Using exam papers effectively

Past exam papers are a valuable learning tool, and I’d like to share a few techniques I’ve tried for making the most out of them.

Pair work

I often ask students to work in pairs, taking it in turns being ‘dictator’ and ‘scribe’. The dictator has to answer the question, but can’t write anything down; the scribe can only write down what the dictator says, and nothing else. So both discuss the problem, but ultimately the dictator has to decide what is written and the scribe has to commit it to paper. This is really effective for making sure that both students in the pair understand the solution before moving on. It’s also great for sharing accountability so the sense of failure is less personal if something goes wrong.

Traffic-light cards

I find the alliteration, ‘Brain, board, book, buddy, boss’ a little patronising, but the principle of exhausting all resources before going to the teacher is a good one. A useful prompt is to ask students to put their revision guides on their desks at the start of the lesson. Without further instruction, they then tend to dip in whenever they need a reminder about a particular topic. Another simple technique is to give each pair a red/green card, initially placed green-side up on their desk. The rule is that when they need support, they should flip the card to red and move onto the next question. This is much better than ‘hands up’, because rather than waving their arms impatiently while you’re attending to other learners, students get on with the rest of the paper and make the most of the lesson time.

Structured feedback

Reviewing answers should be a core part of the learning process, and certainly not a brief plenary activity for the last five minutes of the lesson. One strategy for instantaneous feedback is to indicate ‘checkpoints’ every 4-5 questions through the paper, and then split the mark scheme into corresponding chunks. Each time a pair reaches a checkpoint, they can come to the front to pick up the next piece of the mark scheme and correct their work so far. Where the mark scheme is not easy to follow (perhaps just for one or two questions), it may be worth writing out student-friendly solutions.

A different approach is needed when students have completed a paper individually – perhaps for homework. Sometimes I collect these in to mark myself, so I can give specific feedback, but I do this with caution. In the past there have been times when I’ve collected papers and (despite best intentions) still not marked them two weeks later – which is not good. So usually I go for peer marking in one of two ways:

  1. Students only tick or cross answers to indicate right / wrong. This is much faster than trying to work out method marks.
  2. Questions are collected in separate piles and each student marks one pile. This approach is great for sixth formers, because it gives them time to decipher the mark scheme for their particular question. They can make detailed corrections, and then present a brief overview about common misconceptions at the end.

Diagnostic testing

Exam papers are not the best basis for filling in Personalised Learning Checklists, because some questions are much easier than others, but question-level analysis is useful nonetheless. At my school, we use a software package where students traffic light their results online and a class summary is generated:

This helps us plan follow-up lessons, and also draws students’ attention to their individual development points. I sometimes let sixth formers set their own homework based on questions they got wrong.


Finally, I try to make students feel a sense of responsibility for their progress. There’s nothing wrong with an individual relying on peer support in a lesson or making a mistake that needs correcting, as long as I know they’ll definitely get it right next time. And there’s no better way to ensure this than creating a “next time”. Whenever time allows, I provide a period of consolidation and then give the class the exact same paper again – but this time in test conditions. (I normally say I’ll change the numbers so they don’t just memorise answers, but I rarely do.) This helps to make sure they really assimilate the learning during and beyond the lesson.

Context in the mathematics classroom

I’d like to share my thoughts on a handful of pedagogical statements and their implications for mathematics teaching.

1. “Students must know what they’re learning, not just what they’re doing.”

This really isn’t a big deal in Maths, because a purposeful approach to mathematical tasks is so rarely conditional on metacognitive engagement with an objective. In some subjects, context can be very distracting – students might be writing about a poem but really learning about effective paragraphing. In Maths, if they’re reflecting shapes in diagonal mirror lines then that’s almost certainly their objective too.

As a further illustration, say that two classes are doing the same activity to provoke their thinking about surds. In one group, students can precisely explain that “We are learning to manipulate and compare surds”; in the other, they describe the task: “We’re matching cards with equivalent expressions”. Isn’t the difference chiefly one of articulation, rather than a contrasting quality of learning? By completing a well-designed task, students constantly test and correct their understanding – which is the basis of learning. Even if they can’t identify the title of the topic they’re studying, that doesn’t diminish their powers of pattern recognition which will help them tackle similar problems in the future.

2. “Students should always see the point in what they’re learning.”

This statement is also misplaced because it falsely encapsulates students’ motivations. In my view, we study Maths for four reasons:

  1. Enjoyment
  2. To develop our abstract problem-solving skills
  3. To achieve a qualification
  4. Because in some cases, there will be applications to future study or work

I strongly feel that it’s more helpful to engage students with points 1-3 than to get too caught up in point 4. The reality is that beyond basic numeracy, a lot of what students learn in Maths lessons isn’t going to be of direct use in their later lives. And while pseudo-realistic contexts can be great for sparking interest in otherwise abstract topics, so too can mathematics be a context in itself. I had much more fun today with Year 13 discovering ‘cyclic integration by parts’, than with Year 12 calculating the cross-sectional area of a fictional river.

At this point, I risk coming across as dismissive of the role of context in Mathematics – which is actually pivotal – so I’d like to reframe the above statements into three others which I think are more constructive.

3. “Generalising and specialising are at the heart of mathematics.”

A lesson without the opportunity for learners to generalise is not a mathematics lesson.
– Professor John Mason

It is a centrepiece of subject pedagogy that mathematical thinking consists of conjecturing and testing those conjectures. “Is this always/sometimes/never true? Are there special cases or exceptions to the rule? Does this result apply to a broader set of problems?” In a sense, this eclipses the role of contextualising and de-contextualising learning in other subjects. It also presents a real challenge: do our questions and prompts test the limits of students’ understanding and provoke them to sharpen their internal conjectures? If not, we risk surface learning and uncovered misconceptions.

4. “Students should value every task in the context of mastery.”

Positive outcomes are most readily achieved when students are self-driven to learn – dually through intrinsic curiosity and an unquenchable thirst for progress. We talk about “students taking ownership of their learning”. One of the key ways we can make this happen is by engaging them in the narrative of their progress over time. An experienced colleague on my team uses a ‘GCSE Route Map’ where students shade in topics on a tree of learning as they explore the subject. I am currently leading a disaffected Key Stage 3 group through a perpetual cycle of Diagnosis – Therapy – Testing. The common theme is that by sharing how each learning activity fits into students’ personal journeys towards mastery, we can impact positively on their emotional engagement. The ‘context’ for learning is their own growing understanding.

5. “Feedback focuses students’ attention.”

I have set out my view that tasks and objectives are inherently aligned in Maths, but perhaps this underplays the significance of targeted effort. Feedback is essential because it forces us to challenge specific aspects of our understanding. So actually in any classroom, the apparent learning objective may itself be viewed as a context for the personal developmental experiences of every student. A class may be outwardly ‘learning to draw pie charts’, while its constituent individuals variously attend to their fragmented conceptions of angles, proportion, equivalence and scale.

What are your views on the role of context in mathematics? Please share your thoughts in the comments.

Homework: a culture of high expectations and support

Typical adolescent responses to homework range from a casual “I’ll rush through this on the bus” to a conscientious “I’ll do my best and hand in as much as I can”. The problem is, neither of these approaches are particularly helpful.

If students always give up at the first hurdle, only ever completing as much of the homework as their fragmented grasp of the mathematics allows, then how much is their understanding really consolidated? To what extent are they really taking ownership of their progress towards mastery? Not very much, I would argue.

And for a long time, I allowed this culture. I set the bar low by accepting “I did most of the homework but couldn’t do questions X, Y and Z”. So while students did benefit from the practice, they didn’t get nearly as much out of it as they could have done. We all learn best by getting stuck and unstuck at our own pace, not by giving up and waiting for an expert go through the answers on the board.

So I set about shifting attitudes. Over the course of a few months, one class at a time, I started to raise my expectations of homework and, in equal measure, increase my availability for support. In short, I made it totally clear that “I did most of it but couldn’t do questions 5 and 6” would no longer be good enough. All students would be expected to complete all their homework, all the time, making any necessary shifts in understanding to do so. And most importantly, I would be on hand to help – through an online forum which I set up. The premise was that now when students got stuck and had exhausted their own efforts, they could seek support right away rather than moving on and losing the opportunity to learn.

In the first instance, I tried this with my Year 12 and 13 classes, who eagerly got on board. Here are some examples of the questions they asked.

As you can see, my usual approach is to give only a small hint or explanation so that the learners do as much thinking as possible for themselves. In the following example though, I trusted that the student had already made her own attempt and so offered a full solution:

This next dialogue I find particularly striking, because it shows a student taking true ownership of his learning. The homework was to watch a Khan Academy video in preparation for a new topic – a task which could easily have been be undertaken in a passive way. However, this student wouldn’t settle for a partial understanding and he sought clarification: exactly the growth mindset I had hoped to develop.

There are many more examples of my dialogue with sixth formers – in fact, they now post questions almost every day. Encouraging Key Stage 4 students to do this, however, has been much more challenging. As a starting point, I asked one class to tell me their dream jobs and how these might involve maths. This at least got them logging on to the site.

Since then, they have posted the odd question which has moved their thinking forward. I’d like to get them using the site more and it’s certainly something I’m still working on.

What do you think? How do you make the most of homework with your students?
Please share your thoughts in the comments.

Summary of the changes in GCSE Maths from 2015

The government has just released the new GCSE Maths programme of study for first teaching in 2015. There are significant changes we need to engage with right away to ensure that our current Year 8s are fully prepared by the time they get to KS4.

Changes to grades and tiers

New grades will be numbered 1-9, where 9 is the highest. Falling below the standard required for a 1 will result in a U grade. This means there will be a total of ten possible outcomes, unlike the nine grades of A* to U at present – so a correspondence between the old and new systems is not immediate.

Maths will continue to be tiered, though the split will be different and we cannot assume that the same profile of students should enter for Higher and Foundation as at present. The Foundation Tier will give access to grades 1-5, and the Higher Tier 4-9.

There are currently no grade descriptors, nor any indication of what “expected progress” will look like for students with different starting points. Instead, we will be held to account over our value added against an unpredictable “cohort average”.

So forget moving goalposts – now there are no goalposts at all. Gone are our instruments for self-evaluation: “3 levels”, FFT, A*-C. We’re on our own to determine the right targets (and tiers of entry) for each student.

Changes to the assessment objectives

The three objectives AO1, AO2 and AO3 have not changed but they are weighted differently, with AO3 now accounting for the same proportion of marks as AO2. At Higher Tier, only 40% will be set aside for AO1, placing even greater emphasis on students’ problem-solving skills.

Changes to the provision of formulae

Some formulae previously given in the front of the exam paper will no longer be listed. Students will need to memorise them.

  • Area of a trapezium
  • Volume of a prism
  • Quadratic formula (Higher Tier)
  • Triangle sine, cosine and area rules (Higher Tier)

The cone and sphere formulae will continue to be given, and now also the SUVAT equations – presumably only for rearranging and substitution.

Changes to the course content

The new syllabus is greatly expanded with completely new topics making an appearance at both tiers. But while there are some substantial new challenges for Higher Tier candidates, the greatest shake-up is a vast shift of content from the Higher to the Foundation Tier. It looks like even to secure a Grade 5, concepts will have to be mastered which today’s students targeting A and B grades are struggling to contend with.

The following changes are in comparison with the current AQA Linear specification. Some topics are already tested by other exam boards and these are indicated with an asterisk (*).

Skills to be assessed at Foundation that are currently Higher only

  • Calculate exactly with multiples of π
  • Use standard form
  • Round to any number of significant figures (currently 1 s.f. only)
  • Expand double brackets
  • Factorise quadratics including the difference of two squares
  • Solve quadratic equations by factorising
  • Know the difference between an equation and identity
  • Use y = mx + c to identify parallel lines
  • Sketch quadratic, cubic and reciprocal functions
  • Derive simultaneous equations from real-life situations
  • Solve linear simultaneous equations algebraically and graphically
  • Perform calculations with density, mass and volume
  • Solve problems involving percentage change and reverse percentages
  • Use direct and inverse proportion graphically and algebraically
  • Solve problems involving compound interest
  • Find corresponding lengths in similar shapes
  • Use the congruence criteria for triangles (SSS, SAS, ASA, RHS)
  • Enlarge shapes with fractional scale factors
  • Find the areas and perimeters of compound shapes involving circles, and calculate arc lengths and areas of sectors
  • Use the sin, cos and tan trigonometric ratios for right-angled triangles
  • Use tree diagrams to solve probability questions
  • Infer properties of a population from a sample, while knowing the limitations of sampling

New skills assessed at Foundation and Higher

  • Find the equation of a line through two points or through one point with given gradient
  • Recognise and use sequences of triangular, square and cube numbers, Fibonacci type sequences, quadratic sequences and geometric sequences
  • Calculate compound measures including pressure in numerical and algebraic contexts
  • Express a multiplicative relationship between two quantities as a ratio or a fraction
  • Write a ratio as a linear function
  • Set up, solve and interpret growth and decay problems
  • Use inequality notation to specify error intervals due to rounding
  • Understand the ≠ symbol (not equal)
  • Use the standard convention for labelling sides and angles of polygons
  • Derive the sum of angles in a triangle
  • Know the exact values of sin, cos and tan at key angles (0, 30, 45, 60, 90 degrees)
  • Use Venn diagrams
  • Consider outliers when calculating the range of a distribution
  • Know that correlation does not imply causation

New skills assessed at Higher only

  • Recognise and use the equation of a circle centred at the origin *
  • Find the equation of a tangent to a circle at a given point, using the fact that it is perpendicular to the radius
  • Find approximate solutions using iteration (is this just trial & improvement?)
  • Solve quadratic inequalities
  • Find the nth term of a quadratic sequence
  • Recognise and use geometric sequences where the common ratio may be a surd
  • Apply the concepts of instantaneous and average rates of change by looking at the gradients of tangents and chords to a curve
  • Prove the circle theorems
  • Use the probability “AND” and “OR” rules *
  • Change recurring decimals into their corresponding fractions and vice versa *
  • Find inverse and composite functions
  • Locate turning points of quadratic functions by completing the square *
  • Sketch y = tan x (in addition to sin and cos)
  • Interpret areas under graphs and gradients of graphs in real-life contexts (e.g. recognise that the area under a velocity-time graph represents displacement)

Skills no longer required

  • Design a survey question and identify sources of bias
  • Convert between metric and imperial units
  • Draw and interpret frequency polygons and stem and leaf diagrams
  • Solve equations using trial and improvement (no longer in Foundation Tier)

We’re all in the same boat

It’s worth noting that there will be no alternative mainstream courses. In their guidance on the 2016 performance tables, the DfE explains that qualifications such as iGCSEs will soon stop counting alongside the new, more demanding GCSEs.

[We] will seek evidence that any qualification being proposed for inclusion in performance tables does not have significant overlap with reformed GCSEs. … In time, this will result in the phasing out of academic qualifications from the annual list of non-GCSE qualifications counting towards performance tables.

Next steps

I’m still in the early stages of processing these changes and considering what they mean for our provision at Key Stages 3 and 4. It’s clear that our schemes of work will need to be revised and for Years 7 and 8 this probably can’t wait until next academic year.

I look forward to forming a working group with colleagues at school to engage fully with this information and also with the new Key Stage 3 curriculum which comes into play from 2014. Our first task will be to establish a much clearer picture of how learning can be built up across the five years – responding to the challenge of a deeper syllabus while also avoiding the temptation to “cover” content in an instrumental way.

Another key action will be to explore with SLT the possibility of increased curriculum time. It probably isn’t true that the stakes are raised. From 2016, maths will be double weighted to count for 20% of the Progress 8 floor standard – just as it currently makes up 20% of 5A*CEM. But with one commentator estimating that the syllabus has increased in size by a third, we’ll have lot more learning to fit in. The secretary of state has suggested that schools set aside 7 hours per fortnight for maths, though it isn’t clear for which year groups this advice applies. We return once more to the issue of judging against a cohort average: as soon as some schools change their timetable allocation, we all need to. As a shortage subject though, I suspect it will pay to be a leader in this particular pack.

Key questions

  1. What are the implications of the overwhelming shift in content from Higher to Foundation Tier?
  2. How need we adapt our KS3 schemes of work to ensure the 2017 cohort are fully prepared for the new course?

Please share your thoughts in the comments or tweet me at @jamesgurung.


To support faculty leaders in communicating these changes to their teams, I have created a summary presentation which you are welcome to use.


Expert stickers to facilitate reciprocal learning

It is well known that all students go crazy for stickers, and that this phenomenon does not diminish with age. Many of the best lessons I’ve seen recently have featured “medals and missions”, where learners are rewarded early and often for achieving small milestones in their work. Young people like to see their names on the star board; they like to move their post-its along a big progress chart; frankly they’ll even settle for ticks in their books as recognition that they are doing the right thing.

The challenge is to reconcile this effective motivational technique with the apparently conflicting task of developing independence and resilience in learners. We won’t have served students well if at the end of the course they’re thrown by the prospect of a two-hour exam in an environment totally devoid of support and acknowledgement.

Perhaps you’re hoping I can follow that paragraph with a solution. I can’t. But I’d like to share a lesson which I think struck a bit of a balance.

I provided four worksheets, deliberately designed to test the limits of students’ understanding in topics we had covered recently. The class, a Year 10 group, were to have the whole hour to complete as many as they could. Crucially, I explained that I wasn’t going to be available for answering questions – but that everyone could move around the room to work collaboratively (sharing ideas, not answers). There was a sticker available for each sheet, and stickers would be awarded for fully correct solutions only.

Instantly there was a buzz of activity as students set about on whichever tasks they liked the look of the most. To begin with, I limited my intervention to guiding individuals who hadn’t yet internalised good strategies for getting unstuck. “Have you looked in your revision guide?”, I said a few times. Then when the first few completed sheets came in, I marked them – just ticks and vague circles where things had gone wrong. A couple of students earned their first stickers for perfect work.

Soon it was time to take the activity to the next level. I stopped the class. From now on, I explained, when you finish a sheet you should find someone else who has already earned the corresponding sticker and ask them to check it. I would no longer be marking work, but all students who had earned stickers were given the authority to certify others for the same stickers. This exponentially accelerated the propagation of feedback. I reminded students that they were allowed to give each other hints but not answers.

Rapidly, everyone earned more and more stickers. I spent my time with a couple of learners who were still stuck on their first sheets, helping them to make the necessary shifts in understanding. After about 45 minutes, the first student proudly announced that he had completed all four sheets and brought them to me for checking. I tasked him to earn the one remaining sticker – a “mighty lion” – by genuinely supporting three of his peers and gaining their signatures as evidence.

It was a fun lesson. Students were motivated by the stickers but I like to think that many of them just appreciated the excuse to flaunt their intrinsic enjoyment of the problems, which I had carefully pitched. With everyone working at their own pace and usually met with only right/wrong feedback, resilience was necessary to succeed. And while the environment was abundantly supportive, that support was very rarely coming from me; hence I hope that some independence – at least independence from the teacher – was instilled.

Engaging students in online revision

Last year the day before their Unit 1 exam, I told Year 10 I’d be running an online Q&A that evening where they could ask any last-minute questions. I wasn’t sure if anyone would take me up on the offer, but it seemed worth a try.

I was totally blown away by the response. Over two hours we had 232 comments from 73 readers and the chat was so busy that I had to call a 10 minute break in the middle. The sheer volume of questions asked left no doubt that it had had an impact.

The rules for participating were that students should use their first names only, stay on topic and be polite. There was no need to log in – anyone could simply type their name and write a comment. All messages were queued until I approved them, which made it easier for me to answer one question at a time.

To run the chat I used the excellent CoverItLive app. Commercial users have to pay but when I contacted the company they readily set up a free Unlimited account for my school.

Learning from misconceptions

Recently a colleague reviewed her Year 11 mocks by filling in a fresh copy of the exam paper with “silly mistakes”. For each question, she wrote a response illustrating one or more key misconceptions she had noticed when marking. Students then worked in groups to correct the errors.

Last year I did something similar with my Year 9s. As I was marking their practice SATS papers, I took photos of responses that were partially correct but contained errors. I posted these on our class blog and, for that week’s homework, asked learners to identify the mistakes.

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Using end-of-term tests to move learning forward

The formative use of end-of-term assessments is usually limited. Students sit them in the last week of term, we mark and return them, everyone goes away for the holidays and then the next term we dive straight in to the start of a new unit. The ‘learning from feedback’ stage tends to be constrained to a lesson of going through the answers, after which no one is necessarily motivated to act on their development points.

This week I tried something a bit different. In the 24 hours after my Year 10 class sat their tests…

1. I marked them.

I made a student-friendly mark scheme and instead of writing on the learners’ scripts, I annotated copies of this assessment grid with feedback for each individual. Every time a student dropped a mark, I left a comment about where they had gone wrong.

2. They marked each other’s.

By not writing on the papers themselves, I had preserved the opportunity for peer assessment. The next lesson, everyone got someone else’s paper along with a fresh copy of the mark scheme. Just as I had done the evening before, they wrote feedback onto an assessment grid. The purpose of this exercise was for learners to become familiar with the success criteria for each question and to place themselves in the shoes of the examiner. Every time someone asked “How many marks is this worth?”, I told them to read the mark scheme and use their discretion.

3. They consolidated the feedback.

When each student finished marking, I gave them the sheet with my feedback for the paper they had assessed. I allowed a few minutes for them to reflect on any differences between the marks I had awarded and the marks that they had. To minimise petty debate, I made it clear that my marks were final even if I had made a mistake. However, I asked them to transfer any constructive feedback from their assessment sheet onto mine. Generally my marks were more accurate but the students’ comments were more verbose, so it was helpful to consolidate the feedback onto a single marksheet.

Up until this point, I had insisted that students not tell each other whose papers they had been marking. Hence, when they returned each others’ scripts and feedback, there was a genuine buzz as everyone found out how they had performed. We didn’t talk about marks and grades though.

4. They traffic lighted each topic.

Straight away, I had the learners traffic lighting how well they had demonstrated understanding in each topic.

After this was done on paper, they transferred the information onto a Google Form using our new class set of Chromebooks. This online questionnaire also asked some other questions like how much they had revised and how they felt about their mark. It automatically produced a very helpful summary which I will use to plan the first few lessons of next term.

5. I emailed their parents.

Notoriously, feedback is ineffective unless we create the conditions for students to act on it. To do this, I mail merged each learner’s topic results to their parents. I also provided some suggested revision resources and explained that there would be a re-test next term for learners to demonstrate how much they had progressed.

The response from parents was overwhelmingly positive. One reply read:

I was quite alarmed to see the number of R’s and A’s. We have sat down and talked with him and agreed to a revision timetable over the holidays. I appreciate your email so that we can be proactive over this and not finding out at the last minute.

Overall, my aim has been to move away from perceiving end-of-term tests as terminal summative assessments, and instead to use them formatively as a means for checking current attainment and setting targets.

How do you use end-of-term tests? Do you have creative, high-impact ways of reviewing them and giving feedback? Please share your thoughts in the comments.

Writing on everything

I do a lot of learning walks around my department. I’m genuinely interested in what all our students are up to and their learning experiences in different classrooms. I’m on the hunt for great ideas to steal and, equally, opportunities to support colleagues where I can.

The best thing I’ve seen in loads of places is students writing on everything. They use whiteboard pens to write on desks, windows and even ‘magic whiteboards’ on the walls – and in some cases teachers have embedded this as a totally normal part of their lessons. I’ve blogged already on how amazing whiteboards can be for developing confidence, and it’s awesome to see that colleagues are taking it a stage further.

So I’ve reflected on my own practice and in the last couple of weeks I’ve started:

  • writing on students’ desks with my board pen when I’m working through an example with them individually; and
  • offering them their own board pens to use on the tables (or mini boards, if they prefer).

Everyone seems to like it. Except the cleaner, perhaps.