far from arbitrary.

Hopefully this has shown that the way that the precedence of operations is taught in many of our schools needs urgent revision. But what do we do? What should this revision look like? For me, it has to include the following:

●       Early teaching of the laws of arithmetic, particularly distributivity and its usefulness in factoring and re-factoring calculations.

●       Ensuring properties of operations like “requires the same unit” for addition are explicitly highlighted and linked to the laws of arithmetic. Linked to this, referring to the ones column as “ones” rather than “units” is important; anything can be a unit, but only ones can be ones.

●       Creating time in the curriculum to explore integer calculations in depth with tasks that focus on thinking, reasoning and re-writing rather than simply calculating results, e.g. How many ways can you find to show that ?

●       Ensuring that, as learners progress from calculations in integers to those in rationals and then later irrationals, the same underlying laws and properties are highlighted and linked back to earlier calculations, i.e. learners should recognise that the fact that 2 + 3 = 5 links to 0.2 + 0.3 = 0.5, , , , and .

●       Once security in base 10 integers and arithmetic has been achieved, extending this to other bases so that learners can better understand what changes in a different base (the representations of the numbers) but also what remains unchanged (the properties and laws of arithmetic).

Some of these may be within the remit of individual teachers, schools and trusts. However, some will require wholesale curriculum change and support to create the necessary time to explore these things in depth, and to ensure that they are taught in a “forward-facing” way as learners progress through school-level mathematics so that teachers can have confidence when attempting to build on earlier work that it has been done in such a way as to provide the necessary firm foundations. We can only hope that when the curriculum review currently under way begins to examine the curricula in individual subjects, that they recognise that time must be created for things like this to be explored much more deeply than they are at the minute, so that later learning can proceed at a much faster rate and that more learners can build that deep, connected, conceptual understanding of mathematics that I know is the goal for many teachers of maths across the country.