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In the general pecking order of GCSE subjects, maths enjoys something of a privileged position.
A core subject alongside English and science, maths is a centuries-old discipline that’s evolved to a point where an understanding of basic mathematical concepts is vital in order to successfully live, work and prosper in the modern world.
More specifically, a GCSE maths grade 4 pass and above is required by many post-16 destinations for further study and careers development. It’s one of the most important qualifications your students will be studying for, and they know it - which is why the maths exam (or centre assessed grade, as the case may be) tends to loom large in students’ minds and be seen in a negative light.
As any maths teacher will know, of course, there is beauty and wonder aplenty to be found in the study of maths - not to mention any number of practical life skills that might not readily apparent to your students, but which can be made explicit in a range of creative and imaginative ways.
Students will enter Y10 with a certain set of ingrained attitudes and habits towards maths, but that doesn’t mean they’re set in stone. As the prospect of taking their maths GCSE becomes increasingly real, your students’ may well come to treat the subject with a newfound seriousness that could serve them well when it comes to their revision.
Needless to say, however, 2021 is hardly business as usual in terms of how students will take their GCSEs and how their grades will be awarded. According to guidance issued by the DfE, schools will be able to submit different forms of assessment evidence, including existing student records and coursework, as well as internal tests conducted within school, past papers and new assessment materials that exam boards are due to issue by end of March.
What follows should hopefully apply to this year’s alternative arrangements, as well the more traditional exams Let’s therefore begin with a look at how to build on what your students will have learnt up to that point and bring about a change in attitudes where needed…
The GCSE maths paper your students will, of course, largely consist of abstract questions intended to test their knowledge and understanding of key mathematical concepts, interspersed with a handful of ‘real world’ problems.
By the time they’ve reached KS4, students will know why it’s important to pass their GCSE maths exam, and therefore see the lesson content they’re learning as a means to that end. Most will fail to see why these particular concepts are so important, and struggle to relate algebraic calculations to their everyday lives.
Ideally, your students will have been given a fuller appreciation before now of maths’ daily importance to all our lives, but it’s not too late to lay the groundwork now if needs be.
Some maths topics lend themselves especially well to meaningful, real-world contexts, such as arithmetic, ratio and proportion and geometry. One obvious example is DIY - are there any photos or videos you can show your class to illustrate how you calculated various measurements and positionings?
Consider the following: “Tim has x sweets. John has 3 more than Tim. Together they have 15 sweets. How many does John have?”
The question is essentially asking students to respond with a neat equation, but by trying to present this as a real-world problem that exists within the students’ frame of reference, they may well try and use their own real-world strategies to try and answer it.
Questions like this can be useful - the ability to construct and solve equations for given problems is a crucial skill - but we need to make it clear that that’s what the question is intended to do.
If your students want a ‘real life’ example of when they’ll need to use certain maths skills, the physics lesson they have scheduled later in the day would certainly count. The same may well apply to students’ experiences in geography, D&T and PE lessons.
These links can be made more explicit by devising a school-wide calculation policy, so that different subject teachers can model the processes used in your school’s maths classrooms. That way, students will soon start to recognise just how relevant and versatile the GCSE-level maths they’ve been learning really is.
That said, there is a case to be argued that some elements of the standard GCSE curriculum won’t be especially relevant to your students’ future working lives - not all of them will require knowledge of trigonometric ratios, or the ability to solve quadratic equations.
Even in the process of learning these seemingly arcane subjects, however, your students will still be picking up a range of useful transferable skills. The ability to think logically and follow a specific thought process or method, for example. Critical thinking is another, along with oral and written communication skills.
The tasks students perform when studying algebra – such as solving equations, expanding and factorising – are especially useful in the context of computer programming.
Make it clear to your students that it’s because they possess these kinds of transferable skills that mathematics graduates are highly sought after in a number of different industries.
Your GCSE students will be studying maths in what’s undeniably a target-driven environment - there are specific curriculum areas you’ll need to cover (see below) and certain grades they’ll need to reach at a set time. This can lead to GCSE maths being presented as a series of fragmented, almost arbitrary skills that they ‘need to know’, without it being made clear why.
Even with all this going on, teachers can still model maths as something enjoyable in and of itself. The ideal time to do this is arguably at KS3, where students can be introduced to areas outside the curriculum, such as Fermat’s Last Theorem and set theory in novel and engaging ways.
At KS4, it’s still worth encouraging students to see maths as a valuable tool for engaging with and questioning the world around them – everything from why a thrown ball moves in the way it does, to weather patterns and how graphics are rendered on digital devices.
With your students sufficiently motivated to study and revise, the next step is to ensure that they’re able to comprehend and retain the knowledge they’ll need to succeed come exam time. And for that, it can be useful to…
Whichever form of GCSE maths assessment your students undergo, it will at some point involve them answering some tricky questions that could cause them to come unstuck. As Craig Barton notes, the process of preparing students to confidently tackle a GCSE past paper or similar form assessment should involve developing their readiness to conjecture, hypothesis and problem solve.
This can be done through careful sequencing when it comes to GCSE maths revision. Instead of presenting students with a series of random, disconnected practice questions and working through them, have each question build upon the previous one by keeping key aspects constant and changing just one or two elements - the basis of an approach that Barton calls Reflect, Expect, Check, Explain.
When a student’s GCSE maths revision arrives at fractions, for example, Q1 of their practice questions might ask students to add two fractions together and work out the answer. Q2 might then change one element - perhaps the numerator or denominator of either fraction. The purpose of this is to show students that the questions they’re being asked aren’t unrelated, but that there’s something else at play that they need to pick up on.
The ‘reflect, expect, check explain’ process can be broken down as follows:
Not every student will spot the intended relationships between the questions straight away, but all will benefit from thinking that little bit deeper about the maths revision task they have been set.
You can get a better sense of the intention behind the ‘Reflect, Expect, Check, Explain’ model by contrasting it against more traditional approaches to compiling GCSE maths revision questions.
This will typically involve setting fairly easy questions to start with, which then get progressively harder while simultaneously calling on students to demonstrate more sophisticated - and crucially, more interesting - mathematical techniques.
At first glance this makes sense, with mastery of the basics normally required before moving on to more complex activities. What this means, however, is that every student will encounter the more boring, routine mathematics early on, while only some will reach the part where things get interesting.
Barton’s ‘ Reflect, Expect, Check, Explain’ sequences can and do get gradually harder as they go along, but at every stage they’re prompted to compare the elements of adjacent questions, spot what’s different, interrogate the reasons why that might be and explain their reasoning.
When students work through a series of disconnected maths revision questions, they have little reason to think any further about questions they’ve answered correctly. When the questions are sequenced thoughtfully, every student will have an incentive to consider why they’ve been able to answer a question correctly, and reflect on the substance of what they’re revising more deeply.
We’ve covered how to keep students engaged and structure maths revision activities in a way that encourages deeper engagement with the topic at hand - but what about those traditional class activities that could potentially do more harm than good?
In Craig Barton’s view, these are the maths revision habits and practices that teachers would be best off avoiding:
Working through past papers is one of the first activities many maths teachers will reach for as a matter of course. Setting aside the fact that past papers may have an important role to play in some schools’ actual GCSE assessments this year, the risk of relying on them too much is that students may find themselves falling into a pattern of getting the same questions right and the same questions wrong.
Instead of repeatedly challenging them to take on the five- or six-mark questions of a past paper, class time centred on maths revision may be instead better spent on securing students’ knowledge of the mathematical foundations relevant to the topic at hand.
Allowing students to focus their GCSE maths revision time on the topics of their choosing, based on what they feel they need to work on most is another common practice, but one which assumes that GCSE students are themselves the best judges of their own strengths and weaknesses.
Every GCSE student will appreciate receiving positive scores and feedback from their revision activities - and a surefire way of making sure that happens will be for them to direct their attention to what they’re already good at, thus neglecting the areas they really need to work on most…
It might be tempting to set an activity whereby students complete a practice paper in pairs – but the issue here is that the teacher will often be given little indication as to who did what, and whether either student could have answered the exam questions correctly working on their own.
A more extreme instance of practice number 2, this might see each student assigned a particular GCSE maths topic which they then present to the rest of the class. This is good for encouraging individual students’ engagement with specific mathematical concepts and boosting their confidence; rather less so for the rest of the class, who will likely end up watching a lovingly constructed PowerPoint presentation delivered by a relative novice, while a qualified teacher watches on from a corner of the classroom…
As outlined in the DfE’s Mathematics programmes of study for KS4, your GCSE students’ knowledge will be tested in six distinct subject areas. These are listed below, alongside some advice and resources that you may find helpful.
Subject content here includes - the application of systematic listing strategies; calculations using roots, and with integer (and fractional) indices; exact calculations using fractions, surds and multiples of π and identifying and working with fractions in ratio problems.
Spotting number sequences
A lesson plan in which students explore a situation that gives rise to a seemingly simple pattern of numbers - at least until the fifth term, after which things quickly become rather more complex…
What’s the deal with indices?
Indices is a topic that many maths teachers are fond of, but which students will often struggle with - Jo Morgan looks at why that might be…
Teaching areas covered under algebra include how to - simplify and manipulate algebraic expressions; interpret simple expressions as functions with inputs and outputs; use the form y mx + c = to identify parallel lines; identify and interpret roots, intercepts and turning points of quadratic functions graphically; plot and interpret graphs (including reciprocal graphs); and recognise and use sequences of triangular, square and cube numbers, along with simple arithmetic progressions, Fibonacci-type sequences, quadratic sequences, and simple geometric progressions.
‘x’ marks the spot
Jemma Sherwood offers some advice on the best ways of introducing algebra - which includes not starting too early, providing clarity as to its purpose and giving students prognostic practice…
Algebra simplification, simplified
Collecting like terms in algebra can be a source of considerable confusion for students if you’re not careful, says Peter Mattock - the key to getting them past such difficulties may well lie in their understanding of addition.
The areas students should be taught include how to - compare lengths, areas and volumes using ratio notation and/or scale factors; convert between related compound units in numerical and algebraic contexts; interpret the gradient of a straight line graph as a rate of change; and set up, solve and interpret the answers in growth and decay problems.
‘Simplifying ratios’ treasure hunt
A card-based activity that gives students practice at converting ratios into their simplest form, by dividing both parts - ideal for revision, or as a starter or plenary.
In this lesson, the situation of two crossing ladders is used as an opportunity to work on ideas relating to similar triangles. Students need to identify triangles that are similar, and then construct equal ratios to work out their answers. They will notice interesting properties, such as the fact that the distance between the walls does not affect the height of the crossing point. There are also connections with an average called the harmonic mean.
Students will be expect to, among other things - interpret and use fractional scale factors for enlargements; identify and apply circle definitions and properties; construct and interpret plans and elevations of 3D shapes; interpret and use bearings; calculate arc lengths, angles and areas of sectors of circles; and calculate surface areas and volumes of spheres, pyramids, cones and composite solids.
Exploring geometrical transformations
Traditionally, pupils are taught geometrical transformations by learning a set of skills they can apply to reflecting, rotating and enlarging shapes without giving the process too much thought. But what if they were called upon to use those skills to solve a mathematical problem…?
Interpretation and use of bearings
In this lesson, students make up and decode cryptic messages using bearings - with each word coded as a journey from one letter to the next on a spiral grid of letters. Students will need to be precise about their angles and distances, or they won’t be able to code or decode their message successfully!
Required areas include how to - apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one; use a probability model to predict the outcomes of future experiments; and calculate the probability of independent and dependent combined events, including using tree diagrams and other representations.
Probability and dice games
In this lesson, students are asked to estimate the probability of winning a game in which they throw an ordinary dice and move a counter forward a corresponding number of spaces. To win, they must ‘hit 10’ by landing on the tenth square from the start.
The game will be too complicated for students to work out the probability of winning theoretically, so instead they must estimate it by playing the game several times and seeing how often they win. This will enable them to calculate how much money they might be willing to pay to play the game, assuming the prize for winning is £1.
The outcomes of a simple spinner game
Many real-world situations involve probability, but students’ intuitions can often be wrong. This lesson is based around the scores produced by two easy to make spinners that the students use to test their conjectures. The next step involves constructing a sample space diagram, allowing the students to predict and explain the score outcomes and calculate theoretical probabilities.
Students must understand how to - infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling; interpret and construct tables and line graphs for time series data; apply statistics to describe a population; and use and interpret scatter graphs of bivariate data - while recognising that correlation doesn’t indicate causation, being able to draw estimated lines of best fit, making predictions, interpolating and extrapolating apparent trends, at the same time as being conscious of the risks of doing so.
A data analysis lesson plan with an accompanying guidance sheet that seeks to demonstrate how mathematics can emerge in various everyday scenarios, while showing students how to accurately calculate and analyse information, and examine ‘real’ data for patterns.
Calculating, displaying and interpreting statistics
A series of tutorials that take students on a tour of bar charts and histograms, taking in ways of calculating averages and means along the way.
Everything you need for every subject across Key Stages 3 and 4.