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.

Dictator and scribe

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 bit annoying, 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 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.

Question-level analysis

Exam papers are not the best basis for identifying strengths and development areas, because some exam questions are much easier than others, but an informal analysis can be useful nonetheless. Recording individual question scores is probably not a good use of time, but jotting down whole-class feedback on particular topics can be a great starting point for follow-up planning.

Student accountability

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 rarely relies 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 experience, 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. Students need to experience mathematical ideas in a range of contexts to make important connections and strengthen their powers of generalisation.

4. “Students enjoy being on a journey towards mastery.”

Positive outcomes are most readily achieved when students are self-driven to learn – both through intrinsic curiosity and a 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 diagnostic feedback and practice. 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 importance of targeted effort. Feedback is essential because it forces us to challenge specific aspects of our understanding. The objective itself is merely a context for the personal learning 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? Share your thoughts in the comments.

Summary of the changes in GCSE Maths from 2015

This post was written in 2013 and, while I hope it is still helpful, it reflects the limited information that was available at that time.

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.

Sources

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.

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.

Building self-worth through outward bound residentials

I was recently involved in a youth project working to raise the self-esteem of 21 vulnerable teenagers from local schools. This culminated in taking them to a residential centre for four days where they engaged in a hectic programme of outward bound activities including high ropes, horseriding, rock climbing and zip wires. Needless to say, we all faced challenges (staff included) and returned exhausted. “Thanks for taking us to Wales and letting us do loads of fun things” wrote Connor afterwards, expressing his gratitude on a scrap piece of card.

I’ve done a lot of this sort of thing over the past six years, and in doing so I’ve learned a fair amount about what drives young people and how to get the best out of them. I also think that many schools underestimate the positive impact that high-quality residential and adventurous experiences can have on all their students and particularly the most vulnerable. In this post I’ll offer a few thoughts on these sorts of trips and how to make the most out of them.

Who are our ‘vulnerable’ young people?

I don’t know if there’s an accepted definition, but I tend to think of vulnerable students as those whose low self-worth* puts them at risk of negative behaviours and outcomes. Some such individuals present as timid and introverted; others cover up for their insecurity by acting out or bullying; others try to escape from the pressures of life by truanting, cutting or abusing alcohol and recreational drugs. A few weeks ago I spoke to 15-year-old Ryan about his aspirations. He said he’d like to join the Army next year “unless I’ve got a kid by then”. He had no real concept that this was something under his control.

* It’s almost always about self-worth. This may have been damaged in the past by physical, sexual or emotional abuse, a poor home environment or a history of bullying, or there may have been other factors.

How can time at an outdoor centre benefit these individuals?

There are three key factors.

  1. It’s a great opportunity to achieve. The other day a Year 9 student stormed out of her maths lesson and, seeing me in the courtyard, went into a monologue about how school is boring and the work is irrelevant and teachers are unfair and she had a headache so how was she supposed to put up with someone telling her what to do. Clearly for some, school isn’t the easiest place to thrive. Going out kayaking or abseiling or caving or zip-wiring is different. In this environment, all students are on a level playing field and everyone can genuinely achieve. Even an activity as simple as camping or outdoor cooking can broaden students’ horizons because in many cases they have never even considered that it is something they can do. Thirteen-year-old Joe was afraid of heights but he came on a recent trip and gave every activity a go including climbing and high ropes. On several occasions he broke down in tears, such was his fear, but he always persevered onto the next challenge. Reflecting on the trip, he said he’d never been so brave or experienced such a sense of success before.
  2. Students benefit from the residential setting. Vulnerable adolescents often associate with the wrong crowd in their communities and get into negative patterns of behaviour and routine. Taking them out of their normal environments even for just a few days can be eye-opening for them. They see that there’s no need to drink alcohol to enjoy themselves and discover that they can get up at 6:30am on a Saturday just fine. They help each other out as a team and get to know people they might not normally choose to spend time with. Most of all, they benefit from the leadership of calm, supportive adults who put their best interests first.
  3. The out-of-school context is ideal for developing lasting relationships. A few months ago, 14-year-old Liam came away on a two-week camp and one evening he totally lost it with a member of staff over something I can’t even remember. Various people intervened, but in the end I went after him and we talked for about an hour – not just about what had happened but about life at home and his worries and what was really going on inside his head. As a classroom teacher, I very rarely have a conversation like that in school. Afterwards, Liam said he needed to go for a smoke and when he came back he apologised to all the people he had offended. Away from school, not only do staff and students see each other in a different light, but by supporting our young people we earn their trust and build the foundations of strong professional relationships.

A few thoughts on making trips work

Whenever I come home after a residential, I take time to reflect on what went well and what could be improved. Here are a few of the lessons I’ve learned.

  • Take staff who work with the students on a regular basis: their Heads of Year, teachers, form tutors, mentors. These are the people best placed to bond with them.
  • Plan for every minute. Vulnerable students generally don’t cope well with down time, and giving them an hour to ‘relax’ in the evening is bound to end in trouble. It’s good to have lots of games and activities to keep everyone busy.
  • Model transitions in energy levels. It is a careful balancing act to ensure that participants finish the day tired enough to fall sleep but not so tired that they lose patience with one another and become upset. Be prepared to adapt the programme to achieve this. A group who wake up unusually lethargic may need to be energised before commencing their scheduled activities. Conversely, competitive sports could be substituted for a film if everyone is agitated at the end of a long day. Tone of voice can also go a long way to raising or dissipating a group’s energy.
  • Praise process, not outcomes. “You’re a natural rock climber”, “You’re great at starting camp fires”, “You’re the fastest in the group”. Phrases like these actually discourage learners from stretching themselves because they fear that they will fail and let you down. Instead, praise effort: “I was really proud that you didn’t give up”, or “I’m impressed by how hard you’ve worked on your technique”.
  • Set clear boundaries. Constant positivity is essential, but it needs to be in the context of very clear rules and expectations. If a young person isn’t being compliant, find the right person to talk to them. I’ve been in plenty of situations where a teenager was totally defiant to a colleague but then followed instructions the moment I spoke with them – and of course I have also been in the opposite position. Relationships are key, and staff need to work together to ensure that minor issues are resolved and not escalated. That said, “zero tolerance to bullying, drugs, fighting” must mean exactly that. The ultimate sanction is to send a young person home – and as unappealing as this will always be, it is sometimes the right decision in the interests of the group as a whole.
  • Look for the best in every student. I recently worked with a girl who had a really unpleasant manner. She was totally blunt (“I’m bored, this is shit”) and always came across in a negative way. This immediately set me up to dislike her, and I have to admit it took a few days for me to see past this aggressive front to the kind, sensitive personality beneath. Looking back, it seems likely that she was unconsciously transferring her own sense of worthlessness onto me. She just needed me to be relentlessly patient and positive.
  • Remember that every drama is an opportunity to show support. Help to resolve friendship issues, find some spare toothpaste for the child who forgot theirs, recognise the young lad who is feeling homesick – every positive interaction will earn trust and respect. A few years ago, a student ran up to me and explained that his friend Callum had brought some chlorine water purification tablets and was talking about taking an overdose. When I got to Callum he hadn’t taken anything and it seemed to be a cry for help. So I took him to the office and we had a long conversation about everything; it turned out he was having a tough time at home. Whenever I spoke to him afterwards, it was like we had an understanding.

Schools are a vital safety net for vulnerable children and adolescents, and it is my experience that residential trips can go a long way to building self-worth in those who lack and need it the most.

(All names in this post have been changed.)

Supporting language development in the mathematics classroom

My school is having a big literacy push this year. There’s a ‘literacy focus’ poster in every classroom and we’re all getting used to a standardised marking code for correcting SPAG.

This drive has prompted me to put some thought into language development in my lessons, but to be honest I’ve found it hard to apply much of the whole-school guidance to mathematics. The rarity of extended writing tasks limits opportunity for ‘green penning’, for example, and I’ve yet to find a chance to embed the use of connectives in a lesson. So the question is raised: how can we develop literacy in mathematics?

In their best practice case study, Literacy across the curriculum (2012), Ofsted clarify:

Our message is that individual subjects create different language demands, which need to be identified and addressed within those subjects.

So this is the real question: what are the language demands in maths, and how can we address them? I’ll share my thoughts.

Using technical terms

At every level, students need a mathematical vocabulary to access tasks and questions.

  • Whenever a keyword is introduced, draw attention to it. I usually ask students to copy it down with a definition in their books and put a box around it for emphasis. Peers then check each others’ spelling. It’s best if the definition is in their own words.
  • Avoid simplifying vocabulary, even for students working at lower levels. Language is embedded through frequent use. For example, talk about the numerator and denominator of a fraction, rather than the top and bottom. From an exam point of view, students will run into difficulties if they’re used to talking about the ‘bow tie rule’ of circle theorems and ‘Z angles’ in parallel lines.
  • Be particularly aware of technical terms which have a different meaning in maths to their everyday use – called faux amis by Skemp (1972). For example: origin, axes, base, degree, odd, solution, table, term, division.
  • Circle mis-spelt words in books when marking, and allow lesson time for students to copy down the correct spellings collaboratively.
  • Discuss command words regularly. For example, whenever a question starts with “Evaluate…”, “Show that…” or “Write down…”, nominate a learner to remind you what that means.
  • Praise and reward students when they ask for a word to be defined, to encourage openness in language enquiry.
  • Task students with calling out definitions throughout the lesson: “Every time I say ‘relative frequency’, I want you to say “…which means experimental probability”.
  • Game: ‘Splat!‘. Project a jumble of keywords onto the board and invite two students to come up and play. When you read out a definition, the first person to put their hand on the right word wins.
  • Game: ‘Taboo‘. Produce a set of cards, each with a keyword at the top and then four related words underneath. In teams, students must take turns picking a card from the pack and then trying to describe the top word without using it or any of the other words on the card. In a variation to this game, students can begin by creating their own cards for each other.
  • Display posters around the school with keywords and definitions, coloured according to topic (number, algebra, shape, data). Then set a homework where each student is given a colour and needs to find as many posters as possible of that colour. Provide a worksheet where they can note down each keyword and its definition.

Interpreting text

There are many occasions in maths when students need to scan text to extract information. For example, when answering worded problems they begin by identifying the key facts and considering what the question is asking. Manipulating algebraic expressions, they scan to identify like terms and factors.

  • Create opportunities for students to read numbers and expressions out loud. For example, when introducing the quadratic formula, ask pairs to discuss how they would communicate it verbally to another person.
  • Whenever tackling a test or exam paper, start with a pile of highlighters on each table and allow time for students to highlight all the keywords in each question. Sometimes I use green for ‘I know what this means’ and pink for ‘I need a reminder’.
  • A few moments in to every task, ensure that the instructions are understood: “Tell me what you’re being asked to do”. This approach is particularly useful for students with EAL, those with low reading ages and individuals on the autistic spectrum. It doesn’t just apply to written questions; verbal instructions need confirmation too.

Explaining ideas

As I often remind my classes, it’s no use having great mathematical ideas if you can’t communicate them clearly to others. Students need to be able to do this both verbally and in writing.

  • Model great written communication. Once my students have tried the below exam question, for example, I show them a video of my model solution from which we establish success criteria for them to self-assess their own work.
  • Provide the answers up front, to place emphasis on how students communicate their method. This also means time spent going over the answers can be replaced with peer assessment time to review the clarity of working.
  • Use Kagan Structures such as RallyRobin and Timed Pair Share to maximise the time each student spends speaking and listening.
  • Even if a nominated student can adequately explain a concept to the class, extend an invitation for others to paraphrase the idea.
  • Once or twice each lesson, nominate a student to sum up what has been learned so far, so they get into the habit of explaining concepts in their own words.
  • Create a display board of students’ model answers to exam questions. Give students post-its so they can annotate with specific praise (‘What makes this a great answer?’).
  • When setting extended tasks (e.g. stats projects, painted cube), include success criteria for contextualising answers in plain English.

(Thanks to @amdindependent@MrHillMaths@perryclaire and @awi700 for their great advice which I’ve taken on board in preparing this article.)

What have I missed? Please share your ideas in the comments.

Using mini-whiteboards to transform classroom practice

This year for the first time I have my own classroom, and it’s making a big difference to the way I teach. It’s those few moments between lessons when I can collect myself and set up for the next class, rather than racing between sites. It’s the freedom to update wall displays and rearrange tables whenever I wish, to make the learning environment the best it can be. But it’s also something else that has unexpectedly transformed my practice: it’s having, in the corner of the room, my own supply of mini-whiteboards.

They don’t stay in the corner for very long. The first thing one of my Year 7s asked when he walked into my classroom this afternoon: “Shall I hand out the mini whiteboards, sir?”

There are three reasons I find them invaluable.

  1. Whole-class feedback. Picture this: your 5 minutes of whole-class questioning. You pose a question, pause, pounce on a student to answer. That’s 1/32 of the class’s understanding you just assessed. Bounce it on to a couple of people: “Can you expand on X’s answer?”, “Do you agree with Y?” – now you’re up to 3/32 of the class. When students have mini boards, in the same amount of time you can elicit a response from every single person in the room. That means every learner engaging with the topic and showing you, through the validity of their responses, how much they understand.
  2. Developing confidence. Quite a lot of students don’t like putting pen to paper. This is especially true in maths where there is usually a single right answer and, therefore, a high risk of getting it wrong. I’ve found students much more willing to make a start on mini boards because it’s easy to make corrections and wipe out mistakes. Once they’ve experimented, they can copy out the solution on paper.
  3. Checking individual work. Wandering around the room, too often I find myself drawn towards individual questions (“I’m stuck”) and not really getting a feel for the quality of what students are writing in their books. Some students even go out of their way to cover up work they’re not sure about (“I haven’t finished yet”). Enter mini whiteboards. When students complete the same task on mini boards, they collaborate better by looking at what each other are writing and it’s easy for me to see too. They’re less afraid of someone pointing out a mistake because it’s quick to make corrections.

One great activity is smiles and frowns. I ask students to answer a tricky question on their boards and annotate their solution with a smiley if they’re pretty sure it’s right or a frowny if they’re not so sure. Then everyone leaves their board on their table and moves around the room. ‘Smiley’ students find ‘frowny’ work to peer correct, and ‘frowny’ students find ‘smiley’ work to learn from.

I also enjoy quiz-quiz-trade. To start this game, each learner generates a question on a given topic (e.g. linear equations) and writes it on their mini board. They write the answer on the back. Then they move around the room until they find someone to ‘high five’. This pair now answer each others’ questions. When they get the right answers, they swap boards and continue to move around the room.

Do you have any great ideas for using mini boards in lessons? Please share them in the comments.

mini-whiteboards-why-i-use-them-every-lesson

Comments

Mark: Great post. I love the idea of quiz-quiz-trade. I’ll steal that one and try it tomorrow morning!

Tim: Students draw different unfamiliar graphs accurately. Then one student describes their graph and other sketch it on their whiteboard. They show to the first student who may choose to give further information/amend/remove ambiguity (no gestures allowed) Good for focusing on the key points when sketching a graph (min/max, intercepts, asymptotes, gradient) and communication skills (literacy). Particularly good with secx, cosecx and cotx.

Rachel: When I find mine again I’m definitely using them-you’ve inspired me!

Martyn: I play the front to back game. All students start at the back of the room and answer questions on the whiteboards to move forward step by step. When they get to the front they start scoring points. If they get a question wrong they move backwards one step. You can only score points when at the front. Good fun, great afl and the competition seems to go down quite well

Kate: I’m also a big fan of the mini whiteboards – especially for drafting first sentences or starting ideas without, as you said, the fear of failure that many of our students have. What also has worked well in the past for extended tasks is planning a shared class answer to an essay or exam, allocating groups a paragraph/section of the answer and getting them to write their section onto the table. You can then arrange the tables together to make a complete answer and use the mark scheme or success criteria to edit and assess it. This year I will be getting the students to use a green marker pen for this :-)