What Could A New National Curriculum For Maths Look Like?

Peter Mattock

As I write this, we are roughly two months away from the full report of the Francis Curriculum and Assessment review. The interim report has already been published, and has made clear that the curriculum will undergo “evolution not revolution”. Whilst the review has been going on, the Maths Horizons group has produced their own report detailing their recommendations for changes to the curriculum, which includes calls to “rebalance content from upper primary to lower secondary” (I wrote a response to their interim report here).

I think you would struggle to find a primary school teacher, particularly at KS2, who would disagree with the statement that content needs to be removed from KS2 (or the primary curriculum in general) in order to allow for more time to teach the remaining ideas to sufficient depth. However, what should this content actually be? And how do we move content from primary to secondary without simply shifting the problem up the road, given that the aim is “rebalancing” or “evolution”, which would suggest that removing large swathes of the content is not on the cards?

Whilst I am sure that, behind the scenes, groups like Maths Horizons and others are grappling with this question, I decided to examine the National Curriculum documents across all four key stages and see where I might move (rather than remove) content, taking advantage of some of the cross-over between key stages (particularly between KS2 to KS3 and KS3 to KS4). The only content that I did do away with entirely was Roman numerals at primary school (and I doubt there are many primary teachers that would be sorry to see that go!)

From the outset, I knew there were definitely some things I wanted to change:

1. Moving work on fractions (including using words like “half” and “quarter” in relation to turn and time) out of KS1.
2. Moving the formal examination of multiplication and division out of KS1 (I kept some skip counting in, but slimmed down the rate at which these things are brought in for different values).

The goal with these being that I wanted learners to be able to spend KS1 really focused on developing their understanding of additive relationships in respect to the integers, place value, measures, and shape and space.

3. When finally introducing fractions in year 5 (I brought decimal notation in prior to this in year 4), spending year 5 and 6 focused on equivalence (including simplifying), comparing, and additive relationships within fractions, not involving mixed numbers, multiplication of two fractions (the beginnings of finding a fraction of an amount stayed in KS2) and division of or by a fraction until KS3.
4. The removal of all algebraic notation or proportional work (including percentages) from KS2.

Given that fractions, multiplication and division were being developed later, and at a slower pace, I felt that percentages and other proportional relationships needed more time for these underpinning ideas to mature, and similarly with algebraic notation, learners would need more time to understand the different types of numbers they had begun to encounter (and the similar structures at play with these numbers compared to the integers they had previously worked with) before generalising these relationships.

That isn’t to say that a lot of algebraic thinking shouldn’t be going on from very early days – is this always true, in what ways is this like, and so on – just that the use of the notation to capture this in anything other that thoughts and words can wait.

5. Removal of surds, standard form and the top end percentages problems from KS3.

Given that learners, at this point, had only been introduced to fraction notation in year 5, and not to percentages at all until KS3, I didn’t want any more different representations of numbers clouding things across these three years, and wanted more time to develop basic percentages before looking at things like original value problems (which are always a big feature in KS4 anyway).

6. The removal of conditions for congruence, Pythagoras and trigonometry, constructions, and transformations from KS3.

Again, all of these get significant time at KS4, and I felt that they were better served by including the building blocks at KS3. For the last of these, I am particularly wedded to the idea that learners need a strong grounding in vectors beyond one dimension (1-D vectors should be introduced with the number line) as vectors can be used to understand the effect of all transformations (which then leads very nicely in matrices and transformations beyond school level maths), and so I wanted to replace transformations at KS3 with a basic introduction to 2-D vectors.

Despite knowing some of the things I wanted to change from the outset, this was not an easy process. Making sure that things I decided to cover later than they are currently required a lot of re-checking of what I had removed, re-reading everything I had done to that point, and finding places that content could be moved to in a way that was coherent with what had come before but that left enough time for maturation afterwards.

In the end, I also made some further adaptations that I hadn’t anticipated before starting to work on this rebalancing:

1. Removing the number line from year 1, pushing its introduction back to year 2 (but including 1-D vectors from this point) – the number line can wait until learners are more familiar with object representations of numbers.
2. Moving pictograms and block diagrams out of year 2 so the focus on introductory statistics can be on collecting and processing data rather than representing it (as well as avoiding the multiplicative nature of pictograms until multiplication as an operation is formally introduced).
3. Removing all references to “formal” calculations – there really is no such thing.
4. Moving the introduction of rounding from year 4 to year 5 as I think it is too early to get into the idea of accuracy at that point.
5. Removing reference to continuous data at KS2, as I think this is too early to start to draw the distinction between discrete and continuous.
6. Removing the need to read, write and compare numbers up to 10 000 000 from year 6. Honestly, this just feels like it is there to step up from year 5 where the same is done up to one million and beyond one million the focus should be on the “three-ness” of the pattern of place value block (i.e. unit, 10, 100, new unit, 10, 100, and so on).
7. Removing the need to teach understanding place value or to order numbers at KS3, as this should now be well taken care of in primary school.
8. Removing expanding two or more binomials and rearranging formulae from KS3, placing it explicitly in KS4 – it is always taught in KS4 anyway and so greater focus can be given to the manipulation knowledge and skills that underpin it at KS3.
9. Removal of anything other than linear graphs at KS3 when it comes to drawing or interpreting – again, drawing quadratics and other graphs is always dealt with at KS4 and whilst I don’t think it is too much to ask KS3 pupils to use graphs to find values from a wide array of contexts (tying to data-handling graphs as well), I don’t think we need to focus on anything other than the drawing linear graphs using the algebraic relationship at this stage.
10. Removal of the need to recognise geometric sequences in KS3. I left in the bit about appreciating other non-linear sequences, but being able to recognise geometric sequences at KS3 when you are doing anything else with them seems so arbitrary I think it can safely go in with quadratic sequences and Fibonacci-style sequences at KS4.
11. Using graphical and algebraic relationships of proportion and inverse proportion problems – given that proportion as a concept would now only be introduced at KS3, it feels like this can be left to KS4 (where, again, it is always explored anyway).
12. Removing the need to look at continuous and grouped data, or scatter graphs, from KS3.

All of this (and some other minor bits and pieces) meant adding, in total, 12 lines to the subject content for KS4:

1. distinguish between exact representations of roots and their decimal approximations
2. interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or zero
3. expanding products of two or more binomials
4. rearrange formulae to change the subject {including where the proposed subject appears more than once}
5. reduce a given linear equation in two variables to the form y = mx + c
6. Solve increasingly complex problems involving percentages, including original value problems and problems involving repeated percentage change.
7. identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
8. identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
9. derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle);
10. know and use the criteria for congruence of triangles
11. apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
12. use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

I am sure that anyone familiar with the teaching of GCSE maths will recognise that all of these things are taught, or re-taught, for the majority (if not all) GCSE pupils, and so having them mentioned explicitly in the content list for KS4 does not, in reality, add any extra pressure to the KS4 curriculum as a whole. If you want to see the complete document where I made all of my changes, it can be viewed here: National curriculum PGM adapted KS1 to KS4.docx.

I doubt it is the only way that this re-balancing could be done, and I am sure some people will disagree strongly with the decisions I have taken, but it at least establishes that it can be (in my view) done effectively without having to remove any content from the overall maths curriculum.