For those who weren’t aware, the Maths Horizons project (https://www.mathshorizons.uk/) is a group of educators that have been working to review maths curriculum and assessment in England alongside (but independently of) the governments full curriculum review led by Professor Becky Francis CBE. They have launched a series of consultations in the last few months and have now published a 17 page overview of their findings, including their objectives and recommendations for the next 10 years of maths education in this country.
I think it would be hard for anyone to argue with these
objectives – they seem to cover all the main bases including maths for work and
daily life, problem solving, and more learners studying post-16. I would be
interested to hear of anyone out there in the comments who thinks these don’t
cover what a school-level maths education should be for. The recommendations
then set out a path for how to meet these objectives over the next 10 years, so
let us examine this in a little more detail.
1.
Design a curriculum for mastery
Again, I doubt I would find a
teacher of mathematics out there who doesn’t profess to want pupils to master
mathematical concepts. The report notes, correctly, that the current design of
the curriculum leads to pupils being rushed through content in a “conveyor
belt” (in the words of Mark McCourt) curriculum where, regardless of how secure
pupils are in their current study, the curriculum pushes them on to the next
“unit” with no time to support and correct those that fall a little behind. As
such gaps widen over time, leading to the current state that between age 5 and
age 16 relatively small gaps in children’s prior knowledge become so wide that
some pupils can achieve nearly 240 marks on a higher tier paper, whilst others
can barely score 20 on a foundation paper. The report highlights that in countries
like Singapore, there is much less breadth and much more depth during early
number work, ensuring that the foundations for future learning are well secure.
In working with my trainee teachers this year, I have several times highlighted
that learning about a mathematical idea looks more much like this:
and that if we don’t give enough
time in that slow-burn phase of development, we won’t see that rapid take off
in later learning. The knowledge progression maps that this recommendation
suggests be created in the report need to specify this clearly, including
providing the necessary time for depth in early phases, and how this should
lead to more rapid gains later. This is more work than it sounds though, as a
lot of the knowledge contains in the National Curriculum is not well sequenced
or specified; one example is the idea of gradient, which is only mentioned once
in the KS3 National Curriculum alongside graphs and interpreting gradients of
graphs. However, gradients are an idea in of themselves, having much more
affinity with compound measures and rates of change before being tied to linear
graphs. This needs stripping out and tackling separately before being used with
linear graphs. There are other examples of this in the national curriculum that
will need to be identified and dealt with before these progression maps can be
built.
2.
Rebalancing content from upper primary to
lower secondary
I applaud this statement; it is one that I have long called for myself.
The primary curriculum is woefully overloaded, meaning teachers often have no
choice but to accelerate through content in order to ensure that it is all
“covered” by the time that SATs role around this week. I also applaud the
notion that, broadly speaking, the content of the National Curriculum as a
whole is fit for purpose, and comparable with other countries. The report also
highlights that a lot of the content at primary school is also taught (either
by necessity or design) during secondary school. Indeed, the late, great Dr
Tony Gardiner even suggested a list of this in his epic book “Teaching
Mathematics at Secondary Level” (which is 100% worth reading if you haven’t
and, hopefully, will be called on heavily by the members of the Maths Horizons
project):
·
the extension of place value to decimals;
·
the arithmetic of decimals;
·
work with measures—especially compound measures;
·
the arithmetic of fractions;
·
ratio and proportion;
·
the use of negative numbers;
·
work with coordinates in all four quadrants;
I am sure that there are many
secondary teachers of maths out there that will recognise and agree with these
concepts as serious issues for many pupils progressing from primary school, and
that these topics for a lot of the early work that secondary school mathematics
departments have to do before pupils can progress further in their mathematics.
I do thank that some of these could come off the list (particularly negative
numbers and coordinates) if space were made to deal with the properly at
primary school whilst shifting some of the others completely into secondary
school.
What I am disappointed to note in
the Maths Horizons report, however, is the lack of recognition of a need to
further rebalance some of the lower secondary curriculum content from KS3 into
KS4. The same issues arise here, with content taught at KS3 also having to be
taught at KS4, and the rebalancing of KS2 content into KS3 will only serve to
push the problem into secondary schools, where content will have to be rushed
in order to meet the demands of KS3. A similar list of content that could
simply be left to KS4, whilst KS3 can focus on developing and deepening the
pre-requisite maths so that more rapid progress can be made at KS4 might
include:
·
Significant figures (both the concept and
rounding to them) – to understand this properly takes a depth of appreciation
of the place value system, its invariant properties and the role(s) that 0
plays in the system.
·
Expressing errors as inequalities – this
requires a more flexible knowledge of the use of inequalities, and the
difference between discrete and continuous measures (as well as treating
numbers in both discrete and continuous fashion) than most pupils can achieve
at KS3.
·
Rearranging formulae to change the subject –
this seems to be included so that other concepts can be included (which could
themselves be moved to KS4). Rearrangement requires a depth of knowledge of
equality, and valid manipulations of an equality relationship, that only really
begins to develop using algebraic symbology at KS3. Whilst I am in favour of
learners manipulating all types of equations at KS3, I think the requirement to
do this with a specific goal in mind such as changing the subject can be left
until KS4 to allow the pre-requisite knowledge to mature. It is worth noting as
well that the skills build heavily on equation solving skills (which I would
keep at KS3) and so moving rearranging formulae to KS4 would provide a good
opportunity to revisit and develop these skills in a new context.
·
Reducing a linear equation in two variables to
the form y = mx + c – this is tied to the previous point. Whilst the
relationship between algebraic relationships and their graphical
representations definitely does need to begin to be explored at KS3, the need
to combine all of this knowledge and skills can be left until KS4 when the
knowledge of the separate concepts that come together to make this viable has
matured.
·
Graphical and algebraic representations of
proportion and inverse proportion – beyond the basic graphical representations
of proportional relationships like conversion graphs, I feel that the movement
from proportional representations like Cuisenaire rods™, double-number lines
and ratio tables to more abstract algebraic representations can wait until KS4,
where both algebraic and proportional understanding can be made more mature.
·
Ruler and compass constructions – done well,
these need to be tied directly to properties of shapes, congruence/similarity
and other graphical representations (including graphing inequalities in
two-dimensions). As such, I think these can be left to KS4 alongside some of
these other concepts.
·
Applying transformations to given figures –
Again, if done properly these need to be tied to a basic understanding of
vectors (which I would welcome at KS3 as they are a natural extension of using
a vector representation in one dimension for negative numbers). I also take
issue with the focus on given figures; transformations happen to every point in
the plane (accepting certain invariant points) and so the focus on plane
figures as opposed to individual points is misleading.
·
Identifying and constructing congruent triangles
– Whilst I think the basic idea of congruence as a property of shapes can be
introduced at KS3, the necessary conditions for congruence at triangles can
probably be left to KS4 in favour of other things.
·
Introducing/deriving Pythagoras Theorem –
Whilst, again, Pythagoras Theorem as a property of right-angled triangles could
be introduced to KS3, it is too often simply as a vehicle for forming and
solving equations which would be better placed as an opportunity to revisit
these skills at KS4.
·
Use Pythagoras’ Theorem and trigonometric ratios
to solve problems – as above for Pythagoras, but trigonometry in particular
requires a depth of understanding of similarity, invariance, functionality and
equation solving which should be the focus at KS3 so it can be used for
trigonometry in KS4.
Again, I know all of this is
re-taught at KS4 even when it is taught at KS3 (under the mistaken belief that
a GCSE specification sets out the knowledge that needs to be taught in two
years rather than 11), so I think we can shift this to KS4, making room for the
concepts that will be moved up from KS3 and allowing more time to secure the
fundamental pre-requisites of some of these concepts.
3.
Increase the rigour of mathematical
reasoning and problem solving for all students
Again, I am pleased to see in the report
that proof, reasoning and problem solving should be at the heart of how
mathematics should be taught, rather than as a bolt-on that is taught after
pupils have been taught some procedural maths. In my session at the La Salle
Education Complete Mathematics Conference in March I talked about the
relationship between fluency, reasoning and problem solving, where I shared
this diagram as much more indicative of their interplay:
As part of the recommendation, the report
suggests “creating examples and questions that demonstrate which “familiar
content” should be developed through problem solving, and which types of
problem to use.” My internal jury is out on this part; I think I would need to
see in practice what this looked like. As read, it seems to suggest that
certain content should be taught through problem solving, and different content
not. This is not in line with the diagram I shared above. Rather the examples
and guidance needs to show how each and every concept in maths arises
reasonably (by which I mean “by means of reasoning”) and that this development
allows us access to new questions and problems that we can ask and answer. For
example, when we extend numbers beyond the naturals into the full integers
(i.e. to include negatives) it then becomes perfectly natural to ask “how do
our four basic operations behave with these types of numbers?” This can then be
reasoned out based on our understanding of the properties of these operations
and the models we use for them. Note, I am not saying that pupils should be
left to reason all of this for themselves; teachers will need to take a range
of strategies between reminding, prompting and explicitly teaching the
reasoning that allows us to make sense of how to combine the previously studied
operations with these new types of numbers. If this is what the recommendation
becomes, then I welcome it.
4.
Introduce low-stakes gateway checks of
fundamental knowledge
This is the first recommendation that I am
almost certainly flat-out against, primarily because I cannot see how it is
compatible with recommendations 1 and 2. In a true mastery curriculum, time is
the ultimate variable. It is a simple fact that some pupils will take longer to
achieve the depth of knowledge required, and if we are going to push for all
pupils to do this, we will need to give some more time than others.
Implementing a knowledge check at certain arbitrary points in the learning
journey (such as near the end of certain school years), will have the opposite
effect to that proposed in recommendation 1 – rather than teaching for mastery
it will push teachers to try and rush pupils to this point. Even in a
low-stakes environment (i.e. without the fear of accountability measures),
setting a check at a certain point will communicate to teachers and leaders
that pupils are expected to be at a certain point by a certain time. Planning
will then take this into account, with curricula built to ensure that this
happens, with kids pushed through the curriculum to get to the expected point.
The report highlights the Multiplication Tables Check in year 4 as an example
of such a check (although does recommend reviewing the impact of having the
questions timed and reporting the results). Now, aside from the fact that the
multiplication check doesn’t at all check that pupils really understand
multiplication (it only tests that they can parrot out certain multiplication
“facts”), it also sets in stone the idea that by this age, all pupils should be
at a certain point in their learning journey for multiplication, which may or
may not be true.
The only way I can get on board with
something like this is if schools can choose when individual pupils can sit
such “checks”, in order for it to truly be a check of whether a pupil has
mastered the knowledge necessary to move forward with the curriculum.
5.
Reform the Key Stage 2 SAT exams
Another recommendation to be broadly
applauded, particularly the part that recognises that the “expected standard”
can be reached by securing less than half marks, which doesn’t really help
secondary schools build on what pupils’ study at primary with any real
security. Given that pupils can secure over a third of their marks on the
arithmetic paper alone (the report highlights that many schools will cram the
arithmetic paper material for exactly that reason), a change in the structure of
the papers to integrate the reasoning and problem solving with the arithmetic,
and a change to what is needed to secure the expected standard is probably long
overdue. The only bit I am unsure of here is that, as part of this
recommendation, Maths Horizons have suggested raising the marks required for
the expected standard from 50% to 75%. I am not sure where this figure of 75%
has come from though? There doesn’t appear to be anything in the literature
that suggests that 75% is a good benchmark to aim for to evaluate mastery. The
lowest figure I have seen quoted is 80%, which I think if we are going to strip
down the content would be a better figure to decide if a pupil has mastered the
foundational content present in the primary curriculum.
6.
Reform the GCSE Exams
Again, another recommendation that seems
sensible. The report highlights that, similarly to the KS2 SATs, the benchmark
for achieving a level 2 qualification in maths (GCSE grades 4 or above) is
probably too low. The report talks about the introduction of a “gateway” paper
that all pupils sit with a high threshold to achieve a “standard pass”. It is
unclear what pupils would do after this though, as the report also rejects
suggestions to split the GCSE into “methods” and “applications” (as trialled in
the Linked Pair Pilot back in 2010 – still a great source of exam style
questions though), or to implement the “numeracy” and “mathematics” split that
Wales did in 2015. A big reason for the rejection was the “social sorting” that
happened where some pupils were only given access to one paper, limiting their
chances, and that universities typically ignored the “numeracy” paper (in the
Wales approach). It is difficult to see how the gateway paper would be treated
differently if it were followed with either optional or compulsory further
papers. It also seems, at least on the surface, difficult to make a high
threshold for the gateway sit alongside the previous year’s performance without
making the content very low; how would a standard pass in the year 2030 compare
to a standard pass in the year 2020? Interestingly, the language used is
exclusively “standard pass” rather than “grade 4”, suggesting that maybe the
grading system might change under this proposal. However, this still seems to
make it difficult not to disadvantage the first few years of a cohort that might
sit these reformed GCSEs, who would have to do more to get a standard pass than
previous years.
7.
Explore a maths entitlement for 16- to
19-year-olds
Honestly, I am struggling to see what this
recommendation adds to the whole project. It seems to be a lot of “continue
what you are doing”. Promoting Core Maths is mentioned, alongside building
capacity so that it can be offered in more places. This will bump into the
age-old problem of there not being enough teachers of maths to offer this more
widely, and not enough pupils wanting to take it up. Reviewing the A-Level
content is fine, but generally speaking the A-Level has a good balance of core
and applied elements so unless we went back to optional applied modules, where
students could pick whether to study statistics or mechanics applied modules, I
think the best this can achieve is tweaking around the edges. The idea of a
stand-alone A-Level Further Maths qualification is potentially interesting, but
difficult to see how it would work practically without duplicating a lot of the
work that students complete as part of the normal A-Level qualification, and so
would likely lessen the scope of what is achievable compared to the current
model of the Further Maths A-Level sitting on top of the A-Level Maths, with
students needing to study A-Level Maths in order to also study A-Level Further
Maths. A potentially more useful avenue is to widen access to the current
Further Maths A-Level; but on the whole this recommendation needs to be more
fleshed out to before we can see its merits.