W**hat can we learn from the 2019 GCSE Mathematics papers?**

By Jemma Sherwood

We have now seen the third year of the new GCSE in Mathematics and we are starting to get a feel for the question types, the mix of procedure and application, and the complexity of the problems. It is clearly a different GCSE from that of old, and reading the Examiners’ Reports shows us that while there is a lot to celebrate – pupils can calculate well and apply a range of techniques to very complex problems – we are still seeing many of the same problems we have come to expect.

**Algebra at Foundation**

It seems that Foundation pupils can grasp procedures such as solving linear equations relatively well, but when they are asked to apply algebra in context, perhaps by writing an expression or by solving a geometric problem with algebraic unknowns, they come unstuck.

Problems also result from confusing expressions with equations. Do we spend time clarifying the difference between an expression and an equation? When expected to simplify expressions, some Foundation pupils are conjuring up an equation to be solved.

Perhaps our pupils see algebra and think they need to find a numerical answer, regardless of the nature of the question.

**Reasoning and literacy at both tiers**

Pupils’ explanations are often lacking. This seems to fall into two categories:

- A struggle to clearly explain the mathematics.
- A lack of specific vocabulary that is necessary for a proper explanation.

At the Higher tier questions comparing datasets were poorly answered, with pupils unsure of the need to compare medians *and* quartiles. In both tiers, angle vocabulary such as ‘alternate’ and ‘corresponding’ was often missed or misused, and questions that required specific justification of an answer were not approached clearly.

It can be tempting to gloss over accuracy in vocabulary, but we must remember that those pupils who don’t pick up the vocabulary easily need time practising how to answer these questions, rehearsing what wording is acceptable and what is incomplete (and will therefore earn no marks).

Another issue that cropped up on both tiers was the understanding of highly maths-specific words, such as ‘roots’ and ‘turning points’. Our lessons should be filled with rich mathematical vocabulary, so that these words become as normal as any other.

**Ratio at both tiers**

Ratio is approached quite differently on the new GCSE, frequently forming a part of more complex problems. These questions are badly answered. Pupils show an insecure grasp of scaling ratios to match parts, and drawing bar models or similar representations to help them picture the scenario. With more and more pupils coming from Primary school familiar with the bar model, we should capitalise on this technique and use it to take pupils further. Some pupils attempted to solve ratio problems in a more challenging algebraic way and were almost invariably doomed to failure. If you haven’t already investigated alternative approaches to solving ratio problems, it is well worth your time given the emphasis on this topic at GCSE.

At Higher tier, pupils need more practice on questions that require them to form complex equations from equivalent ratios. It would seem that pupils are very used to sharing in a ratio, or finding parts of a ratio, but knowing that a:b=c:d means that a/b=c/d is not something they are confident with.

**Use of the calculator at both tiers**

Pupils lose marks because they rely too heavily on their calculators. Oftentimes they round answers partway through a solution and lose accuracy for the final answer. This premature rounding permeates reports on both tiers, but poor use of the calculator doesn’t stop there. Pupils need to check the reasonableness of their answers and not assume that what the calculator says is always to be trusted.

Marks are lost because pupils fail to write their method down when they do their working on the calculator. This is fine if the answer is correct, but if they’ve mistyped there are no method marks to be gained.

**Percentages on both tiers**

There are more questions involving compounding percentages on the new GCSE papers. It is no longer sufficient to calculate the percentage of an amount, now pupils must find a percentage of a percentage, both in compound interest questions and in more unusual contexts.

The ability to change fluently between fractions, decimals and percentages in context is an area for obvious improvement, the most apparent of these contexts being probability. It’s important that we interleave certain knowledge into other topics wherever the opportunity arrives, so that our pupils are not taken by surprise when the exam throws something completely unexpected at them.

**Trigonometry at both tiers**

Bringing trigonometry into the Foundation tier was a controversial move, but it would seem that pupils can cope quite well with basic questions. It is when pupils need to use the (supposedly memorised) exact values for 30°, 45° or 60° that they meet difficulties, at both tiers.

It is at the Higher tier that trigonometry causes more significant issues, especially with multi-step questions that require the use of advanced trigonometry as part of a solution. As with everything, our teaching must allow time to expose pupils to such multi-step problems. We must both allow them to become familiar with the idea that they have a bank of techniques that they can use to attack a problem, and give them practice selecting the most appropriate techniques at the right time.

Trigonometry in 3D shapes is also problematic, where pupils find it hard to spot the necessary right-angled triangles.

**Volume and surface area at both tiers**

It seems that formulae become muddled
during exams. Remembering the right formulae, applying correct powers when
having to square or cube, and using the correct dimensions in a correct formula
all caused problems this year. I mourn the disappearance of dimensional
analysis from GCSE in this respect. Understanding why a formula *must*
represent a volume rather than an area, for instance, is hugely powerful in
mitigating these problems. I still, informally, teach dimensional analysis to
my students for this reason. It is another useful tool that improves
understanding and reduces the space for error.

**Advanced algebra at Higher**

Solving simultaneous equations graphically, knowing when to complete the square (when a question does not explicitly use this phrase) and solving quadratic inequalities are three topics that were completed badly this summer. Given the essential nature of each of these to further mathematical study, it is certainly worth devoting more time to them in schemes of work and revision periods.

The final topic that arose as particularly weak this year was algebraic proof. Far too many pupils think that a couple of numerical examples in the positive constitute a proof. Proof, as a concept, is a highly difficult thing to grasp, so there is clearly a lot more work to be done in this area.

Overall, it is clear that our pupils need a combination of more time to gain fluency in many key areas and regular opportunities to solve problems that mix different topics and present content in unexpected ways.

Jemma Sherwood is Head of Mathematics at a Secondary school in Worcestershire, having taught maths for fourteen years. She has a master’s degree in Mathematics Education and spends some of her time training teachers of other subjects who want to convert to teaching mathematics as well as sitting on the governing board of a Primary school.

Jemma has written How to Enhance Your Mathematics Subject Knowledge: Number and Algebra for Secondary Teachers for the Oxford Teaching Guides series, which is a guidebook for new maths teachers, equipping them with the depth of knowledge they need to talk confidently about maths to students at all levels. Shop now via the Oxford Education website or on Amazon.

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